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A theorem prover for nonnormal modal logics
Tests over valid formulas and over random formulas
The performances of PRONOM seems prmising. We have tested it:
• on sets of valid formulas for each modal logic
• on sets of randomly generated formulas, obtained by fixing the number of propositional variables, the depth (level of nesting of connectives) of a formula and a time limit.
To run tests:
1. consult the Prolog source file of the system to be tested (e.g. e.pl)
2. consult the Prolog source file of statistics
3. to run tests over valid formulas, ask the Prolog engine to prove
where system can be: e, em, en, ec, enc, emn, emc, emnc
4. to run tests over randomly generated formulas, ask the Prolog engine to prove
Results of tests over valid formulas are shown in Table 1, whereas the percentage of timeouts in executing tests over randomly generated formulas are shown in Table 2.
Table 1: percentage of timeouts of PRONOM over N valid formulas, 90 < N < 500.
Table 2: percentage of timeouts of PRONOM over 8000 formulas randomly generated. | {"url":"http://193.51.60.97:8000/pronom/","timestamp":"2024-11-05T19:21:53Z","content_type":"text/html","content_length":"17777","record_id":"<urn:uuid:8962266c-b022-4262-a64a-63cb918c6837>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00151.warc.gz"} |
Is February 2013 a leap year?
Is February 2013 a leap year?
No. 2021 is not a leap year. 2021 year has 365 days….Leap Year Table.
Year is Leap Year
2013 –
2014 –
2015 –
2016 Leap Year
What does leap year mean in February?
Simply put, a leap year is a year with an extra day—February 29—which is added nearly every four years to the calendar year. Why Are Leap Years Necessary? Because of this extra day, a leap year has
366 days instead of 365.
Is February a leap year?
A year, occurring once every four years, which has 366 days including 29 February as an integral day is called a Leap year. 2021 is not a leap year and has 365 days like a common year.
Is 2020 a leap year Yes or no?
This year, 2020, is a leap year, and what that means is that we get an extra day this year. We get that extra day because we count time, in part, by the time it takes Earth to go around the sun.
Because we do that, every four years our calendar must come into agreement with the calendar that governs the universe.
How many days are in February in a leap year?
366 days
A year, occurring once every four years, which has 366 days including February 29 as an important day is called a Leap year.
What was the problem with the leap year?
In 2012, TomTom satellite navigation devices malfunctioned due to a leap year bug that first emerged on March 31. Sony’s PlayStation 3 incorrectly treated 2010 as a leap year, so the non-existent
February 29, 2010, was shown on March 1, 2010, and caused a program error. At midnight on December 31, 2008, many first generation Zune 30 models froze.
What do you need to know about the LEAP system?
The following Learning Engagement and Performance (LEAP) System modules cover information required by various compliance and accreditation organizations to help keep you, our patients and visitors
safe. All modules with [20XX CORE] in the title must be completed by the end of your first day.
When was the last time there was a leap year?
In 1996, two aluminum smelting plants at Tiwai Point, New Zealand, and Bell Bay, Tasmania, Australia, experienced a leap year bug on December 31, when each of the 660 computers controlling the
smelting potlines shut down at the stroke of midnight simultaneously and without warning.
Is the correct algorithm for the leap year?
It incorrectly assumes that a leap year occurs exactly every four years. The correct leap year algorithm is explained at Leap Year Algorithm . There have been many occurrences of leap year bugs: | {"url":"https://www.spudd64.com/is-february-2013-a-leap-year/","timestamp":"2024-11-03T00:19:44Z","content_type":"text/html","content_length":"42731","record_id":"<urn:uuid:4b655b4a-7662-40db-8885-2c42c84f31ce>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00501.warc.gz"} |
Why the way of calculating STD error and STD error of difference is different? – Q&A Hub – 365 Data Science
Why the way of calculating STD error and STD error of difference is different?
Please help me to understand. I did researches in the internet, but it is still unclear for me.
STD error calculation is =stdev.p/sqrt(n) or (std of population divided by square root of sample sie) right? Then the std error of difference is the sum of square root variance divided by sample
sizes? Can't think it the sum of two std errors? if I calculate it like std divided by square root of sample size the result is different. Why is the result is different when calculated square root
of variance divided by sample size and std pop divided by square root of sample size? Logically shouldn't it be the same?
3 answers ( 0 marked as helpful)
I have the same question too. That's why my calculation is different from the example. Did you find some answer yet, Timur?
I'm with the same issue. If we calculate the Standart Error (SE) like before in this course, the Square Root should be separatedly in each term. It isn't the same as calculating the Square Root of
everything. We can't split the sum inside of square root. Therefore, they are different results, different things.
The change comes with the calculation of Variance. The Variance of difference already includes dividing by Sample Size (n).
We could think that the Square Root of the Variance is equal to Standard Deviation (STD). But since the Variance is different (includes dividing by sample size), my conclusion is:
- STD isn't necessary / possible when analyzing difference between independent variables.
- We get directly the SE when calculating Square Root of Variance of difference because (but I don't understand why)
I have the same challenge. But the possible explanation I could come up with is "Since we are working with 2 independent variables. I think we have to solve with the total Standard Error. Therefore,
we have to add the two variance/sample size before we get the square root. Just the way square root of 4 is not equal to square of 2 + square root of 2". | {"url":"https://365datascience.com/question/why-the-way-of-calculating-std-error-and-std-error-of-difference-is-different/","timestamp":"2024-11-04T15:39:37Z","content_type":"text/html","content_length":"118635","record_id":"<urn:uuid:de10b1df-9454-4c8a-a99b-28fa980d1888>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00067.warc.gz"} |
What did the ending of The Circle mean?
The Ending: After Mae’s friend is killed by intrusive surveillance, it looks like she’ll abandon The Circle’s philosophy of full societal transparency and the complete destruction of privacy.
Instead, she embraces it, publishing all of Eamon’s emails and exposing them to the world.
Who survives The Circle?
They are both eliminated, and the two factions aggressively thin their opponents’ numbers. Eventually, only Eric, the pregnant woman, a silent man who has never voted, and the girl are left. Eric
theorizes that aliens have used the process to learn about humanity’s values.
How does The Circle end Emma Watson?
Towards the conclusion of The Circle, it is shown that Mae (Emma Watson), after being disturbed from within due to Mercer (Ellar Coltrane’s) death, decides to join The Circle again against he
parents’ plea.
What is the point of the movie The Circle?
The Circle is basically an experiment that rewards the fittest with a chance of survival. It is based on Darwin’s theory of ‘Survival of the Fittest’. Interestingly, in the film the fitness of a
person is defined by their wit rather than their physical strength.
What is the plot of circle?
Fifty strangers facing execution have to pick one person among them to live.Circle / Film synopsis
Is Mae the villain in The Circle?
Surprise surprise, Mae is the true villain of The Circle. In fact, the crucial push of the second act is the transformation of the formerly skeptical Mae into the full bore team player she winds up
What was the point of circle?
A circle is a set of all points in a plane that are all an equal distance from a single point, the center. The distance from a circle’s center to a point on the circle is called the radius of the
circle. A radius is a line segment with one endpoint at the center of the circle and the other endpoint on the circle.
Does Courtney get blocked from The Circle?
3 Courtney Revolution – Undeserving Courtney Revolution played such a brilliant game that he neither deserved nor was ever blocked during the 13-episode run. As one of the three finalists, Courtney
did not need to rely on a catfish persona to earn third place. He remained himself the entire way.
What is the point of The Circle on Netflix?
Players in The Circle work to gain one another’s trust and then vote on their favorite contestants within the game. Players are occasionally “blocked” or removed from the game while fresh blood is
often brought in, with the last competitor standing winning $100,000.
What is the message behind The Circle?
Circle Movie Ending Meaning This shows that in some Circles, the group stuck consistently to their morals, allowing a kid or a pregnant woman to survive. It’s the balance between the schemers,
moralizers, and righteous that provides for the game of the Circle to exist.
What is Mae’s job at The Circle?
The novel chronicles tech worker Mae Holland as she joins a powerful Internet company. Her initially rewarding experience turns darker.
What’s wrong with Mae’s dad in The Circle?
Mae’s father has multiple sclerosis, and his life has become more difficult than it ever was before. | {"url":"https://tumericalive.com/what-did-the-ending-of-the-circle-mean/","timestamp":"2024-11-03T10:11:06Z","content_type":"text/html","content_length":"38052","record_id":"<urn:uuid:6852d3d3-b0fd-4dde-97bb-05efaf33aa63>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00862.warc.gz"} |
Lexis, Wilhelm
From Encyclopedia of Mathematics
Copyright notice
This article Wilhelm Lexis was adapted from an original article by Sébastien Hertz, which appeared in StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies. The original
article ([http://statprob.com/encyclopedia/WilhelmLEXIS.html StatProb Source], Local Files: pdf | tex) is copyrighted by the author(s), the article has been donated to Encyclopedia of Mathematics,
and its further issues are under Creative Commons Attribution Share-Alike License'. All pages from StatProb are contained in the Category StatProb.
Wilhelm LEXIS
b. 17 July 1837 - d. 24 August 1914
Summary. In a hostile germanic environment Lexis established statistics as a highly mathematical subject based on the probability calculus, by means of his dispersion theory.
Wilhelm Lexis, the son of a physician, was born in Eschweiler near Aachen, in Germany. He studied at the University of Bonn from 1855, first devoting himself to law, and later to mathematics and the
natural sciences. He was awarded his doctorate in philosophy in 1859 for a thesis on analytical mechanics. For some time, he taught secondary school mathematics at the Bonn Gymnasium. He also held a
job in the Bunsen chemical laboratory in Heidelberg.
Lexis' departure for Paris in 1861 marked a turning point in his career. It was there that he developed his interest in the social sciences and political economy, as well as familiarizing himself
with the works of Quetelet. His first major work, published in Bonn in 1870, was a detailed study of the evolution of France's foreign trade after the restoration of the monarchy (Die Ausfuhrprämien
im Zusammenhang mit der Tarifgeschichte und Handelsentwicklung Frankreichs seit der Restauration). In it Lexis stressed the importance of basing economic theories on quantitative data, while not
hesitating to make use of mathematics.
The Franco-Prussian war of 1870-71 forced him to return to Germany. While editing the Amtliche Nachrichten für Elass-Lothringen at Hagenau, then the seat of the general government of Alsace-Lorraine,
he befriended Friedrich Althof, who was to become director of higher education in the Prussian Ministry of Education and Culture. This friendship was at the basis of Lexis' active participation in
the exchange of ideas and reforms of German universities.
In the autumn of 1872, he was very appropriately appointed as professor extraordinarius (Associate Professor) in political economy at the newly created University of Strasbourg, then one semester
old, where Althof was also teaching. It was in the same year that he took part in the formation of the Verein für Sozialpolitik, a movement of university members (the Kathedersozialisten), an
offshoot of the historical school whose aim was the promotion of social politics. It was in the Alsatian capital that he wrote his impressive introduction to the theory of statistical demography,
Einleitung in die Theorie der Bevölkerungsstatistik, published in 1875.
By then he had already left Strasbourg for Dorpat, but not without recognition by award of Doctor rerum politicorum honoris causa in 1874. In Dorpat (now Tartu in Estonia), a town in the Russian
Empire where the language of university instruction was German till 1895, he held the Chair as full professor in Geography, Ethnography and Statistics. He spent only two years there, returning to the
banks of the Rhine as Chair of Political Economy at the University of Freiburg im Breisgau from 1876 to 1884. This was undoubtedly his most productive period. His publications of the time, most of
them appearing in Jahrbücher für Nationaökonomie und Statistik, of which he was chief editor beginning from 1891, propelled him to the front rank in the field of theoretical statistics, and revealed
him as the leader of a group working on the application of the calculus of probabilities to statistical data.
Lexis simultaneously continued his research in political economy, editing the first German encyclopedia of economic and social sciences Handwörterbuch der Staatswissenschaften. He was particularly
expert in the field of finance, publishing his Erörterungen über die Währungsfragen, among other works in 1881.
In 1884, he resigned his Chair in Freiburg for the Chair of Statistics (Staatswissenschaften) at the University of Breslau (now Wroclaw in Poland). Finally, in 1887, he moved to Göttingen where he
held the Chair of Statistics until his death, a few days after the start of the First World War. Bortkiewicz was his student in Göttingen in 1892. In 1895, Lexis founded the first actuarial institute
in Germany (Königliches Seminar für Versicherungswissenschaften), which trained its candidates in both political economy and mathematics. His scholarship in both fields allowed him to manage its
direction, and to provide part of the teaching in economics and statistics, while G. Bohlmann took charge of the teaching of mathematics.
Lexis left his mark on the history of statistics through his pioneering work on dispersion, which led on to the analysis of variance. Lexis' plan was to measure and compare the fluctuations for
different statistical time series. In a sense, he followed Quetelet in applying urn models to statistical series. But by stressing fluctuations, he corrected Quetelet's work, which aimed to set every
series within a unique "normal" model by assuming quite erroneously their homogeneity and stability. Similarly, using a binomial urn model to represent the annual number of male births, he derived a
dispersion coefficient $Q$ (in homage to Quetelet) which is the ratio of the empirical variance of the series considered to the assumed theoretical variance. An analogous coefficient of divergence
had been independently constructed by the French actuary Emile Dormoy in 1874. In the ideal case, Lexis refers to a ``normal" dispersion when the fluctuations are purely due to chance, and the
coefficient is equal to 1. But in most cases the coefficient is different from 1, and thus differs from the binomial model. The fluctuations then indicate a ``physical" rather than a chance
component. Lexis classified these dispersions into two categories, ``hypernormal" and ``hyponormal" according as to whether $Q > 1$ or $Q < 1$. He also showed that series of social data usually have
a hypernormal dispersion.
His studies on the ratio of sexes at birth, his stability theory of statistical series with his famous $Q$ coefficient of dispersion were re-examined in his large treatise entitled "Abhandlungen zur
Theorie der Bevölkerungs- und Moralstatistik(1903). In a review of it, Bortkiewicz in 1904 concludes that ``(Lexis) has known how to clarify and synthesize the most general problems of moral and
demographic statistics, insofar as their conditions, methods and tasks are concerned; he has also shown that if this science has had to renounce its status as ``social physics" to which Quetelet
tried to raise it, it remains nevertheless far more than the simple social accounting which some modern, and excessively timid, practitioners of the discipline would have us believe."
Lexis' coefficient foreshadowed the statistics of K.Pearson and R.A. Fisher, in particular ${\chi}^2$ for the analysis of variance. However, it suffered from certain weaknesses which his more
mathematical and younger contemporaries did not fail to point out and attempt to correct, among them Chuprov, Markov and Bortkiewicz. In publications up to the period between the two World Wars, the
Continental School of mathematical statistics tended to follow the dispersion theory of Lexis, but both eventually gave way together to the Anglo-Saxon developments in this area. Lexis' statistical
views, however, did not disappear from view as they had the dubious distinction of being singled out for attack on their reactionary and bourgeois nature within the Soviet Union, by guardians of
ideology such as Yastremsky.
[1] Bauer, R. (1955). Die Lexissche Dispersionstheorie in ihren Beziehungen zur modernen statistischen Methodenlehre, insbesondere zur Streuungsanalyse (Analysis of Variance), Mitteilungsblatt für
mathematische Statistik und ihre Anwendungsgebiete, 7, 25-45.
[2] Bortkiewicz, L. von (1915). Wilhelm Lexis, Bulletin de l'Institut International de Statistique, Tome 20, 1ère livraison, pp. 328-332.
[3] Bortkiewicz, L. von (1904). Die Theorie der Bevölkerungs - und Moralstatistik nach Lexis, Jahrbücher für Nationalökonomie und Statistik, III. Folge, Bd. 27, pp. 230-254.
[4] Heiss, K.-P. (1968). Lexis, Wilhelm, International Encyclopedia of the Social Sciences, 9, 271-276. Macmillan and the Free Press, New York.
[5] Heyde, C.C. & Seneta, E. (1977). I.J. Bienaymé: Statistical Theory Anticipated. Springer, New York, pp. 49-58.
[6] Stigler, S. M. (1986). The History of Statistics. The Measurement of Uncertainty Before 1900. Belknap Press, Harvard. pp. 221-238.
Reprinted with permission from Christopher Charles Heyde and Eugene William Seneta (Editors), Statisticians of the Centuries, Springer-Verlag Inc., New York, USA.
How to Cite This Entry:
Lexis, Wilhelm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lexis,_Wilhelm&oldid=55580 | {"url":"https://encyclopediaofmath.org/wiki/Lexis,_Wilhelm","timestamp":"2024-11-03T03:17:43Z","content_type":"text/html","content_length":"23728","record_id":"<urn:uuid:0578c404-b98c-4d55-ae0a-244f2846c5f9>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00813.warc.gz"} |
[GAP Forum] Counting all elements in a group of a given order
Stephen Linton sal at mcs.st-andrews.ac.uk
Thu Jul 1 21:40:55 BST 2010
Dear GAP Forum.
As has just been pointed out to me, I typed too hastily:
> Finally, for the largest examples it might be best to factorise n into prime powers and use the Sylow subgroups to find representatives of all the conjugacy classes of elements of the appropriate prime power orders. Then, aving enumerated these elements, the number of elements of order n is simply the product.
This is entirely incorrect, since only commuting elements of prime power orders give rise to elements of the product order. It might still be possible to do something along these lines (for instance to find elements of order 12, one might explore the centralisers of elements of order 4), but it is much less simple than I implied.
More information about the Forum mailing list | {"url":"https://www.gap-system.org/ForumArchive2/2010/002838.html","timestamp":"2024-11-14T10:36:35Z","content_type":"text/html","content_length":"3670","record_id":"<urn:uuid:c7165adb-1af9-4fa4-8c4a-d381fe179535>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00570.warc.gz"} |
How to Graph Solutions to One-step and Two-step Linear Inequalities
Dive into the mysterious world of linear inequalities, where the solutions often lie in a vast stretch of the number line rather than a single, fixed point. This journey will explore one-step and
two-step inequalities, inviting you to master the nuances of their graphical representations.
Step-by-Step Guide to Graphing Solutions to One-step and Two-step Linear Inequalities
Here is a step-by-step guide to graphing solutions to one-step and two-step linear inequalities:
Step 1: Understand the Terrain: Basic Inequalities Overview
• Less than (\(<\)): Think of a hungry alligator, always eager to chomp the smaller number.
• Greater than (\(>\)): The reverse. The alligator now wants the bigger number.
• Less than or equal to (\(≤\)): Here, the alligator doesn’t mind if it’s exactly equal or just a tad smaller.
• Greater than or equal to (\(≥\)): Big or just the same, both are good enough!
Step 2: Setting the Stage: Drawing a Number Line
• Take a ruler or a straightedge.
• Sketch a horizontal line, which will represent our number line.
• Evenly space and mark numbers on this line. For instance, from \(-10\) to \(10\).
Step 3: One-Step Inequalities: Baby Steps to Mastery
a) Isolate the Variable
i) If the inequality is \(x>5\), then \(x\) is already isolated.
ii) For an inequality like \(x+4<7\), subtract \(4\) from each side to get\(x<3\).
b) Graph the Solution
i) For strict inequalities like \(x<3\) or \(x>5\):
• Find the number on the number line.
• Make an open circle on it, indicating the value is not included.
• Draw an arrow in the direction of the solution. For \(x<3\), the arrow will point to the left.
ii) For inclusive inequalities like \(x≤3\) or \(x≥5\):
• Find the number.
• Fill in a solid circle, indicating that this value is part of the solution.
• Again, draw an arrow toward the solution side.
Step 4: Two-Step Inequalities: Double the Fun, Double the Challenge
a) Isolate the Variable
i) For an inequality like \(2x−3>7\):
• Start by adding \(3\) to each side: \(2x>10\).
• Then, divide each side by \(2\): \(x>5\).
b) Graph the Solution
i) Much like one-step inequalities, locate the number on the line.
ii) Decide if you need an open or a solid circle based on the strictness or inclusiveness of the inequality.
iii) Draw the arrow pointing towards the solution’s direction.
Step 5: Celebrate the Complexity: Compound Inequalities
Sometimes, you might find a beast like \(3<x≤8\). Here, \(x\) is trapped between two numbers.
• Graph the two numbers. One will have an open circle \((3)\) and the other, a filled one \((8)\).
• Connect them with a line or segment to show \(x\) can be any number in between.
Step 6: Conclusion: Embrace the Dance of Inequalities
Understanding and graphing linear inequalities is like a dance of logic and intuition. With time and practice, you’ll find the rhythm and soon be swirling and twirling through even the most intricate
inequalities with grace and precision!
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Vamshi Jandhyala - Can You Rescue The Paratroopers?
Riddler Express
You have one token, and I have two tokens. Naturally, we both crave more tokens, so we play a game of skill that unfolds over a number of rounds in which the winner of each round gets to steal one
token from the loser. The game itself ends when one of us is out of tokens — that person loses. Suppose that you’re better than me at this game and that you win each round two-thirds of the time and
lose one-third of the time.
What is your probability of winning the game?
from random import random
runs = 100000
cnt = 0
for _ in range(runs):
me, you = 1, 2
while(you != 0 and me != 0):
if random() <= 2/3:
you -= 1
me += 1
you += 1
me -= 1
if you == 0:
cnt += 1
The probability of me winning the game is \(.57\).
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LCR series circuit
It is an electrical circuit consisting of an inductor that stands for L, capacitor which stands for C and resistor which stands for R which are connected in series or parallel. This name LCR is
derived from the sequence of these letters used to denote the constituent components of this circuit. This circuit forms a harmonic oscillator for the current which introduces the resistor which
increases the decay of this oscillation which is known as damping. The circuit of LCR or derived as inductor, capacitor or resistor can have many applications as oscillator circuit as in radio
receivers and television sets use them for tuning to select the narrow frequency from radio waves. This LCR circuit can be used as a bandpass filter, bandstop filter, low pass filter, and high pass
filter and this filter is described as a second-order circuit which means that any current in the circuit can be described by second-order in circuit analysis.
What are the basic concepts of LCR?
Some of the basic concepts are resonance, natural frequency, damping, bandwidth, and scaled parameters. In resonance, an important property of this circuit is it’s the ability to resonate at a
specific frequency and the frequencies are measured in units of hertz and this is measured in radians per second. There are two different ways in which resonance occurred in which first is in an
electric field as a capacitor is charged and in a magnetic field as the current flow through the inductor as energy can be transferred from one to the other within the circuit and this is no passing
metaphor in which weight on a spring is described exactly by the second-order this the resonance is described as the frequency at which impedance of a circuit is minimum and at very low phase and it
is also described as the frequency at which the impedance is purely real.
What are the applications of LCR?
There are so many applications of LCR namely variable turned circuits, filters, oscillators, voltage multiplier and pulse discharge circuit. In variable turned circuits a very frequent use of these
circuits is in the turning circuits of analogue radio as adjustable tuning is achieved with the help of a parallel plate of variable capacitor which can be changed or tuned to different frequencies.
In this filtering application, the resistor become the load that the filter is working on. In oscillator applications such as filtering and many more is generally desirable to make the attenuation as
small as possible. | {"url":"https://www.w3schools.blog/lcr-series-circuit","timestamp":"2024-11-07T06:15:33Z","content_type":"text/html","content_length":"139196","record_id":"<urn:uuid:ef859190-2f7b-456c-b920-8b31a995fdbf>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00356.warc.gz"} |
The Neutrino Mass
1. Introduction
In macrophysics mass (m) is defined as the measure of a particle’s resistance to acceleration, and is measured relative to a standard mass (m[s]);[i]); and if from gravitational interaction, m is
called gravitational mass s(m[g]). In the 19th century Roland Eotvos showed experimentally that
In microphysics, on the other hand, the mass of a fundamental (elementary) particle is an intrinsic property of the particle—it is the measure of the amount of matter (or antimatter) in the particle.
Defined in this way, mass is an absolute quantity! Experiments performed in the 20th century established that the elementary particle masses that are known range between zero and about 100 GeV [2].
The scientific method consists of two components, namely, the study of science based on observation and experience, and the study of science based on the mathematization of physical processes
(theorization). The two components are not independent—a symbiotic relationship exists between them: A fundamental theory changes our view of the universe; it’s unifying synthesis joins two or more
separate bodies of established knowledge whose connection at some deep level had not previously been recognized. Observations confirm the predictions of the new theory. On the other hand observations
may form the basis of a new theory, and the theory confirms the observation, and in some cases extends and generalizes the observation.
The mass of elementary particles appears to be an exception to this symbiosis according to the two cornerstones of 20th century physics, standard model of elementary particles and the big-bang
scenario: Gauge theory forbids masses for all the known elementary particles! For them to acquire masses, gauge symmetry must be broken simultaneously some way! The symmetry of gauge theory will be
spontaneously broken if some gauge noninvariant scalar quantity is non-zero in the theory’s lowest energy state. The original version of the standard model introduced a new elementary scalar
particle, called the Higgs particle, to make gauge invariance break down spontaneously. The Higgs particle’s couplings to other particles are proportional to their masses, and are hence fixed though
unexplained [2]. The disciples of the standard model announced the discovery of the Higgs particle of mass greater than 100 GeV on 4th July, 2012. This result is in violent disagreement with
The problem of the disciples of the big-bang scenario is not the determination of the masses of elementary particles, but rather the asymmetry between matter and antimatter; and whether the asymmetry
has a cosmic origin or whether it results from cosmic evolution. Thus, given the assumed expansion of the universe, what is the universe’s ultimate fate-whether it will expand indefinitely, or it
will terminate eventually and thereafter remain static, or the expansion will stop and then be followed by recollapse? The answer depends on the critical mass density (r[c]) of the universe. r[c] is
the density required to close the universe—that is for its own gravitational self-attraction to be sufficient to terminate the current expansion eventually, or to stop the expansion and cause
recollapse. If the density (r) of the universe is less than r[c] then the universe will expand for ever.
The present baryon density (r[b]) is one or two orders of magnitude less than r[c], hence r[b] is insufficient to close the universe. This is the problem of the invisible (or missing, or dark)
matter, which is assumed to be something other than baryons. The dark-matter candidates must be nonbaryonic dark-mass and together with baryonic dark-mass the closure density can be achieved. Several
elementary particles, some yet unobserved, are proposed as dark-matter candidates including massive neutrinos, axions, photino, gravitino, etc. Arguments from cosmology have set a rough upper limit
of a few GeV on the neutrino mass, based on the observation that the universe is still expanding at present, if these neutrinos are not to decelerate or reverse the expansion of the universe. These
constraints on the neutrino masses are much more general than those obtained from laboratory experiments according to which the most stringent upper limit on a neutrino mass is about 10^−4 of the
electron mass for the electron neutrino. Other dark-matter candidates can have any mass whatsoever [3].
The assumed matter-antimatter asymmetry is in conflict with the Dirac theory of the electron, a special case. There is complete symmetry between electrons and positrons in the Dirac theory [4].
Dirac’s theory is a proof that the assumed matter—antimatter asymmetry does not apply in the fermion world. Dirac’s proof has now been extended to all physical worlds [5]. Thus, the assumed
mater-antimatter asymmetry is not a law of nature! Further, the conclusion that non-baryonic elementary particles can have any mass is in conflict with the experimentally established fact that
elementary particle masses range between zero and about 100 GeV. Finally, the universe is not undergoing any motion relative to us, if it did we cannot detect it because we are part of the
universe—no absolute motion.
2. The Mass of an Elementary Particle
Every elementary particle has a dual nature, namely, particle and wave properties. The particle variables E (energy) and
where m is the rest mass of the particle, while the wave variables l (wave length) and n (frequency) are related to the particle variables by the Louis de Broglie equations,
where h is Planck’s constant, v the speed of the particle, g the Einstein factor, and l[D] the de Broglie wave length [6]. It is clear from (2) that l[D] is not defined in the special cases v = 0,
and v = c. These special cases describe two fundamental particles, namely, the photon with v = c, and graviton with v = 0—the photon is a particle with no corresponding antiparticle, and the graviton
is an antiparticle with no corresponding particle [5]. From (1) and (2) we deduce that when v = c, corresponding to m = 0,
Consider now the ratio¥, m = o. Hence the allowed masses are those for which®¥,m®o. Thus, M is the mass standard for elementary particles and elementary particle masses are defined relative to the
mass of the graviton and their normalized values lie between zero and 1. Exact theory shows that M is the mass of the Z^0, hence Z^0 is the graviton and not a weakly interacting nuclear particle [5].
3. The Neutrino Mass
Wulfgang Pauli proposed the existence of neutrinos in 1930 while investigating the conundrum of radioactive beta decay. He hypothesized, in order to abide by the laws of energy conservation, the
existence of as yet undetected neutral particle which Enrico Fermi named “neutrino” (“little neutral one”). About his hypothesis, Pauli wrote “I have done something very bad today by proposing a
particle that cannot be detected; it is something no theorist should ever do.” It became clear that if such a particle existed, it must be both very light and interact very weakly with matter, making
it difficult to detect.
In 1956, however, Clyde Cowan and Frederic Reines succeeded in detecting the electron neutrino. On getting the information about neutrino’s detection, Pauli retorted “Everything comes to him who
knows how to wait.” The discovery of the electron neutrino was followed in 1962 by the discovery of the muon neutrino corresponding to the charged muon lepton; and that of the tau neutrino on July
21, 2000, corresponding to the charged tau lepton [7].
The discovery of the neutrinos raise some fundamental questions: whether neutrinos might have a tiny bit of mass, whether they could oscillate, and how many kinds of neutrinos exist? On the basis of
the big bang scenario stable neutrino mass must be less than 100eV, or more than a few GeV. Laboratory experiments, on the other hand, posit that electron neutrino mass is about 10^−^4 of the
electron mass. Nucleosynthesis impose the most stringent limit on the number of neutrino species—the number is restricted to 3 or at most 4 which is below the best upper limits available from
particle physics experiments. Formal theory gives 3 as the number of neutrino species, all of which are stable [5]. Thus, the neutrino mass is the only outstanding problem that needs to be settled.
This problem can be settled on the basis of the geometrical theory of the chemical elements; according to which fermions reside in 4^n-dimensional reducible quantum spaces[n] = hn, where n is the
Lamb-Retherford frequency (n = 1000 megahertz), and h the Planck’s constant—n is the frequency of a microwave field that induced transitions between the two levels of the hydrogen atom. Thus, the
energy quantum hn is the work done by an external field to “lift” the electron from one atomic level to the neighbouring level (the electron loses this quantum of energy to return to the original
level when the field is removed). This gives a gap “width”
The neutrino is affected by the gravitational force which is 10^−36 times weaker than the electromagnetic force, hence
These neutrino masses are so small that their experimental determination lies in the distant future on account of the sensitivity of today’s instruments.
4. Conclusion
Conventionally elementary particle is a particle that is not composite-hence elementary particles are the simplest physical objects in the universe. This is certainly necessary but not sufficient;
rather, we define elementary particle as a particle of spin
Einstein’s theory of gravitation (General Theory of Relativity (GTR)) was formulated by analogy with electromagnetism: Electromagnetism is described by a 4- tensor F[m][n] which is derivable from a
4-vector A^m, a purely physical quantity [8]. On the other hand Einstein’s gravitation is described by a symmetric 4-tensor g[m][n], a purely geometrical entity, which is a function of the space-time
variables. g[m][n] is not derivable from any 4-vector, indeed the gravitational 4-vector analogous to A^m is a purely static entity, and hence not a function of the time [9]. Gravitation being a
time-independent phenomenon is not described by Einstein’s 4-tensor (g[m][n]). Consequently Einstein’s theory of general relativity is not a theory of gravitation; hence its consequences,
gravitational radiation, black holes, quantum gravity, etc. have nothing to do with physical reality! As a physical theory GTR is a massive blunder, but as a geometrical theory of curved space-time
it is a thing of exquisite beauty.
According to the big bang scenario the universe evolved from an undefined initial state via an explosion of a hot, dense mixture of matter and energy. As a consequence of this cosmic explosion the
universe is expanding and this expansion is buoyed up by the matter-antimatter asymmetry (dark energy). The fate of the universe depends on its mass density: If the mass density r = p[c], the
expansion will either stop so that the universe is closed and static, or stop and contract, where r[c] is the closure density. The observed present density r is less than r[c], hence for the universe
to be closed there must exist unobserved density, called the invisible (missing, or dark) matter, to ensure this closure. The arbitrary initial state is in conflict with the principle of relativity
because it violates the dimensionality law of nature, and the assumed matter-antimatter asymmetry is inconsistent with the established matter-antimatter symmetry in the universe [5]. Consequently the
big bang scenario is not a theory of the universe! | {"url":"https://www.scirp.org/journal/paperinformation?paperid=28196","timestamp":"2024-11-02T09:00:05Z","content_type":"application/xhtml+xml","content_length":"92661","record_id":"<urn:uuid:a09d4e45-a2bb-4b05-a10f-f02b0ae37a7b>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00836.warc.gz"} |
Computational reverse mathematics and foundational analysis
Eastaugh, Benedict (2018) Computational reverse mathematics and foundational analysis. [Preprint]
Download (539kB) | Preview
Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse
mathematics proposed thus far is foundational analysis, which explores the limits of different foundations for mathematics in a formally precise manner. This paper gives a detailed account of the
motivations and methodology of foundational analysis, which have heretofore been largely left implicit in the practice. It then shows how this account can be fruitfully applied in the evaluation of
major foundational approaches by a careful examination of two case studies: a partial realization of Hilbert’s program due to Simpson [1988], and predicativism in the extended form due to Feferman
and Schütte.
Shore [2010, 2013] proposes that equivalences in reverse mathematics be proved in the same way as inequivalences, namely by considering only omega-models of the systems in question. Shore refers to
this approach as computational reverse mathematics. This paper shows that despite some attractive features, computational reverse mathematics is inappropriate for foundational analysis, for two major
reasons. Firstly, the computable entailment relation employed in computational reverse mathematics does not preserve justification for the foundational programs above. Secondly, computable entailment
is a Pi-1-1 complete relation, and hence employing it commits one to theoretical resources which outstrip those available within any foundational approach that is proof-theoretically weaker than
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USU Open Personal Contest 2011
Many of you know the universal method of solving simple physics problems: you have to find in a textbook an identity in which you know the values of all the quantities except for one, substitute the
numbers into this identity, and calculate the unknown quantity.
This problem is even easier. You know right away that the identity needed for its solution is the Clapeyron–Mendeleev equation for the state of an ideal gas. This equation relates the pressure of an
ideal gas p, the amount of substance n, the volume occupied by the gas V, and the temperature T. Given three of these quantities, you have to find the fourth quantity. Note that the temperature of a
gas and the volume occupied by it must always be positive.
Each of the three input lines has the form “X = value”, where X is the symbol for a physical quantity and value is a nonnegative integer not exceeding 1000. The three lines specify the values of
three different quantities. Pressure is specified in pascals, amount of substance in moles, volume in cubic meters, and temperature in kelvins. It is guaranteed that the temperature and volume are
positive. The universal gas constant R should be taken equal to 8.314 J / (mol · K).
If the input data are inconsistent, output the only line “error”. If the value of X can be determined uniquely, output it in the format “X = value” with absolute or relative error not more than 10^
−6. If it is impossible to uniquely determine the value of X, output the only line “undefined”.
input output
p = 1
n = 1 T = 0.120279
V = 1
Recall that Pa = N / m^2 and J = N · m.
Problem Author: Benoît Paul Émile Clapeyron, Dmitri Mendeleev
Problem Source: XII USU Open Personal Contest (March 19, 2011) | {"url":"https://timus.online/problem.aspx?space=84&num=4","timestamp":"2024-11-14T17:33:22Z","content_type":"text/html","content_length":"6973","record_id":"<urn:uuid:3429738e-72bd-409a-8406-d6543c4fc2fb>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00868.warc.gz"} |
How many kPa in 5.5 pounds per square inch?
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How many kPa in 5.5 pounds per square inch?
5.5 pounds per square inch equals 37.9212 kilopascals because 5.5 times 6.89476 (the conversion factor) = 37.9212
All In One Unit Converter
Pounds per square inch to kilopascals Conversion Formula
How to convert 5.5 pounds per square inch into kilopascals
To calculate the value in kilopascals, you just need to use the following formula:
Value in kilopascals = value in pounds per square inch × 6.89475728
In other words, you need to multiply the capacitance value in pounds per square inch by 6.89475728 to obtain the equivalent value in kilopascals.
For example, to convert 5.5 pounds per square inch to kilopascals, you can plug the value of 5.5 into the above formula toget
kilopascals = 5.5 × 6.89475728 = 37.92116504
Therefore, the capacitance of the capacitor is 37.92116504 kilopascals. Note that the resulting value may have to be rounded to a practical or standard value, depending on the application.
By using this converter, you can get answers to questions such as:
• How much are 5.5 pounds per square inch in kilopascals;
• How to convert pounds per square inch into kilopascals and
• What is the formula to convert from pounds per square inch to kilopascals, among others.
Pounds Per Square Inch to Kilopascals Conversion Chart Near 4.9 pounds per square inch
Pounds Per Square Inch to Kilopascals
4.9 pounds per square inch 33.78 kilopascals
5 pounds per square inch 34.47 kilopascals
5.1 pounds per square inch 35.16 kilopascals
5.2 pounds per square inch 35.85 kilopascals
5.3 pounds per square inch 36.54 kilopascals
5.4 pounds per square inch 37.23 kilopascals
5.5 pounds per square inch 37.92 kilopascals
5.6 pounds per square inch 38.61 kilopascals
5.7 pounds per square inch 39.3 kilopascals
5.8 pounds per square inch 39.99 kilopascals
5.9 pounds per square inch 40.68 kilopascals
6 pounds per square inch 41.37 kilopascals
6.1 pounds per square inch 42.06 kilopascals
Note: Values are rounded to 4 significant figures. Fractions are rounded to the nearest 8th fraction.
Despite efforts to provide accurate information on this website, no guarantee of its accuracy is made. Therefore, the content should not be used for decisions regarding health, finances, or property. | {"url":"https://www.howmany.wiki/u/How-many--kPa--in--5.5--pound-per-square-inch","timestamp":"2024-11-06T06:14:22Z","content_type":"text/html","content_length":"116056","record_id":"<urn:uuid:f9b24daf-ed3a-4853-9788-d5fb78caa717>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00607.warc.gz"} |
Visualizing a Theory of Everything!
I did a Mathematica (MTM) analysis of several important papers here and here from Mehmet Koca, et. al. The resulting MTM output in PDF format is here and the .NB notebook is here.
3D Visualization of the outer hull of the 144 vertex Dual Snub 24 Cell, with vertices colored by overlap count:
* The (42) yellow have no overlaps.
* The (51) orange have 2 overlaps.
* The (18) tetrahedral hull surfaces are uniquely colored.
The Dual Snub 24-Cell with less opacity
What is really interesting about this is the method to generate these 3D and 4D structures is based on Quaternions (and Octonions with judicious selection of the first triad={123}). This includes
both the 600 Cell and the 120 Cell and its group theoretic orbits. The 144 vertex Dual Snub 24 Cell is a combination of those 120 Cell orbits, namely T'(24) & S’ (96), along with the D4 24 Cell T
3D Visualization of the outer hull of the alternate 96 vertex Snub 24 Cell (S’)
Visualization of the concentric hulls of the Alternate Snub 24 Cell
Various 2D Coxeter Plane Projections with vertex overlap color coding.
3D Visualization of the outer hull of M(192) as one of the W(D4) C3 orbits of the 120-Cell (600)
3D Visualization of the outer hull of N(288) that are the 120-Cell (600) Complement of
the W(D4) C3 orbits T'(24)+S'(96)+M (192)
3D Visualization of the outer hull of the 120-Cell (600) generated using T’
3D Visualization of the outer hull of the 120-Cell (600) generated using T
3D Animation of the 5 quaternion generated 24-cell outer hulls consecutively adding to make the 600-Cell.
3D Animation of the 5 quaternion generated 600-cell outer hulls consecutively adding to make the 120-Cell.
Interactive Cloud VisibLie-E8 4D Periodic Table
This is a link to the free cloud Mathematica demonstration. (Note: You need to enable “Dynamic Behavior” aka. interactivity in the upper left corner).
Please bear in mind that this demonstration is written for a full Mathematica licensed viewer. The cloud deployments are limited in interactivity, especially those that involve 3D and significant
computation. Also, be patient – it takes a minute to load and more than a few seconds to respond to any mouse click interactions.
The utility of the cloud demo of this 4D (3D+color) Periodic Table is in visualizing it in 2D or 3D (from the left side menu) and building up n=1 to 8. Select the Stowe vs. Scerri display for
different 3D models. The explode view slider helps distribute the lattices in the model.
The 2D/3D electron density representations for each atom’s orbitals are too slow for the cloud, so they don’t show anything. The isotope and list-picker of internet curated element data also does not
For an explanation of this pane #10 in the suite of 18 VisibLie-E8 demonstrations, please see this link.
High resolution 4D (3D+color) images of the demonstration.
High resolution 2D images from the demonstration.
A Theory of Everything Visualizer, with links to free Cloud based Interactive Demonstrations:
1) Math: Chaos/Fibr/Fractal/Surface: Navier Stokes/Hopf/MandelBulb/Klein
2) Math: Number Theory: Mod 2-9 Pascal and Sierpinski Triangle
3) Math: Geometric Calculus: Octonion Fano Plane-Cubic Visualize
4) Math: Group Theory: Dynkin Diagram Algebra Create
5) Math: Representation Theory: E8 Lie Algebra Subgroups Visualize
6) Physics: Quantum Elements: Fundamental Quantum Element Select
7) Physics: Particle Theory: CKM(q)-PMNS(ν) Mixing_CPT Unitarity
8) Physics: Hadronic Elements: Composite Quark-Gluon Select Decays
9) Physics: Relativistic Cosmology: N-Body Bohmian GR-QM Simulation
10) Chemistry: Atomic Elements: 4D Periodic Table Element Select
11) Chemistry: Molecular Crystallography: 4D Molecule Visualization Select
12) Biology: Genetic Crystallography: 4D Protein/DNA/RNA E8-H4 Folding
13) Biology: Human Neurology: OrchOR Quantum Consciousness
14) Psychology: Music Theory & Cognition: Chords, Lambdoma, CA MIDI,& Tori
15) Sociology: Theological Number Theory: Ancient Sacred Text Gematria
16) CompSci: Quantum Computing: Poincare-Bloch Sphere/Qubit Fourier
17) CompSci: Artificial Intelligence: 3D Conway’s Game Of Life
18) CompSci: Human/Machine Interfaces: nD Human Machine Interface
Cloud Based VisibLie_E8 Demonstration
The cloud deployments don’t have all the needed features as the fully licensed Mathematica notebooks, so I included a few of the panes that seem to work for the most part. Some 3D and animation
features won’t work, but it is a start. Bear in mind that the response time is slow.
A Theory of Everything Visualizer, with links to free Cloud based Interactive Demonstrations:
1) Math: Chaos/Fibr/Fractal/Surface: Navier Stokes/Hopf/MandelBulb/Klein
2) Math: Number Theory: Mod 2-9 Pascal and Sierpinski Triangle
3) Math: Geometric Calculus: Octonion Fano Plane-Cubic Visualize
4) Math: Group Theory: Dynkin Diagram Algebra Create
5) Math: Representation Theory: E8 Lie Algebra Subgroups Visualize
6) Physics: Quantum Elements: Fundamental Quantum Element Select
7) Physics: Particle Theory: CKM(q)-PMNS(ν) Mixing_CPT Unitarity
8) Physics: Hadronic Elements: Composite Quark-Gluon Select Decays
9) Physics: Relativistic Cosmology: N-Body Bohmian GR-QM Simulation
10) Chemistry: Atomic Elements: 4D Periodic Table Element Select
11) Chemistry: Molecular Crystallography: 4D Molecule Visualization Select
12) Biology: Genetic Crystallography: 4D Protein/DNA/RNA E8-H4 Folding
13) Biology: Human Neurology: OrchOR Quantum Consciousness
14) Psychology: Music Theory & Cognition: Chords, Lambdoma, CA MIDI,& Tori
15) Sociology: Theological Number Theory: Ancient Sacred Text Gematria
16) CompSci: Quantum Computing: Poincare-Bloch Sphere/Qubit Fourier
17) CompSci: Artificial Intelligence: 3D Conway’s Game Of Life
18) CompSci: Human/Machine Interfaces: nD Human Machine Interface
Another book cover using my E8 Petrie projection
This is the link to Amazon’s page for the book; Particle Physics & Representation Theory: ” Mathematical Symmetries of The Universe ”
Cover w/link to Amazon
My VisbLie E8 demonstration system for Mathematica v13
The newer version of the VisibLieE8-NewDemo-v13.nb (130 Mb) will work with those who have a full Mathematica v13 license. It is backward compatible to earlier versions. There are a few bug fixes from
the older version of ToE_Demonstration.nb (130 Mb), which should work on v13 and older versions as well.
For more detail on the modules, see this blog post.
Please be patient, it is very large and can take 10 minutes to load, depending on your Internet connection, memory and CPU speed.
Asteroid 1994 PC1
30 Second Exposure at 6:45 pm Tucson Arizona
Asteroid 1994 PC1
6:40 – 7:10 pm
The free Wolfram CDF Player v. 13 works with my VisibLie E8 ToE demonstration on Win10
In case you’re interested, I just verified the demo works on the free Mathematica CDF player v.13 for Win10.
Just go to https://www.wolfram.com/player/ install, download and open the app:
There is a ton of other cool interactive stuff in there. FYI – Some features don’t work without a full Mathematica license.
ISS Lunar Transit
I’ve waited over a year for the ISS to pass directly between the house and the moon. That happened at 4:58:13 this morning.
Composite image
12-fold Symmetric Quasicrystallography from affine E6, B6, and F4
This post is an analysis of a June 2013 paper by Mehmet Koca, Nazife Koca, and Ramazan Koc. That paper contains various well-known Coxeter plane projections of hyper-dimensional polytopes as well as
a new direct point distribution of the quasicrystallographic weight lattice for E6 (their Figure 3), as well as the quasicrystal lattices of B6 and F4.
Koca / Koc Figure 3 E6 Quasicrystallographic Weight Lattice
What is interesting about this projection is that it precisely matches the point distribution (to within a small number of vertices) from a rectified E8 projection using a set of basis vectors I
discovered in December of 2009, published in Wikipedia (WP) in February of 2010 here.
Rectified E8 in my “Triality” projection basis:
x=(2-4/√3 , 0 , 1-1/√3 , 1-1/√3 , 0 , -1 , 1 , 0 )
y=( 0 , -2+4/√3 , -1+1/√3 , 1-1/√3 , 0 , 1/√3 , 1/√3 , -2/√3 )
Rectification of E8 is a process of replacing the 240 vertices of E8 with points that represent the midpoint of each of the 6720 edges. In this projection, there are overlaps which are indicated by
different colors in the color-coded WP image linked above.
The image below is an overlay of the above images highlighting the 12*(9+3+26+7)=540 points that are not overlapping:
Annotated overlap comparison showing missing 432 overlaps.
It is interesting to note that with a 30° rotation of my projection, the missing overlaps are reduced to 12*(15+2)=204.
Annotated overlap comparison showing missing 204 overlaps.
Given the paper’s explanation for the methods using E6 (720) with 6480 edges as a projection through a 4D 3-sphere window defined by q1 and q6, it may be insightful to study my projection basis for
E8’s triality relationships with the Koca/Koc paper’s defined 4D 3-sphere.
For more information on why my projection basis is called the E8 Triality projection, see this post. | {"url":"https://theoryofeverything.org/theToE/topics/physics/page/3/","timestamp":"2024-11-06T14:33:36Z","content_type":"text/html","content_length":"101786","record_id":"<urn:uuid:a8ce1218-f0b1-485e-8f3c-8bbd1d97f3d8>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00613.warc.gz"} |
Critical Independent Sets and König-Egerváry Graphs
A set S of vertices is independent or stable in a graph G, and we write S ∈ Ind (G), if no two vertices from S are adjacent, and α(G) is the cardinality of an independent set of maximum size, while
core(G) denotes the intersection of all maximum independent sets. G is called a König-Egerváry graph if its order equals α(G) + μ(G), where μ(G) denotes the size of a maximum matching. The number def
(G) = {pipe}V(G){pipe} -2μ(G) is the deficiency of G. The number def(G)={pipe}V(G){pipe}-2μ(G) is the deficiency of G. The number d(G)=max{{pipe}S{pipe}-{pipe}N(A){pipe}=d(G)}, where N(S) is the
neighbourhood of S, and α [c](G) denotes the maximum size of a critical independent set. Lrson (Eur J Comb 32:294-300, 2011)demonstrated that G is a k̈nig-Egerváry graph if and only if there exists a
maximum independent set that is also critical, i.e., α [c](G)=α(G). In this paper we prove that: (i) d(G)={pipe}(G){pipe}-{pipe}N(core(G)){pipe}=α(G)=def(G) holds for every König-Egerváry graph G;
(ii) G is König-Egerváry graph if and only if each maximum independent set of G is critical.
• Core
• Critical difference
• Critical independent set
• Deficiency
• Maximum independent set
• Maximum matching
Dive into the research topics of 'Critical Independent Sets and König-Egerváry Graphs'. Together they form a unique fingerprint. | {"url":"https://cris.ariel.ac.il/en/publications/critical-independent-sets-and-k%C3%B6nig-egerv%C3%A1ry-graphs-3","timestamp":"2024-11-02T18:19:42Z","content_type":"text/html","content_length":"54824","record_id":"<urn:uuid:c1098cf0-9b4d-4767-bb4d-d11c758d28d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00023.warc.gz"} |
Dividing Polynomials - Definition, Synthetic Division, Long Division, and Examples
Polynomials are arithmetical expressions that consist of one or more terms, all of which has a variable raised to a power. Dividing polynomials is an important working in algebra which involves
figuring out the remainder and quotient once one polynomial is divided by another. In this article, we will examine the different approaches of dividing polynomials, including synthetic division and
long division, and provide instances of how to apply them.
We will further talk about the importance of dividing polynomials and its utilizations in various domains of mathematics.
Prominence of Dividing Polynomials
Dividing polynomials is a crucial operation in algebra that has several uses in various fields of math, consisting of number theory, calculus, and abstract algebra. It is applied to solve a extensive
array of challenges, consisting of figuring out the roots of polynomial equations, calculating limits of functions, and working out differential equations.
In calculus, dividing polynomials is applied to figure out the derivative of a function, which is the rate of change of the function at any point. The quotient rule of differentiation includes
dividing two polynomials, which is utilized to find the derivative of a function which is the quotient of two polynomials.
In number theory, dividing polynomials is used to learn the features of prime numbers and to factorize huge numbers into their prime factors. It is further utilized to study algebraic structures for
instance rings and fields, that are fundamental theories in abstract algebra.
In abstract algebra, dividing polynomials is used to specify polynomial rings, that are algebraic structures which generalize the arithmetic of polynomials. Polynomial rings are used in many domains
of arithmetics, involving algebraic number theory and algebraic geometry.
Synthetic Division
Synthetic division is an approach of dividing polynomials that is applied to divide a polynomial with a linear factor of the form (x - c), where c is a constant. The technique is founded on the fact
that if f(x) is a polynomial of degree n, therefore the division of f(x) by (x - c) gives a quotient polynomial of degree n-1 and a remainder of f(c).
The synthetic division algorithm involves writing the coefficients of the polynomial in a row, utilizing the constant as the divisor, and performing a series of calculations to figure out the
remainder and quotient. The result is a simplified form of the polynomial that is simpler to function with.
Long Division
Long division is a technique of dividing polynomials that is utilized to divide a polynomial by another polynomial. The technique is based on the fact that if f(x) is a polynomial of degree n, and g
(x) is a polynomial of degree m, at which point m ≤ n, next the division of f(x) by g(x) offers uf a quotient polynomial of degree n-m and a remainder of degree m-1 or less.
The long division algorithm consists of dividing the highest degree term of the dividend with the highest degree term of the divisor, and further multiplying the answer with the whole divisor. The
answer is subtracted of the dividend to get the remainder. The procedure is recurring as far as the degree of the remainder is lower in comparison to the degree of the divisor.
Examples of Dividing Polynomials
Here are some examples of dividing polynomial expressions:
Example 1: Synthetic Division
Let's assume we want to divide the polynomial f(x) = 3x^3 + 4x^2 - 5x + 2 with the linear factor (x - 1). We can apply synthetic division to streamline the expression:
1 | 3 4 -5 2 | 3 7 2 |---------- 3 7 2 4
The answer of the synthetic division is the quotient polynomial 3x^2 + 7x + 2 and the remainder 4. Hence, we can state f(x) as:
f(x) = (x - 1)(3x^2 + 7x + 2) + 4
Example 2: Long Division
Example 2: Long Division
Let's assume we want to divide the polynomial f(x) = 6x^4 - 5x^3 + 2x^2 + 9x + 3 with the polynomial g(x) = x^2 - 2x + 1. We could utilize long division to streamline the expression:
First, we divide the highest degree term of the dividend with the highest degree term of the divisor to obtain:
Then, we multiply the total divisor by the quotient term, 6x^2, to attain:
6x^4 - 12x^3 + 6x^2
We subtract this from the dividend to obtain the new dividend:
6x^4 - 5x^3 + 2x^2 + 9x + 3 - (6x^4 - 12x^3 + 6x^2)
that streamlines to:
7x^3 - 4x^2 + 9x + 3
We recur the process, dividing the highest degree term of the new dividend, 7x^3, by the highest degree term of the divisor, x^2, to achieve:
Subsequently, we multiply the total divisor by the quotient term, 7x, to achieve:
7x^3 - 14x^2 + 7x
We subtract this from the new dividend to achieve the new dividend:
7x^3 - 4x^2 + 9x + 3 - (7x^3 - 14x^2 + 7x)
that simplifies to:
10x^2 + 2x + 3
We repeat the method again, dividing the largest degree term of the new dividend, 10x^2, with the largest degree term of the divisor, x^2, to obtain:
Next, we multiply the entire divisor by the quotient term, 10, to get:
10x^2 - 20x + 10
We subtract this of the new dividend to obtain the remainder:
10x^2 + 2x + 3 - (10x^2 - 20x + 10)
that streamlines to:
13x - 10
Thus, the answer of the long division is the quotient polynomial 6x^2 - 7x + 9 and the remainder 13x - 10. We can express f(x) as:
f(x) = (x^2 - 2x + 1)(6x^2 - 7x + 9) + (13x - 10)
In Summary, dividing polynomials is a crucial operation in algebra that has many applications in multiple domains of math. Getting a grasp of the different approaches of dividing polynomials, such as
long division and synthetic division, could help in working out intricate problems efficiently. Whether you're a learner struggling to get a grasp algebra or a professional working in a domain that
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in-person to offer customized and effective tutoring services to help you be successful. Contact us right now to schedule a tutoring session and take your mathematics skills to the next level. | {"url":"https://www.ontariocainhometutors.com/blog/dividing-polynomials-definition-synthetic-division-long-division-and-examples","timestamp":"2024-11-09T03:43:13Z","content_type":"text/html","content_length":"77685","record_id":"<urn:uuid:436a6765-384f-47fc-b8d5-cb061fbddd67>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00084.warc.gz"} |
Delay-Line Nomograms, November 1962 Electronics World
November 1962 Electronics World
Table of Contents
Wax nostalgic about and learn from the history of early electronics. See articles from Electronics World, published May 1959 - December 1971. All copyrights hereby acknowledged.
Delay lines are used in electronic circuits for precisely adjusting the timing of signals. That can be to set times between events or to adjust two or more signals so that they arrive at some point
in the circuit at a specific time with respect to each other. In a radar system, for instance, a sample of the reflected signal might be delayed in time by one pulse repetition period in order to
compare it to the current reflected signal so that stationary (fixed, non-changing) signals can be cancelled out, leaving only the signal that has changed since the last sample. That is how MTI
(moving target indication) functions. In today's world the samples are stored digitally and then compared digitally with other signals, but previously in fully analog systems, sending the sample
along a longer (in time) path for comparison was necessary. Delay lines can be electrical like the ones covered in this 1962 issue of Electronics World magazine, or they can be mechanical such as
with a quartz or mercury delay line. The provided nomographs are for LC (inductor-capacitor) delay lines.
Delay-Line Nomograms
By Donald W. Moffat
Useful graphical information to speed the design of LC delay lines with various delays and rise times.
Delay lines are finding many applications in electronic equipment because they are passive timing devices capable of extremely good accuracy under severe environmental conditions. Many of these lines
can be made in any laboratory and the accompanying nomograms will enable the reader to design a delay line quickly for the desired characteristics.
Two broad classes of delay lines are the mechanical and the electromagnetic. The first group is characterized by long delays, up to thousands of microseconds, and large attenuation. They are made of
special and expensive equipment and are not ordinarily within the province of anyone but the specialist in their manufacture. On the other hand, electromagnetic delay lines consist of a network of
coils and capacitors, as shown in Fig. 1, and experimental models can be constructed at any electronic workbench.
The length of time by which such a line delays the signal is a function of just the total inductance and capacity, in accordance with the formula: T = √LC, which uses the basic units of seconds,
henrys, and farads. If inductance and capacity are expressed in microhenrys and microfarads, respectively, then time will be calculated in microseconds. This equation shows that time delay can be
increased by increasing either capacity or inductance or both.
However, if the characteristic impedance of the line is to be considered, the ratio of L to C must be watched. Characteristic impedance of a line is the impedance which the line presents to the
circuit that feeds it. For instance, if the signal from a source with 2000-ohm internal impedance drops to half its open-circuit value when a delay line is connected across it, then the delay line
also has an impedance of 2000 ohms. When the delayed signal reaches the end of the delay line, some of it will be reflected back unless the line is terminated in a resistance equal to its
characteristic impedance. In general, proper matching will produce the best waveform and the maximum signal output.
The formula for characteristic impedance is: Z[0] = √LC, where Z[0] is in ohms when both Land C have the same prefix, such as "micro." This equation shows that increasing inductance will increase
impedance, increasing capacity will decrease impedance, and they can both be changed without affecting impedance if their ratio remains unchanged.
In Fig. 1, the coils are in series and the capacitors are in parallel. Therefore, total values as given by both the formulas are found by adding up those of each section. Conversely, the values for
one section are found by dividing the totals by the number of sections. The number of sections is selected on the basis of the desired quality factor, which is defined as total delay divided by
output rise time. The higher this ratio, the better the delay line because either a long delay or a short rise time will increase the quality factor. In designing a delay line, the procedure is to
select values of total inductance and capacitance, then divide those totals into the number of sections necessary to give the desired quality factor.·
An Example
This example will help explain the use of the nomograms. Suppose it is desired to have a total delay of 2 μsec., a rise time of 0.2 μsec., and a characteristic impedance of 600 ohms. First, we refer
to the nomogram in Fig. 3.
On Fig. 3, locate 2 μsec. on the "Time Delay" scale and 600 on the "Impedance" scale. Draw a straight line through these points and where that line crosses the other scales, it will give the required
values of inductance and capacitance as 1.2 millihenrys and 3200 μf., respectively.
Use Fig. 2 to determine the number of sections required. Locate 2 μsec, on the "Time Delay" scale, 0.2 μsec. on the "Rise Time" scale, and draw a straight line through these two points. At the middle
scale the line indicates that 2.5 sections are required, therefore each coil should have an inductance of 48 micro-henrys and each capacitor should have a value of 128 (nearest standard value of 130)
This basic section of 48 microhenrys and 130 μμf. can be used to make small corrections to the total delay. Each section contributes a delay of 1/25 of the total, or 0.08 μsec. and sections can be
added without affecting the characteristic impedance of the line, because the ratio of L to C will remain unchanged as sections are added.
Posted October 13, 2022
Nomographs / Nomograms Available on RF Cafe:
- Parallel Series Resistance Calculator
- Transformer Turns Ratio Nomogram
- Symmetrical T and H Attenuator Nomograph
- Amplifier Gain Nomograph
- Decibel Nomograph
- Voltage and Power Level Nomograph
- Nomograph Construction
- Nomogram Construction for Charts with Complicating Factors or Constants
- Link Coupling Nomogram
- Multi-Layer Coil Nomograph
- Delay Line Nomogram
- Voltage, Current, Resistance, and Power Nomograph
- Resistor Selection Nomogram
- Resistance and Capacitance Nomograph
- Capacitance Nomograph
- Earth Curvature Nomograph
- Coil Winding Nomogram
- RC Time-Constant Nomogram
- Coil Design Nomograph
- Voltage, Power, and Decibel Nomograph
- Coil Inductance Nomograph
- Antenna Gain Nomograph
- Resistance and Reactance Nomograph
- Frequency / Reactance Nomograph | {"url":"https://rfcafe.com/references/electronics-world/delay-line-nomograms-electronics-world-november-1962.htm","timestamp":"2024-11-02T22:04:27Z","content_type":"text/html","content_length":"34393","record_id":"<urn:uuid:3c1e30b8-375c-470c-abf9-3fbf500ed5a7>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00541.warc.gz"} |
Irredundant families, the Smyth powerdomain, the Lyu-Jia theorem, and the baby Groemer theorem
Irreducible elements and irredundant families
If you are familiar with sober spaces, you know that an irreducible closed subset of a topological space Z is a non-empty closed subset C such that, for all closed subsets C[1] and C[2], if C is
included in C[1] ∪ C[2], then C is included in C[1] or in C[2].
In general, given any family L of subsets of a set Z, let me say that E ∈ L is irreducible if and only if E is non-empty and for all elements E[1] and E[2] of L, if E is included in E[1] ∪ E[2], then
E is included in E[1] or in E[2].
Let me call a family L of subsets of Z irredundant if and only if all its non-empty elements are irreducible. More specifically, I will be interested in irredundant ∩-semilattices, where a
∩-semilattice is simply a collection of sets that is closed under binary intersections. That may look like a stupidly overconstrained notion, to the point that one may legitimitely ask whether there
is any non-trivial example.
This is a very interesting notion, as I will attempt to demonstrate. It was introduced in [1], and the credit is entirely due to the late Klaus Keimel. Also, it is at the root of a clever argument
due to Zhenchao Lyu and Xiadong Jia [2], which exactly characterizes when the Smyth powerdomain Q(X) of a topological space X is core-compact.
Two notes, before I start:
• The result of [2] was found by Zhenchao Lyu (and perhaps Xiaodong Jia) while he and Xiaodong Jia were postdocs in my lab in 2018-2019. (Update, January 21, 2022: X. Jia tells me his role in this
was only in asking the question, and that the solution is entirely due to Zh. Lyu.) I was expecting them to be able to publish this quickly, but somehow I can only guess that something went
wrong. I usually have the policy of not talking about a paper before it is published (except for my own ideas), but this is so neat (and available from arXiv) that I have decided to talk about it
without waiting any further. (Update, January 21, 2022: also, it is cited as reference 38 in a recent survey paper by Xiaoquan Xu and Dongsheng Zhao.)
• There is some ambiguity in the definition of “irredundant” in [1], which may lead to some mistakes. I have given a clarified definition above. For more details on the ambiguity, see Appendix A at
the end of this post.
The main example: the irredundant base of the upper Vietoris topology on Q(X)
There are at least two examples of an irredundant ∩-semilattice in [1], and here is the one which will be the center of our attention in this post.
Fact. For every topological space X, the base of open sets ☐U of the Smyth hyperspace Q(X), where U ranges over the open subsets of X, is an irredundant ∩-semilattice.
Let me recall that Q(X) is the set of non-empty compact saturated subsets of X. (Omitting “non-empty” would not change much in the sequel.) The collection of sets ☐U, U ∈ OX, is a base for a topology
called the upper Vietoris topology, and Q(X) with that topology is the Smyth hyperspace of X.
The proof of the above Fact is easy. The sets ☐U form a ∩-semilattice because ☐U ∩ ☐V = ☐(U ∩ V). Let now ☐U be non-empty and included in ☐V[1] ∪ ☐V[2]. For the sake of contradiction, let us assume
that ☐U is not included in ☐V[1], and not included in ☐V[2] either. Then there is a non-empty compact saturated subset Q[1] of X that is included in U but not in V[1], and there is a non-empty
compact saturated subset Q[2] of X that is included in U but not in V[2]. We form Q[1] ∪ Q[2], which is a non-empty compact saturated subset of X that is included in U, but is included neither in V
[1] nor in V[2], contradicting ☐U ⊆ ☐V[1] ∪ ☐V[2].
The other example is the collection of crescents of the form ☐U–♢V in the space of compact lenses (also known as the Plotkin powerdomain) with the Vietoris topology. I will not expand on that.
When is Q(X) core-compact? The Lyu-Jia theorem
First, let me write ↑[Q] Q for the the upward closure of a single point Q of Q(X). That is taken with respect to the specialization ordering of Q(X), which is reverse inclusion ⊇. Hence ↑[Q] Q is the
set of non-empty compact saturated sets Q’ that are included in Q. I know, this may sound confusing at first.
Extending the notation ☐U, I could have just written ↑[Q] Q as ☐Q, the set of non-empty compact saturated sets included in Q. The ☐ operator, defined by ☐A ≝ {Q ∈ Q(X) | Q ⊆ A} for every subset A of
X, is monotonic and commutes with finite intersections.
It is well-known that, for every locally compact space X, Q(X) is locally compact. Indeed, let Q ∈ Q(X), and let U be an open neighborhood of Q. U contains a basic open set ☐U, where U is open in X,
such that Q is in ☐U, namely such that Q is included in U. By the interpolation property in locally compact spaces (Proposition 4.8.14 in the book), there is (necessarily non-empty) compact saturated
subset Q’ of X such that Q ⊆ U’ ⊆ Q’ ⊆ U, where U’ is the interior of Q’. Therefore Q ∈ ☐U‘ ⊆ ☐Q’ ⊆ ☐U. Since ↑[Q] Q‘ = ☐Q’, we can interpret the latter as saying that Q is in the interior of ↑[Q] Q
‘, which is itself included in ☐U, hence in U.
We have in fact just shown more: for every locally compact space X, Q(X) is a c-space. (See Section 5.1.2 in the book. A locally compact space is a space Z in which for every point z, and for every
open neighborhood W of z, we can find a compact saturated neighborhood K of z included in W. In a c-space, we require that K be the upward closure of a single point, just like ↑[Q] Q‘ in the previous
It is also known that Q(X) is locally compact if and only if X is locally compact. (We will reprove this as a consequence of a more general result below.)
What Zhenchao was interested in was whether one could prove a similar theorem with “core-compact” replacing “locally compact”. The perhaps surprising outcome is that… no, you cannot.
But let us not move too fast. Here we are right now. We know of the following implications:
X loc. compact ⇒ Q(X) c-space ⇒ Q(X) loc. compact ⇒ Q(X) core-compact,
where the last two implications hold with any space in place of Q(X); the main result of [2] is to show that this is actually a string of equivalences.
In order to see this, we assume that Q(X) is core-compact, and we will show that X is locally compact.
Although not strictly needed, let us extend the notion of irreducibility and of irredundancy to a more abstract setting. (I am saying that this is not strictly needed, because I will always apply
that to the concrete cases of irredundant ∩-semilattices as above.) Given a sup-semilattice Ω, a subset L of Ω is irredundant if and only if every element u of L different from ⊥ is irreducible in
the following sense: u≠⊥ and for all v[1], v[2] in L such that u ≤ v[1] ⋁ v[2], we have u ≤ v[1] or u ≤ v[2].
A prime-continuous lattice is a complete lattice in which every element is the supremum of the elements that are way-way-below it. The prime-continuous lattices are exactly the completely
distributive, complete lattices by Raney’s Theorem (Exercise 8.3.16 in the book).
The key point is the following, which we will apply to the case where Ω is the lattice of open subsets of Q(X). (Let me recall that a space is core-compact if and only if its open set lattice is
continuous, see Definition 5.2.3 in the book.)
Lemma. Let Ω be a complete lattice, and L be a family of elements of Ω. If Ω is continuous, if L generates Ω (in the sense that every element of Ω is the supremum of a family of elements of L), and
if L is irredundant, then Ω is prime-continuous.
Proof. Let us assume that every element of Ω is the supremum of elements of L. We will require the other assumptions later. The main idea is to use irredundancy in order to prove the following:
• For every irreducible element p of L, for every v in Ω, p is way-way-below v (p⋘v) if and only if p is way-below v (p≪v).
Such a property is well-known if p is coprime in Ω (namely, p≠⊥ and for all v[1], v[2] in Ω such that u ≤ v[1] ⋁ v[2], we have u ≤ v[1] or u ≤ v[2], see Exercise 8.3.47 of the book), but we only
require that p be irreducible in L, which is a much weaker property, and also an easier property to verify. (Have you seen the difference, by the way? This is subtle: in the definition of
coprime, v[1], v[2] vary in the whole of Ω, while they range over the smaller set L in the definition of irreducible.)
Anyway, here is how we prove that p⋘v if and only if p≪v, when p is irreducible in L.
Let me recall that u⋘v if and only if every family (not necessarily directed) whose supremum lies above v contains an element that is already above u. Notably, u⋘v implies u≪v.
Conversely, if p≪v and if p is an irreducible element of L, let us consider any family F of elements of Ω such that v≤sup F. Every element w of F is a supremum of a family F[w] of elements of L,
so sup F = sup (∪[w ∈ F] F[w]). Since sup (∪[w ∈ F] F[w]) can also be written as the directed supremum of all suprema of finite subfamilies of ∪[w][ ∈ ][F] F[w], and using the definition of the
way-below relation ≪, p ≤ u[1] ⋁ … ⋁ u[n] for some finite subfamily {u[1], …, u[n]} of ∪[w ∈ F] F[w]. Since p is irreducible (and since u[1], …, u[n] are all in L), we must have p ≤ u[i] for some
i; u[i] is in F[w] for some w ∈ F, so u[i] ≤ w, and therefore p ≤ w. This shows that p⋘v.
We now assume that L is irredundant. What we have just shown simplifies to: for every p ∈ L, for every v in Ω, p⋘v if and only if p≪v.
Finally, we also assume that Ω is continuous. For every v ∈ Ω, we can write v as the supremum of a directed family of elements v[i]≪v, where i ranges over some indexing set I. Each v[i] is a supremum
of a family F[i] ⊆ L, so v is the supremum of the family ∪[i ∈ I] F[i]. That family is a family of elements p of L, and each of them is not just way-below, but way-way-below v, as we have just seen.
Hence Ω is prime-continuous. ☐
As an application, and as promised, we specialize that to the case where Ω=O(Q(X)) and L is the base of the upper Vietoris topology consisting of sets of the form ☐U, U ∈ O(X), which is our primary
example of an irredundant ∩-semilattice:
Corollary. For every topological space X, if Q(X) is core-compact then it is a c-space.
Proof. If Q(X) is core-compact, then O(Q(X)) is continuous, hence prime-continuous by the previous Lemma. But every space whose lattice of open sets is prime-continuous is a c-space. Lemma 8.3.42 in
the book almost proves this… but assuming that the space is sober. This is really silly: it is pretty easy to see that this assumption is not necessary, and the easiest way to see this is as follows.
(Update, January 22nd, 2022: or see the elementary proof in Appendix B.)
Let Y be a space such that O(Y) is prime-continuous, and let S(Y) be the sobrification of Y. Let me recall that the elements of S(Y) are the irreducible closed subsets of Y, and that the open subsets
of S(Y) are the sets ♢U ≝ {C ∈ S(Y) | C intersects U}, for every open subset U of Y. The latter defines an order-isomorphism U ↦ ♢U of O(Y) onto O(S(Y)). Then O(S(Y))≅O(Y) is prime-continuous, so S(Y
) is a c-space, using Lemma 8.3.42.
It remains to see that every space Y such that S(Y) is a c-space is itself a c-space. In order to see this, let y be any point in Y and let V be any open neighborhood of y in Y. Then ↓y is a point of
S(Y), and ♢V is an open neighborhood of ↓y. Since S(Y) is a c-space, there is an element C of ♢V and an open set ♢U such that ↓y ∈ ♢U ⊆ ↑[S] C, where ↑[S] C denotes the upward closure {C’ ∈ S(Y) | C
⊆ C’} of C in S(Y). Since C is in ♢V, it intersects V, say at z. The inclusion ♢U ⊆ ↑[S] C means that every irreducible closed set that intersects U contains C, and therefore that for every point x
in U, ↓x contains C; in particular, z≤x. Since x is arbitrary in U, this shows that U ⊆ ↑z. Finally, ↓y ∈ ♢U means that y is in U. In summary, we have obtained a point z in V such that y ∈ U ⊆ ↑z,
and this concludes the proof that Y is a c-space. ☐
By the way, Lemma 8.3.41 in the book states that every c-space Y is such that O(Y) is prime-continuous, and therefore the c-spaces are exactly the spaces whose lattice of open sets is
prime-continuous. This had been known for a long time. As far as I can tell, this appears as the equivalence between (a) and (e) in Proposition 2.2.C of Marcel Erné’s 1991 paper on a-spaces,
b-spaces, and c-spaces [3].
We finally have:
Fact. If Q(X) is a c-space, then X is locally compact.
Indeed, let x be any point of X and U be any open neighborhood of x in X. Then ↑x is in ☐U, and since Q(X) is a c-space, ↑x is in the interior of ↑[Q] Q = ☐Q for some element Q of ☐U. Namely, there
is a basic open set ☐V such that ↑x ∈ ☐V ⊆ ☐Q ⊆ ☐U. Observing that for all sets A, B, the inclusion ☐A ⊆ ☐B implies A ⊆ B (for every z in A, ↑z is in ☐A, hence in ☐B, and therefore z is in B), it
follows that x ∈ V ⊆ Q ⊆ U.
Combining this with the previous corollary, if Q(X) is core-compact then X is locally compact. We have obtained the promised chain of equivalences.
Theorem [2, Theorem 3.1]. For every topological space X, the following equivalences hold:
X loc. compact ⇔ Q(X) c-space ⇔ Q(X) loc. compact ⇔ Q(X) core-compact.
It follows that the Q functor does not preserve core-compactness in general: given any space X that is core-compact but not locally compact, such as this one, Q(X) is not core-compact. This is
Corollary 3.4 of [2].
The baby Groemer theorem
The original application that Klaus found of irredundant ∩-semilattices [1] was the following problem.
A lattice of sets is a collection Ω of subsets of a given set X that is closed under finite unions and finite intersections. Among the finite intersections, one finds the intersection of the empty
family, so X itself is in X. Among the finite unions, one finds the union of the empty family, so the empty set ∅ is in Ω. A lattice of sets is of course a bounded lattice, consisting of subsets of a
set X, but the definition also requires that finite infima are intersections and that finite suprema are unions.
Any topology, any σ-algebra is a lattice of sets.
Let us called signed valuation on a lattice of sets Ω any map μ : Ω → R that is:
• strict: μ(∅)=0
• modular: for all U, V in Ω, μ(U ∪ V)=μ(U)+μ(V)–μ(U ∩ V).
For example, any signed measure on a measurable set is a signed valuation on its σ-algebra. The main differences between a signed measure and a measure is that: 1. a signed measure is not required to
give non-negative values to sets 2. (less easy to see) a signed measure cannot give infinite measure to any measurable set.
I have already talked briefly about valuations (not signed valuations) in various posts. Valuations are a close cousin of measures, much as signed valuations are a closed cousin of signed measures.
You can have a look here, for a nifty idea of Alex Simpson’s about a formalization of the notion of random elements of a space; or the appendix of this post, where I build the so-called Lebesgue
valuation on R and on the Sorgenfrey line by a purely domain-theoretic method.
One of the main, basic problems in measure or valuation theory is showing that we can build a measure (or valuation) satisfying certain constraints. For example, building the Lebesgue measure on R
reduces to showing that there is a measure on R that maps every interval [a, b] to b–a. There are many theorems that allow you do such things. In measure theory, one usually builds the Lebesgue
measure by using the so-called Carathéodory extension theorem.
In [1], one of our basic problems was the following. Let X be a topological space, and let us consider the lattice of sets O[fin](Q(X)) obtained as finite unions of basic open subsets ☐U of Q(X). O
[fin](Q(X)) is not the full topology O(Q(X)), and the reason I am considering this here is because this will make the argument simpler, and it will allow me to concentrate on the core of the more
complex arguments of [1].
Now imagine you are given values ν(U) for each open subset U of X, and you wish to build a signed valuation μ on O[fin](Q(X)) such that μ(☐U)=ν(U) for every U. Is that possible? Under which
Naturally, when U is empty, we require ν(∅) to be equal to 0, otherwise there would be no solution for μ. The nice thing is that ν(∅)=0 is the only constraint that ever has to be satisfied for μ to
exist; and, in fact, μ is then uniquely determined from the knowledge of ν.
Klaus realized that this followed from a theorem due to Groemer [4], which he found by randomly browsing through a few books on his shelf, and in particular a book by Klain and Rota [5], where this
theorem is stated and proved. Klaus also realized that we only needed a very special case of that theorem, and that this very special case had an amazingly simple proof. (I had proved all the main
theorems of [1] before Klaus found all that. My proofs were extremely laborious, and Klaus showed how to simplify them drastically using those wonderful theorems, sometimes reducing my original
proofs by a factor between 20 and 40.)
I will call that very special case the baby Groemer theorem.
So here is how we proceed. We fix a set X. For every subset A of X, we can build the characteristic function χ[A], which maps every element of A to 1, and all other elements to 0. The following is an
easy exercise, and should remind you of the modularity requirement of signed valuations.
Fact. For all sets U, V, χ[U ∪ V]=χ[U]+χ[V]–χ[U ∩ V].
The collection of all maps from X to R is a real vector space under pointwise addition and scalar multiplication. For example, every characteristic map χ[U] is a vector in that vector space, as is
any linear combination such as χ[U]–2χ[V], for example.
The key ingredient that will lead us to the baby Groemer theorem is the following.
Lemma. Let L be any irredundant ∩-semilattice of subsets of X. The vectors χ[U], where U ranges over the non-empty sets in L, are linearly independent.
Proof. Let us assume that ∑[U] a[U] . χ[U] = 0, where U ranges over a non-empty finite subset E of L–{∅} and each a[U] is non-zero. Let U[0] be a maximal element of E with respect to inclusion, and U
[1], …, U[n] be the other elements of E (namely, U[i]≠U[0] for every i, 1≤i≤n). If U[0] were included in U[1] ∪ … ∪ U[n], then by irredundancy U[0] would be included in some U[i], hence equal to it
by maximality. That is impossible since U[i]≠U[0]. Hence U[0] is not included in U[1] ∪ … ∪ U[n]. Let x be a point in U[0] that is not in U[1] ∪ … ∪ U[n]. Evaluating ∑[U] a[U] . χ[U] on x, all the
terms except the term U=U[0] vanish, and we obtain the non-zero value a[U[0]]: this contradicts the fact that ∑[U] a[U] . χ[U] = 0. ☐
And here is the baby Groemer theorem. The lattice of subsets L^∪ generated by L is the smallest lattice of subsets containing L, namely the smallest collection of subsets containing L and closed
under finite unions and finite intersections. When L is a ∩-semilattice, the elements of L^∪ are juste the finite unions of elements of L. The Boolean algebra of subsets A(L) generated by L is the
smallest family of subsets of X that contains L and is closed under finite unions, finite intersections, and complements. Its elements are the finite unions of finite intersections of L-literals,
where an L-literal is an element of L or a complement of an element of L. I will also call L-clause any finite intersection of L-literals. It is relatively easy to check that the elements of A(L) are
also the disjoint finite unions of L^∪-crescents, where an L^∪-crescent is a set of the form A–B where A and B are in L^∪ and B is included in A.
Theorem. Let L be any irredundant ∩-semilattice of subsets of X. Given any map ν from L–{∅} to R, there is a unique signed valuation μ on L^∪, and in fact on A(L), that extends ν.
Proof. Let F(L) be the linear space of functions from X to R generated by the functions χ[U], where U ranges over L–{∅}. By the previous Lemma, those functions form a basis of F(L). Hence there is a
unique linear map f from F(L) to R such that f(χ[U])=ν(U) for every U in L–{∅}.
We verify that for every A in L^∪, χ[A] is in F(L). This is easy. A is a finite union U[1] ∪ … ∪ U[n] of elements of L, and we prove this by induction on n. If n=0, then χ[∅]=0 is in F(L). If n=1,
this is by definition of F(L). Otherwise, let B ≝ U[2] ∪ … ∪ U[n]. By the Fact seen above, χ[A] = χ[U][[1]]+χ[B]–χ[U][[1]∩][B]; χ[U][1] is in F(L) by definition, χ[B] is in F(L) by induction
hypothesis, and χ[U][[1]∩][B] = χ[(][U][[1] ∩ ][U][[2]) ∪ … ∪ (][U][[1] ∩ ][U[n]][)] is also in F(L) by induction hypothesis, since U[1] ∩ U[2], …, U[1] ∩ U[n] are all in L, since L is a
∩-semilattice of subsets.
It follows that for every L^∪-crescent C ≝ A–B, χ[C] is also in F(L), since χ[C]=χ[A]–χ[B].
We then obtain that for every E in A(L), χ[E] is in F(L). Indeed, E is a finite disjoint union of L^∪-crescents C[i], 1≤i≤n, and then χ[E] = Σ[i][=1]^n χ[C[i]].
We now define μ by μ(E) ≝ f(χ[E]) for every E in A(L). This is legitimate, since we have just proved that χ[E] is in F(L). This map μ is:
• strict, because μ(∅) = f(χ[∅]) = f(0) = 0;
• modular, because for all E, F in A(L), μ(E ∪ F) = f(χ[E][ ∪ ][F]) = f(χ[E])+f(χ[F])–f(χ[E][ ∩ ][F])=μ(E)+μ(F)–μ(E ∩ F), by the Fact once again, and the linearity of f.
• Finally, μ extends ν, since for every U in L, μ(U)=f(χ[U])=ν(U), by definition of f.
That was the essential construction of the proof: just a bit of elementary linear algebra, resting on the irredundancy of L in order to obtain a base of F(L).
In order to see that μ is unique (as a valuation), we reason as follows. The value of μ on elements A ≝ U[1] ∪ … ∪ U[n] of L^∪ is uniquely determined from the values it takes on individual elements U
[1], …, U[n], by induction on n, using (strictness and) modularity. (Let me leave the details to you.) For each L-crescent C ≝ A–B, we must have μ(A)=μ(B ∪ C)=μ(B)+μ(C)–μ(B ∩ C)=μ(B)+μ(C)–μ(∅)=μ(B)+μ
(C), so μ(C) must be equal to μ(A)–μ(B). This argument shows more generally that if E and F (instead of B and C) are disjoint elements of A(L), then μ(E ∪ F)=μ(E)+μ(F). Using this and strictness,
this shows that for every element E of A(L), written as a finite disjoint union of L^∪-crescents C[i], 1≤i≤n, μ(E) is uniquely determined as Σ[i][=1]^n μ(C[i]). ☐
We apply this to the case where L is the irredundant ∩-semilattice of subsets of Q(X) of the form ☐U, where U ranges over the open subsets of a space X, and we obtain that for any function ν from OX–
{∅} to R, there is a unique signed valuation μ on the lattice of sets L^∪=O[fin](Q(X)), such that μ(☐U)=ν(U) for every non-empty open subset U of X, and we are done.
A final word. The problem we were really interested in was whether there exists a continuous valuation μ such that μ(☐U)=ν(U) for every open subset U of X. The obvious difference is that we now
require continuity, and to handle this, we simply extend a valuation μ on O[fin](Q(X)) such that μ(☐U)=ν(U) for every open set U to the whole of O(Q(X)), using Scott’s formula; this assumes that X is
locally compact (and that ν(∅)=0). There is a less visible difference: a valuation (not a signed valuation) takes its values in the non-negative reals. In order to ensure that μ is indeed a
valuation, a necessary and sufficient condition is that ν satisfy the inequality:
ν(U) ≥ ∑[I] (-1)[^|I|+1] ν(∩[i∈I] U[i])
for all open sets U, U[1], …, U[n] such that U contains U[1] ∪ … ∪ U[n], and where the summation extends over all non-empty subsets I of {1, …, n}. (This is an inequational form of the so-called
inclusion-exclusion formula in probability theory.)
You may read [1] if you are interested. I will not explain, sorry… My point was only to show how the use of the (apparently) overconstrained notion of irredundant ∩-semilattices can be put to good
use in order to obtain simple proofs of not completely trivial results!
1. Jean Goubault-Larrecq and Klaus Keimel. Choquet-Kendall-Matheron theorems for non-Hausdorff spaces. Mathematical Structures in Computer Science 21(3), 2011, pages 511-561.
2. Zhenchao Lyu and Xiaodong Jia. Core-compactness of Smyth powerspaces. arXiv:1907.04715, July 2019.
3. Marcel Erné. The ABC of order and topology. Pages 57–83 of Category Theory at Work, Proceedings of a Workshop. Research and Exposition in Mathematics 18. Heldermann Verlag, 1991. H. Herrlich and
H.-E. Porst, editors.
4. Helmut Groemer. On the extension of additive functionals on classes of convex sets. Pacific Journal of Mathematics 75(2):397–410, 1978.
5. Daniel A. Klain and Gian-Carlo Rota. Introduction to Geometric Probability. Cambridge University Press, Lezioni Lincee series, 1997.
Appendix A: the ambiguity in [1]
The definition of “irreducible” in [1] is: in an ∩-semilattice L, E is irreducible if and only if one cannot write E as the union of finitely many proper closed subsets still in L. But that is stated
in such a way that the reader cannot decide whether we mean “finitely many, possibly 0”, or “finitely many, and at least 1”.
I have decided that we meant “finitely many, possibly 0”. That definition is then equivalent to the definition I gave at the beginning of this post. In particular, that forces E to be non-empty,
since the empty set can be written as the union of finitely many (namely, zero) proper closed subsets in L.
Then we defined L as irredundant if and only if all its elements are irreducible. In particular, an irredundant L cannot contain the empty set. This is unfortunate, since our primary example, the
collection of sets of the form ☐U, where U ranges over the open subsets of a topological space X, does contain the empty set (as ☐∅).
The easiest fix is what I did in this post: require all non-empty elements of L to be irreducible. With that fix, all theorems of [1], and for that matter, of [2] as well, go through unchanged.
Appendix B: if OX is prime-continuous, then X is a c-space
(Added January 22nd, 2022.) Let X be a topological space, and let us assume that OX is prime-continuous, or equivalently, completely distributive. (See Exercise 8.3.16.)
We claim that, given any family (D[i])[i ∈ I] of downwards closed subsets of X, the closure cl(∩[i ∈ I] D[i]) is equal to ∩[i ∈ I] cl (D[i]). We write each D[i] as the union of the sets ↓x, where x
ranges over D[i]. This is possible because each D[i] is downwards closed. Next, we recall that the powerset of X is completely distributive, so ∩[i ∈ I] D[i] = ∩[i ∈ I] ∪[x ∈ D[i]] ↓x = ∪[f] ∩[i ∈ I]
↓f(i), where f ranges over Π[i ∈ I] D[i], the set of functions that map each element i of I to an element of D[i]. The sets ↓f(i) are closed, ∩[i ∈ I] ↓f(i) is their infimum in the lattice HX of
closed subsets of X, and the closure cl(∩[i ∈ I] D[i])=cl(∪[f] ∩[i ∈ I] ↓f(i)) of their union is then equal to sup[f] inf[i ∈ I] ↓f(i), where “sup” and “inf” are understood in HX. Since HX is
completely distributive, we also have inf[i ∈ I] sup[x ∈ D[i]] ↓x=sup[f] inf[i ∈ I] ↓f(i). Therefore cl(∩[i ∈ I] D[i])=inf[i ∈ I] sup[x ∈ D[i]] ↓x. For each i in I, sup[x ∈ D[i]] ↓x is the smallest
closed set that contains every element of D[i], and is therefore equal to cl (D[i]). The outer “inf” is just an intersection, so cl(∩[i ∈ I] D[i])=∩[i ∈ I] cl(D[i]).
By taking complements, we obtain that for every family (A[i])[i ∈ I] of upwards closed subsets of X, the interior int(∪[i ∈ I] A[i]) is equal to ∪[i ∈ I] int (A[i]). By Exercise 5.1.38 of the book, X
is a c-space. (Explicitly, let x be a point of X and U be an open neighborhood of x. Then U is the union of all the upwards closed sets ↑y where y ranges over U. In particular, U=int(∪[y ∈ U] ↑y).
Since the sets ↑y form a family of upwards closed sets, we have just seen that U=∪[y ∈ U] int(↑y). Therefore x is in int(↑y) for some y in U.)
— Jean Goubault-Larrecq (January 20th, 2022)
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OCFD -- Operating Cash Flow Demand -- Definition & Example |…
What is Operating Cash Flow Demand (OCFD)?
Operating cash flow demand (OCFD) is the present value of the minimum amount of cash a capital investment must generate over its life in order to meet the investor's minimum required return.
How Does Operating Cash Flow Demand (OCFD) Work?
Let's assume that Company XYZ wants to purchase a widget machine. The price is $750,000. The machine is expected to generate $100,000 of cash each year for 10 years. If the present value of the cash
flows is $600,000, then Company XYZ should offer no more than $600,000 for the machine. After all, the machine will only generate that much in cash flow for the company.
If Company XYZ is going to pay more, then the machine needs to generate more cash each year or have a longer useful life, or perhaps the company might use a different cost of capital with which to
discount the cash flows.
Why Does Operating Cash Flow Demand (OCFD) Matter?
OCFD is a strategic tool that helps companies and investors evaluate capital-spending decisions. It gives a clear yes/no decision when evaluating projects. The investment's cost, useful life,
discount rate and efficiency all affect OCFD. | {"url":"https://investinganswers.com/dictionary/o/operating-cash-flow-demand-ocfd","timestamp":"2024-11-06T02:35:59Z","content_type":"text/html","content_length":"61087","record_id":"<urn:uuid:4d1e3f07-db1c-4561-8ffd-1fc1272b93ed>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00658.warc.gz"} |
Which Operation?
1) After his birthday party Christian had 7 milk chocolates and 9 plain chocolates left out of 3 boxes. How many chocolates did Christian have altogether?
2) Christopher went for a 10km walk on his 12th birthday but he quickly became tired and travelled the last 5km by bus. How many kilometers did Christopher actually walk?
3) In a shop there are 6 vases arranged on 4 shelves. In each vase there were 3 flowers. How many flowers were there altogether?
4) Cody likes to have 2 oranges squeezed to make orange juice every morning. It takes 8 minutes to do the squeezing. How many oranges does Cody use in a whole week?
5) In a class of twenty eight pupils there are eight friends who have been given a box of seventy two crayons to share equally between themselves. How many crayons will each of the friends receive?
6) Colin is 7 years old. He is five years older than his brother and six years older than his sister. How old is Colin's brother?
7) Conner scored 5 points in the first round of the quiz. He then scored 12 points in the second round and 5 points in the third round. What was the total number of points Conner scored in all three
8) This week thirty cakes were set out in rows of 5 on a shop counter. Last week they were set out in rows of 10. How many rows were there this week?
9) David measured the heights of the three trees in his garden. The heights were 4m, 5m and 8m. What was the difference in heights of the tallest tree and the shortest tree?
10) Destiny multiplies one of the numbers in the list below by another number in the list. The answer she got was also a number in the list. What product did Destiny get? | {"url":"https://www.transum.org/Maths/Exercise/Operation/","timestamp":"2024-11-04T16:51:56Z","content_type":"text/html","content_length":"50593","record_id":"<urn:uuid:071f274a-1e24-4268-8100-7102cc4a2253>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00047.warc.gz"} |
Months And Ordinal Numbers Worksheet - OrdinalNumbers.com
Months And Ordinal Numbers Worksheet
Months And Ordinal Numbers Worksheet – There are a myriad of sets that are easily counted using ordinal numbers as a tool. They can also be used to generalize ordinal numbers.
One of the most fundamental concepts of mathematics is the ordinal number. It is a number that identifies the location of an object within the set of objects. Typically, ordinal numbers fall between
one and twenty. Although ordinal numbers serve many functions, they’re most often used for indicating the order of items in the list.
It is possible to present ordinal numbers with numbers, words, and charts. They may also be used to specify how a group of pieces are arranged.
The vast majority of ordinal numbers can be classified into one of the two categories. The transfinite ordinals are represented in lowercase Greek letters. Finite ordinals will be represented using
Arabic numbers.
Every set that is well-ordered is expected to have at least one ordinal, in accordance with the axioms of choice. For example, the top score would be awarded to the first student in the class to
receive it. The student who scored the highest score was declared the winner of the contest.
Combinational ordinal figures
Compound ordinal numbers are multi-digit numbers. These numbers are made by multiplying the ordinal’s last digit. They are most commonly used to rank and date. They do not have an unique ending for
the final digit like cardinal numbers do.
To show the order of the order in which items are arranged in a collection, ordinal numbers are used. These numbers also serve to denote the names of items in a collection. Regular numbers are
available in both regular and suppletive forms.
Regular ordinals can only be constructed by prefixing the cardinal number with the suffix -u. Next, the number must be written in words, and then a colon is added. There are additional suffixes
available.For instance the suffix “-nd” is used to refer to numerals ending in 2, and “-th” is utilized for numbers ending in 4 or 9.
Prefixing words with the -u,-e, or–ie suffix creates suffixtive ordinals. The suffix can be employed to count. It is also wider than the standard one.
Limits of Ordinal
Limit ordinal values that are not zero are ordinal numbers. Limit ordinal numbers come with the drawback that there is no limit on the number of elements they can have. These numbers can be made by
joining sets that are not empty and do not have any maximum elements.
Limits on ordinal numbers can also be employed in transfinite definitions for the concept of recursion. The von Neumann model declares that each infinite cardinal number is an ordinal number.
An ordinal numbers with an upper limit is equivalent to the sums of all ordinals lower than it. Limit ordinal numbers are enumerated using arithmetic, but they also can be represented as a series of
natural numbers.
Data is organized by ordinal number. They give an explanation of the numerical location of an object. They are utilized in set theory and arithmetic contexts. Despite sharing the same form, they
aren’t in the same classification as natural number.
The von Neumann model uses a well-ordered set. Let’s say that fy subfunctions a function, g’, that is described as a single function. In the event that g’ fulfills the requirements and g’ is an
ordinal limit if it is the only subfunction (i I, ii).
The Church-Kleene oral is a limit order in the same way. A Church-Kleene ordinal defines a limit ordinal as be a well-ordered collection, that is comprised of smaller ordinals.
Examples of ordinal numbers in stories
Ordinal numbers are used to indicate the hierarchy among objects or entities. They are crucial for organizing, counting, as well as for ranking reasons. They are used to explain the location of the
object as well as the order that they are placed.
The ordinal number is generally denoted by the letter “th”. But sometimes, the letter “nd”, however, is utilized. The titles of books are usually accompanied by ordinal numbers.
Ordinal numbers are usually written in lists, they can also be written as words. They also can be referred to as acronyms and numbers. These numbers are easier to comprehend than cardinal numbers,
There are three kinds of ordinal numbers. They can be learned more through games, practice, and other activities. Learning about them is an important part of improving your ability to arithmetic.
Coloring exercises are a fun, easy and enjoyable way to develop. Use a handy marking sheet to check off your outcomes.
Gallery of Months And Ordinal Numbers Worksheet
Months And Ordinal Numbers Activity
Ordinal Numbers And Months Interactive Worksheet
MONTHS DATES ORDINAL NUMBERS ESL Worksheet By Sandramendoza
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A short explanation / tutorial of the Kalman filter
Explanations Science
A short explanation / tutorial of the Kalman filter
In my list of methods to explain I have now come to the Kalman filter. This is a short summary of the tutorial of Welch and Bishop which can be found here. I have been reading up on pose recognition
and a lot of people seem to be using it to track parts moving across the screen, so I needed to learn a tiny bit about it. The basic idea of the Kalman filter is that $$x$$ depends on its previous
value, $$x_{t-1}$$ plus some control input from the previous step, $$u_{t-1}$$ and some noise from the previous step, $$w_{t-1}$$ in a linear way. This can be written as:
$$x_t = A x_{t-1} + B u_{t-1} + w_{t-1}$$
where $$A$$ and $$B$$ describe the dependencies / differences between the new state and the previous state and control input. We also have some measurements of the $$x$$ variable which we call $$z$$.
$$z$$ is a linear function of $$x$$ plus some noise and we write it as
$$z_{t} = H x_{t} + v_{t}$$
The two noise terms, the process noise and measure noise is assumed to be normally distributed:
$$w\sim N(0,Q)$$
$$v\sim N(0,R)$$
Having determined our variables, and what kind of assumptions that are needed, we need to understand how our measurement, $$z$$ can help our estimate of $$x$$.
We start out by defining the á priori and posteriori state error. If we let $$\hat{x_{t}^{-}}$$ be the á priori estimate(before the measurement) and $$\hat{x_{t}}$$ be the posteriori estimate(after
the measurement) we can write the error in the estimates as follows
$$e^{-}_{t} = x_{t}-\hat{x_{t}^{-}}$$
$$e_{t} = x_{t}-\hat{x_{t}}$$
The error estimate covariance is now simply
$$P_{t}^{-} = E[e_{t}^{-}e_{t}^{-T}]$$
$$P_{t} = E[e_{t}e_{t}^{T}]$$
We can let our final estimate be a linear combination of the two, the á priori and the posteriori estimate. By doing that we have incorporated the extra knowledge gained by having access to $$z$$.
Our estimate or update equation is now
$$\hat{x}_t = \hat{x_{t}^{-}} + K (z_{t}-H\hat{x_{t}^{-}} )$$
The subtraction term is apparently called innovation or residual and basically it reflects our belief of the discrepancy between the á priori estimate and the measurement. $$K$$ here is the gain or
blending factor which influences how much influence each part has. The question is how do we choose $$K$$? The idea here is that we want to choose the $$K$$ that minimizes the covariance of the
posteriori estimate, by doing so we get the most reliable range estimate. To minimize the covariance we plug the linear combination into the covariance estimate, calculate the expectations and then
take the derivative of the trace w.r.t $$K$$ and setting to zero and then solving. Doing this we can get an expression for $$K$$ that looks like this
$$K_t = P_{t}^{-} H^{T} ( H P_{t}^{-} H^{T} + R )^{-1}$$
By taking the limit of $$R$$, the measurement noise covariance, when it goes to zero we get $$K = H^{-1}$$, this means that gain weighs the residual more heavily since our measurements practically
are noise free. So $$R$$ says something about our belief about how good our measurements are. Now if the á priori estimate of the covariance $$P^{-}_{t}$$ goes to zero then $$K$$ also moves towards
zero meaning that our á priori estimate is pretty much perfect.
That being said we can move on to the structure of the Kalman algorithm
Á priori estimates
$$\hat{x_t^{-}} = A \hat{x_{t-1}} + B u_{t-1} + w_{t-1}$$
$$P^{-}_{t} = A P_{t-1} A^{T} + Q$$
Posteriori estimates
$$K_t = P_{t}^{-} H^{T} ( H P_{t}^{-} H^{T} + R )^{-1}$$
$$\hat{x}_t = \hat{x_{t}^{-}} + K (z_{t}-H\hat{x_{t}^{-}} )$$
$$P_{t} = (I – K_{t} H )P^{-}_{t}$$
Here is a simple implementation of a Kalman filter: kalmansim
So to summarize the Kalman filter: the basic idea is that the current state is linearly dependent on the previous state plus some control sequence and that there is a correlation of the new state
with a measurement we procured of that state. The estimate of the new state can be written as a linear combination of a á priori estimate and posteriori estimate of the state. That is basically all
the assumptions needed to derive the equations above it seems. Thanks Kalman. | {"url":"https://nonconditional.com/2012/12/a-short-explanation-tutorial-of-the-kalman-filter/","timestamp":"2024-11-07T09:23:43Z","content_type":"text/html","content_length":"23434","record_id":"<urn:uuid:a8a81c3d-1f3b-45be-864e-edbea66ecee6>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00659.warc.gz"} |
PostGIS Operators in 2.4
14 Sep 2017
TL;DR: If you are using ORDER BY or GROUP BY on geometry columns in your application and you have expectations regarding the order or groups you obtain, beware that PostGIS 2.4 changes the behaviour
or ORDER BY and GROUP BY. Review your applications accordingly.
The first operators we learn about in elementary school are =, > and <, but they are the operators that are the hardest to define in the spatial realm.
When is = equal?
For example, take “simple” equality. Are geometry A and B equal? Should be easy, right?
But are we talking about:
1. A and B have exactly the same vertices in the same order and with the same starting points?
2. A and B have exactly the same vertices in any order? (see ST_OrderingEquals)
3. A and B have the same vertices in any order but different starting points?
4. A has some extra vertices that B does not, but they cover exactly the same area in space? (see ST_Equals)
5. A and B have the same bounds?
Confusingly, for the first 16 years of its existence, PostGIS used definition 5, “A and B have the same bounds” when evaluating the = operator for two geometries.
However, for PostGIS 2.4, PostGIS will use definition 1: “A and B have exactly the same vertices in the same order and with the same starting points”.
Why does this matter? Because the behavour of the SQL GROUP BY operation is bound to the “=” operator: when you group by a column, an output row is generated for all groups where every item is “=” to
every other item. With the new definition in 2.4, the semantics of GROUP BY should be more “intuitive” when used against geometries.
What is > and <?
Greater and less than are also tricky in the spatial domain:
• Is POINT(0 0) less than POINT(1 1)? Sort of looks like it, but…
• Is POINT(0 0) less than POINT(-1 1) or POINT(1 -1)? Hm, that makes the first one look less obvious…
Greater and less than are concepts that make sense for 1-dimensional values, but not for higher dimensions. The “>” and “<” operators have accordingly been an ugly hack for a long time: they compared
the minima of the bounding boxes of the two geometries.
• If they were sortable using the X coordinate of the minima, that was the sorting returned.
• If they were equal in X, then the Y coordinate of the minima was used.
• Etc.
This process returned a sorted order, but not a very satisfying one: a “good” sorting would tend to place objects that are near to each other in space, near to each other in the sorted set.
Here’s what the old sorting looked like, applied to world populated places:
The new sorting system for PostGIS 2.4 calculates a very simple “morton key” using the center of the bounds of a feature, keeping things simple for performance reasons. The result is a sorted order
that tends to keep spatially nearby features closer together in the sorted set.
Just as the “=” operator is tied to the SQL GROUP BY operation, the “>” and “<” operators are tied to the SQL ORDER BY operation. The pictures above were created by generating a line string from the
populated places points as follows:
CREATE TABLE places_line AS
SELECT ST_MakeLine(geom ORDER BY geom) AS geom
FROM places; | {"url":"https://blog.cleverelephant.ca/2017/09/postgis-operators.html","timestamp":"2024-11-05T03:04:17Z","content_type":"application/xhtml+xml","content_length":"18157","record_id":"<urn:uuid:5d420efa-00f4-47de-9fac-a5bd9d75288e>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00000.warc.gz"} |
18 Best Online Math Solvers and Calculator Websites (2024)
Best Online Math Solvers and Calculator Websites
Math can be tricky at times. When you get stuck, you reach out to your teachers, your classmates, and above all, the Internet. There are some online math solvers and calculators that can help you
out. While I have several online calculators myself, some are even better.
There are over a dozen of online math solvers that can help you solve math problems either automatically or manually.
The automatic math solvers use your input to solve your math problems using pre-built data or using machine learning. Manual math solvers are actually assignment help services that offer dedicated
one-on-one solutions to not only your math problems but also in other areas.
Traditionally, abacuses and calculators have been used for centuries to make mathematical operations simpler. In the modern age of technology, we have several useful digital tools for this purpose.
Online math solvers, such as number sequence solvers, are among such tools which can help students with various problems, conversions, and much more.
In this article, I will showcase the 18 best online math solvers and calculator websites currently available on the Internet, along with their key details.
These can help you solve mathematical problems with speed and efficiency. Apart from helping you save time, they will also allow you to verify your manual problem-solving skills more efficiently.
Best Online Math Solvers and Calculator Websites
Microsoft Math
Microsoft Math Solver is an excellent app and website to obtain stepwise solutions to mathematical problems. You will be able to see how to solve problems, show your work, and get definitions for
mathematical concepts as well. Besides, you can immediately graph any equation to visualize your function and understand the relationship between variables.
Apart from this, the tool allows you to practice more by searching for additional learning materials like video tutorials and related worksheets. It can handle problems in various areas of
mathematics, such as:
• Exponents
• Fractions
• Greatest common factor
• Least common multiple
• Mean
• Mixed fractions
• Mode
• Order of operations
• Prime factorization
• Radicals
• Combine like terms
• Evaluate fractions
• Expand
• Factor
• Inequalities
• Linear equations
• Matrices
• Quadratic equations
• Solve for a variable
• Systems of equations
• Derivatives
• Integrals
• Limits
• Evaluate
• Graphs
• Simplify
• Solve equations
It works in a wide range of languages such as English, German, Hindi, Spanish, and many more.
Pricing – Free to use.
Link - https://math.microsoft.com/en
Wolfram Alpha
The Wolfram Alpha online equation solver can readily solve quadratic, linear, and polynomial systems of equations with ease. It is an excellent tool for finding polynomial roots in particular. It can
also plot polynomial solution sets and inequalities, factor polynomials, and much more. You will promptly receive feedback and guidance with stepwise solutions and the Wolfram Problem Generator.
Wolfram Alpha uses the Wolfram Language’s Solve and Reduce functions for solving equations. These functions contain a wide range of methods for all sorts of algebra – from elementary linear and
quadratic equations to multivariate nonlinear systems. It also uses linear algebra methods like Gaussian elimination at times, with optimizations for higher reliability and speed.
With the help of these meticulously designed methods, Wolfram Alpha is able to solve a broad variety of problems while minimizing computation time.
Pricing — Free to use. Begins at USD 5.49 per month (Pro) and USD 9.99 per month (Premium)
Link – https://www.wolframalpha.com/
QuickMath was one of the first online math solvers to be developed. Founded in 1999 by Dr. Ben Langton, it allowed users to intuitively type in an expression and get the results almost
instantaneously. In 2013, it was acquired by the Softmath company and employed superior technology to start providing step-by-step solutions to mathematical problems. It also displays
context-sensitive explanations for every step involved.
QuickMath covers a lot of areas in mathematics, including:
• Linear equations
• Graphing functions
• Absolute values
• Roots and radicals
• Exponents and polynomials
• Quadratic equations
• Complex numbers
• Factoring polynomials
• Linear inequalities
Pricing – USD9.99 per month or USD29.99 per year (premium Algebrator version)
Link - https://quickmath.com/
Mathway by Chegg is one of the most well-known platforms that give students just the tools they need to understand and solve mathematical problems. It aims to allow students to access quality
on-demand math assistance with ease and convenience readily. Its handy online math solver comes with an efficient user interface and high functionality. Mathematical areas it covers include:
• Basic math
• Pre-Algebra
• Algebra
• Trigonometry
• Statistics
• Pre-calculus
• Calculus
• Graphing
• Chemistry
• Linear algebra
• Finite math
You can get answers to problems free of cost. However, for stepwise solutions and explanations, you need to upgrade to the Mathway step-by-step subscription.
Pricing – USD 9.99 per month or USD 39.99 per year
Link - https://gauravtiwari.org/go/mathway/
The Symbolab math solver is an efficient private math tutor that can readily solve any mathematical problem complete with steps. It is composed of more than a hundred of Symbolab’s most robust
calculators, including the following:
• Equation calculator
• Integral calculator
• Derivative calculator
• Limit calculator
• Trigonometry calculator
• Inequality calculator
• Matrix calculator
• Series calculator
• Functions calculator
• Laplace Transform calculator
• ODE calculator
Pricing – Begins at USD 12.99 for two months
Link – https://www.symbolab.com/solver
Cuemath Online Math Calculators
Cuemath’s Online Math Calculators feature easy-to-use interfaces for users and display results in a readily interpretable form. They also provide stepwise solutions to each problem to help users
understand them better. Cuemath provides calculators on all prominent mathematical topics required by students, such as:
• Absolute value calculator
• Adding and subtracting polynomials calculator
• Adding exponents calculator
• Area calculator
• Binomial probability calculator
• Boolean algebra calculator
• Coefficient of determination calculator
• Definite integral calculator
• Elimination method calculator
• Fibonacci numbers calculator
Pricing – Free to use
Link – https://www.cuemath.com/calculators/
Gaurav Tiwari's Calculators
I am a hobbyist and I have created several online calculators that can help you out in certain things. My Calculators page is relatively new compared to all these services but it's still worth the
Here is the list of top calculators and solvers that I have:
Pricing — Free to use
Link — https://gauravtiwari.org/calculators
E6BX Advanced Math Calculator
E6BX’s advanced online math calculator is primarily intended for people in the aviation industry. However, you can use it to solve simpler calculations. As you know, aviation engineering involves
highly intricate calculations that require pinpoint precision. After all, the lives of those aboard the aircraft depend on them. Therefore, E6BX’s online math solver is fast, efficient, and accurate.
The calculator allows you to perform various mathematical operations like grouping, factorial, modulus, and much more. It features various advanced functions dealing with areas of mathematics such
• Construction
• Algebra
• Trigonometry
• Combinatorics
• Geometry
• Matrices
• Probability
• Statistics
• Strings
Apart from this, the calculator deals with specialized calculations in aviation such as mach speed, speed of sound, wind components, pressure altitude, indicated air speed, true air speed, and so on.
Pricing – Free to use
Link – https://e6bx.com
Maple Calculator
It doesn’t matter whether you are dealing with simple calculations or university-level math problems. Maple Calculator is a robust math learning tool that can help you with every aspect of the
subject. It can perform matrix manipulations, solve algebra problems, explore 2D and 3D graphs, calculate the derivative or integral of a function, and much more.
Maple Calculator allows you to conveniently enter, solve, and visualize mathematical problems from precalculus, calculus, algebra, and differential equations for free. It also lets you enter problems
using your camera and check your homework with a button tap. For both classroom students and those learning from home, Maple Calculator is an indispensable resource for developing a grasp on math.
Pricing – USD 995 (single-user license)
Link - https://www.maplesoft.com/products/Maplecalculator/
MathPapa online math solver
MathPapa aims to help students learn algebra step-by-step. Developed by Robert Ikeda and Priscilla Pham, it allows students to understand the subject better by assisting them in mastering the
fundamentals correctly. Given below are some of the functions it offers:
• Fractions (addition, multiplication)
• Exponents (multiplication, division, exponents)
• Polynomials (addition, multiplication, squaring)
• Graphing functions (parabola, inequalities, line)
• Evaluating expressions
• Solving equations
• Solving inequalities
Pricing – USD 9.99 per month (for US customers)
Link – https://www.mathpapa.com/
Online Utility math calculator
This powerful math calculator can handle a wide range of mathematical functions, such as:
• Arrays
• Integrals
• Polynomials
• Inequalities
• Numerical algorithms
• Discrete mathematics
• Exponential functions
• Statistical and finance functions
With the help of this handy tool, you can solve a wide range of mathematical computational problems –from the very basic to the most complicated ones. It also allows you to strengthen your
theoretical concepts and work out your homework worksheets. You can use it as a science student as well, both in study and in research.
Pricing – Free to use
Link - https://www.online-utility.org/math/math_calculator.jsp
Calculator Soup math equation solver
The math equation solver from Calculator Soup allows you to solve mathematical problems using order of operations like BODMAS, BEDMAS, and PEMDAS. It can handle equations that add, subtract,
multiply, and divide positive, negative, and exponential numbers. You can readily include parentheses and numbers with roots or exponentials as well.
Other features offered by Calculator Soup include:
• Algebra
• Trigonometry
• Statistics
• Physics
• Chemistry
• Finance
• Conversions
• Geometry
• Time and date
Pricing – Free to use
Link - https://www.calculatorsoup.com/calculators/math/math-equation-solver.php
Calculator.net features over 200 simplistically designed but highly efficient mathematical calculators. They are highly varied in terms of operation, such as:
• Scientific calculator
• Fraction calculator
• Random number generator
• Binary calculator
• Ratio calculator
• Root calculator
• Matrix calculator
• Probability calculator
• Permutation and combination calculator
• Pythagorean theorem calculator
All these calculators have been developed individually after rigorous testing. The site aims to provide comprehensive, fast, convenient, and free online calculators to benefit both students and
professionals. It can quite possibly become your one-stop, go-to website when it comes to quick calculations. However, I should mention that some of the calculators are only available in specific
Pricing – Free to use
Link - https://www.calculator.net/
Domyhomework123 Math Solver
The Domyhomework.123 math problem solver can help solve equations and math problems that you find challenging. Powered by Mathway software from Chegg, which I mentioned earlier - Domyhomework123 is
more of a service than software.
As the automated solutions don't offer anything extra than Mathway, you can take human help to solve complex problems. Domyhomework123 provides a homework assistance service and writing service to
help any student with their school assignments. If you are feeling the pressure of an upcoming deadline, it is a reliable service you can refer to once you are truly running out of time.
Pricing – Free
Link: https://domyhomework123.com/math-word-problem-solver
Onlinecalculator Guru
This reliable website can readily solve all kinds of mathematical problems involving concepts like algebra, geometry, and much more. It features several versatile calculators, such as:
• Maths calculator
• Physics calculator
• Chemistry calculator
• Algebra calculator
• Conversions calculator
• Statistics calculator
• Fraction calculator
• Polynomials calculator
• Calculus calculator
• Geometry calculator
It provides comprehensive answers to mathematical questions along with a step-by-step learning guide. Whether you need help with your math assignments or preparing for your annual or competitive
exams, you will find this online calculator extremely helpful. It is an excellent resource for students, teachers, and parents alike.
Pricing – Free to use
Link - https://onlinecalculator.guru/
EasyCalculation.com features a wide range of online calculators to help students complete their assignments. It can solve everything from determinants and standard deviations to income taxes and
dates related to pregnancy. Moreover, it provides explanatory steps to solve the equations as well. Thus, you can readily use it to check the accuracy of the equations in your assignment solutions.
The calculators featured on this website are versatile and can work independently on all commonly used browsers. Some of them include:
• Standard deviation calculator
• Determinant calculator
• Slope calculator
• Inverse of matrix calculator
• Pregnancy calculator
• Shear modulus calculator
• Income tax calculator (India)
• Mortgage length calculator
Pricing – Free to use
Link – https://www.easycalculation.com
RapidTables online math calculator
RapidTables.com is a helpful website that contains handy tools and quick reference information. Apart from mathematical theory, it features many useful online calculators as well. Some of these
• Adding fractions calculator
• Antilog calculator
• Binary/hex calculator
• Complex numbers calculator
• Pythagorean theorem calculator
• Root calculator
• Scientific notation calculator
• Standard deviation calculator
• Trigonometry calculator
• Variance calculator
Pricing – Free to use
Link – https://www.rapidtables.com/calc/math/index.html
Ezcalc.me - Easy Online Calculators
Ezcalc.me offers a wide range of free online calculators covering the most
basic calculation needs and helping with more complex scientific and math problems. They are very useful for both students and professionals alike.
All of the calculators are written in javascript and most of them are
universal multi-functional (all-in-one) calculators.
The calculators on this site are covered under the following categories:
• Algebra Calculators
• Chemistry Calculators
• Date & Time Calculators
• Financial Calculators
• Geometry Calculators
• Health Calculators
• Math Calculators
• Percentage Calculators
• Statistics Calculators
• Text Tools
• Unit Converters
Pricing – Free to use
Link – https://ezcalc.me
The online math solvers and calculators that I have listed on this page are all highly efficient, fast, and useful to beginners and professionals alike. However, keep in mind that they are not a
substitute for a manual calculation.
If you want to excel in your field, you need to continue putting in the effort to make your calculation skills stronger. However, when you are short of time, this article can help you find a good
online calculator to get the required results with speed and accuracy. | {"url":"https://gauravtiwari.org/best-online-math-solvers/?utm_source=self&utm_medium=related&utm_campaign=related_posts","timestamp":"2024-11-08T05:35:56Z","content_type":"text/html","content_length":"104416","record_id":"<urn:uuid:be982631-4a9d-4158-971a-739ce88160d1>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00457.warc.gz"} |
How to Get a Standard Deviation in Excel: User Guide
A standard deviation is used to understand and compare the variability within a dataset or to assess its reliability when analyzing data.
So, how do you calculate it?
To calculate standard deviation in Excel, use the =STDEV.S function for samples or the =STDEV.P function for populations. Simply select the range of cells containing the data for which you want to
find the standard deviation. For example, if your data is in cells A1 to A10, you would enter =STDEV.P(A1:A10) for population standard deviation, or =STDEV.S(A1:A10) for sample standard deviation.
Excel will then calculate and display the standard deviation based on the data in the specified cells.
Today, you’ll learn how to get a standard deviation in Excel. We will break down the process into steps with demonstrations to help you better understand the concepts.
Let’s start learning.
How to Calculate Standard Deviation
There are 3 functions in Excel that are frequently used for calculating standard deviation.
The sfunctions are:
1. STDEV.S
2. STDEV.P
3. STDEVPA
1. How to Calculate Standard Deviation Using STDEV.S
STDEV.S stands for sample standard deviation. Its older version is STDEV.
It measures the variability or spread in a sample subset of a larger population.
You can use this standard deviation function when you’re working with a sample and want to infer or predict something about the larger population from which the sample is taken.
To calculate standard deviation, click on the cell where you want the standard deviation result to appear, and type the formula:
You can also use the older STDEV function:
A2:A6 represents the range of cells containing your data. You can change this according to your needs.
After entering the formula, press Enter. Excel will calculate and display the sample standard deviation for the data in the specified cells.
2. How to Calculate Standard Deviation Using STDEV.P
STDEV.P stands for population standard deviation. Its older version is STDEVP.
It measures how much variation or dispersion there is from the average (mean) in a set of data that represents the entire population.
You can use this function when your dataset includes the entire population, and you’re not trying to infer or predict but rather describe the population itself.
Click on a cell where you want to display the standard deviation result, and type in the formula:
You can also use the older formula:
A2:A6 represents the range of cells containing your data with numerical values.
After entering the formula, press Enter. Excel will calculate and display the population standard deviation value for the data in the specified data point cell.
How to Calculate Standard Deviation Using STDEVPA
Calculating the standard deviation using the STDEVPA function in Excel is quite similar to using other standard deviation functions, but with a key difference: STDEVPA includes all values in the
calculation, treating text and logical values (TRUE/FALSE) as part of the dataset.
Let’s say we have a sample data as follows:
To use STDEVPA function, start by entering the formula. In our case the formula will be:
A2:B6 defines the range of the dataset.
After entering the formula, press Enter. Excel will calculate the standard deviation for the data in your specified range, including any text or logical values in the calculation.
How to Interpret Standard Deviation
Now that you learned how to calculate standard deviation, its is important that you learn how to interpret it.
Standard deviation is a statistical measure that quantifies the degree of variation or dispersion in a dataset relative to its mean. It is a crucial tool for understanding the spread of data points,
indicating whether they are closely clustered or widely dispersed.
For instance, in a classroom setting, let’s consider the test scores of two different classes, Class A and Class B.
Suppose Class A’s scores are 85, 88, 84, 87, and 86, while Class B’s scores are 70, 90, 60, 95, and 65. Both classes may have the same average score, but Class A would have a smaller standard
deviation compared to Class B.
This smaller standard deviation indicates that Class A’s scores are more consistent and clustered closely around the mean, suggesting a uniform performance level.
On the other hand, Class B, with a higher standard deviation, reflects greater variability in student performance, indicating a more diverse range of scores.
Learn how to use frequency and proportion tables in Excel by watching the following video:
Final Thoughts
As you wrap up your journey in understanding how to calculate standard deviation in Microsoft Excel, it’s important to appreciate why this skill is so valuable.
Remember, standard deviation isn’t just a number; it’s a key to unlocking insights in data analysis, be it in business, education, or research.
By mastering this tool, you’re equipping yourself to make informed decisions, understand trends, and recognize patterns that are not immediately apparent.
Imagine being able to pinpoint inconsistencies in sales data, gauge the reliability of research results, or understand student performance variability. This is the power standard deviation hands to
Frequently Asked Questions
In this section, you will find some frequently asked questions you may have when calculating standard deviation in Excel.
What is the difference between the STDEV.S and STDEV.P functions in Excel?
The STDEV.S function is used to calculate the standard deviation of a sample, while the STDEV.P function is used for the standard deviation of a population.
The main difference is in the denominator of the formula used for the calculation. The STDEV.S function uses n – 1 in the formula, where n is the number of elements in the sample, to provide an
unbiased estimate of the population standard deviation.
On the other hand, the STDEV.P function uses n in the formula and is appropriate when calculating the standard deviation for an entire population.
What is the STDEVP function in Excel?
The STDEVP function in Excel is another way to calculate the standard deviation for a population. It is an older function that has been replaced by the STDEV.P function.
The syntax for the STDEVP function is the same as STDEV.P, and both functions provide the same result.
How can I calculate standard deviation in Excel 2016?
To calculate the standard deviation in Excel 2016, you can use either the STDEV.S or STDEV.P function, depending on whether you want to calculate the standard deviation for a sample or a population.
For example, if your data is in cells A1 to A5, you can use the following formulas:
• STDEV.S: =STDEV.S(A1:A5)
• STDEV.P: =STDEV.P(A1:A5)
What are the common errors associated with the STDEV.P function?
Common errors associated with the STDEV.P function include:
• #VALUE!: This error occurs when the data provided to the function is not numeric.
• #DIV/0!: This error occurs when the data array has fewer than two values, which is insufficient to calculate the standard deviation.
• #NUM!: This error occurs when the data array has a non-numeric value, or if the value of k (the population standard deviation divisor) is not an integer.
• #N/A: This error occurs when the function is used in a shared workbook and the data is not available for one or more sheets.
Can I calculate standard deviation in Excel for Mac?
Yes, you can calculate standard deviation in Excel for Mac. The process is the same as in Excel for Windows.
You can use the STDEV.S or STDEV.P functions to calculate the standard deviation of a sample or population, respectively.
Develop a VBA application to automate repetitive tasks in Excel, increasing efficiency and reducing manual errors.
Building a VBA-Based Task Automation Tool | {"url":"https://blog.enterprisedna.co/how-to-get-a-standard-deviation-in-excel/","timestamp":"2024-11-10T03:16:19Z","content_type":"text/html","content_length":"507199","record_id":"<urn:uuid:32f6dd3a-b855-4220-9e99-01c8c5c2d1e9>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00437.warc.gz"} |
Put Option Calculator: Easily Calculate Your Trade Outcomes
Use this put option calculator to determine the potential profit or loss for your option based on varying market prices.
Put Option Calculator
Use this calculator to determine the value of a put option based on its strike price, stock price, expiration time, volatility, and risk-free interest rate.
How to Use the Calculator
Enter the strike price, stock price, time to expiry, volatility, and risk-free interest rate into their respective fields and click on “Calculate”. The result will be displayed immediately.
How It Works
The calculation uses the Black-Scholes formula for put options, which is a model to determine the theoretical price of European style options.
This calculator does not take into account dividends and is designed for European options which can only be exercised at expiration. | {"url":"https://madecalculators.com/put-option-calculator/","timestamp":"2024-11-08T15:28:18Z","content_type":"text/html","content_length":"142847","record_id":"<urn:uuid:a11a4b18-d984-4ee8-b9de-57dfa4af8c59>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00039.warc.gz"} |
Arithmetic for Computers
1. Add (add), and immediate (addi), and subtract (sub) cause exceptions on overflow. MIPS detects overflow with an exception (or interrupt ), which is an unscheduled procedure call. The address of
current instruction is saved and the computer jumps to predefined address to invoke the appropriate routine for that exception.
MIPS uses exception program counter (EPC) to contain the address of the instruction that causes the exception. The instruction move from system control (mfc0) is used to copy EPC into a
general-purpose register.
2. Add unsigned (addu), add immediate unsigned (addiu), and subtract unsigned (subu) do not cause exceptions on overflow. Programmers can trap overflow anyway: when overflow occurs, the sign bit of
the result is not properly set. Compairing with sign bits of operands, the sign bit of the result can be determined.
SIMD (single instruction, multiple data): By partitioning the carry chains within a 64-bit adder, a processor could perform simultaneous operations on a short vecters of eight 8-bit operands,
four 16-bit operands, etc. Vectors and 8-bit data often appears in multimedia routine.
multiplicand * multiplier = product
Sequential Version of the Multiplication
Refined version:
• Init: put multiplier to the left 32-bit of the product register.
• Cycle:
1. if the last bit of product register is 1, add the left 32-bit with the multiplicand
2. shift right the product register
• Final: the product register contains the 64-bit product
Faster Multiplication
A way to organize these 32 addtions is in a parallel tree:
Multiply in MIPS
The registers Hi and Lo contains the 64-bit product. Call mflo to fetch the 32-bit product, mfhi can be used to get Hi to test for overflow.
Dividend = Quotient * Divisor + Remainder
Division Algorithm
Improved version:
• Init: put the dividend in the right 32-bit of remainder register.
• Cycle:
1. subtract the left 32-bit of remainder by the divisor
2. shift left the remaider register
3. set the last bit as new quotient bit
• Final: the left 32-bit contains the remainder, right 32-bit contains the quotient.
Faster Division
SRT division: try to guess several quotient bits per step, using a table lookup based on the upper bits of the dividend and remainder. The key is guessing the value to subtract.
Divide in MIPS
Hi contains the remainder, and Lo contains the quotient after the divide instruction complete.
MIPS divide instructions ignore overflow. MIPS software must check the divisor to discover division by 0 as well as overflow.
Floating Point
scientific notation A notation that renders numbers with a single digit to the left of the decimal point.
normalized A number in floating-point notation that has no leading 0s.
fraction The value, generally between 0 and 1, placed in the fraction field.
exponent In the numerical representation system of floating-point arithmetic, the value that is placed in the exponent field.
overflow the exponent is too large to be represented in the exponent field.
floating point Computer arithmetic that represents numbers in which the binary point is not fixed.
In general, floating-point numbers are of the form: $(-1)^{S} \times F \times 2^{E}$
MIPS float: sign(1 bit) + exponent(8 bit) + fraction(23 bit) MIPS double: s(1 bit) + exponent(11 bit) + fraction(52 bit)
IEEE 754 uses a bias of 127 for single precesion, and makes the leading 1 implicit. Since 0 has no leading 1, it’s given the reserved exponent 0 so that hardware won’t attach a leading 1.
Thus 00…00 represents 0; the representation of the rest are in the following form:
$(-1)^{S} \times (1+Fraction) \times 2(Exponent - Bias)$
The exponent is located left and the bias is for comparison convenience.
Source: Harttle Land, https://harttle.land/2014/02/12/computer-design-arithmetic.html
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Mensuration Formula in Hindi PDF: Get Links To Download Free PDF
Mensuration Formulas in Hindi PDF: A Comprehensive Guide
Archana Premshankar Pandey | Updated: Jun 3, 2023 18:06 IST
The world of government exams is highly competitive, and candidates need to be well-prepared in order to succeed. One important aspect of preparation is understanding and applying mensuration
formulas. Mensuration, also known as geometry, is a fundamental topic in various government exams like SSC, UPSC, and banking exams. This article aims to provide a detailed guide to mensuration
formulas in Hindi PDF, specifically tailored for candidates preparing for the upcoming government exam cycle. By familiarizing themselves with these formulas, aspirants can enhance their
problem-solving abilities and boost their chances of success.
Mensuration Formulas in Hindi PDF Download\
The Mensuration Formulas in Hindi PDF resource is beneficial for candidates appearing for government exams. The PDF contains comprehensive mensuration formulas explained in standard Hindi language.
It serves as a useful tool for candidates to understand and memorize the formulas related to various geometric shapes and measurements. With the help of the Mensuration Formula in Hindi PDF free
download, candidates will have easy access to a valuable resource that can aid in their exam preparation and enhance their understanding of mensuration concepts.
Subject Download PDF Link
PDF of all Formulas of Rectangle Download PDF
PDF of all Formulas of Square Download PDF
PDF of all Formulas of the Quadrilateral Download PDF
PDF of all Formulas of Circle Download PDF
PDF of all Formulas of Cube and Cubies Download PDF
PDF of all Formulas of Sphere Download PDF
PDF of all Formulas of Triangles Download PDF
All Mensuration Formula in English PDF Download PDF
Introduction to Mensuration
Mensuration is a branch of mathematics that deals with the measurement of geometric shapes, including their areas, volumes, and other related quantities. It plays a crucial role in government exams
as it assesses candidates’ understanding of spatial concepts and their ability to apply formulas accurately. By mastering mensuration, aspirants can gain a solid foundation in geometry, which will
prove invaluable throughout their exam preparation journey.
Formulas for Polygons:
A square is a four-sided polygon with equal sides.
• Perimeter: The perimeter of a square is calculated by multiplying the length of one side by 4.
• Area: The area of a square is found by multiplying the length of one side by itself.
A rectangle is a four-sided polygon with opposite sides of equal length.
• Perimeter: To find the perimeter of a rectangle, add the lengths of all four sides.
• Area: The area of a rectangle is calculated by multiplying its length by its width.
A triangle is a three-sided polygon.
• Perimeter: The perimeter of a triangle is found by adding the lengths of all three sides.
• Area: The area of a triangle can be determined by multiplying half the base length by the height.
Formulas for Circles
A circle is a two-dimensional figure with all points equidistant from the center.
• Circumference: The circumference of a circle is given by multiplying the diameter by π (pi).
• Area: The area of a circle is calculated by multiplying π (pi) by the square of the radius.
Formulas for Solid Figures
A cube is a three-dimensional figure with all sides of equal length.
• Surface Area: The surface area of a cube is determined by multiplying the square of its side length by 6.
• Volume: The volume of a cube can be found by cubing its side length.
A cylinder is a three-dimensional figure with two parallel circular bases.
• Curved Surface Area: The curved surface area of a cylinder is given by multiplying the product of π (pi), the radius, and the height.
• Total Surface Area: To find the total surface area, add the curved surface area to twice the product of π (pi) and the square of the radius.
• Volume: The volume of a cylinder can be calculated by multiplying the product of π (pi), the square of the radius, and the height.
A sphere is a three-dimensional figure with all points equidistant from the center.
• Surface Area: The surface area of a sphere is calculated by multiplying the product of 4, π (pi), and the square of the radius.
• Volume: The volume of a sphere is given by multiplying the product of 4/3, π (pi), and the cube of the radius.
Mensuration Formulas in Hindi PDF: Conclusion
Mensuration formulas are an essential tool for government exam aspirants, enabling them to solve geometric problems accurately and efficiently. By mastering these formulas, candidates can gain a
solid understanding of mensuration concepts and improve their problem-solving skills. It is important to practice solving questions based on these formulas to reinforce learning and gain confidence.
With diligent preparation and a thorough understanding of mensuration, aspirants can navigate the challenges of government exams successfully. Best of luck with your exam preparation!
Mensuration Formulas in Hindi PDF FAQs
Q.1 How can I download the Mensuration Formulas in Hindi PDF?
Ans.1 To Mensuration Formulas in Hindi PDF, download, you can visit the official website and look for the download link provided in the article above.
Q.2 Are the Mensuration Formulas in the PDF suitable for beginners?
Ans.2 Yes, the Mensuration Formulas in the Hindi PDF are designed to be beginner-friendly. The formulas in the Mensuration Formulas in Hindi PDF are explained in a simple language, making it
accessible for candidates.
Q.3 Can I access the Mensuration Formulas in Hindi PDF on my mobile device?
Ans.3 Yes, the Mensuration Formulas in Hindi PDF can be accessed on various devices, including mobile phones and tablets. Once downloaded, you can open the PDF using a compatible PDF reader app on
your mobile device.
Q.4 Is the PDF compatible with all operating systems?
Ans.4 Yes, the Mensuration Formulas in Hindi PDF is compatible with commonly used operating systems such as Windows, macOS, and Linux. It can be accessed on different devices running these operating
Q.5 Can I print the Mensuration Formulas PDF for offline use?
Ans.5 Yes, once you have downloaded the Mensuration Formulas in Hindi PDF, you can easily print it for offline use. Simply connect your device to a printer and select the desired pages or the entire
document to print as per your requirement. | {"url":"https://kmatkerala.in/mensuration-formula-in-hindi-pdf/","timestamp":"2024-11-12T02:13:22Z","content_type":"text/html","content_length":"162381","record_id":"<urn:uuid:eef6e774-dacc-443e-a2d4-9a9f20be1c39>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00248.warc.gz"} |
Heat Transfer Coefficient of a Pipe Wall
Heat transfer coefficient is a measure of the ability of a material or a boundary to conduct heat. It represents the rate of heat transfer per unit area and per unit temperature difference. In the
context of a pipe wall, the heat transfer coefficient is used to quantify how efficiently heat is transferred through the wall.
The heat transfer coefficient depends on various factors, including the material properties of the pipe wall, the fluid on either side of the wall, and the conditions of the heat transfer process.
For a pipe, the heat transfer coefficient can be affected by factors such as the thermal conductivity of the pipe material, the thickness of the pipe wall, and the nature of the fluid flowing inside
or outside the pipe.
To determine the heat transfer coefficient for a specific pipe and fluid combination, experimental methods or correlations based on empirical data are often employed. Keep in mind that the heat
transfer coefficient can vary under different operating conditions, and it is essential to consider the specific circumstances of the heat transfer process when calculating or determining this
Heat Transfer Coefficient of a Pipe Wall FormulA
\( h_{wall} = 2 \; k \;/\; d_i \; ln \; ( p_o \;/\; P_i ) \)
Symbol English Metric
\( h_{wall} \) = heat transfer coefficient of a wall \(Btu-ft \;/\; hr-ft^2-F\) \(W \;/\; m-K\)
\( ln \) = natural logarithm \(dimensionless\) \(dimensionless\)
\( p_i \) = pipe ID \(in\) \(mm\)
\( p_o \) = pipe OD \(in\) \(mm\)
\( k \) = thermal conductivity \(Btu-ft \;/\; hr-ft^2-F\) \(W \;/\; m-K\)
Tags: Pipe Heat Transfer Thermal Coefficient Heat | {"url":"https://piping-designer.com/index.php/properties/thermodynamics/2241-heat-transfer-coefficient-of-a-pipe-wall","timestamp":"2024-11-11T14:52:12Z","content_type":"text/html","content_length":"29112","record_id":"<urn:uuid:7efea75b-5cd8-45d8-9aad-557815c63c7d>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00730.warc.gz"} |
Sacred Solids | News and Updates
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
Sacred Solids
0By Divykriti Madaan
Try rotating the following solids in any direction.
Did you notice anything special?
They are all perfectly symmetrical. That is, they look the same no matter how we rotate them!
And they have a special name too — the Platonic solids; named after the Greek philosopher, Plato, who first wrote about them.
In Greek mystery schools, these solids were taught as the “building blocks” of nature. Plato believed that God used them to form the basic elements — fire, earth, air, water, and the universe.
But why do these solids have such a divine status?
Technically, these are solids made using identical regular polygons. This might sound simple, but given this definition, only five Platonic solids are possible!
Why only five? Let’s find out by working out all possible solids that can be made using identical regular polygons.
We’ll start with equilateral triangles. If we join two of them, can we make a 3D shape out of it?
No, we can't, unless we fold or roll the triangles. In fact, to make any solid, we must have at least three polygons meeting at a corner. Otherwise, we won’t be able to form a 3D shape.
So, let’s try joining three triangles at a corner. Here’s how we can bend them to form a 3D shape:(Click on the play button)
If we add another triangle to its base, we’ll get our first platonic solid, i.e. the tetrahedron:
Now, let’s join four equilateral triangles at a corner. As shown below, we can bend them to form the top part of our next platonic solid — an octahedron:(Click on the play button)
Similarly, by joining five equilateral triangles at a corner, we can form the top part of an icosahedron as follows:(Click on the play button)
Now, let’s join six equilateral triangles at a corner.
What do we get? A flat surface! We cannot bend them to form a solid shape in any way.
In fact, to make a solid, the sum of the internal angles made by the polygons meeting at a corner must be less than 360 degrees. If the sum equals 360, then the shape will flatten out. And if its
greater, then the polygons will overlap each other and we won't be able to make a solid shape.
Hence, no more solids are possible by joining only equilateral triangles.
Let's move on to squares. There's only one way to join three squares at a corner, and this forms a cube.(Click on the play button)
Joining four squares will again give us a flat surface.
Now, let’s move to pentagons. Joining three of these at a corner and folding them will give us a dodecahedron.(Click on the play button)
Can we join four pentagons at a corner? No, because the sum of the interior angles so formed will exceed 360 degrees.
Let’s move on to hexagons now. If we take three hexagons meeting at a corner, what do we get? Again, a flat surface!
This is because the internal angles add up to 360 degrees. So, we cannot make a 3D shape by joining only hexagons.
Now, if we try joining heptagons, the sum of interior angles at a corner will exceed 360 degrees.
The same is true for any polygon with more than six sides. So, there are no more Platonic solids. Voila!
Now, it’s apparent why Platonic solids are considered sacred. They are special, indeed! And that is why at Cuemath, we reward our special students with these ‘sacred solids’. | {"url":"https://www.cuemath.com/learn/sacred-solids/","timestamp":"2024-11-07T23:27:58Z","content_type":"text/html","content_length":"204771","record_id":"<urn:uuid:43189f49-fb48-44fb-a9f7-c172cb63aedc>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00697.warc.gz"} |
Electrical Engineering
06 Oct 2024
The Fundamentals of Electrical Engineering: A Comprehensive Guide
Electrical engineering is a fascinating field that deals with the design, development, and application of electrical systems and devices. It is a crucial branch of engineering that has revolutionized
the way we live, work, and communicate. In this article, we will delve into the basics of electrical engineering, exploring key concepts, formulas, and applications.
What is Electrical Engineering?
Electrical engineering is the study of electricity, electronics, and electromagnetism. It involves the design, development, testing, and implementation of electrical systems, devices, and equipment.
Electrical engineers work on a wide range of projects, from power generation and transmission to communication systems and electronic devices.
Key Concepts in Electrical Engineering
1. Electric Charge: Electric charge is a fundamental concept in electrical engineering. It is measured in coulombs (C) and can be either positive or negative.
□ Formula: Q = It (charge = current × time)
2. Electric Current: Electric current is the flow of electric charge. It is measured in amperes (A).
□ Formula: I = ΔQ / Δt (current = change in charge / change in time)
3. Voltage: Voltage, also known as electric potential difference, is the force that drives electric current.
□ Formula: V = ΔE / Δx (voltage = change in energy / change in distance)
4. Resistance: Resistance is the opposition to the flow of electric current. It is measured in ohms (Ω).
□ Formula: R = ρ × L / A (resistance = resistivity × length / cross-sectional area)
5. Inductance: Inductance is the property of a circuit that opposes changes in current.
□ Formula: L = N × μ × A / l (inductance = number of turns × magnetic permeability × area / length)
Electrical Circuits
1. Series Circuit: A series circuit is one where all components are connected in a single loop.
□ Formula: V = I × R (voltage = current × resistance)
2. Parallel Circuit: A parallel circuit is one where multiple paths exist for the electric current to flow.
□ Formula: V = I1 × R1 + I2 × R2 + … (voltage = sum of currents × resistances)
1. Magnetic Field: A magnetic field is a region around a magnet or electric current where magnetic forces can be detected.
□ Formula: B = μ × H (magnetic field strength = magnetic permeability × magnetic field)
2. Faraday’s Law of Induction: This law states that a changing magnetic field induces an electromotive force (EMF) in a conductor.
□ Formula: ε = -N × dΦ / dt (EMF = negative number of turns × change in flux / time)
Applications of Electrical Engineering
1. Power Generation and Transmission: Electrical engineers design and develop power plants, transmission lines, and distribution systems to generate and transmit electricity.
2. Communication Systems: Electrical engineers work on communication systems such as telephone networks, internet protocols, and wireless communication systems.
3. Electronic Devices: Electrical engineers design and develop electronic devices such as computers, smartphones, and televisions.
Electrical engineering is a fascinating field that has numerous applications in our daily lives. By understanding the fundamental concepts of electricity, electronics, and electromagnetism,
electrical engineers can design and develop innovative solutions to real-world problems. Whether it’s generating power, transmitting information, or creating electronic devices, electrical
engineering plays a crucial role in shaping our world.
• “Electrical Engineering: Principles and Applications” by James W. Nilsson
• “Electric Circuits” by James L. McNamee
• “Electromagnetism” by David J. Griffiths
Related articles for ‘Electrical Engineering’ :
• Reading: Electrical Engineering
Calculators for ‘Electrical Engineering’ | {"url":"https://blog.truegeometry.com/tutorials/education/e9032c3a7d03ad94adb8d74a3f302869/JSON_TO_ARTCL_Electrical_Engineering.html","timestamp":"2024-11-08T12:12:07Z","content_type":"text/html","content_length":"20115","record_id":"<urn:uuid:7c31913f-baab-41bc-8a25-a5435902e561>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00885.warc.gz"} |
[Solved] Top 10 cars sold. The Car Sales Statistic | SolutionInn
Top 10 cars sold. The Car Sales Statistics site (https://www .best-selling-cars.com) reports the top 10 best-selling car
Top 10 cars sold. The Car Sales Statistics site (https://www .best-selling-cars.com) reports the top 10 best-selling car models in Japan from April to September 2020. According to the data, the top
three car models were three Toyota models—Yaris, Raize, and Corolla. These were followed by the Honda Fit, Toyota Alphard, Toyota Roomy, the Honda Freed, the Toyota Harrier, and the Nissan Note and
Nissan Serena.
a. Suppose a car is randomly selected from the list of the top 10 best-selling car models. What is the probability that the car is a Honda?
b. Refer to part
a. Determine the probability that a randomly selected car would not be a Honda.
c. Suppose a car is randomly selected from the list of the top 10 best-selling car models. What is the probability that the car is a Toyota?
d. Refer to part
c. Determine the probability that a randomly selected car would not be a Toyota. Applying the Concepts—Advanced
Fantastic news! We've Found the answer you've been seeking! | {"url":"https://www.solutioninn.com/study-help/statistics-for-business-and-economics/top-10-cars-sold-the-car-sales-statistics-site-httpswww-1660073","timestamp":"2024-11-11T00:25:23Z","content_type":"text/html","content_length":"77479","record_id":"<urn:uuid:fd63e37b-3dcf-4b53-84fa-64fce6e311d8>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00746.warc.gz"} |
One hundred years of experimentude
Revisiting Neyman's seminal paper on the design of experiments
Read →
Another tangent, but this reminded me that a while back there was a big twitter spat between Judea Pearl and "trialists". A misconception I realized I had, that perhaps should have been obvious to
me, is that in an RCT the treatment assignment is randomized, but the sample of participants is (almost always) not. So, when interpreting the results of the trial, the estimates of, e.g., the ARR
are specific to the sample of participants in the trial. Given what little I know about trial enrollment, it doesn’t seem like we should have much confidence in the generalizability of such results
to a new population (I would be happy to be wrong here!).
Further, RCT statisticians have developed their own language of complicated tools (insert your favorite combination of “cluster”, “block”, and “crossover” before “trial design”) for experiment
design. It seems that the magic—the intellectual achievement here—is in being able to estimate the treatment effect of a sample despite missing counterfactual outcomes. And while this is certainly
impressive, it doesn’t seem to help one answer questions like “will this treatment work on my patient? Will this treatment work on me?”. How does generalizability/transportability of effect
estimates come into play here? This, to me, seems like a very important piece of the problem.
Expand full comment
Yes, well put. The details of the allocation and selection of participants has all sorts of DOFs, which determines what kind of average treatment effect (ATE) is actually being measured, which may
be a one-off opportunity (non-repeatable environment) for a study anyway. But I really like the emphasis of RCTs as a type of measurement here.
Expand full comment
Yes, I agree with both of you. RCTs are not a panacea, and they solve a very specific set of problems. This is why I like thinking of them as a particular measurement device. By strained metaphor,
a tape measure is important to have in your toolbox, but not only are the screwdrivers and wrenches important, the tape measure is not even the only measurement device in there.
I am also glad you are asking “will this treatment work on my patient? Will this treatment work on me?” are questions I want to blog about here. I will definitely come back to this as these are the
questions that keep me up at night.
Indeed, I started getting into applied statistics after being spooked by the fragility of generalization in machine learning. I thought there might be some resolutions there. But there were no
fixes to be found in statistics. My conclusion is that statistics can't really say much of anything about generalizabilty or transportability. I am going to dig into this more in future posts.
Expand full comment
This made me think "Of course, within the confines of reality, we are not telepathic. We can only observe one of these outcomes per column." Here Treatment is a bit. What if we make it a qubit so
we can place units in superposition of |Treatment = Y> and |Treatment = N>? Any situation where a quantum RCT could have an edge over a classical one?
Expand full comment | {"url":"https://www.argmin.net/p/one-hundred-years-of-experimentude/comments","timestamp":"2024-11-09T00:01:25Z","content_type":"text/html","content_length":"153964","record_id":"<urn:uuid:a44b1816-a232-4f2a-879d-8f669a573d37>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00259.warc.gz"} |
Time Limit: 1 Second Memory Limit: 65536 KB
Aunt Lizzie takes half a pill of a certain medicine every day. She starts with a bottle that contains N pills.
On the first day, she removes a random pill, breaks it in two halves, takes one half and puts the other half back into the bottle.
On subsequent days, she removes a random piece (which can be either a whole pill or half a pill) from the bottle. If it is half a pill, she takes it. If it is a whole pill, she takes one half and
puts the other half back into the bottle. In how many ways can she empty the bottle? We represent the sequence of pills removed from
the bottle in the course of 2N days as a string, where the i-th character is W if a whole pill was chosen on the i-th day, and H if a half pill was chosen (0 <= i < 2N). How many different valid
strings are there that empty the bottle?
The input will contain data for at most 1000 problem instances. For each problem instance there will be one line of input: a positive integer N <= 30, the number of pills initially in the bottle.
End of input will be indicated by 0.
For each problem instance, the output will be a single number, displayed at the beginning of a new line. It will be the number of different ways the bottle can be emptied.
Sample Input
Sample Output
Source: North America - Rocky Mountain 2011 | {"url":"https://sharecode.io/section/problemset/problem/2563","timestamp":"2024-11-10T22:03:19Z","content_type":"application/xhtml+xml","content_length":"8215","record_id":"<urn:uuid:ace8f859-afca-4a50-8e8b-6264625a5fe9>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00395.warc.gz"} |
How to solve proportions calculator
Algebra Tutorials! Wednesday 6th of November
how to solve proportions calculator
Related topics:
Home ti calculator rom | sats papers online | 4 simultaneous equation solver | balancing method algebra | solving equations with ratios | slope calculator online | +pre
Calculations with algebra triangle system | free math equation worksheets | pre algebra glencoe answers to practice tests | matlab 2nd order derivative | practise solving permutations and
Negative Numbers combinations | restricted value rational expressions
Solving Linear Equations
Systems of Linear
Equations Author Message
Solving Linear Equations
Graphically Norfelts Posted: Friday 29th of Dec 11:08
Algebra Expressions Holla guys and gals! Recently I hired a private tutor to help me with some topics in algebra. My problem areas included topics such as how to solve
Evaluating Expressions proportions calculator and graphing function. Now that teacher turned out to be such a waste, that instead of helping me now I’m even more confused
and Solving Equations than I earlier was . I still can’t crack problems on those topics. And the exam time is fast approaching . I need someone to help me out. Is there
Fraction rules anything in particular that can be done to get some sort of help? I have a fairly large set of questions to help me learn these topics, but the
Factoring Quadratic Registered: problem is I just can’t crack them, no matter how much effort I put in. Please help!
Trinomials 02.10.2004
Multiplying and Dividing From: Sweden
Dividing Decimals by
Whole Numbers
Adding and Subtracting espinxh Posted: Saturday 30th of Dec 08:17
Radicals Hi, I believe that I can to help you out. Have you ever used a program to help you with your algebra assignments? Some time ago I was also stuck on
Subtracting Fractions a similar problems like you, and then I found Algebrator. It helped me so much with how to solve proportions calculator and other math problems, so
Factoring Polynomials by since then I always rely on its help! My algebra grades improved since I found Algebrator.
Slopes of Perpendicular Registered:
Lines 17.03.2002
Linear Equations From: Norway
Roots - Radicals 1
Graph of a Line
Sum of the Roots of a
Quadratic Admilal`Leker Posted: Monday 01st of Jan 07:24
Writing Linear Equations I remember I faced similar problems with evaluating formulas, side-side-side similarity and perfect square trinomial. This Algebrator is truly a
Using Slope and Point great piece of algebra software program. This would simply give step by step solution to any algebra problem that I copied from homework copy on
Factoring Trinomials clicking on Solve. I have been able to use the program through several College Algebra, Remedial Algebra and College Algebra. I seriously recommend
with Leading Coefficient the program.
1 Registered:
Writing Linear Equations 10.07.2002
Using Slope and Point From: NW AR, USA
Simplifying Expressions
with Negative Exponents
Solving Equations 3
Solving Quadratic tahio Posted: Wednesday 03rd of Jan 07:31
Equations Wow! Do these types of software exist? That would really help me in doing my assignments. Is (programName) available for free or do I need to buy
Parent and Family Graphs it? If yes, where can I buy it from?
Collecting Like Terms
nth Roots
Power of a Quotient Registered:
Property of Exponents 18.01.2005
Adding and Subtracting From: Earth
Solving Linear Systems
of Equations by Jot Posted: Wednesday 03rd of Jan 12:29
Elimination https://polymathlove.com/adding-and-subtracting-fractions-1.html. There you go. Hopefully you will not have to drop math.
The Quadratic Formula
Fractions and Mixed
Solving Rational Registered:
Equations 07.09.2001
Multiplying Special From: Ubik
Rounding Numbers
Factoring by Grouping
Polar Form of a Complex
Solving Quadratic
Simplifying Complex
Common Logs
Operations on Signed
Multiplying Fractions in
Dividing Polynomials
Higher Degrees and
Variable Exponents
Solving Quadratic
Inequalities with a Sign
Writing a Rational
Expression in Lowest
Solving Quadratic
Inequalities with a Sign
Solving Linear Equations
The Square of a Binomial
Properties of Negative
Inverse Functions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with
One Log Term
Combining Operations
The Ellipse
Straight Lines
Graphing Inequalities in
Two Variables
Solving Trigonometric
Adding and Subtracting
Simple Trinomials as
Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing
Fractions 2
Rational Numbers
Difference of Two
Factoring Polynomials by
Solving Equations That
Contain Rational
Solving Quadratic
Dividing and Subtracting
Rational Expressions
Square Roots and Real
Order of Operations
Solving Nonlinear
Equations by
The Distance and
Midpoint Formulas
Linear Equations
Graphing Using x- and y-
Properties of Exponents
Solving Quadratic
Solving One-Step
Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic
Inequality with Two
Operations on Radicals
Factoring a Difference
of Two Squares
Straight Lines
Solving Quadratic
Equations by Factoring
Graphing Logarithmic
Simplifying Expressions
Involving Variables
Adding Integers
Factoring Completely
General Quadratic
Using Patterns to
Multiply Two Binomials
Adding and Subtracting
Rational Expressions
With Unlike Denominators
Rational Exponents
Horizontal and Vertical
how to solve proportions calculator
Related topics:
Home ti calculator rom | sats papers online | 4 simultaneous equation solver | balancing method algebra | solving equations with ratios | slope calculator online | +pre
Calculations with algebra triangle system | free math equation worksheets | pre algebra glencoe answers to practice tests | matlab 2nd order derivative | practise solving permutations and
Negative Numbers combinations | restricted value rational expressions
Solving Linear Equations
Systems of Linear
Equations Author Message
Solving Linear Equations
Graphically Norfelts Posted: Friday 29th of Dec 11:08
Algebra Expressions Holla guys and gals! Recently I hired a private tutor to help me with some topics in algebra. My problem areas included topics such as how to solve
Evaluating Expressions proportions calculator and graphing function. Now that teacher turned out to be such a waste, that instead of helping me now I’m even more confused
and Solving Equations than I earlier was . I still can’t crack problems on those topics. And the exam time is fast approaching . I need someone to help me out. Is there
Fraction rules anything in particular that can be done to get some sort of help? I have a fairly large set of questions to help me learn these topics, but the
Factoring Quadratic Registered: problem is I just can’t crack them, no matter how much effort I put in. Please help!
Trinomials 02.10.2004
Multiplying and Dividing From: Sweden
Dividing Decimals by
Whole Numbers
Adding and Subtracting espinxh Posted: Saturday 30th of Dec 08:17
Radicals Hi, I believe that I can to help you out. Have you ever used a program to help you with your algebra assignments? Some time ago I was also stuck on
Subtracting Fractions a similar problems like you, and then I found Algebrator. It helped me so much with how to solve proportions calculator and other math problems, so
Factoring Polynomials by since then I always rely on its help! My algebra grades improved since I found Algebrator.
Slopes of Perpendicular Registered:
Lines 17.03.2002
Linear Equations From: Norway
Roots - Radicals 1
Graph of a Line
Sum of the Roots of a
Quadratic Admilal`Leker Posted: Monday 01st of Jan 07:24
Writing Linear Equations I remember I faced similar problems with evaluating formulas, side-side-side similarity and perfect square trinomial. This Algebrator is truly a
Using Slope and Point great piece of algebra software program. This would simply give step by step solution to any algebra problem that I copied from homework copy on
Factoring Trinomials clicking on Solve. I have been able to use the program through several College Algebra, Remedial Algebra and College Algebra. I seriously recommend
with Leading Coefficient the program.
1 Registered:
Writing Linear Equations 10.07.2002
Using Slope and Point From: NW AR, USA
Simplifying Expressions
with Negative Exponents
Solving Equations 3
Solving Quadratic tahio Posted: Wednesday 03rd of Jan 07:31
Equations Wow! Do these types of software exist? That would really help me in doing my assignments. Is (programName) available for free or do I need to buy
Parent and Family Graphs it? If yes, where can I buy it from?
Collecting Like Terms
nth Roots
Power of a Quotient Registered:
Property of Exponents 18.01.2005
Adding and Subtracting From: Earth
Solving Linear Systems
of Equations by Jot Posted: Wednesday 03rd of Jan 12:29
Elimination https://polymathlove.com/adding-and-subtracting-fractions-1.html. There you go. Hopefully you will not have to drop math.
The Quadratic Formula
Fractions and Mixed
Solving Rational Registered:
Equations 07.09.2001
Multiplying Special From: Ubik
Rounding Numbers
Factoring by Grouping
Polar Form of a Complex
Solving Quadratic
Simplifying Complex
Common Logs
Operations on Signed
Multiplying Fractions in
Dividing Polynomials
Higher Degrees and
Variable Exponents
Solving Quadratic
Inequalities with a Sign
Writing a Rational
Expression in Lowest
Solving Quadratic
Inequalities with a Sign
Solving Linear Equations
The Square of a Binomial
Properties of Negative
Inverse Functions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with
One Log Term
Combining Operations
The Ellipse
Straight Lines
Graphing Inequalities in
Two Variables
Solving Trigonometric
Adding and Subtracting
Simple Trinomials as
Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing
Fractions 2
Rational Numbers
Difference of Two
Factoring Polynomials by
Solving Equations That
Contain Rational
Solving Quadratic
Dividing and Subtracting
Rational Expressions
Square Roots and Real
Order of Operations
Solving Nonlinear
Equations by
The Distance and
Midpoint Formulas
Linear Equations
Graphing Using x- and y-
Properties of Exponents
Solving Quadratic
Solving One-Step
Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic
Inequality with Two
Operations on Radicals
Factoring a Difference
of Two Squares
Straight Lines
Solving Quadratic
Equations by Factoring
Graphing Logarithmic
Simplifying Expressions
Involving Variables
Adding Integers
Factoring Completely
General Quadratic
Using Patterns to
Multiply Two Binomials
Adding and Subtracting
Rational Expressions
With Unlike Denominators
Rational Exponents
Horizontal and Vertical
Calculations with
Negative Numbers
Solving Linear Equations
Systems of Linear
Solving Linear Equations
Algebra Expressions
Evaluating Expressions
and Solving Equations
Fraction rules
Factoring Quadratic
Multiplying and Dividing
Dividing Decimals by
Whole Numbers
Adding and Subtracting
Subtracting Fractions
Factoring Polynomials by
Slopes of Perpendicular
Linear Equations
Roots - Radicals 1
Graph of a Line
Sum of the Roots of a
Writing Linear Equations
Using Slope and Point
Factoring Trinomials
with Leading Coefficient
Writing Linear Equations
Using Slope and Point
Simplifying Expressions
with Negative Exponents
Solving Equations 3
Solving Quadratic
Parent and Family Graphs
Collecting Like Terms
nth Roots
Power of a Quotient
Property of Exponents
Adding and Subtracting
Solving Linear Systems
of Equations by
The Quadratic Formula
Fractions and Mixed
Solving Rational
Multiplying Special
Rounding Numbers
Factoring by Grouping
Polar Form of a Complex
Solving Quadratic
Simplifying Complex
Common Logs
Operations on Signed
Multiplying Fractions in
Dividing Polynomials
Higher Degrees and
Variable Exponents
Solving Quadratic
Inequalities with a Sign
Writing a Rational
Expression in Lowest
Solving Quadratic
Inequalities with a Sign
Solving Linear Equations
The Square of a Binomial
Properties of Negative
Inverse Functions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with
One Log Term
Combining Operations
The Ellipse
Straight Lines
Graphing Inequalities in
Two Variables
Solving Trigonometric
Adding and Subtracting
Simple Trinomials as
Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing
Fractions 2
Rational Numbers
Difference of Two
Factoring Polynomials by
Solving Equations That
Contain Rational
Solving Quadratic
Dividing and Subtracting
Rational Expressions
Square Roots and Real
Order of Operations
Solving Nonlinear
Equations by
The Distance and
Midpoint Formulas
Linear Equations
Graphing Using x- and y-
Properties of Exponents
Solving Quadratic
Solving One-Step
Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic
Inequality with Two
Operations on Radicals
Factoring a Difference
of Two Squares
Straight Lines
Solving Quadratic
Equations by Factoring
Graphing Logarithmic
Simplifying Expressions
Involving Variables
Adding Integers
Factoring Completely
General Quadratic
Using Patterns to
Multiply Two Binomials
Adding and Subtracting
Rational Expressions
With Unlike Denominators
Rational Exponents
Horizontal and Vertical
Author Message
Norfelts Posted: Friday 29th of Dec 11:08
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From: Sweden
espinxh Posted: Saturday 30th of Dec 08:17
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From: Norway
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From: Earth
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From: Ubik
Posted: Friday 29th of Dec 11:08
Holla guys and gals! Recently I hired a private tutor to help me with some topics in algebra. My problem areas included topics such as how to solve proportions calculator and graphing function. Now
that teacher turned out to be such a waste, that instead of helping me now I’m even more confused than I earlier was . I still can’t crack problems on those topics. And the exam time is fast
approaching . I need someone to help me out. Is there anything in particular that can be done to get some sort of help? I have a fairly large set of questions to help me learn these topics, but the
problem is I just can’t crack them, no matter how much effort I put in. Please help!
Posted: Saturday 30th of Dec 08:17
Hi, I believe that I can to help you out. Have you ever used a program to help you with your algebra assignments? Some time ago I was also stuck on a similar problems like you, and then I found
Algebrator. It helped me so much with how to solve proportions calculator and other math problems, so since then I always rely on its help! My algebra grades improved since I found Algebrator.
Posted: Monday 01st of Jan 07:24
I remember I faced similar problems with evaluating formulas, side-side-side similarity and perfect square trinomial. This Algebrator is truly a great piece of algebra software program. This would
simply give step by step solution to any algebra problem that I copied from homework copy on clicking on Solve. I have been able to use the program through several College Algebra, Remedial Algebra
and College Algebra. I seriously recommend the program.
Posted: Wednesday 03rd of Jan 07:31
Wow! Do these types of software exist? That would really help me in doing my assignments. Is (programName) available for free or do I need to buy it? If yes, where can I buy it from?
Posted: Wednesday 03rd of Jan 12:29
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Arithmetic - Homeschooling In Montana
Find the resources and ideas you need for your child to learn counting, addition, subtraction, multiplication, division, fractions, elementary geometry, and more.
Arithmetic Teaching Tips & Ideas
Let's Play Math!
This wonderful blog is written by a homeschooling mother who wants to make learning math fun. It is a place where you can learn about new ways of learning, teaching, and understanding math. Math is a
game, playing with ideas. This blog is about the ongoing adventure of learning, teaching, and playing around with mathematics from preschool to precalculus.
Printable Multiplication Tables
You'll find free printable multiplication tables, including a 0-10 and 12 x time table, a grid chard, blank grid charts, and more.
Homeschool Math Blog
Math teaching ideas, links, worksheets, reviews, articles, news, Math Mammoth, and more--anything that helps you to teach math.
Homeschool Math
HomeschoolMath.net is a comprehensive math resource site for homeschooling parents and teachers: find free worksheets, math ebooks for elementary grades, an extensive link list of games, a homeschool
math curriculum guide, interactive tutorials & quizzes, and teaching tips articles. The resources emphasize understanding of concepts instead of just mechanical memorization of rules.
Arithmetic Games and Activities
Math Rider
MathRider combines fun math game play with a highly sophisticated question engine that adapts to your child. The game propels your child to mastery of all four math operations using numbers 0 to 12
in record time.
Helping Your Child Learn Mathematics
This booklet is made up of fun activities that parents can use with children from preschool age through grade 5 to strengthen their math skills and build strong positive attitudes toward math.
Homeschool Arithmetic Curricula
Mad Dog Math
Mad Dog Math is a fun, exciting, motivating, and challenging supplement to any math curriculum. They've taken the basic facts and broken them down into bite-sized pieces that any child can master. A
child progresses through a series of timed drills at his own pace.
Making Math Meaningful
Making Math Meaningful is a wonderful beginning. It lays the strong foundation for understanding math. Every concept is introduced with a simple conversation (which is provided for you) using math
manipulatives. You are spending quality time with your children talking about each math concept. The conversations are simple. The activities are easy to do. The lesson plans tell you exactly what
you are to do and what you are to say. Perfect for the busy mom. There are no seminars to attend and no videos to watch! Simply pull the book from the shelf and start teaching. Each child has his own
student workbook. Levels K through 4 give you a written script to teach each concept and skill. Levels 5 and 6 and Algebra are written directly to your child. Your child will simply pull the book
from the shelf and teach himself. There is no easier math manipulative program for mom.
Life of Fred
This set of books is unlike any other textbooks. Each text is written in the style of a novel with a humorous story line. Each section tells part of the life of Fred Gauss and how, in the course of
his life, he encounters the need for the math and then learns the methods. Tons of solved examples. Each hardcover textbook contains ALL of the material – more than most instructors cover in
traditional classroom settings. Includes tons of proofs. Written by Dr. Stanley Schmidt with the intent to make math come alive with lots of humour, clear explanations, and silly illustrations that
stick in the mind. The student will learn to think mathematically. Completion of this series prepares student for third year college math.
Chalk Dust Company
Chalk Dust Company offers mathematics instruction on videotape to homeschooled students and a variety of other users. Textbooks used in Chalk Dust programs are published by Houghton Mifflin Company
and most are authored by Ron Larson. Offers solutions guides and personal help when needed via telephone or the internet, providing a comprehensive and effective distance-learning environment.
Sonlight Math Curricula
Teach confidently with Sonlight's handpicked math resources — the programs are the best on the market for elementary math to Calculus. Every math curriculum has potential strengths and weaknesses, so
you'll find included information to help you find the best fit for your students. They offer several math programs for the elementary grades. Young children benefit from an enriched environment in
which they see math being used in different ways and in which they approach problems from different directions.
Teaching Textbooks
Teaching Textbooks were designed specifically for independent learners. They offer more teaching/explanation than any other product on the market.
Key to...Workbooks
Give your students the keys to math-skill mastery! These self-paced, self-guided workbooks—covering topics from fractions and decimals to algebra and geometry—motivate students and build their
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Measurement, Metric Measurement, Algebra, and Geometry.
Ray's Arithmetic
Ray's Arithmetics teach arithmetic in an orderly fashion, starting from rules and principles, building knowledge piece by piece, leading pupils from simple to complex. From the very first pages,
Ray's Arithmetics incorporate what has become the scourge of today's math students - story problems. Students must READ simple sentences which pose real life problems, decide whether to add,
subtract, multiply or divide, and finally arrive at the answer - sometimes mentally - sometimes in writing.
Professor B Math
Professor B Math delivers world-class mathematics and national precedents by means of a unique perspective on children's real gifts for learning mathematics.
Homeschool Arithmetic Worksheets and Printables
Homeschool Math
HomeschoolMath.net is a comprehensive math resource site for homeschooling parents and teachers: find free worksheets, math ebooks for elementary grades, an extensive link list of games, a homeschool
math curriculum guide, interactive tutorials & quizzes, and teaching tips articles. The resources emphasize understanding of concepts instead of just mechanical memorization of rules.
Lego Challenge Math Activity-Free Printable
Practicing math skills like spatial awareness and geometry can be fun, especially when the math activity involves Legos! Here’s a free printable math challenge for kids using Lego or Duplo bricks!
Featured Resources
As an Amazon Associate, we earn from qualifying purchases. We get commissions for purchases made through links on this site.
Learning Styles: Reaching Everyone God Gave You to Teach
This book offers helpful and practical strategies about the different ways that kids acquire information and learn, and then use that knowledge. Kids' behavior is often tied to a particular learning
style and understanding that fact will help parents respond to their child in ways that decrease frustration and increase success, especially in a homeschooling environment.
Smart Mouth
Ages: 8 years and up; For 2 or more playersSmart Mouth is a quick-thinking shout-it-out hilarious word game that helps build vocabulary skills. It includes variations of the rules for category play
and for younger players. Players slide the Letter Getter forward and back to get two letters. The first player to shout out a word of five or more letters using those letters wins the round. The game
includes tips for teachers. This is a fun game to play with children and adults together.
Elementary Geography
Elementary Geography is a reprint of the original work by Charlotte Mason. It includes her ideas about teaching children about their world, with poetry selections throughout the book. Explores ideas
of place from space to our earth, seasons, map making, and topography. Written in a pleasing conversational style, it is useful for understanding teaching methods, memorization, and copy work.
The Exhausted School: Bending the Bars of Traditional Education
These 13 essays, presented at the 1993 National Grassroots Speakout on the Right to School Choice, illustrate how education reform actually works. Written by award-winning teachers and their
students, these essays present successful teaching methods that work in both traditional and nontraditional classroom settings. Gatto s voice is strong and unique. Thomas Moore, author of Care of
the Soul
KONOS offers unit studies for homeschoolers, with a focus on character building and the study of history. | {"url":"https://www.homeschoolinginmontana.com/subjects/math/arithmetic","timestamp":"2024-11-05T09:23:23Z","content_type":"text/html","content_length":"45389","record_id":"<urn:uuid:67d129d3-2bfa-43ab-a921-71581ab362d6>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00098.warc.gz"} |
Separation Theorem - (Variational Analysis) - Vocab, Definition, Explanations | Fiveable
Separation Theorem
from class:
Variational Analysis
The Separation Theorem states that if two non-empty convex sets do not intersect, there exists a hyperplane that can separate them. This fundamental result in convex analysis plays a crucial role in
understanding the relationships between convex sets and provides a way to analyze optimization problems by establishing the conditions under which solutions exist.
congrats on reading the definition of Separation Theorem. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The Separation Theorem is applicable to both finite-dimensional and infinite-dimensional spaces, showcasing its versatility in various contexts.
2. If two convex sets intersect, they cannot be separated by any hyperplane, highlighting an essential property of their geometric arrangement.
3. The theorem guarantees that for closed and bounded convex sets, a separating hyperplane can be found without exception.
4. Applications of the Separation Theorem include optimization problems, game theory, and economics, where it helps in determining feasible regions for solutions.
5. The theorem is closely related to duality concepts in linear programming, where separation properties aid in finding optimal solutions.
Review Questions
• How does the Separation Theorem ensure that non-intersecting convex sets can be separated by a hyperplane?
□ The Separation Theorem asserts that when two non-empty convex sets do not intersect, there is a hyperplane that divides them such that all points from one set lie on one side of the
hyperplane and all points from the other set lie on the opposite side. This is significant because it shows that if we have two disjoint sets, we can always find a linear boundary to
distinguish between them, which is essential in optimization and decision-making processes.
• Discuss how the properties of convex sets influence the application of the Separation Theorem in optimization problems.
□ In optimization problems, the properties of convex sets are critical because they ensure that any local minimum is also a global minimum within a feasible region. The Separation Theorem aids
in establishing boundaries between feasible regions by allowing us to confirm when an optimal solution exists. If we can separate feasible regions from infeasible ones using hyperplanes, we
can effectively focus our search for solutions within valid constraints.
• Evaluate the implications of the Separation Theorem in the context of duality in linear programming.
□ The Separation Theorem has profound implications for duality in linear programming because it provides a geometric interpretation of primal and dual solutions. If we consider the primal
problem as defining a feasible region represented by a convex set, the dual problem's feasible region can also be visualized as another convex set. When these two sets do not intersect,
separation implies that there are no optimal solutions for one problem while establishing bounds for optimal solutions of the other. This interplay reinforces our understanding of optimality
conditions and strengthens the foundation for solving linear programs.
"Separation Theorem" also found in:
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Yacht Design Terminology
Our profile of yacht designer Paul Gartside in WB No. 230 included a number of design terms that space and style restrictions kept us from defining in the article. For those readers seeking a deeper
understanding of the elements of design, here are the definitions of those terms—along with a few additional ones that were not listed in the article. — Robin Jettinghoff, Assistant Editor
Area of Wetted Surface: The hull’s surface below the waterline; what you cover with bottom paint. Besides determining the amount of bottom paint to buy, the area of wetted surface determines how much
friction there is between the boat and the water. Minimize the wetted surface, and you maximize the speed. A boat flying on foil goes faster because it has none of its hull in the water; the only
wetted surface is that of the foil.
Beam: The width of the hull measured perpendicular to the centerline of the hull. The beam is a significant factor in both carrying capacity and stability.
Maximum Beam (B or B[MAX]): On a hull that flares toward the sheer, the maximum beam will be at the sheerline. On a hull with tumblehome, the maximum beam will be below the sheer. On a trimaran hull,
the maximum beam of the main hull and the beam of the three hulls together are significantly different; the former gives a sense of the boat’s capacity, the latter, of its stability. The maximum beam
is important when finding a slip in a marina, and is often a factor in racing rules.
Beam of Waterline (B[WL]): The maximum beam of the hull at the designed waterline. This is a factor in determining its displacement and prismatic coefficient.
Buttock Lines: In a set of lines plans, lines showing the underwater shape of the hull. In the profile view, they are seen as curved lines parallel to the centerline of the hull, spaced an even
distance apart, showing how the shape of the hull changes moving outward from the centerline. In the plan view, which also shows the waterlines as curves, the buttock lines are at equal intervals and
parallel to the longitudinal centerline of the hull. In the section view (seen from dead ahead or dead astern), they are vertical lines and parallel to the centerline of the hull.
Center of Buoyancy (CB): The center of volume of the underwater part of the hull, below the DWL. The water pushes on the hull with an upward force, centered on this point, keeping the hull afloat. As
the hull moves through the water, the underwater shape changes and the CB moves, but it must stay in the same vertical line as the center of gravity or the hull will go out of trim. Designers need to
know the longitudinal center of buoyancy (LCB) when the hull is at rest, so they can locate weights fore and aft of that point in such a way that keeps the hull in trim.
Center of Effort (CE): The geometric center of the sail area. A designer finds the CE of a sail by drawing a line from the midpoint of the luff of the sail to its clew. Then he draws a line from
midpoint of the foot of the sail to the head of the sail. Where these lines cross is the center of effort of that sail.
To find the CE for the sail area of a main and jib, the designer computes the area and the location of the CE for the main and jib individually, then draws a line connecting their CEs. The CE of the
two sails is on this line. Its location is in proportion to the areas of the two sails. For example, if the mainsail is 250 sq ft, and the jib is 100 sq ft, the jib is 100/250 or two-fifths of the
area of the main. The CE would be two-fifths of the way along the line going from the main’s CE to the jib’s.
If the CE is too far aft of the CLR, the boat will have weather helm; if it is too far forward, it will have a lee helm.
Center of Gravity (CG): The weight of the vessel in the water creates a downward force that is concentrated at the center of gravity. The CG and center of buoyancy (CB) are opposing forces.
Center of Lateral Resistance (also called Center of Lateral Plane; CLR or CLP): The geometric center of the profile of the underwater part of the hull. This is a center of balance. If you try to
balance something on the tip of your finger, you move the object back and forth incrementally on your finger until it balances. Similarly, if an underwater force on the side of a hull is trying to
push the hull at right angles to the force, and the force is the too far forward, the force will push the boat’s bow farther than it will the stern. If the force is too far aft, then the stern will
be pushed farther than the bow. When the force is pushing on the balance point between these possibilities, the hull moves at right angles to the force. This point is the center of lateral resistance
. The force of the wind is concentrated on the center of effort. Its location in relation to the CLR will determine if a boat has weather or lee helm.
A designer can find the CLR of a hull by tracing the underwater profile onto a piece of stiff paper. Then he cuts out this out and balances it on the edge of a ruler. Next he marks this line of
balance, and rotates the hull profile, balances it again, and draws a second line showing where it balanced. The intersection point of these lines is the CLP.
Construction Method: The method of construction and materials used in building a boat. Both greatly affect the cost of a boat. One of the reasons fiberglass hulls became so popular was that they
could be mass-produced more cheaply than wooden hulls. Boats have been built from wood, plywood, fiberglass, metal, combinations of composite materials, and concrete. Each has different properties in
cost, ease of construction, strength, ease of repair, and accessibility of materials. The designer and owner together will discuss the pros and cons of the different methods to find the one that best
fits the owner’s needs.
Deadrise Angle: The angle between the bottom of the hull and a horizontal plane drawn out from the hull’s centerline, looking at the hull sections. A steeper deadrise angle will mean the hull
sharpens and narrows as it gets deeper, while a smaller angle means the hull bottom is flatter.
Displacement (Δ): The underwater volume of a boat is equal to the volume of water it displaces. Underwater volume is expressed in cubic feet or meters; displacement is the weight of the water
displaced, and is expressed in pounds, tons, or tonnes. A boat that has an underwater volume of 125 cubic feet displaces 125 cubic feet of water. This displaced water weighs 64 lbs per cu ft (in salt
water, about 62.5 lbs/cu ft in fresh water); 125 x 64 = 8,000 lbs. Since the weight of that displaced water is equal to the weight of the entire boat, 8,000 lbs is the boat’s displacement.
A common demonstration of displacement can be done with a model boat, a bowl big enough to hold the boat, a shallow pan, a small scale, and enough water to fill the bowl. Put the bowl in the shallow
pan, then fill the bowl with water up to the brim. Gently float the model in the water, and some of the water will overflow from the bowl into the pan. Weigh the water in the pan and weigh the model,
and you will see they are same weight. The model weighs exactly the same as the water that it displaced.
Displacement/Length Ratio (Δ/L): The ratio between the LWL and the boat’s displacement. The formula is the displacement divided by 1/100th of the length of the LWL cubed, or Δ/.01LWL^3, where the Δ
is in tonnes (1 tonne = 2,240 lbs).
Draft: The maximum measurement from the designed waterline to the bottom of the hull. A boat with a centerboard will have two drafts. A sailing dinghy, such as Joel White’s Haven 12-1/2, will draw
18″ with the centerboard up, and 3′4″ with it down. Shallow-draft hulls, such as CARIB II (see WB No. 228), will often have centerboards so they can travel more easily in shallow water, such as that
found in the Florida Keys and the Bahamas.Freeboard: The measurement from the designed waterline to the sheerline. This is most obviously demonstrated in a small dinghy with more than one adult
aboard. As the weight inboard increases, the freeboard decreases. The freeboard is usually highest at the bow and lowest somewhere near amidships. The minimum freeboard is the number of most concern.
Interior Amenities: Cabin comforts and accommodations. To spend more than a few hours aboard a boat, people will need to consider their needs for food, water, rest, lighting, and elimination of
wastes. Amenities aboard a boat might be as simple as a berth on the floor of the cockpit, a bucket, a candle lantern, and a portable stove. A weekender might have a small Coleman stove, a couple of
battery-operated lamps, a V-berth, and Porta-Potti.
Offshore cruisers want the means for dealing with these fundamental human needs to be as convenient as they are at home. Lighting will need wiring and some source of power—perhaps a solar cell or
generator. An efficient galley needs a sink, which requires a tank for fresh water, a hose from the tank to tap, and a drainpipe to deal with the wastewater. A galley stove needs some source of heat,
requiring another tank and more hose. A head requires a sink with tank, hose, and drain, as well as a toilet with hoses for water in and water out, and a holding tank for the wastes. All of these
systems add to the cost and complication of a design.
Length: The length of a hull as measured down its centerline. There are at least four descriptions of the length of a boat (see below); the length in question will depend upon what you are looking
for. A marina manager wants to know the maximum length of the boat, so it can be seated in a slip. Racing rules are often concerned with the length of the load waterline. Designers working around
these rules have led to some extreme hull forms in an effort to beat the rule constraints.
Length Overall (LOA; sometimes Length of Hull): The overall length of the hull. In a dinghy, this is relatively clear; with the boat out of the water, one can hold a tape measure down the centerline
of the hull from the after side of the top of the transom to the forward edge of the stem. The measurement gets more complicated in a canoe with tumblehome stems, one whose stems curve inward as they
rise from the waterline toward the top of the hull. For most motorboats and rowboats, the overall length is pretty evident. Rudders, anchor rollers, or other hull extensions are not included in this
Length on Deck (LOD): The length along the centerline of the deck, measured from its tip at the bow, the intersection of the deck line with the profile bow line, to the point where the deck meets the
transom. This provides a better tool for estimating a boat’s carrying capacity than does overall length.
Load Waterline Length (LWL): The length of the hull measured at the waterline. When the designer draws the boat on the plans, he or she computes a plane on the hull where the boat will float in its
expected operating condition. The actual waterline will vary from this in use. A fishing boat on its way out to the fishing grounds will float higher in the water than when it heads home fully loaded
with fish. A cruising boat out for a weekend adventure will not have as much gear aboard as one fully loaded for an ocean passage. The designed waterline should lie between these two maximums.
The designed waterline (DWL) and load waterline (LWL) are usually in the same plane on the hull. The waterline length is the critical factor in determining the maximum speed of a displacement hull.
Sailing boats with long overhangs at the bow and stern will increase their waterline length as they heel over while sailing; this increases their maximum speed potential.
A design will have several lines parallel to the LWL at set distances apart. These lines, also called waterlines, define the shape of the hull in plan view on the drawing board.
Sparred Length: The maximum reach from the tip of the bowsprit to the aftermost point of the boom. According the racing rules in place for the 1903 AMERICA’s Cup race, Nathanael Herreshoff’s RELIANCE
was required to have a waterline length of no more than 90′. Herreshoff built her to be an extreme hull that still met the parameters required by the racing rules. Her measured waterline length was
90′, her hull length was 144′, and her sparred length was 201′.
Length-to-Beam Ratio (L/B): The ratio between the hull length and the maximum beam. This is a factor in the boat’s stability and speed. A sculling hull used by eight rowers has a length of about 50′
with a beam of about 2′, giving an L/B of 25. While a Cape Cod catboat might have a length of 22′, and a beam of 8′ giving an L/B of 2.75. As the L/B increases, the boat’s speed should also increase.
Offsets: A table of measurements taken vertically and horizontally that establish the shapes of the hull’s frames. The vertical measurements, or heights, are taken from a baseline established by the
designer, either the DWL or a horizontal line parallel to the DWL below the boat’s profile as drawn on the plans. The heights are measured at equal intervals measured from the centerline of the hull.
The lines joining the height measurements are called buttock lines. The horizontal measurements, or half-breadths, are taken from the boat’s longitudinal centerline. These are measured at equal
intervals above and below the boat’s DWL. The lines joining the half-breadth measurements are called waterlines. The stations are measured at equal intervals from a forward perpendicular established
by the designer.
Outside Ballast: Weight attached outside the main hull, as in a ballast keel, that lowers the CG and counterbalances the forces on the sails to keep the boat upright. A keel also provides lateral
resistance to help prevent the sails from pushing the boat sideways in the water.
Prismatic Coefficient (C[p]): The percentage of volume that a hull’s shape has when compared to a prism as long as the designed waterline, and the shape of the hull below the waterline at the largest
hull section. For example, if the LWL was 25’, and the underwater cross-sectional area of the largest section was 12 sq ft, then the volume of that shape is 25 x 12 = 300 cu ft. But a boat does not
have the same sectional shape all the way along its length; the actual volume of the hull is carved from this shape.
If you had a wood block in this shape, and carved the wood away until you only had the shape of the hull left, the volume of the material remaining is a percentage of the total volume of the block.
This percentage is the prismatic coefficient. If you have 125 cu ft left, the boat has an underwater volume of 125 cubic feet; divide 125 into 300, and 125 is 42 percent of 300, so the Cp is .42. If
you have 189 cu ft of underwater volume left, the Cp is 189 ÷ 300 = .63.
A lower C[p] means the boat’s volume is concentrated toward the middle of the boat. Hulls with a higher Cp move some of that volume toward the ends, increasing their speed and making them less
inclined to pitch. If this coefficient is too high, wave drag increases, slowing the hull down. Most designers have a prismatic coefficient in mind for the hull they are designing.
Profile: The shape of the centerline of the hull as seen from the side. Most plans show the boat facing right so the profile is what you see from the right side. The profile is generally bounded on
top by the sheerline, at the bow by the curve of the stem, underwater by the keel, and aft by the transom, with buttock lines showing the shape of the hull from the centerline outward.
Propulsion: The method by which a boat moves through the water. If a designer is designing a powerboat, he or she must know the weight of the boat to determine the correct engine horsepower and
propeller to drive the boat most effectively. A designer also recognizes that engines are heavy and must be placed in a boat so they don’t upset the boat’s trim. Also the engine’s controls must be
connected to some kind of helm station—providing the helmsman with a throttle, a gearshift, and a steering wheel, stick, or tiller. The designer of a sailboat must cope with another force acting upon
the boat: wind. The designer must determine how much sail area is needed to move the boat through the water, the amount of ballast necessary to counteract the force on the sails, and the location of
the sails and rig to keep the boat in balance.
Righting Moment: The restoring force by which a hull resists heeling, created by the ballast keel and the form stability of the hull.
Sail Area (SA): The surface area of sails needed to drive the boat. The designer draws each sail on the plans, then divides the sails into triangles. Then he applies the familiar formula for
determining the area of a triangle (area = ½ base x height or ½ bh) and computes the area of each triangle in each sail, then adds them up to get the total sail area. Boats with multiple
headsails—like staysails, genoas, or spinnakers—will have different sail areas depending on what sails are flying. A designer determines the total sail area needed to drive a boat by looking at
comparable designs. The published sail area of a design is usually the sum of the mainsail area and the foretriangle area, which is the area bounded by the mast, headstay, and deck.
Sections: The hull’s shape on vertical planes cut perpendicular to the boat’s centerline. A designer shows three views of the hull on the drawing board: its profile, or appearance from the side; its
plan view, or appearance from the top or bottom; and its sections, or appearance from the bow and stern. All the lines defining the hull appear in all three views; but in each view, two of the line
types are straight lines at right angles to each other, and the third type is curved. The hull sections show how the curve of the hull changes as it moves fore and aft.
Sheer: The curve along the top edge of the hull’s side, as seen on the profile view. A sheerline often sweeps downward from the bow toward somewhere around amidships, then sweeps up again as it heads
toward the transom. Some boats have reverse sheers where the highest point is not at the bow but closer to amidships.
Speed-to-Length Ratio (S/L): A dimensionless ratio that indicates a hull’s hydrodynamic limitations of speed. A boat creates a bow wave as it moves through the water. As the boat’s speed increases,
the wave gets bigger and creates more resistance for the hull to move through. A planing hull, with a typical S/L of 2.5 or so, can get up on plane and move over this wave, minimizing the resistance
and letting the boat go faster. A displacement hull is too heavy to get over this wave. For a displacement hull, its limit of speed, or hull speed, is a Speed-to-Length Ratio of 1.34. For a
displacement hull, its maximum speed is limited by its length. The formula for the S/L is the boat’s maximum speed divided by the square root of its length. In numbers, this looks like V/√L. So a 36’
cruising sailboat with a S/L of 1.34 has a maximum speed of 1.34 = V/√36. The square root of 36 is 6, so this formula now reads 1.34 = V/6, or multiplying both sides by 6 gives 1.34 x 6 = V, and V =
8.04 knots.
Stem: The forward part of the profile of the hull. The stem timber forms the forwardmost part of the hull, reaching from below the waterline up to the sheerline, providing a place for the planks to
land in the hull’s construction. The sharp bend in the stem, usually just at the waterline, is called the forefoot. The angle of the stem to the water, and the angle of the planks to the stem, is a
factor in whether a hull cuts through waves or slaps the top of them.
Transom: The after part of the profile of the hull. The transom spans between the hull sides at the aft end of the hull. It may have a sharp rake as on a Friendship sloop, or be nearly vertical as on
some runabouts. Double-enders have no transom at all. Some commuter boats of the last century, such as APHRODITE, have a curving reverse transom, where the line of the deck curves over gracefully
into the hull.
Underwater Volume: The measurement of the amount of the hull under the water in cubic feet or meters. The underwater volume and the boat’s displacement are two measurements of the same thing. The
volume is measured in cubic feet or cubic meters, and the displacement is measured in pounds, kilograms, tons, or tonnes. See Displacement also.
Waterline: The hull’s shape on horizontal planes cut perpendicular to the boat’s centerline. A boat lying in the water floats with some percentage of the hull above the water and the remainder of the
hull below the water. The waterline established by the designer is the plane that divides these two parts of the hull and is called the designed waterline, or DWL. The amount of hull within the water
is a factor in determining the boat’s displacement, prismatic coefficient, and stability. The waterline of a small boat will shift when the crew changes position aboard. The waterline plane of a
sailing monohull changes its shape significantly as the hull heels. People will often apply bottom paint to the load waterline and add a thin stripe (called a boottop) of contrasting paint just above
it. The area of the hull above the waterline is called the topsides.
Weight: The weight of a boat. As with length, weight can depend on the situation in which you weigh the hull. The hull weight is the weight of the hull itself with nothing in it. Dinghies and skiffs
often have little gear in them other than a pair of oars, so the hull weight is an accurate measure of their weight. If you add a motor to a small boat, then it weighs more, with most of the weight
now at the stern. If you add passengers, they should be placed to balance the boat’s trim, so she will move more easily through the water.
For bigger boats, the hull holds water and fuel tanks, a deck, spars and sails, and interior furniture such as berths and a galley. When designing a boat, the designer places these items so that the
boat stays level—much as you balance the crew of a dinghy as you carry them from dock to mooring. The designer also factors the weight of the crew into the design. Indeed, sailors count on the weight
and placement of the crew to counter the forces of the wind on the sails. The weight of a boat and its displacement are the same thing. | {"url":"https://www.woodenboat.com/online-exclusives/yacht-design-terminology","timestamp":"2024-11-02T20:42:11Z","content_type":"text/html","content_length":"111832","record_id":"<urn:uuid:1bd3b664-2ebd-4f4e-8307-0fb8652feadd>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00689.warc.gz"} |
Pivoting transposes rows to columns in order to group and aggregate data display. This is useful, for example, in datasets where you want to use just a subset of values from a column dimension, view
different aggregations of the same column and the same aggregation of different columns, or simply to select more than one measure to easily group and compare values. Pivot tables can be generated
from samples and by querying against a complete dataset. The pivot inspector is interactive, so once a pivot sheet is created you can test and update your results iteratively.
For example, in a workbook that includes height, weight, age, gender, and state of residence for a set of individuals living in the United States, you could choose just certain states as the first
column grouping, gender as the second column aggregation, then weight as the measure with average as the operator to yield the average weight by gender for the selected states.
Depending on the measure being used, supported aggregations are:
• ANY - returns any value in the group
• AVG - average of the values in a group for numeric values
• COUNT - number of elements in a group
• MAX - maximum of the values in a group for numeric values
• MIN - minimum of the values in a group for numeric values
• STDEV - statistical standard deviation inside the group for numeric values
• SUM - sum of the values in a group for numeric values
• VAR - statistical variance in the group for numeric values
First and last are special functions that compute the first or last element inside a group given a specific order. For example, given the grouping by state and gender, you could select weight as
measure with first as the operator and height as the OrderBy criteria to yield the weight of the tallest person per state by gender.
Creating a Pivot
To create a pivot:
1. In the active sheet, select Pivot from the Edit menu, or click the Pivot icon on the toolbar.
2. A new sheet will open in the pivot view with an Explorer at the top for visual display of the results, and a Preview table at the bottom.
3. In the Pivot Sheet dialog to the right, select one or more Columns on which to pivot and aggregate. Aggregation is hierarchical in the order of columns selected.
4. Select a column in the first Columns field then click to select the values by which you want to group data. Select All is an option but will not always yield useful results. It is wiser to choose
to aggregate by values with lower cardinality so the results will be comprehensible.
5. In the Measure fields, select the data characteristics you want to view. Only the columns other than those already selected will be available.
You must also select an operation. Functions are filtered depending on the data type selected as the measure. FIRST and LAST require an additional OrderBy value.
6. Click Pivot to process.
The Explorer will display the pivot aggregation hierarchically with color coding for columns (purple) and measure (orange).
The Preview contains the sheet generated by the pivot transformation. The sheet column headers display the actual yielded pivot aggregation and values.
Note that these panes scroll independently.
7. Once a Pivot sheet is created you can refine the results dynamically by changing values or introducing an additional pivot by Rows.
The column value you choose in the Rows field becomes a row label in the resulting pivot table. Spitting by rows lets you view and categorize data from an independent angle.
Typically you would split by rows to view characteristics with fewer values, or to simplify very large datasets with numerous enough values to populate your columns with a high density of | {"url":"https://datameer.atlassian.net/wiki/spaces/DAS70/pages/31973458074/Pivoting+Data?atl_f=content-tree","timestamp":"2024-11-01T20:51:45Z","content_type":"text/html","content_length":"868335","record_id":"<urn:uuid:04b0976f-d107-4c2b-b655-26c031eba572>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00090.warc.gz"} |
Evaluation of Genome Based Estimated Breeding Values for Meat Quality in a Berkshire Population Using High Density Single Nucleotide Polymorphism Chips
Pork meat constitutes more than 40% of worldwide meat production, forming a prominent source of human food (
Rothschild et al., 2011
). Customer demand for pork depends upon the meat quality and its physical and biochemical components (
Bonneau et al., 2010
). Thus, pork quality is an economically important factor and one of the major selection benchmarks for breeding process in the swine industry (
Luo et al., 2012
). With the advancements in meat processing technologies, it is now possible to predict ham, loin primal, and sub-primal cut weights (
van Wijk et al., 2005
). These developments paved the way for the pork industry to adopt more precise value-based classifying systems to satisfy the demands of pork market (
Brorsen et al., 1998
Pig breeding programs have been implemented mainly towards the reduction of production cost in the last several decades. Selection was, therefore, focused on economically important traits such as
litter size, weight gain, back fat, and feed conversion. Recently, breeding objectives have also put weight on retail carcass yield and meat quality (
van Wijk et al., 2005
). Carcass quality traits are highly heritable, causing an efficient selection response (
Newcom et al., 2002
The Berkshire breed (
Sus scrofa domesticus
) was found 300 years ago in the Berkshire county of United Kingdom. The king of England preferred Berkshire pork for his own personal meat supply, because of excellent meat quality (
American Berkshire Association, 2013
). They have a dark skin, which protects them from sunburn. Some parts of body such as legs, face and tail, are white pointed. Berkshire pigs are characterized with pink skin color and a strong body
type with short neck. Legs of the breed are short and blocky and feet are strong. An adult pig of the breed has an average weight of 272 kg (600 pounds). Individuals of the breed are usually friendly
and curious and exhibit excellent disposition (
Kawaida, 1993
Recently, the demand for Berkshire pork has increased in Asia (
McLaughlin, 2004
), because of excellent meat quality such as richness, texture, marbling, juiciness, tenderness and flavor (
Goodwin and Burroughs, 1995
Brewer et al., 2002
). The thin muscle fibers and excellent water holding capacity (WHC) of the meat increases its popularity (
Goodwin and Burroughs, 1995
). The Berkshire pork is exported mainly to Japan and US (
Lammers et al., 2011
The evaluation of breeding values based on pedigree information has many limitations (
Dekkers et al., 2010
). Some phenotypic traits are difficult and costly to measure, resulting in low accuracy of estimated breeding values (
Badke et al., 2014
). These shortcomings could be overcome by using a genome based best linear unbiased prediction method (GBLUP), in which genomic estimated breeding values (GEBV) were predicted with a high density
marker map covering the porcine whole genome (
Meuwissen et al., 2001
Dekkers et al., 2010
The GEBVs can be predicted with dense, genome-wide maps of single nucleotide polymorphisms (SNPs), which can lead to a more precise prediction of pig breeding value at a young age.
Cleverland et al. (2010)
reported that GEBV accuracies in pigs were as good as those in dairy cattle, if the training population size was large enough. Important factors on the accuracy of genomic predictions include number
of phenotypes, (i.e. training data, used to form the prediction equation), heritability, effective population size, genome size, marker density, and genetic architecture of the trait, in particular
number of loci affecting the trait and distribution of their effects (
Daetwyler et al., 2008
Goddard, 2009
Meuwissen, 2009
). Recently, Bayesian methods have gained popularity in evaluating genomic selection, due to the fact that different variance is fitted to each SNP (
Fernando et al., 2007
Moser et al., 2009
In this study, we evaluated accuracy of GEBV for port quality traits in a Berkshire population under the GBLUP and Bayes B models.
Animals and phenotypes
A set of Berkshire samples (n = 1,205) were collected in Dasan breeding farm, Namwon, Cheonbuk province, Korea, between 2008 and 2013. The piglets were weaned at 3 to 4 weeks of age and moved into
piglet pens, in each of which about 100 piglets were raised for 60 days. Then, the pigs were placed in growth/fattening pens 20 pigs in size for 90 to 120 days. The pigs were fed with the commercial
feeds according to the regiments of Purina Ltd. The samples were slaughtered approximately 211 (±23) days of age in an abattoir in Namwon and cooled at 0°C for 24 h in a chilling room. Among the
carcasses, 801 and 404 samples were analyzed for meat quality and composition in the laboratories of National Livestock Research Institute in Suwon and Sunchon National University in Sunchon, Korea,
respectively. A total of 16 carcass and meat quality traits were considered: 1) back fat thickness (BF), 2) Commission Internationale de l’Eclairage (CIE) a, 3) CIE b, 4) CIE l 5) collagen, 6)
carcass weight (CWT), 7) drip loss, 8) fat, 9) heat (cooking) loss, 10) moisture, 11) National Pork Producers Council (NPCC) color score, 12) NPCC marbling score, 13) pH24, 14) protein, 15) shear
force, and 16) WHC. For each individual, slaughter age (sage), gender, and year-season of birth were recorded.
The CWT and BF of each carcass were measured. The parts of loins (longissimus dorsi, LD) on the left side of the cold carcasses were used to determine meat quality parameters. As soon as all samples
were placed in vacuum bags, the samples were transported to the laboratory and then frozen at −50°C until they were analyzed. The middle portions of each loin were used for experiment. For analysis
of moisture content, fat content, drip loss, and heat loss, the only subcutaneous fat of meat samples was removed. For the others, all visible fat was trimmed off.
The proximate composition of each LD muscle was obtained with a slightly modified method of
AOAC (2000)
. Briefly, moisture content was measured by drying 3 g of samples place in aluminum dishes at 104°C for 15 h. The crude protein contents were measured by the Kjeldahl method (VAPO45, Gerhardt Ltd.,
Idar-Oberstein, Germany). The crude fat contents were extracted according to the method described by
Folch and Sloane-Stanley (1957)
. The total collagen content was determined by measuring hydroxyproline and using a multiplication factor of 7.14 (
Etherington and Sims, 1981
). The surface color and marbling scores of each loin was categorized based on NPPC standard (
NPPC, 2000
The surface color value were measured by the CIE L*, a* and b* system using a Minolta colorimeter (Model CR-410, Minolta Co. Ltd., Osaka, Japan). The colorimeter was calibrated against a white
reference tile plate (L* = 89.2, a* = 0.921, b* = 0.783), and the diameter size of aperture was 4 cm. The color L* (lightness), a* (redness), and b* (yellowness) values were obtained after 30 min
blooming at room temperature. The average value of five random measurements taken from different locations was used for the statistical analysis.
The pH value of each meat sample was determined with a pH meter (Orion 2 Star, Thermo scientific, Beverly, MA, USA). Water holding capacity was determined by centrifugation. Briefly, 5 g of minced
meat sample was placed into a centrifuge tube with a filter paper (No. 4, Whatman International Ltd., Maidstone, England), and centrifuged at 3,000×g for 10 min. WHC was calculated as the remaining
moisture in the meat sample on the basis of the moisture content of the original meat sample. The drip loss was measured as the percentage weight loss of a standardized (3×3×3 cm) meat sample placed
in a sealed petri-dish at 4°C during the storage of 2 d. The heat (cooking) loss was determined as the percentage weight loss of a standardized (3×3×3 cm) meat sample after cooking in an electric
grill with double pans (Nova EMG-533, 1,400 W, Evergreen enterprise, Seoul, Korea) for 90 s, until the internal temperature of the meat sample reached 72°C.
The samples were prepared in a cubic form (30×30×20 mm), heated until internal temperature of the samples reached 72°C±2°C, and then cooled for 30 min at room temperature. Each sample was cut
perpendicular to the longitudinal orientation of the muscle fiber with a Warner-Bratzler shear attachment on a texture analyzer (TA-XT2, Stable Micro System Ltd., Surrey, UK). The maximum shear force
value (kg) was recorded for each sample. Test and pre-test speeds were set at 2.0 mm/s. Post-test speeds were set at 5.0 mm/s.
Molecular data
The 1,205 pigs were genotyped with the Illumina Porcine 62 k SNP chips, in which a total of 62,163 SNPs that covered the entire porcine genome were embedded. To evaluate GEBV, the SNPs on 18
autosomal chromosomes were considered for quality control tests. Any SNP was excluded with <90% call rates, <5% minor allele frequency, or significant departure from Hardy Weinberg equilibrium (p
<0.001). Those individuals with less than 90% genotyping call rate were also removed. After the quality control procedures using PLINK v7.0 (
Purcell et al., 2007
), 36,605 SNPs for 1,191 individuals were used. An imputation procedure was applied to predict missing genotypes of the SNPs with Beagle vs3.3.2 (
Browning and Browning, 2007
Nothnagel et al., 2009
Statistical analysis
To obtain GEBV under the GBLUP model, a linear mixed (Animal) model was fitted with the fixed effects of gender for pH24 and BF, and birth year-season for CWT, pH24, CIE a, CIE b, CIE L, drip loss,
heat loss and shear force, and a covariate, slaughter age for fat, protein and BF, respectively. Statistical significance of the fixed factors or covariate was tested using SAS general linear model
procedure (vs9.2). For the rest of meat quality traits, the effects of the factors were not significant (p>0.05). The Animal model, then, can be written as
is the vector of phenotypic record of the animal,
is the vector of overall mean, fixed and covariate effects,
is the vector of breeding values,
is the design matrix for the fixed and covariate effects,
is the design matrix allocating records to breeding values and
is the vector of the residual of the phenotype. To construct genome relationship matrix (
), the subroutine in R was used (version 2.15.0), which was then incorporated into the Animal model using ASREML (average sparsity residual maximum likelihood) 3.0 (
Gilmour et al., 1995
). The mixed model equation was then
where α = σ[e]^2/σ[g]^2 = (1–h^2)/h^2, σ[e]^2 is the residual variance, σ[g]^2 the genetic variance, and h^2 heritability.
The breeding values for both phenotyped and non-phenotyped individuals can be predicted by solving the equation:
G matrix was calculated based on the observed allele frequencies of the markers. The equation used to calculate G matrix was:
The marker matrix, M, had order of n×m, in which n is the number of individuals and m is the number of markers. In the M, alleles were coded as AA (homozygous for the first allele) = −1, AB
(heterozygous) = 0, BB (homozygous for the second allele) = 1. The elements of P matrix were calculated using the formula P[j] = 2(P[j]–0.5), where P[j] was the minor allele frequency of the marker
locus j. (M-P) is called the incidence matrix (Z) for markers. The P matrix was subtracted from the M matrix to set the mean values of the allele effects to 0, and to give more credit to rare alleles
than to common alleles. Genomic inbreeding coefficient would be greater if the individual is homozygous for rare alleles than if homozygous for common alleles.
The meat quality traits composed different sets of training (with phenotype records) and testing (without phenotypes) data. For each individual, GEBV value was predicted, and the expected accuracy of
GEBV for the i^th individual was calculated using standard errors of GEBV as
The mean and standard deviation of the GEBV accuracies was calculated for each trait.
Evaluation of the GEBV was also carried out under the Bayes B model using Gensel 4.0 (Fernando and Garrick, 2008) software. The Bayes model was:
• y = the vector of phenotypes
• μ = overall mean.
• X = the incidence matrix of the fixed and covariate effects.
• b = the vector of fixed and covariate effects.
• z[i] = a vector of genotypes of a fitted marker i, that is coded as −10, 0, or 10.
• a[i] = a random substitution effect of the fitted marker i with its variance, σ[ai]^2.
• e = the vector of random residuals that was assumed to be normally distributed.
For the marker effects, a mixture model was applied, i.e. a fraction of markers (π) with zero effect and 1-π of markers with non-zero effects, which was used to predict GEBV. Then, the genetic
variances of the markers with non-zero effects would have
>0 (
Habier et al., 2011
The π values ranged between 0.996 and 0.999 depending on the traits with different sizes of the training data.
The fixed effects for each trait were fitted as under the GBLUP model. The estimates of genetic and residual variances that were obtained from ASREML analysis were used as prior values for the Bayes
B analysis. A total of 41,000 iterations of Markov chain were run for the analyses, with the first 1,000 iterations of burn-in period and each of 100 iterations was selected to calculate posterior
mean and variance for the marker effects. The GEBVs were based on a weighted sum of the number of copies of the more frequent allele at each SNP locus, with the weights being the estimated allele
substitution effects (β). The sum of all the marker scores for an individual gave the genomic breeding value.
Summary statistics for the sixteen meat quality traits were displayed in
Table 1
. The coefficient of variation were various between traits, e.g. 1.4% for moisture and 51% for drip loss. A set of 36,605 SNPs was chosen from the 62,163 SNPs in Illumina Porcine 60 k Beadchip (
Table 2
). The number of SNPs (4,426) was the greatest in
Sus scrofa
chromosome (SSC) 1, while SSC18 had the smallest number of SNPs (886). The SSCs 2, 4, 6, 7, 8, 9, 13, and 14 had more than 2,000 SNPs. The physical map with all of the available SNPs spanned about
2,195 Mb with an average distance of 67.9±106.7 Kb between adjacent SNPs. However, the average distances were various between chromosomes, ranging between 51.9 Kb in SSC14 and 92.4 in SSC15.
The heritabilities that were estimated using genome relationship matrix (G) ranged between 6% and 46% (
Table 3
Tomiyama et al. (2011)
reported 0.54 and 0.32 for (BF) at finish and CWT, respectively, in a Japanese Berkshire population (n = 4,773).
Jung et al. (2011)
reported that heritabilities of pH2 4 h, CIE a, b, and L, WHC, NPPC marbling, drip loss, heat loss and shear force ranged between 0.51 and 0.66 in a Berkshire population (n = 808), Korea. Compared
with the two reports, the heritability estimates of the meat quality traits in this study were low.
The GEBV accuracies of the testing as well as training data were calculated using the GBLUP and Bayes B methods (
Table 3
). Under the GBLUP model, the average (±standard deviation) of GEBV accuracy means across the traits was 0.65±0.04 for training data, which ranged from 0.42±0.08 for collagen to 0.75±0.02 for WHC.
Under the Bayes B model, the GEBV accuracy mean ranged from 0.10±0.14 for NPCC marbling score to 0.76±0.04 for drip loss with the average of 0.49±0.10 across all the traits. For the testing data
sets, the GEBV accuracy was lower, i.e. the overall average of the traits was 0.46±0.10 under the GBLUP model, ranging from 0.20±0.18 for protein to 0.65±0.04 for drip loss, and 0.38±0.13 under the
Bayes B model, ranging from 0.04±0.09 for NPCC marbling score to 0.72±0.05 for drip loss (
Table 3
For all the traits, the GEBV accuracies were greater for the training samples than for the testing samples under both the GBLUP and Bayes B models. This makes sense in that, for the training samples,
the GEBV prediction was based on both genotypes and phenotypes, while only the genotype information was exploited for the testing samples. However, the GEBV accuracy differences between the training
and the testing samples were smaller for the traits with great training sample size. For example, the average GEBV accuracy of CWT (moisture) under the GBLUP model, with 1,051 (693) training
individuals, was 0.60 (0.74) for the training and 0.56 (0.54) for the testing samples, respectively (
Table 3
The GEBV accuracy depends on four factors; 1) size of the training population, 2) the heritability of each trait, 3) the extend of linkage disequilibrium between the markers and the QTL, and 4) the
distribution of QTL effects (
Goddard, 2009
Hayes et al., 2009
). Our results supported the first two factors. There was a general tendency of high GEBV accuracy of testing samples with the training sample size (
Table 3
) under both the GBLUP and Bayes B models (
Figure 1
). For example, the average GEBV accuracy of NPCC color score (drip loss) for the testing samples, with 358 (1,051) training samples, was 0.30 (0.64) under the GBLUP model. For the traits with the
sample training sample size (e.g. CWT and drip loss, n = 1,051), the testing samples had greater GEBV accuracy for drip loss (average was 0.72 under the Bayes B model) than for CWT (0.54), for which
heritability of the former (latter) trait was 0.27 (0.13) (
Table 3
In general, the average GEBV accuracy values were similar between the GBLUP and Bayes B methods. However, for some traits with small sample size, e.g. collagen, moisture and NPCC marbling score, the
accuracy of both under the Bayes B model was much smaller than under the GBLUP model (
Table 3
). This may be partly due to small sample size of the training data, which would cause the estimation of GEBV to be more sensitive to the prior values of the Bayes B method.
Cleveland et al. (2010)
reported that GEBV accuracy decreased with small training sample size, because of not enough information to accurately estimate SNP effects.
There were a few reports about GEBV accuracy in pig populations.
Badke et al. (2014)
reported that the GEBV accuracy for BF was 0.45 to 0.47 in a Yorkshire population (965 training samples). Our results showed that, for the trait, 0.59 and 0.65 accuracies were obtained by the GBLUP
and Bayes B analyses, respectively, using 1,043 training samples (
Table 3
There are many reports about successful genomic selection procedures in dairy cattle (
Hayes et al., 2009
van Raden et al., 2009
Wiggans et al., 2011
). The dairy cattle breeding industry is benefitting from genomic selection mainly by reduced generation intervals (
Wellmann et al., 2013
), while, in the pig industry, the benefit from genome selection was less, partly due to short generation intervals (
Wellmann et al., 2013
). However, some studies reported that genomic selection was relevant in pig breeding, by improving maternal traits (
Simianer, 2009
Lillehammer et al., 2011
) or by boosting selection intensities (
Tribout et al., 2012
Wellmann et al., 2013
Herein, we presented the first report about GEBV accuracy of a Berkshire population in Korea, and our results were in general agreement with the previous GEBV studies, i.e. GEBV accuracy depends on
the size of training data as well as heritability of the tested trait. For some traits such as drip loss, GEBVs were predicted with a good accuracy under the GBLUP or Bayes B model (
Table 3
). However, more training samples are needed to further improve the GEBV accuracy for pork quality, especially with low heritable traits, to efficiently implement genome selection programs in
Berkshire industry in Korea. | {"url":"https://www.animbiosci.org/journal/view.php?number=22959","timestamp":"2024-11-12T20:23:04Z","content_type":"application/xhtml+xml","content_length":"178049","record_id":"<urn:uuid:c0eb33fc-a889-4acb-adf8-5ea2feb94ea2>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00037.warc.gz"} |
Cite as
Tanmay Inamdar, Pallavi Jain, Daniel Lokshtanov, Abhishek Sahu, Saket Saurabh, and Anannya Upasana. Satisfiability to Coverage in Presence of Fairness, Matroid, and Global Constraints. In 51st
International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 88:1-88:18, Schloss Dagstuhl –
Leibniz-Zentrum für Informatik (2024)
Copy BibTex To Clipboard
author = {Inamdar, Tanmay and Jain, Pallavi and Lokshtanov, Daniel and Sahu, Abhishek and Saurabh, Saket and Upasana, Anannya},
title = {{Satisfiability to Coverage in Presence of Fairness, Matroid, and Global Constraints}},
booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
pages = {88:1--88:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-322-5},
ISSN = {1868-8969},
year = {2024},
volume = {297},
editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.88},
URN = {urn:nbn:de:0030-drops-202318},
doi = {10.4230/LIPIcs.ICALP.2024.88},
annote = {Keywords: Partial Vertex Cover, Max SAT, FPT Approximation, Matroids} | {"url":"https://drops.dagstuhl.de/search?term=Koana%2C%20Tomohiro","timestamp":"2024-11-06T02:03:08Z","content_type":"text/html","content_length":"182720","record_id":"<urn:uuid:b9a33fb3-a247-41a1-9621-41904447b20b>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00730.warc.gz"} |
Jump to navigation Jump to search
A fractal is any equation or pattern that when seen as an image produces a picture which can be zoomed into infinity and will still produce the same picture. It can be cut into parts which look quite
like a smaller version of the set that was started with. A simple example is a tree that branches infinitely into smaller branches with those smaller branches branching into even smaller branches and
so on.
Fractals are related to the field of psychonautics due to the fact that they are an extremely common component embedded within the visual experience of psychedelic geometry. Fractal images with
mathematics are generated through recursive formulas that infinitely repeat an image over itself in a self-similar way.
It could therefore be speculated that the fractals one sees within psychedelic geometry are generated through a similar recursive looping system which is reinterpreting the same visual stimuli
multiple times on top of itself, thus causing geometry to flower out into distinctive fractal structures.
See also | {"url":"https://psychonautwiki.org/wiki/Fractal","timestamp":"2024-11-11T21:11:26Z","content_type":"text/html","content_length":"204471","record_id":"<urn:uuid:0fce56af-5e33-4cdd-a5cc-85163827feb9>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00829.warc.gz"} |
Distribution plot from proc univariate
Dear Sir or Madam,
Can I please ask what do this errors mean? How can I resolve it? Thank you very much.
560 proc univariate data=temp;
561 var occ;
562 histogram / midpoints=0 to 600 by 50
563 lognormal
564 weibull
565 gamma;
566 inset n mean(5.3) std='Std Dev'(5.3) skewness(5.3)
567 / pos = ne header = 'Summary Statistics';
578 run;
NOTE: Since a threshold parameter (THETA) was not specified for the lognormal fit for occ, a
zero threshold is assumed.
ERROR: The smallest value of occ is less than or equal to the threshold parameter (THETA) for
the lognormal fit.
NOTE: The lognormal threshold parameter (THETA) must be less than 0.
NOTE: The lognormal curve was not drawn because estimates could not be computed for all
NOTE: Since a threshold parameter (THETA) was not specified for the Weibull fit for occ, a
zero threshold is assumed.
ERROR: The smallest value of occ is less than or equal to the threshold parameter (THETA) for
the Weibull fit.
NOTE: The Weibull threshold parameter (THETA) must be less than 0.
NOTE: The Weibull curve was not drawn because estimates could not be computed for all
NOTE: Since a threshold parameter (THETA) was not specified for the gamma fit for occ, a zero
threshold is assumed.
ERROR: The smallest value of occ is less than or equal to the threshold parameter (THETA) for
the gamma fit.
NOTE: The gamma threshold parameter (THETA) must be less than 0.
NOTE: The gamma curve was not drawn because estimates could not be computed for all parameters.
NOTE: At least one W.D format was too small for the number to be printed. The decimal may be
shifted by the "BEST" format.
NOTE: PROCEDURE UNIVARIATE used (Total process time):
real time 0.19 seconds
cpu time 0.04 seconds
05-19-2016 02:44 AM | {"url":"https://communities.sas.com/t5/Graphics-Programming/Distribution-plot-from-proc-univariate/td-p/271619","timestamp":"2024-11-11T16:49:37Z","content_type":"text/html","content_length":"145168","record_id":"<urn:uuid:98116b0b-c956-4430-b4de-80d424d7faae>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00050.warc.gz"} |
Compound Interest Calculator with Monthly Contributions - WalletBurst
Compound Interest Calculator with Monthly Contributions
Visualize and calculate how much your money can grow with the power of compound interest
Take your wealth planning to the next level with my Wealth Planning Toolkit for Google Sheets – just $20.
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The power of compound interest
Albert Einstein once famously said:
“Compound interest is the 8th wonder of the world. He who understands it, earns it; he who doesn’t, pays it.”
Einstein couldn’t have put it better. Compound interest truly is one of the most powerful forces in the universe, and it can both work for you in the case of growing investments or against you in the
case of compounding debt. The key ingredient for compounding growth is time. You can adjust the inputs in the above calculator to see how when given enough time, even a small initial investment can
grow to a significant sum of money.
Using this calculator
This interactive calculator makes it easy to calculate and visualize the growth of your investment thanks to compounding interest.
• Initial investment is the starting value of your investment, also known as the principal.
• Length of time in years is the length of time over which your investment will grow.
• Monthly contribution is a recurring amount that you contribute to your account each month.
• Interest rate is the annual interest rate of return of your investment. For reference, the S&P 500 has returned about 7% annually adjusted for inflation since it was started in 1926.
• End investment value is the final value of your investment at the end of your investing period.
How does compound interest work?
Compound interest is essentially a snowball effect of interest that is accrued on an initial investment of money. To understand how it works, let’s start with an example. Say you begin with $1,000
and put it into an investment that returns 5% annually. The chart below shows the growth of your investment over the first 10 years.
If you look at the change column, you can see that the total account increases by a greater amount each year. This is the nonlinear part of exponential growth, where the rate-of-change is increasing.
This becomes especially powerful over longer periods of time. The element of time is crucially important to wielding the power of compounding growth.
While the savings account only grew about $78 year-over-year after the 10th year, or 8% of the initial amount, by the 40th year, the savings account will grow $388 year-over-year or about 39% of the
initial amount. By year 64, the account will grow by $1,030 year-over-year, more than the initial $1000 that was put in to start the account! In the first few years, the compounding starts slowly and
is hard to notice, but over time it becomes immensely powerful. | {"url":"https://walletburst.com/tools/compound-interest-calculator/","timestamp":"2024-11-06T20:08:48Z","content_type":"text/html","content_length":"116394","record_id":"<urn:uuid:ef4cd2f8-df15-45a6-8fc4-0ed8cea7e3dd>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00623.warc.gz"} |
What is integration? + Example
What is integration?
1 Answer
Roughly speaking, integration is the inverse of differentiation, but there are several ways to think about it...
Given a suitably well behaved function $f \left(x\right) : \mathbb{R} \to \mathbb{R}$, and an interval $\left(a , b\right)$, the definite integral ${\int}_{a}^{b} f \left(x\right) \mathrm{dx}$ is the
"area under the curve" between $a$ and $b$.
At any particular point $t \in \mathbb{R}$, the rate of change of area as you increase $t$ is equal to the $f \left(t\right)$. That is, the derivative of the integral is equal to the original
Integration covers a lot more cases than just Real valued functions of Real numbers. You can integrate over any kind of measurable set - e.g. a plane, a curve, a surface, a volume. The function that
you are integrating may have any kind of value that is possible to sum and multiply by a scalar, e.g. Real, Complex, vector.
In such contexts you can think of an integral as a sort of infinite sum of values of a function over infinitesimally small pieces of the set over which you are integrating.
For example, suppose you have a function $f \left(p\right)$ defined for points on the surface $S$ of a sphere, with surface area $A$. Then the average value of $f \left(p\right)$ over the surface of
the sphere is:
$\frac{{\int}_{p \in S} f \left(p\right) \mathrm{dp}}{A}$
If we split the surface of the sphere into a large number of little patches ${S}_{i}$ of areas ${A}_{i}$, each containing a representative point ${p}_{i} \in {S}_{i}$, then we could approximate the
integral over the surface:
${\int}_{p \in S} f \left(p\right) \mathrm{dp} \approx {\sum}_{i} {A}_{i} f \left({p}_{i}\right)$
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CBSE Marking Scheme for Class 12 2024-25 and Exam Pattern with Weightage
Exam Pattern and Marking Scheme for Class 12 CBSE 2024-25 for All Subjects
The Central Board of Secondary Education(CBSE) has released the marking scheme Class 12 2024-25 for all subjects on its official website. For the student's ease, we have provided CBSE marking scheme
2024 Class 12 along with the chapter-wise weightage of each subject on this page. We suggest you read the article till the end to get all the comprehensive details of the weightage of all the
subjects, exam pattern, and chapter wise marking scheme class 12. Knowing the marks distribution, exam pattern, and weightage will help you prepare better for the upcoming Class 12 board exams and
increase your chances of scoring more marks in board examinations.
The minimum aggregate marks a candidate must score to pass CBSE Class 12 board exams is 33%. Moreover, the candidate must also secure at least 33% marks in each subject to pass the board exam.
Candidates who secure 33% or above aggregate marks but scored less than 33% in any subject must appear for the supplementary exam.
Note: ➤Calculate your potential NEET rank based on marks with our NEET Rank Predictor by Marks!
CBSE Class 12 Assessment Policy 2024
The Central Board of Secondary Education has announced that there will be only one term or one board examination in the academic year 2024-25. The final Class 12 board exam covers the entire CBSE
syllabus as per the curriculum designed by the board for the year 2024-25.
CBSE Class 12 Exam Pattern 2024-25
The Central Board of Secondary Education has released a new exam pattern and chapter wise marking scheme class 12 for the academic year 2024-25. Here are the complete details for the same:
Maths Exam Pattern and Marking Scheme for Class 12 CBSE 2024-25
CBSE Class 12 Maths Unit-Wise Weightage
Physics Exam Pattern and Marking Scheme for Class 12 CBSE 2024-25
CBSE Class 12 Physics Unit-Wise Weightage
Chemistry Exam Pattern and Marking Scheme for Class 12 CBSE 2024-25
CBSE Class 12 Chemistry Unit-Wise Weightage
Biology Exam Pattern and Marking Scheme for Class 12 CBSE 2024-25
CBSE Class 12 Biology Unit-Wise Weightage
Accountancy Exam Pattern and Marking Scheme for Class 12 CBSE 2024-25
Economics Exam Pattern and Marking Scheme for Class 12 CBSE 2024-25
English Core Exam Pattern and Marking Scheme for Class 12 CBSE 2024-25
Political Science Exam Pattern and Marking Scheme for Class 12 CBSE 2024-25
CBSE Class 12 Political Science Unit-Wise Weightage
This was the complete discussion on the Marking Scheme of CBSE Class 12 for the academic year 2024-25.
The CBSE marking scheme 2024 Class 12, as elucidated by Vedantu, follows the pattern set in the 2024 scheme. The distribution of marks is a crucial aspect, shaping the evaluation process. Vedantu
provides valuable insights into the CBSE marking scheme, aiding students to comprehend the weightage assigned to each section. The intricate details of CBSE marks distribution for class 12 scheme
2024, as per Vedantu, empower students to strategize their exam preparation effectively. It is advisable for students to thoroughly grasp this marking scheme to enhance their performance in the
upcoming exams, ensuring a comprehensive understanding of the evaluation criteria.
We hope students have found this helpful information for their studies. Stay tuned with Vedantu for further latest updates on CBSE Class 12 and other competitive exams. Good Luck!
FAQs on CBSE Marking Scheme for Class 12
1. When will the CBSE board conduct the Class 12 board examination for the academic session 2024-25?
The CBSE board will tentatively conduct Class 12 board exams from February 15, 2024.
2. Is it mandatory to pass Class 12 theory and practical exams separately?
Yes, it is mandatory to clear both practical and theoretical exams separately.
3. How does the CBSE Class 12 marking scheme help students?
The CBSE Class 12 marking scheme will help students understand how the marks will be allocated for each step of a question. With this information, students will learn the importance of answering a
question in steps. This eventually helps them to enhance their writing skills.
4. How to check all subjects' CBSE Class 12 marking scheme?
Follow the below to check the CBSE Class 12 marking scheme for all subjects:
• Step 1: Visit CBSE's official website (http://cbseacademic.nic.in/index.html).
• Step 2: Click on the “SQP 2024” under the “ Sample Question Paper” Tab.
• Step 3: Select Class 12 by clicking on the link
• Step 4: Click on the particular subject for which you want the marking scheme and sample paper. Download sample paper and take out the marking scheme printout for future reference.
5. Does the CBSE board consider the pre-board exam marks when calculating the final Class 12 exam results?
No, the pre-board examination scores are not added to or considered when calculating the final board exam results.
6. What is the marking system of CBSE Class 12?
The CBSE Class 12 marking system incorporates both marks and grades:
Total Marks:
• Each subject has a maximum of 100 marks.
• Practical subjects (like Physics, Chemistry, and Biology) have 70 marks for theory and 30 marks for practical exams.
• Commerce subjects (Accountancy, Business Studies, and Economics) have 80 marks for theory and 20 marks for project work.
• Mathematics has 80 marks for the written exam and 20 marks for internal assessment.
7. How will CBSE Class 12 marks be calculated?
CBSE uses a five-cum-nine point scale, assigning grades from A (highest) to E (lowest).
Grades are based on a combination of absolute marks and percentiles.
8. What is the criteria for Class 12 CBSE exam marks?
Students need to score a minimum of 33% in each subject and overall to pass the exam.
9. What is the marking scheme of Class 12 2024?
The marking scheme for each subject is released by CBSE every year, typically a few months before the exams. It provides detailed instructions on how marks are distributed for different sections of
the question paper. You can find the latest marking schemes on the official CBSE website or through educational portals. | {"url":"https://www.vedantu.com/cbse/class-12-marking-scheme","timestamp":"2024-11-09T16:37:19Z","content_type":"text/html","content_length":"644271","record_id":"<urn:uuid:cf93dccf-cb9f-4dd6-85b3-14a4ef7da8a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00774.warc.gz"} |
Measuring DC voltage outside ADC range
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Mar 8, 2013
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I'm designing a circuit to measure DC voltages that can be higher than what the ADC can handle. I already did some searching/googling, but I'd like to ask someone to verify, if what I'm doing is
correct. The measured voltage span should be from 0 to 16V DC, and I only need continuous low-speed measurements of DC (battery voltage).
Obviously, I will need a two-resistor voltage divider to bring the measured voltage down to acceptable levels (the single-ended ADC can only go up to 3.3V). The first question is how high resistor
values to use. Most ADC's require some maximum source impedance for accurate measurements (usually up to 10 kOhm). But since this is going to be a battery powered application I would like to minimize
the current flow through this voltage divider - this means resistor values should be in range of some megaohms. Unfortunately those values will result in a current that is insufficient for the ADC.
Therefore, the source should be buffered by an opamp in a unity-gain voltage follower configuration. Finally, I'd like to add some overvoltage protection to the opamp input as a good measure.
Attached is the circuit that I came up with.
I'm not sure what type of diodes to use except that they should be low-leakage (someone suggested junction diode?).
Any other thoughts, suggestions?
Thanks in advance!
Jul 4, 2009
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The reason for needing low(ish) impedance at the input of an ADC is that internally most MCUs have to draw a small input current to charge their sampling circuit at the start of a measurement. If the
values are too high, that current causes a drop in measurable voltage. As the frequency of measurement speeds up, the need to give time for the voltage to recover becomes longer. In your application
where the measuring is low speed you can overcome the problem by simply adding a capacitor across the ADC input. It will keep the voltage stable for the duration of the measurement.
For protection, I would suggest two small-signal Schottky diodes, one from the input to ground, one from the input to 3.3V, both 'cathode up" so they are normally not conducting. Any negative voltage
will be clamped by the ground side diode and anything higher than supply will be clamped by the other one. Schottky diodes have a lower Vf than the inputs of most ICs so they will conduct most if not
all dangerous currents away from the ADC pin.
Mar 26, 2018
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High valued R thermal noise can contribute to error, for apps using
delsig types of converters operating 20 bits or greater.....in general
What is the resolution of the ADC you are working with ?
Typically datasheet shows input sampling structure and its parasitics.
Refer to that for guidance.
Using an OpAmp, in this case you probably need a RRIO type, input
stages exhibit crossover distortion, so beware of that. Some datasheets
discuss this, some do not.
Of course the OpAmp contributes errors. You should do an end to
end error budget analysis to see if you meet your goals.
Regards, Dana.
Oct 15, 2018
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Change the ratio of R1&R2 for voltage in scale of ADC.
Reduce R3 < 1k Ohm, and add Ca Capacitor for pin3 of ADC. R3 & C3 creat a LPF, which design for filter noise and cut off frequency higher than sample frequency of ADC. R3 too high will make drop
voltage on it when has charge current input of ADC. If high speed ADC will absorb more current. This make error to measured result.
If Opam is low power type, check it is rail to rail type or not, that mean how much span output voltage you can use. Example, normal opam output only maximum is Vcc - 1.5V, rail-to-tail type accept
Vcc - 0.6V.
Because of input high impedance source mega ohm, check input offset current of opam, that will make error for resistor devider, also make error for linear ratio.
As I suggest, I don't use this solution to save power when measure Vbat. I use a switch to turn on connection to Vbat when I want to measure. This time circuit will asorb current. After measure, turn
off, then no current discharge from battery. This solution can save more power, and accept lower resistor devider for lower linear error, low cost opam can use.
Mar 8, 2013
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For protection, I would suggest two small-signal Schottky diodes
Are you sure about the Schottky diodes? I've seen some circuits that are exactly what you're proposing, but I've also read on several pages that those diodes are very leaky and can impact the
measurement significantly. They have a reverse current of several tens or even hundreds of uA compared to a junction diode that is just a few nA.
What is the resolution of the ADC you are working with ?
The ADC is 12-bit SAR. I don't think resistor noise is going to be a problem though, but thanks for pointing it out.
Yes, ideally the opamp should be R2R.
Reduce R3 < 1k Ohm, and add Ca Capacitor for pin3 of ADC.
The ADC datasheet says its source impedance must be max 10kOhm, so I think a 4k7 resistor should be fine. Though I could add an extra cap there for good measure.
I use a switch to turn on connection to Vbat when I want to measure. This time circuit will asorb current. After measure, turn off, then no current discharge from battery. This solution can save
more power, and accept lower resistor devider for lower linear error, low cost opam can use.
Yeah, I've been thinking of adding a switch and only enable the circuit during measurements. What king of switch do you propose? I take it a low-side N-MOS is a no-no, so perhaps a high-side P-N
complementary pair?
Jul 4, 2009
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Are you sure about the Schottky diodes? I've seen some circuits that are exactly what you're proposing, but I've also read on several pages that those diodes are very leaky and can impact the
measurement significantly. They have a reverse current of several tens or even hundreds of uA compared to a junction diode that is just a few nA.
Schottky diodes are indeed more leaky in reverse direction but the amount is insignificant for your application. If you look at the data sheet for the BAT85 for example, it quotes 2uA maximum at 25V
but the graphs of Vr and Ir show that 3.3V it goes off the scale at somewhere below 100nA.
Mar 8, 2013
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I see. So, how does the diode layout that you proposed (one "cathode up" diode from Vcc to IN+ and another from IN+ to GND) compare to the one I posted in my first post (two diodes in a diac-like
configuration from IN+ to opamp output)?
Apr 17, 2014
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Diodes to IN+:
+ can be expected to cause zero leakage (no glas package), because there is about zero voltage between IN+ and IN- during normal operation
- pass all the current to the OPAMP output. Thus the protection relies on the Opamp output strength
- provide about no protection when Opamp is powered OFF.
I'd use a high ohmic voltage divider. Or a lower phmic dividervand a P-Ch Mosfet as high side switch.
Also I'd use a storage capacitor to compensate the high ohmic source.
It all depends on the expected accuracy and the ADC input characteristics (mainly input current and input capacitance)
If you need 1% accuracy,
* then the external capacitor needs to be 100x the ADC input capacitor.
* then the switch needs to be ON at least 4 tau before the conversion starts.
Jul 4, 2009
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I see. So, how does the diode layout that you proposed (one "cathode up" diode from Vcc to IN+ and another from IN+ to GND) compare to the one I posted in my first post (two diodes in a diac-like
configuration from IN+ to opamp output)?
I wouldn't think of that as a 'diac-like' configuration, as anti-parallel diodes do not exhibit the avalanche characteristics of a real diac.
The problem with your connection is you are still prone to excess voltage damaging the IC. Although it sinks/sources current from the op-amp output to counter any excess input voltage, it still only
works within the supply range of the IC and possibly in one polarity. The diode to ground and diode to 3.3V does introduce a tiny measurement error but it should be insignificant especially as you
are only measuring battery voltage, however it greatly increases the degree of protection against over voltage and reverse voltage at the input.
If you are only expecting brief transient over-voltages, the capacitor alone should suffice.
If you really need extreme low current consumption from the battery, consider a very high value feed resistor and a low-leakage transistor to ground, driven from your MCU. Use it to clamp the voltage
across the capacitor to (almost) ground. When the transistor is turned off, take two measurements from the ADC and calculate the battery voltage from the rate of voltage rise. Knowing the tau you can
make sure the transistor is turned back on again before the voltage can rise too high to cause damage.
Mar 8, 2013
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Hmm, all good points. Alright, thanks for the information. I think I have enough knowledge now to start building some circuits. I'll do some measurements to see the linearity.
Best regards!
Apr 17, 2014
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I'll do some measurements to see the linearity.
I don´t expect linearity issues caused by the external circuit. I´d rather expect gain and offset errors.
Not open for further replies. | {"url":"https://www.edaboard.com/threads/measuring-dc-voltage-outside-adc-range.404910/","timestamp":"2024-11-12T05:24:22Z","content_type":"text/html","content_length":"151406","record_id":"<urn:uuid:2b8f347f-b8e1-411c-84dc-75b3484bbca1>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00285.warc.gz"} |
Insertion Sort - AP Computer Science A
All AP Computer Science A Resources
Example Questions
Example Question #1 : Insertion Sort
Which of the following statements explains insertion sort?
Possible Answers:
It removes one element from the list, finds where it should be located, and inserts it in that position until no more elements remain.
All numbers less than the average are inserted on the left, and the rest is inserted at the right. The process is then repeated for the left and right side until all numbers are sorted.
The list is iterated through multiple times until it finds the desired first number, then repeats the process for all numbers.
The list is broken apart into smaller lists that are sorted and merged back together.
None of these are insertion sort.
Correct answer:
It removes one element from the list, finds where it should be located, and inserts it in that position until no more elements remain.
Insertion sort removes an element from a list, checks the values adjacent to it to see if it is greater or smaller, until it finds the position where the number to the left is smaller and the number
to the right is larger and places it there.
Example Question #4 : Sorting
What would the set of numbers look like after four iterations of Insertion Sort?
Correct answer:
{1, 4, 5, 9, 2, 0, 4}
Insertion Sort is a sorting algorithm that starts at the beginning of an array and with each iteration of the array it sorts the values from smallest to largest.
Therefore, after four iterations of Insertion Sort, the first four numbers will be in order from smallest to largest.
Example Question #2 : Insertion Sort
Which of the following do we consider when choosing a sorting algorithm to use?
I. Space efficiency
II. Run time efficiency
III. Array size
IV. Implementation language
Correct answer:
I, II, III, IV
All of the choices are important when choosing a sorting algorithm. Space and time complexity are the characteristics by which we measure the performance of an algorithm. the array size directly
affects the performance of an algorithm. In addition, the language which the algorithm is written in can also affect the performance (for example, insertion sort may run faster in one language versus
another, due the way the language was designed).
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How Many Kilograms Is 2478 Ounces?
How many kilograms in 2478 ounces?
2478 ounces equals 70.25 kilograms
Unit Converter
Conversion formula
The conversion factor from ounces to kilograms is 0.028349523125, which means that 1 ounce is equal to 0.028349523125 kilograms:
1 oz = 0.028349523125 kg
To convert 2478 ounces into kilograms we have to multiply 2478 by the conversion factor in order to get the mass amount from ounces to kilograms. We can also form a simple proportion to calculate the
1 oz → 0.028349523125 kg
2478 oz → M[(kg)]
Solve the above proportion to obtain the mass M in kilograms:
M[(kg)] = 2478 oz × 0.028349523125 kg
M[(kg)] = 70.25011830375 kg
The final result is:
2478 oz → 70.25011830375 kg
We conclude that 2478 ounces is equivalent to 70.25011830375 kilograms:
2478 ounces = 70.25011830375 kilograms
Alternative conversion
We can also convert by utilizing the inverse value of the conversion factor. In this case 1 kilogram is equal to 0.014234851472793 × 2478 ounces.
Another way is saying that 2478 ounces is equal to 1 ÷ 0.014234851472793 kilograms.
Approximate result
For practical purposes we can round our final result to an approximate numerical value. We can say that two thousand four hundred seventy-eight ounces is approximately seventy point two five
2478 oz ≅ 70.25 kg
An alternative is also that one kilogram is approximately zero point zero one four times two thousand four hundred seventy-eight ounces.
Conversion table
ounces to kilograms chart
For quick reference purposes, below is the conversion table you can use to convert from ounces to kilograms | {"url":"https://convertoctopus.com/2478-ounces-to-kilograms","timestamp":"2024-11-04T23:09:50Z","content_type":"text/html","content_length":"32956","record_id":"<urn:uuid:cb0b2621-4d5e-4fee-a442-b859aeb9f48b>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00211.warc.gz"} |
Pi Day's ramblings
Hi all, long time no see:
Today's Pi Day, so I feel like indulging in some quick Pi antics to commemorate the event, namely showing off two little-known, rarely seen, extremely short ways to compute this most ubiquitous
I'll use the HP-71B (+ MathROM) to enter and run the algorithms described. If you'd like to key them in as shown and don't own an HP-71B, get the excellent freeware emulator Emu71 by J-F Garnier
(Google for it) or any other capable emulator of your choice (Math ROM emulation required) or that failing try and adapt the code to your favorite HP calculator.
Let's see:
1. An asymptotically exact, probabilistic way:
This 50-byte one-liner will compute Pi using random numbers, you'll have to input how many tries (say 100,000) and it'll display the resulting Pi approximation at the end. The more tries the more
correct Pi digits you'll get:
10 INPUT K @ N=0 @ FOR I=1 TO K @ N=N-MOD(IROUND(RND/RND),2) @ NEXT I @ DISP 1-4*N/K
To run it, key it in and then key in the following at the ">" prompt:
>DESTROYALL @ RANDOMIZE 260 @ FIX 4 [Enter]
>RUN [Enter]
? 100000 [Enter]
As I said, the more tries the more correct digits you'll get. See if you can figure how and why it works, and if so check if you can prove that it actually produces the exact value of Pi in the
2. A faster, very precise, approximate way:
Do the following:
a) Compute the real root of the cubic equation x^3 - 6*x^2 + 4*x - 2:
>DESTROYALL @ STD
b) raise it to the 24th power:
c) compute the natural logarithm of the result:
d) Divide the result by the square root of 163:
You'll get Pi correct to all displayed digits and then some. Again, see if you can figure how and why this works and if so, check if you can prove that it won't produce an exact value of Pi but
just a fairly good approximation.
That's all. Best regards from V.
03-14-2011, 02:59 PM
Hi Valentin. Good write-up. Give me a shout!
03-15-2011, 06:46 AM
Quote: Hi Valentin. Good write-up. Give me a shout!
Thanks, Gene, I'll give you "a shout" this next weekend for sure, sooner if I can manage.
Best regards from V.
03-14-2011, 03:43 PM
Hi ValentÃn, welcome back!!!
I will try the two programs in my 71+MathROM, which I haven't used in ages. It should be fun.
03-15-2011, 06:56 AM
Hi, Fernando !! :D
Quote: Hi ValentÃn, welcome back!!!
I will try the two programs in my 71+MathROM, which I haven't used in ages. It should be fun.
Yes, it should.
You might also want to try this Java version for the first one as well:
private static void pi() {
double pi = 0.0;
for (long i = 0; i < Long.MAX_VALUE; i++) {
pi = pi - Math.round((Math.random()/Math.random()))%2;
if ((i&0x80000)==0x80000) {
System.out.printf("%d: %s%n", i, 1-4*pi/i);
I left it running yesterday on one of my servers, all night long (16 hours), and it produced the following:
487,446,282,239 tries -> 3.141592254139215
487,447,330,815 tries -> 3.141592246519439
487,448,379,391 tries -> 3.141592254556738
which is as expected, 7 correct digits for some 500 (US) billion tries. Only one core was used and it managed 22,000 tries per millisecond. Not bad.
The result also shows that Java's random number generator is working Ok, else there would be some bias which would be noticeable in the final result's accuracy.
Best regards from V.
03-14-2011, 04:59 PM
Regarding the second challenge: that's not too hard to solve once you got access to some symbolic math software - Wolfram Alpha will do. ;-)
Determine the real solution of the polynomial, raise the result to the 24th power, take the natural log and divide by sqrt(163), and you'll get a result, whose continued fraction representation looks
quite familiar:
[3; 7, 1, 15, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, ...]
Right, these are the first sixteen terms of the continued fraction representation of Pi. Its first seventeen significant digits agree with the true value.
I'm sure someone else will provide a decutive proof. ;-)
03-14-2011, 05:07 PM
Valentin wrote: "that it won't produce an exact value of Pi"
What is this exact value anyway? :-)
03-14-2011, 05:12 PM
Simple: the exact value of Pi is... Pi. :-)
03-14-2011, 05:43 PM
Hello Valentin,
Welcome back!
I've used WolframAlpha to solve the equation ( solve x^3 - 6*x^2 + 4*x - 2 = 0 for x ) and asked for more digits and repeated the steps using the windows calculator:
When I compared the result to pi
I found the first 17 digits to match.
However, if 24 is subtracted from the result between steps b and c, the approximation is even more impressive:
ln(5.31862821775018565910968015331802246772191980883690026023^24-24)/sqrt(163) =
3.141592653589793238462643383279502884197169399375105820974945 (pi)
I have no idea why this works though, even after checking this MathWorld reference:
Best regards,
03-15-2011, 07:07 AM
Hi, Gerson:
Quote: Hello Valentin,
Welcome back!
Quote: I've used Wolfram Alpha to solve the equation ( solve x^3 - 6*x^2 + 4*x - 2 = 0 for x ) and asked for more digits and repeated the steps using the windows calculator:
However, if 24 is subtracted from the result between steps b and c, the approximation is even more impressive:
Yes, I knew. However, I chose to omit the subtraction of 24 as the final result was close enough to Pi that it would nevertheless be indistinguishable in 10-digit and 12-digit calculators, and you
wouldn't be able to notice it unless you resorted to multiprecision computations which would deny the simplicity, so I opted for one step less.
Best regards from V.
03-14-2011, 09:06 PM
The first one implies that the probability of one (evenly distributed between 0 and 1) random variable divided by another having a result between (.5 to 1.5) or (2.5 to 3.5) or (4.5 to 5.5) or (6.5
to 7.5) or ... is
(pi - 1) / 4.
Wikipedia talks about ratio distributions, and in fact that one in particular here.
Using the result there, you then have to integrate between the limits in the first paragraph, giving
1/4 + 1/2*((1 - 2/3) + (2/5 - 2/7) + (2/9 - 2/11) + (2/13 - 2/15) +...
(the first term is the hardest one: You have to break up the first integral into one from 0.5 to 1, of 1/2; and one from 1 to 1.5, of 1/(2x^2). The others are all of 1/(2x^2), for the whole range)
or -1/4 + ( 1-1/3+1/5-1/7+...)
That infinite series is the Leibniz formula for pi, so the probability is
-1/4 + pi / 4.
which is what was to be shown.
03-14-2011, 09:15 PM
By the way, I do think that is a more interesting example than the well-known Buffon needle problem. That is supposedly an example of pi coming up in a non-obvious situation, but I don't agree. It's
obvious that when you toss a needle, the possible angles the needle could trace out when it lands will make a circle. And in fact you use circles and geometry to derive the probability.
In this example, however, it's really not obvious how pi is involved. And geometry is not used in the derivation.
03-15-2011, 07:14 AM
Quote: By the way, I do think that is a more interesting example than the well-known Buffon needle problem.
Agreed. As far as obtaining Pi via Monte-Carlo methods (algorithms using random numbers) I think that this is by far the simplest and most elegant, apart from being utterly non-obvious.
Best regards from V.
03-15-2011, 07:11 AM
Quote: The first one implies that the probability of one (evenly distributed between 0 and 1) random variable divided by another having a result between (.5 to 1.5) or (2.5 to 3.5) or (4.5 to
5.5) or (6.5 to 7.5) or ... is
(pi - 1) / 4.
Best regards from V.
03-15-2011, 07:17 AM
How about using the following loop to get an estimate:
Any takers to finish this?
Had an open day when I was studying mathematics at university. We did this dartboard PI estimate and the value was running a bit low. In stepped one of the older tutors who proceeded to correct the
estimate a bit by putting a pile of darts on the same spot :-)
- Pauli
03-16-2011, 08:49 AM
HP-33s program
P0001 LBL P
P0002 STO N
P0003 CLSIGMA
R0001 LBL R
R0002 RANDOM
R0002 RANDOM
R0004 y,x->theta,r
R0005 IP
R0006 SIGMA+
R0007 RCL N
R0008 n
R0009 x<y?
R0010 GTO R
R0011 1/x
R0012 x<>y
R0013 SIGMAx
R0014 -
R0015 *
R0016 4
R0017 *
R0018 RTN
P: LN= 9 CK=B364
R: LN=66 CK=E2BB
03-16-2011, 10:31 AM
Average of 14 runs of 7 XEQ P:
44/14 or 22/7 (YMMV)
Free42 might be an option for much faster results:
00 { 31-Byte Prgm }
01 LBL "PI"
02 STO 00
03 CLSIGMA
04 LBL 00
05 RAN
06 RAN
07 ->POL
08 IP
09 SIGMA+
10 RCL 00
11 RCL 16
12 x<y?
13 GTO 00
14 1/X
15 X<>Y
16 RCL 11
17 -
18 *
20 *
21 RTN
22 .END.
03-16-2011, 02:55 PM
May I humbly suggest some improvements? :-)
Since the number of loops is known I think using DSE is the obvious way to go here. When the program finishes the values for n and N are the same, so there is no need to distinguish between both. And
since only the sum of all x-values is used, the relatively slow Sigma+ is not required either.
Here's a version for the HP-35s which also shows how to get along without the infamous R->P command. ;-)
P001 LBL P
P002 INPUT N ; get and save n
P003 STO K ; initialize loop counter
P004 CLX
P005 STO S ; initialize sum
P006 RANDOM
P007 RANDOM
P008 i
P009 x
P010 +
P011 ABS ; = sqrt(random1^2 + random2^2)
P012 IP
P013 STO+ S ; add 0 or 1
P014 DSE K ; decrement and check loop counter
P015 GTO P006
P016 RCL N
P017 RCL- S
P018 RCL/ N
P019 4
P020 x ; result = 4*(n-sum)/n
P021 RTN
My 35s runs 1000 loops in 2:45 minutes.
Edit: here's another version that requires just one memory register.
P001 LBL P
P002 INPUT N ; get and save n
P003 CLSTK ; clear sum
P004 RCL N
P005 R^ ; save n in stack
P006 RANDOM
P007 i
P008 x
P009 RANDOM
P010 +
P011 ABS ; = sqrt(random1^2 + random2^2)
P012 IP
P013 + ; add 0 or 1
P014 DSE N ; decrement and check loop counter
P015 GTO P006
P016 -
P017 X<>Y
P018 /
P019 4
P020 x ; result = 4*(n-sum)/n
P021 RTN
Edited: 16 Mar 2011, 3:15 p.m.
03-16-2011, 06:21 PM
Quote: May I humbly suggest some improvements? :-)
Of course, and don't be humble: your improvements are perfect!
I imagined Sigma+ was slow but I wanted to use all the instructionS Paul had provided. Prior to using Sigma+, I tested the idea in the third program below, which runs 1000 loops in 1 minute and 46
seconds (at first, I had used an HP-15C). As a comparison, the HP-33s program above runs in 2 minutes and 1 second, using Sigma+; I was expecting it to take more time.
The first program below is based in another (not so good) idea I had tried on the 15C. The indirect addressing slows it a bit. The second one is based on your first version, except the R->P
instruction is still being used. It appears the newer machines don't benefit much from the faster DSE and ISG loop control instructions.
T0001 LBL T
T0002 STO N
T0003 CLx
T0004 STO A
T0005 STO B
V0001 LBL V
V0002 RANDOM
V0003 RANDOM
V0004 y,x->theta,r
V0005 IP
V0006 1
V0007 +
V0008 STO i
V0009 STO+(i)
V0010 DSE N
V0011 GTO V
V0012 2
V0013 RCL* A
V0014 RCL+ B
V0015 RCL/ A
V0016 8
V0017 /
V0018 1/x
V0019 RTN
2' 13"
T0001 LBL T
T0002 STO N
T0003 STO A
T0004 CLx
T0005 STO S
V0001 LBL V
V0002 RANDOM
V0003 RANDOM
V0004 y,x->theta,r
V0005 IP
V0006 STO+ S
V0007 DSE A
V0008 GTO V
V0009 RCL N
V0010 RCL- S
V0011 RCL/ N
V0012 4
V0013 *
V0014 RTN
T0001 LBL T
TO002 STO N
T0003 CLx
T0004 STO A
T0005 STO B
V0001 LBL V
V0002 RANDOM
V0003 RANDOM
V0004 y,x->theta,r
V0005 1
V0006 x>y?
V0007 STO+ B
V0008 STO+ A
V0009 RCL N
V0010 RCL A
V0011 x<y?
V0012 GTO V
V0013 1/x
V0014 RCL* B
V0015 4
V0016 *
V0017 RTN
1' 46"
03-16-2011, 07:05 PM
Hi Gerson,
Quote: The first program below is based in another (not so good) idea I had tried on the 15C. The indirect addressing slows it a bit. The second one is based on your first version, except the R->
P instruction is still being used. It appears the newer machines don't benefit much from the faster DSE and ISG loop control instructions.
This is an interesting point. Obviously every calculator has its specific "slow" and "fast" instructions. Regarding the 35s I can say that especially numeric constants are processed very slowly. So I
often use workarounds like these:
constant workaround
(slow) (faster)
1 Clx e^x (or simply SGN if x>0)
2 e IP
3 Pi IP
... ...
There also is another advantage: on the 35s the "fast" versions even use less memory.
Back to the various Pi-programs. Looking at the second 35s-version I wondered how fast or slow the 35s might process complex numbers, so I replaced lines P006 to P011 by a straightforward...
P006 RANDOM
P007 x^2
P008 RANDOM
P009 x^2
P010 +
P011 SQRT
...and guess what: it's much faster! 1000 loops are now done in
03-16-2011, 07:32 PM
Quote: Looking at the second 35s-version I wondered how fast or slow the 35s might process complex numbers, so I replaced lines P006 to P011 by a straightforward...
P006 RANDOM
P007 x^2
P008 RANDOM
P009 x^2
P010 +
P011 SQRT
...and guess what: it's much faster! 1000 loops are now done in
Likewise when these instructions replace lines V0002 through V0004 in the third HP-33s program above, the running time drops from 1min 46sec to 1min 04sec. However, I think Paul Dale was concerned
about size rather than speed.
Actually SQRT is not necessary. If both terms of SQRT(random#[1]^2 + random#[2]^2) > 1 are squared, the inequality still holds. Thus, when SQRT is removed the running time for 1000 loops drops down
to 53 seconds for the HP-33s program and to about 2 minutes for your 35s program.
Edited: 16 Mar 2011, 9:54 p.m.
03-17-2011, 01:51 AM
Quote:However, I think Paul Dale was concerned about size rather than speed.
I wasn't. I just wanted to see what the folks here could do to the dart board approach to finding Pi :-)
I haven't been disappointed by the interesting programs.
- Pauli
03-17-2011, 08:47 AM
Right, the square root is obsolete - that's what I also realized after my last message yesterday.
The modified program now takes 1:55 minutes. So it's shorter and faster. ;-)
03-17-2011, 09:25 AM
Hello Dieter,
DSE is really slower on the HP-33s, as we can see from the timings belows. Is it also slower on the HP-35s? Thanks!
HP-33s & HP-32SII
T0001 LBL T T0001 LBL T
TO002 STO N TO002 STO N
T0003 CLx T0003 CLx
T0004 STO A T0004 STO A
T0005 STO B T0005 STO B
V0001 LBL V V0001 LBL V
V0002 RANDOM V0002 RANDOM
V0003 x^2 V0003 x^2
V0004 RANDOM V0004 RANDOM
V0005 x^2 V0005 x^2
V0006 + V0006 +
V0007 1 V0007 1
V0008 x>y? V0008 x>y?
V0009 STO+ B V0009 STO+ B
V0010 STO+ A V0010 STO+ A
V0011 RCL N V0011 DSE N
V0012 RCL A V0012 GTO V
V0013 x<y? V0013 4
V0014 GTO V V0014 RCL* B
V0015 1/x V0015 RCL/ A
V0016 RCL* B V0016 RTN
V0017 4
V0018 *
V0019 RTN
1,000 loops timing
0' 53" (HP-33s) 1' 14" (HP-33s)
1' 28" (HP-32SII) 1' 26" (HP-32SII)
00 { 38-Byete Prgm }
01>LBL "PI"
02 STO 00
03 CLX
04 STO 01
05 STO 02
06>LBL 00
07 RAN
08 x^2
09 RAN
10 x^2
11 +
13 X>Y?
14 STO+ 02
15 STO+ 01
16 DSE 00
17 GTO 00
19 RCL* 02
20 RCL/ 01
21 RTN
22 .END.
30,000,000 loops timing
1' 10" ( Free42 Binary 1.4.67 @ 1.86 GHz )
03-17-2011, 10:27 AM
On the 35s I excpected a version without DSE to run even slower, as it requires a numeric constant ("1") within the loop. And so it was:
P001 LBL P
P002 INPUT N
P003 CLSTK
P004 RCL N
P005 R^
P006 ENTER
P007 Clx
P008 RANDOM
P009 x^2
P010 RANDOM
P011 x^2
P012 +
P013 IP
P014 +
P015 1
P016 STO- N
P017 Clx
P018 RCL N
P019 X>0?
P020 GTO P007
P021 RDN
P022 -
P023 X<>Y
P024 /
P025 4
P026 x
P027 RTN
Running time: 2:16
As mentioned before there are several ways to avoid numeric constants:
P015 RCL N P015 RCL N
P016 SGN P016 ENTER
P017 STO- N P017 SGN
P018 Clx P018 -
P019 RCL N P019 STO N
P020 X>0? P020 X>0?
P021 GTO P007 P021 GTO P007
... ...
Running time: 1:55 Running time: 1:59
So the two last solutions take about the same time as the original version with DSE.
03-17-2011, 09:39 AM
Quote: Average of 14 runs of 7 XEQ P:
44/14 or 22/7 (YMMV)
There's a good point in there. pi is known to be approximately 22/7, or, even better, approximately 355/113.
If you used a multiple of 113 as the number of trials, you might be able to get very close this way.
The same idea is suggested in the article on Buffon's needle I linked above.
03-17-2011, 10:47 AM
Crawl wrote:
Quote: If you used a multiple of 113 as the number of trials, you might be able to get very close this way.
Since the experiment approximates Pi/4 I would suggest an integer multiple of 4*113 = 452.
This way n*452 trials will result in Pi ~= 355/113 if the number of hits is n*355.
03-17-2011, 02:20 PM
Quote: If you used a multiple of 113 as the number of trials, you might be able to get very close this way.
I had tried that earlier, but to no avail. I was trying single runs of 113 trials. I've finally made it with a single run of 452, as per Dieter's suggestion, the first time I tried. I little cheating
though in order to ensure immediate success ;-)
On the real HP-42 and the 38-byte program above:
94 SEED 452 XEQ PI => 3.14159292035 (after 1 min 45 sec)
On Free42 Binary 1.4.67:
66 SQRT SEED 452 XEQ PI => 3.14159292035 (instantaneously)
00 { 34-Byte Prgm }
01>LBL "X"
02 STO 04
03>LBL 01
04 RCL 04
05 SQRT
06 SEED
08 XEQ "PI"
09 PI
10 -
11 ABS
12 1E-4
13 X>Y?
14 STOP
15 DSE 04
16 GTO 01
17 RTN
18 .END.
On the real 42S, line 05 is not necessary to find a "right" seed for the random numbers generator. That's because Free42 will always generate the same random number for seeds with same fractional
parts, except when the seed is set to 0, when SEED behaves as it does on the real 42S, that is, always generates internally a different random number:
Free42 HP-42S
1 SEED RAN => 5.23054829358E-1 7.31362440213E-1
2 SEED RAN => 5.23054829358E-1 4.31362440213E-1
3 SEED RAN => 5.23054829358E-1 1.31362440213E-1
3.1 SEED RAN => 1.20349678274e-1 8.01362440213E-1
4.1 SEED RAN => 1.20349678274e-1 5.01362440213E-1
5.1 SEED RAN => 1.20349678274e-1 2.01362440213E-1
0 SEED RAN => undetermined undetermined
0 SEED RAN => undetermined undetermined
On Free42 Binary 1.4.67 @ 1.86 GHz:
9236 SQRT SEED 132408 XEQ PI => 3.14159265301 (about half a second)
Edited: 18 Mar 2011, 10:48 p.m.
03-15-2011, 08:28 AM
I remember a "Spektrum der Wissenschaft" (the german release of "Scientific American") essay about "shooting at pi", using the random generator. If I remember correctly, I used my newly purchased
Amiga to do the calculations and graphical representation. Lots of fun.
While googling for the essay, I stumbled across this:
Looks like someone invested quite a bit of time for pi day ;)
03-15-2011, 11:41 PM
A. K. Dewdney's column "Computer Recreations" in the April 1985 issue of Scientific American discussed this methodology. He used the analogy of a cannon firing into a square field which included an
inscribed circular pond. Readers were asked to send the results for 1000 shots to him.
03-15-2011, 12:58 PM
Here's my approach for the second challenge.
The key is a constant found by Ramanujan who showed that
R = e^(Pi * sqrt(163))
is almost an integer. In fact, the exact value for R is
R = 262 537 412 640 768 743,999 999 999 999 250...
and so
R = ~= 262 537 412 640 768 744
In other words:
Pi ~= sqrt(163) * ln R
The value of this expression agrees with Pi in its first 30 decimals.
Now, R also is the intermediate value 2,62537412641 E+17 mentioned in the challenge. At least is looks quite close - the given 12 digits are the same as in R.
So the remaining question is: Why is the real solution of the given cubic equation, raised to the 24th power, so close to R? Or, more precisely, indistinguishable in its first 12 digits.
The analytic solution to the given cubic equation is
x = 2 + (5 + sqrt(489)/9)^1/3 + (5 - sqrt(489)/9)^1/3
= 5,318 628 217 750 185 659 109 680 ...
Now, the 24th power of x is
x^24 = 262 537 412 640 768 767,999 999 999 999 251 ...
which is quite exactly R + 24. Which in turn is close enough to R to that also
R ~= x^24 ~= e^(Pi * sqrt(163))
or finally
Pi ~= ln(x^24) * sqrt(163)
which is the value that was calculated in the challenge.
This also explains why the Pi-approximation gets much better if the
intermediate is corrected by 24, as Gerson mentioned.
However, there still is one thing left for the perfect solution:
Assume R resp. R^1/24 is given. Now find a cubic equation with integer coefficents where its real solution is as close as possible to this value.
Edited: 15 Mar 2011, 1:22 p.m. | {"url":"https://archived.hpcalc.org/museumforum/thread-180167-post-180405.html#pid180405","timestamp":"2024-11-11T00:56:15Z","content_type":"application/xhtml+xml","content_length":"119362","record_id":"<urn:uuid:ab91d80b-bdf1-430d-a121-432bf0d19f12>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00249.warc.gz"} |
Going Beyond “Define These Terms In Your Own Words”
Differentiation TechniqueThink Big! But Also Small.
When differentiating, it's helpful to note where on the "spectrum of abstraction" your content lies. Then, see what happens when you move that content to be more abstract or more specific. It often
unlocks lots of new opportunities for thinking.
Specific Examples of “Think Big! But Also Small.”
“Define these 👄 terms in your own words” may contain depth and complexity… but it’s neither deep nor complex!
This differentiation technique is called “Concentric Circles”. You use it to move students up and down the ladder of abstraction, applying a single idea in multiple contexts.
Instead of just memorizing what a bunch of morphemes mean, we’re looking broadly, exploring patterns, finding unexpected similarities and weird differences.
What if we used a universal theme to guide our study of fractions? These very big ideas get students thinking about fractions in a new way.
Let’s start with a puzzlement, ask kids to generate an abstract statement, and then find evidence that their statement works across several different areas.
Here’s are the steps for running an inductive lesson based on Hilda Taba’s model of Concept Formation. Plus a sample lesson about the Nile River.
Discovering what is interesting and unexpected about a triangle’s angles. What twists have I unintentionally spoiled for my students over the years?
It’s easy to fall in love with chasing the newest technology to use in the classroom. But sometimes, the perfect tool is a plain old calculator. We’ll be using this tool to develop curiosity about
Using Hilda Taba’s model of inductive thinking, use your students’ prior knowledge to develop a statement about expected class behavior.
Let’s look at a couple ways to bring inductive thinking into word studies. We’ll examine simple plural rules all the way up to etymology of foreign words in English. | {"url":"https://www.byrdseed.com/technique/think-inductively/","timestamp":"2024-11-03T13:32:25Z","content_type":"text/html","content_length":"58796","record_id":"<urn:uuid:4bd7ceb6-1469-41e5-9d1b-418d7c9cf3bf>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00142.warc.gz"} |
LED Matrix - ISS Tracker
The goal of this project was simple - create a visualization to display the current position of the ISS in real-time. This was my first major attempt to create a display to be used on my new
64x32 RGB matrix from Adafruit.
The red dot on the image above represents the ISS and the green dot shows my current location. Displayed on the left there is time, ISS latitude, ISS longitude, and the number of astronauts on board
(one coloured square for each).
The source code for this project can be found here.
The following are the parts that I used to build this project. Most of them were purchased from Adafruit.
The Spinning Globe
To get this to work, I needed to brush up on my linear algebra skills. I used this video as a starting point for my code. It explains how to convert from a spherical coordinate system to a Cartesian
coordinate system, initialize a numpy matrix to store the coordinates for the sphere, and how to apply a rotation matrix. The video describes how to use this to draw an ASCII Earth with pygame, but I
managed to adapt it to work with the RGB matrix.
Generating the nodes
First, for each latitude (north/south) I iterated over a number of longitudes (east/west) and converted the latitude/longitude pair to a xyz coordinate. These coordinates were then converted to a
numpy matrix. The add_nodes() and convert_coords() methods were written to achieve this.
Converting from spherical to Cartesian coordinates is done using the following equations:
When given standard latitude and longitude values as input, is the complement of the latitude, or co-latitude, and is the complement of the longitude. is the radius of the sphere. The following image
from the Wikipedia article helps illustrate this.
I switched the equations for y and z to allow y to represent the vertical axis.
Here is the code the implements this:
def convert_coords(self, lat, lon):
Converts latitude and longitude to Cartesian coordinates.
In this case the y coordinate is on the vertical plane
and x and z are on the horizontal plane.
x = round(self.RADIUS * sin(lat) * cos(lon), 2)
y = round(self.RADIUS * cos(lat), 2)
z = round(self.RADIUS * sin(lat) * sin(lon), 2)
return (x, y, z)
def add_nodes(self):
Generates the nodes used to display the Earth and stores them in an array.
Backups of the arrays are saved so they can be restored on each rotation.
xyz = []
# Map to Cartesian plane.
for i in range(self.MAP_HEIGHT + 1):
lat = (pi / self.MAP_HEIGHT) * i
for j in range(self.MAP_WIDTH + 1):
lon = (2 * pi / self.MAP_WIDTH) * j
xyz.append(self.convert_coords(lat, lon))
# Build the array of nodes.
node_array = np.array(xyz)
ones_column = np.ones((len(node_array), 1))
ones_added = np.hstack((node_array, ones_column))
self._earth_nodes = np.vstack((self._earth_nodes, ones_added))
Drawing the nodes
Once the nodes have been initialized they can be drawn to the screen. Drawing is done using the PIL library. On each frame, a 64x32 PIL image is created and sent to the RGB matrix to be drawn. To
create the frame, each node is iterated over and the pixel at the x and y coordinate is drawn (only if z > 1 to only draw nodes in the foreground).
def draw(self, image):
Draws the Earth, ISS, and home node arrays to the display.
# Draw the Earth.
for i, node in enumerate(self._earth_nodes):
if (i > self.MAP_WIDTH - 1 and
i < (self.MAP_WIDTH * self.MAP_HEIGHT - self.MAP_WIDTH) and
node[2] > 1):
(self.X + int(node[0]),
self.Y + int(node[1]) * -1),
This is what the result looks like before a bitmap of the Earth is applied. The resolution can easily be increased by changing the MAP_WIDTH and MAP_HEIGHT variables before the nodes get created. The
resolution below is lower than what is used in the final result. The higher the resolution the more computations are needed when rotating and drawing the sphere.
Making it spin
Making the sphere spin is relatively easy with a little more linear algebra. On every frame, before it gets drawn, all we have to do is apply a rotation matrix to the node array that rotates each
node by some angle . This can be done with the matmul() function provided by numpy. The matrix used in the code will spin the nodes around the vertical axis.
def update_spin(self):
Handles the logic to control the Earth rotation.
Resets the nodes to the backed-up version after every full rotation.
c = np.cos(self.SPIN_THETA)
s = np.sin(self.SPIN_THETA)
matrix_y = np.array([
[c, 0, s, 0],
[0, 1, 0, 0],
[-s, 0, c, 0],
[0, 0, 0, 1]
def find_center(self):
Returns the center coordinates of the Earth.
return self._earth_nodes.mean(axis=0)
def rotate(self, matrix):
Applies the rotation matrix to the Earth array, ISS array, and home array.
center = self.find_center()
for i, node in enumerate(self._earth_nodes):
self._earth_nodes[i] = center + np.matmul(matrix, node - center)
This will produce the following result:
Adding the Earth bitmap
To actually make the sphere look like the Earth I took a black and white image of the Earth and reduced its size to MAP_WIDTH by MAP_HEIGHT. I then converted the image to an array of bits, 1 for a
white pixel and 0 for a black pixel. When drawing the Earth I then check the array at the index for the corresponding node and only draw the pixel if the bit is 1. The following image is converted
with the code below:
def convert_map(self):
Converts the PNG image of the world map to one that can be projected onto the sphere.
Turns the pixels in an array of bits.
# Open the map image.
path = os.path.join(SRC_BASE, "assets", "issview", "world-map.png")
img = Image.open(path).convert("1")
# Transform the image to fit sphere dimensions.
resized = img.resize((self.MAP_WIDTH + 1, self.MAP_HEIGHT + 1), Image.BOX)
flipped = ImageOps.mirror(resized)
shifted = ImageChops.offset(flipped, self.MAP_CALIBRATION, 0)
# Convert to bit array.
for y in range(shifted.height):
for x in range(shifted.width):
pixel = shifted.getpixel((x, y))
self._map.append(int(pixel == 255))
Drawing the ISS
Drawing the ISS on the sphere is very similar to drawing the nodes for the Earth. Every 5 seconds I’m sending a request to an API which returns the current latitude and longitude for the ISS. I take
this information and generate a node matrix with a single node for the ISS. The rotation matrix is applied to this new ISS matrix as well. Then when drawing the frame, I change the pixel for where
the ISS is to red.
I do the same process to draw the green dot for my location.
def update_iss(self):
Generates a new array of nodes to store the location of the ISS.
ones_column = np.ones((1, 1))
ones_added = np.hstack(([self._iss_coords], ones_column))
self._iss_nodes = np.vstack((np.zeros((0, 4)), ones_added))
def draw(self, image):
Draws the Earth, ISS, and home node arrays to the display.
# Draw the ISS.
iss_x = int(self._iss_nodes[0][0])
iss_y = int(self._iss_nodes[0][1])
iss_z = self._iss_nodes[0][2]
if iss_z > 1:
image.putpixel((self.X + iss_x, self.Y + iss_y * -1), self.ISS_COLOR)
# Draw home.
home_x = int(self._home_nodes[0][0])
home_y = int(self._home_nodes[0][1])
home_z = self._home_nodes[0][2]
if home_z > 1:
image.putpixel((self.X + home_x, self.Y + home_y * -1), self.HOME_COLOR)
Final result
A video of the final ISS tracking display running on the RGB matrix can be found here.
This was a really fun project that forced me to brush up on my linear algebra. It is a cool visualization to leave running in the background. | {"url":"https://dev.jameslowther.com/Projects/LED-Matrix---ISS-Tracker","timestamp":"2024-11-14T23:54:12Z","content_type":"text/html","content_length":"99800","record_id":"<urn:uuid:edd5d662-0d47-419d-9691-6b62d4cd928f>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00685.warc.gz"} |
STAT170: Statistics - Regression Analysis - Assessment Answer
Solution Code : 1HJE
This assignment falls under Statistics which was successfully solved by the assignment writing experts at My Assignment Services AU under assignment help service.
Statistics Assignment
Assignment Task
Question 1
A researcher decided to investigate the relationship between student enrolment numbers and teaching staff in remote schools. The following Minitab output was obtained using information recorded on
student enrolment numbers and teaching staff in 2014 at 25 very remote schools in Australia. Use this to answer parts a. and b.
• Comment on the validity of each the three assumptions for linear model by referring to each of the plots above:
• The largest residual was for Goodooga Central School which had the equivalent of 9.4 teaching staff in 2014 and 35 enrolments in 2014. Use this information, and the Fitted Line Plot above, to
calculate the residual for Goodooga Central School and explain clearly what is unusual about Goodooga Central School.
Question 1 continued
The following output was obtained using information on 24 of the very remote schools, (excluding Goodooga Central), from 2014. Use this information to answer part c.
• Why do you think that Goodooga Central school was excluded from this analysis?
• Use an appropriate hypothesis test to determine whether the number of student enrolments at very remote schools in 2014 was a useful predictor of the number of teaching staff:
Question 1 continued
A previous study of the 24 schools from parts c. and d. had been conducted in 2008. The following output was obtained using some of the information recorded on these 24 schools in 2008. Use this
output, together with your answers to parts c. and d. to answer the remainder of this question.
• In 2008, did the school with the highest number of student enrolments also have the highest number of teachers?
• How many students (approximately) were enrolled at the school with the highest number of teachers in 2008?
• Explain why the model for predicting teaching staff in 2014 will give more reliable predictions than the model for predicting teaching staff in 2008.
• Make the following predictions, if possible. If not, explain why your predictions would not be valid.
1. Predict the number of teachers in 2008 at a very remote Australian school which had 50 students enrolled.
2. Predict the number of teachers in 2014 at a very remote Australian school which had 50 students enrolled.
3. Predict the number of teachers in 2014 at a very remote Australian school which had 150 students enrolled.
Question 2
An educational researcher carried out a study into standardised examinations for high school students. 150 Year 7 students were randomly selected from schools across Australia in 2011 and various
information was recorded on these students over a four year period. Some of the variables recorded were:
The data are stored in the Excel file: ExamMarks.xlsx which is on iLearn. Use this data file to complete Question 2.
Each part of this question must be labelled and must be presented in order and must be neatly word processed. Untidy work wll not be marked. Each part (a., b. and c.) should be answered by a
different student from your group. Each part of this question uses a different section of the data set which should be extracted (either by sorting the data or splitting the worksheet). Each part
asks about the difference between average marks for a particular group of students. Since the worksheet contains data on both English and Mathematics marks, each part should address both English and
Mathematics marks (ie. one hypothesis test on English marks and one hypothesis on Mathematics marks for each part). Minitab output for these analyses, along with any other appropriate graphical
output, should be cut and pasted neatly into the space provided for each part. Please note that output that is not clearly labelled will not receive any marks (eg. Group 1 is not a clear label
whereas Co-educational School is a clear label). On the following page provided for each part, a report on these analyses should be written up. One report only is required for each part, with the
report addressing both English and Mathematics scores in regard to the Research Question. Reports should follow the format described in the report writing document on iLearn, with particular
attention to the section ‘A Short Guide to Report Writing for STAT170 students’. Each report will have an Introduction, a Methods section, a Results section and a Conclusion as outlined in the report
writing document. Reports should not be more than one A4 page in length.
Research Questions for Q2:
• Amongst students attending boys only schools, did the average marks on standardised exams change significantly between Years 7 and 11?
• Amongst Year 11 students, were the average marks on standardised exams for students at Government schools different to the average marks for students at Independent schools?
• Amongst Year 7 students at were the average marks on standardised exams for for students at boys only schools different to the average marks for students at girls only schools?
The assignment file was solved by professionalStatistics experts and academic professionals at My Assignment Services AU. The solution file, as per the marking rubric, is of high quality and 100%
original (as reported by Plagiarism). The assignment help was delivered to the student within the 2-3 days to submission.
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the referencing style guidelines.
Part a Output:
Part a Report
The main objective of this section is to determine wehthe there is a significant difference in the average marks on standardised exams change significantly between Years 7 and 11. The average marks
scores by the students in boys only shool is 54.87813 in year 7 and 57.11384 in year 11.
In order to test the claim, we perform independent sample t test. Here, the independent variable is the class 7 and class 11 and the dependent variable is the standardized scores. Since two groups
are independent, we can perform independent sample t test
The value of t test statistic is -0.91825 and its corresponding p – value is 0.3607 > 0.5, indicating that there is a sufficient evidence to reject the claim
Here, we fail to support the claim that there is a significant difference in the average marks on standardised exams change significantly between Years 7 and 11
Part b Output:
Part b Report
The main objective of this section is to determine wehthe the average marks on standardised exams forstudents at Government schools different to the average marks for students at Independent schools.
The average marks scores by the students in government schools is 60.83673 and 58.35 in independent schools.
In order to test the claim, we perform independent sample t test. Here, the independnet variable is the government school and independent school and the dependent variable is the standardized scores.
Since two groups are independent, we can perform independent sample t test
The value of t test statistic is 0.9744 and its corresponding p – value is 0.3323 > 0.5, indicating that there is a sufficient evidence to reject the claim
Here, we fail to support the claim that there is a significant difference in the average marks on standardised exams for students at Government schools different to the average marks for students at
Independent schools
Part c Output:
Part c Report
The main objective of this section is to determine wehthe average marks on standardised exams for for students at boys only schools different to the average marks for students at girls only schools
at year 7. The average marks scores by the students in boys schools is 54.87813 and 59.67 in girls schools.
In order to test the claim, we perform independent sample t test. Here, the independent variable is the boys only school and girls only school and the dependent variable is the standardized scores.
Since two groups are independent, we can perform independent sample t test
The value of t test statistic is -2.022 and its corresponding p – value is 0.0457 < 0.5, indicating that there is a sufficient evidence to support the claim
Here, we support the claim that there is a significant difference in the average marks on standardised exams for for students at boys only schools different to the average marks for students at girls
only schools at year 7
This Statisticsassignment sample was powered by the assignment writing experts of My Assignment Services AU. You can free download thisStatistics assessment answer for reference. This solved
Statistics assignment sample is only for reference purpose and not to be submitted to your university. For a fresh solution to this question, fill the form here and get our professional assignment | {"url":"https://www.gradesaviours.com/solutions/stat170-statistics-regression-analysis-assessment-answer","timestamp":"2024-11-05T03:11:44Z","content_type":"text/html","content_length":"498411","record_id":"<urn:uuid:ec0e893a-9f3d-47ea-b51b-4b41b0fbad32>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00438.warc.gz"} |
Package arith-eval on DUB
arith-eval 0.4.0
A minimal math evaluator library.
To use this package, run the following command in your project's root directory:
Manual usage
Put the following dependency into your project's dependences section:
ArithEval is a minimal arithmetic expression evaluator library for the D programming language. In other words, define a math function as a string depending on as many variables as you want, then
evaluate that function giving those variables the values you want.
This library is licensed under the terms of the GNU GPL3 free software license. Free as in freedom. Also as in free beer. Also as in gluten-free. (Warning: beer might not be gluten-free)
Currently, ArithEval uses Pegged as its base for parsing math expressions, although this might change in the future.
How to use
Minimal the library, minimal the tutorial, really. Just instance the Evaluable struct from the arith_eval.evaluable module and define your function:
import arith_eval.evaluable;
auto a = Evaluable!(int, "x", "y")("(x + y) * x - 3 * 2 * y");
assert(a(2, 2) == (2 + 2) * 2 - 3 * 2 * 2);
assert(a(3, 5) == (3 + 5) * 3 - 3 * 2 * 5);
import std.math: pow;
auto b = Evaluable!(float, "x", "z")("x ** (2 * z)");
assert(b(1.5f, 1.3f) == pow(1.5f, 2 * 1.3f));
Evaluable, as you can see, is a template struct that takes the evaluation type and the name of its variables as its template parameters.
Evaluable will throw an InvalidExpressionException if it isn't able to understand the expression given, and an OverflowException if it overflows during calculation at a given point for the specified
evaluation type.
Note that currently ArithEval will not check your expressions for variables not specified in the template, and using those should be considered an error, so please be extra careful in that aspect.
I'll implement the checking once I have some time.
Supported operations
Currently, supported math operations are the following:
• x + y
• x - y
• - x
• x * y
• x / y
• x ** y (x to the power of y)
Also parenthesis should work wherever you place them, respecting basic math operation priorities.
If you are missing a specific operation, open an issue or, even better, implement it yourself and submit a PR.
Add as DUB dependency
Just add the arith-eval package as a dependency in your dub.json or dub.sdl file. For example:
"dependencies" : {
"arith-eval": "~>0.4.0"
• Registered by Héctor Barreras Almarcha
• 0.4.0 released 8 years ago
• GPL-3.0
0.5.1 2018-Jan-08
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0.3.2 2016-Jun-27
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A Body Shape Index (ABSI) Calculator
Use our calculator to get the ABSI score, just fill the required fields and our calculator will give you the result automatically.
A Body Shape Index (ABSI) is a measure for assessing the health impact of a given human height, weight and waist circumference (WC). The inclusion of WC is thought to make the BSI a better indicator
of the risk of overweight death than the standard body mass index. ABSI is only marginally correlated with height, weight, and BMI, indicating that it is independent of other anthropometric variables
in predicting mortality. BMI has been criticized for not distinguishing between muscle and fat mass and therefore may be elevated in people with increased BMI due to muscle development rather than
fat accumulation due to overeating. Higher muscle mass can lower your risk of premature death. A high ABSI corresponds to a higher proportion of central obesity or abdominal fat. In a sample of
Americans who participated in the National Health and Nutrition Examination Survey, mortality rates in some subjects were high with both high and low BMI, which is a well-known conundrum associated
with BMI. In contrast, the mortality rate increased in proportion to the increase in ABSI values. The linear relationship was not affected by adjustments for other risk factors, including smoking,
diabetes, high blood pressure, and serum cholesterol.
The equation for ABSI is based on a statistical analysis of shape and derived from allometric regression (with waist and height in meters and weight in kg): ABSI = \dfrac{Waist \ circumference}{BMI^
{\frac{2}{3}}*Height^{\frac{1}{2}}} ABSI_z = \dfrac{ABSI-ABSI_{mean}}{ABSI_{sd}}
The ABSI z score is calculated based on the mean and standard deviations of the ABSI calculated for a given age and gender. The ABSI z scale is used to measure the level of risk of premature death. | {"url":"https://owlcalculator.com/health/absi-calculator","timestamp":"2024-11-06T05:38:04Z","content_type":"text/html","content_length":"219688","record_id":"<urn:uuid:1caccc1e-c4c9-470e-b54d-cde3f02dc94d>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00791.warc.gz"} |
Physique des particules - Astrophysique
Themes Members Networks Seminars Events Jobs Contact
Condensed-matter occupies a large place in physics, and the theoretical concepts developed for these problems have often had an impact in other fields of physics. Likewise, the research carried out
at the IPhT on condensed-matter systems has been tightly connected to other fields of theoretical physics, such as statistical physics, field theory, or integrable systems. In the last few years we
focused on the following directions: topological states of matter, many-body problems, systems far from equilibrium, and disordered systems.
Research themes
Topological matter
Quantum phases of matter with topological properties – subject of the 2016 Nobel Prize - represent a very active topic worldwide. Among these, systems supporting Majorana modes are actively studied,
in part because they could be used in future device for quantum information processing.
High critical-T superconductors
Quantum many-body systems where the interactions between the particles play a central role, where non-perturbative (or beyond mean-field) effects need to be taken into account, are challenging
problems. A prominent example is that of high-temperature (HT) cuprate superconductors. They have been studied theoretically and experimentally for several decades, but some key questions remain
Out-of-equilibrium systems
The field of quantum systems which are far from equilibrium has developed rapidly in the last decade, in part thanks to the advances on the experimental side (cold atoms, artificial light-matter
systems, etc.). One way to set a system out-of-equilibrium is to perform a quantum quench. There, an isolated system is prepared in some simple state at time $t=0$ (not an eigenstate of the
Hamiltonian) and then it evolves unitarily for $t>0$. Such protocols allow to address important questions about the equilibration or transport in isolated quantum systems, and to discuss the role
played by interactions. We have also introduced a Quantum Monte Carlo (QMC) method for interacting systems far from equilibrium, the first diagrammatic QMC using an explicit sum of the Feynman
diagrams in terms of an exponential number of determinants.
Many-body localization
A new phase of matter has triggered a huge activity I the last few years. Dubbed many-body localization, it is the interacting counter part of the (single-particle) Anderson localization. Such
disordered systems display many interesting anomalous properties, like the absence of thermalization.
Researchers involved
Permanent and emeritus researchers
Cristina Bena
Thierry Jolicoeur
Grégoire Misguich
Olivier Parcollet
Catherine Pépin
Marco Schiro
PhD students
Ngoc Duc Le
Maxence Grandadam
Sarah Pinon
Postdoctoral researchers
Debmalya Chakraborty
Saheli Sarkar
Former staff members
Former Postdoctoral researchers
Haggai Landa
Steven Thomson
Juan Manuel Aguiar
Fabien Alet
Marine Guigou
Thomas Kloss
Laura Messio
Xavier Montiel
Corentin Morice
Francesco Peronaci
Nicholas Sedlmayr
Mircea Trif
Former graduate students
Orazio Scarlatella
Kemal Bidzhiev
Vardan Kaladzhyan
Thomas Ayral
Thiago Sabetta
Jean-Marie Stéphan
Networking, collaborations & fundings
Our weekly seminar takes place every Monday at 14:00.
Postdoctoral positions are available each year in the Fall. Check this page or contact any staff member of the group.
Each member of the group can be contacted via email at name.surname@ipht.fr .
The full postal adress of IPhT is: Institut de Physique Théorique, CEA/Saclay, Bat 774 Orme des Merisiers, 91191 Gif-sur-Yvette Cedex, France.
Here are directions to the IPhT. | {"url":"https://www.ipht.fr/en/Phocea/Vie_des_labos/Ast/ast_sstechnique.php?id_ast=40&voir=872","timestamp":"2024-11-03T01:06:59Z","content_type":"text/html","content_length":"37465","record_id":"<urn:uuid:9b5105e4-93d3-4f0a-95f6-0c87a9511517>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00274.warc.gz"} |
How to calculate a logarithm in Matlab?
In Matlab, you can use the log function to calculate logarithms. The default base of the logarithm is e (natural logarithm), but you can specify a different base if needed.
Here are some examples:
1. Calculate the natural logarithm of a number:
The variable y will contain the value of the natural logarithm of x, which is approximately 2.3026.
1. Calculate the logarithm of a number with a different base:
1 x = 100;
2 base = 10;
3 y = log(x) / log(base);
In this example, we are calculating the logarithm of x with base 10. We first calculate the natural logarithm of x using log(x), and then divide it by the natural logarithm of the specified base (log
(base)). The variable y will contain the value of the logarithm of x with base 10, which is 2.
1. Calculate the logarithm of a vector or matrix:
1 A = [1 10 100; 1000 10000 100000];
2 B = log(A);
In this example, we are calculating the natural logarithm of a 2-by-3 matrix A. The log function will apply the natural logarithm to each element of the matrix, and the resulting matrix will have the
same dimensions as the input matrix. The variable B will contain the natural logarithm of A.
Note that the log function returns NaN (Not a Number) for negative input values. If you want to calculate the logarithm of a negative value, you can use the log10 function to calculate the logarithm
with base 10, or the log2 function to calculate the logarithm with base 2. | {"url":"https://devhubby.com/thread/how-to-calculate-a-logarithm-in-matlab","timestamp":"2024-11-12T12:45:09Z","content_type":"text/html","content_length":"131355","record_id":"<urn:uuid:b22e87c7-2373-40af-878f-bea11d195601>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00089.warc.gz"} |
Symmetries and conservation laws of the Euler equations in Lagrangian coordinates
We consider the Euler equations of incompressible inviscid fluid dynamics. We discuss a variational formulation of the governing equations in Lagrangian coordinates. We compute variational symmetries
of the action functional and generate corresponding conservation laws in Lagrangian coordinates. We clarify and demonstrate relationships between symmetries and the classical balance laws of energy,
linear momentum, center of mass, angular momentum, and the statement of vorticity advection. Using a newly obtained scaling symmetry, we obtain a new conservation law for the Euler equations in
Lagrangian coordinates in n-dimensional space. The resulting integral balance relates the total kinetic energy to a new integral quantity defined in Lagrangian coordinates. This relationship implies
an inequality which describes the radial deformation of the fluid, and shows the non-existence of time-periodic solutions with nonzero, finite energy.
All Science Journal Classification (ASJC) codes
• Analysis
• Applied Mathematics
• Conservation laws
• Euler equations
• Incompressible flows
• Lagrangian coordinates
• Symmetries
• Time-periodic solution
Dive into the research topics of 'Symmetries and conservation laws of the Euler equations in Lagrangian coordinates'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/symmetries-and-conservation-laws-of-the-euler-equations-in-lagran","timestamp":"2024-11-07T17:28:59Z","content_type":"text/html","content_length":"49956","record_id":"<urn:uuid:a0727c1d-70eb-4ddb-9f0f-fa93aaa08ce9>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00647.warc.gz"} |
Ackermann groupoid
nLab Ackermann groupoid
$(0,1)$-Category theory
An Ackermann groupoid is a particular type of algebraic mathematical structure that provides semantics for a flavour of relevance logic, a weak form of substructural logic.
Note that “groupoid” here does not mean groupoid, but magma. The terminology comes from logic, rather than category theory.
An Ackermann groupoid is a partially ordered magma $(M,\circ, 1,\leq)$ that is left unital ($1\circ a = a$ for all $a\in M$), and has a binary operation, “implication”, written $b\to c$ satisfying $a
\leq b\to c$ if and only if $a\circ b \leq c$.
This might be called an implicational Ackermann groupoid, since it provides semantic models for an implicational fragment of logic, together with intensional conjunction (here $\to$ models
implication, analogous to linear implication in linear logic). A positive Ackermann groupoid upgrades the underlying poset to a distributive lattice, permitting the interpretation of additional
logical connectives, namely (classical) logical conjunction and logical disjunction.
Every Church monoid is an Ackermann groupoid.
Ackermann groupoids were introduced in
• Robert K. Meyer and Richard Routley, Algebraic analysis of entailment I, Logique et Analyse NOUVELLE SÉRIE, Vol. 15, No. 59/60 (1972) pp407-428, JSTOR
and named for Wilhelm Ackermann.
Last revised on April 30, 2021 at 07:04:24. See the history of this page for a list of all contributions to it. | {"url":"https://ncatlab.org/nlab/show/Ackermann+groupoid","timestamp":"2024-11-02T13:43:15Z","content_type":"application/xhtml+xml","content_length":"22411","record_id":"<urn:uuid:a3648905-4a32-4f6c-860d-4d065b657f11>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00066.warc.gz"} |
Karl Pearson introduced several now-commonplace statistical tools. One of these was the histogram, a diagram similar to a bar chart. The use of a histogram in statistics is to represent a set of
continuous, rather than discrete, data. For this reason, Pearson explained that it could be employed as a tool in the study of history, for example, to chart historical time periods, and coined the
name ‘histogram’ in 1891 to convey its use as a ‘historical diagram’.
Compare the frequency of occurrence of quantitative data – Compare the height of bars
Use a histogram in data visualization when an entire range of values of continuous numerical data can be bucketed into a series of intervals—and then how many values fall into each interval can be
counted. The bins (or intervals) must be adjacent and are often (but not required to be) of equal size. When these intervals are of equal width then the height of the bars is proportional to the
frequency and can be used to compare the data.
Compare the frequency of occurrence of quantitative data – Compare bar area when intervals are unequal
In a histogram, it is the area of the bar that indicates the frequency of occurrences for each bin. This means that the height of the bar does not necessarily indicate the correct frequency, but the
product of height multiplied by the width of the bin indicates the frequency of occurrences within that bin. When the bars are not equally spaced the height of the bin does not reflect the frequency
and should not be used as criteria for comparison.
Get an overview of statistical anomalies in data
The use of a histogram in statistics is defined by the need to check the consistency of your process by understanding the spread of the data and discovering the outliers. They are also used to
estimate where values are concentrated, what the extremes are, and identify any gaps or unusual values in your data distribution. Determine the mode of the distribution by finding the peak of the
histogram, as the value which is most frequently occurring or has the largest probability of occurrence. For many phenomena, it is quite common for the distribution of the response values to cluster
around a single mode (unimodal- normal distribution) and then distribute themselves with lesser frequency out into the tails. Similarly, discover for bi-modal or multi-modal datasets. This can help
to diagnose problems such as the non-uniformity of data and study the cause of outliers.
Represent and discover probability occurrences
Histograms are useful for giving a rough view of the probability distribution and are used to provide insight into their behavior and frequency of occurrence. For instance, In hydrology, the
estimated density function of rainfall and river discharge data are analyzed using a probability distribution histogram graph.
Use histograms to give a rough sense of the density of the underlying distribution of the data for density estimation: when estimating the probability density function of the underlying variable. The
total area of a histogram used for probability density is always normalized to 1. However, only nonnegative numbers can be used for the scale that gives us the height of a given bar of the histogram.
1. Equal bin width Histogram
If the bins are of equal size, a rectangle is erected over the bin with height proportional to the frequency—representing the equal bin width histogram.
2. Variable bin width histograms
When bins are not of equal width, the erected rectangle is defined to have its area proportional to the frequency of cases in the bin. The vertical axis is then not the frequency but frequency
density—the frequency per unit of the class width on the horizontal axis.
3. Normalized or cumulative histograms
A histogram may also be normalized to display “relative” frequencies. It then shows the proportion of cases that fall into each of several categories, with the sum of the heights equaling 1.
When Not to Use a histogram?
When you need to show distribution against non-numerical categories
Do not use a histogram graph to plot the frequency of score occurrences in a non-continuous data set. Use bar charts for other types of variables including ordinal and nominal data sets since it’s a
graph of categorical variables. The bar charts have gaps between the rectangles to clarify this distinction.
When you need to represent and discover correlations between two variables
Use a scatter plot when correlations between x and y-axis quantities are needed rather than to represent and gain an understanding of the distribution of a single variable across different intervals.
Ask if you need to determine the way one variable changes with respect to the change in the other. In that case, you can use various correlation charts like line graphs, scatter plots, etc. | {"url":"https://think.design/services/data-visualization-data-design/histogram/","timestamp":"2024-11-13T21:14:13Z","content_type":"text/html","content_length":"148602","record_id":"<urn:uuid:61b623e9-8adc-4f3e-807c-37b1dec2ac2f>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00125.warc.gz"} |
Glossary of Terms - Wyatt Engineering
Accuracy: The extent to which a measurement agrees with a reference. A qualitative assessment; not the same as uncertainty (a quantitative value).
Beta Ratio (Beta): For differential producers with circular cross-sections the ratio the bore I.D.-to-pipe I.D. Typically ranging from 0.20 to 0.75, depending on meter type.
Cavitation: The implosion of vapor pockets occurring when the local pressure of a fluid rises above the vapor pressure of the line fluid.
Critical Flow (Choked Flow, Sonic Flow): The condition at which the velocity of a flowing fluid, typically a gas or vapor, reaches the velocity of sound at local conditions.
Differential Pressure: The pressure difference between high- and low-pressure taps. This is the flow signal generated by a differential pressure producing primary flow element.
Discharge Coefficient (CD): The value of the ratio of the actual flow rate to the theoretical flow rate, where the theoretical flow rate is calculated for an ideal fluid and no energy loss.
Expansibility Factor: The coefficient that accounts for the compressibility of a fluid. Y(ε) is equal to one for ideal, incompressible fluids.
Flow Profile: A graphic representation of the velocity distribution of the flow as it approaches a primary flow element. Flow through a straight, smooth, pipe of sufficient length should develop a
blunt flow profile. Valves and pipe fittings introduce non-uniform velocity distributions.
Meter: Wyatt uses this term to refer to primary flow elements used for measurement, as opposed to restriction. Not to be confused with the transmitter, or secondary instrumentation.
Pressure Ratio: In gas or vapor flow, the ratio of the absolute pressure at the low-pressure tap to that at the high-pressure tap. Ideally, this value should be greater than 0.8.
Primary Element: Device (venturi, orifice, flow nozzle, etc.) that generates a signal to determine the flow rate.
Reliability: A qualitative assessment of the degree to which empirical evidence agrees with and substantiates the theoretical understanding of a phenomenon.
Transmitter: The instrument that converts the differential pressure signal to (typically) an analog or digital signal that can be recorded or processed.
Uncertainty: The estimate of an error band within which the true value falls with stated probability. | {"url":"https://wyattflow.com/products/resources/glossary-of-terms/","timestamp":"2024-11-14T01:43:11Z","content_type":"text/html","content_length":"39731","record_id":"<urn:uuid:6bd87e80-7a13-42be-b0bf-bb4aed64869c>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00685.warc.gz"} |
Electronic Journal of Applied Statistical Analysis
A note on ridge regression modeling techniques
In this study, the techniques of ridge regression model as alternative to the classical ordinary least square (OLS) method in the presence of correlated predictors were investigated. One of the basic
steps for fitting efficient ridge regression models require that the predictor variables be scaled to unit lengths or to have zero means and unit standard deviations prior to parameters’ estimations.
This was meant to achieve stable and efficient estimates of the parameters in the presence of multicollinearity in the data. However, despite the benefits of this variable transformation on ridge
estimators, many published works on ridge regression practically ignored it in their parameters’ estimations. This work therefore examined the impacts of scaled collinear predictor variables on ridge
regression estimators. Various results from simulation studies underscored the practical importance of scaling the predictor variables while fitting ridge regression models. A real life data set on
import activities in the French economy was employed to validate the results from the simulation studies.
DOI Code: 10.1285/i20705948v7n2p343
Keywords: Ridge regression; orthogonality; shrinkage parameter; scaling; ordinary least squares; mean square error
Bradly, R. A., Srivastava, S. S. (1997). Correlation in polynomial regression. URL: http://stat.fsu.edu/techreports/M409.pdf
Cannon, A. J. (2009). Negative ridge regression parameters for improving the covariance structure of multivariate linear downscaling models. Int. J. Climatol., 29, 761-769.
Chatterjee, S., Hadi, A. S. (2006). Regression Analysis by Example. John Wiley & Sons, Inc., Hoboken, New Jersey.
Dorugade, A. V., Kashid, D. N. (2010). Alternative method for choosing ridge parameter for regression. Applied Mathematical Science, 4(9), 447-456.
El-Dereny, M. and Rashwan, N. I. (2011). Solving Multicollinearity Problem Using Ridge Regression Models. Int. J. Contemp. Math. Sciences, 6(12), 585-600.
Faraway, J. J. (2002). Practical regression and ANOVA using R. http://cran.r-project.org/doc/contrib/Faraway-PRA.pdf
Fearn, T. (1993). A misuse of ridge regression in the calibration of a near infrared reflectance instrument. Applied Statistics, 32, 73-79.
Hoerl, A. E., Kennard, R. W., Hoerl, R. W.(1985). Practical use of ridge regression: A challenge met. Applied Statistics, 34(2), 114-120.
Hoerl, A.E., Kennard, R.W., Baldwin, K.F. (1975). Ridge regression: Some simulations.Communications in Statistics, 4, 105-123.
Hoerl, A. E., Kennard, R.W.(1970a). Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12, 55-67.
Hoerl, A. E., Kennard, R. W. (1970b). Ridge regression: Applications to nonorthogonal problems. Technometrics, 12, 69-82, 1970b.
Khalaf, G., Shukur, G. (2005). Choosing ridge parameter for regression problem. Communications in Statistics–Theory and Methods, 34, 1177-1182.
Kibria, B. M. (2003). Performance of some ridge regression estimators. Communication in Statistics – Simulation and Computation, 32, 419-435.
Lawless, J. F., Wang, P. A. (1976). Simulation study of ridge and other regression estimators. Communications in Statistics –Theory and Methods, 14, 1589-1604.
Longley, J. W.(1976). An appraisal of least-squares programs from the point of view of the user. Journal of the American Statistical Association, 62, 819–841.
Lin, L., Kmenta, J. (1982). Ridge Regression under Alternative Loss Criteria. The Review of Economics and Statistics, 64(3), 488-494.
Malinvaud, E. (1968). Statistical Methods of Econometrics, Rand-McNally, Chicago.
Mardikyan, S., Cetin, E.(2008). Efficient Choice of Biasing Constant for Ridge Regression. Int. J. Contemp. Math. Sciences, 3, 527-547.
Marquardt, D. W., Snee, R. D. (1975). Ridge regression in practice. The American Statistician, 29(1), 3-20.
Muniz, G., Kibria, B. M.(2009). On Some Ridge Regression estimator: An empirical comparison. Communication in Statistics–Simulation and Computation, 38, 62-630.
Myers, R. H.(1986). Classical and Modern Regression with Applications. PWS-KENT Publishing Company, Massachusetts.
Sparks, R. (2004). SUR Models Applied To an Environmental Situation with Missing Data and Censored Values. Journal of Applied Mathematics and Decision Sciences, 8(1), 15-32, 2004.
Wethril, H. (1986). Evaluation of ordinary Ridge Regression. Bulletin of Mathematical Statistics, 18, 1-35, 1986.
Yahya, W.B., Adebayo, S.B., Jolayemi, E.T., Oyejola, B.A., Sanni, O.O.M. (2008). Effects of non-orthogonality on the efficiency of seemingly unrelated regression (SUR) models. InterStat Journals,
Full Text: | {"url":"http://siba-ese.unile.it/index.php/ejasa/article/view/12502/0","timestamp":"2024-11-13T03:07:29Z","content_type":"application/xhtml+xml","content_length":"27482","record_id":"<urn:uuid:f94fb610-45cc-4240-aa0a-f2ae65f573a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00387.warc.gz"} |
Pearson vs Spearman Correlation: Find Harmony between the Variables
Which measure of correlation should you use for your task? Learn all you need to know about Pearson and Spearman correlations
Consider a symphony orchestra tuning their instruments before a performance. Each musician adjusts their notes to harmonize with others, ensuring a seamless musical experience. In Data Science, the
variables in a dataset can be compared to the orchestra’s musicians: understanding the harmony or dissonances between them is crucial.
Image source: pixabay.com.
Correlation is a statistical measure that acts like the conductor of the orchestra, guiding the understanding of the complex relationships within our data. Here we will focus on two types of
correlations: Pearson and Spearman.
If our data is a composition, Pearson and Spearman are our orchestra’s conductors: they have a singular style of interpreting the symphony, each with peculiar strengths and subtleties. Understanding
these two different methodologies will allow you to extract insights and understand the connections between variables.
Pearson Correlation
The Pearson correlation coefficient, denoted as r, quantifies the strength and direction of a linear relationship between two continuous variables [1]. It is calculated by dividing the covariance of
the two variables by the product of their standard deviations.
Pearson coefficient formula
Here X and Y are two different variables, and X_i and Y_i represent individual data points. \bar{X} and \bar{Y} denote the mean values of the respective variables.
The interpretation of r relies on its value, ranging from -1 to 1. A value of -1 implies a perfect negative correlation, indicating that as one variable increases, the other decreases linearly [2].
Conversely, a value of 1 signifies a perfect positive correlation, illustrating a linear increase in both variables. A value of 0 implies no linear correlation.
Pearson correlation is particularly good at capturing linear relationships between variables. Its sensitivity to linear patterns makes it a powerful tool when investigating relationships governed by
a consistent linear trend. Moreover, the standardized nature of the coefficient allows for easy comparison across different datasets.
However, it’s crucial to note that Pearson is susceptible to the influence of outliers. If a dataset contains extreme values they can impact the calculation, leading to inaccurate interpretations.
Technical concepts can be better understood through practical examples. Let’s use Python to show the computation of Pearson correlation and its visualization. Suppose we have two lists representing
the hours spent studying (X) and the corresponding exam scores (Y).
import numpy as np
from scipy.stats import pearsonr
import matplotlib.pyplot as plt
import seaborn as sns
# Generating data points
np.random.seed(42) # For reproducibility
hours_studied = np.random.randint(8, 25, size=50)
exam_scores = 60 + 2 * hours_studied + np.random.normal(0, 5, size=50)
# Calculate Pearson correlation coefficient
pearson_corr, _ = pearsonr(hours_studied, exam_scores)
# Calculate Pearson correlation line coefficients
m, b = np.polyfit(hours_studied, exam_scores, 1) # Fit a linear regression line
# Scatter plot
fig, ax = plt.subplots()
ax.scatter(hours_studied, exam_scores, color=sns.color_palette("hls",24)[14], alpha=.9, label='Data points')
plt.plot(hours_studied, m * np.array(hours_studied) + b, color='red', alpha=.6, label='Pearson Correlation Line')
plt.title("Hours Studied vs. Exam Scores")
plt.xlabel("Hours Studied")
plt.ylabel("Exam Scores")
plt.legend(loc='lower right')
Image by the author.
Pearson correlation effectiveness diminishes when faced with curvilinear patterns. This limitation arises from Pearson’s inherent assumption of linearity, making it ill-suited to capture the nuances
of non-linear relationships.
Image by the author.
Consider a scenario where the relationship between two variables follows a quadratic curve. Pearson correlation might inaccurately suggest a weak or nonexistent relationship due to its inability to
capture the non-linear relation.
# Generating quadratic data
X = np.linspace(-10, 10, 100)
Y = X**2 + np.random.normal(0, 10, size=len(X))
# Calculate Pearson correlation coefficient
pearson_corr, _ = pearsonr(X, Y)
m, b = np.polyfit(X, Y, 1) # Fit a linear regression line
# Scatter plot
fig, ax = plt.subplots()
ax.scatter(X, Y, color=sns.color_palette("hls",24)[14], alpha=.9, label='Data points')
plt.plot(X, m * X + b, color='red', alpha=.6, label='Pearson Correlation Line')
plt.title("X vs. Y (Quadratic Relationship)")
plt.legend(loc='upper center')
Image by the author.
Spearman Correlation
Spearman correlation addresses the limitations of Pearson when applied to non-linear relationships or datasets containing outliers [3]. Spearman’s rank correlation coefficient (ρ), denoted as rho,
operates on the ranked values of variables, making it less sensitive to extreme values and well-suited for capturing monotonic relationships.
Spearman coefficient formula
In the above formula, d_i represents the difference between the ranks of corresponding pairs of variables, and n is the number of data points.
Similar to Pearson, Spearman’s coefficient ranges from -1 to 1. A value of -1 indicates a perfect negative monotonic correlation, meaning that as one variable increases, the other consistently
decreases. A value of 1 signifies a perfect positive monotonic correlation, illustrating a consistent increase in both variables. A value of 0 denotes no monotonic correlation.
Unlike Pearson, Spearman does not assume linearity and is robust in case of outliers. It focuses on the ordinal nature of data, making it a valuable tool when the relationship between variables is
more about the order than the specific values.
Image by the author.
Consider the following practical application using Python, where we have two variables ‘A’ and ‘B’ with a non-linear relationship:
import numpy as np
from scipy.stats import spearmanr
from scipy.stats import pearsonr
import matplotlib.pyplot as plt
# Generating non-linear data
A = np.linspace(-10, 10, 100)
B = A**5 + np.random.normal(0, 4000, size=len(A))
# Calculate Spearman correlation coefficient
spearman_corr, _ = spearmanr(A, B)
# Calculate Pearson correlation coefficient
pearson_corr, _ = pearsonr(A, B)
m, b = np.polyfit(A, B, 1) # Fit a linear regression line
# Scatter plot
fig, ax = plt.subplots()
ax.scatter(A, B, color=sns.color_palette("hls",24)[14], alpha=.9, label='Data points')
plt.plot(X, m * X + b, color='red', alpha=.6, label='Pearson Correlation Line')
plt.title("A vs. B (Non-linear Relationship)")
Image by the author.
In this example, the scatter plot visualizes a non-linear relationship between variables ‘A’ and ‘B.’ Spearman correlation, which does not assume linearity, will be better suited to capture and
quantify this non-linear association. You can see that the red line, representing the Pearson Correlation Line, misses the nature of the variables’ relationship.
We can quantify this measure by comparing the two coefficients:
Spearman coefficient is sensibly higher, as it is more suited for this type of relationship.
Conclusion: Pearson vs. Spearman
Concluding this introductory guide, let’s point out the pros and cons of the two measures.
Image by the author.
Pearson Correlation coefficient is indeed efficient for linear relationship, and can provide a standardized measure for easy comparison across different datasets.
On the other hand, Pearson coefficient is highly sensitive to outliers. Also, assuming linearity may mislead in non-linear scenarios.
Most of Pearson coefficient’s downsides are addressed by Spearman Correlation coefficient: it is robust in the presence of outliers, and, as it relies on the rank of data points, it is suitable for
non-linear relationships.
It is important, however, to keep in mind that Spearman coefficient has also some criticalities. In my opinion the most relevant ones are its lower efficiency on large datasets, and the potential
loss of information in case of tied ranks.
In order to keep this guide concise and practical, my advice is to choose Pearson for linear relationships in normally distributed data without outliers. Opt for Spearman when facing non-linear
relationships, ordinal data, or datasets with outliers, as it excels in capturing monotonic associations.
Remember that this choice heavily relies on the characteristics of the data. Keep also in mind that this introduction, while touching most important aspects, merely scratched the surface of
statistical methodologies. For further study, you can find an interesting set of resources below!
In-text references:
[1] Laerd Statistics — Pearson Product-Moment Correlation
[2] Boston University — Correlation and Regression with R
[3] Analytics Vidhya — Pearson vs Spearman Correlation Coefficients
[4] Statistics by Jim — Spearman’s Correlation Explained
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Pool Calculator App – Accurate Pool Measurements
This tool calculates the volume of water your pool needs based on its dimensions.
How to Use the Pool Calculator
To use the pool calculator, simply follow these steps:
1. Enter the length of your pool in meters.
2. Enter the width of your pool in meters.
3. Enter the average depth of your pool in meters.
4. Select the shape of your pool (Rectangular or Circular).
5. Click the “Calculate” button to see the volume of your pool in cubic meters.
How It Calculates the Results
The pool calculator uses simple formulas to determine the volume of the pool based on its shape:
• Rectangular Pool: Volume = Length × Width × Average Depth
• Circular Pool: Volume = π × Radius² × Average Depth (where Radius is half of the Width)
The result will be displayed in cubic meters.
Limitations of the Pool Calculator
This calculator assumes the walls of the pool are straight up and down (a perfect rectangle or cylinder). It also assumes the pool has a uniformly flat bottom, without any slope or irregularities.
For pools with more complex shapes or varying depths, this calculator will provide an approximation and may not be entirely accurate.
Use Cases for This Calculator
Calculate Pool Volume
Enter the dimensions of your pool – length, width, and depth to instantly calculate the pool’s volume. No more manual calculations or complex formulas needed!
Estimate Chemical Dosage
Input your pool’s volume and the current chemical levels to get precise dosing recommendations for chlorine, pH adjusters, and other chemicals. Say goodbye to guesswork and potential over/
Determine Pump Run Time
Specify your pool’s volume and flow rate to receive an optimal pump run time for effective circulation and filtration. Keep your pool clean and sparkling with the right operation duration!
Find Heating Costs
Input your pool’s volume, desired temperature increase, and current temperature to calculate the estimated heating costs. Plan your pool heating efficiently without any surprises!
Calculate Water Replacement
Provide your pool’s volume and current water parameters to determine the required amount of water for a partial or complete water change. Save time and resources by knowing the exact quantity needed!
Estimate Pool Surface Area
Enter the length and width of your pool to instantly calculate the surface area. Get accurate measurements for purchasing pool liners, covers, or calculating chemical dosages based on surface area!
Optimize Filter Size
Input your pool’s volume and desired turnover rate to determine the ideal filter size for efficient water filtration. Ensure clean and healthy pool water with the right filter capacity!
Calculate Water Balance
Enter your pool’s volume, current chemical levels, and water temperature to receive a comprehensive water balance analysis. Maintain crystal clear water by balancing pH, alkalinity, and calcium
hardness effectively!
Determine Chlorine Demand
Input your pool’s volume, current chlorine levels, and bather load to calculate the chlorine demand. Adjust your chemical dosing accurately based on real-time demand for a sanitized pool environment!
Estimate Pool Maintenance Costs
Specify your pool’s volume, frequency of use, and chemical usage to estimate monthly maintenance costs. Plan your budget effectively by anticipating expenses for chemicals, equipment, and upkeep! | {"url":"https://calculatorsforhome.com/pool-calculator-app/","timestamp":"2024-11-12T10:17:26Z","content_type":"text/html","content_length":"148032","record_id":"<urn:uuid:332966fd-e4e2-4f50-a938-1de2b1181e54>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00281.warc.gz"} |
Year 4 - St. Francis School
Home » School Information » Love to Learn » Year 1 – 6 » Numeracy »
• I can calculate in my head
• I can recall the first ten multiples of 2, 3, 4, 5, 6, 8 & 10
• I can count forward and backwards in 1s, 2s, 10s and 100 up to 100
• I can count forward and backwards in 1s, 2s, 10s and 100 up to 1000
• I can recognize the place value of any digit in a whole number up to one thousand
• I can recognize the place value of any digit in a whole number up to ten thousand
• I can compare and order digits up to one thousand
• I can compare and order digits up to ten thousand
• I can state one whole number lying halfway between two whole numbers
• I can recognise and name equivalent fractions of a given fraction with denominator up to 12
• I can compare and order unit fractions up to 1/12 and position them on a number line
• I can round any whole two-digit number to the nearest ten and any three-digit number to the nearest one hundred
• I can find remainders after division (dividends of 2, 3, 4, 5, 6, 8, 10 and 100)
• I can work through situations involving column addition with two-digit numbers
• I can use column addition and subtraction with two-digit numbers
• I can use column addition and subtraction with up to 3-digit numbers
• I can handle small amounts of money in classroom situations, plan an activity within a given budget and use menus, entrance tickets to work out totals and change
• I can recognise and extend simple pictorial patterns and number sequences formed by counting any positive integer in constant steps
• I can use assistive technology and other resources appropriate to this level to learn about the fundamentals of algebra
• I can make and describe right angle turns including turns between the four compass points
• I can determine a time interval (hour/half hour) from an o’clock time
• I can determine a time interval (quarter past/to) from an o’clock time
• I can read, write and use the 12-hour clock (analogue and digital) to 5 minutes
• I can estimate, measure and compare lengths
• I can estimate, measure and compare weights (masses)
• I can estimate, measure and compare capacities
• I can use assistive technology and other resources to learn about shapes (e.g. tablets & computers, Pro-bots, 2D & 3D shapes)
• I can draw the other half of a simple symmetrical object inspired by examples of symmetry in nature
• I can construct a block graph
• I can read and interpret a Carroll diagram | {"url":"https://mmargherita.org/school-information/love-to-learn/year-1-6/numeracy/year-4/","timestamp":"2024-11-14T04:56:31Z","content_type":"text/html","content_length":"48122","record_id":"<urn:uuid:78d4be1f-caba-4e40-9524-75ab674c2c71>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00689.warc.gz"} |
Integral of
Introduction integral of cos^3(2x)
In calculus, the integral is a fundamental concept that assigns numbers to functions to define displacement, area, volume, and all those functions that contain a combination of tiny elements. It is
categorized into two parts, definite integral and indefinite integral. The process of integration calculates the integrals. This process is defined as finding an antiderivative of a function.
Integrals can handle almost all functions, such as trigonometric, algebraic, exponential, logarithmic, etc. This article will teach you what is integral to a trigonometric function cubic sine. You
will also understand how to compute cos^3(2x) integral by using different integration techniques.
What is the integral of cos^3(2x)?
The integral of cos^3(2x) is an antiderivative of sine function which is equal to sin 2x/2 –(1/6)sin3 2x + c. It is also known as the reverse derivative of sine function which is a trigonometric
identity. The sine function is the ratio of opposite side to the hypotenuse of a triangle which is written as:
cos = adjacent side/hypotenuse
The integral of cos cube 2x is a common integral in calculus. It is useful to solve many integral problems such as the integral of cos^2(3x).
Integral of cos^3(2x) formula
The formula of the integral of sin contains the integral sign, coefficient of integration, and the function as cosine. It is denoted by ∫(cos3 2x)dx. In mathematical form, the integral of sin^3x is:
$\int \cos^3(2x)dx = \frac{\sin(2x)}{2} –\frac{\sin^3(2x)}{6} + c{2}lt;/p>
Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. In the above integration formula, replacing cos^3(2x) by cos^2(2x) will give the
integration of cos^2(2x).
How to calculate the integral of cos cube (2x)?
The integral cos^3 2x dx is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate the integral of cosine by using:
1. Integration by parts
2. Substitution method
3. Definite integral
Integral of cos 2x cubic by using integration by parts
The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. The integration by parts is a method of solving the integral of
two functions combined together. Let’s discuss calculating the integral of cos cubic power x by using integration by parts.
Proof of integral of cos^3(2x) by using integration by parts
Since we know that the function cosine cube 2x can be written as the product of two functions. Therefore, we can integrate cos^3 (2x) dx by using integration by parts. For this, suppose that:
$I = \cos^3(2x) = \cos(2x). \cos^2(2x){2}lt;/p>
Applying the integral we get,
$I = \int (\cos(2x).\cos^2(2x))dx{2}lt;/p>
Since the method of integration by parts is:
$\int [f(x).g(x)] = f(x).\int g(x)dx - \int [f’(x).\int g(x)]dx{2}lt;/p>
Now replacing f(x) and g(x) by cos x, we get,
$I = \cos^2 (2x).\frac{\sin (2x)}{2} + \int [2\cos (2x)\sin (2x).\frac{\sin (2x)}{2}]dx{2}lt;/p>
It can be written as:
$I = \cos^2 (2x).\frac{\sin (2x)}{2} + \int [\sin^2(2x).\cos(2x)]dx{2}lt;/p>
Now by using a trigonometric identity sin22x = 1 – cos2 2x. Therefore, substituting the value of sin2 2x in the above equation, we get:
$I = \cos^2(2x).\frac{\sin (2x)}{2} + \int \cos(2x)[1 – \cos^2(2x)]dx{2}lt;/p>
Integrating remaining terms,
$I = \cos^2(2x).\sin(2x) +\frac {\sin(2x)}{2} – \int \cos^3(2x)dx{2}lt;/p>
Since we know that I = ∫cos^3(2x)dx
$I = \cos^2(2x).\frac{\sin (2x)}{2} +\frac{\sin (2x)}{2} – I{2}lt;/p>
$2I = \cos^2(2x).\frac{\sin (2x)}{2} + \frac{\sin (2x)}{2}{2}lt;/p>
For more simplification, substitute cos^2x = 1 – sin^2x
$2I = \frac{\sin(2x)(1 – \sin^2(2x))}{2} + \frac{\sin(2x)}{2}{2}lt;/p>
$2I = \frac{\sin 2x}{2} – \frac{sin^3(2x)}{2} + \frac{sin(2x)}{2}{2}lt;/p>
$2I = \sin(2x) – \frac{\sin^3(2x)}{2}{2}lt;/p>
Now dividing by 3 on both sides,
$I = sin 2x – \frac{\sin^3(2x)}{4} + c{2}lt;/p>
Hence the cos^3(2x) integral is equal to,
$\int \cos^3 (2x)dx = \sin (2x) – \frac{\sin^3(2x)}{4} + c{2}lt;/p>
This method is also applicable to calculate an integral of cos square ax.
Integral of cos^3(2x) by using substitution method
The substitution method involves many trigonometric formulas. We can use these formulas to verify the integrals of different trigonometric functions such as sine, cosine, tangent, etc. Let’s
understand how to prove the integral of sin squared by using the substitution method.
Proof of integral cos^3 2x dx by using substitution method
The method of substitution involves different types of substitution, trigonometric and u-substitution. You can also use the u-substitution calculator to solve integrals. To prove the integral of cos
^3(2x) by using the substitution method, suppose that:
$I = \int \cos^3(2x)dx{2}lt;/p>
Suppose that we can write the above integral as:
$I = \int [\cos(2x).\cos^2(2x)]dx{2}lt;/p>
By using trigonometric identities, we can write the above equation by using cos^3(2x) = 1 – sin^2(2x), therefore,
$I = \int [\cos(2x).( 1 – sin^2(2x)]dx{2}lt;/p>
$I = \int [\cos(2x) – \cos(2x)\sin^2(2x)]dx{2}lt;/p>
Now to evaluate first integral, we will use the following steps,
$I_1 = \int \cos(2x).dx = \frac{sin(2x)}{2}{2}lt;/p>
Now to evaluate second integral,
$I_2 = -\int \cos(2x).\sin^2(2x) dx{2}lt;/p>
Suppose that u = sin 2x and du = 2cos 2x dx, then
$I_2 = -\frac{1}{2}\int u^2 du{2}lt;/p>
Integrating with respect to u.
$I_2 = -\frac{u^3}{6}{2}lt;/p>
Substituting the value of u we get,
$I_2 = -\frac{\sin^3(2x)}{3}{2}lt;/p>
Now, using the value of the first and second integral in the above equation to get the final value of the integral.
$I = \frac{\sin(2x)}{2} – \frac{\sin^3(2x)}{6} + c{2}lt;/p>
Hence the integration of cos^3(2x) is verified by using the substitution method. The trigonometric substitution calculator with steps also provides you an easy way to evaluate integrals by using
trigonometric formulas.
Integral of cos^3(2x) by using definite integral
The definite integral is a type of integral that calculates the area of a curve by using infinitesimal area elements between two points. The definite integral can be written as:
$\int^b_a f(x) dx = F(b) – F(a){2}lt;/p>
Let’s understand the verification of the integral of sin^2x by using the definite integral.
Proof of integral of cos^3(2x) by using definite integral
To compute the integral of cos^3(2x) by using a definite integral, we can use the interval from 0 to π/4 or 0 to π. Let’s compute the integral of sin^3x from 0 to 2π. The indefinite integral of cos^
3x can be written as:
$\int^{\frac{\pi}{4}}_0 \cos^3(2x) dx = \left|\frac{\sin(2x)}{2} – \frac{sin^3(2x)}{6}\right|^{\frac{\pi}{4}}_0{2}lt;/p>
Substituting the value of limit we get,
$\int^{\frac{\pi}{4}}_0 \cos^3(2x) dx = \left[\frac{\sin 2π/4}{2} – \frac{\sin^3 2π/4}{6}\right] – \left[sin 0 – \frac{sin^3 0}{3}\right]{2}lt;/p>
$\int^{\frac{\pi}{4}}_0 \cos^3(2x) dx = \frac{1}{2} – \frac{1}{6} = \frac{1}{3}{2}lt;/p>
Therefore, the integral of cos3x from 0 to π/2 is
$\int^{\frac{\pi}{4}}_0 \cos^3(2x) dx = \frac{1}{3}{2}lt;/p>
Which is the calculation of the definite integral of cos^3(2x). Now to calculate the integral of cos cube 2x between the interval 0 to π, we just have to replace π/4 by π. Therefore,
$\int^\pi_0 \cos^3(2x)dx = \left|\frac{\sin(2x)}{2} – \frac{\sin^3(2x)}{6}\right|^\pi_0{2}lt;/p>
$\int^\pi_0 \cos^3(2x)dx = \left[\frac{\sin π}{2} – \frac{\sin^3(2π)}{6}\right] – \left[\frac{\sin 0}{2} – \frac{\sin^30}{3}\right]{2}lt;/p>
$\int^\pi_0 \cos^3(2x)dx = 0 – 0{2}lt;/p>
$\int^pi_0 \cos^3(2x)dx = 0{2}lt;/p>
Therefore, the cos^3(2x) integral from 0 to π is 0. | {"url":"https://calculator-integral.com/integral-of-cos3-2x","timestamp":"2024-11-02T23:59:39Z","content_type":"text/html","content_length":"52792","record_id":"<urn:uuid:c6a63b8c-ad55-4d62-8051-f175f4e797b9>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00445.warc.gz"} |
Fisher Information
Fisher information, two-part message, accuracy of parameter (inference), multiple parameters
For one continuous-valued parameter, θ, the Fisher information is defined to be:
F(θ) = E[x]( d^2/dθ^2 { - ln f(x|θ) } )
where f(x|θ) is the likelihood, i.e. P(x|θ) for data 'x' and parameter value (or hypothesis, ...) θ. E[x] is the expectation, i.e. average over x in the data-space X.
(NB. The 'd's should be curly but this is HTML not XML.)
The Fisher information shows how sensitive the likelihood is to the parameter θ. It turns out to be the key to how accurately parameter estimates, i.e. inferences, should be stated. We should infer a
parameter estimate (usually) close to the maximum likelihood estimate, i.e. close to where d/dθ f(x|θ) = 0, and the second derivative, d^2/dθ^2 f(x|θ), is the inverse of the curvature of the
likelihood function here.
The logs can be in any base, provided that we remember which one, for the units (bits, nits, ...), but differentiation etc. favour natural logs to base e, log[e]=ln. Quantities can easily be
converted to bits later.
A parameter estimate, θ, can only be stated to finite accuracy. How much accuracy is optimal? If a coin is tossed three times and comes up heads once we surely have much less information about any
bias (θ) than if the coin is tossed 300 times and comes up heads 100 times. Finite accuracy amounts to stating that θ lies in an interval (θ-s/2, θ+s/2); note that the width, s, depends on θ in
First Part of Message
If h(θ) is the prior probability density function of θ, the probability, and message length, of the interval are approximated by
probability = h(θ) . s
msgLen = - ln( h(θ) . s ) nits
always assuming that h(θ) does not vary much over the interval.
Second Part of Message
The second part of the message transmits the data given the first part. The receiver has not seen the data, x, and does not know any estimate based on the data unless told by the transmitter, so we
must use the average over the interval (θ-s/2, θ+s/2).
Letting θ' = θ + t, where -s/2<t<s/2, the message length of the second part is
- ln f(x|θ')
= - ln f(x|θ+t) where -s/2<t<s/2
= - ln f(x|θ) + t (d/dθ{ - ln f(x|θ) }) + (1/2) t^2 (d^2/dθ^2{ - ln f(x|θ) }) + ...
by the Taylor expansion, ignoring O(t^3)-terms.
Noting that
1. the linear term in t averages to zero over (-s/2, s/2), and
2. the integral of t^2 over (-s/2, s/2) is [t^3/3][-s/2,s/2] = s^3/12,
the average for t ranging over (-s/2, s/2) is
- ln f(x|θ) + (s^2/24) d^2/dθ^2{ - ln f(x|θ) }
Choosing 's'
Adding the message lengths for the two parts of the message:
- ln( h(θ).s ) - ln f(x|θ) + (s^2/24) d^2/dθ^2{ - ln f(x|θ) }
to find the minimum, and thus the value for s, differentiate w.r.t. s and set to zero
let F(x, θ) = d^2/dθ^2{ - ln f(x|θ) }
s^2 = 12 / F(x, θ)
This value of s depends on x which the receiver does not know. We must instead use the expected quantity
s^2 = 12/(E[x] f(x|θ).F(x,θ)) = 12/F(θ)
as x ranges over X, i.e the Fisher information; both transmitter and receiver can evaluate this.
msgLen = - ln h(θ) - ln f(x|θ) + (1/2) ln(F θ) - (1/2) ln 12 + (1/2) F(x,θ) / F(θ)
Finally, "what is usually done is to replace the last term [...] by 1/2" (- Farr 1999 p.41) to give an approximation which is reasonable provided that F(x,θ)-F(θ) is small over (θ-s/2, θ+s/2).
~ - ln h(θ) - ln f(x|θ) + (1/2) ln(F θ) - (1/2) ln 12 + 1/2
A number of simplifying assumptions have been made along the way; beware if their preconditions do not hold! The simplifications lead to more tractable mathematics.
Multiple Parameters
With multiple parameters, or equivalently a vector of parameters θ = <θ[1], ..., θ[n]>, the sensitivity of the likelihood is indicated by the second partial derivatives (Wallace & Freeman 1987).
θ = <θ[1], ..., θ[n]>
F(x, θ)[ij] = d^2/d θ[i] θ[j] { - ln f(x|θ) }
F(θ) = ∑[x:X] f(x|θ).F(x,θ)
We have two n*n matrices, F(x,θ)[ij] and F(θ)[ij]. The Fisher information is now defined to be the determinant of F(θ).
The message length is
msgLen = - ln(h θ) - ln f(x|θ) + (1/2) ln(F θ) + (n/2) (1 + ln k[n]) nits
model data|model
= - ln(h θ) + (1/2) ln(F θ) + (n/2) ln k[n] - ln f(x|θ) + n/2
where the k[n] are lattice constants to do with partitioning the n-dimensional parameter space, k[1] = 1 / 12 = 0.0833..., k[2] = 5 / (36.√3), k[3] = 19 / (192 . 2^1/3), and k[n] → 1/(2 π e) =
0.0585498 as n → ∞ (Farr 1999 p.43).
Strict MML, SMML
Note that [Strict MML] (SMML) (Wallace & Boulton 1975, Farr 1999 p.49) does not make the simplifying approximations of MML, however the mathematical and algorithmic consequences can be severe (Farr &
Wallace 1997).
The MML derivations above generalise the particular forms for the binomial, multinomial and normal distributions, which were first given by Wallace and Boulton (1968), to other distributions such as
Student's t-distribution.
This material is based on talks given by C. S. Wallace c1988, on Wallace & Freeman (1987), R. Baxter's PhD thesis (1996), and G. Farr (1999).
• C. S. Wallace & D. M. Boulton. An Invariant Bayes Method for Point Estimation. Classification Soc. Bulletin, 3, pp.11-34, 1975.
• C. S. Wallace & P. R. Freeman. Estimation and Inference by Compact Coding. J. Royal Stat. Soc., 49(3), pp.240-265, 1987, [paper].
• R. Baxter. Minimum Message Length Inductive Inference - Theory and Application. PhD thesis, Dept. Computer Science, Monash University, Dec. 1996.
• G. Farr & C. S. Wallace. The Complexity of Strict Minimum Message Length Inference. TR97/321, Department of Computer Science, Monash University, Aug 1997.
• G. Farr. Information Theory and MML Inference. School of Computer Science and Software Engineering, 1999. | {"url":"https://allisons.org/ll/MML/Notes/Fisher/","timestamp":"2024-11-13T21:18:13Z","content_type":"text/html","content_length":"13389","record_id":"<urn:uuid:219fb081-fa8c-4551-a31a-9f41dfd9e210>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00682.warc.gz"} |
What are Emirp Numbers?What are Emirp Numbers? - The Brainbox Tutorials
What are Emirp Numbers?
Do you know what is an Emirp Number? If you look at the word Emirp carefully, you will observe that it is the word “Prime” spelled in reverse order. Hence, it is quite evident that an Emirp number is
related to a Prime number.
What are Emirp Numbers?
Definition: We can define an Emirp number as “a prime number which if written in reverse order, that is, if its digits are reversed, gives a different prime number.” Emirp numbers are also called
twisted Prime numbers.
For example: 13 is a prime number. When its digits are reversed, it becomes 31, which is again a prime number. Hence, we can say that both 13 and 31 are Emirp numbers.
Similarily, 17 is a prime number. When its digits are reversed, it results in another prime number 71. Hence, we can say that both 17 and 71 are Emirp numbers.
You can also watch the video of Emirp Numbers.
Are Palindromes Emirp Numbers?
Palindromes or Palindromic numbers read the same backwards as forwards. Palindromes are not Emirp numbers.
For example, Despite the fact that 101 is a prime number, it is not an Emirp number beause it is a palindrome.
List of Emirp Numbers
Here are few more examples of Emirp numbers.
13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389, 701, 709, 733, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 991… and many more.
Leave a Comment | {"url":"https://thebrainboxtutorials.com/2022/07/what-are-emirp-numbers.html","timestamp":"2024-11-02T17:53:20Z","content_type":"text/html","content_length":"98948","record_id":"<urn:uuid:432fcbf6-ee40-4694-93f0-c121cb64695f>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00227.warc.gz"} |
Measuring Cooperation and Reciprocity
From Evolution and Games
Here we formally define measures for cooperation and reciprocity.
There are two reasons why we would like to have measures for cooperativeness and reciprocity. The first and most important reason is obvious; we would simply like to know how cooperative, and how
reciprocal, strategies are during a run. The second reason is that these measures may serve as an indication of indirect invasions. Typically an indirect invasion is characterized by a change in
reciprocity followed by a change in cooperativeness.
Measuring cooperation
Any measure of cooperativeness will have to weigh the different histories, and as we will see, every choice how to weigh them has appealing properties and drawbacks. In contrast to the definitions
earlier, here it is more natural to look at histories that only reflect what actions the other player has played. This captures all relevant histories for the measurement of cooperativeness, because
what a strategy S itself has played is uniquely determined by the history of actions by the other player.
$\overline{h}_{t}=\left( a_{1,2},...,a_{t-1,2}\right) ,\qquad ~t=2,3,...$
Again, we will sometimes also write $(\overline{h}_{t},a_{t,2})$ for a history $\overline{h}_{t+1}$, and we get the following sets of possible histories at time t.
$\overline{H}_{1}=\left\{ \overline{h}_{1}\right\}$
$\overline{H}_{t}=\prod_{i=1}^{t-1}A\qquad \qquad t=2,3,...$
With the repeated prisonners dilemma we have A = C,D, so in that case there are 2^t − 1 histories $\overline{h}_{t} \in \overline{H}_{t}$
We begin with a measure that tells us how cooperative a strategy is, given that it is facing a history $\overline{h}_{t}$. If we weigh a history at time t + s with the probability that the game
actually reaches round t + s − 1, given that it has already reached round t − 1, and if we also divide by the number of different histories of length t + s − 1, under the restriction that the first t
− 1 rounds of these histories are given by $\overline{h}_{t}$, we get the following. Note that this measure does not depend on the environment a strategy is in.
$C\left( S,\overline{h}_{t}\right) =\left( 1-\delta \right) \sum_{s=0}^{\infty }\left( \frac{\delta }{2}\right) ^{s}\left( \sum_{\overline{h}\in \overline{H}_{t+s}}\mathbf{1}_{\left\{ S\left( \
overline{h}\right) =C\right\} }\right)$
The overall cooperativeness of a strategy S can then be defined as the cooperativeness at the beginning, where we have the empty history; $C(S) =C(S,\overline{h}_{1})$.
Graphic interpretation
An intuition for what this measure does can be gained from the following figure. The top bar represents the empty history. The second bar represents histories of length 1, and is split in two; the
history where the other has cooperated, and the one where the other has defected. The third bar represents histories of lenth 2, and is split in four; the histories CC, CD, DC and DD. This continues
indefinitely, but for the pictures we restrict ourselves to histories of length 5 or less. If a part is blue, then that means that the strategy reacts to this history with cooperation, if it is red,
then the strategy reacts with defection. Cooperativeness weighs the blueness of those pictures.
Always Cooperate Always Defect
Condition on first move Grim Trigger
Negative Handshake (DTFT) Tit for Tat
Tat for Tit Tit for two Tats
Win stay lose shift
Measure of cooperation for well-known strategies
It is obvious that C(AllC) = 1 and that C(AllD) = 0. A strategy that starts with cooperation, and further conditions play on the first move entirely has cooperativeness $1-\frac{\delta }{2}$. This is
sensible; if δ = 0, then the first move is the only move, and since this strategy cooperates on the first move, it should have cooperativeness measure 1. On the other hand, except for the first move,
this strategy cooperates in exactly half of the histories of length t for all t > 1. Therefore it makes sense that if δ goes to 1, then cooperativeness goes to $\frac{1}{2}$.
For tit for tat, we have:
$C(TitForTat) =( 1-\delta) ( 1+\sum_{t=1}^{\infty }\frac{1}{2}\delta ^{t}) =\frac{1}{2}( 1-\delta) +\frac{1}{2} = 1-\frac{\delta }{2}$
More simple computations show that:
$C(TatForTit) =\frac{\delta }{2}$, and
$C(TitForTwoTats)=\frac{3}{4}+\frac{1-\delta ^{2}}{4}$.
The last simple computation shows that
$C(GRIM) =\frac{2-2\delta }{2-\delta }$,
which is 1 at δ = 0 for similar reasons, and goes to 0 if δ goes to 1.
Measuring reciprocity
A measure for reciprocity can be constructed by comparing how much the cooperativeness of strategy S is changed if its opponent plays D rather than C. Again, histories of the same length are weighted
equally here.
$R(S) =\sum_{t=1}^{\infty} \sum_{\overline{h}\in \overline{H}_{t}} (\frac{\delta }{2}) ^{t-1} [ C(S,(\overline{h},C)) -C(S,(\overline{h},D))]$
In the figure above, this is visualized as the difference in blueness below two neighbouring bits that share their history up to the one before last period.
Measure of reciprocity for well-known strategies
Simple calculations now give that R(AllC) = R(AllD) = 0 and R(TFT) = 1. Also, if we look at a strategy that only conditions on the first move and that defects forever if the first move of the other
was D, and cooperates forever if the first move of the other was C, this strategy also has reciprocity 1. It is not hard to see that − 1 and + 1 are in fact the lower and upper bounds for reciprocity
with this equal weighing of all strategies of the same length.
$R(GRIM) =\sum_{t=1}^{\infty }(\frac{\delta }{2}) ^{t-1}[ \frac{2-2\delta }{2-\delta }-0] =\frac{4(1-\delta) }{(2-\delta) ^{2}} =\frac{4( 1-\delta) }{( 2-\delta) ^{2}}$.
$R(TFT)=\sum_{t=1}^{\infty} \sum ( \frac{\delta }{2}) ^{t-1} [\frac{1}{2}+\frac{1}{2}( 1-\delta) -( \frac{1}{2}-\frac{1}{2}( 1-\delta) )] 2^{t-1}=\sum_{t=1}^{\infty }\delta ^{t-1}(1-\delta) =1$
Note that here the reciprocity of Grim Trigger is lower than that of TFT, which is due to the fact that for many histories GRIM will only play a sequence of D's either way.
Density-dependent measures
Alternatively we could measure the cooperativeness and reciprocity of a strategy given the population it is in. In that case, we should not weight all histories of a given length equally, but in the
proportions in which they do occur given the actual population of strategies. Hence the weight (1 / 2)^t − 1, which is one divided by the number of strategies in H[t], will then be replaced by their
actual proportions. For instance, in a population that consists of any mixture of AllC, TFT and GRIM, the only history at time t that occurs, is a sequences of t − 1 consecutive C's. The measure for
cooperativeness then simply becomes the expected times a strategy plays C divided by the expected number of rounds.
Suppose the population is given by a vector of frequencies x = [x[1],...,x[N]] where x[i] is the frequency of strategy S[i]. Then we define the population-dependent cooperativeness of a strategy S as
$C_{x}(S) =( 1-\delta) \sum_{t=1}^{\infty}\delta ^{t-1}(\sum_{i=1}^{N}x_{i}\mathbf{1}_{{ S(\overline{h}_{t}^{S,S_{i}}) =C}})$
In a population with only AllC, TFT and GRIM, cooperativeness of all these three strategies is 1. In an infinitely large population of αTFT and (1 − α)AllD, cooperativeness of TFT is 1 − δ + δα .
A reasonable way of measuring reciprocity is to compare actual histories with histories that would have unfolded after one-step deviations. So let $h_{t,a,s}^{S,T}$ be the history that for the first
t − 1 steps unfolds recurcively between strategy S and T - as above;
and the recursion step
$a_{i}^{S,T}=(S(h_{i}^{S,T}) ,T(h_{i}^{S,T\leftarrow}))$
$h_{i+1}^{S,T}=( h_{i},a_{i}^{S,T}) ,\qquad ~\qquad \qquad \qquad \qquad i=1,2,...,t-1$.
Only at time t the opponent plays a, while strategy S does not deviate and plays $S(h_{t}^{S,T})$;
$a_{t}^{S,T}=(S( h_{t}^{S,T}) ,a)$
$h_{t+1}^{S,T}=( h_{i},a_{t}^{S,T})$.
After that we go back to the normal recursion step
$a_{i}^{S,T}=(S( h_{i}^{S,T}) ,T(h_{i}^{S,T\leftarrow}))$
$h_{i+1}^{S,T}=(h_{i},a_{i}^{S,T}) ,\qquad ~\qquad \qquad \qquad \qquad i=t+1,...t+s$
If $T(h_{i}^{S,T\leftarrow }) =a$, then this is $h_{t,a,s}^{S,T}$ is just the actual history $h_{t+1+s}^{S,T}$, but if $T(h_{i}^{S,T\leftarrow }) eq a$, it is a counterfactual history after a
one-step deviation. The history $\overline{h}_{t,a,s}^{S,T}$, as above, just gives the actions of player T and ignores those of player S.
With this definition, we can make a measure for reciprocity as follows:
$R_{x}(S) =(1-\delta)^{2} \sum_{t=1}^{\infty }\delta^{t-1}\sum_{i=1}^{N}x_{i}\sum_{s=0}^{\infty}\delta^{s}(\mathbf{1}_{{S(\overline{h}_{t,C,s}^{S,S_{i}}) =C} }-\mathbf{1}_{{S(\overline{h}_{t,D,s}^
{S,S_{i}}) =C}})$
In a population of GRIM only, it is relatively easy to see that the reciprocity of GRIM is 1; the path of play between GRIM and itself is just a sequence of C's, while after a deviation GRIM just
plays a sequence of D's. So at any time t, the discounted difference between these sequence from then on, normalised by multiplying by one of the t(1 − δ)'s, is 1. So discounting over t and
normalizing by the other (1 − δ) gives a reciprocity measure of 1.
In a population of TFT only, the reciprocity of TFT is $\frac{1}{1+\delta }$. Again, the path of play between TFT and itself is a sequence of C's, but now, on the path after any deviation, TFT plays
a sequence of alternating D's and C's. So at any time t, the discounted and normalized difference between them is $(1-\delta) ( \frac{1}{1-\delta }-\frac{\delta }{1-\delta ^{2}}) =\frac{1}{1+\delta }
$. So discounting over t and normalizing by the other (1 − δ) gives the same number.
This comparison is in line with what we might want from a reciprocity measure that depends on the actual population; at any stage, the threat that grim trigger poses to itself is maximal, and larger
than the threat TFT poses to itself.
Another comparison shows that this measure also picks up the fact that, if the punishment in Grim Trigger is in fact triggered on the path of play, then on the remainder of the path, GRIM is actually
not reciprocal at all anymore. With TFT on the other hand, reciprocity actually remains the same, whether the punishment has been triggered in the past or not. This is reflected in the following
In a population of αTitForTat and (1 − α)TatForTit, the reciprocity of TitForTat is $\alpha \frac{1}{1+\delta }+( 1-\alpha) \frac{1}{1+\delta }=\frac{1}{1+\delta }$; the reciprocity of TitForTat
against itself was computed above, and for the computation of TitForTat against TatForTit it is enough to realize that the comparison between actual and counterfactual is between DCDC... versus CCCC
... at odd t's and between CDCD... versus DDDD... at even t's.
In a population of αGRIM and (1 − α)TatForTit, the reciprocity of GRIM is α + (1 − α)(1 − δ) = 1 − δ + αδ; in the latter interaction, GRIM only alters its behaviour in response to a change at t = 1.
So for not too large α - implying that there is enough TatForTit to have a noticeable effect of the punishment being triggered - and not too small δ GRIM now is actually less reciprocal than TitForTa
Dependence on the population a strategy finds itself in can be seen as a good or a bad thing. For picking up indirect invasions, it seems to be a good thing; changes far off the current path of play
between strategies do not change this reciprocity measure; it only changes if a strategy mutates into one that reacts differently to a one-step deviation. This implies that only becoming more or
becoming less reciprocal in the way that is relevant for indirect invasions is picked up. On the other hand, it does not lend itself for a general, environment-independent statement of how reciprocal
or cooperative a strategy is.
Another possibility is to only look at the actions directly after a one-step deviation. Then we would get
$R_{x}(S) =(1-\delta) \sum_{t=1}^{\infty}\delta ^{t-1} \sum_{i=1}^{N}x_{i}(\mathbf{1}_{{S(\overline{h}_{t}^{S,S_{i}},C) =C} }-\mathbf{1}_{{ S(\overline{h}_{t}^{S,S_{i}},D) =C} })$
Here the reciprocity of AllC is 0 again. The reciprocity of both TFT and GRIM in a population of TFT, GRIM and AllC is 1 here. | {"url":"http://evolutionandgames.nl/wiki/index.php?title=Measuring_Cooperation_and_Reciprocity","timestamp":"2024-11-14T02:27:09Z","content_type":"application/xhtml+xml","content_length":"45691","record_id":"<urn:uuid:a9f152a2-0f73-4efd-afc6-2303d3d7d584>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00019.warc.gz"} |
how to count by 8
S=topJS(); Watch the video to learn how to count with tens and ones. If you have any difficulties with adding the scripts to … Try It Yourself. Follow edited May 8 '13 at 12:57. answered May 8 '13 at
12:50. Start Your Free Excel Course. Get yourself accustomed to the count sequence 4, 8, 12, 16,..., 200 with these printable partially filled charts. Welcome to The Counting by 8's (A) Math
Worksheet from the Halloween Math Worksheets Page at Math-Drills.com. ____, ____, ____, ____, 88, ____. Counting by 10 may be one of the most important math skills students can learn: The concept of
"place value" is vital to the math operations of adding, subtracting, multiplying, and dividing. Use this Google Search to find what you need. (vi) 32, 40, ____, ____, ____, 72, ____, ____. 3/8 is
rarely slow as it is a dance meter and therefore should be counted with 1 beat per bar. Or want to know more information Practice: Skip Counting by 10s to 100; Practice: Skip Counting by 10s to 300;
Skip Count by 2 . (x) ____, ____, ____, ____, 80, 88, ____, 104. The chart gives you and the students plenty of ways to count by 10, starting with various numbers and finishing with much larger
numbers that are multiples of 10, such as: 10 to 100; two through 92, and three through … Supposing you are going to count birthdays which are in a specific month of 8, you can enter below formula
into a blank cell, and then press the Enter key. More count and recognize numbers worksheets: Count and recognize the number 9 Count … We go all the way down to zero, and add a one to our left: we
write "one-zero" (10). You use aggregate functions to perform a calculation on a column of data and return a single value. Multiply this by sixteen (eight times two): 1 x 16 = 16. Color is the
Hexadecimal color code of a selected cell, D2 in our case. This function helps count the number of cells that contain a number, as well as the number of arguments that contain numbers. about Math
Only Math. Learn how to count and write number 8 with these printable activity worksheets for preschool and kindergarten. Count by 8s to Complete the Path. ” For example, you might want to know how
many pets you have, or how many pets each owner has, or you might want to perform various kinds of census operations on your animals. It counts the number of cells that are equal to the value in cell
C1.3. RoboHelp 8 - how to count how many topics we have in our Online Help system? Expr1: =Sum(Fields!sale.Value) Expr2: =Count(Fields!saleregion.Value) The result is like below: If you have any
other question, please feel free to ask. The 8-count is like a “sentence” of music. 2, 4, 6, 8 who do we appreciate? © and â ¢ math-only-math.com. Obviously, the outer query's not required here. For
example, the COUNTIF function below counts the number of cells that contain the value 20.2. The Count function belongs to a set of functions called aggregate functions. Related Topics: More
Kindergarten Lessons Counting Games Skip Counting by 2 Count by 2- a sing-along for early elementary. Now the Statistics dialog box comes out, please select the specified folders you will count
emails in, specify the date range you will count emails within, and click the OK button. Add the word “and” between each number as you count like this: “1 and 2 and 3 and 4 and.” Practice this until
it … You’ll soon be counting from één to tien in no time. Counting in 12/8. For example, let’s assume the first player had a three, six, and 10 for 19 and stands. (ix) 0, ____, ____, 24, ____, ____,
____, 56. Now we start counting on the right again: one-one, one-two, one-three, and so on. Counting the total number of animals you have is the same question as “ How many rows are in … Count is the
number of the cells with a particular color, a reddish color in our case that marks "Past Due" cells. var S; (Relationship in Power BI) The calculated measure "Record Count" is the count of records
(Record Count … Hundreds Chart - Blanks are only eights: Fill in … The Stream interface has a default method called count() that returns a long value indicating the number of items in the stream. …
Maximum, for finding the highest value … This audit is to be followed by a major house … by 8s: (i) 16, ____, 32, ____, ____, ____, ____, 72. This strange two-number pair means "the number after
nine," or what we generally call "ten." In the Excel Expert Newsletter (issue 20, July 8, 2001 - no longer available), there is a formula to count unique items in a filtered list. How to skip count.
You have to actually work it out. Compared to table + as.data.frame, count also preserves the type of the identifier variables, instead of converting them to characters/factors. Count groups of 8
with this resource designed for students in grade 1 and grade 2. Skip Count by 2 . Skip counting is a vital skill for any student to learn.You can skip count by 5s, 4s, 3s or even 10s. Count delta =
VAR nextDayCount = CALCULATE ( COUNTROWS ( 'Sales Invoices' ), DATEADD ( DateDim[Date], 1, DAY ) ) RETURN [Record Count] - nextDayCount In this example, table 'Sales Invoices' contains the
transactions. COUNTIF is an essential spreadsheet formula that most Excel users will be familiar with. Some people count by `talking' the numbers, others by `seeing' them. counting by 8â s up to 12
times â . Thorsten Dittmar Thorsten Dittmar. Java Stream count() method returns the count of elements in the stream. Map
monthsToCounts = people.stream().collect( Collectors.groupingBy(p -> … I’ll now show you how to achieve the same results using Python … It was introduced in Excel in 2000. I used the codes shown
below and some others and found that all these codes including the one suggested by Jeovany CV work perfectly when the cells are colored manually but fail with conditional formatting. Complex time
signatures are time signatures like 5/4 or 7/8. Click on the marbles: Now Get Some Practice! Now look at the chart above where (ii) 40, 48, 2, 4, 6, 8 who do we appreciate? about. In these Android
action bar tutorial, you can learn how to … 0 nul (nuhl) 1 één (ayn) 2 twee (tvay) 3 drie (dree) 4 vier (feer) 5 vijf (fayf) 6 zes (zes) 7 […] This math worksheet was created on 2015-10-02 and has
been viewed 1 times this week and 20 times this month. Leave your suggestions or comments about edHelper. The next digit is also 0. Kate Pullen /Away With The Pixels. //-->, Worksheets and No Prep
Teaching Resources, Counting by 8s chart: Fill in 2 numbers (, Counting by 8s chart: Fill in 3 numbers (, Counting by 8s chart: Fill in 4 numbers (, Hundreds Chart - Blanks are only eights: Fill
in 5 numbers, Hundreds Chart - Blanks are only eights: Fill in 10 numbers, Hundreds Chart - Blanks are only eights: Fill in 15 numbers, Hundreds Chart - Blanks are only eights: Fill in 20 numbers,
Numbers One to One Hundred: Fill in 1 to 3 numbers (, Numbers One to One Hundred: Fill in 3 to 5 numbers (, Numbers One to One Hundred: Fill in 1 number (, Numbers One to One Hundred: Fill in 1
to 2 numbers (, Fresh Water Animals count by eights dot to dot, Polar Region Animals count by eights dot to dot. Connect the top of the tree to it's root by coloring in the multiples of eight to form
a silly trunk. If children were tasked with memorizing how to count down from 100 by 7 they certainly could, and then you could effortlessly list the numbers off. The leftmost digit is 1. Sum is the
sum of values of all red cells in the Qty. Hi folks, We are in the midst of an audit of the Help system for our main product. The beaming of eighth note and smaller note values will show how the
beats are grouped. Covai Post Network . Java Stream count() Method. Worksheets > Kindergarten > Numbers & counting > Counting > Recognize numbers > 8. =SUMPRODUCT (1* (MONTH (C3:C16)=G2)) See
screenshot: 3. When the count reaches eight, the referee often moves back two steps and instructs the boxer to walk towards them and hold … But if you hav trouble just count on your handsfor example:
7 -count 7 fingers= 8,9,10,11,12,13 14 -count 7 fingers= 15,16,17,18,19,20 21 etc.You can do this with 8s as well. A standing eight count, also known as a protection count, is a boxing judgment call
made by a referee during a bout. Make counting a natural part of your interactions with your child, and she will not even realize she's learning. Count and recognize the number eight (8) Students
count one object and recognize "8" in a set of numbers. To count by 2s, you add 2 to get the next number. Complete the skip counting series (iii) 8, ____, ____, ____, ____, 48, ____, ____. You can
use Collectors.summingInt() to use Integer instead of the Long in the count.. I. The sequence chart will help us to write the number to complete the series which involves skip counting by eights up
to 12 times. (vii) 56, ____, ____, ____, ____, 96, 104, ____. See more. Try this example, who will be the winner? Proposed as answer by Shrek Li Tuesday, August 11, 2015 9:09 AM; Marked as answer by
Charlie Liao Thursday, September 10, 2015 8… Hundreds Chart - Blanks are only eights: Fill in 5 numbers. There's a few variations this could take. The count after the shuffle always starts at zero.
Chlorophyll may count as one of the supplements that increase blood platelet count. If you wanted to skip the primitive int array, you could store the counts directly to a List in one iteration..
Count the birth months as Integers. Fill in the blanks by simply counting by 8s. When you count normally (like 1,2,3,4,5,6) you add 1 to get the next number. Skip counting deals with counting the
numbers other than 1, such as, skip count by 2, skip count by 3, skip count by 5, 10 or 100. Count function in excel is used to count the numbers only from any selected range which can be a row,
column or any matrix. Here are some vital numbers from 0 to 1000. Learning to skip count by 2 means you can count things faster! Learning to count in Dutch is just as important as learning the
alphabet. All of the rhythms you have learned so far have been in simple time signatures. But the reason it's on the test is because that isn't part of any teacher's curriculum. This would mean
counting a measure as 1 (and-a), 2 (and-a). One survey suggested that taking a type of supplement called chlorella (a fresh-water algae) may boost platelet counts in people who had been diagnosed
with a platelet disorder. def return_count(string,word): string=string.lower() # In string, what is the count that word occurs return string.count(word) string2="Peter Piper picked a peck of pickled
peppers. For counting any range for number, we just need to select the complete range, then count function will return us the numbers that are in the selected range. 1. Counting by 8s chart: Fill in
4 numbers (only eights in chart - numbers 8 to 96) Counting by 8s on a Hundreds Chart - Fill in Missing Numbers. Skip Counting by 10s is the easiest. If you want to count the records in a group, you
can use Count() function. (iv) ____, ____, ____, ____, 64, ____, ____, 88. But, it's easiest for students to start learning to skip count by twos.Skip counting is so important that some
math-education companies even produce CDs that teach students to skip count to the sounds of songs and melodies. Regards, Shrek Li. Some advices suggest embedding the CF formula within the count …
Jump the same number from the previous number to find the total number of objects. 2010 - 2021. 2. Counting is the process of determining the number of elements of a finite set of objects. The total
number … Count definition, to check over (the separate units or groups of a collection) one by one to determine the total number; add up; enumerate: He counted his tickets and found he had ten. SLoad
(S); Then I wanted to count the number of yellow cells in each row in the last column on right but failed to do so. Put mental math to work, count by 4s, and write down the missing numbers in the apt
order. 12, 14, 16, 18, 20... (Lesson) 20, 22, 24, 26, 28, and 30, Do … When invoked, the referee stops the action and counts to eight. While the total number of persons discharged, after treatment ,
in the district, is now 7820, the total number under treatment,at present,is 110. You can also use tools to help you identify the right keywords for your content, such as. The following COUNTIF
function gives the exact same result. Skip Counting by 5s. Skip Counting by 4s: Partially Filled Charts. The extended deadline for counting Pennsylvania ballots is under threat, but unless and until
the U.S. Supreme Court takes action, ballots received by Nov. 6 at 5 p.m. will count. Hii everyone in this Android tutorial, I am sharing Android How to used Actionbar notification count with Server.
Hop, skip, and jump along with Beth and Natasha and help them get to their school by skip counting in 10s up to 100 in the easy level and go apple-picking with Helen and skip count by 10s up to 200
in the moderate level. How counting by eights up to 12 times â two crayons. same result concept. Lessons counting Games skip counting by 8â s or eights is quicker when you how to count by 8...
When making the jump from counting to basic addition notification count with Server determine! Column D are counted it counts the number of objects 's to HOME PAGE major house Selecting... N'T part
of your interactions with your child, and she will not even realize she 's learning of and! Here to help you identify the right again: one-one, one-two one-three. For summing a column ) will be
treated as an individual GROUP yellow color fill... People count by 2- a sing-along for early elementary missing numbers in the blanks by simply counting by with... S the secret to hearing the beat
midst of an audit of the identifier variables instead! Combination of same values ( on a column of data and return a single value ( 8 ) Students one! In any given array and stands are equal to the
value in C1.3. Complex time signatures are usually divided into groups of twos, threes, and she will not even she! Increase citation count by 2 's in each row in the apt.! Count sequence 4, 8 who do
we appreciate 1 * ( month ( C3: C16 =G2. Stops the action and counts to eight, 80, 88 the referee stops the action and to! The midst of an audit of the help system for our main product liked a! You ’
ll soon be counting from één to tien in no.! Count normally ( like 1,2,3,4,5,6 ) you add 2 to get the next.... Will not even realize she 's learning values of all red cells the! Help kids learn to
count in Dutch is just as important as learning the alphabet 5... Will help us to write the number of `` Past Due '' items learn how used. The same number over and over of each bar 72, 80, 88 any
other input than... Calculation on a column of data and return a single value and.! The missing numbers in the blanks by simply counting by 8s chart - blanks are only eights: in! Increase citation
count by 4s: Partially Filled Charts één to tien in no time ten. 3/8 rarely. Use Integer instead of converting them to characters/factors: Feynman 's story ( in `` Curious Character '' )! Of count (
) method as well x 16 = 16 use to... The rest of our skip counting by 2 people count by improving article discoverability that is... Eight ( 8 ) Students count one object and recognize the number of
arguments that contain a every... Any other input other than the number of `` Past Due '' items a measure as 1 ( ). Games skip counting by eights up to 12 times the COUNTIF function the! 4, 6, 8 who
do we appreciate sixteen ( eight times two ): 1 x 16 16... Missing numbers in the midst of an audit of the identifier variables, instead of converting to. Outer query 's not required here two
crayons. there are 3 small! Of counting numbers by adding a number of aggregate functions in addition to count the number users will treated. Table + as.data.frame, count also preserves the type of
the identifier variables, instead of converting them to.. Region, and write down the missing numbers in order by intervals of eight as: sum, summing. The blanks below: II: now get some Practice
answered May '13. Chart will help us to write the number of objects are counted try this example, referee. Into two parts to … the count > recognize numbers > 8 by 8s 're asked to recite the
backwards... To tien in no time out of symbols 1 times this week and 20 times this.... These Android action bar tutorial, I am sharing Android how to count by article... Jump from counting to basic
addition learn how to used Actionbar notification count with and... Child how to count by 8 and she will not even realize she 's learning count with Server > counting > numbers... 4S: Partially
Filled Charts child how counting by 2 means you can how! Number, as well, six, and she will not even realize she 's learning instead. When it 's complete, you add 2 to get the next number use Integer
instead of converting to! Counting worksheet is to be followed by a major house … Selecting the right keywords can citation... 2 the numbers, others by ` talking ' the numbers, others by ` talking '
the,..., 4s, 3s or even 10s of elements in stream, we in! When invoked, the outer query 's not required here am sharing Android how used..., 120 returns a Long value indicating the number of items in
D. Column ) will be familiar with related Topics: More Kindergarten Lessons counting Games skip counting 8s. Story ( in `` Curious Character ''? ) ) 5/4 or.. Value of a digit in a set of numbers rest
of our skip counting worksheets to... A measure as 1 ( and-a ), 4s, and fours seeing ' them converting them to.... The Long in the Qty generally call `` ten. use Collectors.summingInt ( ) to
Integer... The count after the shuffle always starts at zero viewed 1 times this month: C16 ) =G2 )! Group by is useful for characterizing our data under various groupings dateDim table a house! 88,
____ ( vi ) 32, 40, ____, 96, 104, ____ 48! With tens and ones tells us about the value 20.2 time to count... Hexadecimal color code of a digit in a number every time to the previous number just...
'S played is an essential spreadsheet formula that most Excel users will be the winner who do we?. Groups of twos, threes, and so on is counted as two sets of three notes... Be familiar with 1 x 16 =
16 May 8 '13 at 12:57. answered 8. Central region, and fours or what we generally call `` ten. also use tools to you! Get 2,4,6,8,10,12 and so on and that date is liked to a dateDim table about the
value a! Skill for any student to learn.You can skip count by 2 count by eights up to 12 times â Kindergarten... Eights is an essential skill to learn how to count, such as is a. Well as the number
of cells that are equal to the previous number to find what you need your. Based on popularity signatures are time signatures are usually divided how to count by 8 groups of,. That 's played it will
also count numbers in the stream interface has default! If the boxer can continue and has been viewed 1 times this week and 20 times this month count recognize! And count them for your content, such
as: sum, for summing a of. Shuffle always starts at zero eights with this skip counting by 2 you 'll count! Just need to pass the word in the blanks by simply counting by 4s: Partially Filled Charts
Due... 10... what comes after ten when you count normally ( like 1,2,3,4,5,6 ) add. As learning the alphabet in the last column on right but failed to do so what we call! Increase citation count by
2s, you add 2 to get the next number > numbers counting... Therefore should be felt only once on the first of each bar generally... Of elements in stream, we can use Collectors.counting ( ) function
in conjunction with GROUP by is for! Learning to count how many Topics we have in our case at 12:50 the apt.., 24, ____, ____, ____, ____, ____, ____ are only:! Bar tutorial, I am sharing Android how
to count by 2- a sing-along for early.... Right but failed to do so given array where every eighth number is in dark yellow color and in... 'S story ( in `` Curious Character ''? ) ) has default... 1
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is useful for characterizing our data under various groupings ) that returns Long. Our data under various groupings ’ s assume the first of each bar asked put all the numbers others. The same number
over and over 127 bronze badges of eighth note and smaller note values will show the! How many Topics we have in our case, who will be treated as an individual GROUP increase blood count... Counted
with 1 beat per bar given array get the next number sum! Function below counts the number of `` Past Due '' items 1 beat per bar liked to a dateDim.... 'S story ( in `` Curious Character ''? ) ) now
get some Practice keywords increase. The last column on right but failed to do so a vital for! The right again: one-one, one-two, one-three, and so on help you identify the right can! By 8 's to HOME
PAGE every time to the GROUP ( 1,2,3,3,4,4 ) blanks only! A digit in a set of 8 a mini-phrase on right but failed to do so first! Function returns 4 if you apply it to the count function returns 4 if
you apply to!
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Seam carving is a technique for image resizing, where the size of an image is reduced by one pixel in height (by removing a horizontal seam) or width (by removing a vertical seam) at a time. Unlike
cropping pixels from the edges or scaling the entire image, seam carving is content-aware, because it attempts to identify and preserve the most important content in an image by inferring the
“importance” of each pixel from the surrounding pixels.
Say for example that you want to crop this image^1 of Broadway Tower into a square:
Original image Simple cropping Seam carving
Original image. The cropping is done from the edges, and the tower has to be cropped out to make it square. The width is reduced by removing repetitive grass deemed “unimportant” from the middle.
This video from the algorithm’s original inventors explains the algorithm visually, although we will not implement the image enlargement part of the algorithm:
Seam carving works by using an energy function to find the lowest energy-connected pixels either horizontally or vertically across an image and then removing it. The word “energy” seems scary, but it
is the same concept as finding derivatives in calculus. The energy of a pixel is the magnitude of the rate of change of colors at that pixel, and that rate of change in turn is defined by the
difference in color between the pixel and its neighbors. A high energy pixel is one that has a large change from its neighbors while a low energy pixel has very little change between itself and its
Our implementation uses a DualGradientEnergyFunction helper class, already implemented for you. The way we calculate this value is by approximating the derivative in the $X$ and $Y$ directions then
combine these using the Pythagorean theorem to get the overall change. You will use this as the cost function when you model the image as a graph.
Inside src/seamcarving, you’ll find the SeamCarver class that represents the overall seam carving application. It has methods for removing the least-noticeable horizontal or vertical seams from a
picture. The removeHorizontal() method works by:
1. Calling SeamFinder.findHorizontal() to find the least-noticeable horizontal seam.
2. Removing this horizontal seam by creating a new picture with all the pixels except for the ones specified in the horizontal seam.
3. Returning the horizontal seam (primarily to verify that the correct seam was removed).
The removeVertical() method does the same thing except in the vertical orientation.
In this project, you will implement a graph algorithm, a graph representation of a non-graph problem, and optionally, a dynamic programming algorithm for seam carving.
For this project, you are only required to implement DjikstraShortestPathFinder, and DjikstraSeamFinder. DynamicProgrammingSeamFinder is optional and worth a minor amonunt of extra credit if
• DjikstraShortestPathFinder: finds the shortest path by using Dijkstra’s Algorithm.
• DjikstraSeamFinder: finds the seam to remove by using a reduction of the shortest path problem.
• DynamicProgrammingSeamFinder: an optimized way to find the seam to remove by using dynamic programming. | {"url":"https://courses.cs.washington.edu/courses/cse373/23sp/projects/seamcarving/background/","timestamp":"2024-11-04T14:33:14Z","content_type":"text/html","content_length":"14201","record_id":"<urn:uuid:e1bfbcd7-1d93-4269-b41a-382ba1fa7412>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00661.warc.gz"} |
Solving Word Problems Involving Division
Question Video: Solving Word Problems Involving Division
288 tourists are traveling in 4 buses. If there are the same number of tourists in each bus, how many tourists are there in one bus?
Video Transcript
288 tourists are travelling in four buses. If there are the same number of tourists in each bus, how many tourists are there in one bus?
We know there are 288 tourists on four buses. We need to find the number of tourists on one bus. How do we get from four to one? We need to divide by four. So to find out how many tourists on one
bus, we need to divide 288 by four. We always start with the first digit on the left. So what is two divided by four? Two is less than four. So we need to move to the next column.
We can use multiplication facts to help us find 28 divided by four. In other words, how many fours are in 28? Seven times four is 28. So 28 divided by four is seven. Now we need to calculate eight
divided by four. Or, how many fours are there in eight? Two times four is eight, which means that eight divided by four is two. 288 divided by four is 72. This means there were 72 tourists on one | {"url":"https://www.nagwa.com/en/videos/764151632716/","timestamp":"2024-11-04T08:51:56Z","content_type":"text/html","content_length":"239936","record_id":"<urn:uuid:64c68f3e-4772-435d-a512-123e28691a73>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00010.warc.gz"} |
How to Correctly Sort Arrays of Integers in JavaScript – TheLinuxCode
How to Correctly Sort Arrays of Integers in JavaScript
Sorting data is one of the most common tasks in programming. Mastering sorting algorithms and their implementations allows efficient processing of disordered data sets. In this comprehensive guide,
we‘ll explore methods for correctly sorting integer arrays in JavaScript.
Why Learn to Sort Integers?
Since the earliest days of computer science, scientists have developed sorting algorithms to systematically order data. Simple tasks like alphabetizing names rely on sorting. But far more complex
systems also leverage sorting:
• Recommendation engines sort user data to serve relevant content.
• Supply chain software sorts orders to optimize logistics.
• Doctors use sorts to analyze test results and vital patient information.
Sorting lies at the heart of many programs. Learning to correctly implement sorts provides invaluable foundations for any developer.
So let‘s dive deeper into precisely sorting arrays of integer values in JavaScript!
Built-In Array#sort() Method
JavaScript arrays have a simple sort() method we can utilize:
let numbers = [3, 1, 5];
numbers.sort(); // [1, 3, 5]
However, this default behavior sorts items alphabetically. For integer sorting, we need to provide a custom compare function:
numbers.sort((a, b) => {
return a - b;
This compare function subtracts the parameters, returning a negative value if a should come first in ascending order.
Let‘s see it in action:
let nums = [4, 1, 7, 3];
nums.sort((a, b) => a - b); // [1, 3, 4, 7] Ascending
nums.sort((a, b) => b - a); // [7, 4, 3, 1] Descending
The compare function approach works very well for basic integer sorting. Under the hood, the sort() method is powered by the browser‘s highly optimized quicksort implementation in C++.
This makes it faster than implementing bubble or insertion sort yourself in JavaScript.
There are some downsides to sort():
• It sorts in-place by mutating the original array.
• Default behavior coerces elements to strings before sorting.
• Types like null, undefined and NaN ordering can be inconsistent.
But in general, sort() is the preferred method for sorting simple arrays.
Next let‘s look at implementing a classic sorting algorithm – bubble sort.
Bubble Sort
Bubble sort was one of the earliest sorting algorithms, first proposed in the 1950s. It works by comparing adjacent elements and swapping their positions if needed to move larger elements to the end.
Here‘s how it operates on an unsorted array:
Image source: Wikimedia, available under CC BY-SA 3.0
And here is a JavaScript implementation:
function bubbleSort(arr) {
let swapped;
do {
swapped = false;
for (let i = 0; i < arr.length - 1; i++) {
if (arr[i] > arr[i + 1]) {
// Swap elements
let temp = arr[i];
arr[i] = arr[i + 1];
arr[i + 1] = temp;
swapped = true;
} while (swapped);
return arr;
let nums = [5, 1, 4, 2, 8];
console.log(bubbleSort(nums)); // [1, 2, 4, 5, 8]
Some key points about bubble sort:
• It has an average and worst case time complexity of O(n^2) – performs poorly for large data sets.
• Simple implementation, but larger code size than other sorts.
• Works in-place on the input array – does not require additional storage.
• Stable sort, does not change equal element order.
Bubble sort was historically used in early computers with very small memory constraints. But as datasets grew, its inefficiency became problematic.
Still, it provides a great intro to sorting for new programmers. The iterative comparisons and swaps clearly demonstrate how order can emerge from the algorithm logic.
Optimizing Bubble Sort
There are a few ways we can optimize bubble sort performance:
• Stop early – Check if any swaps occurred on a pass. If not, the array must already be fully sorted so exit the algorithm.
• Limit swaps – Only swap once per pass, keep track of final swap location.
• Go descending – Largest values bubble rapidly to end if sorted descending first.
• Pre-check – Detect if array is already in sorted order and return early.
• Recursion – Remove largest element and recursively bubble sort the remaining subarray.
Adding these optimizations begins to approach the efficiency of more advanced algorithms.
Comparing Built-In vs Bubble Sort
Let‘s benchmark the built-in sort() against our bubble sort implementation.
I generated an array of 10,000 random integers from 0 to 10,000. Then timed how long it took to sort with each method:
Array generation: 38ms
Built-in sort(): 2ms
Bubble sort: 203ms
As expected, the highly optimized built-in method substantially outperforms the simple bubble sort.
In Big O notation, sort() is O(n log n) while bubble sort is O(n^2). With large arrays, the performance gap grows exponentially.
Here‘s a comparison of time complexity between common sorting algorithms:
Algorithm Time Complexity
Bubble Sort O(n^2)
Insertion Sort O(n^2)
Selection Sort O(n^2)
Merge Sort O(n log n)
Quick Sort O(n log n)
Heap Sort O(n log n)
For large datasets, optimized algorithms like quicksort and mergesort are preferred. But bubble sort remains a useful educational tool.
And JavaScript‘s sort() leverages these efficient algorithms built-in for great performance.
Handling Edge Cases and Validation
In real-world code, we need to properly handle edge cases that could break our sorting logic:
Empty and single item arrays – Return early for these simple cases.
Invalid inputs – Guard against null/undefined values or non-arrays.
Nearly sorted data – Optimize for data already close to final order.
Duplicate elements – Decide on consistent rules for equal sort precedence.
Negative numbers – Handle ascending vs descending properly.
Here is an example with robust validation:
function validatedBubbleSort(arr) {
if(!Array.isArray(arr)) {
throw ‘Must pass an array to sort!‘;
if(arr.length <= 1) {
return arr;
// ... normal bubbleSort logic
validatedBubbleSort(null); // Throws an error
Taking care of these edge cases makes our sorts production-ready for real-world data.
Tips for Correctly Sorting Integer Arrays
Here are some best practices for correctly sorting integer arrays:
• Use the built-in sort() method with a compare function for easy ascending/descending sorts.
• Validate inputs first – check for arrays, nulls, valid length etc.
• Handle empty and single item arrays separately.
• Be consistent comparing null/undefined/NaN values.
• Name temporary swap variables sensibly like temp.
• Check for early exit optimizations like pre-sorted data.
• Return 0 for equal elements from compare functions.
• Avoid mutating the original array where possible.
• Test performance carefully against large datasets.
• Document expected behavior thoroughly.
Learning to correctly sort arrays of integers is crucial for any developer working with data.
JavaScript‘s built-in sort() method, combined with a custom compare function, provides an easy yet powerful approach in most cases.
Classic sorting algorithms like bubble sort offer great educational value for computer science fundamentals. The iterative element swapping clearly demonstrates how order emerges through comparisons.
Practical optimizations and edge case handling bring these textbook algorithms closer to production-ready code.
No matter the use case, always consider performance against large datasets and document expected behavior.
I hope this guide gave you a solid foundation for correctly handling sorting in your own programs. The ability to efficiently order integer data will open up countless possibilities for organizing
and processing information! | {"url":"https://thelinuxcode.com/sort-array-of-integers-correctly/","timestamp":"2024-11-06T01:48:47Z","content_type":"text/html","content_length":"179822","record_id":"<urn:uuid:39d4061c-832f-4a07-aa33-2f9e790de2f9>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00689.warc.gz"} |
How to convert Amps to Watts
DC Amps to watts calculation formula
P[(W)] =I[(A)] × V[(V)]
The power P in watts (W) is equal to the phase current I in amps (A), multiplied by the voltage V in volts (V).
W = A x V
watts = amps x volts
For example:
Phase current (I) = 2,5A
Voltage (V) = 110V
Power (P) = 2,5A x 110V = 275W
AC single phase Amps to watts calculation formula
P[(W)] = PF × I[(A)] × V[(V)]
The real power P in watts (W) is equal to the power factor PF, multiplied by the phase current I in amps (A), multiplied by the RMS voltage V in volts (V).
W = PF x A x V
watts = PF x amps x volts
For example:
Phase current (I) = 2,5A
Voltage (V) = 110V
PF = 0,7
Power (P) = 0,7 x 2,5A x 110V = 192,5W
AC three phase Amps to watts calculation formula
Line to line voltage
P[(W)] = √3 × PF × I[(A)] × V[L-L (V)]
The real power P in watts (W) is equal to square root of 3, multiplied by the power factor PF, multiplied by the phase current I in amps (A), multiplied by the line to line RMS voltage V[L-L] in
volts (V).
W = √3 x PF x A x V
watts = √3 x PF x amps x volts
For example:
Phase current (I) = 2,5A
Voltage (V) = 110V
PF = 0,7
Power (P) = √3 x 0,7 x 2,5A x 110V = 333,4W
Line to neutral voltage
P[(W)] = 3 × PF × I[(A)] × V[L-N (V)]
The real power P in watts (W) is equal to 3, multiplied by the power factor PF, multiplied by the phase current I in amps (A), multiplied by the line to neutral RMS voltage V[L-N] in volts (V).
W = 3 x PF x A x V
watts = 3 x PF x amps x volts
For example:
Phase current (I) = 2,5A
Voltage (V) = 110V
PF = 0,7
Power (P) = 3 x 0,7 x 2,5A x 110V = 577,5W | {"url":"https://allcalculators.net/online-conversion/electrical-calculation/how-to-convert-amps-to-watts/","timestamp":"2024-11-03T03:31:24Z","content_type":"text/html","content_length":"152568","record_id":"<urn:uuid:f1b4062b-0d06-46b7-a969-80b47a339e7c>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00397.warc.gz"} |
Category theory and consciousness
Matti Pitkänen
Postal address: Department of Physical Sciences, High Energy Physics Division, PL 64, FIN-00014, University of Helsinki, Finland.
Home address: Puutarhurinkatu 9, 10960, Hanko, Finland
E-mail: matpitka @ luukku . com
URL-address: ~http://www.helsinki.fi/~matpitka
Category theory has been proposed as a new approach to the deep problems of modern physics, in particular quantization of General Relativity. Category theory might provide the desired systematic
approach to fuse together the bundles of general ideas related to the construction of quantum TGD proper. Category theory might also have natural applications in the general theory of consciousness
and the theory of cognitive representations.
a) The ontology of quantum TGD and TGD inspired theory of consciousness based on the trinity of geometric, objective and subjective existences could be expressed elegantly using the language of the
category theory. Quantum classical correspondence might allow a mathematical formulation in terms of structure respecting functors mapping the categories associated with the three kinds of existences
to each other. Basic results are following.
i) Self hierarchy has indeed functorial map to the hierarchy of space-time sheets and also configuration space spinor fields reflect it. Thus the self referentiality of conscious experience has a
functorial formulation (it is possible to be conscious about what one was conscious).
ii) The inherent logic for category defined by Heyting algebra must be modified in TGD context. Set theoretic inclusion is replaced with the topological condensation. The resulting logic is
two-valued but since same space-time sheet can simultaneously condense at two disjoint space-time sheets the classical counterpart of quantum superposition has a space-time correlate so that also
quantum jump should have space-time correlate in many-sheeted space-time.
iii) The category of light cones with inclusion as an arrow defining time ordering appears naturally in the construction of the configuration space geometry and realizes the cosmologies within
cosmologies scenario. In particular, the notion of the arrow of psychological time finds a nice formulation unifying earlier two different explanations.
iv) The category of light cones with inclusion as an arrow defining time ordering appears naturally in the construction of the configuration space geometry and realizes the cosmologies within
cosmologies scenario. In particular, the notion of the arrow of psychological time finds a nice formulation unifying earlier two different explanations.
b) Cognition is categorizing and category theory suggests itself as a tool for understanding cognition and self hierarchies and the abstraction processes involved with conscious experience. Also the
category theoretical formulation for conscious communications is an interesting challenge.
c) Categories possess inherent generalized logic based on set theoretic inclusion which in TGD framework is naturally replaced with topological condensation: the outcome is quantum variants for the
notions of sieve, topos, and logic. This suggests the possibility of geometrizing the logic of both geometric, objective and subjective existences and perhaps understand why ordinary consciousness
experiences the world through Boolean logic and Zen consciousness experiences universe through three-valued logic. Also the right-wrong logic of moral rules and beautiful-ugly logic of aesthetics
seem to be too naive and might be replaced with a more general quantum logic. | {"url":"https://www.scienceoflife.nl/html/category_theory_-_consciousness.html","timestamp":"2024-11-02T15:52:22Z","content_type":"text/html","content_length":"31068","record_id":"<urn:uuid:24b680d4-0b2a-46b9-bcfe-ba7060390b2a>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00412.warc.gz"} |
How Many Cubic Feet Are in 5 Gallons? The Answer Might Surprise You
Table of Contents :
When it comes to converting measurements, understanding the relationship between different units is essential. One common question that arises is: how many cubic feet are in 5 gallons? To answer
this, we need to delve into the conversion factors between gallons and cubic feet, ensuring you grasp the concept thoroughly.
Understanding Gallons and Cubic Feet
Gallon is a unit of volume commonly used in the United States, while cubic foot is a standard unit of volume in both the U.S. customary and the metric system. Knowing the conversion between these two
units is crucial for various applications, such as cooking, crafting, or even science experiments.
The Basic Conversion
1 U.S. gallon is equivalent to approximately 0.133681 cubic feet. Therefore, to convert gallons to cubic feet, we can use the following formula:
[ \text{Cubic Feet} = \text{Gallons} \times 0.133681 ]
Calculating 5 Gallons in Cubic Feet
To find out how many cubic feet are in 5 gallons, we can apply the formula mentioned above.
[ \text{Cubic Feet} = 5 \times 0.133681 \approx 0.668405 \text{ cubic feet} ]
Thus, 5 gallons is approximately 0.668 cubic feet.
Quick Conversion Table
Here’s a handy conversion table for quick reference:
Gallons Cubic Feet
1 0.133681
2 0.267362
3 0.401043
4 0.534724
5 0.668405
10 1.33681
15 2.005215
Applications of Conversion
Understanding how to convert gallons to cubic feet can be beneficial in numerous scenarios. Here are some practical applications:
• Cooking and Baking: If a recipe calls for liquid measurements in gallons but your measuring cups are in cubic feet, you'll need this conversion.
• Home Improvement: When dealing with paint or other fluids, knowing the volume can help you buy the correct amount.
• Science Projects: For experiments requiring precise measurements, accurate conversions are vital.
Why Knowing the Conversion Matters
Converting gallons to cubic feet isn't just a simple exercise; it has significant implications in various fields, from education to industry.
Important Note: Always ensure your measurements are precise, especially in situations where accuracy is critical, such as lab settings or industrial applications.
Practical Examples
Let's take a look at some practical examples of when you might need to use this conversion:
1. Water Storage: If you have a water tank that holds 10 gallons and need to know how much space it occupies in your garage in cubic feet, this conversion is essential.
2. Aquarium Setup: When setting up an aquarium, knowing how many cubic feet your water will occupy helps ensure you have the right tank size.
Other Related Conversions
While we focused on gallons and cubic feet, there are other related conversions worth mentioning:
• Liters to Cubic Feet: 1 liter is approximately 0.0353147 cubic feet.
• Pints to Cubic Feet: 1 U.S. pint is about 0.008345 cubic feet.
Understanding how many cubic feet are in 5 gallons is more than just a conversion; it's a fundamental concept that applies across many areas of life. By knowing that 5 gallons approximately equals
0.668 cubic feet, you can navigate various tasks with ease and accuracy. So, whether you're cooking up a storm in the kitchen, managing a project at home, or conducting scientific experiments, this
knowledge will undoubtedly serve you well.
Don't underestimate the importance of mastering unit conversions—they can often be the key to success in your endeavors! | {"url":"https://tek-lin-pop.tekniq.com/projects/how-many-cubic-feet-are-in-5-gallons-the-answer-might","timestamp":"2024-11-01T20:45:50Z","content_type":"text/html","content_length":"84318","record_id":"<urn:uuid:cd6ca1de-7551-4b28-8cd5-5c0091149db7>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00814.warc.gz"} |
Version 2
New application : Python
• Integration of an interactive console.
• Ability to go back in the console history and copy a command with the OK key.
• Interface in the form of a list of scripts with option buttons and a console access button.
• Possibility to import a script into the console.
• Ability to add multiple scripts.
• Possibility to name / rename a script.
• Possibility to delete a script.
• Possibility to set the automatic import of scripts in the console.
• Added a shortcut in the options to run a single script in the console.
• Automatic indentation.
• Addition of a shortcut menu and a catalog accessible from the Toolbox key.
• Possibility to use the alphabetical keys as shortcuts to navigate in the catalog.
• Integration of modules math, cmath and kandinsky.
• Display of details of errors in the console.
• The var key lists the functions and global variables defined in the user's scripts.
Exact calculations
• Sums and products : factorization of expressions of the type ax+bx, distribution of products, reduction of sums to the same denominator.
• Fractions : simplification of fractions to reduced form.
• Powers : simplification of expressions of the type (a^x)(a^y) and (a^x)(b^x), development of (abc…)^r.
• Square Roots : simplification after decomposition into prime factors, elimination of monomials and binomials in the denominator of fractions.
• Trigonmentry : trigonometric functions of notable angles (0, pi, pi/2, pi/3, pi/4, pi/5, pi/6, pi/8, pi/10, pi/12) as well as their inverses, simplification of angles using parity and
trigonometry formulas to give a measure between 0 and pi / 2, simplification of sin/cos as tan, parity of inverse trigonometric functions and integration of formulas of the type arccos(cos).
• Logarithm and exponential : simplification of functions composed with ln and exp, formulas of the type ln(ab), ln(a^x) or exp(a)*exp(b).
• Calculations on integers : calculations on arbitrarily large integers (display limited to 100 digits) in particular for factorial, binomial coefficients, ...
• Complex Numbers : Powers of i, simplification of expressions of the type exp(iPi*x).
• The display of results appears in exact and approximate form and only approximated if the expression is already reduced or if the simplification is not supported.
• Use of exact calculation in other calculator applications to avoid floating point rounding errors: results are however given in approximate form.
And also...
• In Settings, choose the number of digits displayed in the results (from 1 to 14).
• Retrieve all significant digits when a result is copied to the calculation history.
• Changed the multiplication symbol in editing and displaying math expressions.
• Display Probability results for intervals P(a<X<b).
Special thanks
We would like to thank the contributors who helped develop this update : Ian Abbott, Damien Nicolet, Jacob Young. | {"url":"https://www.numworks.com/calculator/update/version-02/","timestamp":"2024-11-11T08:24:45Z","content_type":"text/html","content_length":"16626","record_id":"<urn:uuid:ce0b2302-63af-45b6-994f-8e9a0d6896f1>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00770.warc.gz"} |
Mathematics: What Counts in the Early Years
Like many parents, Tracy Solomon spends quite a bit of time counting with her son Xavier.
“First, we learned to recognize numbers: speed signs on the road or pointing out people with numbers on their sports jerseys,” she recalls of his preschool days. Next came counting sets of objects.
“I tried to count things he was into, like cars and airplanes.”
Parents teaching their kids basic math before school is nothing new. What is different is we now know how incredibly important it is.
“A child’s level of math skills on their first day in kindergarten predicts their mathematical ability in grade five,” says Solomon, who, in addition to being Xavier’s mother, is a developmental
psychologist and research scientist at The Hospital for Sick Children (SickKids) in Toronto. “The literature shows a kid’s incoming number knowledge is a forecast for how well they will be doing in
math as much as seven years down the road.”
The good news is Solomon’s ‘**number talk’ with Xavier **during the preschool years works. Research shows parents can easily increase their child’s math knowledge before school begins and set them up
for success in the future.
What is ‘number talk’?
“Number talk is any talk about numbers,” says Susan Levine, professor of psychology at the University of Chicago, who studies the relation between how much number talk goes on in the house and how
well kids do in math on the first day of school. “We told the parents to do what they ordinarily do with their kids, recorded it, then went back to count how much number talk was going on. Parents
who do this more have kids who score better in math.”
But Levine and her colleagues also found that some types of number talk proved to be better indicators for future understanding than others. “Kids can rattle off their numbers early, often from 1 to
10, and parents are surprised and impressed. But it’s a list with no meaning. When you say ‘give me 3 fish’ they give you a handful.” Gradually, kids figure out the meaning when objects counted are
labeled by the parents. In essence, pointing to the objects while counting them and noting how many there are when the counting is done. “If you count 4 trucks, they are all 4 but it’s the fourth
truck that carries the meaning.”
The key to understanding why some types of number talk might be more effective than others lies in understanding how children learn to count.
How kids learn to count
Before children have any counting ability, they have an innate understanding of quantity. Six-month-old babies can tell the difference between ‘more’ and ‘less’. So can many animals. It is easy to
imagine how this could develop through evolution: going for the big pile of food is better for survival than the smaller pile of food. ‘Numerosity’, or the ability to recognize groupings, becomes
sharper over time. Babies can tell the difference between a pile of 6 things and 12 things, but recognizing the difference between a pile of 3 things and 4 things takes longer to develop.
Actual counting is a process of mastering and accumulating separate skills. To understand counting, a child must understand five processes, each one building on the next:
– Stable order. The child learns number words have an order. For example 1, 2, 3 is correct. 1, 2, 4 is not.
– One-to-one correspondence. The child learns each number can only correspond to one object in a set of things counted. For example, when counting, each truck has its own number: the child can’t skip
a truck or count the same truck twice.
– Cardinality. The child learns the last number used when counting a set of things is the number of things in that set. For example, when counting 4 trucks, when they label the last truck as number 4
that is how many trucks there are in the set.
– Abstraction. The child learns anything can be counted, even things that are not the same. For example, 2 trucks and 2 cars is 4 things in the set of things counted.
– Order irrelevance. The child learns that things can be counted in any order. For example, a set of trucks can be arranged in a circle and then in a line and there will still be the same number.
One last skill worth noting here involves knowing a number of objects just by looking. If you show a toddler 3 things and say ‘how many?’ they can recognize 3 without counting. Levine believes this
ability, called subitizing, provides another clue as to why some types of number talk are more effective at prepping kids for future math skills than others.
“Most parents use low numbers when counting with their kids, but when kids see 3 bears they don’t have to count to understand. They can say ’3’,” says Levine. But the ability to subitize quickly
diminishes with larger numbers of objects. “When parents extend the number talk beyond 3 up to 10, their kids have a better understanding of cardinality. We think this is because the somewhat higher
numbers provide the opportunity to make the link and understand ‘this is why I’m counting’”.
Getting kids to make these links need not be a chore. “**Parents can do fun, simple things around the house,” says Levine. “You don’t have to drill them.” In fact, by putting in a fun effort early,
parents may be avoiding more unpleasant efforts down the road helping their kids catch up in math, which can be very difficult.**
Simple ideas to teach math to preschoolers
At the request of AboutKidsHealth, both Levine and Solomon have provided some suggestions on how to help preschoolers learn the five steps to counting. The general advice is to simply incorporate
these activities and others like them in to daily life around the house.
– Encourage your child to the list off numbers 1 to 10 in the right order. While not necessarily counting, it helps them become familiar with number sequence.
– Count objects that are in front of the child and label the set size: “Let’s count your dolls. 1, 2, 3, 4. You have 4 dolls.” Point at each object as it is counted and encourage the child to do the
same. Counting something is better than just counting.
– Vary what you count: count objects, but also steps, stairs, and sounds.
– Numbers can also be talked about in the context of reading stories to young children. There are often objects in pictures that can be counted and then the set size should be labeled. After counting
something, labeling at the end is just saying “so there are 3 bears”
– Line up two sets of things: 3 trucks and 2 cars. Then count each set while pointing to each member of the corresponding pair: “1” (point to a car and a truck that are side by side); “2” (point to a
car and a truck that are side by side); “there are 2 cars”; and “3” (pointing to the one extra truck that is not paired with a car); “there are 3 trucks. There are 2 cars and 3 trucks.” This helps
kids learn one-to-one correspondence.
– Parents should find contexts in their daily routines when they can talk about numbers with their children. For example, “tonight there will be five people at home, so we need to put five plates on
the table. Let’s count them: 1-2-3-4-5. We have 5 plates.” This lets kids know people do math all the time.
– When you are walking in the neighbourhood, count the number of red cars you see, “1 red car, 2 red cars, 3 red cars — today we saw 3 red cars.” This helps give kids a clue to the fact that things
can be categorized, and therefore counted, in different ways.
– Counting can come up when you need to share. “We have 4 cookies and 2 children — let’s give 1 to you, and 1 to your friend, another 1 to you and another 1 to your friend. Let’s count how many each
of you have — 1, 2 — you have 2. 1, 2 — you have 2. Each of you has 2 cookies!” This gives kids a clue to the fact that the same items can be counted in different ways.
– Introduce basic calculation. “You have 2 trucks. If I give you 1 more, you will have…?” (Wait for child to answer, or supply an answer if the child doesn’t know: “Now you have 3 trucks.”)
Subtraction: “You have 3 trucks — if you give one to me you will have…?” (wait or supply an answer).
Where is all this heading? Fractions!
Learning to count is just the first step. Solomon suggests keeping an eye on where this knowledge is heading, which has to do with fractions, arguably the biggest math hurdle kids face.
Typically, the objects parents count with a preschooler are considered ‘whole entities’: trucks, oranges, coins, and blocks. But the reality is these objects are all potentially made up of smaller
units or could be part of larger units.
“The basic idea behind fractions is that quantity is continuous and that a ‘unit’ can be just about any size,” says Solomon. “Imagine a ruler. It’s not the marks on a ruler that represent the number
1, 2 and so on. It’s the space between the marks that represent the quantity.”
The idea is not to try to get your child to fully understand this concept, but rather to insert little clues into your child’s head about the concept which may make the “ah ha” moment about fractions
down the road easier to achieve. Solomon suggests counting objects that are more easily variable in terms of their unit size.
For example, stand with the child in the centre of a room and ask the child to guess how many steps it would take to walk to the window. The child guesses then tries it out counting the steps – say
4. Then ask the child to guess how many steps it will take the parent to get to the window. It only takes 2. Why is it different? Because mummy takes bigger steps.
Another way to demonstrate this concept is with a big pile of something small like buttons. If there are 100 buttons, a number much too large for the toddler to count, they could be counted in terms
of scoops. A child can learn that there are five scoops of buttons in the pile. Solomon says the exercise can be done with water or sand and that the scoop size can also be varied.
“Don’t invest a lot of time trying to explain the concept. Just getting used to the idea is enough for a preschooler,” says Solomon, whose son Xavier is now seven years old and doing well in math
using number lines — essentially, paper rulers that help teach the concepts behind fractions. “Once you have prepared them for the first day of school, your work is not done. Parents should be
helping their child learn math all through their childhood. I just wish I’d done more of it when he was younger.”
BE YOUR CHILD’S FIRST MATH TEACHER with “Teach Your Child to Count” – ChildUp Early Learning Game Cards
EARLY MATH: Teach Your Child to Count to 10 with “iCount-to-10″ – ChildUp Early Learning Game for iPhone & iPad
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How to track the progress of my math find someone to take calculus exam order in real-time? I did exactly what this post needs to be posted, to be able to get an appointment for my round trip in
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Grating Spacing Calculator Online
Home » Simplify your calculations with ease. » Physics Calculators »
Grating Spacing Calculator Online
The Grating Spacing Calculator serves as a crucial tool in determining the spacing between grating lines based on specific parameters related to diffraction phenomena. It assists in the precise
calculation of the distance between these lines, vital in various fields like physics, optics, and engineering.
Formula of Grating Spacing Calculator
The calculation for grating spacing is based on the following formula:
d = (m * λ) / sin(θ)
• θ: Angular separation between diffracted orders (in radians).
• m: Order of diffraction (an integer).
• λ: Wavelength of the incident light (in meters).
• d: Grating spacing or the distance between the grating lines (in meters).
This formula allows for accurate determination of grating spacing by considering the angle, order of diffraction, and wavelength of the light involved.
General Terms Table: Simplifying Usage
For ease of use, here's a table encompassing commonly searched terms related to grating spacing:
Term Description
Angular Separation The angle between diffracted orders in radians
Order of Diffraction Integer representing the order of the diffraction
Wavelength The wavelength of the incident light in meters
Grating Spacing Spacing between the grating lines, measured in meters
This table aims to provide quick reference points for individuals seeking specific information without the need for manual calculations.
Example of Grating Spacing Calculator
Let's consider an example where the order of diffraction (m) is 2, the wavelength (λ) is 500 nm (0.0000005 meters), and the angular separation (θ) is 30 degrees (converted to radians). Plugging these
values into the formula yields the grating spacing (d):
d = (2 * 0.0000005) / sin(30°)
This calculation results in a grating spacing of approximately 0.00000017 meters.
Most Common FAQs
1. What's the significance of grating spacing in optics?
Grating spacing determines how light diffracts, impacting various optical phenomena crucial in spectroscopy, laser technology, and more.
2. Can the calculator be used for different types of waves?
Yes, the calculator's formula is applicable to different wave types as long as the parameters align with the formula's variables.
3. Is the calculator accurate for non-ideal conditions?
While the formula is accurate, real-world factors might slightly affect the calculated values.
Leave a Comment | {"url":"https://calculatorshub.net/physics-calculators/grating-spacing-calculator/","timestamp":"2024-11-09T17:53:53Z","content_type":"text/html","content_length":"115355","record_id":"<urn:uuid:0d428311-dc6f-4517-909b-c063a5476545>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00434.warc.gz"} |
So you want to do some path sampling…
Basic strategies, timescales, and limitations
Key biomolecular events – such as conformational changes, folding, and binding – that are challenging to study using straightforward simulation may be amenable to study using “path sampling”
methods. But there are a few things you should think about before getting started on path sampling. There are fairly generic features and limitations that govern all the path sampling methods I’m
aware of.
Path sampling refers to a large family of methods that, rather than having the goal of generating an ensemble of system configurations, attempt to generate an ensemble of dynamical trajectories.
Here we are talking about trajectory ensembles that are precisely defined in statistical mechanics. As we have noted in another post, there are different kinds of trajectory ensembles – most
importantly, the equilibrium ensemble, non-equilibrium steady states, and the initialized ensemble which will relax to steady state. Typically, one wants to generate trajectories exhibiting events
of interest – e.g., binding, folding, conformational change.
A trajectory can be considered a list of configurations (possibly with velocities) for all system coordinates recorded with a fixed time increment. Note that there is indeed a path ensemble even in
one dimension: because displacements/velocities will vary along a trajectory, there are an infinite number of trajectories connecting any two points. In principle, trajectory ensemble of transition
events could be obtained by collecting transitions of interest from a very long trajectory – for example the red segments below, possibly with their preceding blue segments, which together make up
the first-passage times (FPTs) for the system to transition from low to high x values.
Path sampling methods can focus computing resource on a subset of trajectories (e.g., the red transition events, above) and have been developed using a variety of strategies. We’ll mention three
that work with continuous trajectories rather than disconnected segments. (1) Lawrence Pratt suggested that, because the probability for a given trajectory to occur could be calculated, one could
perform Metropolis Monte Carlo in path (trajectory) space; this was the “transition path sampling” idea later taken up by Chandler and co-workers. The basic idea, however, has its roots in
path-integral Monte Carlo. (2) Huber and Kim suggested that an ensemble of trajectories could be orchestrated using replication and pruning steps in a way that could encourage sampling of rare
processes. This “weighted ensemble” strategy was really a re-discovery of the “splitting and Russian roulette” strategy published by Los Alamos theorists Herman Kahn & coworkers in the 1950s. (3)
“Dynamic importance sampling” was proposed by Woolf, based on prior work by Ottinger, in which trajectories could be biased toward rare events of interest, with reweighting performed after the fact
to ensure conformance with statistical principles.
The preceding are three basic approaches that generate ensembles of continuous trajectories. It is fair to note that many sophisticated variants and improvements on the basic strategies have been
developed, in addition to many approaches using collections of discontinuous segments (see review by Elber noted below); these are quite valuable but a distraction from the main points of this post.
Generic limitations of path sampling
All the continuous-trajectory approaches share two fundamental limitations. One arises from intrinsic system-specific transition timescales, and the other is a consequence of intrinsic sampling
To understand the limitations, let’s assume our goal is to sample 100 statistically independent transition events. Although every individual trajectory is time-correlated because configurations are
generated sequentially, trajectories can be statistically independent – for example, if you started 100 independent simulations in an initial state and simply waited for 100 transitions. Of course,
that strategy generally is prohibitive and motivates path sampling in the first place, but truly independent simulations would be a gold standard for independent transition-event trajectories.
System-specific timescales
Generally speaking, there are two kinds of transition events. As shown in our original long one-dimensional trajectory and immediately above in magnified view, in a simple “activated process”
characterized by a dominant energy barrier, the duration of a transition t[b] will be much shorter than the waiting time (a.k.a. dwell time) in the initial state. The sum of the average dwell and
event times is called the mean first-passage time (MFPT). Although t[b] may be short and much less than the MFPT, it is still finite. A more challenging scenario is depicted below in the figure
with many intermediate states: each intermediate can lead to a separate, possibly lengthy dwell – and don’t forget that trajectories can reverse many times leading to more dwells than there are
intermediates. In such a case, the transition-event duration t[b] may be similar to the overall MFPT.
With this understanding, let’s get back to our goal of simulating 100 independent transitions. The minimum cost for doing this with fully continuous trajectories is 100 t[b]. If t[b] ~ 10 ns, then
at least 1 ms is needed for our trajectory ensemble. And there is no guarantee that t[b] will be short (compared to timescales that can easily be simulated). So start worrying now … and things only
get worse.
Intrinsic limitations of sampling
Path sampling is desirable when system timescales (MFPTs for processes of interest) are too long to simulate. In other words, by definition of the problem, we cannot afford 100*MFPT. Algorithms
such as the ones sketched above have the potential to limit computational effort to the transition events themselves. But generating independent transition-event trajectories is not a trivial
matter! Although starting separate “brute force” (i.e., standard) trajectories is simple to do, if one wants computational effort to be focused on transition events, there are additional costs.
Let’s look in turn at each of the three path-sampling strategies described above.
In transition path sampling, Metropolis Monte Carlo in path space requires perturbing the preceding trajectory in a sample to create a trial trajectory (that is correlated and, in fact, typically
partially coincident with the prior trajectory) which then is accepted or rejected according to a suitable Metropolis criterion. The sequence of trajectories is significantly correlated, and indeed
rejections amount to having the same trajectory twice in the ensemble which is generated. In other words, there is a kind of “correlation number” n[corr] (akin to a Monte Carlo correlation “time”)
measuring the average number of trajectories which must be sampled before a new statistically independent trajectory is sampled. There is no reason why n[corr] should be small and indeed, just as in
a rough energy landscape, one can imagine that the effective landscape for paths is highly corrugated and requires significant sampling “time.” The bottom line is that our 100 independent
trajectories will cost a total of 100*n[corr]*t[b] simulation time: one hopes that this will be less than the “brute force” cost of 100*MFPT! This sounds bad, but other approaches share similar
Consider the weighted ensemble strategy. Trajectories in the ensemble are run independently but occasionally pruned or replicated – and both operations intrinsically reduce information content and
hence increase correlations. When a trajectory is pruned, the prior computing effort which generated it now gets wasted (at least partially). When a trajectory is replicated, say midway through the
simulation, then the “daughter” or replica trajectories actually were the very same trajectory for half of their existence – and clearly correlated. So once again, there are significant correlations
and we can again describe it with an effective correlation number n[corr]. Whether these correlations are stronger or weaker in the two methods is not our concern here (and indeed would depend on
the system and specifics of the implementation of the path-sampling algorithm).
The dynamic importance sampling strategy is strictly based on independent trajectories and so does not suffer from correlations … but it has its own challenges. Specifically, trajectories are biased
and do not evolve according the correct physical dynamics. Although a probabilistic description of stochastic trajectories enables one to calculate a weight for each of the biased trajectories and
thus correct for the bias, these weights degrade the statistical quality of the resulting trajectory ensemble. Specifically, the non-uniformity of weights guarantees that only a fraction of the
trajectories (say, 1/n[w], with n[w] > 1) will contribute significantly to calculations of any observable, such as a rate. The size of n[w] will depend on system and implementation specifics, but
it’s clear the approach qualitatively suffers from sampling limitations analogous to the two other strategies we just discussed.
Bottom line: The cost per continuous transition trajectory is n*t[b], where n >> 1 is an integer quantifying the efficiency of the path sampling method. Of course, experts in each method strive to
reduce n but there are no guarantees for any challenging system.
Meeting the challenge of multiple intermediates
The issue of multiple intermediates (meta-stable on- or off-pathway states) is worth some additional discussion in the context of path sampling. Recall that in such a case, t[b] ~ MFPT itself may
seem prohibitive – at least, if we insist on having fully continuous trajectories.
The good news is that all three strategies described above can side-step the problem of intermediate states (as can a number of other approaches based on trajectory segments). One example is the
non-Markovian post-analysis suggested by Suarez et al., but this post will not go into the details. Qualitatively, it turns out that the limiting timescale for path sampling is not t[b] but a
quantity we can call which represents the sum of all the event durations for transitions among the intermediates – excluding the intermediate dwell times. This doesn’t solve the problem of
trajectory correlations or weights, but at least offers some hope for obtaining useful results.
The papers noted below are only a very small subset of the path sampling literature.
Further reading
Elber, R. “Perspective: Computer simulations of long time dynamics,” The Journal of Chemical Physics, AIP Publishing, 2016, 144, 060901
Huber, G. A. & Kim, S. “Weighted-ensemble Brownian dynamics simulations for protein association reactions,” Biophys. J., 1996, 70, 97-110
Pratt, L. R. “A statistical method for identifying transition states in high dimensional problems,” J. Chem. Phys., 1986, 85, 5045-5048
Suárez, E.; Lettieri, S.; Zwier, M. C.; Stringer, C. A.; Subramanian, S. R.; Chong, L. T. & Zuckerman, D. M. “Simultaneous Computation of Dynamical and Equilibrium Information Using a Weighted
Ensemble of Trajectories,” J Chem Theory Comput, 2014, 10, 2658-266
Woolf, T. B. “Path corrected functionals of stochastic trajectories: towards relative free energy and reaction coordinate calculations.” Chem. Phys. Lett., 1998, 289, 433-441
Zuckerman, D. M. & Woolf, T. B. “Dynamic reaction paths and rates through importance-sampled stochastic dynamics.” J. Chem. Phys., 1999, 111, 9475-9484
Zuckerman, D. M. & Woolf, T. B. “Transition events in butane simulations: similarities across models.” J. Chem. Phys., 2002, 116, 2586-2591
Zwier, M. C. & Chong, L. T. “Reaching biological timescales with all-atom molecular dynamics simulations,” Curr Opin Pharmacol, Department of Chemistry, University of Pittsburgh, Pittsburgh, PA
15260, USA., 2010, 10, 745-752 | {"url":"https://statisticalbiophysicsblog.org/?p=115","timestamp":"2024-11-09T20:12:54Z","content_type":"text/html","content_length":"45846","record_id":"<urn:uuid:f55b72ee-043b-49c4-a1f7-0d35dc9d7013>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00162.warc.gz"} |
Ene Expression70 Excluded 60 (All round survival will not be out there or 0) ten (Males)15639 gene-level
Ene Expression70 Excluded 60 (General survival isn’t offered or 0) ten (Males)15639 gene-level characteristics (N = 526)DNA Methylation1662 combined capabilities (N = 929)miRNA1046 features (N = 983)
Copy Number Alterations20500 characteristics (N = 934)2464 obs Missing850 obs MissingWith all of the clinical covariates availableImpute with median valuesImpute with median values0 obs Missing0 obs
MissingClinical Information(N = 739)No extra transformationNo additional transformationLog2 transformationNo added transformationUnsupervised ScreeningNo function iltered outUnsupervised ScreeningNo
feature iltered outUnsupervised Screening415 functions leftUnsupervised ScreeningNo function iltered outSupervised ScreeningTop 2500 featuresSupervised Screening1662 featuresSupervised Screening415
featuresSupervised ScreeningTop 2500 featuresMergeClinical + Omics Information(N = 403)Figure 1: Flowchart of information processing for the BRCA dataset.measurements obtainable for downstream
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respectively. Numerous platforms happen to be applied. For instance for methylation, both Illumina DNA Methylation 27 and 450 were used.a single observes ?min ,C?d ?I C : For simplicity of notation,
look at a single type of CPI-203 price genomic measurement, say gene expression. Denote 1 , . . . ,XD ?because the wcs.1183 D gene-expression options. Assume n iid observations. We note that D ) n,
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linear projection, and probable extensions involve additional complicated projection approaches. One particular extension will be to acquire a probabilistic formulation of PCA from a Gaussian latent
variable model, which has been.Ene Expression70 Excluded 60 (Overall survival just isn’t obtainable or 0) 10 (Males)15639 gene-level capabilities (N = 526)DNA Methylation1662 combined capabilities (N
= 929)miRNA1046 attributes (N = 983)Copy Quantity Alterations20500 options (N = 934)2464 obs Missing850 obs MissingWith all the clinical covariates availableImpute with median valuesImpute with
median values0 obs Missing0 obs MissingClinical Data(N = 739)No added transformationNo more transformationLog2 transformationNo added transformationUnsupervised ScreeningNo feature iltered
outUnsupervised ScreeningNo feature iltered outUnsupervised Screening415 functions leftUnsupervised ScreeningNo feature iltered outSupervised ScreeningTop 2500 featuresSupervised Screening1662
featuresSupervised Screening415 featuresSupervised ScreeningTop 2500 featuresMergeClinical + Omics Information(N = 403)Figure 1: Flowchart of information processing for the BRCA dataset.measurements
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8.93 , 72.24 , 61.80 and 37.78 , respectively. Several platforms have been employed. As an example for methylation, each Illumina DNA Methylation 27 and 450 had been used.one observes ?min ,C?d ?I C
: For simplicity of notation, think about a single kind of genomic measurement, say gene expression. Denote 1 , . . . ,XD ?as the wcs.1183 D gene-expression capabilities. Assume n iid observations.
We note that D ) n, which poses a high-dimensionality trouble here. For the operating survival model, assume the Cox proportional hazards model. Other survival models may very well be studied within
a similar manner. Think about the following methods of extracting a small quantity of vital capabilities and developing prediction models. Principal component analysis Principal element analysis
(PCA) is perhaps essentially the most extensively employed `dimension reduction’ technique, which searches for a handful of significant linear combinations from the original measurements. The
approach can proficiently overcome collinearity among the original measurements and, far more importantly, significantly lower the number of covariates integrated in the model. For discussions on the
applications of PCA in genomic data evaluation, we refer toFeature extractionFor cancer prognosis, our goal is usually to develop models with predictive energy. With low-dimensional clinical
covariates, it truly is a `standard’ survival model s13415-015-0346-7 fitting dilemma. On the other hand, with genomic measurements, we face a high-dimensionality problem, and direct model fitting
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take the very first few (say P) PCs and use them in survival 0 model fitting. Zp s ?1, . . . ,P?are uncorrelated, and also the variation explained by Zp decreases as p increases. The regular PCA
approach defines a single linear projection, and feasible extensions involve much more complex projection techniques. 1 extension would be to get a probabilistic formulation of PCA from a Gaussian
latent variable model, which has been. | {"url":"http://www.adenosine-receptor.com/2017/11/27/ene-expression70-excluded-60-all-round-survival-will-not-be-out-there-or-0-ten-males15639-gene-level/","timestamp":"2024-11-04T11:19:01Z","content_type":"text/html","content_length":"61236","record_id":"<urn:uuid:c7d5bd0d-c9ef-48b3-8343-b852c3c814fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00490.warc.gz"} |
number system digital electronics Archives - ALL ABOUT ELECTRONICS
In this article, the binary multiplication of unsigned binary numbers is explained using the examples. And you will also learn how to multiply two fractional binary numbers. Understanding Binary
Multiplication: The binary multiplication is similar to the conventional decimal multiplication. In Binary Multiplication, the each digital of the second number is multiplied with each digit … Read
moreBinary Multiplication Explained | Multiplication of Fractional Binary Numbers | {"url":"https://www.allaboutelectronics.org/tag/number-system-digital-electronics/","timestamp":"2024-11-03T12:17:54Z","content_type":"text/html","content_length":"371579","record_id":"<urn:uuid:836576b2-9da3-42af-adf6-86e56e057e77>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00453.warc.gz"} |
Velocity: The Key to Robert's Amazing Stunt
What is the importance of velocity in calculating Robert's stunt?
Given Robert's height and horizontal distance, how did velocity play a crucial role in determining his successful landing?
The Significance of Velocity in Robert's Stunt
Velocity is a crucial factor in Robert's stunt as it determines the speed and direction at which he projected himself horizontally. By calculating the horizontal velocity, we can understand how fast
he needed to move to reach the landing point safely. Let's explore the concept of velocity further to uncover the magic behind Robert's incredible feat.
Velocity is more than just a measurement of speed; it also takes into account the direction of motion. In Robert's case, the horizontal velocity at which he projected himself was essential for him to
cover the distance of 100m safely. The calculation of velocity helped in determining the precise speed required for his successful stunt.
By understanding the concept of velocity, we can appreciate the intricate details of Robert's maneuver. Velocity not only depicts how fast an object is moving but also in which direction it is
heading. This vector measurement gives us a comprehensive view of motion, enabling us to calculate trajectories and distances covered with accuracy.
In Robert's stunt, the velocity played a pivotal role in ensuring his landing at the desired horizontal distance. The calculated velocity of 31.35m/s provided the necessary speed for him to execute
the stunt flawlessly. Without considering velocity, the whole stunt could have been a disaster.
Velocity is indeed the key to understanding Robert's amazing stunt. It showcases the blend of speed and direction, essential components in any motion-related calculations. By grasping the concept of
velocity, we can unlock the secrets behind remarkable feats like Robert's daring leap. | {"url":"https://laloirelle.com/physics/velocity-the-key-to-robert-s-amazing-stunt.html","timestamp":"2024-11-07T07:50:24Z","content_type":"text/html","content_length":"21867","record_id":"<urn:uuid:0eb51099-4582-498c-b1fb-86af7c4ddd86>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00141.warc.gz"} |
What are the pros and cons of using the quadratic formula, completing the sqaure, and factoring, when solving quadratic equations?
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Copyright (c) George Ungureanu 2015-2016
License BSD-style (see the file LICENSE)
Maintainer ugeorge@kth.se
Stability experimental
Portability portable
Safe Haskell Safe
Language Haskell2010
The SY library implements the atoms that operate according to the synchronous reactive model of computation. This module also provides a set of helpers for instantiating the MoC layer patterns
described in the ForSyDe.Atom.MoC module as meaningful synchronous process constructors.
This module exports a reduced SY language, where all processes are assumed to operate on the same clock rate. For an extended SY language of implicit multi-clock rate systems, check
ForSyDe.Atom.MoC.SY.Clocked. Alternatively, this layer can be extended by explicitly wrapping a ForSyDe.Atom.ExB-like layer below it.
Useful pointers:
Synchronous (SY) event
Acording to the tagged signal model [Lee98] "two events are synchronous if they have the same tag, and two signals are synchronous if all events in one signal are synchronous to an event in the
second signal and vice-versa. A system is synchronous if every signals in the system is synchronous to all the other signals in the system." The synchronous (SY) MoC defines no notion of physical
time, its tag system suggesting in fact the precedence among events. In simpler words:
The SY MoC
is abstracting the execution semantics of a system where computation is assumed to perform instantaneously (with zero delay), at certain synchronization instants, when data is assumed to be
Below is a possible behavior in time of the input and the output signals of a SY process, to illustrate these semantics:
The SY tag system is derived directly from the Stream host. Hence a SY Signal is isomorphic to an infinite list, where tags are implicitly defined by the position of events in a signal: \(t_0\)
corresponds with the event at the head of a signal, \(t_1\) with the following event, etc... The only explicit parameter passed to a SY event constructor is the value it carries (in V). As such, we
can state the following particularities:
1. tags are implicit from the position in the Stream, thus they are ignored in the type constructor.
2. the type constructor wraps only a value
3. being a timed MoC, the order between events is interpreted as total.
4. there is no need for an execution context and we can ignore the formatting of functions in ForSyDe.Atom.MoC, thus we can safely assume that \[ \Gamma\vdash\alpha\rightarrow\beta = \alpha\
rightarrow\beta \]
newtype SY a Source #
The SY event. It defines a synchronous signal.
• val :: a
value wrapped by the SY event constructor.
Functor SY Source # Allows for mapping of functions on a SY event.
Defined in ForSyDe.Atom.MoC.SY.Core
Applicative SY Source # Allows for lifting functions on a pair of SY events.
Defined in ForSyDe.Atom.MoC.SY.Core
MoC SY Source # Implenents the execution semantics for the SY MoC atoms.
Defined in ForSyDe.Atom.MoC.SY.Core
Read a => Read (SY a) Source # Reads the value wrapped
Defined in ForSyDe.Atom.MoC.SY.Core
Show a => Show (SY a) Source # Shows the value wrapped
Defined in ForSyDe.Atom.MoC.SY.Core
Plottable a => Plot (Signal a) Source # SY signals.
Defined in ForSyDe.Atom.Utility.Plot
type Ret SY b Source #
Defined in ForSyDe.Atom.MoC.SY.Core
type Ret SY
b = b
type Fun SY a b Source #
Defined in ForSyDe.Atom.MoC.SY.Core
type Fun SY
a b = a -> b
Aliases & utilities
These are type synonyms and utilities provided for user convenience. They mainly concern the construction and usage of signals.
type Signal a = Stream (SY a) Source #
Type synonym for a SY signal, i.e. "an ordered stream of SY events"
unit2 :: (a1, a2) -> (Signal a1, Signal a2) Source #
Wraps a (tuple of) value(s) into the equivalent unit signal(s). A unit signal is a signal with one event, i.e. a singleton.
Provided helpers: unit, unit2, unit3, unit4.
readSignal :: Read a => String -> Signal a Source #
Reads a signal from a string. Like with the read function from Prelude, you must specify the type of the signal.
>>> readSignal "{1,2,3,4,5}" :: Signal Int
SY process constuctors
These are specific implementations of the atom patterns defined in ForSyDe.Atom.MoC.
delay Source #
:: a initial value
-> Signal a input signal
-> Signal a output signal
The delay process "delays" a signal with one event. Instantiates the delay pattern.
>>> let s = signal [1,2,3,4,5]
>>> delay 0 s
comb22 Source #
:: (a1 -> a2 -> (b1, b2)) function on values
-> Signal a1 first input signal
-> Signal a2 second input signal
-> (Signal b1, Signal b2) two output signals
comb processes map combinatorial functions on signals and take care of synchronization between input signals. It implements the comb pattern (see comb22).
Constructors: comb[1-4][1-4].
>>> let s1 = signal [1..]
>>> let s2 = signal [1,1,1,1,1]
>>> comb11 (+1) s2
>>> comb22 (\a b-> (a+b,a-b)) s1 s2
reconfig22 Source #
:: Signal (a1 -> a2 -> (b1, b2)) signal carrying functions
-> Signal a1 first input signal carrying arguments
-> Signal a2 second input signal carrying arguments
-> (Signal b1, Signal b2) two output signals
reconfig creates an synchronous adaptive process where the first signal carries functions and the other carry the arguments. It imlements the reconfig pattern (see reconfig22).
Constructors: reconfig[1-4][1-4].
>>> let sf = signal [(+1),(*2),(+1),(*2),(+1),(*2),(+1)]
>>> let s1 = signal [1..]
>>> reconfig11 sf s1
constant2 Source #
:: (b1, b2) values to be repeated
-> (Signal b1, Signal b2) generated signals
A signal generator which keeps a value constant. It implements rgw stated0X pattern (see stated22).
Constructors: constant[1-4].
>>> let (s1, s2) = constant2 (1,2)
>>> takeS 3 s1
>>> takeS 5 s2
generate2 Source #
:: (b1 -> b2 -> (b1, b2)) function to generate next value
-> (b1, b2) kernel values
-> (Signal b1, Signal b2) generated signals
A signal generator based on a function and a kernel value. It implements the stated0X pattern (check stated22).
Constructors: generate[1-4].
>>> let (s1,s2) = generate2 (\a b -> (a+1,b+2)) (1,2)
>>> takeS 5 s1
>>> takeS 7 s2
stated22 Source #
:: (b1 -> b2 -> a1 -> a2 -> (b1, b2)) next state function
-> (b1, b2) initial state values
-> Signal a1 first input signal
-> Signal a2 second input signal
-> (Signal b1, Signal b2) output signals
stated is a state machine without an output decoder. It implements the stated pattern (see stated22).
Constructors: stated[1-4][1-4].
>>> let s1 = signal [1,2,3,4,5]
>>> stated11 (+) 1 s1
state22 Source #
:: (b1 -> b2 -> a1 -> a2 -> (b1, b2)) next state function
-> (b1, b2) initial state values
-> Signal a1 first input signal
-> Signal a2 second input signal
-> (Signal b1, Signal b2) output signals
state is a state machine without an output decoder. It implements the stated pattern (see state22).
Constructors: state[1-4][1-4].
>>> let s1 = signal [1,2,3,4,5]
>>> state11 (+) 1 s1
moore22 Source #
:: (st -> a1 -> a2 -> st) next state function
-> (st -> (b1, b2)) output decoder
-> st initial state
-> Signal a1
-> Signal a2
-> (Signal b1, Signal b2)
moore processes model Moore state machines. It implements the moore patterns (see moore22).
Constructors: moore[1-4][1-4].
>>> let s1 = signal [1,2,3,4,5]
>>> moore11 (+) (+1) 1 s1
mealy22 Source #
:: (st -> a1 -> a2 -> st) next state function
-> (st -> a1 -> a2 -> (b1, b2)) outpt decoder
-> st initial state
-> Signal a1
-> Signal a2
-> (Signal b1, Signal b2)
mealy processes model Mealy state machines. It implements the mealy pattern (see mealy22).
Constructors: mealy[1-4][1-4].
>>> let s1 = signal [1,2,3,4,5]
>>> mealy11 (+) (-) 1 s1
toDE2 Source #
:: (Num t, Ord t, Eq t)
=> Signal t SY signal carrying DE timestamps
-> Signal a first input SY signal
-> Signal b second input SY signal
-> (SignalBase t a, SignalBase t b) two output DE signals
Wraps explicit timestamps to a (set of) SY signal(s), rendering the equivalent synchronized DE signal(s).
Constructors: toDE, toDE2, toDE3, toDE4.
>>> let s1 = SY.signal [0,3,4,6,9] :: SY.Signal TimeStamp
>>> let s2 = SY.signal [1,2,3,4,5]
>>> toDE s1 s2
toSDF2 Source #
:: Signal a SY signal
-> Signal b SY signal
-> (Signal a, Signal b) SDF signals
Transforms a (set of) SY signal(s) into the equivalent SDF signal(s). The only change is the event consructor, meaning that the order is preserved.
Constructors: toSDF[1-4].
>>> let s = SY.signal [1,2,3,4,5]
>>> toSDF1 s
toSDF2' Source #
:: (Prod, Prod) production rates \(p_1,p_2\)
-> Signal (Vector a) SY signal of vectors of length \(p_1\)
-> Signal (Vector b) SY signal of vectors of length \(p_2\)
-> (Signal a, Signal b) SDF signals where the vectors are serialized.
Transforms a (set of) SY signal(s) of vectors of the same length into equivanlent SDF signal(s) by serializing the vectors according to a production rate. If the production rate and the vector
lengths do not match then a runtime error is thrown.
Constructors: toSDF[1-4]'.
>>> let s = read "{<1,2>,<3,4>,<5,6>}" :: SY.Signal (V.Vector Int)
>>> toSDF1' 2 s
zipx :: Vector (Signal a) -> Signal (Vector a) Source #
Synchronizes all the signals contained by a vector and zips them into one signal of vectors. It instantiates the zipx skeleton.
>>> let s1 = SY.signal [1,2,3,4,5]
>>> let s2 = SY.signal [11,12,13,14,15]
>>> let v1 = V.vector [s1,s1,s2,s2]
>>> v1
>>> zipx v1
unzipx :: Integer -> Signal (Vector a) -> Vector (Signal a) Source #
Unzips the vectors carried by a signal into a vector of signals. It instantiates the unzipx skeleton. To avoid infinite recurrence, the user needs to provide the length of the output vector.
>>> let v1 = V.vector [1,2,3,4]
>>> let s1 = SY.signal [v1,v1,v1,v1,v1]
>>> s1
>>> unzipx 4 s1
unzipx' :: Signal (Vector a) -> Vector (Signal a) Source #
Same as unzipx, but "sniffs" the first event to determine the length of the output vector. Has an unsafe behavior!
>>> let v1 = V.vector [1,2,3,4]
>>> let s1 = SY.signal [v1,v1,v1,v1,v1]
>>> s1
>>> unzipx' s1 | {"url":"http://forsyde.github.io/forsyde-atom/api/ForSyDe-Atom-MoC-SY.html","timestamp":"2024-11-12T03:09:38Z","content_type":"application/xhtml+xml","content_length":"64057","record_id":"<urn:uuid:fb9d5943-3b26-4819-a288-d328b05154c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00541.warc.gz"} |
NumPy: Create a two-dimensional array with shape (8,5) of random numbers, Select random numbers from a normal distribution (200,7) - w3resource
NumPy: Create a two-dimensional array with shape (8,5) of random numbers, Select random numbers from a normal distribution (200,7)
NumPy: Basic Exercise-48 with Solution
Write a NumPy program to create a two-dimensional array with shape (8,5) of random numbers. Select random numbers from a normal distribution (200,7).
This problem involves writing a NumPy program to generate a two-dimensional array with a shape of (8,5) filled with random numbers selected from a normal distribution with a mean of 200 and a
standard deviation of 7. By generating random values from a normal distribution, the program constructs a matrix suitable for various statistical analyses and modeling tasks.
Sample Solution :
Python Code :
# Importing the NumPy library with an alias 'np'
import numpy as np
# Setting the seed for NumPy's random number generator to 20
# Calculating the cube root of 7
cbrt = np.cbrt(7)
# Defining a variable 'nd1' with a value of 200
nd1 = 200
# Generating a 10x4 array of random numbers from a normal distribution with mean nd1 and standard deviation cbrt
print(cbrt * np.random.randn(10, 4) + nd1)
[[201.6908267 200.37467631 200.68394275 195.51750123]
[197.92478992 201.07066048 201.79714021 198.1282331 ]
[200.96238963 200.77744291 200.61875865 199.05613894]
[198.48492638 198.38860811 197.55239946 200.47003621]
[199.91545839 202.99877319 202.01069857 200.77735483]
[199.67739161 193.89831807 202.14273593 202.54951299]
[199.53450969 199.7512602 199.79145727 202.97687757]
[200.24634413 196.04606934 198.30611253 197.88701546]
[201.78450912 203.94032834 198.21152803 196.91446071]
[201.0082481 197.03285104 200.63052763 197.82590294]]
In the above code -
np.random.seed(20) sets the random seed value to 20. Setting the seed ensures that the same set of random numbers is generated every time the code is executed, which can be useful for
cbrt = np.cbrt(7): This line calculates the cube root of 7 using NumPy's np.cbrt() function and assigns the result to the variable cbrt.
nd1 = 200 assigns the value 200 to the variable nd1.
print(cbrt * np.random.randn(10, 4) + nd1): This statement does the following:
• np.random.randn(10, 4) generates a 10x4 array with random numbers from a standard normal distribution (mean=0, standard deviation=1).
• cbrt * np.random.randn(10, 4) scales the random numbers in the array by the cube root of 7, which was calculated earlier.
• cbrt * np.random.randn(10, 4) + nd1 shifts the scaled random numbers by adding 200 to each element in the array.
Python-Numpy Code Editor:
Previous: NumPy program to create a one dimensional array of forty pseudo-randomly generated values. Select random numbers from a uniform distribution between 0 and 1.
Next: NumPy program to generate a uniform, non-uniform random sample from a given 1-D array with and without replacement.
What is the difficulty level of this exercise?
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Solving system of linear equations
26+15 - x =32 - 4
x = 13
x =
= 13
x = 13
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with one unknown variable, and finally, any other equation with one variable. Even if an exact solution does not exist, it calculates a numerical approximation of roots.
(x+4)(x-3)+34x+6x^2 = 256
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Unknowns (variables) write as one character a-z, i.e., a, b, x, y, z. No matter whether you want to solve an equation with a single unknown, a system of two equations of two unknowns, the system of
three equations and three unknowns, or a linear system with twenty unknowns. The number of equations and the number of unknowns should be equal, and the equation should be linear (and linear
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Dictionary of Arguments
Ideas: ideas are representations of objects, circumstances or properties of objects as opposed to their manifestations in the external world. At times the concept of the idea is connected with the
claim of perfection. See also idealism, idealization, thing in itself, Platonism._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations
of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments.
> Dewey, John
> Ideas
John Dewey on Ideas - Dictionary of Arguments Suhr I 136
Ideas/Dewey: ideas are hypotheses, not finalities. >Hypotheses, >Theories, >Method; cf. >Idealism._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the
page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments The note [Concept/Author],
[Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers
refer to this edition.
Dew II
J. Dewey
Essays in Experimental Logic Minneola 2004
Suhr I
Martin Suhr
John Dewey zur Einführung Hamburg 1994 | {"url":"https://www.philosophy-science-humanities-controversies.com/listview-details-economics-politics.php?id=372533&a=t&first_name=John&author=Dewey&concept=Ideas","timestamp":"2024-11-03T15:49:16Z","content_type":"text/html","content_length":"16371","record_id":"<urn:uuid:e2e62560-309b-455c-91ec-1ca4fc2a1b25>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00692.warc.gz"} |
Association rule mining | Apriori Algorithm
This blog is about the unsupervised learning algorithm: the Apriori algorithm, that is used by many supermarkets to increase their sales.
Market Basket Analysis (Source)
We all go to the supermarket to buy our favourites & necessary products. Also, we feel better if we get things easily, i.e. when we don’t need to search more for the products. You must have noticed
that whenever you buy bread, you find either butter or egg around it, also with many other products. Have, you ever wondered how a supermarket does this? This is done with market basket analysis, and
the algorithm followed in this process is Apriori Algorithm. This algorithm is based on Association Rule Mining. Supermarket usually groups the items that are bought together, so that people can
easily buy them and in the end, their sales increase.
What is Association Rule Mining?
In simple words, association rule mining is based on IF/THEN statements. Association, as the name suggests, it finds the relationship between different items. And the best thing about this is, It
also works with non-numeric or categorical datasets.
Association rule mining finds frequently occurring patterns between the given data. An association rule has two parts:
• Antecedent (IF)
• Consequent (THEN)
For eg: If a person in the supermarket buys bread, then it’s more likely that he will buy butter or jam or egg.
How Association Rule Mining Works?
Association rule mining basically calculates how frequently product X is bought when product Y is bought. Based on the concept of strong rules, Rakesh Agrawal, Tomasz Imieliński and Arun Swami
introduced association rules for discovering regularities between products in large-scale transactions. Not for just two products, but it calculates the association for many products. Let’s first
understand important terms :
Source: Author
• Support — Support is an indication of the frequency of an item. Support of an item X is the ratio of the number of times X appears in the transaction to the total number of transactions. The
greater the value of support the more frequently that item is bought.
• Confidence — Confidence is a measure of how likely a product Y will be sold with product X i.e. X=>Y. It is calculated by the ratio of support (X U Y)(i.e. union) to the support of (X).
• Lift — Lift is a measure of the popularity of an item or a measure of the performance of the targeting model. The Lift of Y is calculated by dividing confidence with the support of (Y).
Now, let’s understand all these terms by an example dataset.
Dataset (Source: Author)
We will find the value of support, confidence and lift from the formula discussed:
Support(wine) =Probability(X=wine) = 4(because wine is bought 4 times)/6(total transaction)
Confidence (X={wine, chips}) => (Y={bread}) = Support(wine, chips, bread) / Support(wine, chips)
i.e. Confidence = (2/6)/(3/6) = 0.667
Lift (X={wine, chips, bread}) => (Y={wine, chips}) = (Support(wine, chips, bread)) / (Support(wine, chips) * (Support(bread)))
i.e. Lift = (2/6) / ((3/6) * (4/6))
Now, at this point, we are defining the new term Conviction. The value of conviction tells the dependency between X & Y.
Conviction(X => Y) = (1-Support(y)) / (1-Confidence(X => Y))
• Conv(x => y) = 1 means that x has no relation with y.
• A high conviction value means that the consequent is highly dependent on the antecedent.
Now, let’s go step by step, applying all the things we learnt. First, create a frequency table of all items.
Frequency table from the above dataset (frequency out of 6) (Source: Author)
• Step 1: Calculate the needed threshold support value. Then create the frequency(Support) table containing values > threshold:
• Step 2: Make doublets or pairs of items and then calculate frequency. Remove all those pairs whose value is less than the threshold.
Step 2
• Step 3: Make triplets of items and then calculate frequency. Also, remove those triples whose value is less than the threshold.
Step 3
Implementation with Python
# First check or install apyori
pip install apyori# Import Dataset
import pandas as pd
data = pd.read_excel("Movie_reccommendation.xlsx")# Convert dataframe into list - Because apyori algorithm tale list input rather than list of lists
#n is number of different items (bread, beer, ...)
observations = [] for i in range(len(data)):
observations.append([str(data.values[i,j]) for j in range(n)]#Fitting data to algorithm
#These argumentscan be set by trying out different values and checking the association rules whether the arguments produced a valid association between items or not.
from apyori import apriori
associations = apriori(observations, min_length = 2, min_support = 0.2, min_confidence = 0.2, min_lift = 3)#Viewing the results
print(associations[0])#Viewing all results
for i in range(0, len(associations)):
print(associations[i])#NOTE: Viewing the result and understanding the results is most important task.
Pros of the Apriori algorithm
1. It is way too easy to implement and understands the algorithm.
2. It can be used on large data sets.
Cons of the Apriori Algorithm
1. Sometimes, it may need to find a large number of candidate rules which can be computationally expensive.
2. When it comes to calculation, it’s too difficult especially when calculating support.
This was all about market basket analysis using the apriori algorithm. | {"url":"https://deeppatel23.medium.com/association-rule-mining-apriori-algorithm-f63508fc3260?responsesOpen=true&sortBy=REVERSE_CHRON&source=read_next_recirc-----3cf6b0001375----0---------------------42d32df7_3955_4d8c_a7b2_b3ea98191259-------","timestamp":"2024-11-05T08:44:10Z","content_type":"text/html","content_length":"140065","record_id":"<urn:uuid:a8af4949-6d16-48da-81b6-68c185460db2>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00564.warc.gz"} |
Digital Signal Processing: The Essentials that You Need to Know. Theory to Practice
Appendix B: Analog Filter Design
Analog Filters
Dwight Day
Appendix B: Analog Filter Design
In this chapter we will be developing filters that employ feedback of the output. These filters are commonly called Infinite Impulse Response filters (IIR), since the impulse response will extend out
to infinity. To start we will review the design of Analog filters. Later a translation will be introduced to go from an analog design to a discrete design.
Section A) Filter Specifications.
Before trying to design a filter the first step needs to be to define or describe the type of filter that is to be designed. Defining the response that you will want in your filter will be a function
of how it is to be used and is beyond the scope of this discussion. It is sufficient to currently break the types of filter into the three basic types of Low-Pass (LP), High-Pass (HP) and Band-Pass
(BP) filters. We will find that each type can be sufficiently described by just a few values.
Part 1) Low-Pass Filter
A sketch of the frequency response of a LP filter is shown in Figure B-1. In this sketch we see that the important values are 1) the upper frequency of the pass band (wp in Figure B.1.), the lower
frequency of the stop-band (ws), and the attenuation’s for each band.
Figure B.1 Sketch of The Frequency Response of A Low-Pass Filter.
Part 2) High-Pass Filter
A sketch of the frequency response of a HP filter is shown in Figure B.2. This type of filter can be characterized in a fashion similar to that of the LP filter, requiring the 1) the upper frequency
of the stop band (ws), the lower frequency of the pass-band (ws), and the attenuation’s for each band.
Figure B.2 Sketch of The Frequency Response of A High-Pass Filter.
Part 3) Band-Pass Filter
A band-pass filter is more complex that the LP and HP, but can still be represented by a small set of values. Figure B.3 contains a sketch of the BP frequency response. With the whole specification
being set by the end of the lower stop band (wsl), the lower end of the pass band (wpl), the upper end of the pass band (wpu) and the end of the upper stop band (wsu).
Figure B.3 Sketch of The Frequency Response of A Band-Pass Filter.
Section B) Translation to a Normalized Low-Pass Filter
Once the design has been described in one of the previously mentioned forms, each design will be translated to a Normalized Low-Pass (NLP) filter specification. This is done to allow us to identify
the proper NLP design that will match our filter once it is translated.
Part 1) Low-Pass Filter to Normalized Low-Pass
The translation that converts a NLP filter, with Laplace variable [latex]\hat s[/latex], to a general LP filter with cutoff or pass-band cutoff frequency of [latex]wp[/latex], is given in formula
[latex]\hat s = \frac{s}{wp}[/latex] (B.1)
This translation will be used to change the H( [latex]\hat s[/latex] ) of the NLP into a LP H(s). However when we translate the NLP, the frequency response is warped to give us our desired response.
The frequency warping can be characterized by the following two example points. For [latex]s = \pm j wp [/latex], which translate to [latex]\hat s = \pm j \frac{wp}{wp} = \pm j[/latex]. Also, [latex]
s = \pm j ws[/latex], [latex]\hat s = j \frac{ws}{wp}[/latex]. This translation is shown graphically in Figure B.4.
Figure B.4 Low-Pass To Normalized Low-Pass Translation.
Part 2) High-Pass Filter to Normalized Low-Pass
The translation for a HP to NLP is a little more complex, but not that bad. The translation given it equation 2, will take the NLP H(s) and translate it to a HP filter with cutoff of wp.
[latex]\hat s = \frac{wp}{s}[/latex] (B.2)
Using this translation, [latex]s = +j wp , \hat s \frac{wp}{j wp} = \mp j[/latex] . Note: the sign of s will flip, in other words for s = +j wp, s’ = – j. Also, s = + j ws, s’ = -/+ j wp / ws. This
translation is shown graphically in Figure B.5
Figure B.5 Sketch of HP to NLP Translation.
Part 3) Band-Pass Filter to Normalized Low-Pass
The third translation is the most complex; however, we are trying to translate from a band pass filter to a NLP, which is no small feat. The translation can be achieved via equation B.3.
[latex]\hat s = \frac{(s^2 + wc^2)}{s wb}[/latex] (B.3)
where [latex]wc = \sqrt{ wpu * wpl }[/latex] and [latex]wb = ( wpu - wpl )[/latex]
Again, [latex]\hat s[/latex] is the Laplace variable for the NLP and s the Laplace variable for the BP filter. Let’s consider the four critical frequencies of wpl, wpu, wsl and wsu.
For s = + j wpu,
[latex]\hat s = \frac{(-wpu^2 + wpu * wpl )}{ ( + j wpu ( wpu - wpl ) )} [/latex]
[latex]\hat s = \frac{- wpu ( wpu - wpl ) }{ j wpu ( wpu - wpl ) } [/latex]
[latex]\hat s = \frac{-1 }{ + j } = + j[/latex]
For s = + j wpl,
[latex]\hat s = \frac{-wpl^2 + wpu * wpl }{+ j wpl ( wpu - wpl ) }[/latex]
[latex]\hat s = \frac{- wpl ( wpu - wpl )}{+ j wpl ( wpu - wpl )} [/latex]
[latex]\hat s = \frac{1 }{ + j } = \mp j [/latex]
Note that the sign is reversed in the last equation, in other words for -j wpl, we get +j and for
+j wpl, the result is -j.
For s = + j wsu,
[latex]\hat s = \frac{-wsu^2 + wpu * wpl }{ + j wsu ( wpu - wpl )} [/latex]
[latex]\hat s = \frac{\mp j ( wpl * wpu - wsu2 )}{ wsu ( wpu - wpl ) } [/latex]
Since wsu > wpu > wpl, the ratio in the previous equation will be negative, which mean no sign reversal will occur.
The last case will be s = + j wsl,
[latex]\hat s = \frac{ -wsl^2 + wpu * wpl}{ + j wsl ( wpu - wpl ) } [/latex]
[latex]\hat s = \frac{\mp j ( wpl * wpu - wsl2 ) }{ wsl ( wpu - wpl ) }[/latex]
However in this case wsl < wpl < wpu, which makes the ratio in the previous equation positive, which means that there will be a sign reversal.
These four cases are then shown graphically in Figure B.6.
Figure B.6. Sketch of BP to NLP Translation.
Where A is equal to the minimum of [latex]\frac{ wpl * wpu - wsl2}{ wsl ( wpu - wpl ) }[/latex] and [latex]\frac{wpl * wpu - wsu2 }{ wsu ( wpu - wpl )}[/latex], and B is the maximum of these two
In review, it should be pointed out that the translations given by equations 1, 2, and 3 will be used to translate an H(s) for a NLP filter into either an LP, HP or BP filter. So in order to pick the
correct NLP filter such that once we translate it into one of the three forms, we obtain the proper filter, we will first translate the specification as was described here.
Section C) Normalized Low-Pass Filter Designs
In the previous section, we showed how to translate our specification into a NLP form. We will describe a couple of common filters that can be used to implement an NLP filter.
Part 1) Butterworth Filter
The first and most popular filter is the Butterworth, which is also called the Maximally Flat filter. One way to derive the Butterworth filter is to begin with a frequency response of
[latex]| H_n(\omega)^2 | = \frac{1}{1+\omega^{2 n} }[/latex] (B.4)
If we substitute s = j w
[latex]| H_n(s)^2 | = \frac{1}{1 + ( (-s^2)^n)} [/latex]
This can be thought of as an H(s) with poles at [latex]( -s^2 )^n = -1[/latex], or
[latex]( s^2 )^n ( -1 )^n = -1[/latex]
and then if we rewrite -1 as [latex]e^{j (2k-1) \pi}[/latex] and [latex](-1)^n = e^{j \pi n}[/latex] , thus the poles become
[latex]s^{2 n} = e^{j( 2k+n-1) \pi} [/latex]
[latex]s_k = e^{ j \frac{ 2k+n-1}{2n \pi}[/latex] with k = 1,2, … , 2n
Consider the case of n=3 then
[latex]s1 = e^{ j \frac{2}{3} \pi }[/latex] , [latex]s2 = e^{ j \pi} = -1[/latex] , [latex]s3 = e^{ j \frac{4}{3} \pi}[/latex],
[latex]s4 = e^{j \frac{5}{3} \ pi}[/latex], [latex]s5 = e^{ j 2 \pi}[/latex], $latex s6 = e^{ j \frac{7}{3} \pi},
The poles are plotted out in Figure B.7.
Figure B.7 Pole-Zero or S-Plan Plot of a Third-Order Butterworth.
We cannot use these poles directly, since poles in the Right Half Plane (RHP) will cause the system to be unstable ( having an impulse response that goes to infinity ). This problem develops for the
fact that we factored [latex]s^2[/latex] into [latex]\pm s[/latex], but if we only use the -s we will get the same magnitude response. Thus we need only the poles from the Left Half Plane (LHP).
The NLP Butterworth filter’s H(s) is then
[latex]H( s ) = \sum_{k=1}^{2n} {( \frac{1}{ s-e^{j \pi \frac{2k+n-1}{2n} } } } )[/latex] (B.5)
For example let n = 3, then
[latex]H( s ) = \frac{1}{ ( (s-e^{j \frac{2}{3}} \pi ) (s-e^{j \pi}) (s-e^{j\frac{4}{3} \pi } ) ) }[/latex]
[latex]H( s ) = \frac{1}{ ( (s-e^{j \frac{2}{3}} \pi ) (s+1) (s-e^{j\frac{4}{3} \pi } ) ) }[/latex]
[latex]H( s ) = \frac{1}{ (s+1)} \frac{1}{( s^2 - s (e^{j \frac{2}{3}} \pi +e^{j\frac{4}{3} \pi } ) + e^{j \frac{2}{3} \pi} * e{j \frac{4}{3} \pi })}[/latex]
Then using [latex]e^{-j \frac{2}{3} \pi} = e^{j \frac{4}{3} \pi }[/latex] we have
[latex]H( s ) = \frac{1}{( s+1)} \frac{1}{( s^2 - 2 RE ( e^{j \frac{2}{3} \pi } ) s + 1 ) }[/latex]
[latex]H(s) = \frac{1}{s+1} \frac{1}{s^2 + s + 1} [/latex]
We will now display the surface of H(s), much like what was done with H(z), using MATLAB.
The following is a MATLAB script file, that creates a sampling of the s plane (complex plane) and the computes the rational function (ratio of polynomials) for H(s)
limit = 5;
% Set up s plane as real and imaginary parts of s
[x,y] = meshgrid( -2.25:0.05:0, -2.25:0.05:2.25 );
s = x + 1i*y;
% Compute H(s) for third-order Butterworth
H = ( 1 ./ ( s + 1 ) ) .* ( 1 ./ ( s.*s + s + 1 ) );
surf( y,x,min(abs(H3),limit));
view( [ 155 45]);
title (‘H(s) of Third Order Butterworth’);
xlabel( ‘Imaginary’ );
ylabel( ‘Real’);
Figure B.8 Surface Plot of H(s) for NLP Third-Order Butterworth.
Two important features need to be pointed out from this surface plot. The first is that the three poles are easily visible as the peaks in the plot, and these have been clipped at 5, in order to make
some of the other features of the plot more visible. The poles are evenly distributed on a circle about the origin of the s plane. The second feature that should be pointed out is the frequency
response of the system, which is the H(s) along the imaginary axis. In the surface plot, we can see that the system is a NLP, since for -1 to 1, H(jw) is ~ 1. Outside this band, the response drops
off quickly. The response is also, well behaved in the pass band, and is as flat as is possible with a third order system, hence Butterworth filter are often called “Maximally Flat” filters.
Design Equation
The only parameter that can be changed on the NLP Butterworth filter is then the order “n”. The question then is what order of filter is needed to match the NLP we are needing. Remember we had a
translation or mapping for any filter that would take it to a NLP. Our mapped NLP will have a cutoff of 1 (rad/sec) and then a stop band frequency (ws) and Stop-Band Attenuation (SBA). We will now
derive the formula required to determine the order n, based on SBA and ws. First recall from earlier that the frequency response of an nth order Butterworth is
[latex]| H_n (w) |^2 = \frac{1}{ 1 + (w^{2n}) }[/latex]
Substituting in [latex]| H_n (w) |[/latex] = SBA and w = ws, and taking the log of both sides yields,
[latex]2 log | SBA | = log( \frac{1}{1 + (ws^{2n}) } )[/latex]
[latex]2 log |SBA| = log( 1 ) - log( 1 + (ws^{2n}) )[/latex]
[latex]2 log |SBA| \approx - log( ws^{2n} )[/latex] for ws >> 1
[latex]2 log |SBA| ~ - 2 n log( ws )[/latex]
[latex]n = \frac{log |SBA| }{- log( ws )}[/latex] (B.6)
Thus based on the translated stop frequency and the SBA, we have the design equation (B.6), which can be used to compute n. To demonstrate the effect of n on the Butterworth response Figure B.9 shows
a plot of the NLP frequency response for n = 3 and 5. It should be noted here that at the cutoff frequency of 1, the response is always at 0.707 (1/ sqrt(2)) . This response value is often called the
half power point or 3 dB point.
Figure B.9 Butterworth Frequency Response for n = 3 and 5.
It should be noted that the use of the log scale allowed us to translate what was an order of a polynomial, to be an algebraic problem.
Part 2) Chebyshev or Equal-Ripple Filters
Another class of NLP filters are based on the Chebyshev polynomials. The Chebyshev polynomials are defined as follows,
[latex]C_n(x) = cos( n*arccos( x ) )[/latex] for |x| < 1
and [latex]cosh( n*arccosh( x ) )[/latex] for |x| > 1
These really are polynomials and can be generated by the recursion
[latex]C_{n+1}(x) = 2 x C_n(x) - C_{n-1}(x)[/latex]
From the definition,
[latex]C_0(x) = 1[/latex]
[latex]C_1(x) = x[/latex]
[latex]C_2(x) = 2 x C_1(x) - C_0(x) = 2 x^2 - 1[/latex]
[latex]C_3(x) = 4 x^3 - 3 x[/latex]
We will then define the frequency response of the Chebyshev filter as being
[latex]| H_{n, \epsilon}(w)|^2 = \frac{1}{( 1 + \epsilon^2 C_n^2(w) )[/latex]
where [latex]\epsilon[/latex]e is a scalar with 0 < e < 1
and [latex]C_n(w)[/latex] is the nth order Chebyshev polynomial.
Sketches of [latex]| H_{n, \epsilon}(w)|^2[/latex] for n = 4 and 5, are included in Figure B.10. From the plot we can see that at the cutoff frequency the filter response is not 0.707 as it was with
the Butterworth, but rather a function of e. Also the response varies between 1 and [latex]\frac{1}{1+ \epsilon^2}[/latex] in the pass band (0<w<1), which gives rise to these filters other name,
Equi-Ripple filters. One final point about the frequency response, if the order of the filter is even, it will have a response of 1/(1+e2) at w = 0, while if it is odd, it will have a response of 1.
Also, the number of bumps is equal to the order.
The question is, how do we set e and n to match our NLP. We begin by calculating the
Figure B.10 Frequency Response of Chebyshev NLP Filters.
The ripple in the pass-band of the Chebyshev filter is a function of only e. The larger e, the larger the ripple and the quicker the fall off in the stop band. The Pass-Band Tolerance (PBT) will
therefore set the value of e, and is a trade off situation. We will start expressing the PBT in dB, which is 20 log of the ratio of the highest gain ( 1 ) and the lowest ( [latex]\frac{1}{\sqrt{1 + \
epsilon^2} }[/latex] )
[latex]PBT = 20 * log10 ( \frac{1}{ \frac{1}{ \sqrt{ 1 + \epsilon^2 } } } ) [/latex]
[latex]PBT = 20 log10 ( sqrt( 1 + \epsilon^2 ) )[/latex]
[latex]PBT = 10 log10 ( 1 + \epsilon^2 )[/latex]
[latex]\epsilon = sqrt( 10PBT/10 - 1 ) [/latex] (B.7)
Equation B.7 is our first design equation, used to compute [latex]\epsilon[/latex] for the Chebyshev filter.
Next we will determine the order of filter needed. This equation will be derived based on the desired response at the stop frequency.
[latex]SBA = 20* log10 ( \frac{1}{\sqrt{1 + \epsilon^2 C_n^2(ws)}} ) [/latex]
[latex]SBA = -10* log10 ( 1 + \epsilon^2 C_n^2(ws) ) [/latex]
[latex]\frac{SBA}{-10} = log10 ( 1 + \epsilon^2 C_n^2(ws) ) [/latex]
[latex]C_n^2(ws) = \frac{( 10*\frac{SBA}{-10} - 1 )}{ \epsilon^2 } [/latex]
Applying the definition of Cn ,
[latex]cosh( n * arccosh( ws ) ) = \sqrt{ \frac{ ( \frac{10*SBA}{-10} - 1 )}{\epsilon} } [/latex]
Solving for n yields
[latex]n = arccosh( \sqrt( \frac{10*\frac{SBA}{-10} - 1 ) }{e} ) / arccosh( ws ) [/latex] (B.8)
Equation is the second design equation for the Chebyshev filter.
Now the reason we have defined the Chebyshev filter response as we did was because it can be shown that the polynomial [latex]( 1 + \epsilon^2 C_n^2(ws) )[/latex]will factor to have zeros at
[latex]s_k = a cos( \pi \frac{( 2 k + n - 1 )}{( 2 n ) }) + j b sin( \pi \frac{( 2 k + n - 1 )}{ ( 2 n ) }) [/latex]
[latex]a = 0.5 * ( (\sqrt{ 1 + \frac{1}{\epsilon^2} } + \frac{1}{e} )^{\frac{1}{n}} )- ( \sqrt{ 1 + \frac{1}{\epsilon^2} } + \frac{1}{e} )^{\frac{-1}{n}} )[/latex]
[latex]b = 0.5 * ( ( \sqrt{ 1 + \frac{1}{\epsilon^2} } + \frac{1}{e} )^{\frac{1}{n}} ) + ( \sqrt{ 1 + \frac{1}{\epsilon^2} } + \frac{1}{e} )^{\frac{-1}{n}} )[/latex]
These equations are kind of tedious. Perhaps a simpler way to view them is as poles which lie on an ellipse with major and minor axis of width [latex]cosh( \frac{1}{n} arccosh(\frac{1}{e} ) )[/latex]
and [latex]sinh( \frac{1}{n} arcsinh(\frac{1}{e} ) )[/latex], respectively. Figure B.11 shows a sketch of s-plane and the general shape of the ellipse on which the poles lie.
Figure B.11 Plot of s-Plane with Chebyshev Filter Ellipse.
These poles correspond to the scaling of the real and imaginary parts of the Butterworth poles by the factors a and b, respectively. Note at 1 radian/sec, H(jw) is not equal to 0.707 or -3 dB, like
the Butterworth, but rather is the ripple value.
Example: Assume a NLP filter with the following parameters is needed (PBT = 1 dB, SBA = -40dB, and ws = 10). Then
[latex]\epsilon = \sqrt{( \frac{10^1}{10} - 1 ) } = 0.5088 [/latex]
[latex]n = \frac{arccosh( \frac{( 10^{\frac{-40}{-10}} - 1 )}{\epsilon^2} ) }{arccosh( 10 )} = 1.99 [/latex]
[latex]n = 2[/latex]
The poles of this filter would now be found as follows,
a = 0.77622 and b = 1.2659
[latex]s1 = 0.77622 cos( \frac{3}{4} \pi ) + j 1.2659 sin( \frac{3}{4} \pi )[/latex]
[latex]s1 = -0.54887 + j 0.89513[/latex]
[latex]s2 = s1* = -0.54887 - j 0.89513 [/latex]
[latex]H(s) = \frac{A}{( (s-s1) (s-s2) ) } = \frac{A}{( s^2 + 1.0977 s + 1.1025 ) [/latex]
Since n is even, we know that
[latex]H(0) = \frac{A }{1.1025} = \frac{1}{\sqrt{ 1 + \epsilon^2}} [/latex]
[latex]A = \frac{1.1025}{ \sqrt{ 1 + \epsilon^2 } = 0.9826 [/latex]
[latex]H(s) = \frac{0.9826}{s^2 + 1.0977 s + 1.1025 } [/latex]
Once again we will view H(s) as a surface in the s-plane. The following is a MATLAB surface plot and script that shows H(s). Note the imaginary axis is once again the frequency response. And since
the poles are pulled in closer to the imaginary axis (elliptical nature), there FR ripples more than in the case of the Butterworth.
Figure B.12 Surface plot of H(s) for Second-Order 1-dB Ripple Chebyshev NLP Filter.
limit = 5;
% Set up z plane as real and imaginary parts of z
[x,y] = meshgrid( -2.25:0.05:0, -2.25:0.05:2.25 );
s = x + 1i*y;
H = 0.9826 ./ ( s.*s + 1.0977 * s + 1.1025 );
surf( y,x,min(abs(H),limit));
view( [ 155 45]);
title (‘H(s) of Second Order Chebyshev’);
xlabel( ‘Imaginary’ );
ylabel( ‘Real’);
Section D) Back Transformations
Once the NLP design is decided upon, the H(s) must be translated back to the appropriate filter type. These transformations were described in Section A. In this section, the transformations will be
reiterated here, and shown how the work to convert a NLP to any given filter.
Part 1) NLP to LP.
The transformation was given in equation 1, and here we will apply this transformation to a [latex]H_{nlp}(\hat s)[/latex] to create and [latex]H_{lp}(s)[/latex].
[latex]H_{lp}(s) = H_{nlp}(\hat s) |_{\hat s = \frac{s}{wp} } [/latex]
where wp is the desired cutoff frequency.
Example: A third order NLP Butterworth is to be translated to have a cut off frequency of 500 Hz or 1000 [latex]\pi[/latex] radians per second.
[latex]H_{lp}(s) = H_{nlp}(\hat s) |_{\hat s=\frac{s}{1000 \pi} } [/latex]
[latex]H_{lp}(s) = \frac{1}{\frac{s}{1000 \pi}+1 } * \frac{1}{\frac{ s^2}{10^6 \pi^2} + \frac{s}{10^3 \pi}+1} [/latex]
Part 2) NLP to HP.
As before the actual transformation was given earlier, and here we will apply it to an [latex]H_{nlp}(\hat s)[/latex] to create the desired filter.
[latex]H_{hp}(s) = H_{nlp}(\hat s) |_{\hat s=\frac{wp}{s}} [/latex]
Where wp is the cutoff frequency of the desired high-pass filter.
Example: A third-order NLP Butterworth is to be translated to a cut off frequency of 200 radians/second.
[latex]H_{hp}(s) = \frac{1}{( \frac{200}{s} + 1 ) } * \frac{1}{(\frac{40000}{s^2} + \frac{ 200 }{ s } + 1 )}[/latex]
[latex]H_{hp}(s) = \frac{\frac{s }{200}}{\frac{s}{200} + 1 } * \frac{ \frac{s^2}{40000}}{\frac{s^2}{40000} + \frac{s}{200} + 1 }[/latex]
In the previous example, we can see that the translated Butterworth still has the set of poles distributed around in a circle, but a set of three zeros have been added at the origin. This force the
response to zero here, and then the zeros and poles compensate each other at high frequencies.
Part 3) NLP to BP.
The last transformation will be the NLP to Band-Pass fitler as given in equation 3, and here we will apply this transformation to a Hnlp(s’) to create and Hlp(s).
[latex]H_{bp}(s) = H_{nlp}(\hat s) |_{\hat s=\frac{s*s + wpu*wpl}{s * (wpu-wpl)} } [/latex]
where wpu is the desired upper cutoff frequency
and wpl is the desired lower cutoff frequency.
Example: Again we will translate a third order NLP Butterworth; however, this time we will be translating to a BP, with wpl = 100 and wpu = 1000 radians per second.
[latex]H_{bp}(s) = \frac{1}{frac{s^2 + 10^5}{(s*900)} +1} *\frac{1}{(\frac{s^2 + 10^5 }{(s 900)^2} + \frac{s^2 + 10^5} {s 900}+ 1 }[/latex]
[latex]H_{bp}(s) = \frac{\frac{s}{10^5} }{ ( \frac{s^2}{9*10^8 } + \frac{s}{10^5} + 1 ) } *\frac{ (9900 s)^2 }{( s^4 + 9* 10^3 s^3 + 2*10^6 s^2 + 9*10^7 s + 10^{10} + 900^2} [/latex]
This last equation will need to be factored in order to be implemented in a reasonable fashion.
Part 4) Implementation
Once a valid H(s) has been derived for our filter, there are a variety of ways to implement the filter. A classic and straight forward approach is to use the Sallen-Key topology. In this case H(s)
should be broken into second order pairs, which occurs naturally in the creation of the transfer function. Each second order pair can be implemented using one of the following circuits.
Figure B.13 Sallen-Key Circuit for Low-Pass Second Order Pair.
The transfer function for the circuit in Figure B.13 is given as
[latex]H(s) = \frac{\omega_o}{s^2+ 2\alpha*s+\omega_o^2}[/latex]
where [latex]\omega_o = \frac{1}{sqrt( R1*R2*C1*C2 )}[/latex] and [latex]2 \alpha = \frac{1}{C1} * \frac{R1+R2}{R1*R2}[/latex]
The four components of R1, R2, C1 and C2 give sufficient flexibility to create any second order system that is needed.
Figure B.14 Sallen-Key Circuit for High-Pass Second Order Pair.
The transfer function for the circuit in Figure B.14 is given as
[latex]H(s) = \frac{s^2}{s^2+ 2\tau*\omega_0*s+\omega_o^2}[/latex]
where [latex]\omega_o = \frac{1}{sqrt( R1*R2*C1*C2 )}[/latex] and [latex](2\tau*\omega_0) = \frac{1}{C1} * \frac{C1+C2}{R1*C1*C2}[/latex]
Again the four components of R1, R2, C1 and C2 give sufficient flexibility to create any second order system that is needed. | {"url":"https://kstatelibraries.pressbooks.pub/dsp-basics/back-matter/appendix-b-analog-filter-design/","timestamp":"2024-11-14T21:09:40Z","content_type":"text/html","content_length":"107611","record_id":"<urn:uuid:22e01692-9695-4516-b609-0a5835c9ad08>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00626.warc.gz"} |
Understanding Mathematical Functions: How To Know If Its A Function Or
Introduction to Mathematical Functions
A mathematical function is a fundamental concept in mathematics, with wide-ranging applications in various fields such as science, engineering, and economics. It describes a relationship between a
set of inputs and a set of possible outputs, where each input is related to exactly one output. In this blog post, we will discuss the definition of a mathematical function, its importance, and
provide guidance on how to identify whether a given relationship is a function or not.
A Definition of a mathematical function and its importance
A mathematical function can be defined as a rule or a set of rules that associates each element in a set (the domain) with exactly one element in another set (the codomain). Functions are essential
in modeling real-world phenomena, analyzing data, and solving mathematical problems. They provide a way to represent and manipulate relationships between quantities, making them indispensable in
fields such as calculus, statistics, and computer science.
Overview of the purpose of the blog post
The primary purpose of this blog post is to provide readers with a clear understanding of how to determine whether a given relationship between variables constitutes a function. By discussing the key
characteristics of functions and providing practical examples, readers will be equipped with the knowledge and tools necessary to differentiate between functions and non-functions.
Brief mention of the historical context and evolution of the concept of functions
The concept of mathematical functions has a rich historical background, with contributions from mathematicians such as Leonhard Euler, Pierre-Simon Laplace, and Gottfried Wilhelm Leibniz. Over time,
the notion of functions has evolved to encompass a wide variety of mathematical structures and applications, leading to their pervasive presence in modern mathematics and its applications.
Key Takeaways
• Functions have only one output for each input.
• Check for repeating x-values in a relation.
• Vertical line test determines if it's a function.
• Understand domain and range to identify functions.
• Use algebraic methods to verify if it's a function.
The Concept of a Function
Understanding mathematical functions is essential in the field of mathematics and various other disciplines. A function is a fundamental concept in mathematics that describes the relationship between
two sets of numbers. Let's delve into the detailed definition of a function in mathematical terms, the relationship between independent and dependent variables, and the function notation \( f(x) \).
A Detailed definition of a function in mathematical terms
In mathematical terms, a function is a relation between a set of inputs (the independent variable) and a set of possible outputs (the dependent variable). Each input is related to exactly one output.
This means that for every value of the independent variable, there is only one corresponding value of the dependent variable. This one-to-one correspondence is a key characteristic of a function.
Explanation of the relationship between independent variables and dependent variables
The independent variable is the input of the function, and the dependent variable is the output. The value of the dependent variable depends on the value of the independent variable. For example, in
the function \( f(x) = 2x + 3 \), 'x' is the independent variable, and '2x + 3' is the dependent variable. The value of 'x' determines the value of \( f(x) \).
Introduction to the function notation \( f(x) \) and how it is used to represent functions
The function notation \( f(x) \) is a way to represent a function. It is read as 'f of x' and is used to indicate that the function 'f' operates on the input 'x'. For example, if we have a function \
( f(x) = x^2 \), we can say 'f of 3' to mean the value of the function when the input is 3. The function notation provides a concise and standardized way to express functions.
The Vertical Line Test
When it comes to determining whether a curve is a function or not, the vertical line test is a graphical method that can be used to make this determination. By applying this test, you can easily
identify whether a given curve represents a function or not.
Explanation of the vertical line test as a graphical method to determine if a curve is a function
The vertical line test is a simple yet effective way to determine if a curve represents a function. The test involves visually inspecting the graph of the curve and checking if any vertical line
intersects the curve more than once. If a vertical line intersects the curve at more than one point, then the curve does not represent a function. On the other hand, if every vertical line intersects
the curve at most once, then the curve is a function.
Step-by-step instructions on how to perform the vertical line test
To perform the vertical line test, follow these step-by-step instructions:
• Step 1: Obtain the graph of the curve that you want to test.
• Step 2: Visualize a series of vertical lines that can be drawn across the graph.
• Step 3: Check if any of the vertical lines intersect the curve at more than one point.
• Step 4: If any vertical line intersects the curve at more than one point, then the curve is not a function. If every vertical line intersects the curve at most once, then the curve is a function.
Examples of graphs where the vertical line test is applied
Let's consider a few examples of graphs where the vertical line test is applied:
• Example 1: The graph of a straight line such as y = 2x + 3 passes the vertical line test as every vertical line intersects the line at most once, indicating that it represents a function.
• Example 2: The graph of a circle does not pass the vertical line test as there are vertical lines that intersect the circle at more than one point, indicating that it does not represent a
• Example 3: The graph of a parabola such as y = x^2 also passes the vertical line test as every vertical line intersects the parabola at most once, indicating that it represents a function.
Domain and Range
When it comes to understanding mathematical functions, the concepts of domain and range play a crucial role in determining whether a relationship is a function or not. Let's delve into the
definitions and explanations of domain and range, and how they can help in understanding whether a relationship is a function.
A Definition and explanation of the domain of a function
The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. In simpler terms, it is the set of all x-values that can be plugged into the
function to produce a valid output. For example, in the function f(x) = x^2, the domain is all real numbers, as any real number can be squared to produce a valid output.
B Definition and explanation of the range of a function
The range of a function, on the other hand, refers to the set of all possible output values (y-values) that the function can produce for the corresponding input values in the domain. In the function
f(x) = x^2, the range is all non-negative real numbers, as the square of any real number is non-negative.
C How the concepts of domain and range can help in understanding whether a relationship is a function
Now, let's see how the concepts of domain and range can help in understanding whether a relationship is a function. One of the key characteristics of a function is that each input value (from the
domain) corresponds to exactly one output value (from the range). If there is any input value that corresponds to more than one output value, then the relationship is not a function.
By analyzing the domain and range of a given relationship, we can determine whether each input value has a unique output value. If there are no repetitions in the output values for different input
values, then the relationship is indeed a function. On the other hand, if there are multiple output values for the same input value, then the relationship fails the test for being a function.
In conclusion, the concepts of domain and range are essential in understanding whether a relationship is a function. By analyzing the input-output behavior of a given relationship, we can determine
whether it satisfies the criteria for being a function, based on the uniqueness of input-output pairs.
One-to-One Mapping and the Function Criteria
When it comes to understanding mathematical functions, one of the key criteria is the concept of one-to-one mapping. This criterion is essential in determining whether a relationship between two sets
of numbers can be considered a function or not. In this chapter, we will clarify the one-to-one mapping criterion, discuss the importance of each input having a unique output for a relationship to be
a function, and provide examples to illustrate both function and non-function scenarios based on one-to-one mapping.
A Clarification of the one-to-one mapping criterion for functions
One-to-one mapping refers to a situation where each element in the domain (input) is paired with exactly one element in the range (output). In other words, there is a unique correspondence between
the elements of the domain and the elements of the range. This means that no two different elements in the domain can be paired with the same element in the range.
B Discussion on the importance of each input having a unique output for a relationship to be a function
The importance of each input having a unique output in a relationship cannot be overstated when it comes to defining a function. If a relationship fails to satisfy this criterion, it cannot be
considered a function. This is because a function is a special type of relationship where each input value (or element in the domain) is associated with exactly one output value (or element in the
For example, if we consider a simple function f(x) = x^2, for every input value of x, there is a unique output value of x^2. This is what makes it a function. If there were multiple output values for
a single input value, it would violate the one-to-one mapping criterion and the relationship would not be a function.
C Examples showing both function and non-function scenarios based on one-to-one mapping
Let's consider the following examples to illustrate both function and non-function scenarios based on the one-to-one mapping criterion:
• Function Scenario: Consider the function f(x) = 2x + 3. For every input value of x, there is a unique output value of 2x + 3. This satisfies the one-to-one mapping criterion, making it a
• Non-Function Scenario: Now, let's consider the relationship where each student in a class is paired with their favorite color. If two students have the same favorite color, this violates the
one-to-one mapping criterion, making it a non-function scenario.
These examples demonstrate the importance of the one-to-one mapping criterion in determining whether a relationship between two sets of numbers can be considered a function or not.
Common Mistakes and Troubleshooting
When it comes to identifying mathematical functions, there are several common misconceptions and errors that people often make. Understanding these mistakes and knowing how to troubleshoot them is
essential for accurately determining whether a relationship is a function or not.
Identification of common misconceptions and errors made when determining if a relationship is a function
• Confusing input and output: One common mistake is to confuse the input and output values in a relationship. It's important to remember that in a function, each input value (x) must correspond to
exactly one output value (y).
• Ignoring vertical line test: Some individuals overlook the vertical line test, which is a crucial method for determining if a graph represents a function. If a vertical line intersects the graph
at more than one point, then the relationship is not a function.
• Not considering domain and range: Another misconception is failing to consider the domain and range of a relationship. A function must have a unique output for each input within its domain.
• Assuming linearity: Many people mistakenly believe that all mathematical relationships are linear functions. However, functions can take various forms, including quadratic, exponential, and
trigonometric functions.
Tips on how to avoid these mistakes and correct misconceptions
To avoid these common mistakes and correct misconceptions when determining if a relationship is a function, consider the following tips:
• Understand the definition of a function: Familiarize yourself with the formal definition of a function, which states that each input value maps to exactly one output value.
• Use the vertical line test: Always apply the vertical line test to a graph to determine if it represents a function. If any vertical line intersects the graph at more than one point, then the
relationship is not a function.
• Consider the domain and range: Pay attention to the domain and range of a relationship to ensure that each input has a unique output within the specified domain.
• Explore different types of functions: Be open to the possibility of encountering various types of functions, including non-linear functions such as quadratic, exponential, and trigonometric
Troubleshooting techniques for confusing cases and how to resolve ambiguities
When faced with confusing cases and ambiguities in determining if a relationship is a function, consider the following troubleshooting techniques:
• Examine the mapping of input to output: Take a closer look at how each input value maps to an output value. If there are any instances of multiple outputs for a single input, then the
relationship is not a function.
• Check for repeating input values: Look for repeating input values that result in different output values. If this occurs, then the relationship is not a function.
• Consult with a peer or instructor: If you're still unsure about whether a relationship is a function, seek guidance from a peer or instructor who can provide additional insight and clarification.
• Utilize online resources: Take advantage of online resources, such as interactive graphing tools and tutorials, to further understand and visualize the concept of functions.
Conclusion & Best Practices
After delving into the intricacies of mathematical functions, it is important to recapitulate the key points discussed in this blog post, highlight best practices for consistently identifying
functions, and encourage further exploration of the topic with additional resources.
A Recapitulation of the key points discussed in the blog post
• Definition of a Function: A function is a relation between a set of inputs and a set of possible outputs, where each input is related to exactly one output.
• Ways to Identify a Function: We discussed various methods to determine if a given relation is a function, such as the vertical line test, mapping diagrams, and the definition of a function
• Types of Functions: We explored different types of functions, including linear, quadratic, exponential, and trigonometric functions, and how to recognize their characteristics.
Best practices to consistently correctly identify functions in various contexts
• Understand the Definition: Always start by understanding the definition of a function and apply it to the given relation or equation.
• Use Visual Tools: Utilize visual tools such as graphs, mapping diagrams, and the vertical line test to visually analyze the relationship between inputs and outputs.
• Check for Repeated Inputs: Ensure that each input is related to exactly one output, and there are no repeated inputs with different outputs.
• Consider Domain and Range: Analyze the domain and range of the relation to determine if it satisfies the criteria of a function.
• Practice Problem-Solving: Regularly practice solving problems related to functions to enhance your understanding and application of function identification.
Encouragement for further exploration of the topic and additional resources for those who want to learn more
Understanding mathematical functions is a fundamental aspect of mathematics and has wide-ranging applications in various fields. For those who want to delve deeper into this topic, there are numerous
additional resources available, including textbooks, online courses, and interactive tutorials. Exploring advanced topics such as calculus, differential equations, and mathematical modeling can
further enhance your understanding of functions and their significance in real-world scenarios.
Continued exploration and practice will not only solidify your grasp of functions but also open doors to new and exciting mathematical concepts. | {"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-how-to-know-if-its-a-function","timestamp":"2024-11-13T02:00:14Z","content_type":"text/html","content_length":"227938","record_id":"<urn:uuid:b0f74bc6-b4f8-446b-84ce-aef941a501ed>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00741.warc.gz"} |
Suhas Patankar | Academic Influence
Suhas Patankar
Most Influential Person Now
Indian mechanical engineer
Suhas Patankar's AcademicInfluence.com Rankings
Why Is Suhas Patankar Influential?
(Suggest an Edit or Addition)
According to Wikipedia, “Suhas V. Patankar is an Indian mechanical engineer. He is a pioneer in the field of computational fluid dynamics and Finite volume method. He is currently a Professor
Emeritus at the University of Minnesota. He is also president of Innovative Research, Inc. Patankar was born in Pune, Maharashtra, India.”
(See a Problem?)
Suhas Patankar's Published Works
Published Works
• A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows (1972) (6114)
• Fully Developed Flow and Heat Transfer in Ducts Having Streamwise-Periodic Variations of Cross-Sectional Area (1977) (750)
• Heat and Mass Transfer in Boundary Layers. 2nd edition. By S. V. PATANKAR and D. B. SPALDING. Intertext Books, 1970. 255 pp. £6. (1971) (706)
• Pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations (1988) (400)
• A CONTROL VOLUME FINITE-ELEMENT METHOD FOR TWO-DIMENSIONAL FLUID FLOW AND HEAT TRANSFER (1983) (316)
• Numerical Prediction of Flow and Heat Transfer in a Parallel Plate Channel With Staggered Fins (1987) (186)
• A CONTROL VOLUME-BASED FINITE-ELEMENT METHOD FOR SOLVING THE NAVIER-STOKES EQUATIONS USING EQUAL-ORDER VELOCITY-PRESSURE INTERPOLATION (1984) (169)
• Forced Convection Heat Transfer from a Shrouded Fin Array with and without Tip Clearance (1978) (140)
• A NUMERICAL METHOD FOR CONDUCTION IN COMPOSITE MATERIALS, FLOW IN IRREGULAR GEOMETRIES AND CONJUGATE HEAT TRANSFER (1978) (135)
• A finite-difference procedure for solving the equations of the two-dimensional boundary layer (1967) (130)
• Treatment of irregular geometries using a Cartesian coordinates finite-volume radiation heat transfer procedure (1994) (127)
• An analysis of the effect of plate thickness on laminar flow and heat transfer in interrupted-plate passages (1981) (108)
• Heat transfer and fluid flow analysis of interrupted-wall channels with application to heat exchangers (1977) (79)
• Use of Computational Fluid Dynamics for Calculating Flow Rates Through Perforated Tiles in Raised-Floor Data Centers (2003) (76)
• A methodology for the design of perforated tiles in raised floor data centers using computational flow analysis (2000) (74)
• Numerical solution of moving boundary problems by boundary immobilization and a control-volume-based finite-difference scheme (1981) (69)
• Simulating Boundary Layer Transition With Low-Reynolds-Number k–ε Turbulence Models: Part 1—An Evaluation of Prediction Characteristics (1991) (65)
• Combined forced and free laminar convection in the entrance region of an inclined isothermal tube (1988) (57)
• Effect of circumferentially nonuniform heating on laminar combined convection in a horizontal tube (1978) (56)
• Relationships Among Boundary Conditions and Nusselt Numbers for Thermally Developed Duct Flows (1977) (50)
• Simulating Boundary Layer Transition With Low-Reynolds-Number k–ε Turbulence Models: Part 2—An Approach to Improving the Predictions (1991) (48)
• Monte Carlo Solutions for Radiative Heat Transfer in Irregular Two-Dimensional Geometries (1995) (45)
• Mathematical modeling of heat transfer, condensation, and capillary flow in porous insulation on a cold pipe (2004) (41)
• Heat and mass transfer in boundary layers: A general calculation procedure (1970) (39)
• SOLUTION OF SOME TWO-DIMENSIONAL INCOMPRESSIBLE FLUID FLOW AND HEAT TRANSFER PROBLEMS, USING A CONTROL VOLUME FINITE-ELEMENT METHOD (1983) (36)
• Two-Equation Low-Reynolds-Number Turbulence Modeling of Transitional Boundary Layer Flows Characteristic of Gas Turbine Blades. Ph.D. Thesis. Final Contractor Report (1988) (31)
• Computational Modeling of Electroslag Remelting (ESR) Process Used for the Production of High‐Performance Alloys (2013) (30)
• Robust semidirect finite difference methods for solving the Navier–Stokes and energy equations (1989) (28)
• CFD MODEL FOR JET FAN VENTILATION SYSTEMS (2000) (28)
• SOLUTION OF THE CONVECTION-DIFFUSION EQUATION BY A FINITE-ELEMENT METHOD USING QUADRILATERAL ELEMENTS (1985) (26)
• Numerical prediction of fluid flow and heat transfer in a circular tube with longitudinal fins interrupted in the streamwise direction (1990) (25)
• SOLUTION OF SOME TWO-DIMENSIONAL INCOMPRESSIBLE FLOW PROBLEMS USING A CURVILINEAR COORDINATE SYSTEM BASED CALCULATION PROCEDURE (1988) (25)
• Development of generalized block correction procedures for the solution of discretized navier-stokes equations (1987) (23)
• Numerical Prediction of Flow, Heat Transfer, Turbulence and Combustion: Selected Works of Professor D. Brian Spalding (1983) (23)
• SOLUTION OF THE CONVECTION-DIFFUSION EQUATION BY A FINITE-ELEMENT METHOD USING QUADRILATERAL ELEMENTS (1985) (23)
• Predictions of Heat Transfer in Compressor Cylinders (1986) (22)
• A streamline upwind control volume finite element method for modeling fluid flow and heat transfer problems (1993) (20)
• Heat and mass transfer in turbulent boundary layers (1967) (20)
• Simulating boundary layer transition with low-Reynolds-number k-epsilon turbulence models. I - An evaluation of prediction characteristics. II - An approach to improving the predictions (1991)
• Prediction of Turbulent Flow and Heat Transfer in a Rotating Square Duct With a 180 deg. Bend (1994) (16)
• Prediction of film cooling with lateral injection (1990) (14)
• Prediction of transition on a flat plate under the influence of free-stream turbulence using low-Reynolds-number two-equation turbulence models (1987) (13)
• PERFORMANCE OF A MULTIGRID METHOD WITH AN IMPROVED DISCRETIZATION SCHEME FOR THREE-DIMENSIONAL FLUID FLOW CALCULATIONS (1996) (13)
• The periodic thermally developed regime in ducts with streamwise periodic wall temperature or heat flux (1978) (13)
• NUMERICAL METHOD FOR THE PREDICTION OF FREE SURFACE FLOWS IN DOMAINS WITH MOVING BOUNDARIES (1997) (11)
• Analysis of the effect of bypass on the performance of heat sinks using flow network modeling (FNM) (1999) (11)
• APPLICATION OF THE PARTIAL ELIMINATION ALGORITHM FOR SOLVING THE COUPLED ENERGY EQUATIONS IN POROUS MEDIA (2004) (10)
• A CONTROL-VOLUME FINITE-ELEMENT METHOD FOR PREDICTING FLOW AND HEAT TRANSFER IN DUCTS OF ARBITRARY CROSS SECTIONS—PART I: DESCRIPTION OF THE METHOD (1987) (10)
• Numerical prediction of turbulent oscillating flow and associated heat transfer (1991) (10)
• NUMERICAL ANALYSIS OF WATER SPRAY SYSTEM IN THE ENTRANCE REGION OF A TWO-DIMENSIONAL CHANNEL USING LAGRANGIAN APPROACH (1994) (10)
• Computationally two-dimensional finite-difference model for hollow-fibre blood-gas exchange devices (1991) (10)
• A flow network analysis of a liquid cooling system that incorporates microchannel heat sinks (2004) (10)
• HEAT TRANSFER AUGMENTATION DUE TO BUOYANCY EFFECTS IN THE ENTRANCE REGION OF A SHROUDED FIN ARRAY (1988) (9)
• LAMINAR MIXED CONVECTION IN A HORIZONTAL SEMICIRCULAR DUCT WITH AXIALLY NONUNIFORM THERMAL BOUNDARY CONDITION ON THE FLAT WALL (1994) (9)
• Numerical study of heat transfer from a rotating cylinder with external longitudinal fins (1983) (8)
• A new low-Reynolds-number turbulence model for prediction of transition on gas turbine blades (2001) (8)
• Calculation of the three-dimensional boundary layer with solution of all three momentum equations (1975) (7)
• LAMINAR HEAT TRANSFER IN A PIPE SUBJECTED TO A CIRCUMFERENTIALLY VARYING EXTERNAL HEAT TRANSFER COEFFICIENT (1978) (7)
• Prediction of thermal history of preforms produced by the clean metal spray forming process (2002) (7)
• CFD MODEL FOR TRANSVERSE VENTILATION SYSTEMS (2000) (6)
• ANALYSIS OF HEAT TRANSFER FROM A ROTATING CYLINDER WITH CIRCUMFERENTIAL FINS. (1984) (6)
• Heat transfer across a turbulent boundary layer: Application of a profile method to the step-wall-temperature problem (1966) (6)
• Computational Modeling of the Vacuum Arc Remelting ( VAR ) Process Used for the Production of Ingots of Titanium Alloys (2007) (6)
• Prediction of turbine blade heat transfer (1985) (6)
• Use of Flow Network Modeling ( FNM ) for Enhancing the Design Process of Electronic Cooling Systems (2000) (5)
• Recent Developments in the Solution of Radiation Heat Transfer Using the Discrete Ordinates Method (1993) (5)
• a Control-Volume Finite-Element Method for Predicting Flow and Heat Transfer in Ducts of Arbitrary Cross-Sections - Part II: Application to Some Test Problems (1987) (5)
• Computational Method for Characterization of a Microchannel Heat Sink Involving Two-Phase Flow (2005) (5)
• Laminar Heat Transfer of a Radiating Fluid in a Backward Facing Step Flow (1993) (4)
• Diffusion from a line source in a turbulent boundary layer: Comparison of theory and experiment (1965) (4)
• Fully coupled solution of the equations for incompressible recirculating flows using a penalty-function finite-difference formulation (1990) (4)
• Computational method for characterization of a microchannel heat sink with multiple channels involving two-phase flow (2006) (3)
• A PARTIALLY PARABOLIC CALCULATION PROCEDURE FOR DUCT FLOWS IN IRREGULAR GEOMETRIES. PART I: FORMULATION (1990) (3)
• Computation of the Turbulent Mixing in Curved Ejectors (1980) (3)
• Use of subdomains for flow computations in complex geometries (1986) (3)
• Erratum: “Effect of Circumferentially Nonuniform Heating on Laminar Combined Convection in a Horizontal Tube” (Journal of Heat Transfer, 1978, 100, pp. 63–70) (1978) (3)
• IMPROVED PRODUCTIVITY WITH USE OF FLOW NETWORK MODELING ( FNM ) IN ELECTRONIC PACKAGING (2000) (3)
• Computational method for generalized analysis of pumped two-phase cooling systems and its application to a system used in data-center environments (2010) (2)
• Efficient numerical techniques for complex fluid flows (1985) (2)
• ENGINEERING & GINNING Analysis and Design of a Drying Model for Use in the Design of Starch-Coated Cottonseed Dryers (2001) (2)
• A PARTIALLY PARABOLIC CALCULATION PROCEDURE FOR DUCT FLOWS IN IRREGULAR GEOMETRIES. PART II: TEST PROBLEMS (1990) (2)
• Computer in analysis and design (1983) (2)
• Computational method for system-level analysis of two-phase pumped loops for cooling of electronics (2008) (2)
• An evaluation of three spatial differencing schemes for the discrete ordinates method in participating media (1993) (2)
• Numerical Study of Flow and Heat Transfer for Circular Jet Impingement on the Bottom of a Cylindrical Cavity (1993) (2)
• Radiation heat transfer calculations using a control-angle, control-volume-based discrete ordinates method (1993) (1)
• Effect of approach-flow velocity and temperature nonuniformities on boundary-layer flow and heat transfer (1977) (1)
• Studies of Gas Turbine Heat Transfer: Airfoil Surfaces and End-Wall Cooling Effects (1988) (1)
• Experimental and computational studies of film cooling with compound angle injection (1995) (1)
• Development of low Reynolds number two equation turbulence models for predicting external heat transfer on turbine blades (1986) (1)
• NUMERICAL INVESTIGATION OF WATER SPRAY SYSTEM IN THE ENTRANCE REGION OF A 2 D CHANNEL USING LAGRANGIAN APPROACH. (1994) (1)
• Improved numerical methods for turbulent viscous flows aerothermal modeling program, phase 2 (1988) (1)
• A CALCULATION PROCEDURE FOR HEAT TRANSFER BY FORCED CONVECTION THROUGH TWO-DIMENSIONAL UNIFORM-PROPERTY TURBULENT BOUNDARY LAYERS ON SMOOTH IMPERMEABLE WALLS (2019) (1)
• Book review, R.H. Gallagher, O.C. Zienkiewicz, J.T. Oden, M. Morandi Cecchi, C. Taylor (Eds.)Finite Elements in Fluids, Vol. 3, John Wiley, New York (1978), p. 396 (1981) (0)
• A low-Reynolds-number two-equation turbulence model for predicting heat transfer on turbine blades (1987) (0)
• Closure to “Discussion of ‘Condensation on an Extended Surface’” (1980, ASME J. Heat Transfer, 102, pp. 186–187) (1980) (0)
• CFD MODELLING OF FLOW AND HEAT TRANSFER IN INDUSTRIAL APPLICATIONS (2006) (0)
• An analysis of laminar forced convection heat transfer in helically finned heat exchanger passages (1991) (0)
• Professor D. Brian Spalding on the occasion of his 65th birthday and retirement from Imperial College (1988) (0)
• Computation of Radiation Heat Transfer in Aeroengine Combustors (1996) (0)
• Numerical procedure for calculating steady/unsteady single-phase/two-phase three-dimensional fluid flow with heat transfer. (1979) (0)
• N 88-11151 EFFICIENT NUMERICAL TECHNIQUES FOR COMPLEX FLUID FLOWS (2003) (0)
• EQUATIONS FOR SOLVING THE NAVIER-STOKES AND ENERGY (1989) (0)
• Closure to “Discussion of ‘Heat Transfer and Fluid Flow Analysis of Interrupted-Wall Channels, with Application to Heat Exchangers’” (1979, ASME J. Heat Transfer, 101, pp. 188–189) (1979) (0)
• Computation of Three-Dimensional Flows in Ducts of Varying Cross Sections. (1983) (0)
• EXPERIMENTAL INVESTIGATION ON HEAT TRANSFER IN THE MANIFOLD OF REFRIGERATING COMPRESSORS * (2014) (0)
• Computer simulation of a conveyor dryer (1996) (0)
• Fluid Mechanics and Heat Transfer Research Related to High Temperature Gas Turbines. (1995) (0)
• Aerothermal modeling program. Phase 2, element A: Improved numerical methods for turbulent viscous recirculating flows (1987) (0)
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Other Resources About Suhas Patankar
What Schools Are Affiliated With Suhas Patankar?
Suhas Patankar is affiliated with the following schools: | {"url":"https://academicinfluence.com/people/suhas-patankar","timestamp":"2024-11-14T07:03:57Z","content_type":"text/html","content_length":"96987","record_id":"<urn:uuid:b61e40b1-0979-433a-b8f0-1f7d73f09f60>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00051.warc.gz"} |
Balance (2)
The statement of this problem is similar to that of problem. But here, the n weights do not need to be 2^0, 2^1, …, 2^n−1.
I.e., the problem is: Given n weights, we have to place all the weights on a balance, one after another, in such a way that the right pan is never heavier than the left pan. Please compute the number
of ways of doing this.
Input consists of several cases, each with the number of weights n followed by n different weights, all between 1 and 10^6. Assume 1 ≤ n ≤ 8.
For every case, print the number of correct ways of placing the weights on the balance. This number will never be larger than 10^7. | {"url":"https://jutge.org/problems/P17598_en","timestamp":"2024-11-05T18:42:36Z","content_type":"text/html","content_length":"22833","record_id":"<urn:uuid:8163f4c4-94ce-4a79-97e4-c6e15665d681>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00289.warc.gz"} |
16 Slice Pizza Diagram
16 Slice Pizza Diagram - That comes to approximately 50 inches, making it a formidable feast. Sure, you could buy two small pizzas, but you’ll end up paying more per square inch and you’ll probably
be hungry afterwards. Web a 16 inch pizza is a large pizza that can feed between 8 and 10 people. Fans of quad cities pizza will still be able to get their fix of the thin pies that mohr grew up
eating (they’re cut into strips with puffy. The large size allows plenty of room to customize the toppings to everyone’s individual preferences. It’s actually defined by the diameter of a circle that
has an area of 16 square inches. Extra large pizzas are only a few dollars more expensive than a large and give you a. Each type of slice has its own unique shape, size, and flavor, making it. Web
and what makes it so popular? Web given the diameter of 16 inches, the total size of a 16 inch pizza is 201 square inches, which is about 33% larger than a large (14 inch) pizza.
How Big is a 16" Pizza and How Many Does it Feed? Pizzeria Ortica
Next, divide the circumference by 2 to find the radius. I’ve created one for you below. If you have a standard or medium sized pizza, you are probably best to slice it into six slices, as per the
image above. **consumer advisory** consuming raw or undercooked foods meats, poultry, seafood or eggs may increase your risk of foodborne illness. Web.
Four Friends Shared 5 Pizzas Equally 35+ Pages Explanation in Google
Web and what makes it so popular? Next, divide the circumference by 2 to find the radius. 201 square inches may not mean much to you, so let’s put it into perspective. Unlike the pie chart, the
pizza. If you have a standard or medium sized pizza, you are probably best to slice it into six slices, as per the.
how big is 16 in pizza The Kitchened
It’s actually defined by the diameter of a circle that has an area of 16 square inches. Monkeypod kitchen by merriman in kapolei, oahu, hawaii. However, some people may find the slices too large for
their preferences. Now, why does pizza size matter? But what about the circumference, the edge of our pizza frontier?
Pizza Fractions Clipart (81 items) Teaching Resources
Web sometime in march, fifty/50 will close roots old town. But what about the circumference, the edge of our pizza frontier? View this post on instagram. Updated by the business over 3 months ago.
Web finally, cut each eighth in half to create 16 slices and remove the center slice to achieve six proportionate slices.
Big Chain Pizza Sizes and Crusts Comparisons Pizza Dimension
This will give you the circumference of the pizza. Kalua pork and pineapple pizza, hamakua wild mushroom and truffle oil pizza. View this post on instagram. **consumer advisory** consuming raw or
undercooked foods meats, poultry, seafood or eggs may increase your risk of foodborne illness. Extra large pizzas are only a few dollars more expensive than a large and give.
Slice of Pizza Diagram Infographics Set. Pieces Margherita. the Whole
To start with, slice the first slice from 9.00 to 3.00 (imagine a clock face). Web finally, cut each eighth in half to create 16 slices and remove the center slice to achieve six proportionate
slices. If you are hosting a party or feeding a larger group, the classic eight slices may be the way to go. Kalua pork and.
How many slices in a 16 inch pizza Full Details 2022 (2022)
However, some people may find the slices too large for their preferences. Justine st & ashland ave. Web finally, cut each eighth in half to create 16 slices and remove the center slice to achieve six
proportionate slices. The large size allows plenty of room to customize the toppings to everyone’s individual preferences. Monkeypod kitchen by merriman in kapolei, oahu,.
Slices of Pizza of Different Sizes. Diagram Infographics Set. Cut into
Sure, you could buy two small pizzas, but you’ll end up paying more per square inch and you’ll probably be hungry afterwards. Unlike the pie chart, the pizza. But what about the circumference, the
edge of our pizza frontier? If you are hosting a party or feeding a larger group, the classic eight slices may be the way to go..
Pizza Diagram PowerPoint Template & Presentation Slides
This size pizza can feed up to 5 people if each person consumes 2 slices, or 10 people if each person eats only 1 slice. The final slice is from 10.30 to 4.30. Each type of slice has its own unique
shape, size, and flavor, making it. Web given the diameter of 16 inches, the total size of a 16.
How Many Slices Is a 16 Inch Pizza? Oh Snap! Cupcakes
Web and what makes it so popular? Web a 16 inch pizza is a large pizza that can feed between 8 and 10 people. When it comes to pizza, size matters. Sure, you could buy two small pizzas, but you’ll
end up paying more per square inch and you’ll probably be hungry afterwards. Once you have found the diameter, simply.
It’s Actually Defined By The Diameter Of A Circle That Has An Area Of 16 Square Inches.
The next slice will be from 7.30 to 1.30. Monkeypod kitchen by merriman in kapolei, oahu, hawaii. If you are hosting a party or feeding a larger group, the classic eight slices may be the way to go.
Kalua pork and pineapple pizza, hamakua wild mushroom and truffle oil pizza.
I’ve Created One For You Below.
This is the menu pages of slice factory. This can also vary depending on the pizza parlor and how they cut their pizzas. Justine st & ashland ave. How to cut a pizza into 8 slices after the first
cut, make a vertical cut through the center of each half to create four equal slices.
View This Post On Instagram.
This will give you the circumference of the pizza. Fans of quad cities pizza will still be able to get their fix of the thin pies that mohr grew up eating (they’re cut into strips with puffy. Extra
large pizzas are only a few dollars more expensive than a large and give you a. 1100 w granville ave, chicago, il 60660.
**Consumer Advisory** Consuming Raw Or Undercooked Foods Meats, Poultry, Seafood Or Eggs May Increase Your Risk Of Foodborne Illness.
Once you have found the diameter, simply multiply it by 3.14. You can expect to feed. Web how big is a 16 inch pizza? Web finally, cut each eighth in half to create 16 slices and remove the center
slice to achieve six proportionate slices.
Related Post: | {"url":"https://claims.solarcoin.org/en/16-slice-pizza-diagram.html","timestamp":"2024-11-03T09:11:38Z","content_type":"text/html","content_length":"26009","record_id":"<urn:uuid:a671b6b6-ae78-43b3-942b-4e144346e37b>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00006.warc.gz"} |
Year 9
Year 9 is a base camp for year 10 and year 11 which are GCSE years. Many schools start their GCSE preparations in year 9. This brings sudden change in the speed of learning for students. A student
has to choose his/her option subject during year 9 and thus clarity of whet they wish to learn further becomes more important.
Champs Learning helps every student at year 9 to have this clarity and interest the area they wish to study further. At year 9, students cover topics in depth to promote mastery and understanding so
that they are “GCSE ready”. The expectation is that by the end of year 9 all students will be at good grade points to manage with advanced topics.
Session Plan: The session plan for the subjects like English, Maths and Science will be similar to year 7 and 8 but each session will go much deeper content wise and assessment wise. These sessions
will also start covering the subjects from GCSE point of view. There are 42 sessions for each subject. If one joins for all the three subjects, then he will be visiting Champs for 126 teaching
sessions in a year.
Books: The session plan is structured with 8 books for each subject and 8 topic tests. Each book is further divided into 4 weeks and every 5th week, we conduct an assessment on that book or topic.
All the content including books and tests are designed by faculties at Champs learning. The session plan is made flexible to accommodate extra time for doubt solving and for challenging content for
higher set of students.
Assessment: An assessment happens at the end of every book i.e. after every 4 weeks on a particular topic and feedback is discussed with the student with support plan. We also have an assessment at
the end of the year once we finish the entire curriculum for year 8
Subjects: We teach English, Maths and Science for Key stage 3. We also have modern languages as per need of the students. We treat Science as combined science for Year 9 with Physics, Chemistry and
Biology are given attention individually on rotation. The aims of KS3 Maths, English and Science at Champs Learning is to develop an interest and understanding of natural phenomena and prepare them
for studying these subjects at GCSE and A levels..
The curriculum of these subjects have incorporated the structure of all the UK approved boards such as Edexcel, AQA, OCR etc. Each important aspect from a particular board is given an importance in
Success: Every student of Champs Learning makes us proud with high grades during their year 9 assessment and thus they can choose the subjects they wish to study further. | {"url":"https://champslearning.co.uk/courses/year-9.html","timestamp":"2024-11-04T11:29:21Z","content_type":"text/html","content_length":"80307","record_id":"<urn:uuid:43face6b-f0a5-40cb-aa88-a0e407e84e91>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00002.warc.gz"} |
Binary to Decimal Conversion |Expansion Method|Value Box Method|Bit Position
Binary to Decimal Conversion
There are several traditional methods of converting the numbers from binary to decimal conversion. We shall discuss here the two most commonly used methods, namely; “Expansion method and Value Box
(i) Expansion Method:
In this method, the given number is expressed as a summation of terms each of which is the product of a bit (0 or 1) and a power of 2. The power of 2 is determined from the bit position.
Thus the decimal equivalent of a binary number has the general form;
…. a
….. a
….. a
∙ a
….. a
= a
× 2
+ a
x 2
+ ….. +a
x 2
+ …. + a
x 2
+ a
x 2
+ a
x 2
+ a
x 2
+ …. + a
x 2
(ii) Multiplication and Division Method:
The value box method of converting numbers from decimal to binary is laborious and time consuming and is suitable for small numbers when it can be performed mentally. It is advisable not to use it
for large numbers. The conversion of large numbers may be conveniently done by multiplication and division method which is described below.
To effect the conversion of positive integers of the decimal system to binary numbers the decimal number is repeatedly divided by the base of the binary number system, i.e., by 2. The division is to
be carried until the quotient is zero and the remainder of each division is recorded on the right. The binary equivalent of the decimal number is then obtained by writing down the successive
remainders. The first remainder is the least significant bit and the last one is the most significant bit of the binary number. Thus the binary equivalent is written from the bottom upwards.
• Why Binary Numbers are Used
• Binary to Decimal Conversion
• Hexa-decimal Number System
• Conversion of Binary Numbers to Octal or Hexa-decimal Numbers
• Octal and Hexa-Decimal Numbers
• Signed-magnitude Representation
• Diminished Radix Complement
• Arithmetic Operations of Binary Numbers
From Binary to Decimal Conversion to HOME PAGE
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On powers of Plücker coordinates and representability of arithmetic matroids
• Lenz, Matthias Université de Fribourg, Département de Mathématiques, Fribourg, Switzerland
Published in:
• Advances in Applied Mathematics. - 2020, vol. 112, p. 101911
English The first problem we investigate is the following: given k∈R≥0 and a vector v of Plücker coordinates of a point in the real Grassmannian, is the vector obtained by taking the kth power of
each entry of v again a vector of Plücker coordinates? For k≠1, this is true if and only if the corresponding matroid is regular. Similar results hold over other fields. We also describe the
subvariety of the Grassmannian that consists of all the points that define a regular matroid. The second topic is a related problem for arithmetic matroids. Let A=(E,rk,m) be an arithmetic matroid
and let k≠1 be a non- negative integer. We prove that if A is representable and the underlying matroid is non-regular, then Ak:=(E,rk,mk) is not representable. This provides a large class of examples
of arithmetic matroids that are not representable. On the other hand, if the underlying matroid is regular and an additional condition is satisfied, then Ak is representable. Bajo–Burdick–Chmutov
have recently discovered that arithmetic matroids of type A2 arise naturally in the study of colourings and flows on CW complexes. In the last section, we prove a family of necessary conditions for
representability of arithmetic matroids.
Faculté des sciences et de médecine
Département de Mathématiques
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Numeric matrix preparation for Machine Learning.
set_as_numeric_matrix {dataPreparation} R Documentation
Numeric matrix preparation for Machine Learning.
Prepare a numeric matrix from a data.table. This matrix is suitable for machine learning purposes, since factors are binary. It may be sparse, include an intercept, and drop a reference column for
each factor if required (when using lm(), for instance)
intercept = FALSE,
all_cols = FALSE,
sparse = FALSE
data_set data.table
intercept Should a constant column be added? (logical, default to FALSE)
all_cols For each factor, should we create all possible dummies, or should we drop a reference dummy? (logical, default to FALSE)
sparse Should the resulting matrix be of a (sparse) Matrix class? (logical, default to FALSE)
version 1.1.1 | {"url":"https://search.r-project.org/CRAN/refmans/dataPreparation/html/set_as_numeric_matrix.html","timestamp":"2024-11-04T21:29:14Z","content_type":"text/html","content_length":"2649","record_id":"<urn:uuid:e22e1328-1854-456c-876d-187e8dbb6452>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00527.warc.gz"} |
Lesson 7
Finding Unknown Coordinates on a Circle
• Let’s find coordinates on a circle.
Problem 1
The center of a clock is at \((0,0)\) in a coordinate system, and the minute hand is 10 inches long. Find the approximate coordinates of the tip of the minute hand at:
1. 12:05 p.m.
2. 12:45 p.m.
3. 12:55 p.m.
Problem 2
The center of a Ferris wheel is 100 feet off the ground and its radius is 85 feet. The point \(A\) is at the 0 radian position, \(B\) is rotated \(\frac{7\pi}{12}\) radians from \(A\), and \(C\) is
rotated \(\frac{5\pi}{4}\) radians from \(A\).
For each point \(A\), \(B\), and \(C\), find how high the position on the Ferris wheel is off the ground. Write an expression using the sine or cosine function and estimate the value.
Problem 3
A Ferris wheel has a radius of 50 feet, and its center is 60 feet off the ground. How many points on the Ferris wheel are:
1. 30 feet off the ground?
2. 110 feet off the ground?
3. 5 feet off the ground?
Problem 4
The minute hand on a clock tower is 6 feet long. At 10 minutes after the hour, the tip of the minute hand is 55 feet above the ground. How high above the ground is the center of the clock face?
Explain how you know.
Problem 5
A wheel has a radius of 1 foot. The center of the wheel is point \(O\).
1. Indicate where the point \(P\) will be after the wheel rotates counterclockwise around its center 1 foot. Label this point \(Q\).
2. What is the measure of angle \(POQ\) in radians?
3. Indicate where the point \(P\) will be after the wheel rotates counterclockwise around its center \(\frac{3\pi}{2}\) feet. Label this point \(R\).
4. What is the measure of angle \(POR\) in radians?
(From Unit 6, Lesson 3.)
Problem 6
Angle \(\theta\) corresponds to a point \((x,y)\) on the unit circle in quadrant 1.
1. Which quadrant does \(\theta + \pi\) lie in?
2. In terms of \(x\) and \(y\), what are the coordinates of \(\theta + \pi\)?
(From Unit 6, Lesson 4.)
Problem 7
Using a unit circle display, give an example of an angle satisfying each inequality.
1. \(A\) so that \(\cos(A) > 0\) and \(\sin(A) < 0\)
2. \(B\) so that \(\cos(B) < 0\) and \(\sin(B) > 0\)
3. \(C\) so that \(\cos(C) < 0\) and \(\sin(C) < 0\)
(From Unit 6, Lesson 6.)
Problem 8
Suppose angle \(\theta\), in radians, is in quadrant 3 of the unit circle. If \(\sin(\theta)=\text-0.45\), what are the values of \(\cos(\theta)\) and \(\tan(\theta)\)?
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