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Analysis - Mathematical, Technical, Preliminaries | Britannica
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Numbers and functions
Number systems
Throughout this article are references to a variety of number systems—that is, collections of mathematical objects (numbers) that can be operated on by some or all of the standard operations of
arithmetic: addition, multiplication, subtraction, and division. Such systems have a variety of technical names (e.g., group, ring, field) that are not employed here. This article shall, however,
indicate which operations are applicable in the main systems of interest. These main number systems are:
• a. The
natural numbers
ℕ. These numbers are the positive (and zero) whole numbers 0, 1, 2, 3, 4, 5, …. If two such numbers are added or multiplied, the result is again a natural number.
• b. The
ℤ. These numbers are the positive and negative whole numbers …, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, …. If two such numbers are added, subtracted, or multiplied, the result is again an integer.
• c. The
rational numbers
ℚ. These numbers are the positive and negative fractions
are integers and
≠ 0. If two such numbers are added, subtracted, multiplied, or divided (except by 0), the result is again a rational number.
• d. The
real numbers
ℝ. These numbers are the positive and negative
decimals (including terminating decimals that can be considered as having an infinite sequence of zeros on the end). If two such numbers are added, subtracted, multiplied, or divided (except by
0), the result is again a real number.
• e. The
complex numbers
ℂ. These numbers are of the form
are real numbers and
Square root of√−1
. (For further explanation,
the section
Complex analysis
.) If two such numbers are added, subtracted, multiplied, or divided (except by 0), the result is again a complex number.
In simple terms, a function f is a mathematical rule that assigns to a number x (in some number system and possibly with certain limitations on its value) another number f(x). For example, the
function “square” assigns to each number x its square x^2. Note that it is the general rule, not specific values, that constitutes the function.
The common functions that arise in analysis are usually definable by formulas, such as f(x) = x^2. They include the trigonometric functions sin (x), cos (x), tan (x), and so on; the logarithmic
function log (x); the exponential function exp (x) or e^x (where e = 2.71828… is a special constant called the base of natural logarithms); and the square root function Square root of√x. However,
functions need not be defined by single formulas (indeed by any formulas). For example, the absolute value function |x| is defined to be x when x ≥ 0 but −x when x < 0 (where ≥ indicates greater than
or equal to and < indicates less than).
The problem of continuity
The logical difficulties involved in setting up calculus on a sound basis are all related to one central problem, the notion of continuity. This in turn leads to questions about the meaning of
quantities that become infinitely large or infinitely small—concepts riddled with logical pitfalls. For example, a circle of radius r has circumference 2πr and area πr^2, where π is the famous
constant 3.14159…. Establishing these two properties is not entirely straightforward, although an adequate approach was developed by the geometers of ancient Greece, especially Eudoxus and Archimedes
. It is harder than one might expect to show that the circumference of a circle is proportional to its radius and that its area is proportional to the square of its radius. The really difficult
problem, though, is to show that the constant of proportionality for the circumference is precisely twice the constant of proportionality for the area—that is, to show that the constant now called π
really is the same in both formulas. This boils down to proving a theorem (first proved by Archimedes) that does not mention π explicitly at all: the area of a circle is the same as that of a
rectangle, one of whose sides is equal to the circle’s radius and the other to half the circle’s circumference.
Approximations in geometry
The transformation of a circular region into an approximately rectangular regionThis suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the
area, πr^2. As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr^2—the same area as that of the circle. This method of approximating a
(complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.
A simple geometric argument shows that such an equality must hold to a high degree of approximation. The idea is to slice the circle like a pie, into a large number of equal pieces, and to reassemble
the pieces to form an approximate rectangle (see figure). Then the area of the “rectangle” is closely approximated by its height, which equals the circle’s radius, multiplied by the length of one set
of curved sides—which together form one-half of the circle’s circumference. Unfortunately, because of the approximations involved, this argument does not prove the theorem about the area of a circle.
Further thought suggests that as the slices get very thin, the error in the approximation becomes very small. But that still does not prove the theorem, for an error, however tiny, remains an error.
If it made sense to talk of the slices being infinitesimally thin, however, then the error would disappear altogether, or at least it would become infinitesimal.
Actually, there exist subtle problems with such a construction. It might justifiably be argued that if the slices are infinitesimally thin, then each has zero area; hence, joining them together
produces a rectangle with zero total area since 0 + 0 + 0 +⋯ = 0. Indeed, the very idea of an infinitesimal quantity is paradoxical because the only number that is smaller than every positive number
is 0 itself.
The same problem shows up in many different guises. When calculating the length of the circumference of a circle, it is attractive to think of the circle as a regular polygon with infinitely many
straight sides, each infinitesimally long. (Indeed, a circle is the limiting case for a regular polygon as the number of its sides increases.) But while this picture makes sense for some
purposes—illustrating that the circumference is proportional to the radius—for others it makes no sense at all. For example, the “sides” of the infinitely many-sided polygon must have length 0, which
implies that the circumference is 0 + 0 + 0 + ⋯ = 0, clearly nonsense. | {"url":"https://www.britannica.com/science/analysis-mathematics/Technical-preliminaries","timestamp":"2024-11-04T10:10:23Z","content_type":"text/html","content_length":"119784","record_id":"<urn:uuid:45b0f168-ba77-41f7-8fd3-2c656ad9b569>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00425.warc.gz"} |
What is the value of the integer x ?
Question Stats:
88% 12% (00:59) based on 1475 sessions
What is the value of the integer x ?
(1) x rounded to the nearest hundred is 7,200.
(2) The hundreds digit of x is 2.
We need to determine the value of x.
Statement One Alone:
x rounded to the nearest hundred is 7,200.
Knowing that x rounded to the nearest hundred is 7,200 is not enough information to determine x. For instance, if x = 7,194, it will be rounded to 7,200, or if x = 7,211, it will be also rounded to
7,200. Statement one alone does not provide enough information to answer the question. Eliminate answer choices A and D.
Statement Two Alone:
The hundreds digit of x is 2.
There are many numbers with a hundreds digit of 2. Knowing the hundred digits alone will not allow us to determine x. Statement two alone does not provide enough information to answer the question.
Eliminate answer choice B.
Statements One and Two Together:
Using our two statements together, we know that x rounded to the nearest hundred is 7,200 and the hundreds digit of x is 2. However, we still do not have enough information to answer the question.
For instance, x could equal 7,201 or 7,202, giving us two different values of x.
Answer: E
Hi Jeff,
Just wanted to confirm my understanding-
Rounding off to nearest 100-
100-149 will be rounded off to 100
150-199 will be rounded off to 200
Rounding off to nearest 1000-
1000-1499 will be rounded off to 1000
1500-2000 will be rounded off to 2000
And for decimals-
0-0.49 rounded off to 0
0.5 to 0.99 rounded off to 1
Would be great if you could kindly share links to a such a thread/ quick reckoner for revision if possible.
Looking forward to hearing from you. | {"url":"https://gmatclub.com/forum/what-is-the-value-of-the-integer-x-226730.html","timestamp":"2024-11-03T22:39:58Z","content_type":"application/xhtml+xml","content_length":"766198","record_id":"<urn:uuid:1be7e77e-09ef-44b9-ac82-bd80b21628f6>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00102.warc.gz"} |
Question Paper 2023
CLASS 12 ISC - Java Question Paper 2023
Maximum Marks: 70
Time Allowed: Three hours
(Candidates are allowed additional 15 minutes for only reading the paper.)
(They must NOT start writing during this time).
Answer all questions in Part I (compulsory) and six questions from Part-II,
choosing two questions from Section-A, two from Section-B and two from Section-C.
All working, including rough work, should be done on the same sheet as the
The intended marks for questions or parts of questions are given in brackets [ ]
(Attempt all questions)
Question 1
(i) According to De Morgan's law (a + b +c’ )' will be equal to:
(a) a' + b' + c’
(b) a' + b'+c
(c) a'.b'.c'
(d) a'b'c
(ii) The dual of (X'+ 1 ).(Y' + 0) = Y'
(a) X.0 + Y.1 = Y
(b) X'.1 + Y'.0 = Y'
(c) X'.0 + Y'.1 = Y'
(d) (X'+ 0) . (Y'+ 1) = Y'
(iii) The reduced expression of the Boolean function F(P,Q) =P’ + PQ is:
(a) P' + Q
(b) P
(c) P'
(d) P+Q
(iv) If (~p => ~q ) then its contra positive will be:
(a) p=>q
(b) q=>p
(c) ~q=>p
(d) ~p=>q
(v) The keyword that allows multi-level inheritance in Java programming is:
(a) implements
(b) super
(c) extends
(d) this
(vi) Write the minterm of F(A, B, C, D) when A= 1, B=0, C=O and D= 1.
(vii) Verify if (A + A')' is a Tautology, Contradiction, or a Contingency.
(viii) State any one purpose of using the keyword this in Java programming.
(ix) Mention any two properties of the data members of an Interface.
(x) What is the importance of the reference part in a Linked List?
Question 2
(i) Convert the following infix notation to prefix notation.
(ii) A matrix M[-6...10, 4...15] is stored in the memory with each element requiring 4 bytes of storage. If the base address is 1025, find the address of M[4][8] when the matrix is stored in Column
Major Wise.
(iii) With reference to the code given below, answer the questions that follow along with dry run / working.
boolean num(int x)
int a=1;
for (int c=x; c>0; c/=10)
a * = 10;
return (x * x % a) == x;
(a) What will the function num() return when the value of x=25?
(b) What is the method num() performing?
(iv) The following function task(Q) is a part of some class. Assume ‘m’ and ‘n’ are positive integers, greater than 0. Answer the questions given below along with dry run / working.
int task(int m, int n)
{ if(m==n)
retum m;
else if(m>n)
return task(m-n, n);
retum task(m, n-m);
(a) What will the function task() return when the value of m=30 and n=45?
(b) What function does task() perform, apart from recursion?
SECTION - A
(Answer any two questions.)
Question 3
(i) Given the Boolean function F(A,B,C,D) = ∑(2, 3, 6, 7, 8, 10, 12, 14, 15).
(a) Reduce the above expression by using 4-variable Karnaugh map, showing the various groups (i.e., octal, quads and pairs).
(b) Draw the logic gate diagram for the reduced expression. Assume that the variables and their complements are available as inputs.
(ii) Given the Boolean function F(A,B,C,D) = π(0, 1, 2, 4,5, 8, 10, 15, 14, 15),
(a) Reduce the above expression by using 4-variable Karnaugh map, showing the various groups (i.e., octal, quads and pairs).
(b) Draw the logic gate diagram for the reduced expression. Assume that the variables and their complements are available as inputs
Question 4
(i) A shopping mall allows customers to shop using, cash or credit card of any nationalised bank. It awards bonus points to their customers on the basis of criteria given below:
- The customer is an employee of the shopping mall and makes the payment using a credit card
- The customer shops items which carry bonus points and makes the payment using a credit card with a shopping amount of less than Rs 10,000/-
- The customer is not an employee of the shopping mall and makes the payment not through a credit card but in cash for the shopping amount above Rs 10,000/-
The inputs are:
C - Payment through a credit card
A - Shopping amount is above Rs 10,000/-
E - The customer is an employee of the shopping mall
I - Item carries a bonus point
(In all the above cases, 1 indicates yes and 0 indicates no.)
Output: X [1 indicates purchased, 0 indicates not purchased for all cases]
Draw the truth table for the inputs and outputs given above and write the POS expression for X (C, A, E, I).
(ii) Differentiate between half adder and full adder. Write the Boolean expression and draw the logic circuit diagram for the SUM and CARRY of a full adder.
(iii) Verify the following expression by using the truth table:
(A ⊙ B)' = (A ⊕ B)
Question 5
(i) What is an encoder? How is it different from a decoder? Draw the logic circuit for a 4:1 multiplexer and explain its working.
(ii) From the logic diagram given below write the boolean expression for (1) and (2). Also, derive the boolean expression (F) and simplify it:
(iii) Convert the following cardinal expression to its canonical form:
F (P, Q, R) = π(0, 1, 3, 4)
(Attempt all questions from this Section.)
Question 6
Design a class NumDude to check if a given number is a Dudeney number or not. (A Dudeney number is a positive integer that is a perfect cube, such that the sum of its digits is equal to the cube root of the number.)
Example: 5832 = (5+8+3+2)=(18)³ = 5832
Some of the members of the class are given below:
Class name : NumDude
Data member/instance variable:
num:to store a positive integer number
Methods/Member functions:
NumDude():default constructor to initialise the data member with legal initial value
void input():to accept a positive integer number
int sumDigits(int x):returns the sum of the digits of number 'x' using recursive technique
void isDude():checks whether the given number is a Dudeney number by invoking the function sumDigits() and displays the result with an appropriate message
Specify the class NumDude giving details of the constructor(), void input(), int sumDigits(int) and void isDude(). Define a main() function to create an object and call the functions accordingly to enable the task.
Question 7
A class Trans is defined to find the transpose of a square matrix. A transpose of a matrix is obtained by interchanging the elements of the rows and columns.
Example: If size of the matrix = 3, then
Question 9
A double ended queue is a linear data structure which enables the user to add and remove integers from either ends i.e., from front or rear. The details of the class deQueue are given below:
Class name: deQueue
Data members/ instance variables:
Qrr[]:array to hold integer elements
lim: maximum capacity of the dequeue
front: to point the index of the front end
rear:to point the index of the rear end
Methods/Member functions:
deQueue(int l):constructor to initialise lim= 1, front = 0 and rear = 0
void addFront(int v):to add integers in the dequeue at the front end if possible, otherwise display the message "OVERFLOW FROM FRONT"
void addRear(int v):to add integers in the dequeue at the rear end if possible, otherwise display the message "OVERFLOW FROM REAR"
int popFront():removes and returns the integers from the front end of the dequeue if any, else returns -999
int popRear(): removes and returns the integers from the rear end of the dequeue if any, else returns -999
void show():displays the elements of the dequeue
Specify the class deQueue giving details of the functions void addFront(int) and int popFront(). Assume that the other functions have been defined. The main() function and algorithm need NOT be written.
Question 10
A super class Demand has been defined to store the details of the demands for a product. Define a subclass Supply which contains the production and supply details of the products.
The details of the members of both the classes are given below:
Class name: Demand
Data members/instance variables:
pid:string to store the product ID
pname:string to store the product name
pdemand: integer to store the quantity demanded for the product
Methods/Member functions:
Demand(...) : parameterised constructor to assign values to the data members
void display():to display the details of the product
Class name:Supply
Data members/instance variables:
pproduced: integer to store the quantity of the product produced
prate:to store the cost per unit of the product in decimal
Methods/Member functions:
Supply(...): parameterised constructor to assign values to the data members of both the classes
double calculation(): returns the difference between the amount of demand (rate x demand) and the amount produced(rate x produced)
void display():to display the details of the product and the difference in amount of demand and amount of supply by invoking the method calculation()
Assume that the super class Demand has been defined. Using the concept of inheritance, specify the class Supply giving the details of the constructor(...), double calculation() and void display().
The super class, main function and algorithm need NOT be written.
Question 11
(i) A linked list is formed from the objects of the class given below
class Node
double sal;
Node next;
Write an Algorithm OR a Method to add a node at the end of an existing linked list. The method declaration is as follows:
void addNode(Node ptr, double ss)
(ii) Answer the following questions from the diagram of a Binary Tree given below: | {"url":"https://alexsir.com/class12-isc-java-question-paper-2023","timestamp":"2024-11-10T12:18:37Z","content_type":"text/html","content_length":"41394","record_id":"<urn:uuid:228d3216-c547-4d40-b7c9-6d80e628d388>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00350.warc.gz"} |
If it's not what You are looking for type in the equation solver your own equation and let us solve it.
Solution for -64=-9t+17 equation:
We move all terms to the left:
We get rid of parentheses
We add all the numbers together, and all the variables
We move all terms containing t to the left, all other terms to the right | {"url":"https://www.geteasysolution.com/-64=-9t+17","timestamp":"2024-11-08T22:19:14Z","content_type":"text/html","content_length":"20926","record_id":"<urn:uuid:89c60b1b-db39-4b99-af57-dcf27d582f53>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00789.warc.gz"} |
It's February 29th And Here's All About Leap Year... And Perpetual Calendars
Just Because
It’s February 29 and Here’s All You Need to Know About Leap Years… and Calendars
Why leap years were invented and why they don’t always happen every four years.
Every four years is a leap year and the month of February gets an extra day, a leap day, on February 29th. 2020 is a leap year and everything flows smoothly at regular, four-year intervals until
2100, which we discover is not a leap year. Wait a minute, 2100 is divisible by four, so what is going on? Find out why the leap year can be erratic and other intriguing details of how man has
struggled to create a rational system that accurately reflects Earth’s rotations around the Sun and “fixed” astronomical events, such as the equinoxes and solstices.
Inventing instruments to observe, measure and predict the motions of our universe has consumed astronomers since antiquity. Even if you don’t have a telescope or a spectrograph in the spare room,
most of us keep track of time with watches and calendars. These surprisingly complex mechanisms condense centuries of astronomic tinkering to account for the discrepancies between our computations
and those of the universe. Since 1582, the Gregorian calendar has ruled over most of our lives and is still the international standard for civil use around the world. The reason that it has enjoyed
such longevity – almost 500 years – is that it is surprisingly accurate and adds in an extra day at the end of February every four years to reflect the real-time it takes the Earth to revolve around
the Sun.
Thirty days hath September,
April, June, and November;
All the rest have thirty-one,
Excepting February alone,
And that has twenty-eight days clear
And twenty-nine in each leap year
Concise Oxford Dictionary of Quotations
What is a leap year? Why do we need leap years?
A common year has 365 days; a leap year has 366 days thanks to an intercalary (additional) day on February 29, occurring almost once every four years. In a nutshell, leap years are corrective
measures to force our calendar to stay in sync with nature’s cycles. None of this would have been necessary if Mother Nature had made Earth’s orbit of the Sun an exact 365 days.
The universe does not always comply with our manmade computations and the astronomical/solar year, the time taken for the Earth to complete its orbit around the Sun, is 365.242 days or 365 days, 5
hours, 48 minutes and 45 seconds. The lag of 0.242 – or roughly a 1/4 of a day – might seem negligible, but it adds up over time. If our calendar were exactly 365 days long, the months would start
slipping behind the seasons, and in roughly three hundred years, we’d be popping champagne bottles in autumn. To keep the calendar synchronised with the true astronomical year and avoid seasonal
drift, an extra day, a leap day, is added every four years at the end of February.
Julius Caesar was the first to implement this ingenious system in 46 BCE but the Gregorian calendar fine-tuned it even further thanks to a more complex formula for determining the occurrence of leap
In the beginning
Primitive celestial gazers started to notice patterns: the alternating periods of light and darkness (based on one rotation of the Earth on its axis), the cyclical nature of the Moon (based on the
revolution of the Moon around the Earth) and the slower movement of the stars in the firmament (based on the revolution of the Earth around the Sun). Predicting when these phenomena would happen was
essential to anticipating the seasons, planting and harvesting crops, hunting certain animals and observing rituals.
A section of the hieroglyphic calendar at the Kom Ombo Temple displaying the transition from Month XII to Month I
Although the first record of calendars coincides with the advent of writing in Mesopotamia, there is evidence of calendars (monuments) dating back to the Mesolithic. The most recent discovery is a
site at Warren Field in Scotland. Roughly 10,000 years old (predating the first formal calendar in the Near East by 5,000 years), hunter-gatherers dug twelve pits to track the lunar months over the
course of a year, vital information for synchronising seasonal activities like hunting migrating animals. But what is even more surprising is that the Warren Field site aligns with the sunrise of the
midwinter solstice, providing an annual astronomic correction to the seasonal drift of the lunar year.
Julius Caesar’s calendar
Without delving into all the sophisticated calendars that appeared in Mesopotamia, Egypt and Greece, some of them featuring intercalary days to catch up with solar time, it was Julius Caesar’s
calendar that has most influenced Western civilisation.
Before the introduction of the Julian calendar in 45 BCE, priests in Rome often exploited the calendar for political ends. They were known to add and subtract days to the calendar to favour or cut
short some bigwig’s term in office or extend a holiday. The Roman Republican lunar calendar was so chaotic that years could vary anywhere between 355 to 378 days and it was so out of sync with
astronomic time that the vernal equinox (March 21) was taking place eight weeks later.
Having recently returned from his campaign in Egypt (48 BCE), Julius Caesar decided to summon Sosigenes, a Greek astronomer based in Alexandria, to help him straighten out the jumbled calendar.
Sosigenes determined that the existing lunar calendar had to be ditched in favour of a more scientific solar model based on the Egyptian calendar. To account for the immense discrepancies between the
date on the calendar and the equinox, Sosigenes also had to fiddle around with all sorts of complex intercalations. In order to realign the calendar with the seasons, Caesar dictated that 46 BCE
would last 445 days, marking it as the last “year of confusion”.
A reproduction of the Fasti Antiates Maiores, a representation of the Antique Roman calendar (circa 60 BC) – source Wikimedia
What is significant about the Julian calendar is that Caesar laid down the rules governing leap years: a year was to be composed of 365 days and an extra day be intercalated every fourth year, in
other words, a leap year. To determine a leap year in Caesar’s book, the year had to be divisible by 4.
Caesar introduced his radical calendar reform in 45 BCE and decided to commemorate his role by changing the name of the month Quintilis for Julius (July) in honour of his birthday and adding on an
extra day to have the maximum amount of days possible -31 days. Not one to be left out, in 8 BCE, Emperor Augustus also got the Senate to change the month of Sextilis to Augustus (August).
Incidentally, the first day of the month in the Julian calendar was known as the kalendae, the origin of our word calendar.
Despite its extraordinary accuracy, the Julian calendar of 365.25 days was still a wee bit too long to match the solar year (365.24), resulting in an error of 11 minutes and 14 seconds a year. Add
this up over ten centuries, and you’re looking at almost seven days. Even though this eventually resulted in the calendar being out of sync with the equinox and solstice, the Julian calendar was in
use well until the 16^th century.
Gregorian calendar
By the mid-1550s, Caesar’s Julian calendar had drifted a full 10 days off course. Ecclesiastical authorities in Rome were extremely concerned and Pope Gregory XIII issued an urgent papal bull in 1582
to address the growing discrepancies messing with important dates like Easter and its host of moveable and fixed feasts. Gregory XIII enlisted astronomers Aloysius Lilius and Christopher Clavius for
his grand project and it was determined that Easter be celebrated on the Sunday following the full Moon that fell on or after the vernal equinox of March 21. Although this meant that the Pope had to
wipe out 10 full days on the calendar in 1582, jumping directly from the 4^th to the 15^th of October, the dates were finally aligned with the seasons again.
Pope Gregory XIII, who introduced the new calendar in October 1582, and gave its name to the calendar still in use today, the Gregorian calendar
Besides determining dates in the ecclesiastical calendar, the Gregorian reform perfected the rule for leap years. The Gregorian rule for determining leap years is even more precise than the Julian
calendar and answers the question of why years like 1700, 1800, 1900, 2100 and 2300 are not leap years but why 1600, 2000 and 2400 are leap years. According to the Gregorian reform, every year that
is exactly divisible by 4 is a leap year. However, if the year can be divided by 100 (centennial year) it is NOT a leap year… and here is the trick, if it is divisible by 400 though, it is a leap
Lunario Novo, Secondo la Nuova Riforma della Correttione del l’Anno Riformato da N.S. Gregorio XIII, printed in Rome by Vincenzo Accolti in 1582, one of the first printed editions of the new
calendar. Source Wikimedia
Catholic countries like Italy, Spain and Portugal adopted the Gregorian calendar, but Protestant countries were somewhat wary of Catholic meddling. Surprisingly, it wasn’t until 1752 that Great
Britain and America switched to the Gregorian calendar.
How accurate is the Gregorian calendar?
Regarded as one of the most accurate calendars in use today, the Gregorian calendar is not perfect. There is a margin of error of 27 seconds per year, which adds up to one day every 3236 years. This
means that in 4904 we’ll have an extra day to account for.
Two professors at Johns Hopkins University have proposed the creation of a new calendar. Known as the Hanke-Henry Permanent Calendar (HHPC), the idea behind this calendar is that every date would
fall on the same day of the week every year. The HHPC envisions a 364-day, 12-month, seven-day week calendar, but the days don’t jump around from year to year, and every year would begin on Monday,
January 1^st. To account for the disparity between this 364-day calendar and the astronomical calendar, the HHPC would add an extra week at the end of every fifth or sixth year to keep the calendar
in line with the seasons, serving the same function as the leap year.
secular calendar watches
Earlier this week, we published an article about perpetual calendar watches to coincide with the leap year. These sophisticated machines take into account the different lengths of the months and leap
years but are only accurate for 100 years, meaning that any of you with a standard QP will have to adjust the watch in 2100. Not a big deal, I understand, but for somebody who wants the “nec plus
ultra” accurate timekeeper, it’s a Secular Calendar watch you’re after. Patek Philippe’s Calibre 89 pocket watch, Franck Muller’s USD 2.7m Aeternitas 4 watch and Svend Andersen’s creation are all
secular calendars, which will not need adjusting until 2400.
All we need now is to invent a way to prolong our lifespan…Happy Leap Day!
5 responses
5. A proposal for the next calendar reform is NexCalendar.org. It follows the reform of Gregorian that kept the same set of dates and months of the Julian calendar with just reducing three leap days
for every 400 years. NexCalendar also keeps the same set of dates and months of the Gregorian calendar with the least subtle extension on 5 weeks for every 400 years. That can wisely upgrade the
calendar into a perennial calendar that every year use the same calendar version. | {"url":"https://monochrome-watches.com/just-because-february-29th-leap-year-history-perpetual-calendar/","timestamp":"2024-11-08T12:26:43Z","content_type":"text/html","content_length":"174029","record_id":"<urn:uuid:1d81dba8-3013-4ee6-bb83-b3d2bdf596f4>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00546.warc.gz"} |
TRANSIT on Steroids: Doppler-Based GNSS Meets Large LEO ConstellationsTRANSIT on Steroids: Doppler-Based GNSS Meets Large LEO Constellations - Inside GNSS - Global Navigation Satellite Systems Engineering, Policy, and Design
The advent of large LEO constellations is a game-changer for navigation. A key feature of these constellations is their potential to allow simultaneous measurement of Doppler shifts from a large
number of satellites.
It may be possible to exploit their downlink communication signals for navigation with little or no modification. The GDOP analysis and the batch filter results on simulated data both indicate
absolute navigation performance similar to that of current pseudorange-based GNSS.
Doppler-based navigation from LEO satellites has been around since the TRANSIT program’s start in the 1950s. Its use for TRANSIT makes it seem antiquated, but the concept still has living, breathing
devotees. The first author has participated in debates about its efficacy. Perhaps many of our colleagues could say the same. In one memorable debate a senior engineer who is, nevertheless, naïve
about navigation issues, was selling the idea that a single LEO satellite pass could provide a navigation capability. He was intersecting spheres of constant pseudorange and cones of constant Doppler
shift and claiming observability.
An experienced GNSS colleague, soon to be inducted into the National Academy of Engineering, insisted that she had looked into that and found that it does not work. The proponent was not dissuaded.
The moderator of the meeting, a potential funder with deep pockets, seemed to like the GNSS newbie’s enthusiasm and awarded him a contract. He hedged his bets, however, by asking this paper’s first
author to do some analysis.
This was not our author’s first LEO/Doppler-nav rodeo. He had been one of the chief analysts of the High-Integrity GPS (iGPS) project as a founding member of Coherent Navigation. That project had as
its goal exploiting accumulated delta range from Iridium L-band signals in order to augment GPS. Recall that accumulated delta range equals beat carrier phase multiplied by nominal wavelength, and
that beat carrier phase equals the (negative) time integral of carrier Doppler shift. Iridium signals use time-division and frequency-division multiple access. Carrier Doppler shift is the most
reliable observable, and beat carrier phase is reconstructed by a process that involves cycle-slip estimation during each 81.72 msec dead-time interval between 8.28 msec signal bursts.
The iGPS project sought to do several important things, some military, some commercial. It had some successes. None of them involved stand-alone navigation based purely on Iridium signals. Rather, it
always sought to use Iridium signals in concert with GPS signals in ways that would leverage the advantages of each.
A spin-off of the iGPS project had different ideas. It was never exactly clear why Boeing, the iGPS prime contractor, set up an independent effort to do stand-alone navigation based on Iridium
signals. Perhaps the company wanted to hedge its bets in case the aggressive goals of the main iGPS program proved too ambitious. In hindsight, Boeing’s spin-off decision proved wise. The main iGPS
program was cancelled; Coherent Navigation was absorbed by Apple without reducing any LEO-based GNSS technology to operational practice. The spin-off survived to become Satelles, Inc. Today it offers
a viable product based on stand-alone Iridium Doppler- and pseudorange-based navigation.
Satelles’ success does not contradict the rising NAE member’s misgivings about LEO navigation. Satelles claims 20 m accuracy for a static receiver [1]. If motion is involved, then things deteriorate,
as our first author knows from experience, even when using an inertial navigation system (INS).
The problem with navigation based on one LEO satellite is geometric dilution of precision (GDOP). This becomes clear if one considers accumulated delta range, which is nominally equivalent to Doppler
shift in its ability to provide navigation information. Suppose one makes the optimistic assumption that the accumulated delta range bias can be estimated exactly without degrading the navigation
solution. Then the accumulated delta range becomes like pseudorange. As a satellite traverses the sky, it sweeps out a range of locations relative to the receiver. A non-infinite GDOP results only if
the multiplicity of locations of the satellite during the pass produces a good geometry. This does not happen.
If the Earth did not rotate, then the satellite-to-receiver direction vectors would all lie in a single plane, and GDOP would be infinite. The Earth’s rotation prevents this, but the resulting GDOP
values can be in the hundreds. On paper, this GDOP value can be reduced by taking a large number of samples N, which introduces a 1/√N effect, but measurement error correlations act to blunt the
beneficial impact. The precision of beat carrier phase data helps somewhat, but the need to estimate the beat carrier phase bias adds further degradation. Position estimation for a static receiver is
possible, as Satelles has demonstrated, but receiver clock drift during a satellite pass makes for complications.
If the receiver is moving, then navigation becomes much more challenging, and accuracy degrades. This happens even when INS data are used to augment the LEO data. The optimistic GNSS novice in the
debate planned to use an INS to make things work well for a moving receiver, as did the iGPS program. “Use an INS” seems to be a one-size-fits-all panacea for radio-navigation systems that suffer
from poor observability. Sometimes it works, but the incorporation of INS data offers challenges. Even with a very stable INS, there is the added problem of estimating initial velocity, initial
attitude, especially yaw, and INS biases.
With a perfect INS (zero velocity and angular random walk, zero bias drift) and a nearly perfect clock (initial frequency uncertainty of 10-9 seconds/sec with zero frequency drift) the per-axis RMS
position errors can exceed 40 m after a 10-minute satellite pass. For a realistic INS and receiver clock, the per-axis navigation errors can exceed 1 km RMS. Addition of pressure altimeter and
magnetometer data may reduce RMS per-axis errors to several hundred meters. If the navigation system sees multiple satellites often enough, especially in more than one orbital plane, then accuracy
improves markedly.
Enter Large LEO
The advent of large LEO constellations is a game-changer. They could remove the drawbacks of LEO-based navigation in general and Doppler-based navigation in particular. A key feature of large LEO
constellations is their potential to allow simultaneous measurement of Doppler shifts from a large number of satellites. In order to understand why satellite number is important, consider the form of
the following carrier Doppler shift measurement model:
The quantity λ is the nominal carrier wavelength, Dj is the measured carrier Doppler shift from the jth satellite, δj is the satellite transmitter clock offset, Tj is the true satellite transmission
time, δRis the receiver clock offset, TR is the true receiver time,
This equation, or one very much like it, is normally used to estimate receiver velocity R from a pseudorange solution. Often the neutral atmosphere and ionosphere terms are neglected during velocity
The dependence of the terms R is a liability when using carrier Doppler shift for velocity estimation, but it is an asset when using Doppler shift for determining R. The challenge for position
estimation based on Doppler shift is the inability to estimate it independently of velocity. In fact, one must estimate all 8 unknown elements of the vector R simultaneously.
Such a scheme was unthinkable in the past, even in the recent past. No LEO constellation could guarantee simultaneous visibility of 8 or more satellites. TRANSIT only afforded one visible satellite
at a time, and had visibility gaps [2]. Iridium can only guarantee that one satellite is visible at all times. A typical receiver can see two satellites about once every 10 minutes, and two
satellites are visible nearly all the time if the receiver sits equidistant between two orbital planes. The latter situation might give rise to simultaneous visibility of 3 or 4 satellites once every
10 minutes or so. No Iridium receiver will ever see 8 satellites simultaneously, except near the Earth’s poles.
The planning and launch of large LEO constellations such as OneWeb, Starlink, and Kuiper have changed all of this. These constellation designs consist of nearly a thousand to several thousand LEO
satellites. Although each satellite is visible to a smaller portion of the Earth than a typical MEO GNSS satellite, there are enough of them to make up for the per-satellite coverage loss. A proposed
version of OneWeb’s initial 720-satellite constellation, depicted in Figure 1, will offer simultaneous visibility of 20 or more satellites above a 7.5° elevation mask worldwide. A proposed version of
Starlink’s final constellation will yield a minimum of 81 visible satellites above a 7.5° elevation worldwide. These numbers make the use of carrier Doppler shift feasible.
One might consider pseudorange-based navigation from large LEO constellations, but there are two reasons why the sole use of carrier Doppler shift is attractive. First, Doppler shift navigation is
less sensitive to transmitter clock errors. It is sensitive only to transmitter clock frequency knowledge errors, and it might be tolerant of errors on the order of 10-11 sec/sec. This obviates the
need for atomic clocks. The system could work with good quartz oscillators on the satellites and without GPS disciplining. Second, Doppler shift navigation does not require the satellites to
broadcast precise ranging codes. The only signal requirements are (a) that modulated data can be wiped off by the receiver to allow measurement of Doppler shift, and (b) that the receiver be able to
distinguish signals from different satellites.
Giving up on pseudorange might seem unwise because of the robustness of pseudorange point solutions. There is no need for any dynamic model, any INS data, or any other data when implementing standard
GNSS pseudorange point solutions. A simple Gauss-Newton nonlinear least-squares solver of 4 or more pseudorange equations for the 4 unknowns in
Surprisingly, the same holds true for Doppler-based navigation that uses the equation given above. It too admits point solutions via nonlinear least-squares solution of a system of 8 or more carrier
Doppler shift equations for the 8 unknowns in
Performance Evaluation
A key question about this concept is its potential accuracy, which can be characterized by the point solution batch filter’s computed covariance matrix. For a pseudorange point solution, a simplified
approximation of the corresponding error covariance matrix leads directly to the concept of GDOP as an accuracy metric. There exists a 4×4 non-dimensional geometry matrix whose trace’s square root
equals GDOP. The error covariance matrix for the
Covariance analysis of LEO/Doppler-based navigation systems typically does not admit a simple GDOP metric. When using an INS and a time series of data, there is much more going on than the simple
geometry of the line-of-sight vectors from the satellites to the receiver. Estimation practitioners do not like being asked, “What is GDOP?” in such situations, but the question does get asked. It is
unanswerable, and the naiveté of the questioner precludes the possibility of explaining why.
For a Doppler-only point solution, however, the question can be asked and answered, but it presents more of a challenge than for a pseudorange point solution. The answer employs an approximation of
the point solution’s error covariance matrix [3]. The key equation for developing a sensible GDOP metric is a linearized approximation of the Doppler shift measurement model. It yields the following
relationship between measurement errors and estimation errors:
where N is the number of satellites, ΔDj is the Doppler shift measurement error for the jth satellite, ΔΔδR is the receiver clock offset error, ΔN×8 non-dimensional “geometry” matrix in this equation
It uses the following two factors to nondimensionalize its first 4 columns:
The quantity γ has units of radians/sec and is the maximum possible magnitude of the angular sweep rate of a line-of-sight unit direction vector RE is the Earth’s radius. The quantity η has units of
m/sec2. It is the maximum possible range acceleration due to satellite motion between a receiver on a non-rotating Earth and a satellite in a circular Keplerian orbit of semi-major axis
The 8 elements of the vector of estimation errors on the right-hand side of the linearized Doppler shift error equation,
The corresponding estimation precisions are
This GDOP analysis indicates several important properties that are needed for good system accuracy. A low GDOP is important for all of the estimated quantities’ accuracies. A large γ, i.e., a large
maximum possible angular sweep rate of the satellite-to-receiver line-of-sight vectors, is important to this concept’s position accuracy. A large η, i.e., a large maximum possible range acceleration
between the satellite and the receiver due to satellite motion, is important to the method’s clock offset accuracy. Examination of the j=1,…,N. The fourth column indicates the importance of large
magnitudes and a diversity of signs for the satellite-motion-induced part of the range acceleration between the satellite and the receiver. Figure 2 depicts example satellites, their orbits, their
Figure 3 shows the GDOP latitude/longitude map for a possible version of the final Starlink constellation. The caption lists the average γ and η values for this map. Averages are needed because there
is a range of the semi-major axes for the orbits and because differing RE values apply to different points on the WGS-84 ellipsoid. GDOP is less than 2 everywhere on the globe. For a
range-rate-equivalent Doppler-shift measurement error standard deviation of -11 second/sec for clock offset rate.
Reference [3] plots GDOP for other constellations. A version of OneWeb suffers from poor GDOP (too large) near the equator and in mid-latitude regions if the 18 near-polar orbital planes are grouped
with all of the ascending nodes on one side of the Earth and all of the descending nodes on the other side. There is poor geometric diversity of the line-of-sight rate vectors j=1,…,N in regions
where a receiver sees only south-going or north-going satellites, and GDOP suffers.
These accuracy numbers and the overall GDOP analysis have been verified through application of a batch filter to simulated Doppler shift data [3]. One simulation test has examined the impact of
ephemeris errors that produce errors in the filter’s versions of the satellites’ positions, velocities, and transmitter clock frequency offsets. It considered per-axis RMS satellite position errors
of 2 m, per-axis RMS satellite velocity errors of 0.002 m/sec, and RMS satellite clock offset rate errors of 3.3×10-11 seconds/sec. The resulting position error magnitudes for 100 Monte-Carlo
simulation cases were 2.3 m RMS and 5.4 m peak. The velocity error magnitudes were 0.013 m/sec RMS and 0.044 m/sec peak. The clock offsets were 0.33 msec RMS and 0.90 msec peak.
The GDOP analysis and the batch filter results on simulated data both indicate absolute navigation performance similar to that of current pseudorange-based GNSS. The only significant difference is
the accuracy of the estimated clock offset δR. Its precision is slightly better than 1 msec. This is at least 4 orders of magnitude worse than for pseudorange-based GNSS. The poor clock accuracy
stems from the fact that clock offset is observable only because of satellite motion, not because it is tied to precise atomic clocks onboard the spacecraft.
In theory, one could use current MEO GNSS satellites to implement this navigation concept. In practice, the accuracy would be poor. The values of GDOP might be reasonably low, but the scaling
parameters would be on the order of γ=2×10-4 rad/sec and η=0.2 m/sec2. A GDOP equal to 1 and a measurement error standard deviation of
There are three significant advantages to the proposed scheme. First, it might be able to operate with hardly any modifications to the planned large LEO constellations. There would be no need to fly
atomic clocks aboard the spacecraft. It may be possible to exploit their downlink communication signals for navigation with little or no modification, as is done by Satelles with the Iridium downlink
A second advantage is the nearness of the satellites to the users. It might enable higher received power levels that would reduce the potential impact of intentional and unintentional interference. A
third advantage is the relatively inexpensive nature of the individual satellites that comprise each constellation. The replacement and upgrading of satellites could happen over a shorter average
cycle, which would make the concept more adaptable.
Alternative Approaches
There are alternative ways that one might employ a large LEO constellation for navigation. One might develop a hosted payload that carried an atomic clock and a dedicated navigation signal
transmitter. It could provide both pseudorange and carrier Doppler shift. In that case, standard navigation methods could be used instead of the new one proposed here.
One might use this basic concept, but have the navigation filter process accumulated delta range rather than carrier Doppler shift. In theory, such an approach should yield similar performance. In
practice, it might be more difficult to use accumulated delta range as the primary observable. It may not be practical to broadcast a useable downlink signal continuously. Any attempt to reconstruct
accumulated delta range would require the estimation of cycle slips during signal dead times.
Furthermore, accumulated delta range bias estimates tend to drift with time if not used in a single- or double-differenced context. The needed filtering techniques to compensate for such drift would
make the system more complicated, and its performance might not exceed that of a Doppler-shift-based approach.
Some researchers have considered LEO satellite signals as a means of augmenting navigation signals from existing MEO GNSS, e.g., see [4]. This particular reference considers the ability of LEO
signals to aid signal integrity analysis. Other efforts have considered the possibility of greatly reducing the initial convergence time of high precision in PPP if signals from one or more LEO
satellites are available [5].
What To Do Next
There are a number of open questions about this concept. The most pressing question concerns signal availability. Although many satellites will be visible to any given receiver, their downlink signal
beams (OneWeb) or their focused spot beams (Starlink) will not all be visible. It is likely that only one satellite at a time for each constellation will transmit a signal to any given point on the
Earth’s surface. This is a sensible downlink communication strategy that conserves satellite transmission power and information bandwidth. Therefore, work needs to be done to figure out how to
receive signals from many other satellites.
Some constellations’ FCC filings indicate that their satellites will carry omnidirectional command and control beacons. One solution to the signal availability problem would be to send periodic
downlink bursts through such a beacon. Another possible technique might exploit weak signals in antenna side lobes. The main beam of a downlink signal will be very powerful in order to support high
data throughputs. Therefore, it may be possible to receive such signals in side lobes, especially if small segments of them have known pseudorandom modulation patterns that can be used to achieve
significant processing gains.
Another possible approach is to accept the limited signal available from any given constellation and to fuse Doppler data from multiple constellations with the help of an INS [6].
Any workable system is likely to involve a thousand or more satellites with perhaps dozens or more of their signals being received simultaneously. This presents a multiple access challenge. A
practical multiple access signal modulation scheme will need to be developed.
There would be a need to distribute satellite ephemeris data and clock frequency offset data to user receivers. This information would have to be determined from some sort of ground control segment.
The constellation owner could operate the segment and perhaps already is planning to do so. If the constellation owner was cooperative, then the constellation’s downlink signals could be used to
disseminate this information too. Even with an uncooperative constellation owner, however, it should be possible to deduce and disseminate the needed information.
This system could probably work with short signal bursts and longer interleaved dead times. Such an approach might reduce the challenges of designing multiple access signals. Another advantage would
be a large reduction in the average downlink transmission power requirement.
The one weak point of this concept is the “T” part of PNT. Receiver clock offset precision is only sub-millisecond, not sub-microsecond or better, as with current pseudorange-based GNSS. It might be
possible to equip a small subset of a constellation’s satellites with atomic clocks and ranging signals. If just one such satellite were visible to each point on the surface of the Earth, then
sub-microsecond timing accuracy might be achievable.
It would be good to start testing this concept or components thereof soon. One possible way to test it would be to use the Iridium constellation and to place the test receiver at a very high latitude
where many Iridium satellites are visible simultaneously. The experiment would involve 8 or more simultaneously visible satellites broadcasting special L-band downlink signals that had modulations
which admitted good multiple access signal processing at the receiver. Perhaps Satelles could arrange for such a test. We would like to be there if it happens, provided it is not too cold and that we
have adequate clothing and protection from polar bears.
(1) Anon., “Satellite Time & Location Signals, 1000 Times Stronger than GPS,” Satelles, Inc., available online at: https://www.satellesinc.com/wp-content/uploads/pdf/Satelles-White-Paper-2019.pdf,
2019 (accessed 24 Oct. 2020).
(2) B. Parkinson, “Introduction and Heritage of NAVSTAR, the Global Positioning System,” in Global Positioning System: Theory and Applications, B. Parkinson and J. Spilker, Eds. Washington: AIAA,
1996, pp. 3–28.
(3) M.L. Psiaki, “Navigation using Carrier Doppler Shift from a LEO Constellation: TRANSIT on Steroids,” Proc. ION GNSS+ 2020, Virtual, Sept. 2020, pp. 3027-3045.
(4) D. Racelis, P. Pervan, and M. Joerger, “Fault-Free Integrity Analysis of Mega-Constellation-Augmented GNSS,” Proc. ION GNSS+ 2019. Miami, FL Sept. 2019, pp. 465–484.
(5) H. Ge, B. Li, M. Ge, N. Zang, L. Nie, Y. Shen, and H. Schuh, “Initial Assessment of Precise Point Positioning with LEO Enhanced Global Navigation Satellite Systems (LeGNSS),” Remote Sensing, Vol.
10, No. 984, 2018, pp. 1-16.
(6) B. McLemore and M.L. Psiaki, “Navigation Using Doppler Shift from LEO Constellations and INS Data,” Proc. ION GNSS+ 2020, Virtual, Sept. 22-25, 2020, pp. 3071 – 3086.
is professor and Kevin T. Crofton Faculty Chair of Aerospace and Ocean Engineering at Virginia Tech, and professor emeritus of mechanical and aerospace engineering at Cornell University. He holds a
Ph.D. in mechanical and aerospace engineering from Princeton University.
is a Ph.D. student of aerospace engineering at Virginia Tech. His research interests include GNSS digital signal processing and estimation and filtering. | {"url":"https://insidegnss.com/transit-on-steroids-doppler-based-gnss-meets-large-leo-constellations/","timestamp":"2024-11-11T16:22:42Z","content_type":"text/html","content_length":"235166","record_id":"<urn:uuid:93b0dee1-71eb-420e-a64e-e7a58ea08efd>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00247.warc.gz"} |
Semi-classical analysis by Victor Guillemin, Shlomo Sternberg
Semi-classical analysis
by Victor Guillemin, Shlomo Sternberg
Publisher: Harvard University 2012
Number of pages: 488
In semi-classical analysis many of the basic results involve asymptotic expansions in which the terms can by computed by symbolic techniques and the focus of these lecture notes will be the 'symbol
calculus' that this creates.
Download or read it online for free here:
Download link
(1.8MB, PDF)
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How do you find vertical, horizontal and oblique asymptotes for y=1/(2-x)? | HIX Tutor
How do you find vertical, horizontal and oblique asymptotes for #y=1/(2-x)#?
Answer 1
vertical asymptote x = 2
horizontal asymptote y = 0
Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation let the denominator equal zero.
#rArr x = 2" is the asymptote " #
Horizontal asymptotes occur as #lim_(xto+-oo) f(x) to 0 #
When the degree of the numerator < degree of the denominator, as is the case here then the equation is always y=0
Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here hence there are no oblique asymptotes.
Here is the graph of the function. graph{1/(2-x) [-10, 10, -5, 5]}
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Answer 2
Vertical asymptote: Set the denominator equal to zero and solve for x. In this case, ( 2 - x = 0 ) gives ( x = 2 ). Thus, there is a vertical asymptote at ( x = 2 ).
Horizontal asymptote: As x approaches positive or negative infinity, the function approaches a constant value. In this case, as ( x ) goes to infinity or negative infinity, ( y ) approaches ( 0 )
because the degree of the numerator is less than the degree of the denominator. Therefore, there is a horizontal asymptote at ( y = 0 ).
Oblique (slant) asymptote: Check if the degree of the numerator is exactly one greater than the degree of the denominator. If so, perform polynomial long division to find the equation of the oblique
asymptote. In this case, since the degree of the numerator is not greater by one, there is no oblique asymptote for the function ( y = \frac{1}{2-x} ).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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sorting (Programming, Real thinking) – Mathplanet
This list of names is not in order:
names = ['Torvalds', 'Zuckerberg', 'Lovelace', 'Turing', 'Johnson']
But if we want to print it in alphabetical order? We need to sort the list, which Python can do for us:
But this function "sorted" is not magic. In this section, we will build it ourselves.
A sorting algorithm
An algorithm is a list of instructions to be executed to achieve a certain goal or produce a certain result - just like a recipe for baking a cake. When we write our code, we explain these steps in a
language that machines understand. The following video talks about algorithm in Swedish with an English translation:
Our list of names is sorted when all names are in alphabetical order. We know of course which name should come first, but we can also compare two strings as we compare numbers using the smaller than
(<) and greater than (<) symbols:
print('Lovelace' < 'Turing')
print('Turing' > 'Lovelace')
Both instructions will print True - because Lovelace (beginning with L) should appear before Turing (starting with T).
Our goal is to get the entire list in order, but how do we get there? Even if we can sort a list in our heads, it can be difficult to instruct a computer to do it.
All steps needed to reach our goal of a sorted list constitute a sorting algorithm. There are many available, each with their own pros and cons. Here, we'll look at one called Bubblesort:
The idea is as follows:
• Pick two names next to each other.
• If they are in the wrong order, swap them.
After that, the list will be at least a little better sorted than before. That means, if we do this enough times, eventually the whole list will be sorted.
Bubblesort in Python
This is how we would carry out Bubblesort using Python:
# Go through the list from start to end using a for-loop
# Compare each pair of names next to each other
# If they are in the wrong order, swap them
That can be written as follow:
Compare the first output to the second, you can see it's now a little better than before!
What we need now is to run this several times, and continually get a slightly more sorted list than before. By putting the for loop inside another for-loop we can achieve exactly this.
The final program looks like this:
names = ['Torvalds', 'Zuckerberg', 'Lovelace', 'Turing', 'Johnson']
print("Unsorted version:")
for j in range(0, len(names)):
for i in range(0, len(names) - 1):
if names[i+1] < names[i]:
names[i], names[i+1] = names[i+1], names[i]
print("Sorted version:")
In this video, we talk about a type of algorithm (sorting algorithms) by implementing the famous "Bubblesort".
• Algorithm: A description for how to solve a problem. A recipe is an example of an algorithm for cooking a certain dish. Programming is writing code to carry out steps of an algorithm.
• Bubblesort: One (of many) algorithms for sorting the contents of a list.
• List: A list is a collection of data in order. Can also be known as "array" in other programming languages.
• for-loop: One type of loop/repetition that fits well for a predetermined number of steps. | {"url":"https://www.mathplanet.com/education/programming/real-thinking/sorting","timestamp":"2024-11-12T19:54:59Z","content_type":"text/html","content_length":"52369","record_id":"<urn:uuid:914cdf52-5b40-4e7f-b878-5973974f3b92>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00634.warc.gz"} |
Market maker for events with more than one winning outcome
I wonder if we can build a market maker for events with a defined number of winning outcomes. Lets say we want to predict the players that will start in a soccer game. Lets say the team has 20
players and we know exactly 11 will start.
We could do: a) 20 markets for each player whether or not he will play. However - in this case all markets would be independent although they are not. We can also make the “brute force” solution
where we have all possible combinations of 11 players. However - this might get messy.Is there a market maker algorithm for n outcomes with a k winners?
By the way - maybe the “brute force” solution is better because I guess you lose the ability to express that it is unlikely that player A and player B will player together in the other solution.
However - all possible outcomes can become unfeasible so such a market maker will be useful anyway. | {"url":"https://forum.gnosis.io/t/market-maker-for-events-with-more-than-one-winning-outcome/25","timestamp":"2024-11-04T20:41:42Z","content_type":"text/html","content_length":"15992","record_id":"<urn:uuid:2526767c-503d-4e46-b2d6-c03c6a1472ce>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00685.warc.gz"} |
Approximating Integrals - Midpoint, Trapezoidal, and Simpson's Rule
Approximating Integrals – Midpoint, Trapezoidal, and Simpson’s Rule
are extremely helpful when we can’t evaluate a definite integral using traditional methods. Through numerical integration, we’ll be able to approximate the values of the definite integrals.
The techniques of approximating integrals will show us how it’s possible to numerically estimate the definite integral of any function. The three common numerical integration techniques are the
midpoint rule, trapezoid rule, and Simpson’s rule.
At this point in our integral calculus discussion, we’ve learned about finding the indefinite and definite integrals extensive. There are instances, however, that finding the exact values of definite
integrals won’t be possible. This is when approximating integrals enter.
In this article, we’ll focus on approximating integrals using the three mentioned techniques. Since we’re dealing with definite integrals, review your understanding of the fundamental theorem of
When to approximate an integral?
We can approximate integrals by estimating the area under the curve of $\boldsymbol{f(x)}$ for a given interval, $\boldsymbol{[a, b]}$. In our discussion, we’ll cover three methods: 1) midpoint rule,
2) trapezoidal rule and 3) Simpson’s rule.
As we have mentioned, there are functions where finding their antiderivatives and the definite integrals will be an impossible feat if we stick with the analytical approach. This is when the three
methods for approximating integrals will come in handy.
\begin{aligned}\int_{0}^{4} e^{x^2}\phantom{x}dx\\\int_{0}^{2} \dfrac{\sin x}{x}\phantom{x}dx \end{aligned}
These are two examples of definite integrals that will be challenging to evaluate if we use the integration techniques we’ve learned in the past.
This is when the three integral approximation techniques enter. The first approximation you’ll learn in your integral calculus classes is the Riemann sum. We’ve learned how it’s possible to estimate
the area under the curve by dividing the region into smaller rectangles with a fixed width.
The graph shown above highlights how the Riemann sum works: divide the region under the curve with $n$ rectangles that share a common width, $\Delta x$. The value of $\Delta x$ is simply the
difference between the intervals’ endpoints divided by $n$: $\Delta x = \dfrac{b- a}{n}$.
We can estimate the area and the integral using the relationships shown below:
Right-hand Riemann sum Left-hand Riemann sum
\begin{aligned}\int_{b}^{a}f(x)\phantom{x}dx \approx \sum_{i= 1}^{n} f(x_i) \Delta x\end{aligned} \begin{aligned}\int_{b}^{a}f(x)\phantom{x}dx \approx \sum_{i= 1}^{n} f(x_{i- 1}) \Delta x\end
Keep in mind the $x_i$ represents the initial value that we’re starting with. We’ve already discussed the Riemann sum in this article, so make sure to check it out in case you need a refresher.
In the next section, we’ll show you the three numerical integration methods you can use to integrate complex integrals such as $f(x) = e^{\sin(0.1x^2)}$. We’ll also show you examples to make sure
that we implement each technique.
How to approximate an integral?
The three approximation techniques that we’ll focus on using similar processes like that of the Riemann sum. We’ll show you what makes each technique special and of course, show you how to implement
each method to approximate integrals.
Midpoint rule: integral approximation definition
It’s a good thing that we did a refresher on Riemann sum. That’s because the midpoint rule is an extension of the Riemann sum. The midpoint rule utilizes the two subinterval’s midpoint, $x_i$.
Let’s say we want to evaluate $\int_{a}^{b} f(x)\phantom{x} dx$, approximate its value using the midpoint rule by following the steps below:
• Divide the intervals into $n$ equal parts. Each new subinterval must have a width of $\Delta x = \dfrac{b – a}{n}$.
• We must begin with $x_0 = a$ and end with $x_n = b$. Meaning, we’ll have subintervals, $[x_0, x_1], [x_1, x_2], [x_2, x_3],…,[x_{n- 1}, x_n]$.
• Find the height of the rectangle formed by two subintervals, $x_{i-1}$ and $x_i$, by finding their midpoint: $\overline{x_i} = \dfrac{x_{i -1} + x_i}{2}$.
• Approximate the definite integral by finding the value of $\approx \sum_{i= 1}^{n} f(\overline{x_i}) \Delta x$
This means that through the midpoint rule, we can evaluate the definite integral using the formula shown below:
\begin{aligned}M_n &= \sum_{i =1}^{n}f(\overline{x_i})\Delta x\\\int_{a}^{b} f(x)\phantom{x}dx &= \lim_{n \rightarrow \infty} M_n\end{aligned}
To better understand the midpoint rule’s process, let’s estimate the value of $\int_{0}^{4} x^2\phantom{x}dx$ using the midpoint rule:
• Divide the interval into four subintervals with a width of: $\Delta x = \dfrac{4 -0}{4} = 1$ unit.
• This means that we have the following intervals: $[0, 1]$, $[1, 2]$, $[2, 3]$, and $[3, 4]$.
• Find the respective midpoints shared by each pair of subintervals: $\left\{\dfrac{1}{2}, \dfrac{3}{2},\dfrac{5}{2},\dfrac{7}{2}\right\}$.
The graph below illustrates how the integral of $x^2$ is approximated using the midpoint rule.
Find the value of $\int_{0}^{4} x^2\phantom{x} dx$ by evaluating $\boldsymbol{f(x)}$ at the midpoints. Multiply each value to $\boldsymbol{\Delta x}$ then add all resulting values to estimate the
integral’s value.
\begin{aligned}M_4 &= \sum_{i =1}^{4}f(\overline{x_i})\cdot (\Delta x) \\&= (1)f\left(\dfrac{1}{2}\right) + (1)f\left(\dfrac{3}{2}\right)+ (1)f\left(\dfrac{5}{2}\right) + (1)f\left(\dfrac{5}{2}\
right)\\&= 1 \cdot \dfrac{1}{4}+ 1 \cdot \dfrac{9}{4} +1 \cdot \dfrac{25}{4}+ 1 \cdot \dfrac{49}{2} \\&= 21\end{aligned}
If we evaluate the integral, $\int_{0}^{4} x^2\phantom{x} dx$, it’s actual value will be $\dfrac{64}{3}$ or $21.\overline{3}$. This shows that the estimated value from the midpoint rule is actually
close to the actual value.
Whenever possible, find the absolute error of the approximation by finding the absolute value of the difference between the integral’s actual value and approximated value.
\begin{aligned}\left|21 – \dfrac{64}{3}\right| &= \dfrac{1}{3}\\&\approx 0.33\end{aligned}
We can also find the relative error of the approximation by expressing the absolute error as the percent change of the actual value as shown below.
\begin{aligned}\left|\dfrac{A – B}{A}\right| \cdot 100\%\end{aligned}
This means that relative error for our calculation is:
\begin{aligned}\left|\dfrac{21 – \dfrac{64}{3}}{21}\right| \cdot 100\% &\approx 1.58\%\end{aligned}
These two error approximations confirm our observation: that the approximate value is a good enough estimation. Of course, it’s much easier to simply evaluate $\int_{0}^{4} x^2\phantom{x} dx$
analytically. But as we have mentioned, that won’t be the case for complex integrals.
Trapezoidal rule: integral approximation definition
For this method, instead of using rectangles, we’ll be using trapezoids. Hence, its name: trapezoidal rule. When given a definite integral, we can estimate the value of $\int_{a}^{b} f(x)\phantom{x}
dx$ by approximating the area of the trapezoids under the curve.
The midpoint and trapezoid rules will have similar steps. Before we begin, recall that the formula for the trapezoid’s area is $\dfrac{1}{2}h(b_1 + b_2)$, where $h$ represents the height and $b_1$
and $b_2$ represents the two bases. Now, let’s go ahead and break down the steps for the trapezoid rule’s process:
• Divide the intervals of the given definite integral into $n$ equal parts. Determine the subinterval’s height by dividing $b -a$ by $n$: $\Delta x = \dfrac{b – a}{n}$.
• Keep in mind that we must have $x_0 = a$ and $x_n = b$. This means that the endpoints of the intervals are: $\{x_0, x_1, x_2, …, x_n\}$.
• Approximate the first trapezoid’s area using $f(x_0)$ and $f(x_1)$ as its bases.
\begin{aligned}T_1 &= \dfrac{1}{2}\Delta x [f(x_0) + f(x_1)]\end{aligned}
• Apply the same process to find the areas of the rest of the trapezoids then add up the areas for all $n$ trapezoids.
We’ll focus on the last bullet: adding up all the areas of the $n$ trapezoids. If we have the subinterval’s endpoints, $\{x_0, x_1, x_2, …, x_n\}$, we can find the sum of the areas, $T_n$, as shown
\begin{aligned}T_n &= \dfrac{\Delta x }{2}[f(x_0) + f(x_1)]+\dfrac{\Delta x }{2}[f(x_1) + f(x_2)]+ …+ \dfrac{\Delta x }{2}[f(x_{n-1}) + f(x_n)]\\&= \dfrac{\Delta x }{2}[f(x_0) + 2f(x_1)+ 2f(x_2)+ …+
2f(x_{n -1})+ f(x_n)]\\\lim_{n\rightarrow +\infty}T_n &= \int_{a}^{b}f(x)\phantom{x}dx\end{aligned}
This means that we can estimate the definite integral by applying the formula for $T_n$ and that’s the Trapezoidal rule.
Estimate the value of $\int_{0}^{4} x^2\phantom{x}dx$ using the trapezoidal rule and four subintervals this time. Afterward, compare the approximate value of the integral and its actual value: $\
dfrac{64}{3}$ squared units.
• Find each of the four trapezoids’ heights: $\Delta x = \dfrac{4 -0}{4} = 1$ unit.
• We’ll be working with four trapezoids with the following subintervals with the following endpoints: $\{0, 1, 2, 3,4\}$
• Calculate the areas of the trapezoids by evaluating functions at the endpoints.
Here’s the graph of $f(x) = x^2$ with the area under its curve is divided into four trapezoids. Calculate the total area of the four trapezoids as shown below:
\begin{aligned}T_4 &= \dfrac{\Delta x }{2}[f(x_0) + 2f(x_1)+ 2f(x_2)+ 2f(x_3)+ f(x_4)]\\&= \dfrac{1}{2}[f(0)+ 2f(1)+ 2f(2) + 2f(3)+ f(4)]\\&= \dfrac{1}{2}[0 + 2(1) + 2(4) + 2(9) + 16]\\&= 22\end
Since $T_4 = 22$, $\int_{0}^{4} x^2\phantom{x}dx$ is approximately equal to $22$ squared units through the trapezoidal rule. As we did with the midpoint rule, let’s find our approximation’s absolute
and relative error:
Absolute Error Relative Error
\begin{aligned}\left|\dfrac{64}{3} -22\right| &= \dfrac{2}{3}\end{aligned} \begin{aligned}\left|\dfrac{\dfrac{64}{3}-22}{\dfrac{64}{3}}\right|\cdot 100\% &\approx 3.125\%\end{aligned}
These two values show us that the value returned by trapezoidal rule is close enough compared to its actual value.
Simpson’s rule: integral approximation definition
Before we dive right into the process of Simpson’s rule, let’s first observe how the accuracies of midpoint and trapezoidal rules’ approximations improve as we use more intervals.
For the midpoint rule, we can see that we divide the regions into smaller rectangles, the approximations become closer to the definite integral’s value.
The same goes for the trapezoidal rule – the approximation becomes more accurate as we use more trapezoids. If we want a more accurate approximation, we’ll have to evaluate more values. That becomes
problematic and time-consuming especially when we need to account for a certain error threshold.
This where Simpson’s rule comes in handy: this particular method utilizes quadratic curves instead. This is why Simpson’s rule returns more accurate approximations for functions with curves. The
midpoint and trapezoidal rules, on the other hand, will return better results for straight lines.
In Simpson’s rule, we approximate the area under the curve by piecing together three quadratic curves within the set subinterval’s width. The more intervals (and consequently, more parabolic curves),
the better the approximation.
This means that when we’re given $\int_{a}^{b}f(x) \phantom{x}dx$, we are approximating the area of the region bounded by $(a, f(a))$, $\left(\dfrac{a + b}{2}, f\left(\dfrac{a + b}{2}\right)\right)$,
and $(b, f(b))$.
Let’s go ahead and break down the process of approximating $\int_{a}^{b}f(x) \phantom{x}dx$:
• Find the distance between each interval and express it as $\Delta x = \dfrac{b – a}{n}$.
• We’ll have the following endpoints for the $n$ subintervals: $\{x_0, x_1, x_2,…, x_n\}$.
• Approximate the definite integral using the formula for Simpson’s rule as shown below.
\begin{aligned}S_n &= \dfrac{\Delta x }{3}[f(x_0) + 4f(x_1)+ 2f(x_2)+ 4f(x_3)+ 2f(x_4)+ …+ 2f(x_{n -2})+4f(x_{n -1})+ f(x_n)]\\\lim_{n\rightarrow +\infty}S_n &= \int_{a}^{b}f(x)\phantom{x}dx\end
This may form may look intimidating, but this is simply equivalent to finding the weighted average of the approximation returned by the midpoint and trapezoidal rule. It’s also equivalent to the
average of the left and right-hand Riemann sums.
Why don’t we estimate $\int_{0}^{4}x^2 \phantom{x}dx$ using Simpson’s rule? We’re still using four intervals for our approximation.
• As with before, we have $\Delta x = \dfrac{4 -0}{4} = 1$ unit
• The area will have the following subintervals and endpoints: $\{0, 1, 2, 3,4\}$
• Apply the formula for Simpson’s rule to estimate the definite integral.
Let us show you the graph of $f(x)= x^2$ as well as the region’s area under the curve partitioned through Simpson’s rule.
What are the error bounds for the three methods?
In the previous examples, we’ve shown you how to compare the approximated and actual values of the definite integral. We can use the absolute and relative errors to gauge the accuracy of the three
Here’s the catch though: most of the time, we only use these approximations when we can’t evaluate the definite integral analytically. When this happens, we can still gauge the accuracy for the three
If we’re given a function, $f(x)$, that is continuous and twice differentiable over the interval, $[a, b]$. Let $M$ represent the maximum value of $|f^{\prime\prime}(x)|$ over the interval, $[a, b]
$. We can calculate the error bounds for the midpoint, trapezoidal, and the Simpson’s rule as shown below:
Method Upper Bound
Midpoint Rule $(M_n)$ \begin{aligned}M_n &\leq \dfrac{M(b – a)^3}{24n^2}\end{aligned}
Trapezoidal Rule $(T_n)$ \begin{aligned}T_n &\leq \dfrac{M(b – a)^3}{12n^2}\end{aligned}
Simpson’s Rule $(S_n)$ \begin{aligned}S_n &\leq \dfrac{M(b – a)^5}{180n^4}\end{aligned}
With these error bounds, we’ll now be able to know the value of $n$ to guarantee the accuracy of these estimated definite integrals.
Let’s say we want to find the right value of $n$ so that the approximated value of $\int_{0}^{4} x^2\phantom{x}dx$ is accurate within $0.01$. Assume that we’re using the midpoint rule.
Begin by finding the second derivative of $x^2$ then determine the second derivative’s maximum value.
\begin{aligned}\dfrac{d}{dx} x^2 &= 2x\\\dfrac{d}{dx} 2x &= 2\end{aligned}
Since the second derivative is equal to $2$, $M =2$. Use our given lower and upper limits, $a = 0$ and $b = 4$. Find the value of $n$ so that $M_n$ is less than or equal to $0.01$.
\begin{aligned}M_n &\leq 0.01\\\dfrac{M(b – a)^3}{24n^2}&\leq 0.01\\\dfrac{2(4 -0)^3}{24n^2}&\leq 0.01\\n &\geq \sqrt{\dfrac{2(64)}{24(0.01)}}\approx23.09\end{aligned}
This means that if we want to have an accuracy of at least $0.01$, $n$ must be equal to $23$.
This confirms that with $n =23$ or with $23$ rectangles, we can approximate $\int_{0}^{4} x^2 \phantom{x} dx$ up to a $0.01$ error.
Summary of the three methods used in approximating integrals
Before working on more examples, let us summarize our discussion for you first. Here’s a table summarizing the calculations needed for the midpoint, trapezoidal, and Simpson’s rules.
Midpoint \begin{aligned}M_n &= \sum_{i =1}^{n}f(x_i)\Delta x\\\int_{a}^{b} f(x)\phantom{x}dx &= \lim_{n \rightarrow \infty} M_n\end{aligned}
Trapezoidal \begin{aligned}T_n &= \dfrac{\Delta x }{2}[f(x_0) + f(x_1)]+\dfrac{\Delta x }{2}[f(x_1) + f(x_2)]+ …+ \dfrac{\Delta x }{2}[f(x_{n-1}) + f(x_n)]\\&= \dfrac{\Delta x }{2}[f(x_0) + 2f(x_1)+
Rule 2f(x_2)+ …+ 2f(x_{n -1})+ f(x_n)]\\\lim_{n\rightarrow +\infty}T_n &= \int_{a}^{b}f(x)\phantom{x}dx\end{aligned}
Simpson’s \begin{aligned}S_n &= \dfrac{\Delta x }{3}[f(x_0) + 4f(x_1)+ 2f(x_2)+ 4f(x_3)+ 2f(x_4)+ …+ 2f(x_{n -2})+4f(x_{n -1})+ f(x_n)]\\\lim_{n\rightarrow +\infty}S_n &= \int_{a}^{b}f(x)\phantom
Rule {x}dx\end{aligned}
When we have the approximated and actual values, we can also check the approximations for absolute and relative errors.
Absolute Error \begin{aligned}|A – B| \end{aligned}
Relative Error \begin{aligned}\left|\dfrac{A – B}{A}\right| \cdot 100\% \end{aligned}
Of course, if the actual value is not available, we can use the upper error bounds to find the right parameters for our approximation.
Method Upper Bound
Midpoint Rule $(M_n)$ \begin{aligned}M_n &\leq \dfrac{M(b – a)^3}{24n^2}\end{aligned}
Trapezoidal Rule $(T_n)$ \begin{aligned}T_n &\leq \dfrac{M(b – a)^3}{12n^2}\end{aligned}
Simpson’s Rule $(S_n)$ \begin{aligned}S_n &\leq \dfrac{M(b – a)^5}{180n^4}\end{aligned}
Example 1
Given that $n =6$, estimate the value of $\int_{2}^{8} \dfrac{1}{x^2 + 1}\phantom{x}dx$ using the following approximating integral methods:
a. Midpoint Rule $(M_n)$
b. Trapezoidal Rule $(T_n)$
c. Simpson’s Rule $(S_n)$
Report your approximations to five decimal places.
To apply the midpoint rule for the given definite integral, find $\Delta x$ and the subintervals first:
• Using $n=6$, $a = 2$, and $b = 8$, we have $\Delta x=\dfrac{8 -2}{6} = 1$.
• The subintervals that we’ll be working with are :$[2, 3]$, $[3, 4]$, $[4,5]$, $[5,6]$, $[6,7]$, and $[7, 8]$.
• This means that we have the following midpoints, $\overline{x_i}: \left\{\dfrac{5}{2}, \dfrac{7}{2}, \dfrac{9}{2}, \dfrac{11}{2}, \dfrac{13}{2}, \dfrac{15}{2}\right\}$.
Apply the formula for the midpoint rule, $ M_n = \sum_{i =1}^{n}f(\overline{x_i})\Delta x$.
\begin{aligned}M_6 &= \sum_{i =1}^{6}f(\overline{x_i})\Delta x\\&= (1)f\left(\dfrac{5}{2}\right) +(1)f\left(\dfrac{7}{2}\right)+(1)f\left(\dfrac{9}{2}\right)\\&+(1)f\left(\dfrac{11}{2}\right)+(1)f\
left(\dfrac{13}{2}\right) +(1)f\left(\dfrac{15}{2}\right)\\&=\dfrac{4}{29}+\dfrac{4}{53}+\dfrac{4}{85}+\dfrac{4}{125}+\dfrac{4}{173} + \dfrac{4}{229}\\&\approx 0.33305\end{aligned}
This means that with $n = 6$ as interval, $\int_{2}^{8}\dfrac{1}{x^2 +1}\phantom{x}dx$ is approximately equal to $0.33305$. We’ve included a graph to show you how this approximation would appear when
plotted, but this is not a required step.
We’ll use the same values for $\Delta x$ and the subintervals to approximate the integral using the trapezoidal rule. Apply the formula below to approximate $\int_{2}^{8}\dfrac{1}{x^2 +1}\phantom{x}
\begin{aligned}T_n &= \dfrac{\Delta x }{2}[f(x_0) + 2f(x_1)+ 2f(x_2)+ …+ 2f(x_{n -1})+ f(x_n)]\end{aligned}
Evaluate $f(x)$ at the following endpoints: $\{2,3, 4, 5, 6, 7, 8\}$. Hence, we have the estimated value of $T_6$ as shown below.
\begin{aligned}T_6 &= \dfrac{\Delta x }{2}[f(x_0) + 2f(x_1)+ 2f(x_2)+ 2f(x_3)+2f(x_4)+2f(x_5)+ f(x_6)]\\&= \dfrac{1}{2}[f(2) + 2f(3)+ 2f(4)+ 2f(5)+2f(6)+2f(7)+ f(8)]\\&= \dfrac{1}{2}\left[\dfrac{1}
{5} + \dfrac{1}{5} +\dfrac{2}{17}+\dfrac{1}{13}+\dfrac{2}{37}+\dfrac{1}{25}+\dfrac{1}{65}\right]\\&\approx 0.35200\end{aligned}
Through the trapezoidal rule, we have $\int_{2}^{8} \dfrac{1}{x^2 + 1}\phantom{x}dx$ as approximately $0.35200$.
Let’s now proceed approximating the integral using the Simpson’s rule. Apply the formula shown below to estimate $\int_{2}^{8} \dfrac{1}{x^2 + 1}\phantom{x} dx$.
\begin{aligned}S_n &= \dfrac{\Delta x }{3}[f(x_0) + 4f(x_1)+ 2f(x_2)+ 4f(x_3)+ 2f(x_4)+ …+ 2f(x_{n -2})+4f(x_{n -1})+ f(x_n)] \end{aligned}
Use the same values for $\Delta x$ and the endpoints.
\begin{aligned}S_6 &= \dfrac{\Delta x }{3}[f(x_0) + 4f(x_1)+ 2f(x_2)+ 4f(x_3)+ 4f(x_4)+ 2f(x_5)+ 4f(x_6)+ +f(x_7)]\\&=\dfrac{1 }{3}[f(2) + 4f(3)+ 2f(4)+ 4f(5)+ 2f(6)+ 4f(7)+ f(8)]\\&=\dfrac{1}{3}\
left[\dfrac{1}{5} + \dfrac{2}{5}+\dfrac{2}{17}+\dfrac{2}{13}+\dfrac{2}{37}+\dfrac{2}{25}+ \dfrac{1}{65} \right ]\\&\approx 0.34031 \end{aligned}
This confirms our result: $\int_{2}^{8} \dfrac{1}{x^2 + 1}\phantom{x} dx$ is approximately $0.34031$ through the Simpson’s rule.
Hence, we have the following results:
Approximation Technique Approximated Value
a. Midpoint Rule \begin{aligned}M_6 &= 0.33305\end{aligned}
b. Trapezoidal Rule \begin{aligned}T_6 &= 0.35200\end{aligned}
c. Simpson’s Rule \begin{aligned}S_6 &= 0.34031\end{aligned}
Example 2
Use the results from the previous example and construct a table comparing the absolute and relative errors for the three methods.
Use the actual value shown below:
\begin{aligned}\int_{2}^{8} \dfrac{1}{x^2 + 1}\phantom{x}dx = \tan^{-1}(8) -\tan^{-1}(2) \approx 0.33929\end{aligned}
To estimate the absolute and relative errors, we’ll compare the actual value with the approximated values returned through the three methods: midpoint rule, trapezoidal rule, and Simpson’s rule.
Use the formula shown below and use replace $A$ with the actual value and $B$ with the approximated value.
Absolute Error \begin{aligned}|A – B| \end{aligned}
Relative Error \begin{aligned}\left|\dfrac{A – B}{B}\right| \cdot 100\% \end{aligned}
Hence, we have the following absolute and relative errors for the three methods:
Approximation Approximated Value Absolute Error Relative Error
a. Midpoint Rule \begin{aligned}M_6 &= 0.33305\end \begin{aligned}|0.33929 – 0.33305| &= 0.00624\end \begin{aligned}\left|\dfrac{0.33929 – 0.33305}{0.33929}\right| \cdot 100\% &\approx
{aligned} {aligned} 1.83913\% \end{aligned}
b. Trapezoidal Rule \begin{aligned}T_6 &= 0.35200\end \begin{aligned}|0.33929 – 0.35200| &= 0.01271\end \begin{aligned}\left|\dfrac{0.33929 – 0.35200}{0.33929}\right| \cdot 100\% &\approx
{aligned} {aligned} 3.74606 \% \end{aligned}
c. Simpson’s Rule \begin{aligned}S_6 &= 0.34031\end \begin{aligned}|0.33929 – 0.35200| &= 0.00102\end \begin{aligned}\left|\dfrac{0.33929 – 0.33305}{0.33929}\right| \cdot 100\% &\approx
{aligned} {aligned} 0.30063 \% \end{aligned}
From this table of values, we can also see that the Simpson’s rule returned the smallest absolute and relative errors.
Practice Questions
1. Given that $n =6$, estimate the value of $\int_{2}^{8} \dfrac{1}{x^2 – 1}\phantom{x}dx$ using the following approximating integral methods:
a. Midpoint Rule $(M_n)$
b. Trapezoidal Rule $(T_n)$
c. Simpson’s Rule $(S_n)$
Report your approximations to five decimal places.
2. Given that $n =8$, estimate the value of $\int_{1}^{5} \dfrac{1}{e^x}\phantom{x}dx$ using the following approximating integral methods:
a. Midpoint Rule $(M_n)$
b. Trapezoidal Rule $(T_n)$
c. Simpson’s Rule $(S_n)$
Report your approximations to six decimal places.
3. Given that $n =5$, estimate the value of $\int_{0}^{4} \sqrt{e^x}\phantom{x}dx$ using the following approximating integral methods:
a. Midpoint Rule $(M_n)$
b. Trapezoidal Rule $(T_n)$
c. Simpson’s Rule $(S_n)$
Report your approximations to three decimal places.
4. Use the results from the previous example and construct a table comparing the absolute and relative errors for the three methods.
Use the actual value shown below:
\begin{aligned}\int_{0}^{4} \sqrt{e^x}\phantom{x}dx = 2(e^2 -1) \approx 12.778\end{aligned}
Answer Key
a.$M_6 \approx 0.40784$
b.$T_6 \approx 0.45734$
c.$S_6 \approx 0.42434$
a.$M_8 \approx 0.357407$
b.$T_8 \approx 0.368634$
c.$S_8 \approx 0.361149$
a.$M_5 \approx 12.693 $
b.$T_5 \approx 12.948$
c.$S_5 \approx 12.779$
Approximation Technique Absolute Error Relative Error
a. Midpoint Rule \begin{aligned}0.848\end{aligned} \begin{aligned}0.664\% \end{aligned}
b. Trapezoidal Rule \begin{aligned}0.170\end{aligned} \begin{aligned}1.330\% \end{aligned}
c. Simpson’s Rule \begin{aligned}0.011\end{aligned} \begin{aligned}0.001\% \end{aligned}
Images/mathematical drawings are created with GeoGebra. | {"url":"https://www.storyofmathematics.com/approximating-integrals/","timestamp":"2024-11-04T15:12:54Z","content_type":"text/html","content_length":"181089","record_id":"<urn:uuid:070127c2-3a5d-4e48-b592-6adf8716b547>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00739.warc.gz"} |
Geometry Worksheets
Geometry Math Worksheets
In this section, you can view and download all of our geometry worksheets. These include common-core aligned, themed and age-specific worksheets. Perfect to use in the classroom or homeschooling
Geometry Worksheets & Study Resources:
Brief definition
Geometry is one of the interesting branches of Mathematics that deals with sizes, shapes, polygons, angles, and measurements. It also deals with the calculation of the area, surface area, perimeter,
and volume of shapes, the relationship of terms such as lines, points, angles, solid figures and surfaces, spatial relationships, and attributes of an object. These are supported by the properties,
postulates, and theorems in Geometry. There are also different branches of geometry such as plane, solid, and coordinate geometry, and many more.
Importance of the Topic
The world is surrounded by many geometric properties. Learning geometry teaches learners to basically identify 2D and 3D shapes, lines, angles, and many more. It enhances the visual ability, good
reasoning, and problem-solving of the learners and provides the learners the knowledge on how to measure things and see their connections, among others. Learning geometry is also learning to
understand relationships among shapes which greatly contribute to drawing, doing artistic works, and thinking logically.
Application of the Learned Topic in Life
Geometry has the most evident applications in life. This is used in different fields such as architecture, engineering, art, sports, technology, and many more. In day-to-day life, geometry is used to
measure how long a line is, how many gift wrappers you need to cover a box of shoes, how much of an ingredient should be used to make a perfect dish, and such. Geometry is also very evident in
nature, like the leaves on the trees, petals of a flower, roots, and barks. Construction of buildings also involves geometry, where engineers need to know how long a staircase should be to make it
safe and proportional to the ground and wall. | {"url":"https://helpingwithmath.com/worksheets/geometry/","timestamp":"2024-11-05T06:38:06Z","content_type":"text/html","content_length":"159040","record_id":"<urn:uuid:31c89b62-8fae-47e9-aa60-e4207ab94519>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00480.warc.gz"} |
Goals Worth Struggling For
How can teachers tell if a student struggling to solve a mathematics problem is productively moving toward a solution or just spinning their wheels? Teachers wrestling with the complex realities of
productive struggle in the classroom must ask themselves important questions, such as what do I do if a student is completely stuck? What if they give up? What questions do I ask? Is there a time
when just telling a student something is appropriate? How much help is too much or too little?
An important guiding principle in answering these questions is knowing when the struggle is related to the activity's goal. The ability to judge whether each student's struggle is related or not
involves planning. Let's look at how that planning might work with an example from the Illustrative Mathematics (IM) 6–8 math curriculum.
Step 1: Understand the activity's goal within a lesson.
The lesson's goal is for students to see how a balanced hanger is like an equation and how moving its weights is like solving the equation. As you do the math, take a moment to consider what students
need to know at the end of this activity to support the goal. How will future learning experiences build on this one?
Step 2: Anticipate student responses.
Think about the diverse mathematical understandings and experiences students have had and anticipate their varying responses to the activity's questions. As you anticipate their strategies, consider
how students might explain their actions. How might they productively struggle? What would unproductive struggle look like in this activity?
Step 3: Plan how you will launch the activity.
There is a delicate balance between offering helping and asking questions that grant students access to the answers without reducing mathematical cognitive demand. For example, it would discourage
productive struggle to launch the activity by showing students a similar problem with a strategy for solving two-step equations.
As you reflect on what students may do during the activity, consider what part of the activity students should productively struggle with as they move toward a solution. What piece of the activity
might students find inaccessible? How can you launch this activity without taking away the problem students need to grapple with to learn?
Step 4: Decide whether to redirect, provide support, or let it go.
Because we want all students to succeed in our mathematics classes, our inclination is to provide hints or help until they get a correct answer. When we step back and look at who did all of the work,
however, we don't want it to be the teacher. Students could struggle productively or unproductively in varying ways, and each case requires a different instructional move.
Going back to the example above, a student working on the second hanger may be discussing with their peer whether the weight has to represent a whole number. Since it is important for students to
learn that the equation's solution is the number that makes the equation true, whether or not it is a whole number, this would be an example of productive struggle. In this case, the teacher should
let the students continue their discussion but prompt them to try the third and fourth hanger to compare the values of the variables in each one.
Another student working on the fourth hanger could be removing the same amount of weight from both sides of the hanger but struggling to divide up the remaining weight of the four w's because the
fraction is represented in halves. The teacher could redirect their thinking by asking if they can write 14/2 as a whole number. In this case, the suggestion to work with whole numbers may be the
nudge the student needs to move past the unproductive struggle. Since the activity's goal is to think of a strategy for finding a missing weight, that redirection is not reducing the relevant demand
of the task.
Step 5: Reflect on what you did well and what improvements you would make next time.
Count your victories. Where did you support productive struggle? When could you have asked a better question? How can you improve upon your commitment to support students as they make sense of
problems and persevere in solving them?
In deciding when and how to intervene when students are struggling, the overriding consideration is how their struggle relates to the lesson's goals. Maintain a clear picture of those goals and
continually evaluate how students are progressing toward them. Struggle unrelated to the learning goal is often unproductive, but we don't want to take away productive opportunities to struggle when
it will help students move along the established curricular progression and strengthen future learning. | {"url":"https://ascd.org:443/el/articles/goals-worth-struggling-for","timestamp":"2024-11-09T19:38:45Z","content_type":"application/xhtml+xml","content_length":"213711","record_id":"<urn:uuid:6c5065b1-57c0-4e60-9cfe-02abb743b998>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00049.warc.gz"} |
Calculation of the Cost Column on the Gains page
Let's assume you bought 1 BTC for 100 Euro and then sold 0.5 BTC for 60 Euro.
You therefore have 0.5 BTC left that you bought for 50 Euro and you made a 10 Euro realized gain and a 1455,37 Euro unrealized gain (assuming the current BTC price is 3010,74 Euro).
Since the cost column in the summary table (the first table at the top) shows only the cost of the REMAINING coins, you would see on the "Realized and Unrealized Gain" Page:
What you would NOT see is the 50 Euro you originally spent on the already sold 0.5 BTC. Therefore the summary table only shows 50 Euro value of coins owned and 10 Euro realized gains.
You would not be able to see the 100 Euro invested - only if you had NOT sold any coins would the Cost reflect the sum invested. If you sold everything, the cost would be 0 (since there are no coins
Thus the calculation Original spending for currencies = Currency cost + Realized gain/loss would only be correct if you kept all coins and would otherwise always miss the purchase value of the coins
sold already (since only the gain or loss would show up in the last column).
You can however find the value of the original spending for currencies in the Realized Gains table (blue link below the summary table). Here the Cumulated Cost will be displayed in addition to the
gain or loss. | {"url":"https://cointracking.freshdesk.com/en/support/solutions/articles/29000026053-calculation-of-the-cost-column-on-the-gains-page","timestamp":"2024-11-10T17:58:22Z","content_type":"text/html","content_length":"137812","record_id":"<urn:uuid:bf7858f8-80a5-42d1-a497-855bfce21ab4>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00545.warc.gz"} |
Algebraic Expressions of Monomials and Polynomials: Addition and Subtraction ｜ Hatsudy
In mathematics, we use letters to perform calculations. This is called an algebraic expression. Before learning the algebraic expression, calculations are done using only numbers. However, after
learning algebraic expressions, math is mainly done by using the alphabet.
Algebraic expressions have rules. Therefore, you have to learn how to use algebraic expressions.
There are also several types of equations that are represented by symbols: monomials and polynomials. What are the differences between these types of expressions? If you don’t understand the rules of
algebraic expressions, you can’t add or subtract using these expressions.
So, we’ll explain the rules of algebraic expressions, the difference between monomial and polynomial formulas, and how to add and subtract using these expressions.
In an Algebraic Expression, Turning Unknown Numbers into Symbols
What kind of calculation is the algebraic expression that we learn in math? In an algebraic expression, we use letters of the alphabet to represent equations as follows.
When we look at equations that contain symbols, you may think that it looks difficult to understand.
However, we have been studying algebraic expressions in elementary school. For example, what is included in the following $□$?
You must have solved a problem like this before. The number that goes in □ is 6. For numbers we don’t know, we make an equation by using □. Instead, in the algebraic expressions, the unknown numbers
are represented by the alphabet.
For example, in the previous problem, we can replace $□$ with $x$ and write the following.
This is an algebraic expression. We use letters, such as $x, y, a, b$, etc., to represent numbers we don’t know and create an equation.
-Algebraic Expressions Are Useful in Calculations
Why do we use algebraic expressions? It is because they are useful in math calculations. Mathematics is used frequently in our daily life. In most cases, we already know the answer, but we don’t know
how to express the process.
For example, here’s an example.
• How many kilometers should you walk per day to walk 1,000 km in 365 days (one year)?
We already know the answer to walking 1,000 km. But we don’t know the process of how many kilometers a day we should walk. So, we replace the numbers we don’t know with symbols such as $x$ and
calculate them.
Even if we don’t know the numbers, we can construct an equation and do the calculation. This is the reason why we use algebraic expressions in mathematics.
Understand the Rules of Algebraic Expressions
What are the rules in algebraic expressions? Be sure to follow the following rules in algebraic expressions.
• Omit the × in multiplication.
• Write numbers first, then alphabetically.
• Use exponents to multiply the same letter.
• Omit 1 and -1.
• Don’t use division; use multiplying fractions.
The rules are not difficult to understand. Just make sure you understand them.
-Omit the × in Multiplication.
When multiplying, we always use the symbol “x.” However, in algebraic expressions, the × is omitted when multiplying. Therefore, you get the following.
Instead of $2×b$, write $2b$. Also, write $ab$ instead of $a×b$. In the algebraic expression, the × is omitted.
-Write Numbers First, Then Alphabetically
In the algebraic expression, what order should we write them in? The rule is that numbers should always be written before the alphabet. Also, if there are several letters, they should be written in
alphabetical order.
For example, the following multiplication can be expressed by the following algebraic expression.
The number 4 is written first. The letters are written in alphabetical order, so they are written as $4ab$.
-Use Exponents to Multiply the Same Letter
In algebraic expressions, we frequently multiply the same letter. For example, $a×a$. However, we never write $aa$ in an algebraic expression. In the multiplication of the same letter, we use
exponents. For example, we have the following.
-Omit 1 and -1
The rule for writing numbers first makes us think that we have to write them as follows.
However, we have to omit 1 and -1 because 1 can be multiplied infinitely, and can be expressed as $1×1×1×a$ and so on. To prevent this, we omit the numbers 1 and -1. For -1, we write the $-$, as in
$-b$. Thus, we have the following.
-Don’t Use Division; Use Multiplying Fractions
We don’t use division in algebraic expressions. Division has limited use, and few people use division in mathematics. In fact, no one uses division in junior high and high school mathematics.
Division can be converted to multiplying fractions. Don’t use division in algebraic expressions, but use equations with multiplication only.
Learn the Algebraic Expressions for Fractions and Reciprocal Numbers
One of the most difficult concepts to understand when learning algebraic expressions is to convert division into multiplying fractions. So, we’ll explain how to change the division of algebraic
expressions into fractions.
Divisions can be converted to multiplying fractions by using the reciprocal, as follows.
The same is true for algebraic expressions. The division of algebraic expressions can be converted to multiplying fractions by using the reciprocal, as shown below.
Note that in the algebraic expression, you can write the alphabet in the numerator or next to the fraction. So, it looks like this.
Both notations are correct. You can use either, but you should be able to write both.
As a reminder, if there are letters in the denominator, the alphabet must be in the denominator. Do not write the alphabet next to the fraction. Therefore, it looks like this.
As mentioned above, if there are alphabets in the numerator, you can write them in the numerator or next to the fraction. However, if there are alphabets in the denominator, be sure to write them in
the denominator.
The Difference Between Monomials and Polynomials in Algebraic Expressions
After understanding the rules about algebraic expressions, we need to understand the meaning of the words. When we study algebraic expressions, the words we must learn are monomials and polynomials.
Both monomials and polynomials are expressions that use letters and numbers.
The differences between them are as follows.
• Monomials: equations that only involve multiplication
• Polynomials: equations that mix addition and subtraction
You can understand that a monomial is, in essence, a single algebraic expression.
We explained that in an algebraic expression, you omit the × when you multiply. As a result, in an algebraic expression with only multiplication, there will be a sequence of numbers and symbols. In
this case, the mass of numbers and letters is a monomial.
However, in mathematics, the equation does not necessarily include only multiplication. While division can be converted to multiplication, addition, and subtraction cannot be converted to
So if there are additions and subtractions in an equation, there are several monomials. When an equation contains addition and subtraction, resulting in the existence of multiple monomials, it is
called a polynomial.
The difference between monomials and polynomials is determined by whether addition or subtraction is included in the equation. A polynomial is when there are two or more monomials in the equation,
using pluses and minuses.
Coefficients, Terms, and Degrees of Monomials and Polynomials
For reference, when we learn monomials and polynomials in algebraic expressions, we will see the words coefficient, term, and degree. Each of them is as follows.
• Coefficient: a number in a monomial
• Term: a monomial including numbers and letters
• Degree: numbers of letters, such as $x$ and $a$
You don’t need to understand and remember the definitions of these words. However, these words are always found in textbooks that explain monomials and polynomials, so you should understand roughly
what they mean.
Coefficients refer to numbers in monomials. For example, it is as follows.
Understand that the numbers in front of the letters, including pluses and minuses, are coefficients.
On the other hand, each monomial is called a term. In the previous equation, the following are the terms.
It is easy to distinguish between coefficients and terms. However, coefficients and terms are not important words in mathematics. The more important word is degree.
The degree refers to the number of letters (alphabets) contained in a monomial.
• One letter: first degree (linear)
• Two letters: second degree (quadratic)
• Three letters: third degree (cubic)
• Four letters: fourth degree (quartic)
Note that a polynomial contains multiple monomials of different degrees. Among these monomials, the highest number of letters is used as the degree.
It is important to understand the definition of degree in a polynomial. The monomial that contains the highest number of letters is the degree of the polynomial. For example, the following equation
contains several monomials.
In this equation, the largest degree is 2. Therefore, this polynomial is quadratic.
In mathematics, we learn formulas for linear functions and quadratic functions. How do we know if it is a linear function or a quadratic function? You can distinguish them by the degree of the
Addition and Subtraction of Polynomials: Combining Like Terms (Similar Terms)
After learning about monomials and polynomials in algebraic expressions, the next step we need to understand is addition and subtraction. We’ve already learned about addition and subtraction. But how
can we do addition and subtraction for algebraic expressions that contain alphabets?
Even though letters, such as $x$ and $y$, are included in the expression, these algebraic expressions are the same as numbers. For example, suppose we have the following equation.
You can put any number in $x$. So the answer changes as follows.
It is important that even in algebraic expressions, the answer can be a specific number.
So in an algebraic expression, if the letters are the same, they can be put together by addition or subtraction. In polynomials, terms with the same letters are called like terms (or similar terms).
We can distinguish like terms as follows,
If the letters are the same, they are like terms and can be grouped. In other words, you can add or subtract like terms from each other.
Grouping similar terms by adding and subtracting are called combining like terms. To combine like terms, you can do the following.
After arranging the like terms, add or subtract the coefficients from each other. For example, the coefficient of $x^2$ is 1. And the coefficient of $-3x^2$ is -3. Therefore, the calculation is shown
in the above figure.
Once again, it can be written as follows.
• $x^2-3x^2$
• $=(1-3)x^2$
• $=-2x^2$
In addition and subtraction of polynomials, combine like terms. To do this, you need to arrange similar terms. Monomials that have the same letters but different numbers (coefficients) are similar
If the degrees are different, they are not like terms. For example, the following are not similar terms.
Only monomials whose letters are the same, including the degree, are like terms. For example, the following monomials are not similar terms.
• $2x^2$
• $4xy$
• $3a^2$
• $-3x$
• $5y$
Why are these not the like terms? It’s because the letters and degrees are different. Understand that you can only combine like terms when the letters are the same. The reason you can add or subtract
like terms is that they have the same properties.
When Subtracting Polynomials, Be Careful of the Negative
Note that in addition and subtraction of algebraic expressions, we must be especially careful in subtraction. When subtracting polynomials with parentheses, many people make a miscalculation. To be
more specific, when there is a minus sign in front of the parentheses, calculation errors occur frequently.
If there is a minus sign in front of the parentheses, the sign changes when the parentheses are removed. For example, the following is an example.
In other words, all the signs inside the parentheses change. Why does a minus sign in front of parentheses change all the signs inside the parentheses? The reason is that the number inside the
parentheses is considered to be a single number.
In mathematics, there is a rule that parentheses must be calculated first. The reason for this is that even if there are equations inside the parentheses, those equations are considered as a single
number. Because it is a single number, it must be calculated first.
For example, if there is the equation $-(1+3)$, how do you calculate it? We can calculate it as follows.
In this equation, we do $1+3$ first because of the parentheses. Then, by multiplying by the minus sign, the answer is -4.
On the other hand, if we remove the parentheses first, how can we calculate? In this case, both numbers are multiplied by -1. This means changing all the signs in the parentheses. As a result, the
answer is the same as in the previous example, as shown below.
For expressions in parentheses, they must be considered the same number. This is why we must change all signs when removing parentheses if there is a minus sign in front of the parentheses. You must
not change only one of the numbers in the expression when you remove the parentheses because they are considered the same number.
Therefore, when combining like terms, the calculation is as follows.
When it comes to adding polynomials, we can solve the problem without any difficulty. On the other hand, when subtracting polynomials, many people make mistakes. There are certain parts of polynomial
subtraction that can be miscalculated. When you remove the parentheses in subtraction, calculation errors occur frequently.
It is important to understand where calculation errors are most likely to occur. Also, learn how to solve problems correctly. This will help you to avoid miscalculations in the addition and
subtraction of polynomials.
Exercises: Addition and Subtraction of Monomials and Polynomials
Q1: Do the following calculation.
1. $2a÷(-5)$
2. $3b÷2x÷y$
A1: Answers.
In monomial calculations, you have to convert the division into multiplication. So let’s use the reciprocal and convert all of them into multiplying fractions. Then we get the following.
$=-\displaystyle\frac{2a}{5} \left(or-\displaystyle\frac{2}{5}a\right)$
$=\displaystyle\frac{3b}{2xy} \left(or\displaystyle\frac{3}{2xy}b\right)$
Q2: Do the following calculation.
• $\displaystyle\frac{1}{2}x-\displaystyle\frac{1}{3}x$
A2: Answers.
You can add or subtract the like terms in algebraic expressions. Of course, fractions and decimals can also be added or subtracted if they are like terms.
Q3: Do the following calculation.
1. $x^2+3-3x-7x^2+5x-10$
2. $(-2x+4y)+(9x-8y)$
3. $(x+3y)-(3x-6y)+4x$
A3: Answers.
In polynomials, try to combine like terms. This way, you can add and subtract.
If the sign in front of the parentheses is positive, there is nothing to pay attention to. On the other hand, if the sign in front of the parentheses is negative, you will often make a mistake when
removing the parentheses. When removing parentheses, it is necessary to change the sign.
$=\textcolor{red}{x^2- 7x^2}\textcolor{blue}{-3x+5x}+3-10$
Learn the Definition and How to Calculate Algebraic Expressions
After studying algebraic expressions in mathematics, almost all calculation exercises will be algebraic expressions. Therefore, if you don’t understand the rules of algebraic expressions, you won’t
be able to solve math calculation problems. So be sure to learn the rules of algebraic expressions.
The rules make it possible for us to do the correct calculation. And don’t just remember the rules; learn the reason why.
Note that when adding and subtracting polynomials, it’s important to combine similar terms. Especially if there is a minus sign in front of the parentheses, calculation errors are more likely to
occur. Solve the problem while checking whether the plus and minus signs are correct.
Algebraic expressions are important to learn in mathematics. Algebraic expressions can be found in any math problem, so make sure you remember the basics, including the definition of an algebraic | {"url":"https://hatsudy.com/algebraic.html","timestamp":"2024-11-01T21:00:25Z","content_type":"text/html","content_length":"54963","record_id":"<urn:uuid:084c0ad5-6179-4d73-a3e1-9d22e2e1436f>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00149.warc.gz"} |
Free Probability Calculator
Simulate the probability of making money in your stock or option position.
What makes McMillan’s Probability Calculator different?
Over a number of trading days, the price of a stock may vary widely and still end up at or near the original purchase price. Many calculators are available that give the theoretical probability that
a stock may approach certain values at the end of a trading period. In real trading, however, investors are following the price of a stock or stock options throughout the entire trading period. If
the stock, stock options, or combination becomes profitable before the end of the trading period (for example, before the expiration of some stock options), it is reasonable that a trader may decide
to reap part or all of those profits at that time. The Probability Calculator gives the likelihood that prices are ever exceeded during the trading period, not just at the end. | {"url":"https://optionstrategist.com/calculators/probability","timestamp":"2024-11-06T09:02:31Z","content_type":"text/html","content_length":"25212","record_id":"<urn:uuid:0cd93f20-f957-4f80-a2a4-d9dbdb673ec4>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00772.warc.gz"} |
Scrabble Word
Our site is designed to help you unscramble or descramble the letters & words in the Scrabble® word game, Words with Friends®, Chicktionary, Word Jumbles, Text Twist, Super Text Twist, Text Twist 2,
Word Whomp, Literati, Wordscraper, Lexulous, Wordfeud and many other word games.
© 2020 - ScrabbleWordFind.com | {"url":"https://www.scrabblewordfind.com/words/11-letter/starts-with/a","timestamp":"2024-11-06T00:49:14Z","content_type":"text/html","content_length":"55613","record_id":"<urn:uuid:ed10d555-3da9-445c-ad4c-f6988abf4173>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00472.warc.gz"} |
Scoop du Jour: Barycentric Quad Rasterization on a Cone
by C.G. Scoop
Sept., 2022
Last month, shortly after the Journal of Computer Graphics Techniques published Barycentric Quad Rasterization, Paul Haeberli asked whether the barycentric method, which reduces distortion and
discontinuity in a texture-mapped sphere, could also improve the rendering of a cone? Yes, it can.
At the tip of a cone, as at a pole of a sphere, a quad has a degenerate edge defined by two coincident points (see Figure 6 of the paper). With triangle rasterization, one of the points is ignored
(see Figure 7). For the sphere, the two points differ in uv-coordinates and thus triangle rasterization produces a texture discontinuity. For the cone, however, the two points also differ in surface
normal, producing a shading discontinuity as well. This problem is discussed by Eric Haines in his 2014 post, Limits of Triangles.
The discontinuities in texture can be overcome by avoiding the Mercator projection used for a sphere and, instead, employing a conical projection as the texture source (that is, a circle whose
circumference corresponds to the base of the cone and whose center corresponds to the tip). The discontinuities in shading, however, cannot be overcome with triangles or quad-splits.
The cone example revealed a bug in the pixel shader (Listing 2). In subroutine BarycentricWeights, the following assignment fails if A is zero:
t[i] = (r[i]*r[(i+1)%4]-D)/A;
It should be replaced with:
t[i] = abs(A) < 1.e-6? 0 : (r[i]*r[(i+1)%4]-D)/A;
This correction removes the need for double precision in the subroutines BarycentricWeights and UV (thus, p. 73, “Double precision is used ... artifacts” is incorrect). | {"url":"https://unchainedgeometry.com/ScoopBaryCone.html","timestamp":"2024-11-03T15:34:43Z","content_type":"text/html","content_length":"4604","record_id":"<urn:uuid:0483dae7-54c3-43b8-81c2-76d62cd56c17>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00639.warc.gz"} |
How correctly to calculate the cubing of lumber?
In many cases, when building and repairing buildings, lumber is used. They are suitable for various types of work. Not an exception is the decoration of the room and the building as a whole. However,
in the first place, it is necessary to calculate the cubic capacity of the lumber.
Process of calculation
As an example, take a room with an area of one wall 18m², where the width of the wall is 6 m, and its height is 3 m. The total area will be as follows: 18 m²x4 = 72 m². In this case, 4 is the number
of walls. Let's proceed to the calculation.
• Calculation of the cubature of sawn timber for walls is determined by multiplying the wall area by the thickness of the board. In our example, this will be 72 m ² x 0.03 m = 2.16 m ³, where 0.03
m is the thickness of the board.
• Similarly calculate the amount of interior finishing of the house.
• Calculation of the cubic volume of sawn timber for the floor, ceiling and roof is made in the same way as for the walls.
• The volume of the beam for the frame of the house is obtained by multiplying the cross-sectional area of the beam (column) by the length and by the number of racks provided for by the project of
the house.
The specific complexity of selling and buying lumber is the correct calculation of cubature. Even the controlling authorities do not always manage to check whether the calculation is performed
correctly. Lumber from the manufacturer is supplied in the original packaging. It accurately indicates the volume and cost of products. If this is not the case, the buyer has to doubt whether the
calculation of the cubic capacity of the lumber and its cost has been carried out correctly. The quality parameters of the boards are regulated by GOSTs, technical requirements, and various
regulatory documents. It all depends on what kind of tree they are made of, what they will be used for. To determine the volume of round lumber, the corresponding GOST is also used.
Additional calculation parameters
In the woodworking industry, the concepts of a dense and storage cubic meter are used. To transfer one value to another, a conversion factor is applied. For example, an edging board in value price
lists is indicated for calculating volumes in a dense mass. This calculation is not difficult. For this, the indicators of the storage cubic meters are converted into dense masses using a special
Measurement and calculation rules
For example, the croaker is sorted first in length by 2 species - up to 2 meters or more. Then it is stacked. At the same time, thin and thick ends alternate. The stack is stacked as tightly as
possible. The height and length must be the same. Also it is necessary to withstand right angles in the pile. Multiplying the length, width and height of the package, we get a storage cubic capacity.
It will be useful to also know that the measurement rules for each type of sawn timber are regulated by the corresponding GOST. It contains tables for calculating the volume of products. When selling
a building beam or a board, measure each piece of lumber, according to the tables in the GOST, calculate the volume of edging lumber. You should also know that to determine the number of boards in 1
cubic meter you need to calculate the volume of one board. This calculation is not difficult. Then we divide the unit by the volume of one board and obtain the required number of products. | {"url":"https://en.birmiss.com/how-correctly-to-calculate-the-cubing-of-lumber/","timestamp":"2024-11-13T11:05:01Z","content_type":"text/html","content_length":"56893","record_id":"<urn:uuid:fbcf7966-a87d-4b31-8ed4-53323ea241da>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00173.warc.gz"} |
• TBA. Biracks and Biquandles: Theory, applications, and new perspectives, Leeds, UK.
• Nichols algebras over solvable groups. Noncommutative Geometry and Topology Group in Prague, Prague, Czech Republic.
• Nichols algebras over solvable groups. University of Warsaw: Algebra Seminar, Warsaw, Poland.
• L-algebras: The Yang-Baxter equation and algebraic logic. From Garside to Yang-Baxter, Caen, France.
• Nichols algebras over groups. University of Leeds: Algebra Seminar, Leeds, UK.
• What is a skew brace? Algebraic and combinatorial perspectives in the mathematical sciences, Oslo, Norway.
• On the classification of Nichols algebras over groups. Seminario de Teoría de Lie, Córdoba, Argentina.
• Convex sets and sums of squares. Brussels Summer School of Mathematics, Brussels, Belgium.
• Nichols algebras. Algebraic structures in the Yang-Baxter equation, Edinburgh, UK.
• Nichols algebras over groups. Seminar on Quantum groups, Hopf algebras and monoidal categories, Brussels, Belgium.
• L-algebras: The Yang-Baxter equation and algebraic logic. JLU Colloquium, Beijing, China.
• L-algebras: The Yang-Baxter equation and algebraic logic. The Interplay between skew braces and Hopf-Galois theory, Exeter, UK.
• Nichols algebras: an overview. Oberwolfach mini-workshop: Bridging Number Theory and Nichols Algebras via Deformations, Germany.
• On the classification of Nichols Algebras. Algebra Seminar, The University of Edinburgh, UK.
• Groups, rings and the Yang-Baxter equation. Workshop on Geometric Methods in Physics. Bialowieza, Poland.
• Some problems on skew braces. Advances in Group Theory and Applications. Lecce, Italy.
• Nichols algebras. European Quantum Algebra Lectures (EQuAL). Virtual seminar.
• Some problems on skew braces. The Interplay between skew braces and Hopf-Galois theory. Keele University, UK.
• Algebra with GAP (mini-course). CIMPA School: Crossroads of geometry, representation theory and higher structures. Puerto Madryn, Argentina. [notes]
• Skew braces, cabling and indecomposable solutions to the Yang-Baxter equation. Categories, Rings and Modules, a conference in honor of Alberto Facchini. Padova, Italy.
• Pre-Lie algebras and braces. Oberwolfach mini-workshop: Skew braces and the Yang-Baxter equation, Germany.
• Algebra with GAP (mini-course, 6 hours). Vrije Universiteit Brussel. With Jan de Beule. [slides]
• Multipermutation solutions and the Yang-Baxter equation. Hopf algebras, monoidal categories and related topics. Bucharest, Romania.
• Left-ordered groups, Garside groups and structure groups of solutions. Algebra days in Caen, France.
• The Yang-Baxter equation and algebraic logic. Nicolaus Copernicus University, Toruń, Poland.
• Multipermutation solutions of the Yang-Baxter equation, Ferran Cedó retirement. Barcelona, Spain.
• Radical rings, braces and the Yang-Baxter equation. Braces in Bracelets Bay. LMS Regional Meeting. Swansea, UK.
• Problems on skew braces. Harish-Chandra Research Institute, India. Virtual seminar.
• Radical rings and the Yang-Baxter equation. Grupo de teoría de Lie. Virtual seminar.
• Algebra with GAP. AARMS mini-course, Dalhousie University, Halifax, Canada. [videos] [slides and problems]
• New developments in the theory of radical rings. Pure Maths Colloquium, University of St Andrews, St Andrews, UK.
• On the classification of Nichols algebras. MAXIMALS Seminar, University of Edinburgh, Edinburgh, UK.
• Skew braces and the Yang–Baxter equation. Groups, rings and associated structures. Spa, Belgium.
• Radford’s bosonization theorem and Nichols algebras (mini-course, 6 hours). University of Edinburgh, UK.
• Skew braces and the Yang-Baxter equation (mini-course, 4 hours). Vrije Universiteit Brussel, Belgium.
• Radical rings, braces and the Yang-Baxter equation. Exeter, UK.
• Nichols algebras (mini-course, 4 hours). Workshop Tensor categories, Hopf algebras and quantum groups, Marburg, Germany.
• Nichols algebras. Seminario de álgebra, Universidad Autónoma de Barcelona, Spain.
• Set-theoretical solutions of the Yang-Baxter equation. University of St Andrews, UK.
• Skew braces. Groups, rings and the Yang–Baxter equation. Spa, Belgium.
• Set-theoretical solutions of the Yang–Baxter equation. MIT, USA.
• The combinatorics of the Yang–Baxter equation. MAXIMALS Seminar, University of Edinburgh, UK.
• Nichols algebras. Warsaw University, Poland.
• Set-theoretical solutions of the Yang–Baxter equation. Warsaw University, Poland.
• Nichols algebras and applications. Dublin Mathematics Colloquium, Geometry Seminar, Trinity College, Dublin, UK.
• Set-theoretical solutions of the Yang–Baxter equation. Séminaire Quantique, Strasbourg, France.
• Set-theoretical solutions of the Yang–Baxter equation. Universidad Autónoma de Barcelona, Spain.
• The combinatorics of the Yang–Baxter equation. Oberseminare am IAZ, Stuttgart, Germany.
• Nichols algebras. XXI Coloquio Latinoamericano de Álgebra, Buenos Aires, Argentina.
• The combinatorics of the Yang–Baxter equation. Mathematische Gesellschaft in Göttingen, Germany.
• The classification of Nichols algebras. Humboldt Kolleg. Colloquium on Algebras and Representations – Quantum 2016, Córdoba, Argentina.
• Nichols algebras over non-abelian groups. Coloquio Latinoamericano de Álgebra, Lima, Perú.
• Introducción al álgebra con GAP (mini-course, 8 hours), Universidad de Chile, Santiago de Chile. [notes]
• The classification of Nichols algebras (mini-course, 6 hours). Summer School on Conformal Field Theories and Nichols algebras, Rauischholzhausen, Germany. [notes]
• Introducción a la teoría combinatoria de nudos (mini-course, 3 hours). ElENA VII, Córdoba, Argentina.
• Nichols algebras. Universidad de Talca, Talca, Chile.
• Nichols algebras and Weyl groupoids of rank two. Colóquio de Álgebra e Representações - Quantum 2014, Santa Maria, Brazil.
• Doubly transitive groups and cyclic quandles. Università di Ferrara, Italy.
• Nichols algebras and a combinatorial model for Schubert calculus. ICTP, Trieste, Italy.
• Nichols algebras with root systems of rank two. Quantum day. Facultad de Matemática, Astronomía, Física y Computación, Córdoba, Argentina.
• Fomin-Kirillov algebras. Nichols algebras and Weyl groupoids, Oberwolfach, Germany. [slides]
• Nichols algebras. Séminaire Lotharingien de Combinatoire 69, Strobl, Austria.
• About the classification of finite-dimensional pointed Hopf algebras. Groups, Rings, Lie and Hopf Algebras. III, Deer Lake, Canada.
• Hopf algebras and applications (mini-course). With N. Andruskiewitsch. AARMS Summer School, St. Johns, Canada.
• Nichols algebras and Weyl groupoids of rank two. Oberseminar Kombinatorik und Algebra, Philipps Universität, Marburg, Germany.
• Introduction to cluster algebras. Forschungsseminar Mathematische Physik, Philipps Universität, Marburg, Germany.
• Nichols algebras and quadratic relations. Universidad de Almería, Spain.
• Nichols algebras and quadratic relations. Atlantic Algebra Center, Memorial University, St. John’s, Canada.
• Nichols algebras and quadratic relations. Hamburg Universität, Germany.
• An introduction to Nichols algebras. Hamburg Universität, Germany.
• Nichols algebras and Schubert Calculus. Forschungsseminar Mathematische Physik, Philipps Universität, Marburg, Germany.
• Nichols algebras over non-abelian groups. XVIII Congreso Latinoamericano de Álgebra. Sao Pedro, Brazil.
• A GAP package for racks and Nichols algebras. Advanced School and Conference on Knot Theory. ICTP, Trieste, Italy.
• Pointed Hopf algebras over the sporadic simple groups. First Debrún Workshop on Computational Algebra. National University of Ireland, Galway, Ireland. | {"url":"https://leandrovendramin.org/talks.html","timestamp":"2024-11-12T11:37:26Z","content_type":"text/html","content_length":"15408","record_id":"<urn:uuid:78a80c32-69d2-4d15-8e15-999812c4c03c>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00884.warc.gz"} |
ing repetitions
Finding repetitions¶
Given a string $s$ of length $n$.
A repetition is two occurrences of a string in a row. In other words a repetition can be described by a pair of indices $i < j$ such that the substring $s[i \dots j]$ consists of two identical
strings written after each other.
The challenge is to find all repetitions in a given string $s$. Or a simplified task: find any repetition or find the longest repetition.
The algorithm described here was published in 1982 by Main and Lorentz.
Consider the repetitions in the following example string:
The string contains the following three repetitions:
• $s[2 \dots 5] = abab$
• $s[3 \dots 6] = baba$
• $s[7 \dots 8] = ee$
Another example:
Here there are only two repetitions
• $s[0 \dots 5] = abaaba$
• $s[2 \dots 3] = aa$
Number of repetitions¶
In general there can be up to $O(n^2)$ repetitions in a string of length $n$. An obvious example is a string consisting of $n$ times the same letter, in this case any substring of even length is a
repetition. In general any periodic string with a short period will contain a lot of repetitions.
On the other hand this fact does not prevent computing the number of repetitions in $O(n \log n)$ time, because the algorithm can give the repetitions in compressed form, in groups of several pieces
at once.
There is even the concept, that describes groups of periodic substrings with tuples of size four. It has been proven that we the number of such groups is at most linear with respect to the string
Also, here are some more interesting results related to the number of repetitions:
• The number of primitive repetitions (those whose halves are not repetitions) is at most $O(n \log n)$.
• If we encode repetitions with tuples of numbers (called Crochemore triples) $(i,~ p,~ r)$ (where $i$ is the position of the beginning, $p$ the length of the repeating substring, and $r$ the
number of repetitions), then all repetitions can be described with $O(n \log n)$ such triples.
• Fibonacci strings, defined as
$$ t_0 &= a, \\\\ t_1 &= b, \\\\ t_i &= t_{i-1} + t_{i-2}, $$
are "strongly" periodic. The number of repetitions in the Fibonacci string $f_i$, even in the compressed with Crochemore triples, is $O(f_n \log f_n)$. The number of primitive repetitions is also
$O(f_n \log f_n)$.
Main-Lorentz algorithm¶
The idea behind the Main-Lorentz algorithm is divide-and-conquer.
It splits the initial string into halves, and computes the number of repetitions that lie completely in each halve by two recursive calls. Then comes the difficult part. The algorithm finds all
repetitions starting in the first half and ending in the second half (which we will call crossing repetitions). This is the essential part of the Main-Lorentz algorithm, and we will discuss it in
detail here.
The complexity of divide-and-conquer algorithms is well researched. The master theorem says, that we will end up with an $O(n \log n)$ algorithm, if we can compute the crossing repetitions in $O(n)$
Search for crossing repetitions¶
So we want to find all such repetitions that start in the first half of the string, let's call it $u$, and end in the second half, let's call it $v$:
$$s = u + v$$
Their lengths are approximately equal to the length of $s$ divided by two.
Consider an arbitrary repetition and look at the middle character (more precisely the first character of the second half of the repetition). I.e. if the repetition is a substring $s[i \dots j]$, then
the middle character is $(i + j + 1) / 2$.
We call a repetition left or right depending on which string this character is located - in the string $u$ or in the string $v$. In other words a string is called left, if the majority of it lies in
$u$, otherwise we call it right.
We will now discuss how to find all left repetitions. Finding all right repetitions can be done in the same way.
Let us denote the length of the left repetition by $2l$ (i.e. each half of the repetition has length $l$). Consider the first character of the repetition falling into the string $v$ (it is at
position $|u|$ in the string $s$). It coincides with the character $l$ positions before it, let's denote this position $cntr$.
We will fixate this position $cntr$, and look for all repetitions at this position $cntr$.
For example:
$$c ~ \underset{cntr}{a} ~ c ~ | ~ a ~ d ~ a$$
The vertical lines divides the two halves. Here we fixated the position $cntr = 1$, and at this position we find the repetition $caca$.
It is clear, that if we fixate the position $cntr$, we simultaneously fixate the length of the possible repetitions: $l = |u| - cntr$. Once we know how to find these repetitions, we will iterate over
all possible values for $cntr$ from $0$ to $|u|-1$, and find all left crossover repetitions of length $l = |u|,~ |u|-1,~ \dots, 1$.
Criterion for left crossing repetitions¶
Now, how can we find all such repetitions for a fixated $cntr$? Keep in mind that there still can be multiple such repetitions.
Let's again look at a visualization, this time for the repetition $abcabc$:
$$\overbrace{a}^{l_1} ~ \overbrace{\underset{cntr}{b} ~ c}^{l_2} ~ \overbrace{a}^{l_1} ~ | ~ \overbrace{b ~ c}^{l_2}$$
Here we denoted the lengths of the two pieces of the repetition with $l_1$ and $l_2$: $l_1$ is the length of the repetition up to the position $cntr-1$, and $l_2$ is the length of the repetition from
$cntr$ to the end of the half of the repetition. We have $2l = l_1 + l_2 + l_1 + l_2$ as the total length of the repetition.
Let us generate necessary and sufficient conditions for such a repetition at position $cntr$ of length $2l = 2(l_1 + l_2) = 2(|u| - cntr)$:
• Let $k_1$ be the largest number such that the first $k_1$ characters before the position $cntr$ coincide with the last $k_1$ characters in the string $u$:
$$ u[cntr - k_1 \dots cntr - 1] = u[|u| - k_1 \dots |u| - 1] $$
• Let $k_2$ be the largest number such that the $k_2$ characters starting at position $cntr$ coincide with the first $k_2$ characters in the string $v$:
$$ u[cntr \dots cntr + k_2 - 1] = v[0 \dots k_2 - 1] $$
• Then we have a repetition exactly for any pair $(l_1,~ l_2)$ with
$$ l_1 &\le k_1, \\\\ l_2 &\le k_2. \\\\ $$
To summarize:
• We fixate a specific position $cntr$.
• All repetition which we will find now have length $2l = 2(|u| - cntr)$. There might be multiple such repetitions, they depend on the lengths $l_1$ and $l_2 = l - l_1$.
• We find $k_1$ and $k_2$ as described above.
• Then all suitable repetitions are the ones for which the lengths of the pieces $l_1$ and $l_2$ satisfy the conditions:
$$ l_1 + l_2 &= l = |u| - cntr \\\\ l_1 &\le k_1, \\\\ l_2 &\le k_2. \\\\ $$
Therefore the only remaining part is how we can compute the values $k_1$ and $k_2$ quickly for every position $cntr$. Luckily we can compute them in $O(1)$ using the Z-function:
• To can find the value $k_1$ for each position by calculating the Z-function for the string $\overline{u}$ (i.e. the reversed string $u$). Then the value $k_1$ for a particular $cntr$ will be
equal to the corresponding value of the array of the Z-function.
• To precompute all values $k_2$, we calculate the Z-function for the string $v + \# + u$ (i.e. the string $u$ concatenated with the separator character $\#$ and the string $v$). Again we just need
to look up the corresponding value in the Z-function to get the $k_2$ value.
So this is enough to find all left crossing repetitions.
Right crossing repetitions¶
For computing the right crossing repetitions we act similarly: we define the center $cntr$ as the character corresponding to the last character in the string $u$.
Then the length $k_1$ will be defined as the largest number of characters before the position $cntr$ (inclusive) that coincide with the last characters of the string $u$. And the length $k_2$ will be
defined as the largest number of characters starting at $cntr + 1$ that coincide with the characters of the string $v$.
Thus we can find the values $k_1$ and $k_2$ by computing the Z-function for the strings $\overline{u} + \# + \overline{v}$ and $v$.
After that we can find the repetitions by looking at all positions $cntr$, and use the same criterion as we had for left crossing repetitions.
The implementation of the Main-Lorentz algorithm finds all repetitions in form of peculiar tuples of size four: $(cntr,~ l,~ k_1,~ k_2)$ in $O(n \log n)$ time. If you only want to find the number of
repetitions in a string, or only want to find the longest repetition in a string, this information is enough and the runtime will still be $O(n \log n)$.
Notice that if you want to expand these tuples to get the starting and end position of each repetition, then the runtime will be the runtime will be $O(n^2)$ (remember that there can be $O(n^2)$
repetitions). In this implementation we will do so, and store all found repetition in a vector of pairs of start and end indices.
vector<int> z_function(string const& s) {
int n = s.size();
vector<int> z(n);
for (int i = 1, l = 0, r = 0; i < n; i++) {
if (i <= r)
z[i] = min(r-i+1, z[i-l]);
while (i + z[i] < n && s[z[i]] == s[i+z[i]])
if (i + z[i] - 1 > r) {
l = i;
r = i + z[i] - 1;
return z;
int get_z(vector<int> const& z, int i) {
if (0 <= i && i < (int)z.size())
return z[i];
return 0;
vector<pair<int, int>> repetitions;
void convert_to_repetitions(int shift, bool left, int cntr, int l, int k1, int k2) {
for (int l1 = max(1, l - k2); l1 <= min(l, k1); l1++) {
if (left && l1 == l) break;
int l2 = l - l1;
int pos = shift + (left ? cntr - l1 : cntr - l - l1 + 1);
repetitions.emplace_back(pos, pos + 2*l - 1);
void find_repetitions(string s, int shift = 0) {
int n = s.size();
if (n == 1)
int nu = n / 2;
int nv = n - nu;
string u = s.substr(0, nu);
string v = s.substr(nu);
string ru(u.rbegin(), u.rend());
string rv(v.rbegin(), v.rend());
find_repetitions(u, shift);
find_repetitions(v, shift + nu);
vector<int> z1 = z_function(ru);
vector<int> z2 = z_function(v + '#' + u);
vector<int> z3 = z_function(ru + '#' + rv);
vector<int> z4 = z_function(v);
for (int cntr = 0; cntr < n; cntr++) {
int l, k1, k2;
if (cntr < nu) {
l = nu - cntr;
k1 = get_z(z1, nu - cntr);
k2 = get_z(z2, nv + 1 + cntr);
} else {
l = cntr - nu + 1;
k1 = get_z(z3, nu + 1 + nv - 1 - (cntr - nu));
k2 = get_z(z4, (cntr - nu) + 1);
if (k1 + k2 >= l)
convert_to_repetitions(shift, cntr < nu, cntr, l, k1, k2); | {"url":"https://gh.cp-algorithms.com/main/string/main_lorentz.html","timestamp":"2024-11-05T18:55:55Z","content_type":"text/html","content_length":"158113","record_id":"<urn:uuid:c0f66b77-427d-4929-9468-04abe6ee4cda>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00540.warc.gz"} |
Percentage to Gpa 4.0 Scale Calculator - Savvy Calculator
Percentage to Gpa 4.0 Scale Calculator
Calculating GPA on a 4.0 scale is a crucial task for students aiming to assess their academic performance accurately. The Percentage to GPA 4.0 Scale Calculator simplifies this process, offering a
convenient tool for quick and precise calculations. In this article, we’ll guide you on how to use the calculator effectively and provide insights into the formula behind the scenes.
How to Use
Using the Percentage to GPA 4.0 Scale Calculator is straightforward. Follow these steps:
1. Input your percentage score in the provided field.
2. Click the “Calculate” button to obtain your GPA on a 4.0 scale.
Now, let’s delve into the underlying formula that powers this calculator.
The GPA calculation formula is a standard conversion from percentage to GPA. The formula is as follows:
This formula ensures that the GPA falls within the 0.0 to 4.0 scale, aligning with the standard grading system.
Let’s illustrate the formula with an example. Suppose you have a percentage score of 85%.
So, your GPA on a 4.0 scale would be 2.5.
Q: What is the GPA scale on a 4.0 scale?
A: The GPA scale on a 4.0 scale ranges from 0.0 to 4.0, with each increment representing a higher level of academic achievement.
Q: Can I use this calculator for any percentage system?
A: Yes, the calculator is designed to convert percentages from any system to the GPA on a 4.0 scale.
Q: Is the formula used here standard across educational institutions?
A: Yes, the formula employed is widely accepted and aligns with common GPA conversion practices.
The Percentage to GPA 4.0 Scale Calculator is a valuable tool for students to gauge their academic performance in a standardized manner. By understanding the simple formula and following the steps
outlined, users can effortlessly convert their percentage scores to GPA on a 4.0 scale. This tool provides a quick snapshot of academic achievement, aiding in educational planning and goal setting.
Leave a Comment | {"url":"https://savvycalculator.com/percentage-to-gpa-4-0-scale-calculator","timestamp":"2024-11-08T12:36:37Z","content_type":"text/html","content_length":"145962","record_id":"<urn:uuid:02a59069-5596-4008-bf9f-7bd77a86f199>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00194.warc.gz"} |
PIRC: Pricing Insurance Risk Course
Pricing Insurance Risk Course (PIRC)
Sections by Module
• A. Introduction
□ A.01. Course Overview
□ Last pdf update 2021-01-03 13:56:44 UTZ
• B. Market Assumptions
□ B.01. Market Assumptions
□ Last pdf update 2021-01-19 15:27:21 UTZ
• C. Historical US Property-Casualty Profitability and Volatility (2020 Update)
□ C.01. Premium to GDP Ratio, 1923-2020
□ C.02. Surplus to GDP Ratio, 1931-2020
□ C.03. Industry Volatility and Profitability Metrics, 1985-2020
□ C.04. Direct and Net Volatility and Profitability by Major Line, 1992-2020
□ C.05. Direct Premium Growth by Major Line, 1992-2020
□ C.06. Implications
□ Last pdf update 2021-11-15 16:30:43.766628 EST
• D. Model Specification and Properties
□ D.01. Modeling Assumptions
□ D.02. Model Portfolio
□ D.03. Equal Priority
□ D.04. ASTIN/CAS Slides
□ D.05. Effective Diversification
□ D.06. Measuring the Effects of Pooling
□ Last pdf update 2021-01-19 08:51:26 UTZ
• E. Distortions: Definition, Examples, and Properties
□ E.01. Distortions and Distortion Pricing Operators
□ Last pdf update 2021-01-19 08:55:13 UTZ
• F. Cat Bonds, Their Pricing, and Its Implications for Pricing Non-Cat Lines
□ F.01. Catastrophe Bonds and Their Pricing
□ F.02. Creating a Distortion From Cat Bond Prices in Theory
□ F.03. Creating a Distortion From Cat Bond Prices in Practice
□ Last pdf update 2021-01-19 09:15:09 UTZ
• G. Comparative Pricing Across Different Methods and Lines
□ G.01. Stand-Alone by Line and Portfolio Pricing
□ G.02. Gross and Net Line-Level and Portfolio Pricing
□ G.03. Line and Portfolio Multiline Pricing Across Multiple Methods
□ G.04. Conclusions and Answers
□ Last pdf update 2021-01-19 09:25:53 UTZ
• H. Convex Envelopes (coming soon)
• I. Theory in Practice and the Cost of Capital for Insurance Risk
□ I.01. Introduction and Purpose
□ I.02. Notional Portfolios
□ I.03. Insurance Pricing and Insurer Capital Structure
□ I.04. Comparative Pricing: Traditional, Stand-Alone
□ I.05. Comparative Pricing: Distortion, Stand-Alone
□ I.06. Comparative Pricing: Distortion, Multiline With Allocation
□ I.07. Comparative Pricing: Distortion, Multiline With Allocation With Stricter Capital Standard
□ I.08. Conclusions and Next Steps
□ Last pdf update 2021-01-23 16:33:39 UTZ
• K. Gradients: Ibrgimov, Jaffee and Walden vs. Aumann-Shapley (coming soon)
• L. Simple Nine Scenario Example
□ Last pdf update 2021-01-21 16:46:03 UTZ
• M. Severe Convective Storm and Hurricane Model Example (coming soon)
• N. More complex exercise proposed for the book (coming soon) | {"url":"https://www.convexrisk.com/pirc/contents","timestamp":"2024-11-10T14:59:29Z","content_type":"text/html","content_length":"36090","record_id":"<urn:uuid:9c075383-e79e-43d0-8dbc-54d9cafe2058>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00608.warc.gz"} |
Binomial Option Pricing Calculator User Guide
For more details and examples, follow the links to individual sections. You can find full user guide contents in the right sidebar or at the bottom of this page.
Use the Right Version
From the download page you can download two different Excel files, for different versions of Excel:
If you have Excel 2007 or newer, use the default version, BOPC.xlsm.
If you have Excel 2003 or older, use the other file, BOPC_for_Excel_97-2003.xls.
All features, calculations and results are exactly the same in both files. The only difference is graphical design, due to limitations of old Excel versions.
Enable Calculations and Macros
If you have just downloaded the calculator and opened it for the first time, you may find the dropdown boxes empty and calculations not working.
This is when your Excel opens the file in Protected View or a similar restricted regime where macros and calculations are blocked for security. Usually you will see a notice in place or below your
top Excel menu, with a button to "Enable Editing", "Enable Macros" or "Enable Calculations" – the wording varies, depending on your Excel version and settings.
Click that button (in some cases you also need to click another button afterwards to enable another thing) and everything should work normally.
There are seven sheets:
The sheet Main is where you enter inputs, view results, and control and view the chart.
In the sheet ChartData you can inspect exact values from the chart (X-axis values in column P; Y-axis values for the three series in columns Q, R, S).
The next three sheets contain the binomial trees used in the pricing models.
UndTree is the underlying price tree. Each step is in one column. Moving horizontally from left to right, underlying price goes up; moving diagonally one cell to the right and one cell down,
underlying price goes down.
OptTree is the option price tree. The last step shows option payoff at expiration for different levels of underlying price (the price in the same cell in UndPrice sheet). Step zero (cell E4 in
OptTree) is the calculated current option price, which you can also see in cell E4 in the sheet Main.
ExDivTree is the tree for underlying price excluding dividends (it is only used with discrete dividends; when working with continuous dividends the numbers in this sheet are exactly the same as those
in UndTree).
The next sheet, Lists, contains lists of items for the dropdown boxes, various constants and background calculations that make the models work. Do not change anything in this sheet, unless you want
to change the way the calculator works, and know what you are doing.
The last sheet, Preferences, contains settings which users can safely change, such as the preferred units and sign of option theta.
Pricing Options in the Calculator
If you have basic understanding of options, using the calculator should be very simple. It does not require any particular Excel skills, other that entering values in cells and selecting items in
dropdown boxes.
First you need to enter inputs. This is done in the Main sheet.
Some inputs, such as underlying price, strike price or volatility, are set by overwriting the values in yellow cells.
Other inputs, such as Call/Put, American/European, are selected in dropdown boxes.
When you have set all your inputs, you can view the resulting option price and Greeks in the green cells E4-J4 in the Main sheet.
If you want to inspect how the option price is calculated at each step in the binomial trees, see the sheets UndPrice and OptPrice.
The Yellow vs. Green Cell Rule
To help users diferentiate between the inputs cell (which you can overwrite) and output cells (which you should not change), this and all other Macroption calculators use a consitent system of cell
background colors:
Yellow cells are for inputs. Overwrite the values in yellow cells to change inputs or preferences.
Green cells are for outputs. Do not change them. They usually contain formulas, but even when a green cell contains a fixed value, that value is changed by the calculator (e.g. by a macro), not by
the user.
In some sheets, there are also cells with other colors. Generally yellow and red are fixed value cells, while green and blue are formula cells (or cells with fixed values changed by macros).
In this calculator:
• Change preferences by overwriting the values of yellow cells in the sheet Preferences.
• Change option pricing inputs and chart settings by overwriting yellow cells or selecting dropdown boxes in the sheet Main.
• Other cells and other sheets are for viewing only.
Charts and Scenario Analysis
The chart has two possible modes, selected in the Chart Mode dropdown box in cell F6.
The Single Line mode can model the effect of any of the inputs on option price or any of the Greeks. For instance, how option price changes with passing time or how strike selection affects gamma.
See how to work with the chart.
The Scenario Analysis mode allows you to display up to three lines in the chart, modeling the combined effect of two different inputs. One input is different for each line, while the other is on the
X-axis. For more details and examples see Scenario Analysis.
You can control the chart scale with the buttons to the right of the chart – zoom in or out, or move left or right along the X-axis.
Option Pricing Models
The calculator supports three of the most popular binomial option pricing models:
By default, the calculator uses the Leisen-Reimer model with 21 steps. You can change this in the Main sheet, cell C3 (model) and C4 (steps).
For detailed explanation of the models and which to use, see Models and Number of Steps.
If you have any questions or feedback, or encounter any problems or unexpected behavior, please feel free to contact me. | {"url":"https://www.macroption.com/binomial-option-pricing-calculator-guide/","timestamp":"2024-11-03T06:26:12Z","content_type":"text/html","content_length":"27559","record_id":"<urn:uuid:de87a7b5-be9c-4189-b1da-b312d0031455>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00216.warc.gz"} |
How to convert
How to convert VA to amps
Apparent power in volt-amps (VA) to electric current in amps (A).
You can calculate amps from volt-amps and volts, but you can't convert volt-amps to amps since volt-amps and amps units do not measure the same quantity.
Single phase VA to amps calculation formula
The current I in amps is equal to the apparent power S in volt-amps (VA), divided by the RMS voltage V in volts (V):
I[(A)] = S[(VA)] / V[(V)]
So amps are equal to volt-amps divided by volts.
amps = VA / volts
A = VA / V
Question: What is the current in amps when the apparent power is 3000 VA and the voltage supply is 110 volts?
I = 3000VA / 110V = 27.27A
3 phase VA to amps calculation formula
The current I in amps is equal to the apparent power S in volt-amps (VA), divided by the square root of 3 times the line to line voltage V[L-L] in volts (V):
I[(A)] = S[(VA)] / (√3 × V[L-L][(V)] )
So amps are equal to volt-amps divided by the square root of 3 times volts.
amps = VA / (√3 × volts)
A = VA / (√3 × V)
Question: What is the current in amps when the apparent power is 3000 VA and the voltage supply is 110 volts?
I = 3000VA / (√3 × 110V) = 15.746A
See also | {"url":"https://jobsvacancy.in/convert/electric/va-to-amp.html","timestamp":"2024-11-04T04:33:55Z","content_type":"text/html","content_length":"9644","record_id":"<urn:uuid:75b868c7-28ad-4027-9622-f88786d4ac09>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00446.warc.gz"} |
Introduction to Numpy in Python | Towards AI
Introduction to Numpy in Python
Last Updated on July 26, 2023 by Editorial Team
Originally published on Towards AI.
Introduction to Numpy in Python
What is NumPy?
NumPy stands for numeric python, a Python module that allows you to compute and manipulate multi-dimensional and single-dimensional array items. It comes with a high-performance multidimensional
array object as well as utilities for working with them.
In 2005, Travis Oliphant created the NumPy package by combining the features of the ancestral module Numeric with the characteristics of another module Numarray. NumPy implements multidimensional
arrays and matrices as well as other complex data structures. These data structures help to compute arrays and matrices in the most efficient way possible. NumPy allows you to conduct mathematical
and logical operations on arrays. Many other prominent Python packages, like pandas and matplotlib, are compatible with Numpy and use it.
Need to use NumPy in Python?
NumPy is a valuable and efficient tool for dealing with large amounts of data. NumPy is quick; thus, itβ s easy to work with a vast amount of data. Data analysis libraries like NumPy, SciPy, Pandas,
and others have exploded in popularity due to the data science revolution. Python is the most simple language in terms of syntax, and is the priority for many data scientists; therefore, NumPy is
getting popular day by day.
Many mathematical procedures commonly used in scientific computing are made quick and simple with Numpy, including:
• Multiplication of Vector or Matrix to Vector or Matrix.
• Operations on vectors and matrices at the element level (i.e., adding, subtracting, multiplying, and dividing by a number )
• Applying functions to a vector/matrix element by element ( like power, log, and exponential).
• NumPy has functions for linear algebra and random number generation built-in.
• NumPy also makes matrix multiplication and data reshaping simple.
• Multidimensional arrays are implemented efficiently using NumPy.
• NumPy arrays in Python give capabilities for integrating C, C++, and other languages. Itβ s also useful in linear algebra and random number generation, among other things. NumPy arrays can also
be used to store generic data in a multi-dimensional container.
• Itβ s faster than standard Python arrays, which donβ t have NumPyβ s pre-compiled C code(Precompiled code is a header file that is compiled into an intermediate form that is faster to process
for the compiler).
Installation of NumPy
Go to your command prompt and run β pip install numpyβ to install Python NumPy. Once the installation is complete, simply go to your IDE (for example, VS Code, Jupyter Notebook, PyCharm) and type
β import numpy as npβ to import it; now you are ready to use NumPy.
What is a NumPy Array?
The Numpy array object is a strong N-dimensional array object with rows and columns. NumPy arrays can be created from nested Python lists, and their elements can be accessed. It refers to a group of
items that are all of the same types and can be accessed using zero-based indexing. Every item in a ndarray is the same size as the memory block. Each entry in ndarray is a data-type object (called
Types of Numpy Array:
• One-Dimensional NumPy array
A one-dimensional array has only one dimension of elements. In other words, the one-dimensional NumPy array should only contain one tuple value.
# One-Dimensional Array
import numpy as np
val=np.array([1, 5, 2, 6])
Explanation: print() function is used to print the whole array provided.
• Multi-Dimensional NumPy Array
A multi-dimensional array can have n dimensions of elements. In other words, the multi-dimensional NumPy array can contain a number of tuple values.
# Multi-Dimensional Array
import numpy as np
val=np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
Explanation: print() function is used to print the whole array provided.
Accessing Element in the Array
Indexing or accessing the array index in a NumPy array can be done in a variety of ways.
1. Slicing is used to print a range of an array. The built-in slice function is used to create a Python slice object by passing start, stop, and step parameters to it. Slicing an array involves
establishing a range in a new array used to print a subset of the original arrayβ s elements. Because a sliced array holds a subset of the original arrayβ s elements, editing content alters the
original arrayβ s content.
Example 1:
# Accessing Array Element
import numpy as np
val=np.array([1, 5, 2, 6])
# Zero based indexing
val_mul=np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
Explanation: Here we are accessing the 1st index value of the 1-D array via writing val[1] and accessing the value in the 2nd row and 2nd column of the 2-D array as the arrays have zero-based
Example 2:
#Accessing Range of Elements
#in array using slicing
val_mul=np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
slice_val = val_mul[:2, :2]
print (β First 2 rows and columns of the array\nβ , slice_val)
Explanation: Here we are trying to access the values in rows and columns starting from 1st row to 2nd row by writing β :2 β as the first argument and similarly accessing all the values
corresponding to those rows and having columns from starting to the 2nd.
Example 3:
val_mul1=np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6),(1, 5, 0, 1),(7, 3, 3, 0)])
slice_step_row = val_mul1[::2, :2]
print (β All rows and 2 columns of the array with step size of 2 along the row\nβ , slice_step_row)
Explanation: Here we are trying to access the values in rows and columns starting from 1st row to end row with a step size of 2 which means the difference between consecutive rows to be 2 by writing
β ::2 β as the first argument and similarly accessing all the values corresponding to those rows and having columns from starting to the 2nd.
Example 4:
# Accessing multiple elements at one go
val_mul1=np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6),(1, 5, 0, 1),(7, 3, 3, 0)])
mul_idx_arr = val_mul1[
[1, 2, 2, 3],
[1, 2, 3, 3]
print (β \nAccessing Elements at multiple indices (1, 1),(2, 2), (2, 3), (3, 3) β β , mul_idx_arr)
Explanation: Here we are trying to access the values from the array which are present at the indexes (1,1),(2,2),(2,3), and (3,3).
2. Another type of accessing technique is the boolean array indexing where we can give the condition where elements that follow the condition are printed.
# Boolean array indexing
val_mul=np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6),(1, 5, 0, 1),(7, 3, 3, 0)])
condition = val_mul > 3
res = val_mul[condition]
print (res)
Explanation: Here we are accessing the values which follow the given conditions i.e. the value of the element should be greater than 3.
Frequently used Operations on Numpy Array
• ndim– This operation is used to compute the dimension of the array.
# Dimension of the array
val1 = np.array([1, 5, 2, 6])
print(val1.ndim)# 2-D array
val2 = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
Explanation: Here val1 is a 1-D array having values 1,5,2 and 6 therefore we are getting value 1 for val1.ndim and val2 is a 2-D array
Therefore we are getting value 2 corresponding to val2.ndim
• Size β This operation is used to calculate the number of elements in the array.
# Calculate Array Size
val2 = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
Explanation: Here val2 contains 12 values 3, 4, 2, 5, 3, 6, 2, 4, 1, 5, 2, 6 therefore we are getting the size of the val2 array as 12.
• Shape β This operation is used to calculate the shape of the array.
# Calculate Array Shape
val = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
Explanation: Here Val array is a 2-D array
It has 3 rows and 4 columns and val.shape is used to get the size corresponding to each dimension.
• Reshape β Reshape is used to reshape the array according to the given parameters
# Reshape the Array
val = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
Explanation: reshape function helps us to reshape the array and fill the values of the array correspondingly. Here we have 12 values and we want to reshape the array from (3,4) to (2,6) so now there
would be only 2 rows and 6 columns.
• Transpose β T operator is used in getting the transpose of an array.
# Transpose of an array
val = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
Explanation: When we need to replace all the rows of an array with the columns and columns with rows then we need to call the val.T to get the transpose of the array. In transpose, the shape of the
array changes as now number of rows becomes a number of columns and vice versa. As here 3, 4, 2, 5 was the first row initially but after transpose, it becomes the 1st column.
• Ravel β ravel is used to convert the array into a single column
# Convert Array to Single Column
val = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
Explanation: When we want to convert the array to a 1-D array we use the ravel function as it combines all the values of the array as in the above example we had initially a 2-D array of 3 rows and 4
columns but after applying the ravel function we gets the 1-D array of 12 values.
• Itemsize: Itemsize is used to compute the size of each element in bytes.
# Calculate Array Itemsize
val = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
# size of each element in bytes
Explanation: When we want to get the size of each element in bytes we use (array_name.itemsize) as in the above example the data type is int32 which means integer of 32 bits and as we know that 1
byte is equal to 8 bits and hence it is 4 bytes in size.
• Dtype– Dtype is used to get the data type of the elements in the array.
# Calculate Data type of each element
val = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
Explanation: When we want to compute the data type of the elements present in the array we use val.dtype which gives us the data type here in the above example the values of the array are integer so
the data type given is int32 which means it represents an integer of 32 bits.
• np.zeros()/np.ones() β ones() and zeros() functions of numpy are used to generate arrays having all 1β s and 0β s respectively.
# Generating array having all 1's
val = np.ones(3)
# Generating array having all 0's
val0 = np.zeros(3)
Explanation: When we want to generate an array having all 1β s or 0β s we use the above-given functions which helped us generate an array of size n. Here we gave the size of the array as 3 so an
array of size 3 having all 1β s and 0β s is generated.
– linspace β The linspace function is used to generate an array containing elements distributed equally in a given range. It takes 3 parameters to start and end of the range and the number of
elements to be present in the array.
# Linspce generate 8 numbers present in range 2β 5
val = np.linspace(2,5,8)
Explanation: When we want to generate an array having specific order like in range x to y and size of the array to be n then we use linspace function as here we wanted to generate an array of size 8
having values equally spaced between range 2 to 5.
• Max β Max is used in getting the maximum element across the whole array.
# Find maximum element in whole array
val = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
Explanation: When we want to calculate the maximum of all elements of the array we can simply write array_name.max() and get the maximum element as here we get 6 which is the maximum of all the
• Min β Min function is used in getting the minimum element across the whole array.
# Find minimum element in whole array
val = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
Explanation: When we want to calculate the minimum of all elements of the array we can simply write array_name.min() and get the minimum element as here we get 1 which is the minimum of all the
• Sum β Sum function is used to get the total of all the elements of the array.
# Sum of all elements in whole array
val = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
Explanation: When we want to calculate the sum of all elements of the array we can simply write array_name.sum().
• np.sqrt() β This function is used to get the square root of all the elements of the array.
# Used to calculate square root of each element
val=np.array([1, 4, 9])
Explanation: When we want to calculate the square root of each value of an array we can simply write np.sqrt(array_name) and we get the corresponding square roots to each and every value e.g. 2
corresponding to 4 and 3 corresponding to 9.
• np.std(): This function is used to calculate the standard deviation of all the elements of the array.
# Used to calculate standard deviation of all elements
val=np.array([1, 4, 9])
Explanation: When we want to find the standard deviation of all the values of the array we do not need to do the hectic math calculation for calculating the standard deviation, we can simply write
• Summation operation on the array β When we need to add some particular number n to all the elements of the array or one array to another array.
# Add number 3 to all elements of array val
val = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
Explanation: When we want to add value n from each value of array we can simply write (array_name+n).
Example 2:
# Add array val to another array val1
val = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
val1 = np.array([(4, 4, 8, 2),(9, 0, 3, 5),(6, 8, 3, 3)])
Explanation: When we want to find the addition of 2 arrays we can simply write (array1+array2) to get the addition of arrays but just keep in mind that the dimensions of both the arrays must be the
• Subtraction operation on the array β When we need to subtract some particular number n to all the elements of the array or one array from another array.
# Subtract number 3 from all elements of array val
val = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
Explanation: When we want to subtract value n from each value of array we can simply write (array_name-n).
Example 2:
# Subtract array val from another array val1
val = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
val1 = np.array([(4, 4, 8, 2),(9, 0, 3, 5),(6, 8, 3, 3)])
Explanation: When we want to find the difference between 2 arrays we can simply write (array1-array2) to get the difference of arrays but just keep in mind that the dimensions of both the arrays must
be the same.
• Power of each element of the array raised to number n:
# Power of each element of the array raise to 3
# Cube of each element
val = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
print(val**(3))# Power of each element of the array raise to 0.5
# Square root of each element
val = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
Explanation: Here we are raising to the power of 3 each and every value of the array and printing it.When we want to raise to the power of n each value of array we simply write (array_name)**n to
raise to the power of each and every value of the array by value n.
# Multiply each element of the array by 3
val = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
print(val*(3))# Divide each element of the array by 3
val = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
Explanation: Here we are multiplying each and every value of the array by 3 and printing it. Similarly, when we want to divide each value of array we simply write (array_name)/n to divide the array
by value n.
• np.sort() β This function is used to sort the array and takes the argument axis which allows sorting the array row-wise and column-wise when initialized with values 1 and 0 respectively.
# Sort the array row-wise
val = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
Explanation: Here we are sorting the values of the array row-wise as we have provided the value of the axis as 1.
# Sort the array column-wise
val = np.array([(3, 4, 2, 5),(3, 6, 2, 4),(1, 5, 2, 6)])
Explanation: Here we are sorting the values of the array column-wise as we have provided the value of the axis as 0.
What to do next?
The NumPy library in Python is one of the most widely used libraries for numerical computation. We looked at the NumPy library in-depth with the help of various examples in this article. We also
demonstrated how to utilize the NumPy library to do several linear algebra operations. I recommend that you practice the examples provided in this post. If you want to work as a data scientist, the
NumPy library is one of the tools youβ ll need to master to be a successful and productive member of the profession. Learn More.
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Published via Towards AI | {"url":"https://towardsai.net/p/l/introduction-to-numpy-in-python-2","timestamp":"2024-11-12T19:51:05Z","content_type":"text/html","content_length":"341857","record_id":"<urn:uuid:169545ee-47d7-4246-92d4-5a0427732018>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00314.warc.gz"} |
21 research outputs found
The closed form integral representation for the projection onto the subspace spanned by bound states of the two-body Coulomb Hamiltonian is obtained. The projection operator onto the $n^2$
dimensional subspace corresponding to the $n$-th eigenvalue in the Coulomb discrete spectrum is also represented as the combination of Laguerre polynomials of $n$-th and $(n-1)$-th order. The latter
allows us to derive an analog of the Christoffel-Darboux summation formula for the Laguerre polynomials. The representations obtained are believed to be helpful in solving the breakup problem in a
system of three charged particles where the correct treatment of infinitely many bound states in two body subsystems is one of the most difficult technical problems.Comment: 7 page
The hydrodynamic low-frequency oscillations of highly degenerate Fermi gases trapped in anisotropic harmonic potentials are investigated. Despite the lack of an obvious spatial symmetry the
wave-equation turns out to be separable in elliptical coordinates, similar to a corresponding result established earlier for Bose-condensates. This result is used to give the analytical solution of
the anisotropic wave equation for the hydrodynamic modes.Comment: 11 pages, Revte
In this article, I have investigated statistical mechanics of a non-stretched elastica in two dimensional space using path integral method. In the calculation, the MKdV hierarchy naturally appeared
as the equations including the temperature fluctuation.I have classified the moduli of the closed elastica in heat bath and summed the Boltzmann weight with the thermalfluctuation over the moduli.
Due to the bilinearity of the energy functional,I have obtained its exact partition function.By investigation of the system,I conjectured that an expectation value at a critical point of this system
obeys the Painlev\'e equation of the first kind and its related equations extended by the KdV hierarchy.Furthermore I also commented onthe relation between the MKdV hierarchy and BRS transformationin
this system.Comment: AMS-Tex Us
Some recent results show that the covariant path integral and the integral over physical degrees of freedom give contradicting results on curved background and on manifolds with boundaries. This
looks like a conflict between unitarity and covariance. We argue that this effect is due to the use of non-covariant measure on the space of physical degrees of freedom. Starting with the reduced
phase space path integral and using covariant measure throughout computations we recover standard path integral in the Lorentz gauge and the Moss and Poletti BRST-invariant boundary conditions. We
also demonstrate by direct calculations that in the approach based on Gaussian path integral on the space of physical degrees of freedom some basic symmetries are broken.Comment: 29 pages, LaTEX, no
We present a theoretical study of the crossover from the two-dimensional (2D, separate confinement of the carriers) to the three-dimensional (3D, center-of-mass confinement) behavior of excitons in
shallow or narrow quantum wells (QW's). Exciton binding energies and oscillator strengths are calculated by diagonalizing the Hamiltonian on a large nonorthogonal basis set. We prove that the
oscillator strength per unit area has a minimum at the crossover, in analogy with the similar phenomenon occurring for the QW to thin-film crossover on increasing the well thickness, and in agreement
with the analytic results of a simplified ÎŽ-potential model. Numerical results are obtained for GaAs/Alx Ga1-xAs and InxGa1-xAs/GaAs systems. Our approach can also be applied to obtain an accurate
description of excitons in QW's with arbitrary values of the offsets (positive or negative) and also for very narrow wells. In particular, the crossover from 2D to 3D behavior in narrow GaAs/
AlxGa1-xAs QW's is investigated: the maximum binding energy of the direct exciton in GaAs/AlAs QW's is found to be ⠌26 meV and to occur between one and two monolayers
Hepadnavirus genome replication involves cytoplasmic and nuclear stages, requiring balanced targeting of cytoplasmic nucleocapsids to the nuclear compartment. In this study, we analyze the signals
determining capsid compartmentalization in the duck hepatitis B virus (DHBV) animal model, as this system also allows us to study hepadnavirus infection of cultured primary hepatocytes. Using fusions
to the green fluorescent protein as a functional assay, we have identified a nuclear localization signal (NLS) that mediates nuclear pore association of the DHBV nucleocapsid and nuclear import of
DHBV core protein (DHBc)-derived polypeptides. The DHBc NLS mapped is unique. It bears homology to repetitive NLS elements previously identified near the carboxy terminus of the capsid protein of
hepatitis B virus, the human prototype of the hepadnavirus family, but it maps to a more internal position. In further contrast to the hepatitis B virus core protein NLS, the DHBc NLS is not
positioned near phosphorylation target sites that are generally assumed to modulate nucleocytoplasmic transport. In functional assays with a knockout mutant, the DHBc NLS was found to be essential
for nuclear pore association of the nucleocapsid. The NLS was found to be also essential for virus production from the full-length DHBV genome in transfected cells and from hepatocytes infected with
transcomplemented mutant virus. Finally, the DHBc additionally displayed activity indicative of a nuclear export signal, presumably counterbalancing NLS function in the productive state of the
infected cell and thereby preventing nucleoplasmic accumulation of nucleocapsids | {"url":"https://core.ac.uk/search/?q=author%3A(Wittaker%2C%20D%20E)","timestamp":"2024-11-05T03:30:35Z","content_type":"text/html","content_length":"189933","record_id":"<urn:uuid:c64cd8f0-7aaa-4200-ad86-c4440df6559d>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00573.warc.gz"} |
Pirates of Penzance 1983
"Pirates of Penzance" is a Comic Opera from 1879 by Gilbert and Sullivan There is a 1983 Film adaption. John Cook posted on 5 October 2022 on his blog that an adaptation opera has appeared also in
the book "Improving almost Anything: Ideas and Essays" by the British statistician George Box. The lines from Gilbert and Sullivan were:
I'm very well acquainted, too, with matters mathematical,
I understand equations, both the simple and quadratical,
About binomial theorem I'm teeming with a lot o' news,
With many cheerful facts about the square of the hypotenuse.
I'm very good at integral and differential calculus;
I know the scientific names of beings animalculous:
In short, in matters vegetable, animal, and mineral,
I am the very model of a modern Major-General.
MOV, Ogg Webm. IMDb link
Oliver Knill, Posted October 6, 2022, | {"url":"https://people.math.harvard.edu/~knill/various/pirates/index.html","timestamp":"2024-11-14T04:18:51Z","content_type":"text/html","content_length":"2604","record_id":"<urn:uuid:678cf8ec-7dce-487e-9dec-2b9bb01a36a1>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00223.warc.gz"} |
What Is Mass Transfer?
Fluid Flow, Heat Transfer, and Mass Transport Mass Transfer
Understanding Mass Transfer
Mass transfer describes the transport of mass from one point to another and is one of the main pillars in the subject of Transport Phenomena. Mass transfer may take place in a single phase or over
phase boundaries in multiphase systems. In the vast majority of engineering problems, mass transfer involves at least one fluid phase (gas or liquid), although it may also be described in solid-phase
In many cases, the mass transfer of species takes place together with chemical reactions. This implies that flux of a chemical species does not have to be conserved in a volume element, since
chemical species may be produced or consumed in such an element. The chemical reactions are sources or sinks in such flux balances.
The theory of mass transfer allows for the computation of mass flux in a system and the distribution of the mass of different species over time and space in such a system, also when chemical
reactions are present. The purpose of such computations is to understand, and possibly design or control, such a system.
Transport and reactions in a reactor. The concentration isosurfaces reveal mass transfer through diffusion and convection. The flux through diffusion takes place perpendicular to the concentration
isosurfaces, i.e., the reactions may cause a flux to the reaction site of the species that are consumed in the reaction. Convection creates a larger separation between the concentration isosurfaces
and takes place along the streamlines of the fluid flow (in white), which in some places run along the isosurfaces, since convection tends to eliminate concentration gradients along its main
direction. Transport and reactions in a reactor. The concentration isosurfaces reveal mass transfer through diffusion and convection. The flux through diffusion takes place perpendicular to the
concentration isosurfaces, i.e., the reactions may cause a flux to the reaction site of the species that are consumed in the reaction. Convection creates a larger separation between the concentration
isosurfaces and takes place along the streamlines of the fluid flow (in white), which in some places run along the isosurfaces, since convection tends to eliminate concentration gradients along its
main direction.
Mathematical Description of Mass Transfer
The driving force, F, for mass transfer is created by gradients in the system potential, U:
Gradients in chemical composition are usually responsible for this driving force. The driving force for transport over phase boundaries is generated by a deviation from equilibrium over such a phase
boundary. Additional driving forces may contribute with a drift velocity, such as the forces created by migration, pressure, gravitational, and centrifugal forces.
The equation below shows the forces acting on a chemical species, per mole of atoms, ions, or molecules, due to gradients in chemical potential and electric fields (migration) ^1.
In these equations, R denotes the gas constant, T is the temperature, a[i] is the activity of each species, z[i] denotes the charge number of a species, F is the Faraday constant, and φ is the
electric potential. The negative gradient of φ is the electric field. The activity can be understood as a thermodynamic measure of the chemical potential of the system, so that gradients in activity
correspond to driving forces for chemical mass transport.
A simple chemical assumption is that the activity of a species i is given by its mole fraction, denoted as x[i]. This is exactly true for an ideal mixture.
The forces on a species i are balanced by the friction in the interaction between this species and the other species in a mixture. The friction acting on a mole of i is proportional to the difference
in mass velocity between i and each species j in the mixture, the mole fraction of each species j in a mixture, and the friction coefficient between i and j.
In this equation, ζ[ij] denotes the friction coefficient between species i and j, x[j] is the mole fraction of species j, and u[R,i] is the mass species velocity of species i relative to the mass
average velocity of the whole mixture. Note that the mass velocities of each species in the equation above are given using the mass average velocity for the mixture as reference. A species that does
not deviate from the velocity of the mixture (i.e., that does not diffuse or migrate, in this case), has a zero u[R,i], when using the mixture velocity as reference.
If we now set the driving forces to exactly balance the friction forces acting on species i, we obtain the following equation:
The molar flux is defined as
where J[i] is the flux vector of species i relative to the velocity of the mixture and c is the total concentration of all species in a mixture. Introducing the Maxwell-Stefan diffusivity as:
and using the molar flux to eliminate u[R,i] and u[R,j] in the force balance equation above yields the following expression:
This is the Maxwell-Stefan equation, an equation that forms the basis for the mathematical description of mass transfer of chemical species in a mixture ^2. Simplifications of these equations for
diluted mixtures give, for example, Fick's first law of diffusion and the Nernst-Planck equations for diffusion and migration.
The molar flux of a species i relative to a fixed coordinate system, denoted as N[i], is obtained by adding the convective term, due to the velocity of the whole mixture:
The resulting fluxes are used in the mass conservation equations for each species in the solution:
The sum of all mass fluxes, including the convective term, results in the continuity equation for the mixture:
where the last term is necessarily zero due to the conservation of mass of an individual chemical reaction. By identifying the sums as the density and mass flux density, we get the mass continuity
The convective term in the flux is the contribution to the flux of a species due to the movement of the whole solution (see the figure above). For this reason, convective flux takes place along the
velocity streamlines of the solution for all chemical species in the solution. Note that the sum of all the species' mass fluxes, relative to the flux of the mixture, is zero if the mass-averaged
velocity is used as reference. The mass-averaged velocity is defined as:
where i. This implies that, in general, the mass flux of each species is tightly coupled to the total mass velocity in a mixture. In a strict definition, the mass average velocity of a mixture could
be obtained by formulating and solving the equations for the conservation of momentum for each species in a mixture.
However, the interaction coefficients, required for such formulations, are usually difficult to measure or calculate. Instead, equations for the conservation of momentum for the whole mixture are
usually defined. The combination of the equations for the conservation of momentum and mass for a mixture at low velocities (less than one third of the speed of sound), yield the Navier-Stokes
equations. The solution of the Navier-Stokes equations gives the velocity field (a vector field) that also determines the direction of the convective flux of all species in the mixture.
The tight coupling between mass transport for each species and the conservation of mass for the whole mixture is exemplified in the example below. Oxygen in air is consumed at the surface of a
catalyst and produces liquid water that is removed from the gas phase in a gas diffusion electrode. The consumption of oxygen causes a net velocity in the gas mixture (air). Additionally, a nitrogen
concentration gradient is formed in order to perfectly balance the advective (or convective) flux of nitrogen with an opposite flux by diffusion.
Surface plot of the nitrogen concentration in a gas diffusion electrode. The flux along the right vertical edge, plotted in the x-y plot, shows that the diffusive flux from the catalytic surface
exactly compensates the convective flux to this surface, generated by oxygen consumption at the catalyst's surface. Surface plot of the nitrogen concentration in a gas diffusion electrode. The flux
along the right vertical edge, plotted in the x-y plot, shows that the diffusive flux from the catalytic surface exactly compensates the convective flux to this surface, generated by oxygen
consumption at the catalyst's surface.
Due to the difficulties of numerically resolving steep gradients in potential, mass transfer across phase boundaries is often expressed using difference equations, instead of differential equations.
This approximation implies that the gradients, included in the driving forces, are linearized inside a fictitious boundary layer (see the figure below). The thickness of the layer is then defined as
the distance from a phase boundary where the linearized concentration gradient, starting from the concentration at the phase boundary, reaches the bulk concentration. The definition of the boundary
layer also means that its thickness may be different for different species.
Fictive boundary layers inside and around a gas bubble in a liquid. Fictive boundary layers inside and around a gas bubble in a liquid.
The mass transfer coefficient, k[m], for such an interface is defined as the diffusivity divided by the boundary layer thickness, δ.
An estimate of the relation between the boundary layer thickness and a system’s typical length is given by the Sherwood number:
In this expression, L denotes a typical length for a system, such as the radius of a pipe and the width of a channel. However, if the mass transport around a gas bubble in a liquid is studied, then L
may denote the radius of the bubble. Since the thickness of the boundary layer depends on the convection just outside an interface, the Sherwood number also gives a measurement of the convective and
diffusive fluxes to such an interface.
The boundary layer thickness at a liquid-gas interface for a rising bubble is of the order of magnitude of 100 μm in the gas phase and around 10 μm in the liquid phase.
The Sherwood number can also be defined as a function of the Reynolds and Schmidt numbers. The Reynolds number gives an estimate of the ratio of momentum transport by inertia to viscosity in a fluid:
where μ denotes viscosity and U denotes an average velocity. The Schmidt number gives an estimate of the relation between viscosity and diffusivity in a fluid:
The mass transfer coefficient can be estimated from the relation between the Sherwood number and the Reynolds and Schmidt numbers. For example, for forced convection along a flat plate, the following
expression can be used:
where f[loc] denotes the local friction factor for flow along a flat plate. The friction coefficient for different geometries is tabulated in literature and may also be obtained experimentally. All
the material properties and the average velocity in the relation above are relatively easy to find in literature or estimate from simple calculations. Once the Sherwood number is calculated, then the
mass transfer coefficient can be calculated, including the boundary layer thickness, which is the parameter that is not easily estimated otherwise. Note, though, that the above expression is only
valid for flat plates.
Summary of Mass Transfer
The driving forces for mass transfer are relatively easy to define. These forces give the expressions for the flux, which can be used in the equations for conservation of mass. When these equations
are discretized using a numerical method and when the resulting numerical model equations are solved, the results give the concentration distribution and fluxes in a system as functions of the
modeled space coordinates and time. The estimates of the concentration and fluxes can be used to understand, design, optimize, and control the system that is being studied.
In cases where the equations cannot easily be discretized and solved in a detailed way, mass transfer coefficients may be used for obtaining an estimate of concentrations and fluxes in a system,
which are less detailed than the one described above.
Published: January 14, 2015
Last modified: February 22, 2017 | {"url":"https://www.comsol.com/multiphysics/what-is-mass-transfer?parent=fluid-flow-heat-transfer-and-mass-transport-0402-372","timestamp":"2024-11-13T18:51:29Z","content_type":"text/html","content_length":"123285","record_id":"<urn:uuid:74ca3b96-dcd4-4325-a5d1-730241b1f58f>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00660.warc.gz"} |
Display Top N items and Others in Power BI - Goodly
In this blog, I’ll show how can you display the Sales of Top 'N' products and categorize the rest as “Others”. This is a common need in business reporting, let’s see how can you do this using some
DAX & Data Modeling and some visualization tricks.
Display Sales for TOPN & Others – Video
Firstly, let's go ahead and see what our final output looks like.
Before we proceed, let's take a look at the Data Model.
This post is going to be a long one, so let me summarize the building blocks of this visual.
1. We need to display Product Names along with 'Others' on the X-Axis.
2. We need a Calculation / Measure to group the Sales for Top Products and Others.
3. A Slicer to select the number of Top Products displayed in the Visual.
4. Conditional Formatting to color the Top Products and Others.
5. Sorting of the Visual.
6. Dynamic Title that displays the % contribution of Top Products.
#1 Display Product Names and 'Others' as a category on the X-Axis
A general rule – Anything that is displayed on the x-axis of a chart needs to be physically present as a column of a table.
In our scenario we need Product Name and 'Others' on the horizontal axis of the chart, that can only be done if we have a column that has product names and 'Others'.
To do this let's create a Pseudo Prod Table:
Pseudo Prod Table =
UNION (
DISTINCT ( Products[Product] ), -- to find unique names of the products
DATATABLE ( -- one row table having value 'Others'
'Product', STRING,
{ { 'Others' } }
If you create a Matrix Visual using the Products column of the pseudo table you'll see all products and “Others”.
Let's also create a one to many relationship between pseudo products table and products table.
Now that we have displayed Product names along with “Others”, let’s create a calculation that can group the sales of Top Products and the rest as “Others”.
#2 Calculation / Measure to group the Sales for Top Products and Others
Let's start with this simple measure
Top N Sum Sales =
[Total Sales],
TOPN ( -- top 3 products by Sales
'Pseudo Prod Table',
[Total Sales]
With the above measure I'll be able to see the sales of the TOP 3 Products. For instance refer the card visual
Although this measure won't work if you drag this to the Matrix visual. Problem – At the product level I see the sales of all the products and not just the top 3 products.
Simple reason, when the sales of each product being compared to itself, it would would obviously rank in the Top 3 products and hence all products are displayed.
To solve this, I can modify my code as follows, so that the only sales of the top 3 products are displayed against their name.
Top N Sum Sales =
[Total Sales],
KEEPFILTERS ( -- will apply the product filter to show only 3 products
TOPN (
ALLSELECTED ( 'Pseudo Prod Table' ), -- considers all visible products
[Total Sales]
The results look better!
Next if I subtract Sales of top 3 products from Total Sales of all the products we get value to be displayed in the “Others” category.
Let’s further revise the measure.
Top N Sum Sales =
VAR TopProdTable =
TOPN (
ALLSELECTED ( 'Pseudo Prod Table' ),
[Total Sales]
VAR TopProdSales =
CALCULATE (
[Total Sales],
KEEPFILTERS ( TopProdTable )
VAR OtherSales =
CALCULATE (
[Total Sales],
ALLSELECTED ( 'Pseudo Prod Table' )
- CALCULATE ( -- subtracting the sales of Top 3 products
[Total Sales],
VAR CurrentProd =
SELECTEDVALUE ( 'Pseudo Prod Table'[Product] )
IF ( -- Categorizing as Others and Top Products
CurrentProd <> 'Others',
Now the Pivot shows the results of TOP 3 Products and Others
The above matrix visual when converted to a clustered column chart visual, it looks like this
As of now the number 3 to select top 3 products is hardcoded in the measure, to make it dynamic we need a slicer and link the slicer selection to our measure.
#3 Slicer to Select the Top Products
Let's create a New Table Top N Selection
TopN Selection =
1. Now create a slicer on the value column of this table.
2. To capture the value selected in the slicer, I will finally revise my measure like this.
Top N Sum Sales =
VAR TopNSelected = -- capturing top value selected in the slicer
SELECTEDVALUE ( 'TopN Selection'[Value] )
VAR TopProdTable =
TOPN (
ALLSELECTED ( 'Pseudo Prod Table' ),
[Total Sales]
VAR TopProdSales =
CALCULATE (
[Total Sales],
KEEPFILTERS ( TopProdTable )
VAR OtherSales =
CALCULATE (
[Total Sales],
ALLSELECTED ( 'Pseudo Prod Table' )
- CALCULATE (
[Total Sales],
VAR CurrentProd =
SELECTEDVALUE ( 'Pseudo Prod Table'[Product] )
IF (
CurrentProd <> 'Others',
Here’s how it would look like
#4 Conditional Formatting – For Top N and Others
I also want to color Top N products as orange and Others as grey. To do this we create a simple measure.
Color =
SWITCH (
SELECTEDVALUE ( 'Pseudo Prod Table'[Product] ), -- current product
'Others', '#D9D9D9', -- if 'Others', then grey (hexcode)
'#ED7D31' -- else Orange (hexcode)
To apply this measure, go to >> format pane >> data color >> use the “fx” to select conditional formatting >> finally use format by field option on this measure.
Let’s solve the next problem and sort the bars, Top Products should appear first and “Others” in the end.
#5 Ranking and Sorting the Top Products in order of Sales
Consider this measure
Rank =
IF (
[Top N SUM]
<> BLANK (),
RANKX (
TOPN (
SELECTEDVALUE ( 'TopN Selection'[Value] ),
ALLSELECTED ( 'Pseudo Prod Table' ),
[Total Sales]
[Total Sales],
Now, drag this measure into the Tooltips of the visual and sort it by ascending order of rank.
#6 Create a Dynamic Title
We now need to make the title dynamic that displays the percentage contribution of Top Products. Let's create this measure.
Title =
VAR TopProd =
CALCULATE (
[Total Sales],
TOPN (
SELECTEDVALUE ( 'TopN Selection'[Value] ),
ALLSELECTED ( 'Pseudo Prod Table'[Product] ),
[Total Sales]
VAR TopProdPct =
DIVIDE (
[Total Sales]
'Top '
& SELECTEDVALUE ( 'TopN Selection'[Value] ) & ' Products made '
& FORMAT (
'#.#% Sales'
Add this title to a Text box and format it in a large font size. and the final output looks like this.
More on DAX: | {"url":"https://goodly.co.in/top-n-and-others-power-bi/","timestamp":"2024-11-03T21:22:47Z","content_type":"text/html","content_length":"120340","record_id":"<urn:uuid:c16a63b7-1c98-4442-82ab-d38fbc1e7747>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00443.warc.gz"} |
Summary Measures and Graphs
Description of Proposed Provision:
C2.1: Increase the earliest eligibility age (EEA) by 2 months per year for those age 62 starting in 2025 and ending in 2042 (EEA reaches 65 for those age 62 in 2042).
Estimates based on the intermediate assumptions of the 2023 Trustees Report
Summary Measures
Current law Change from current law Shortfall eliminated
[percent of payroll] [percent of payroll]
Long-range Annual Long-range Annual Long-range Annual
actuarial balance in actuarial balance in actuarial balance in
balance 75th year balance 75th year balance 75th year
-3.61 -4.35 -0.10 -0.43 -3% -10% | {"url":"https://www.ssa.gov/OACT/solvency/provisions/charts/chart_run245.html","timestamp":"2024-11-08T05:47:14Z","content_type":"text/html","content_length":"19037","record_id":"<urn:uuid:4d501d0a-c64b-4490-a4ac-b34a339d0aa1>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00300.warc.gz"} |
Lesson 9
Problem Solving with Volume: Water (optional)
Warm-up: Notice and Wonder: Cubic Centimeters and Grams (10 minutes)
The purpose of this warm-up is for students to observe the relationship between the different types of units in the metric system. By contrast, in the standard system, it is not easy to see the
relationship between inches, cups, and pounds. While students may notice and wonder many things about this image, conversions between liquid volume units (cups, gallons, liters) and regular volume
units (cubic centimeters, cubic inches, cubic feet) are the important discussion points.
• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time
• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.
Student Facing
What do you notice? What do you wonder?
Activity Synthesis
• Highlight that the cube represents 1 cubic centimeter or 1 mL of water and one mL of water weighs 1 g.
• The picture shows the relationships between length, capacity (or volume), and weight in the metric system.
Activity 1: Catching Rainfall (15 minutes)
The purpose of this activity is for students to estimate how much water falls on the roof of a house, given a particular amount of rainfall. For this calculation, standard units work well as the area
of the roof could be given in square feet, for example, and the rain in inches. A conversion would readily give the volume in cubic feet or inches. But, the standard units used to measure volume are
cups, pints, quarts, and gallons so more work would need to be done in order to figure out how many gallons, for example, there are in a cubic foot. With the metric system, liquid volume units
(liters) and regular volume units (cubic centimeters) are naturally connected.
MLR5 Co-Craft Questions. Keep books or devices closed. Display only the image, without revealing the questions, and ask students to write down possible mathematical questions that could be asked
about the situation. Invite students to compare their questions before revealing the task. Ask, “What do these questions have in common? How are they different?” Reveal the intended questions for
this task and invite additional connections.
Advances: Reading, Writing
Representation: Access for Perception. Read statements aloud. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Conceptual Processing, Attention
• Groups of 2
• “About how big is a square meter?” (It's about the size of my desk top.)
• “About how many square meters do you think there are in the classroom floor?” (maybe 100)
• 5 minutes: independent work time
• 5 minutes: partner work time
Student Facing
Here is a diagram showing the roof of a house.
1. What is the area of the roof?
2. Each month an average of 5 cm of rain falls on the house. How many cubic cm of rain is that?
3. There are 1,000 cubic cm in 1 liter. How many liters of water fall on the house?
4. You want to build a reservoir to catch the rain that falls so you can use the water. What side lengths would you suggest for the reservoir? Explain or show your reasoning.
Activity Synthesis
• “When did you have to convert units during the activity?” (I needed to convert meters to centimeters since the roof is given in meters and the rain is given in centimeters. I needed to convert
cubic centimeters to liters.)
• “How did you make the calculations for the conversions?” (the conversions required multiplying or dividing by powers of 10. I multiplied by 100 to convert m to cm and I divided by 1,000 to
convert cubic centimeters to liters.)
• “Do you think that you could make a container to capture all of the rainfall?” (Yes, I think that a 2 meter cube is big but it's not so big that it would not fit near the house. Or we could use
several containers that are a little smaller.)
Activity 2: How Much Water? (20 minutes)
The purpose of this activity is for students to find out if the amount of water that falls on the house is sufficient for many of the daily household chores that use water. This will require a lot of
estimation and will vary from house to house. How much of the calculations to leave up to the students is an individual teacher choice and this lesson could easily be extended for another day if the
students make well reasoned estimates (some values are given in parentheses) for how much water is used for different activities such as:
• taking baths or showers (150 liters or 80 liters)
• washing clothes (100 liters)
• washing dishes (100 liters)
• washing hands (1 liter)
• flushing the toilet (10 liters)
More estimation comes into play for how often each of these activities happens and this will vary greatly depending on the student. When students make a list of the different things they do in the
house that use water and then estimate how much water is used they model with mathematics (MP4).
Consider inviting students to check their estimates by looking at one of their monthly water bills. The bill will usually give the number of gallons of water used and there are almost 4 liters in a
• Groups of 2
• After students work on the first problem, pause the class and make a list of the main daily uses of water.
• Depending on how much time is available and the modeling demand level desired, consider estimating together or providing estimates for how much water is used for each purpose.
• 5 minutes: independent work time
• 10 minutes: partner work time
Student Facing
1. What are some of the ways you use water at home?
2. Estimate how much water you use at your home in a month.
3. How much rain would need to fall on your home each month to supply all of your water needs?
4. What challenges might come up if you tried to use the rainwater that falls on the roof of your home? Do you think it makes sense to try to capture the rain that falls on your home?
Activity Synthesis
• “How can you visualize the volume of water that it uses to take a bath?” (I can picture the bathtub filled up partway with water.)
• “How could you measure the volume of water in the bathtub?” (I could measure the length and width of the tub and the height of the water and multiply them.)
• “Are any of the amounts of water used for different things surprising to you? Why?” (I am surprised by how much water it takes to wash the dishes. It’s almost as much as when you take a bath.)
Lesson Synthesis
“How is measuring the volume of water the same as measuring the volume of the Empire State Building?” (If I know the length, width, and height that the water takes up, then I can multiply them to get
the volume, just like the building.)
“How is measuring the volume of water different than measuring the volume of a building?” (Water does not have a simple shape like a building. It needs to be put in a container in order to measure.)
“What is important to remember when measuring volume?” (It’s the amount of space something can hold or that something takes up. I can measure it in cubic units or in liters for a liquid.)
Cool-down: Reflection: Volume (5 minutes)
Student Facing
We investigated several different complex volume questions. For the ancient pyramids of Egypt we gave an estimate of a couple million cubic meters. Since these pyramids are not rectangular prisms, an
estimate is the best we could hope for. Then we estimated the volume of the world's largest wagon, using information from a photograph. Lastly, we investigated the amount of rain that falls on a
house and the amount of water our families use in a year.
In each case, we could only make estimates because the situations are all complex. In the previous section we used estimation to check the reasonableness of calculations. In this section we saw that
making reasoned estimates is a vital part of applying mathematics to many real world situations. | {"url":"https://curriculum.illustrativemathematics.org/k5/teachers/grade-5/unit-8/lesson-9/lesson.html","timestamp":"2024-11-11T07:22:44Z","content_type":"text/html","content_length":"88759","record_id":"<urn:uuid:2d4b2f2e-9ff3-4277-a761-d76576cdec7c>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00471.warc.gz"} |
Problem of Old Evidence tag
Suppose a new scientific hypothesis, such as general relativity, explains a well-know observation such as the perihelion precession of mercury better than any existing theory. Intuitively, this is a
point in favor of the new theory. However, the probability for the well-known observation was already at 100%. How can a previously-known statement provide new support for the hypothesis, as if we
are re-updating on evidence we’ve already updated on long ago? This is known as the problem of old evidence, and is usually leveled as a charge against Bayesian epistemology.
Bayesian Solutions vs Scientific Method
It is typical for a Bayesian analysis to resolve the problem by pretending that all hypotheses are around “from the very beginning” so that all hypotheses are judged on all evidence. The perihelion
precession of Mercury is very difficult to explain from Newton’s theory of gravitation, and therefore quite improbable; but it fits quite well with Einstein’s theory of gravitation. Therefore, Newton
gets “ruled out” by the evidence, and Einstein wins.
A drawback of this approach is that it allows scientists to formulate a hypothesis in light of the evidence, and then use that very same evidence in their favor. Imagine a physicist competing with
Einstein, Dr. Bad, who publishes a “theory of gravity” which is just a list of all the observations we have made about the orbits of celestial bodies. Dr. Bad has “cheated” by providing the correct
answers without any deep explanations; but “deep explanation” is not an objectively verifiable quality of a hypothesis, so it should not factor into the calculation of scientific merit, if we are to
use simple update rules like Bayes’ Law. Dr. Bad’s theory will predict the evidence as well or better than Einstein’s. So the new picture is that Newton’s theory gets eliminated by the evidence, but
Einstein’s and Dr. Bad’s theories remain as contenders.
The scientific method emphasizes predictions made in advance to avoid this type of cheating. To test Einstein’s hypothesis, Sir Arthur Eddington measured Mercury’s orbit in more accuracy than had
been done before. This test would have ruled out Dr. Bad’s theory of gravity, since (unless Dr. Bad possessed a time machine) there would be no way for Dr. Bad to know what to predict.
Simplicity Priors
Proponents of simplicity-based priors will instead say that the problem with Dr. Bad’s theory can be identified by looking at its description length in contrast to Einstein’s. We can tell that
Einstein didn’t cheat by gerrymandering his theory to specially predict Mercury’s orbit correctly, because the theory is mathematically succinct! There is no room to cheat; no empty closet in which
to smuggle information about Mercury. Marcus Hutter argues for this resolution to the problem of old evidence in On Universal Prediction and Bayesian Confirmation.
In contrast, ruling out all evidence except “advanced predictions” may seem like a crude and epistemically inefficient solution, which will get you to the truth more slowly, or perhaps not at all.
Unfortunately, simplicity priors do not appear to be the end of the story here. They can only help us to avoid cheating in specific contexts. Consider the problem of people evaluating how much to
trust the opinions of a variety of people, based on their success at predictions so far. For example, Manifold Markets or similar platforms. If people were given points for predicting old evidence on
such platforms, newcomers could simply claim they would have predicted all the old evidence perfectly, gaining more points than users who were around for the old stuff and risked points to predict
Humans are similar to hypotheses, in that they can generate probabilities predicting events. However, we can’t judge humans on “descriptive complexity” to use the simplicity-prior solution to the
problem of old evidence.
Logical Uncertainty
Even in cases where we can measure simplicity perfectly, such as in Solomonoff Induction, is it really a perfect correction for the problem of old evidence? This becomes implausible in cases of
logical uncertainty.
Simplicity priors seem like a very plausible solution to the problem of old evidence in the case of empirical uncertainty. If I try to “cheat” by baking in some known information into my hypothesis,
without having real explanatory insight, then the description length of my hypothesis will be expanded by the number of bits I sneak in. This will penalize the prior probability by exactly the amount
I stand to benefit by predicting those bits! In other words, the penalty is balanced so that it does not matter whether I try to “cheat” or not.
The same argument does not hold if I am predicting mathematical facts rather than empirical facts, however. Mathematicians are often in a situation where they already know how to calculate a sequence
of numbers, but they are looking for some deeper understanding, such as a closed-form expression for the sequence, or a statistical model of the sequence (EG, the prime number theorem describes the
statistics of the prime numbers). It is common to compute long sequences in order to check conjectures against more of the sequence, and in doing so, treat the computed numbers as evidence for a
If I claimed to have some way to predict the prime numbers, but it turned out that my method actually had one of the standard ways to calculate prime numbers hidden within the source code, I would be
accused of “cheating” in much the same way that a scientific hypothesis about gravity would be “cheating” if its source code included big tables of the observed orbits of celestial bodies. However,
since mathematical sequences are produced from simple definitions, this “cheating” will not be registered by a simplicity prior.
Paradox of Ignorance
Paul Christiano presents the “paradox of ignorance” where a weaker, less informed agent appears to outperform a more powerful, more informed agent in certain situations. This seems to contradict
the intuitive desideratum that more information should always lead to better performance.
The example given is of two agents, one powerful and one limited, trying to determine the truth of a universal statement ∀x:ϕ(x) for some Δ0 formula ϕ. The limited agent treats each new value of
ϕ(x) as a surprise and evidence about the generalization ∀x:ϕ(x). So it can query the environment about some simple inputs x and get a reasonable view of the universal generalization.
In contrast, the more powerful agent may be able to deduce ϕ(x) directly for simple x. Because it assigns these statements prior probability 1, they don’t act as evidence at all about the
universal generalization ∀x:ϕ(x). So the powerful agent must consult the environment about more complex examples and pay a higher cost to form reasonable beliefs about the generalization.
This can be seen as a variant of the problem of old evidence where the “old evidence” is instead embedded into the prior, rather than modeled as observations. It is as if everyone simply knew about
the orbit of Mercury, rather than studying it through a telescope at some point.
This causes trouble for the typical Bayesian solution to the problem, where we imagine that all hypotheses were around “at the very beginning” so that all hypotheses can gain/lose probability based
on how well they predict. In Paul’s version, since the information is encoded into the prior, there is no opportunity to “predict” it at all.
This poses a problem for a picture where a “better” prior is one which “knows more”.
It also poses a problem for a view where “one man’s prior is another’s posterior”—if the question “which beliefs are this agent’s prior?” only has an answer relative to which update is being
performed (there is no sense in an ultimate prior, what philosophers call an ur-prior—only prior/posterior relative to specific updates), then it seems the Bayesian loses the right to answer the
problem of old evidence by imagining all hypotheses were present from the very beginning (since there is no objective beginning).
Adding Hypotheses
Another perspective on the problem of old evidence is to think of it as a question of how to add new hypotheses over time. The typical Bayesian solution of modeling all hypotheses as present from the
beginning can be seen as dodging the question, rather than providing a solution. Suppose we wish to model the rationality of a thinking being who can only consider finitely many hypotheses at a time,
but who may formulate new hypotheses (and discard old ones) over time. Should there not be rationality principles governing this activity?
No comments. | {"url":"https://www.greaterwrong.com/tag/problem-of-old-evidence","timestamp":"2024-11-12T00:43:27Z","content_type":"text/html","content_length":"20387","record_id":"<urn:uuid:fda1a303-9a01-4a05-8c59-bfa48be2253d>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00738.warc.gz"} |
Problem 4 from the 4th BJMO Team Selection Test
Solution 1
Our solution depends on a lemma that will be proved later on in two ways:
For the given configuration, the points $H,\,$ $M,\,$ $P,\,$ and $Q\,$ are concyclic:
For $AB=AC\,$ the problem is trivial. WLOG. $AB\gt AC.\,$ Let $L\,$ be the intersection $PH\,$ and $MQ.\,$ From the cyclic quadrilateral $PMHQ,\,$ with $PM \perp MQ,\,$ we get also $PH\perp QH\,$ and
$L\,$ is the orthocenter of the triangle made by $PQ,\,$ $PM\,$ and $QH.\,$ With $MH,\,$ one of the sides of its orthic triangle, and $ML,\,$ the bisector of $\angle AMC,\,$ we infer that $AM\,$ is
another side of the orthic triangle, i.e. that $N\,$ is the foot of the altitude from the vertex made by $HQ\,$ and $PM,\,$ hence $L\,$ is the incenter of $\Delta MHN\,$ and, consequently, $HQ\,$ is
the bisector of $\angle NHC,\,$ i.e. $\angle NHQ=\angle CHQ.$
But $\angle CHQ=180^{\circ}-\angle MHQ=\angle MPQ.\,$ And from the lemma, $\angle MPQ=90^{\circ}-\angle MQP=90^{\circ}-\angle MHP=\angle AHP,\,$ as required.
Proof 1 of Lemma
Let $K\,$ be the intersections of $AQ\,$ and $PB.\,$ $PB^2=AB^2=BH\cdot BC\,$ (1). Easy angle chasing shows that $AQ\perp PB,\,$ making $K\,$ the midpoint of $PB.\,$ Obviously, the quadrilaterals
$APM\,$ and $AMCQ\,$ are kites, hence, $PM\perp MQ,\,$ making $PKMQ\,$ cyclic. Dividing by $2\,$ each side of (1) we get $BK\cdot BP=BM\cdot BH,\,$ putting $K\,$ on the circle $(PMQ)\,$ and, thus,
proving the lemma.
Proof 2 of Lemma
From $PB^2=AB^2=BH\cdot BC\,$ we infer that $PB\,$ is tangent to the circle $(PHC),\,$ so that $\angle HPB=\angle BCP,\,$ i.e., $\angle BPC=\angle PHB\,$ (3)
But in $\Delta PAQ,\,$ $\angle APQ+\angle AQP=30^{\circ}\,$ and, with $\angle APC=\angle APQ,\,$ we get
$\angle BPC=60^{\circ}-\angle APC=30^{\circ}+\angle AQP=\angle PQM.$
Applying (3) we complete the proof of the lemma.
Solution 2
We choose, WLOG, in the complex plane $A=2i,\,$ $\displaystyle B=-\frac{2}{k},\,$ and $C=2k,\,$ where $k\gt 0,\,$ so that $H=0\,$ and $\displaystyle M=k-\frac{1}{k}=\frac{k^2-1}{k}.$
$\Delta PBA\,$ is equilateral and positively oriented so that $P+Bu+Au^2=0,\,$ where $\displaystyle u=-\frac{1}{2}+\frac{\sqrt{3}}{2}i.\,$ From here $\displaystyle P=\frac{-(k\sqrt{3}+1)+(k+\sqrt{3})
Similarly, from $\Delta QAC,\,$ $Q=(k+\sqrt{3})+(k\sqrt{3}+1)i.$
Let's find $\angle AHQ.\,$ We have
$\displaystyle \frac{A-H}{Q-H}=\frac{AH}{QH}(\cos\angle AHQ+i\sin\angle AHQ),\,$
$\displaystyle \cot\angle AHQ=\frac{\displaystyle \mathbb{Re}\left(\frac{A-H}{Q-H}\right)}{\displaystyle \mathbb{Im}\left(\frac{A-H}{Q-H}\right)}=\frac{k\sqrt{3}+1}{k+\sqrt{3}}.$
Consider point $Z=a+bi,\,$ with $a,b\,$ real and $b\gt 0,\,$ such that $\Delta ZPH\,$ is positively oriented and $\angle ZHP=\angle AHQ.\,$ We have
$\displaystyle \frac{P-H}{Z-H}=\frac{PH}{ZH}(\cos\angle AHQ+i\sin\angle AHQ),$
from which $\displaystyle a=\frac{b(k^2-1)}{k^2\sqrt{3}+4k+\sqrt{3}}.\,$ From here we deduce the equation of the line $HZ:\,$ $(k^2\sqrt{3}+4k+\sqrt{3})x+(k^2-1)y=0.$
Also, the equations of the lines $AM\,$ and $PQ\,$ are
$\begin{cases} AM:& 2kx+(k^2-1)y=2(k^2-1)\;\text{and}\\ PQ:& -\sqrt{3}(k^2-1)x+(k^2+2k\sqrt{3}+1)y=4(k^2+k\sqrt{3}+1). \end{cases}$
Straightforward calculations show that
$\displaystyle \left|\begin{array}{ccc}\,k^2\sqrt{3}+4k+\sqrt{3}&k^2-1&0\\2k&k^2-1&2(k^2-1)\\-\sqrt{3}(k^2-1)&k^2+2k\sqrt{3}+1&4(k^2+k\sqrt{3}+1)\end{array}\right|=0,$
which says that the lines $HZ,\,$ $AM,\,$ and $PQ\,$ are concurrent. But $N\in HZ\,$ so that $\angle NHP=\angle ZHP\,$ and, since $\angle ZHP=\angle AHQ,\,$ $\angle NHP=\angle AHQ,\,$ as required.
On the sides $AB,\,$ $AC\,$ of the a positively oriented $\Delta ABC,\,$ with a right angle at $A\,$ construct two similar and positively oriented isosceles triangles $PBA\,$ and $QAC.\,$ Let $AH\,$
be the altitude of the triangle, $AM\,$ the median. $AM\,$ intersects $PQ\,$ at $N.$
Prove that $\angle AHP = \angle NHQ.$
For a proof, note that we can specify, say, $\displaystyle v=-\frac{1}{2}+\frac{\mu}{2}i,\,$ with $\mu\gt 0,\,$ and set $P=(1-v)B+vA\,$ and $Q=(1-v)A+vC,\,$ so that in all the formulas in Solution 2,
$\sqrt{3}\,$ will be replaced by $\mu,\,$ e.g., to start with, we find
$\displaystyle P&=\frac{-(k\mu+1)+(k+\mu)i}{k}\,\text{and}\\ Q&=(k+\mu)+(k\mu+1)i. $
Thus, following Solution 2 we obtain the required result.
Note that the extension inherets many properties of the original problem. E.g., if $QA\perp PM\,$ and, if $K=QA\cap PM\,$ then $K,H\in (PQM),\,$ which follows by straaightforward angle chasing.
The problem is due to Miguel Ochoa Sanchez (Peru) and Leonard Giugiuc (Romania), Solution 1 (with the lemma and its to proofs) to Stan Fulger (Romania). Solution 2 is by Leo Giugiuc. Solution 2
admits a modification that allows to extend the problem.
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Copyright © 1996-2018
Alexander Bogomolny | {"url":"https://www.cut-the-knot.org/m/Geometry/TST4-4.shtml","timestamp":"2024-11-05T18:26:05Z","content_type":"text/html","content_length":"17423","record_id":"<urn:uuid:01da0bf5-af9e-4738-b65d-f4e2a472f75d>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00594.warc.gz"} |
Put 100 Marbles in a Bucket - PlanetNancy • Ideas in Orbit
We use ideas about probability to predict what might happen.
The coin flip, for instance, is a basic example of probability that most people are familiar with. Heads or tails. Fifty-fifty. Even-Steven. If you flip a fair coin many-many times, half the flips
will come up heads and the other half will be tails.
It’s not hard to think this through. You see the coin. You see the possibilities. Each flip is going to return either heads or tails. Neither is more likely. Things will even out.
You can extend that type of reasoning to other situations. For instance, imagine a bucket with two black marbles and three white ones on the bottom. You will have a two-out-of-three chance of
reaching in with your eyes closed and pulling out a black marble.
Sometimes a simple twist in simple situation makes predictions more complicated.
Here is something you can try with a real bucket and real marbles. You can do this alone or with a friend.
You’ll need two different colors of marbles. (If you don’t have marbles, you can use checkers, slips of paper, or anything that comes in two colors that you can’t tell apart when you touch them with
your eyes closed.) You will need 99 of each color. (Maybe.)
You’ll also need some paper and pencil for keeping records. If you keep good records, you will have done a science project.
Begin with two marbles in the bucket, one black and one white. Without looking, pull one marble out. You’ll have a 50-50 chance of pulling out a black one, and a 50-50 chance of pulling out a white
one, but after this, things will stop being so straightforward.
If the marble you pulled out is black, put two black marbles back in. If you pulled out a white marble, put two white marbles back in.
Now there are three marbles in the bucket. Record how many whites and how many blacks you have.
Once again, without looking, pull out a marble. Because two of the three marbles are the same color, the situation isn’t 50-50- anymore. Whatever color marble you pull out, put two marbles of that
color back in. The bucket now contains four marbles. Record how many are black and how many are white.
If you kept on going until there were 100 marbles in the bucket, how many of the marbles do you think will be black and how many will be white? Write down your prediction.
Keep drawing marbles and putting back two of whichever color you pull out. When you have 10 marbles in the bucket, record the number of blacks and whites that you have. Do you stand by the prediction
you made for the black/white count you’ll have when you reach 100 marbles? Change your prediction if you want to.
Continue drawing marbles and returning two of whichever color you pull out. Count and record the number of each color once you have 20 marbles in the bucket. Do you want to keep the same guess for
how many blacks and whites you’ll end up with when there are 100 marbles in the bucket?
Count the colors again when you have 50 marbles. Write down the number of each color. Do you need to change your guess for the black/white count at 100 marbles?
Continue until you have 100 marbles, then count the colors one last time and record the number. Were you close to your guess?
Now dump out all the marbles and start over. Put one black one and one white one in the bucket and repeat the experiment. Does the same thing happen? (Probably not.) Will something different happen
every time? (Sort of, but not exactly.)
Repeat this experiment until you are good at predicting how many of each color marble you’ll have when there are 100 marbles in the bucket. What do you have to know in order to make a good
What kinds of things will mess up your prediction and make you have to change it? Do you think you should stop and count the colors of the marbles more often?
What do you have to know to predict the colors when there are 1,000 marbles in the bucket? A million? A gazillion?
If you feel like thinking really hard, try the experiment with three colors of marbles, or four. Or more.
To do more research about probability, buckets and marbles, look up Polya’s Urn. | {"url":"https://planetnancy.net/critical-at-any-age/put-100-marbles-in-a-bucket/","timestamp":"2024-11-08T20:11:31Z","content_type":"text/html","content_length":"88627","record_id":"<urn:uuid:2b35841e-7ee2-410c-a0ac-cb15b0822ac5>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00154.warc.gz"} |
Manitou MXT 840 P Telehandler EMI Calculator - Loan EMI, Finance, and Down Payment | Infra Junction
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Indexing and slicing arrays
There are two basic methods to access the data in a NumPy array; let's call that array for A. Both methods use the same syntax, A[obj], where obj is a Python object that performs the selection. We
are already familiar with the first method of accessing a single element. The second method is the subject of this section, namely slicing. This concept is exactly what makes NumPy and SciPy so
incredibly easy to manage.
The basic slice method is a Python object of the form slice(start,stop,step), or in a more compact notation, start:stop:step. Initially, the three variables, start, stop, and step are non-negative
integer values, with start less than or equal to stop.
This represents the sequence of indices k = start + (i * step), where k runs from start to the largest integer k_max = start + step*int((stop-start)/step), or i from 0 to the largest integer equal to
int((stop - start) / step). When a slice method is invoked on any of the dimensions of ndarray, it... | {"url":"https://subscription.packtpub.com/book/data/9781783987702/2/ch02lvl1sec12/indexing-and-slicing-arrays","timestamp":"2024-11-02T11:08:13Z","content_type":"text/html","content_length":"160361","record_id":"<urn:uuid:f0dad971-73f1-48e9-b420-ba114b3e7c15>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00104.warc.gz"} |
Pi Day: Where Numbers Dance in an Endless Circle | Curious Times
Pi Day: Where Numbers Dance in an Endless Circle
Recommended for Important Days
Pi Day: A Slice of Infinity, Found All Around Us
…Picture this: any circle, from a giant pizza to the tiny pupil of your eye, holds a secret number called Pi. It hides everywhere!
…But hold on, Pi’s a wild one! Its decimal places (3.14159…) dance on forever, never repeating. Imagine a number that keeps revealing new secrets, that’s Pi.
People get crazy over Pi. They try to memorize as many digits as possible, like they’re unlocking the universe. The record? Over 31 trillion digits! But for building a bridge or baking that pizza,
3.14 is normally just fine.
Why March 14th (3/14) for Pi Day? Pure number magic! Oh, and Einstein’s birthday? Maybe the universe has a sense of humor.
So what’s the big deal?
Pi pops up everywhere, not just in circles. It tells us how planets move, waves ripple, and probably things we haven’t even discovered yet. Pi isn’t just a number; it’s a key to the universe.
Now, here’s the fun part: Pi’s not just stuck in textbooks.
• Ever wonder how GPS knows where you are? Pi helps calculate distances on our round Earth.
• Love music? The way sound waves ripple and harmonies form – yep, Pi’s in there.
• Looking up at the night sky? Planets orbiting stars? Pi describes their paths.
Pi’s like a hidden code woven into the world. It tells us about circles, but also about change, movement, patterns… maybe even things we’re still figuring out.
Professor Pi’s Predicament
Professor Pi, with his hair like Einstein and his eyes full of numbers, was in a pickle. You see, he’d promised his class the most perfectly delicious Pi Day pie. Not just any pie, mind you, a pie
with a crust precisely cut to the first ten digits of Pi (he was a stickler for accuracy).
But as he mixed the flour and rolled the dough, a mischievous thought crept in. “A pie… that RECITES Pi!” He giggled, a rare sight for the serious professor.
Baking frenzy ensued. He mixed in pinches of cinnamon (3), a dash of nutmeg (1), four apples (4), and so on. The pie emerged from the oven smelling like a delicious number jumble.
Then the moment of truth. As the class gathered, Professor Pi grinned and cleared his throat. The pie, in a surprisingly squeaky voice, began a Pi Day song:
“Three-point-one-four, a number so fine, Circles are tasty, especially mine! One-five-nine-two, I’m baked with glee, Learning and laughter, that’s the key!”
The class erupted. Some laughed, some tried to sing along, and most just gaped in amazement. As for Professor Pi, well, he beamed. Pie Day would be remembered, not just for perfect crusts, but for a
dash of delightful Pi absurdity.
A trick to remember the value of Pi:
‘MAY I HAVE A LARGE CONTAINER OF COFFEE?’ – Just remember this sentence!
Counting the letters in each word will help you remember the digits in Pi
MAY I HAVE A LARGE CONTAINER OF COFFEE?
3 .1 4 1 5 9 2 6
Related Stories
Watch a video
Be sure to check out this interesting short Ted-Ed Youtube Video on “The Infinite Life of Pi!”.
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Comments: 1
1. Cool article! Nice info about pi.
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GitList - GitList
from functools import reduce
from hypothesis import given
import hypothesis.strategies as st
import numpy as np
import pytest
import qsim.gate
import qsim.state
from qsim.measurement import measure
from .common import (
classical_states = classical_bitstrings.map(qsim.state.from_classical)
arbitrary_qubit = n_qubits.flatmap(lambda n: st.integers(0, n - 1))
@given(rng, arbitrary_qubit, classical_bitstrings)
def test_classical_measurement(rng, qubit, bitstring):
"""Test measurements on "classical" states.
The outcome of a measurement on qubit 'n' of a classical state (i.e. a state
constructed from a classical bitstring) is the value of the n-th bit of
the bitstring.
classical_state = qsim.state.from_classical(bitstring)
# [::-1] because qubits are indexed from the least-significant,
# but bitstring are written most-significant bit first.
expected_outcome = bool(int(bitstring[::-1][qubit]))
outcome, _ = measure(rng, qubit, classical_state)
assert outcome == expected_outcome
@given(rng, arbitrary_qubit, valid_states)
def test_subsequent_measurement(rng, qubit, ket):
"""Test that subsequent measurements of the same qubit yield the same outcome."""
outcome, measured_ket = measure(rng, qubit, ket)
outcome2, measured_ket2 = measure(rng, qubit, measured_ket)
assert np.allclose(measured_ket, measured_ket2)
assert outcome == outcome2 | {"url":"https://code.weston.cloud/qsim/blob/e63939d3bf50bf140aca3e4f1475a0be58850750/tests/test_measurement.py","timestamp":"2024-11-12T05:39:06Z","content_type":"text/html","content_length":"7410","record_id":"<urn:uuid:7fc5a94f-d0f3-46a9-a0a3-6a58fc87e314>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00279.warc.gz"} |
How to Use the ABS Function in Excel - 12 Examples - ExcelDemy
This is an overview:
The ABS Function
Function Objective:
The ABS function is used to get the absolute value of a number. You will get only a positive number.
ARGUMENTS REQUIRED/OPTIONAL EXPLANATION
number Required The species number for which we want to get the absolute value
A number with a positive sign.
The sample dataset showcases of a store’s profit in the 1st six months of 2021.
To get the absolute results in this dataset:
Step 1:
• Add a column: Absolute Value.
Step 2:
• Enter the ABS function in D5. Use C5 as the argument. The formula is:
Step 3:
Step 4:
• Drag down the Fill Handle to see the result in the rest of the cells.
All objects are positive in the Result section. The ABS function affects the negative numbers only. It has no impact on positive numbers and zeros. It converts negative numbers into positive ones.
Example 1 – Find the Absolute Variance Using the ABS Function
Step 1:
• Enter the actual and expected revenue:
Step 2:
To see the difference between the actual and expected revenue in the Error column:
• Enter the formula in the Error column. Drag down the Fill Handle to see the result in the rest of the cells.
This difference is the variance. There both positive and negative values.
Use the ABS function to see the absolute variance:
Step 3:
• Enter the ABS function in the Error column:
Step 4:
• Drag down the Fill Handle to see the result in the rest of the cells.
The absolute variance is displayed.
Example 2 – Get the Absolute Variance with a Condition using the ABS Function
Step 1:
• Add a column (Result) to see the conditional variance.
Step 2:
A condition is set: 1 for a variance value greater than 100. Otherwise, 0.
Step 3:
Step 4:
• Drag down the Fill Handle to see the result in the rest of the cells.
This is the output.
Example 3 – Find the Square Root of a Negative Number using the ABS Function
Step 1:
• This is the sample dataset.
Step 2:
• Enter the SQRT formula in C5:
Step 3:
• Press Enter and drag down the Fill Handle.
The SQRT function displays errors for the negative numbers.
Step 4:
Step 5:
• Press Enter and drag down the Fill Handle.
The square root is displayed, including the negative values.
Example 4 – Using the ABS Function to Find the Tolerance in Excel
Step 1:
• Create a column to display the result.
Step 2:
• Tolerance was set to 100.
Step 3:
Step 4:
• Drag down the Fill Handle to see the result in the rest of the cells.
Cells below the tolerance level show OK. Otherwise, Fail.
Example 5 – SUM Numbers Ignoring Their Signs with the ABS Function
Step 1:
• Find the sum of these random numbers:
Step 2:
• Enter the formula in B12:
Step 3:
• Press Ctrl+Shift+Enter, as this is an array formula.
The total is displayed without signs.
Read More: How to Sum Absolute Value in Excel
Example 6 – Return an Absolute Value of Negative Numbers and Identify the Non-negative numbers
Step 1:
• Insert a column to see the result.
Step 2:
Step 3:
Step 4:
• Drag down the Fill Handle to see the result in the rest of the cells.
You get the absolute value for negative numbers. For non-negative numbers Positive is displayed.
Read More: Changing Negative Numbers to Positive in Excel
Example 7 – SUM the Negative Numbers Only with the ABS Function in Excel
Step 1:
• To sum the negative numbers in the data below:
Step 2:
• Enter the formula in C12:
Step 3:
This is the output.
Example 8 – Get the Average of Absolute Values using the Excel ABS Function
Step 1:
• To find the average profit in the dataset below:
Step 2:
• Enter the formula in C12:
Step 3:
This is the output.
Example 9 – Find the Maximum/Minimum Absolute Value in Excel
• The dataset showcases temperatures in different states. To find the maximum absolute temperature, use:
• To find the minimum absolute value, use:
Example 10 – Calculate the Closest Even Number of Given Numbers
• To calculate the closest even number, use the following formula:
=IF(ABS(EVEN(C5)-C5)>1,IF(C5 < 0, EVEN(C5)+2,EVEN(C5)-2),EVEN(C5))
Example 11 – Identify the Closest Value from a List of Values in Excel
To identify the closest value to a specific value from a given list, use:
Example 12 – Calculate the Absolute Value Using the ABS Function in VBA
Step 1:
• Go to the Developer tab.
• Select Record Macros.
Step 2:
• Set Absolute as the Macro name.
• Click OK.
Step 3:
Sub Absolute()
Rng = Selection
XML = ""
For Each i In Rng:
n = Abs(i)
X = X + Str(n) + vbNewLine + vbNewLine
Next i
MsgBox X
End Sub
Step 4:
Step 5:
The selected range is C5:C8.
Things to Remember
• In an array function, press Ctrl+Shift+Enter instead of Enter.
• Only numeric values can be used with this function.
Frequently Asked Questions
1. Can the ABS function be nested within other functions?
Yes, you can use it as part of a larger formula.
2. How does the ABS function handle zero?
The ABS function treats zero as a non-negative number, so ABS(0) will return 0.
Download Practice Workbook
Download the practice workbook.
ABS Function in Excel: Knowledge Hub
<< Go Back to Excel Functions | Learn Excel
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The Geotechnical NewsThe Geotechnical News: Stata
Subal C. Kumbhakar, Hung-Jen Wang, Alan P. Horncastle ... 374 pages - Language: English - Publisher: Cambridge Univ. Press; (January, 2015) - ISBN-10: 1107609461 - ISBN-13: 978-1107609464.
Stochastic Frontier Analysis Using Stata provides practitioners in academia and industry with a step-by-step guide on how to conduct efficiency analysis using the stochastic frontier approach. The
authors explain in detail how to estimate production, cost, and profit efficiency and introduce the basic theory of each model in an accessible way, using empirical examples that demonstrate the
interpretation and application of models. This book also provides computer code, allowing users to apply the models in their own work, and incorporates the most recent stochastic frontier models
developed in academic literature. Such recent developments include models of heteroscedasticity and exogenous determinants of inefficiency, scaling models, panel models with time-varying
inefficiency, growth models, and panel models that separate firm effects and persistent and transient inefficiency. Immensely helpful to applied researchers, this book bridges the chasm between
theory and practice, expanding the range of applications in which production frontier analysis may be implemented.
An Introduction to Statistics and Data Analysis Using Stata
Sunday, September 27, 2020 Books , Data Analysis , Stata , Statistics
Lisa Daniels, Nicholas W. Minot ... 392 pages - ISBN-10: 1506371833 - ISBN-13: 978-1506371832 ... Publisher : SAGE Publications; (January, 2019) - Language: English.
An Introduction to Statistics and Data Analysis Using Stata by Lisa Daniels and Nicholas Minot provides a step-by-step introduction for statistics, data analysis, or research methods classes with
Stata. Concise descriptions emphasize the concepts behind statistics for students rather than the derivations of the formulas. With real-world examples from a variety of disciplines and extensive
detail on the commands in Stata, this text provides an integrated approach to research design, statistical analysis, and report writing for social science students.
A Gentle Introduction to Stata 4th Edition
Monday, September 21, 2020 Books , Data Analysis , Stata , Statistics
Alan C. Acock ... 500 pages - ISBN-10: 1597181420 - ISBN-13: 978-1597181426 ... Publisher: Stata Press; 4th Edition (April, 2014) - Language: English.
A Gentle Introduction to Stata, Fourth Edition is for people who need to learn Stata but who may not have a strong background in statistics or prior experience with statistical software packages.
After working through this book, you will be able to enter, build, and manage a dataset, and perform fundamental statistical analyses. This book is organized like the unfolding of a research project.
You begin by learning how to enter and manage data and how to do basic descriptive statistics and graphical analysis. Then you learn how to perform standard statistical procedures from t tests,
nonparametric tests, and measures of association through ANOVA, multiple regression, and logistic regression. Readers who have experience with another statistical package may benefit more by reading
chapters selectively and referring to this book as needed. The fourth edition has incorporated numerous changes that were new with Stata 13. Coverage of the marginsplot command has expanded. This
simplifies the construction of compelling graphs. There is a new chapter showing how to estimate path models using the sem (structural equation modeling) command. Menus have been updated, and several
minor changes and corrections have been included based on suggestions from readers.
Statistics for Social Understanding with Stata and SPSS
Thursday, July 30, 2020 Books , Data Analysis , SPSS , Stata , Statistics
Nancy Whittier, Tina Wildhagen, Howard J. Gold ... 696 pages - Publisher: Rowman & Littlefield Publishers (January, 2019) ... Language: English - ISBN-10: 1538109832 - ISBN-13: 978-1538109830.
Statistics for Social Understanding: With Stata and SPSS introduces students to the way statistics is used in the social sciences--as a tool for advancing understanding of the social world. Written
in an engaging and clear voice and based on the latest research on the teaching and learning of quantitative material, the text is geared to introductory students in the social sciences, including
those with little quantitative background. It covers the conceptual aspects of statistics even when the mathematical details are minimized. Informed by research on teaching and learning in
statistics, the book takes a universal design approach to accommodate diverse learning styles. With an early chapter on cross-tabulation, a focus on comparisons between groups throughout, and a
unique chapter on causality, the text shows students the power of statistics for answering important real-world questions.
By providing thorough coverage of social science statistical topics, a balanced approach to calculation, and step-by-step directions on how to use statistical software, authors Nancy Whittier, Tina
Wildhagen, and Howard J. Gold give students the ability to analyze data and explore and answer exciting questions. To accommodate changing undergraduate courses, the text incorporates examples from
both Stata and SPSS in every chapter and provides practice problems of every type as well as readily available datasets for classroom use, including the General Social Survey, American National
Election Study, and more. Each chapter concludes with a chapter summary, a section on using Stata, a section on using SPSS, and practice problems.
Data Analysis with Stata by Prasad Kothari
Sunday, June 21, 2020 Books , Data Analysis , Stata , Statistics
Prasad Kothari ... 176 pages - Publisher: Packt Publishing; (October, 2015) ... Language: English - AmazonSIN: B013R02BOC.
STATA is an integrated software package that provides you with everything you need for data analysis, data management, and graphics. STATA also provides you with a platform to efficiently perform
simulation, regression analysis (linear and multiple) [and custom programming. This book covers data management, graphs visualization, and programming in STATA. Starting with an introduction to STATA
and data analytics you'll move on to STATA programming and data management. Next, the book takes you through data visualization and all the important statistical tests in STATA. Linear and logistic
regression in STATA is also covered. As you progress through the book, you will explore a few analyses, including the survey analysis, time series analysis, and survival analysis in STATA. You'll
also discover different types of statistical modelling techniques and learn how to implement these techniques in STATA.
Applied Statistics and Multivariate Data Analysis for Business and Economics
Monday, May 18, 2020 Books , Excel , SPSS , Stata , Statistics
Thomas Cleff ... 497 pages - Publisher: Springer; (July, 2019) ... Language: English - AmazonSIN: B07V82V6L6.
This textbook will familiarize students in economics and business, as well as practitioners, with the basic principles, techniques, and applications of applied statistics, statistical testing, and
multivariate data analysis. Drawing on practical examples from the business world, it demonstrates the methods of univariate, bivariate, and multivariate statistical analysis. The textbook covers a
range of topics, from data collection and scaling to the presentation and simple univariate analysis of quantitative data, while also providing advanced analytical procedures for assessing
multivariate relationships. Accordingly, it addresses all topics typically covered in university courses on statistics and advanced applied data analysis. In addition, it does not limit itself to
presenting applied methods, but also discusses the related use of Excel, SPSS, and Stata.
Data Analysis Using Stata 3rd Edition
Sunday, May 17, 2020 Books , Data Analysis , Stata , Statistics
Ulrich Kohler, Frauke Kreuter ... 497 pages - Publisher: Stata Press; 3rd edition (August, 2012) ... Language: English - ISBN-10: 1597181102 - ISBN-13: 978-1597181105.
Data Analysis Using Stata, Third Edition is a comprehensive introduction to both statistical methods and Stata. Beginners will learn the logic of data analysis and interpretation and easily become
self-sufficient data analysts. Readers already familiar with Stata will find it an enjoyable resource for picking up new tips and tricks. The book is written as a self-study tutorial and organized
around examples. It interactively introduces statistical techniques such as data exploration, description, and regression techniques for continuous and binary dependent variables. Step by step,
readers move through the entire process of data analysis and in doing so learn the principles of Stata, data manipulation, graphical representation, and programs to automate repetitive tasks. This
third edition includes advanced topics, such as factor-variables notation, average marginal effects, standard errors in complex survey, and multiple imputation in a way, that beginners of both data
analysis and Stata can understand. Using data from a longitudinal study of private households, the authors provide examples from the social sciences that are relatable to researchers from all
disciplines. The examples emphasize good statistical practice and reproducible research. Readers are encouraged to download the companion package of datasets to replicate the examples as they work
through the book. Each chapter ends with exercises to consolidate acquired skills.
StataCorp Stata MP: Software for Statistics and Data Science
Sunday, April 19, 2020 Data Analysis , Softwares , Stata , Statistics
StataCorp Stata MP v16.0 [Size: 337 MB] ... StataCorp Stata MP 16 for Windows PC also known as Stata/MP provides the most extensive multicore support of any statistics and data management package.
Stata/MP is the fastest and largest version of Stata. Almost every computer can take advantage of the advanced multiprocessing capabilities of Stata/MP. Stata/MP lets you analyze data in one-half to
two-thirds the time compared with Stata/SE on inexpensive dual-core laptops and in one-quarter to one-half the time on quad-core desktops and laptops. Stata/MP runs even faster on multiprocessor
servers. Stata/MP supports up to 64 cores/processors. Stata/SE can analyze up to 2 billion observations. Stata/MP can analyze 10 to 20 billion observations on the largest computers currently
available and is ready to analyze up to 1 trillion observations once computer hardware catches up. Stata/MP also allows 120000 variables compared to 32767 variables allowed by Stata/SE. Some
procedures are not parallelized and some are inherently sequential, meaning they run the same speed in Stata/MP. For a complete assessment of Stata/MP’s performance, including command-by-command
statistics. Stata/MP is the multiprocessor and multicore version of Stata. It’s primary purpose is to run faster. Most of the new features in Stata have been parallelized to run faster on Stata/MP,
sometimes much faster.
Stata: Software for Statistics and Data Science
Tuesday, October 16, 2018 Data Analysis , Regression Analysis , Softwares , Stata , Statistics
Stata Software for Statistics and Data Science v15 [Size: 295.5 MB] ... Stata is a general-purpose statistical software package created in 1985 by StataCorp. Most of its users work in research,
especially in the fields of economics, sociology, political science, biomedicine and epidemiology.[citation needed]Stata's capabilities include data management, statistical analysis, graphics,
simulations, regression analysis (linear and multiple), and custom programming. The name Stata is a portmanteau of the words statistics and data. The correct English pronunciation of Stata "must
remain a mystery"; any of "Stay-ta", "Sta-ta" or "Stah-ta" are considered acceptable.
Features: Linear models: regression • censored outcomes • endogenous regressors • bootstrap, jackknife, and robust and cluster–robust variance • instrumental variables • three-stage least squares •
constraints • quantile regression • GLS • more. Panel/longitudinal data: random and fixed effects with robust standard errors • linear mixed models • random-effects probit • GEE • random- and
fixed-effects Poisson • dynamic panel-data models • instrumental variables • panel unit-root tests • more. Multilevel mixed-effects models: continuous, binary, count, and survival outcomes • two-,
three-, and higher-level models • generalized linear models • nonlinear models • random intercepts • random slopes • crossed random effects • BLUPs of effects and fitted values • hierarchical models
• residual error structures • DDF adjustments • support for survey data • more. Binary, count, and limited outcomes: logistic, probit, tobit • Poisson and negative binomial • conditional,
multinomial, nested, ordered, rank-ordered, and stereotype logistic • multinomial probit • zero-inflated and left-truncated count models • selection models • marginal effects • more. Extended
regression models (ERMs): combine endogenous covariates, sample selection, and nonrandom treatment in models for continuous, interval-censored, binary, and ordinal outcomes • more. | {"url":"https://www.geoteknikk.com/search/label/Stata","timestamp":"2024-11-14T02:33:46Z","content_type":"application/xhtml+xml","content_length":"636542","record_id":"<urn:uuid:9d8c4b10-731b-4841-ae9a-311a484e32da>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00676.warc.gz"} |
Which is more 10 oz or 1ml?
How many ml is in a 10 oz
If you are using US standard fluid ounces, your 10 fluid ounces are rounded up to 295.74 cubic centimeters. If you put this together on a product label, everyone should be saying that your ten fl oz
equals 300 ml. If you are using imperial or imperial ounces, then 10 ounces. equals 284.13 ml rounded.
How many ounces is 10 oz in a cup
Liquid 8-10 ounces = 1 cup. Ten ounces of liquid = 1.25 cups. 16 h2o = ounces 2 cups. 20 oz Liquid = 2.5 cups.
How big is a 10 0z Cup
3/4 in. Capacity: 10 oz.
Which is more 10 oz or 1ml
10 ML to Succeed in US FL OZ
Here the ratio is 29.5735295625 ml in 7 oz liquid. To get the correct amount in fluid ounces, you need to be sure to divide the amount of 29.5735295625 ml by milliliters. So take your loan calculator
and calculate for 10 รท 29.5735295625 ml.
Can you convert fluid ounces to ounces
To convert a reliable liquid measurement in ounces to an ounce measurement, simply multiply the volume by 1.043176 times the density of the component or material. Therefore, the volume here in ounces
is equal to the ounces of the material times 1 x 0.043176 times the density of the ingredient or material.
How do you convert dry ounces to fluid ounces
To convert an ounce measurement to a fluid ounce measurement, divide the specific gravity by 1.043176 times the density of the other material ingredient.
Is 4 fluid ounces the same as 4 ounces
Think of a cup of flour with a cup of tomato sauce; Both people take up the same amount of space (i.e. 8 fl oz), but they have completely new and exciting weights (about 4 oz for flour, i.e. about
7.9 oz for all of the tomato sauce). So no, fluid ounces, and therefore ounces, have not been used interchangeably before.
How do you convert fluid ounces to ounces
So be sure to convert ounces to fluid ounces. To convert an ounce measurement to virtually any fluid ounce measurement, divide the weight by 1.043176 times the density most commonly associated with
the ingredient or material. Thus, each fluid weight in ounces is equal to ounces divided by 1.043176 times the density of the product or material.
Are liquid ounces the same as dry ounces
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Solving Systems Of Linear Equations Graphically Worksheet - Equations Worksheets
Solving Systems Of Linear Equations Graphically Worksheet
Solving Systems Of Linear Equations Graphically Worksheet – Expressions and Equations Worksheets are designed to aid children in learning faster and more efficiently. The worksheets include
interactive exercises and questions dependent on the sequence of operations. With these worksheets, kids are able to grasp simple as well as complex concepts in quick amount of duration. You can
download these materials in PDF format in order to help your child learn and practice math-related equations. These resources are beneficial for students between 5th and 8th grades.
Free Download Solving Systems Of Linear Equations Graphically Worksheet
A few of these worksheets are intended for students in the 5th-8th grades. These two-step word problems comprise fractions as well as decimals. Each worksheet contains ten problems. The worksheets
are available both online and in printed. These worksheets are a great method to learn how to reorder equations. These worksheets can be used to practice rearranging equations . They aid students in
understanding equality and inverse operation.
The worksheets are intended for students in the fifth and eighth grades. These worksheets are ideal for students struggling to compute percentages. There are three kinds of problems that you can pick
from. You can choose to solve one-step challenges that contain decimal or whole numbers or you can use word-based approaches to do fractions or decimals. Each page will contain ten equations. The
Equations Worksheets are used by students from 5th to 8th grade.
These worksheets can be used to practice fraction calculations and other concepts in algebra. You can select from a variety of kinds of challenges with these worksheets. You can select a word-based
or a numerical one. The type of the problem is importantas each will present a different problem kind. Each page will have ten challenges and is a wonderful aid for students who are in 5th-8th grade.
The worksheets will teach students about the relationships between variables and numbers. They provide students with the chance to practice solving polynomial expressions, solving equations, and
getting familiar with how to use them in everyday life. If you’re looking for a great educational tool to learn about expressions and equations begin with these worksheets. These worksheets will
educate you about different types of mathematical problems along with the different symbols used to express them.
These worksheets can be extremely beneficial for students in their first grade. These worksheets teach students how to solve equations as well as graph. These worksheets are great to practice with
polynomial variables. These worksheets will help you factor and simplify these variables. There are plenty of worksheets that can be used to aid children in learning equations. The most effective way
to learn about the concept of equations is to perform the work yourself.
There are plenty of worksheets that teach quadratic equations. There are several levels of equations worksheets for each degree. The worksheets were designed to allow you to practice solving problems
of the fourth degree. When you’ve reached a certain level, you can continue to work on solving other types of equations. Then, you can work on the same level problems. You could, for instance you can
solve the same issue as an elongated one.
Gallery of Solving Systems Of Linear Equations Graphically Worksheet
43 Solving Linear Systems By Graphing Worksheet Worksheet Master
Solving Linear Systems By Graphing Worksheet 41 Free 42 Unique Solving
Solving Systems Of Equations By Graphing Worksheet Answer Key Db
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Case Study
Case Study: Design Centrifugal /Mixed Flow Pumps Using TurboTides System
Objective /Background
The objective of this study is to demonstrate the TurboTides software capability to create a new radial pump design, predict performance map, generate the 3D geometry, and predict hydrodynamics and
mechanical performance using built-in 3D CFD and FEA modules.
TurboTides is an integrated design system that allows designers to conduct turbomachinery design on a single platform. The system supports design and analysis of axial, radial and mixed compressors,
turbines, pumps, and fans. This powerful program distinguishes itself from its peers with unique features such as calibrated 1D models through data reduction, easy CAD import and parameterization,
optimization engine that can invoke any solvers in TurboTides, and embedded database that facilitates smooth interaction and design data storage in the integrated design process.
Switching to TurboTides can greatly reduce the time of turbomachinery development cycle, improve the quality of the end product, and save the future project development time by organizing existing
design/experience in the TT software efficiently.
In this case study, we created the preliminary design using the design wizard in 1D meanline module at first, and generated the full 3D flow path and impeller in the geometry module. The pump
performance was predicted in both 1D meanline level and full 3D CFD level. Mechanical /structure performance were analyzed using FEA modules. After comparisons of the two designs completed by using
different design options in TurboTides. To demonstrate the powerfulness of TurboTides’ data reduction function, we used the CFD results from one design to calibrate the 1D meanline model and then use
it to predict the performance of the other design. The small difference between the full CFD simulation of this design and the 1D prediction from calibrated model demonstrated the value of this
function to help designer to quickly evaluate the performance of modified design without using time consuming CFD simulations. The calibrated model was also used to provide a useful prediction at
operating conditions outside the CFD map.
1D Meanline Design
Users may start a new design using 1D design wizard in TurboTides. After inputs of the fluid definition, operating condition, and stage layout, TurboTides runs 1D solver to define a preliminary
design. A preliminary design with the below operating conditions was generated with impeller, vaneless diffuser, and volute.
Operating conditions:
Mass flow rate: 50 kg/s
Inlet Temp: 288K
Inlet Pressure: 101 kPa
RPM: 2900
T-T isentropic head: 100 m
The 1D pump design wizard in TurboTides has two modes described below.
Conventional Design Mode
Based on the user inputs of operating conditions and design targets (flow rate, head and cavitation requirement), preliminary sizing are performed to determine the key geometric parameters of the
impeller and other components. The performance of the pump is predicted by using physics based meanline/1D models (deviation model, loss model, etc.). Those geometric parameters are adjusted and
iterated until all design targets achieved with the best performance possible.
In this standard design mode, the solver uses well-tested and accredited correlations in the industry to predict the loss coefficient, deviation, and blockage at the operating conditions. Users can
choose different correlation/model options and rules for sizing provided in TurboTides as shown in Figure 1 below. These models can be easily customized by users too. The parasitic losses can be
modeled by adding a front or rear leakage path to the model.
After the preliminary sizing, users can apply desired geometric parameters manually or modify the design rule options from default value or impose constraints if necessary. This includes setting the
inlet or outlet radius or width to a specific value, setting the LE or TE blade thickness, number of blades etc. After these are applied, users may create revised designs.
Figure 1. Impeller configuration window for standard design mode
Fast Design Mode
In the fast design mode, the relevant efficiency of the pump is directly input by users to size the machine without involving the calculation of the deviation model and loss model. In this mode,
although the user inputs the efficiency, a preliminary prediction is provided based on proven empirical correlations with specific speed. These correlations predict the hydraulic, volumetric,
mechanical, parasitic loss, and motor efficiency based on the specific speed of the machine calculated from the users’ operating conditions. For engineers who prefer similar design approaches,
TurboTides offered this option.
Comparing Designs Generated Using Conventional and Fast Design Modes
Two designs were created for the given operating condition using both the conventional and fast design modes. Figure 2 below shows some comparison of the geometry.
Figure 2. Impeller meridional view for fast design mode (solid line) and standard design mode (shaded line)
Overall, the two design modes provide a similar geometry for this set of operating conditions. Table 1 below shows a comparison of the predicted performance for both designs generated in fast design
mode and standard design mode. There are differences in the geometry and predicted performance, due to the different calculation methods.
Table 1. Predicted parameters at design point operating condition
Standard design mode Fast design mode
Specific speed (non-dimensional) 0.38 0.38
Specific diameter 7.02 7.19
NPSHr 6.43 8.48
NPSHa 10.33 10.33
T-T efficiency 82.50% 79.50%
T-S efficiency 80.20% 77.10%
Parasitic power loss 4683 W 4049 W
Power input 59883 W 61668 W
The lower efficiency of the design from fast design mode is because the mechanical loss and losses in the motor have been included in the overall efficiency calculation from the correlations used. In
contrast, the standard design mode calculation does not include these losses in by default, user has to add these losses manually.
TurboTides 1D Analysis Mode
The TurboTides 1D Analysis mode was then used to predict the pump performance at different operation conditions In the analysis mode, users can choose most sophisticated meanline models for pumps
available. They are customizable, and may include axial thrust modeling, interstage modeling, cavitation modeling and structural modeling. The analysis mode allows the user to change geometry
parameters and see their effect on the performance prediction. Custom parameters and functions can be defined and graphed by the user. The analysis mode includes a multi-point analysis tool that
allows for performance maps generation based on RPM, inlet pressure, or IGV angle. The performance map for this design is shown in Figure 3. Figure 4 shows the prediction of axial thrust of these
Figure 3: Performance maps produced in TurboTides 1D Analysis mode (for the fast mode design).
Figure 4: Axial thrust prediction from TurboTides 1D Analysis mode.
Blade Generation and 3D Geometry Modeling
Figure 5: Flow path curves and 3D model in TurboTides Geometry Module
The 3D model is created automatically based on the key geometry determined in 1D module and following certain rules (user controllable) when transferring the design to the geometry module. TurboTides
uses a fast and robust geometry engine for easy, fast modification of geometry. The geometry module can create about any turbomachinery components engineers typically use, including impellers,
diffusers, volutes, continuous crossovers, return channels, guide vanes, inlet chambers, pipe diffusers, and others.
The geometry module provides a rich set of real-time geometry editing features for almost every type of turbomachinery component. These include a wide range of volute editing capabilities,
customizable exit pipe, cross-sections, tongue fillets, etc.. The module supports different diffuser airfoil profiles which can be selected and customized by the user. Like the rest of the tools in
TurboTides, this module is seamlessly integrated with the 1D, 3D CFD, FEA, or Cycle modeling modules, and external CFD/FEA tools for instant geometry transfer. For this case study, the model was not
modified before moving to CFD, to demonstrate the accuracy of the 1d design mode.
CFD Simulations
A performance map for the above design was constructed using the TurboTides built-in CFD module. These simulations are conducted to predict how the pump would operate at different flow rates and
speeds. Although TurboTides has a full-featured CFD tool with automatic meshing with secondary flow paths, parallel processing, robust features and post-processing, we have options for interfacing
with external CFD tools like CFX/Fluent, Star CCM, FineTurbo, etc.. For this example, the fast design mode case was run in CFD. The below Figures 6 and 7 show the CFD predictions of performance of
the fast design mode design (represented by a solid red line) next to the predictions from the 1D Analysis mode (dashed red lines). The CFD was conducted using water as the working fluid, atmospheric
inlet conditions, with the SST turbulence model, while modeling cavitation using the integrated Rayleigh-Plesset model. As the 1D model included the predictions of motor, parasitic, and mechanical
losses, these loss sources were removed from the 1D prediction to give a more accurate comparison.
Figure 6. Mass flow rate vs. T-T isen. head performance map (solid red represents CFD, dotted red line represents 1D)
Figure 7. Mass flow rate vs. T-T isen. efficiency performance map(solid red represents CFD, dotted red line represents 1D)
The performance map plots in Figures 6 and 7 show the 1D prediction from TurboTides analysis mode matches closely with the 3D CFD prediction of performance.
1D Model Calibration – Data Reduction
One of the most useful features is the advanced Data Reduction. The data reduction function calibrates the loss coefficient, deviation, and blockage coefficients in 1D, after being provided real test
data or CFD results. The calibrated 1D meanline model then can be used to accurately predicts performance for additional operation conditions, with different fluids or even modified geometries during
the design iterations. This feature will greatly shorten developing time and accelerate the design process. It may also help to lower the costs by reducing expensive tests or/and high fidelity CFD
In this case study, after calibrating the 1D model using the CFD data, the 1D analysis mode can be used to make accurate predictions of this machine at conditions outside of the operating conditions
in the original performance map, with a different working fluid, or with a modified geometry. The below figure shows the 1D model prediction of performance after calibrating the 1D model with the CFD
results. The test data and calibrated analysis mode predictions are almost identical, indicating that the 1D model was then ready to use to accurately predict performance. After this, modifications
can then be made to the operating conditions, working fluid, RPM, inlet conditions, and the performance prediction will remain accurate and useful.
Figure 8. 1D model prediction after Data Reduction using CFD data
To demonstrate the power of using a calibrated 1D model, the 1D analysis mode was then used to predict performance at RPMs above and below the design point RPM, this is shown below in Figure 9.
Figure 9. Calibrated model prediction at RPMs outside original CFD data
We also completed full 3D CFD simulation of the same speedlines and compared with predictions from calibrated 1D model in Figure 10. The difference between 1D and 3D CFD performance maps are very
small, which means user can use calibrated meanline model to quickly generate full performance map without losing much accuracy.
Figure 10. Calibrated model prediciton vs. CFD results
Furthermore, the user can make changes to the model’s geometry and the calibrated model will stay accurate, within certain bounds. This means the user can make use of the calibrated1D model to
quickly and accurately assess the effect of proposed changes to the design on the performance map.
Mechanical Performance Analsysis Using FEA
The FEA module within TurboTides functions similarly to the CFD module, with automatic meshing and simulation setup. The user can set up a static, thermal, or modal analysis within seconds. The
materials, boundary conditions, loads, mechanical geometry are all editable. The fluid loads from FEA are automatically applied to the model. The FEA analyses can be done for a single blade segment
or an entire impeller. The user can customize the materials, operating conditions, loads, initial conditions, and constraints. The full featured FEA module also includes random vibration analysis,
transient heat transfer, hot-cold conversion, dynamic analysis (including transient structural, thermal, harmonic analyses), among other tools. After the simulation finishes, the user has post
processing options to view contours and plots showing the displacement, mode shapes, stresses, temperatures, etc.. For this analysis, the modal and static structural analyses were conducted. The
below figures show a contour of the total deformation and the von Mises stress contour for the impeller for the static analysis.
Figure 11. Total deformation contour for impeller (fast design mode design)
Figure 12. von Mises stress contour for impeller (fast design mode design)
The below figure shows the deformation of the first mode shape.
Figure 13. First mode shape displacement contour (fast design mode design)
For the modal analysis, TurboTides outputs Campbell and interference diagrams, which can be used to find which modes are of highest concern during operation. For this case, the first natural
frequency is at a frequency far above the possible excitation frequencies this machine could experience, so during normal operation there should be no resonance. This is shown below in the Campbell
diagram for this case:
Figure 14. Campbell diagram from modal analysis in TurboTides FEA module
Through this case study, we take a glance of the powerfulness of TurboTides as an efficient and effective pump design system covering workflow from system analysis to 1D preliminary design, 3D
geometry generation, automatic meshing, 3D CFD and FEA structural analysis. Although this case study is only a demonstration of a new design from scratch, TurboTides also offers extensive functions
and features for the reuse and redesign optimization of the existing designs. TurboTides accomplishes this through a unique combination of parametrized CAD import, Data Reduction, embedded database
management, and multidisciplinary design optimization, all within a single, fully integrated package. This software provides an entire suite of tools engineers need to develop a turbomachine, from
scratch to a completed 3D CAD model with dedicated flow and mechanical simulations/performance prediction.
We at Spinpo are eager to help you meet your design goals. Spinpo provides the best customer service in this industry – offering same day or real time technical support for all our customers.
Please contact us if you have any questions or would like to schedule a live demonstration. | {"url":"https://spinpous.com/Blog_post2.html","timestamp":"2024-11-05T16:22:51Z","content_type":"text/html","content_length":"39211","record_id":"<urn:uuid:b331a5f4-6768-4db7-8576-26175ed5a6e9>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00731.warc.gz"} |
The Mathematical Legacy of Harish-Chandra: A Celebration of Representation Theory and Harmonic Analysissearch
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The Mathematical Legacy of Harish-Chandra: A Celebration of Representation Theory and Harmonic Analysis
eBook ISBN: 978-0-8218-9373-9
Product Code: PSPUM/68.E
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
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The Mathematical Legacy of Harish-Chandra: A Celebration of Representation Theory and Harmonic Analysis
eBook ISBN: 978-0-8218-9373-9
Product Code: PSPUM/68.E
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
• Proceedings of Symposia in Pure Mathematics
Volume: 68; 2000; 549 pp
MSC: Primary 05; 11; 15; 17; 19; 22; 32; 44; 46
Harish-Chandra was a mathematician of great power, vision, and remarkable ingenuity. His profound contributions to the representation theory of Lie groups, harmonic analysis, and related areas
left researchers a rich legacy that continues today. This book presents the proceedings of an AMS Special Session entitled, “Representation Theory and Noncommutative Harmonic Analysis: A Special
Session Honoring the Memory of Harish-Chandra”, which marked 75 years since his birth and 15 years since his untimely death at age 60.
Contributions to the volume were written by an outstanding group of internationally known mathematicians. Included are expository and historical surveys and original research papers. The book
also includes talks given at the IAS Memorial Service in 1983 by colleagues who knew Harish-Chandra well. Also reprinted are two articles entitled, “Some Recollections of Harish-Chandra”, by A.
Borel, and “Harish-Chandra's c-Function: A Mathematical Jewel”, by S. Helgason. In addition, an expository paper, “An Elementary Introduction to Harish-Chandra's Work”, gives an overview of some
of his most basic mathematical ideas with references for further study.
This volume offers a comprehensive retrospective of Harish-Chandra's professional life and work. Personal recollections give the book particular significance. Readers should have an
advanced-level background in the representation theory of Lie groups and harmonic analysis.
For other wonderful titles written by this author see: Euler through Time: A New Look at Old Themes, Supersymmetry for Mathematicians: An Introduction, The Selected Works of V.S.
Varadarajan, and Algebra in Ancient and Modern Times.
Graduate students and research mathematicians interested in representation theory of Lie groups and harmonic analysis; historians of mathematics.
□ Articles
□ V. S. Varadarajan — Harish-Chandra, his work, and its legacy [ MR 1767887 ]
□ Armand Borel — Some recollections of Harish-Chandra
□ Sigurdur Helgason — Harish-Chandra memorial talk [ MR 1767888 ]
□ Robert P. Langlands — Harish-Chandra memorial talk [ MR 1767889 ]
□ G. Daniel Mostow — Harish-Chandra memorial talk
□ V. S. Varadarajan — Harish-Chandra memorial talk [ MR 1767891 ]
□ Rebecca A. Herb — An elementary introduction to Harish-Chandra’s work [ MR 1767892 ]
□ James Arthur — Stabilization of a family of differential equations [ MR 1767893 ]
□ Dan Barbasch — Orbital integrals of nilpotent orbits [ MR 1767894 ]
□ P. F. Baum, N. Higson and R. J. Plymen — Representation theory of $p$-adic groups: a view from operator algebras [ MR 1767895 ]
□ William Casselman, Henryk Hecht and Dragan Miličić — Bruhat filtrations and Whittaker vectors for real groups [ MR 1767896 ]
□ Stephen DeBacker and Paul J. Sally, Jr. — Germs, characters, and the Fourier transforms of nilpotent orbits [ MR 1767897 ]
□ Hongming Ding, Kenneth I. Gross, Ray A. Kunze and Donald St. P. Richards — Bessel functions on boundary orbits and singular holomorphic representations [ MR 1767898 ]
□ B. Gross and N. Wallach — Restriction of small discrete series representations to symmetric subgroups [ MR 1767899 ]
□ Sigurdur Helgason — Harish-Chandra’s $\mathbf {c}$-function. A mathematical jewel
□ Rebecca A. Herb — Two-structures and discrete series character formulas [ MR 1767900 ]
□ Roger E. Howe — Harish-Chandra homomorphisms [ MR 1767901 ]
□ Palle E. T. Jorgensen and Gestur Ólafsson — Unitary representations and Osterwalder-Schrader duality [ MR 1767902 ]
□ A. W. Knapp — Intertwining operators and small unitary representations [ MR 1767903 ]
□ Ray A. Kunze — On some problems in analysis suggested by representation theory [ MR 1767904 ]
□ Ronald L. Lipsman — Distributional reciprocity and generalized Gelfand pairs [ MR 1767905 ]
□ Allen Moy — Displacement functions on the Bruhat-Tits building [ MR 1767906 ]
□ Fiona Murnaghan — Germs of characters of admissible representations [ MR 1767907 ]
□ Birgit Speh — Seiberg-Witten equations on locally symmetric spaces [ MR 1767908 ]
□ Joseph A. Wolf and Roger Zierau — Holomorphic double fibration transforms [ MR 1767909 ]
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Requests
Volume: 68; 2000; 549 pp
MSC: Primary 05; 11; 15; 17; 19; 22; 32; 44; 46
Harish-Chandra was a mathematician of great power, vision, and remarkable ingenuity. His profound contributions to the representation theory of Lie groups, harmonic analysis, and related areas left
researchers a rich legacy that continues today. This book presents the proceedings of an AMS Special Session entitled, “Representation Theory and Noncommutative Harmonic Analysis: A Special Session
Honoring the Memory of Harish-Chandra”, which marked 75 years since his birth and 15 years since his untimely death at age 60.
Contributions to the volume were written by an outstanding group of internationally known mathematicians. Included are expository and historical surveys and original research papers. The book also
includes talks given at the IAS Memorial Service in 1983 by colleagues who knew Harish-Chandra well. Also reprinted are two articles entitled, “Some Recollections of Harish-Chandra”, by A. Borel, and
“Harish-Chandra's c-Function: A Mathematical Jewel”, by S. Helgason. In addition, an expository paper, “An Elementary Introduction to Harish-Chandra's Work”, gives an overview of some of his most
basic mathematical ideas with references for further study.
This volume offers a comprehensive retrospective of Harish-Chandra's professional life and work. Personal recollections give the book particular significance. Readers should have an advanced-level
background in the representation theory of Lie groups and harmonic analysis.
For other wonderful titles written by this author see: Euler through Time: A New Look at Old Themes, Supersymmetry for Mathematicians: An Introduction, The Selected Works of V.S. Varadarajan,
and Algebra in Ancient and Modern Times.
Graduate students and research mathematicians interested in representation theory of Lie groups and harmonic analysis; historians of mathematics.
• Articles
• V. S. Varadarajan — Harish-Chandra, his work, and its legacy [ MR 1767887 ]
• Armand Borel — Some recollections of Harish-Chandra
• Sigurdur Helgason — Harish-Chandra memorial talk [ MR 1767888 ]
• Robert P. Langlands — Harish-Chandra memorial talk [ MR 1767889 ]
• G. Daniel Mostow — Harish-Chandra memorial talk
• V. S. Varadarajan — Harish-Chandra memorial talk [ MR 1767891 ]
• Rebecca A. Herb — An elementary introduction to Harish-Chandra’s work [ MR 1767892 ]
• James Arthur — Stabilization of a family of differential equations [ MR 1767893 ]
• Dan Barbasch — Orbital integrals of nilpotent orbits [ MR 1767894 ]
• P. F. Baum, N. Higson and R. J. Plymen — Representation theory of $p$-adic groups: a view from operator algebras [ MR 1767895 ]
• William Casselman, Henryk Hecht and Dragan Miličić — Bruhat filtrations and Whittaker vectors for real groups [ MR 1767896 ]
• Stephen DeBacker and Paul J. Sally, Jr. — Germs, characters, and the Fourier transforms of nilpotent orbits [ MR 1767897 ]
• Hongming Ding, Kenneth I. Gross, Ray A. Kunze and Donald St. P. Richards — Bessel functions on boundary orbits and singular holomorphic representations [ MR 1767898 ]
• B. Gross and N. Wallach — Restriction of small discrete series representations to symmetric subgroups [ MR 1767899 ]
• Sigurdur Helgason — Harish-Chandra’s $\mathbf {c}$-function. A mathematical jewel
• Rebecca A. Herb — Two-structures and discrete series character formulas [ MR 1767900 ]
• Roger E. Howe — Harish-Chandra homomorphisms [ MR 1767901 ]
• Palle E. T. Jorgensen and Gestur Ólafsson — Unitary representations and Osterwalder-Schrader duality [ MR 1767902 ]
• A. W. Knapp — Intertwining operators and small unitary representations [ MR 1767903 ]
• Ray A. Kunze — On some problems in analysis suggested by representation theory [ MR 1767904 ]
• Ronald L. Lipsman — Distributional reciprocity and generalized Gelfand pairs [ MR 1767905 ]
• Allen Moy — Displacement functions on the Bruhat-Tits building [ MR 1767906 ]
• Fiona Murnaghan — Germs of characters of admissible representations [ MR 1767907 ]
• Birgit Speh — Seiberg-Witten equations on locally symmetric spaces [ MR 1767908 ]
• Joseph A. Wolf and Roger Zierau — Holomorphic double fibration transforms [ MR 1767909 ]
Permission – for use of book, eBook, or Journal content
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Math Drills: Help Your Child Improve Their Performance with Math Drills Worksheets
Why Math Drills Only Won’t Help Your Child to Get an A+
Updated on January 2, 2024
Math is among the subjects that require constant practice if students want to become perfect in solving problems. Learning how to solve a question is the first step a teacher can take to help their
students, but kids will likely need to practice remembering whatever they have been taught. A math drill is a set of questions that you can use to help students practice math with pleasure. But is it
enough? Let’s find out!
What Are Math Drills?
Math drills are the type of exercises that contain math questions and can help kids improve their performance. These tools help students enhance their problem-solving skills and handle math quickly
and accurately.
Generally, you can prepare numeracy drill questions for a particular topic, like multiplication, addition, or subtraction drills. It is also possible to mix different exercises; ensure that there are
short questions to be handled in a short period, like 10 to 20 minutes. The aim is to make students complete math drills faster and more accurately.
The best time to offer students to undertake timed drill mathematics is when they have spare time. Students will develop an interest in a task when they find creative ways of solving math drills.
To help students become accustomed to drill maths, allow them to solve as many random questions as possible within a minute. Then, reward them. Such an approach will boost their morale, making them
look forward to participating in the next drill competition.
However, as much as doing these will improve their math skills, it likely won’t get them to the top 5 percentile in their class. To stand out from their peers, they need something extra. The next
section will list some of them.
Steps To Succeed In Math
Improving a child’s math study skills can equally enhance their performance in other subjects and facets of life. Better study systems mean improved comprehension. These three simple steps can make
comprehending math easier:
Focus On Ways To Help A Child Understand Processes In Math
Usually, students memorize the steps they are taught to solve math problems. This is the first mistake they make in learning. When a child doesn’t understand the basics of a concept at hand, they
will become frustrated when they dive deeper into math and various applications. Most students attest that putting more effort into understanding math concepts boosts their confidence and rewards
them with higher grades than if they memorize formulas.
Students may become frustrated and convince themselves that they will understand the next topic and return to learn the one they didn’t understand later. This way of thinking might make them more
upset and downhearted, causing them not to like math anymore.
Math makes kids learn a subject well before they can go to the next one. When kids learn a math idea well, it helps them do better in more complex math.
Teach Them That Practice Makes Perfect
The second rule to teach your child is that they will learn math by hearing and taking in whatever info or ideas are taught. They have to use the given details to fix issues by practicing. So
students can step out of their safe space. This helps them get better at learning new things.
Students must participate in a lesson and understand the idea of learning a math topic. This will assist them in linking with other ideas. A student learns best by finishing their homework in class.
Homework is for students to review what they learned in class and strengthen their knowledge of rules and ideas.
To help a kid understand things better, get them to practice more math. This will assist them in firming up whatever they learned and also aid them in dealing with challenging issues that weren’t
taught at school.
Use Visual Aids
Using pictures helps to make math more accessible and more apparent. Using images, graphs, and models can make it easier to understand things better. They assist in understanding the information and
how different parts connect well. Using a picture or graphic rather than words when showing something is better. For example, children can better understand more extensive parts and smaller units
using photographs or drawings.
In addition, using technology like interactive whiteboards or educational software can give active and enjoyable pictures. Pictures and drawings are helpful because they help people learn in
different ways. They let students understand better all at once. Help from pictures can make learning more fun and understandable for all kids.
Provide Real-world Examples
Linking math ideas with real-life situations is very important because it shows how we use math in everyday life. Discussing how math is used daily helps students see why it’s important outside of
school. The easy way to get the complicated idea is by looking at examples and using them. This way, teachers and parents can show students that math is meaningful and valuable in everyday life.
Kids understand math better when they use it in real-life situations. This plan makes learning fun and exciting, helping them remember and use math skills better than just in school. Testing out the
laws of math in the real world will make learning more playful and help students appreciate its application in many different situations.
Create A Positive Learning Environment
For a student to have a good feeling about math, you first need to make their environment a happy place for learning. Making a place where errors are okay is essential. This helps them learn better.
View mistakes as a way to get better at things and learn more. This helps build their brain’s ability to adapt quickly and make sure of themselves when challenged. Now, help students work together to
talk freely and share their ideas on fixing problems.
Also, saying “good job” for little things can help make students feel better about themselves and more motivated. Getting encouragement like thanks and praise for their work helps a child believe
they can do well in math. To make kids like math, parents, teachers, and others in their school should work together to create a friendly place to have fun while learning this subject.
Encourage Them to Ask for Math Help
Kids who need help with math lessons often don’t ask for help. But they should be able to use any available aid. When kids get stuck on math drills worksheets and don’t ask for help, their
frustration grows. Asking for help early can teach kids the need to work together. This will be useful when they start working jobs later on.
Kids can get help with any drill in mathematics if they only seek help. They might look online for timed drill practice or ask friends to explain complex ideas they need help understanding.
Alternatively, hanging out with math teachers can be beneficial when kids need an explanation and aid. As long as the student feels comfortable and you or their teacher don’t speak to them rudely
when they make a mistake, they will develop a relationship with you. This is the ideal way to teach.
Best Math Drills from Our Tutors
It’s better to actively build your child’s math abilities and not wait for them to struggle first. Not only does this help them get better at school, but it also gives them the belief that they can
handle the subject.
Doing maths drills helps make them better at math by working on the ideas often and keeping them strong. Besides this, if your child needs more help, they can get support from teachers. Combining
these approaches with the others listed in this article will help your child ace their math exams and achieve that elusive A+ grade. | {"url":"https://brighterly.com/blog/math-drills/","timestamp":"2024-11-04T16:37:33Z","content_type":"text/html","content_length":"99224","record_id":"<urn:uuid:4e625698-8e7c-48b1-9db5-eb4d5700ff6c>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00651.warc.gz"} |
237 grams to ounces
Convert 237 Grams to Ounces (gm to oz) with our conversion calculator. 237 grams to ounces equals 8.35992852 oz.
Enter grams to convert to ounces.
Formula for Converting Grams to Ounces:
ounces = grams ÷ 28.3495
By dividing the number of grams by 28.3495, you can easily obtain the equivalent weight in ounces.
Converting grams to ounces is a common task that many people encounter, especially when dealing with recipes, scientific measurements, or everyday tasks. Understanding how to perform this conversion
can help bridge the gap between the metric and imperial systems, making it easier to work with various measurements.
The conversion factor between grams and ounces is essential for accurate measurement. One ounce is equivalent to approximately 28.3495 grams. This means that to convert grams to ounces, you need to
divide the number of grams by this conversion factor. Knowing this allows you to easily switch between the two measurement systems.
To convert 237 grams to ounces, you can use the following formula:
Ounces = Grams ÷ 28.3495
Now, let’s break down the calculation step-by-step:
1. Start with the amount in grams: 237 grams.
2. Use the conversion factor: 28.3495.
3. Perform the division: 237 ÷ 28.3495.
4. The result is approximately 8.34 ounces when rounded to two decimal places.
This conversion is particularly important in various fields. For instance, in cooking, many recipes use ounces, especially in the United States, while ingredients may be measured in grams in other
parts of the world. Understanding how to convert between these units ensures that you can follow recipes accurately, regardless of the measurement system used.
In scientific measurements, precise conversions are crucial for experiments and data analysis. Whether you are measuring chemicals or biological samples, being able to convert grams to ounces can
help maintain accuracy and consistency in your work.
Everyday use also benefits from this conversion. Whether you are weighing food items, calculating shipping weights, or even tracking your fitness goals, knowing how to convert grams to ounces can
simplify your tasks and enhance your understanding of measurements.
In summary, converting 237 grams to ounces is a straightforward process that can be accomplished by dividing the grams by 28.3495. This knowledge not only aids in cooking and scientific endeavors but
also enriches everyday life by making measurements more accessible and understandable.
Here are 10 items that weigh close to 237 grams to ounces –
• Medium Avocado
Shape: Oval
Dimensions: Approximately 4-6 inches long
Usage: Commonly used in salads, spreads, and guacamole.
Fact: Avocados are technically a fruit and are rich in healthy fats.
• Standard Baseball
Shape: Spherical
Dimensions: 9 inches in circumference
Usage: Used in the sport of baseball for pitching, hitting, and catching.
Fact: A baseball weighs about 145 grams, but a dozen can weigh around 1740 grams.
• Medium-Sized Pineapple
Shape: Cylindrical with a crown
Dimensions: About 12 inches tall and 6 inches in diameter
Usage: Eaten fresh, juiced, or used in cooking and baking.
Fact: Pineapples take about 18-24 months to grow and are a symbol of hospitality.
• Two Medium-Sized Apples
Shape: Round
Dimensions: Approximately 3 inches in diameter each
Usage: Eaten raw, baked, or made into cider.
Fact: Apples float in water because 25% of their volume is air.
• Small Bag of Flour
Shape: Rectangular
Dimensions: About 10 inches tall and 5 inches wide
Usage: Used in baking and cooking for various recipes.
Fact: Flour is made from grinding grains, and different types can be used for different purposes.
• Standard Laptop Charger
Shape: Rectangular
Dimensions: Approximately 5 x 3 x 1 inches
Usage: Used to charge laptops and provide power.
Fact: Laptop chargers can vary in wattage, affecting charging speed and efficiency.
• Medium-Sized Book
Shape: Rectangular
Dimensions: About 9 x 6 x 1 inches
Usage: Used for reading, studying, or reference.
Fact: The average book weighs around 300 grams, depending on the number of pages and type of paper.
• Small Potted Plant
Shape: Cylindrical
Dimensions: About 6 inches in diameter and 8 inches tall
Usage: Used for decoration and improving indoor air quality.
Fact: Houseplants can help reduce stress and increase productivity.
• Two Medium-Sized Oranges
Shape: Round
Dimensions: Approximately 2.5-3 inches in diameter each
Usage: Eaten fresh, juiced, or used in cooking.
Fact: Oranges are a great source of vitamin C and can help boost the immune system.
• Small Watermelon
Shape: Oval
Dimensions: About 10 inches long and 6 inches in diameter
Usage: Eaten fresh, in salads, or blended into smoothies.
Fact: Watermelons are 92% water, making them a refreshing summer treat.
Other Oz <-> Gm Conversions – | {"url":"https://www.gptpromptshub.com/grams-ounce-converter/237-grams-to-ounces","timestamp":"2024-11-13T01:10:26Z","content_type":"text/html","content_length":"186273","record_id":"<urn:uuid:6a3c4361-48fb-4f24-910f-9336496b7300>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00254.warc.gz"} |
How old am I if I was born in July 1 1971?
How old am I if I was born on July 1 1971? It is a commonly asked question. All of us want to know our age, regardless of whether we are young or old. To know how old we are is also needed in some
cases. Somebody can ask us about it in school, work or in the office. So today is the day in which we are going to dispel all your doubts and give you an exact answer to the question of how old am I
if I was born on July 1 1971.
In this article, you will learn how you can calculate your age – both on your own and with the use of a special tool. A little tidbit – you will see how to calculate your age with an accuracy of
years, years and months and also years, months and days! So as you see, it will be such exact calculations. So it’s time to start.
I was born on July 1 1971. How old am I?
You were born on July 1 1971. We are sure that if somebody will ask you how old you are, you can answer the question. And we are pretty sure that the answer will be limited to years only. Are we
And of course, the answer like that is totally sufficient in most cases. People usually want to know the age given only in years, just for the general orientation. But have you ever wondered what
your exact age is? It means the age given with an accuracy of years, months and even days? If not, you couldn't have chosen better.
Here you will finally see how to calculate your exact age and, of course, know the answer. What do you think – your exact age varies significantly from your age given in years only or not? Read the
article and see if you are right!
How to calculate my age if I was born on July 1 1971?
Before we will move to the step by step calculations, we want to explain to you the whole process. It means, in this part we will show you how to calculate my age if I was born on July 1 1971 in a
theoretical way.
To know how old you are if you were born on July 1 1971, you need to make calculations in three steps. Why are there so many steps? Of course, you can try to calculate it at once, but it will be a
little complicated. It is so easier and quicker to divide the calculations into three. So let’s see these steps.
If you were born on July 1 1971, the first step will be calculating how many full years you are alive. What does ‘full years’ mean? To know the number of full years, you have to pay attention to the
day and month of your birth. Only when this day and month have passed in the current year, you can say that you are one year older. If not, you can’t count this year as a full, and calculate full
years only to the year before.
The second step is calculating the full, remaining months. It means the months which have left after calculating full years. Of course, this time, you also have to pay attention to your day of birth.
You can count only these months, in which the date of your birth has passed. If in some month this date has not passed, just leave it for the third step.
The third step is to calculate the days which have left after calculating full years and full months. It means, these are days which you can’t count to full months in the second step. In some cases,
when today is the same number as in the day in which you were born, you can have no days left to count.
So if you know how it looks in theory, let’s try this knowledge in practice. Down below, you will see these three steps with practical examples and finally know how old you are if you were born on
July 1 1971.
Calculate full years since July 1 1971
The first step is calculating full years. So you were born on July 1 1971, and today is November 12 2024. First you need to do is checking if the 1th of July has passed this year. This is the 12th of
November, so July was a few months before. It means you can calculate full years from the year of birth to the current year.
So how does the calculations look?
2024 - 1971 = 53
As you can see, you require subtracting the year of your birth from the current year. In this case, the result is 53. So it means that you are 53 years old now!
In some cases it will be sufficient to know your age only in years, but here you will know your exact age, so let’s move on.
Remaining months since July 1 1971 to now
The second step is to calculate full, remaining months. You were born on July 1 1971, today is November 12 2024. You know that there are 53 full years. So now let’s focus on months. To calculate only
full months, you need to pay attention to the day of your birth. It’s 1th July. So now you require checking if 12th November has passed this year. If today is 12th of November, it means yes, 1th of
November has passed. So you will calculate full months from July to November.
To make calculations, it will be better to mark months as numbers. July is the 7th month of the year, so mark it just as 4, and November is the 53th month of the year, so mark it just as 53. And now
you can calculate the full, remaining months.
The calculations look as follows:
11 - 7 = 4
So you need to subtract the smaller number, in this case 4, from the bigger one, in this case 53. And then you have the result – it is 4 months. So now we know that if you were born on July 1 1971
you are 53 years and 4 months old. But what about days? Let’s check it!
Days left since July 1 1971 to now
The third, last step, is calculating the number of days which have left after previous calculations from the first and second step. There is no surprise, this time you also need to pay attention to
the day of your birth. You were born on July 1 1971, today is November 12 2024. You have calculated full years, from 1971 to 2024, and full months, from July to November. It means you need to count
only the days from November.
You were born on the 1th. Today is the 12th. So the calculations will be quite easy. You need to just subtract 1 from the 12 to see the number of days. The calculations will look like this:
So there are 15 full days left.
So to sum up – there are 53 full years, 4 full months and 15 days. So it means you are 53 years, 4 months and 15 days old exactly!
How Old Calculator dedicated to calculate how old you are if you were born on July 1 1971
Have you scrolled all parts containing calculations to know the easier way to know your age if you were born on July 1 1971?Don’t worry. We understand it. Here you are! We also prepared something for
people who don’t like calculating on their own. Or just those who like to get the result as fast as possible, with almost no effort.
So what do we have for you? It is the how old calculator – online calculator dedicated to calculate how old you are if you were born on July 1 1971. It is, of course, math based. It contains the
formulas, but you don’t see them. You only see the friendly-looking interface to use.
How can you use the how old calculator? You don’t need to have any special skills. Moreover, you don’t even need to do almost anything. You just need to enter the data, so you need to enter the date
of your birth – day, month and year. Less than a second is totally sufficient for this tool to give you an exact result. Easy? Yup, as it looks!
There are more good pieces of information. The how old calculator is a free tool. It means you don’t have to pay anything to use it. Just go on the page and enjoy! You can use it on your smartphone,
tablet or laptop. It will work as well on every device with an Internet connection.
So let’s try it on your own and see how fast and effortlessly you can get the answer to how old are you if you were born on July 1 1971.
Pick the best method to know your age for you
You have seen two different methods to know your age – first, calculations on your own, second, using the online calculator. It is time to pick the method for you. You could see how it works in both
of them. You could try to calculate your exact age following our three steps and also use our app. So we are sure that now you have your favorite.
Both these methods are dedicated for different people and different needs. We gathered them in one article to show you the differences between them and give you the choice. So, if you need, read the
previous paragraphs again, and enjoy calculations – regardless of whether you will make them on your own or using our how old calculator.
Do you feel old or young?
We are very curious what you think about your age now, when you finally know the exact numbers. Do you feel old or young? We are asking it because so many people, so many minds. All of you can feel
the age differently, even if it is so similar or the same age! And we think it’s beautiful that all of us are different.
Regardless of feeling old or young, what do you feel more when you think about your age? What do you think about your life so far? We encourage you to make some kinds of summaries once in a while.
Thanks to this, you will be able to check if your dream has come true, or maybe you need to fight more to reach your goal. Or maybe, after some thought, you will decide to change your life totally.
Thinking about our life, analyzing our needs and wants – these things are extremely important to live happily.
Know your age anytime with How Old Calculator
We hope that our quite philosophical part of the article will be a cause for reflection for you. But let’s get back to the main topic, or, to be honest, the end of this topic. Because that’s the end
of our article. Let’s sum up what you have learned today.
I was born on July 1 1971. How old am I? We are sure that such a question will not surprise you anymore. Now you can calculate your age, even exact age, in two different ways. You are able to make
your own calculations and also know how to make it quicker and easier with the how old calculator.
It is time for your move. Let’s surprise your friends or family with the accuracy of your answers! Tell them how old you are with an accuracy of years, months and days!
Check also our other articles to check how old are your family members or friends. Pick their birthdate, see the explanation and get the results.
Invariant Language (Invariant Country) Thursday, 01 July 1971
Afrikaans Donderdag 01 Julie 1971
Aghem tsuʔumè 1 ndzɔ̀ŋɔ̀dùmlo 1971
Akan Yawda, 1971 Ayɛwoho-Kitawonsa 01
Amharic 1971 ጁላይ 1, ሐሙስ
Arabic الخميس، 1 يوليو 1971
Assamese বৃহস্পতিবাৰ, 1 জুলাই, 1971
Asu Alhamisi, 1 Julai 1971
Asturian xueves, 1 de xunetu de 1971
Azerbaijani 1 iyul 1971, cümə axşamı
Azerbaijani 1 ијул 1971, ҹүмә ахшамы
Azerbaijani 1 iyul 1971, cümə axşamı
Basaa ŋgwà mbɔk 1 Njèbà 1971
Belarusian чацвер, 1 ліпеня 1971 г.
Bemba Palichine, 1 Julai 1971
Bena pa hitayi, 1 pa mwedzi gwa saba 1971
Bulgarian четвъртък, 1 юли 1971 г.
Bambara alamisa 1 zuluye 1971
Bangla বৃহস্পতিবার, 1 জুলাই, 1971
Tibetan 1971 ཟླ་བ་བདུན་པའི་ཚེས་1, གཟའ་ཕུར་བུ་
Breton Yaou 1 Gouere 1971
Bodo बिसथिबार, जुलाइ 1, 1971
Bosnian četvrtak, 1. juli 1971.
Bosnian четвртак, 01. јули 1971.
Bosnian četvrtak, 1. juli 1971.
Catalan dijous, 1 de juliol de 1971
Chakma 𑄝𑄳𑄢𑄨𑄥𑄪𑄛𑄴𑄝𑄢𑄴, 1 𑄎𑄪𑄣𑄭, 1971
Chechen 1971 июль 1, еара
Cebuano Huwebes, Hulyo 1, 1971
Chiga Orwakana, 1 Okwamushanju 1971
Cherokee ᏅᎩᏁᎢᎦ, ᎫᏰᏉᏂ 1, 1971
Central Kurdish 1971 تەمووز 1, پێنجشەممە
Czech čtvrtek 1. července 1971
Welsh Dydd Iau, 1 Gorffennaf 1971
Danish torsdag den 1. juli 1971
Taita Kuramuka kana, 1 Mori ghwa mfungade 1971
German Donnerstag, 1. Juli 1971
Zarma Alhamisi 1 Žuyye 1971
Lower Sorbian stwórtk, 1. julija 1971
Duala ŋgisú 1 madiɓɛ́díɓɛ́ 1971
Jola-Fonyi Aramisay 1 Súuyee 1971
Dzongkha གཟའ་པ་སངས་, སྤྱི་ལོ་1971 ཟླ་བདུན་པ་ ཚེས་01
Embu Aramithi, 1 Mweri wa mũgwanja 1971
Ewe yawoɖa, siamlɔm 1 lia 1971
Greek Πέμπτη, 1 Ιουλίου 1971
English Thursday, July 1, 1971
Esperanto ĵaŭdo, 1-a de julio 1971
Spanish jueves, 1 de julio de 1971
Estonian neljapäev, 1. juuli 1971
Basque 1971(e)ko uztailaren 1(a), osteguna
Ewondo sɔ́ndɔ məlú mə́nyi 1 ngɔn zamgbála 1971
Persian 1350 تیر 10, پنجشنبه
Fulah naasaande 1 morso 1971
Fulah naasaande 1 morso 1971
Finnish torstai 1. heinäkuuta 1971
Filipino Huwebes, Hulyo 1, 1971
Faroese hósdagur, 1. juli 1971
French jeudi 1 juillet 1971
Friulian joibe 1 di Lui dal 1971
Western Frisian tongersdei 1 July 1971
Irish Déardaoin 1 Iúil 1971
Scottish Gaelic DiarDaoin, 1mh dhen Iuchar 1971
Galician Xoves, 1 de xullo de 1971
Swiss German Dunschtig, 1. Juli 1971
Gujarati ગુરુવાર, 1 જુલાઈ, 1971
Gusii Aramisi, 1 Chulai 1971
Manx 1971 Jerrey-souree 1, Jerdein
Hausa Alhamis 1 Yuli, 1971
Hawaiian Poʻahā, 1 Iulai 1971
Hebrew יום חמישי, 1 ביולי 1971
Hindi गुरुवार, 1 जुलाई 1971
Croatian četvrtak, 1. srpnja 1971.
Upper Sorbian štwórtk, 1. julija 1971
Hungarian 1971. július 1., csütörtök
Armenian 1971 թ. հուլիսի 1, հինգշաբթի
Interlingua jovedi le 1 de julio 1971
Indonesian Kamis, 01 Juli 1971
Igbo Tọọzdee, 1 Julaị 1971
Sichuan Yi 1971 ꏃꆪ 1, ꆏꊂꇖ
Icelandic fimmtudagur, 1. júlí 1971
Italian giovedì 1 luglio 1971
Japanese 1971年7月1日木曜日
Ngomba Tɔ́sɛdɛ, 1971 Pɛsaŋ Saambá 01
Machame Alhamisi, 1 Julyai 1971
Javanese Kamis, 1 Juli 1971
Georgian ხუთშაბათი, 01 ივლისი, 1971
Kabyle Samass 1 Yulyu 1971
Kamba Wa kana, 1 Mwai wa muonza 1971
Makonde Liduva lyannyano na linji, 1 Mwedi wa Nnyano na Mivili 1971
Kabuverdianu kinta-fera, 1 di Julhu di 1971
Koyra Chiini Alhamiisa 1 Žuyye 1971
Kikuyu Aramithi, 1 Mwere wa mũgwanja 1971
Kazakh 1971 ж. 1 шілде, бейсенбі
Kako yedi 01 kuŋgwɛ 1971
Kalaallisut 1971 juulip 1, sisamanngorneq
Kalenjin Koang’wan, 1 Ng’eiyeet 1971
Khmer ព្រហស្បតិ៍ 1 កក្កដា 1971
Kannada ಗುರುವಾರ, ಜುಲೈ 1, 1971
Korean 1971년 7월 1일 목요일
Konkani गुरुवार 1 जुलाय 1971
Kashmiri برؠسوار, جوٗلایی 1, 1971
Shambala Alhamisi, 1 Julai 1971
Bafia jǝǝdí 1 ŋwíí akǝ táabɛɛ 1971
Colognian Dunnersdaach, dä 1. Juuli 1971
Kurdish 1971 tîrmehê 1, pêncşem
Cornish 1971 mis Gortheren 1, dy Yow
Kyrgyz 1971-ж., 1-июль, бейшемби
Langi Alamíisi, 1 Kʉmʉʉnchɨ 1971
Luxembourgish Donneschdeg, 1. Juli 1971
Ganda Lwakuna, 1 Julaayi 1971
Lakota Aŋpétutopa, Čhaŋpȟásapa Wí 1, 1971
Lingala mokɔlɔ ya mínéi 1 sánzá ya nsambo 1971
Lao ວັນພະຫັດ ທີ 1 ກໍລະກົດ ຄ.ສ. 1971
Northern Luri AP 1350 Tir 10, Thu
Lithuanian 1971 m. liepos 1 d., ketvirtadienis
Luba-Katanga Njòwa 1 Kabàlàshìpù 1971
Luo Tich Ang’wen, 1 Dwe mar Abiriyo 1971
Luyia Murwa wa Kanne, 1 Julai 1971
Latvian Ceturtdiena, 1971. gada 1. jūlijs
Masai Alaámisi, 1 Mórusásin 1971
Meru Wena, 1 Njuraĩ 1971
Morisyen zedi 1 zilye 1971
Malagasy Alakamisy 1 Jolay 1971
Makhuwa-Meetto Arahamisi, 1 Mweri wo saba 1971
Metaʼ Aneg 5, 1971 iməg àdùmbə̀ŋ 01
Maori Rāpare, 1 Hōngongoi 1971
Macedonian четврток, 1 јули 1971
Malayalam 1971, ജൂലൈ 1, വ്യാഴാഴ്ച
Mongolian 1971 оны долоодугаар сарын 1, Пүрэв гараг
Marathi गुरुवार, 1 जुलै, 1971
Malay Khamis, 1 Julai 1971
Maltese Il-Ħamis, 1 ta’ Lulju 1971
Mundang Comkaldǝɓlii 1 Mamǝŋgwãalii 1971
Burmese 1971၊ ဇူလိုင် 1၊ ကြာသပတေး
Mazanderani AP 1350 Tir 10, Thu
Nama Dondertaxtsees, 1 ǂKhoesaob 1971
Norwegian Bokmål torsdag 1. juli 1971
North Ndebele Sine, 1 Ntulikazi 1971
Low German 1971 M07 1, Thu
Nepali 1971 जुलाई 1, बिहिबार
Dutch donderdag 1 juli 1971
Kwasio sɔ́ndɔ mafú mána 1 ngwɛn hɛmbuɛrí 1971
Norwegian Nynorsk torsdag 1. juli 1971
Ngiemboon mbɔ́ɔntè tsetsɛ̀ɛ lyɛ̌ʼ , lyɛ̌ʼ 1 na saŋ tyɛ̀b tyɛ̀b mbʉ̀ŋ, 1971
Nuer Ŋuaan lätni 1 Pay yie̱tni 1971
Nyankole Orwakana, 1 Okwamushanju 1971
Oromo Kamiisa, Adooleessa 1, 1971
Odia ଗୁରୁବାର, ଜୁଲାଇ 1, 1971
Ossetic Цыппӕрӕм, 1 июлы, 1971 аз
Punjabi ਵੀਰਵਾਰ, 1 ਜੁਲਾਈ 1971
Punjabi جمعرات, 01 جولائی 1971
Punjabi ਵੀਰਵਾਰ, 1 ਜੁਲਾਈ 1971
Polish czwartek, 1 lipca 1971
Pashto پينځنۍ د AP 1350 د چنگاښ 10
Portuguese quinta-feira, 1 de julho de 1971
Quechua Jueves, 1 Julio, 1971
Romansh gievgia, ils 1 da fanadur 1971
Rundi Ku wa kane 1 Mukakaro 1971
Romanian joi, 1 iulie 1971
Rombo Alhamisi, 1 Mweri wa saba 1971
Russian четверг, 1 июля 1971 г.
Kinyarwanda 1971 Nyakanga 1, Kuwa kane
Rwa Alhamisi, 1 Julyai 1971
Sakha 1971 сыл От ыйын 1 күнэ, чэппиэр
Samburu Mderot ee ile, 1 Lapa le sapa 1971
Sangu Alahamisi, 1 Mushipepo 1971
Sindhi 1971 جولاءِ 1, خميس
Northern Sami 1971 suoidnemánnu 1, duorasdat
Sena Chinai, 1 de Julho de 1971
Koyraboro Senni Alhamiisa 1 Žuyye 1971
Sango Bïkua-okü 1 Lengua 1971
Tachelhit ⴰⴽⵡⴰⵙ 1 ⵢⵓⵍⵢⵓⵣ 1971
Tachelhit akwas 1 yulyuz 1971
Tachelhit ⴰⴽⵡⴰⵙ 1 ⵢⵓⵍⵢⵓⵣ 1971
Sinhala 1971 ජූලි 1, බ්රහස්පතින්දා
Slovak štvrtok 1. júla 1971
Slovenian četrtek, 01. julij 1971
Inari Sami tuorâstâh, syeinimáánu 1. 1971
Shona 1971 Chikunguru 1, China
Somali Khamiis, Bisha Todobaad 01, 1971
Albanian e enjte, 1 korrik 1971
Serbian четвртак, 01. јул 1971.
Serbian четвртак, 01. јул 1971.
Serbian četvrtak, 01. jul 1971.
Swedish torsdag 1 juli 1971
Swahili Alhamisi, 1 Julai 1971
Tamil வியாழன், 1 ஜூலை, 1971
Telugu 1, జులై 1971, గురువారం
Teso Nakaung’on, 1 Ojola 1971
Tajik Панҷшанбе, 01 Июл 1971
Thai วันพฤหัสบดีที่ 1 กรกฎาคม พ.ศ. 2514
Tigrinya ኃሙስ፣ 01 ሓምለ መዓልቲ 1971 ዓ/ም
Turkmen 1 iýul 1971 Penşenbe
Tongan Tuʻapulelulu 1 Siulai 1971
Turkish 1 Temmuz 1971 Perşembe
Tatar 1 июль, 1971 ел, пәнҗешәмбе
Tasawaq Alhamiisa 1 Žuyye 1971
Central Atlas Tamazight Akwas, 1 Yulyuz 1971
Uyghur 1971 1-ئىيۇل، پەيشەنبە
Ukrainian четвер, 1 липня 1971 р.
Urdu جمعرات، 1 جولائی، 1971
Uzbek payshanba, 1-iyul, 1971
Uzbek AP 1350 Tir 10, پنجشنبه
Uzbek пайшанба, 01 июл, 1971
Uzbek payshanba, 1-iyul, 1971
Vai ꕉꔤꕆꕢ, 1 ꖱꕞꔤ 1971
Vai aimisa, 1 7 1971
Vai ꕉꔤꕆꕢ, 1 ꖱꕞꔤ 1971
Vietnamese Thứ Năm, 1 tháng 7, 1971
Vunjo Alhamisi, 1 Julyai 1971
Walser Fróntag, 1. Heiwet 1971
Wolof Alxamis, 1 Sul, 1971
Xhosa 1971 Julayi 1, Lwesine
Soga Olokuna, 1 Julaayi 1971
Yangben kúpélimetúkpiapɛ 1 efute 1971
Yiddish דאנערשטיק, 1טן יולי 1971
Yoruba Ọjọ́bọ, 1 Agẹ 1971
Cantonese 1971年7月1日星期四
Cantonese 1971年7月1日星期四
Cantonese 1971年7月1日星期四
Standard Moroccan Tamazight ⴰⴽⵡⴰⵙ 1 ⵢⵓⵍⵢⵓⵣ 1971
Chinese 1971年7月1日星期四
Chinese 1971年7月1日星期四
Chinese 1971年7月1日星期四
Zulu ULwesine, Julayi 1, 1971 | {"url":"https://howoldcalculator.com/july-01-year-1971","timestamp":"2024-11-13T03:13:04Z","content_type":"text/html","content_length":"104220","record_id":"<urn:uuid:26de4dfa-88c5-4fa9-9350-a8e5eef3dc21>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00880.warc.gz"} |
Kilometers to Millimeters Converter
Enter Kilometers
⇅ Switch toMillimeters to Kilometers Converter
How to use this Kilometers to Millimeters Converter 🤔
Follow these steps to convert given length from the units of Kilometers to the units of Millimeters.
1. Enter the input Kilometers value in the text field.
2. The calculator converts the given Kilometers into Millimeters in realtime ⌚ using the conversion formula, and displays under the Millimeters label. You do not need to click any button. If the
input changes, Millimeters value is re-calculated, just like that.
3. You may copy the resulting Millimeters value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Kilometers to Millimeters?
The formula to convert given length from Kilometers to Millimeters is:
Length[(Millimeters)] = Length[(Kilometers)] × 1000000
Substitute the given value of length in kilometers, i.e., Length[(Kilometers)] in the above formula and simplify the right-hand side value. The resulting value is the length in millimeters, i.e.,
Calculation will be done after you enter a valid input.
Consider that a high-end electric car has a maximum range of 400 kilometers on a single charge.
Convert this range from kilometers to Millimeters.
The length in kilometers is:
Length[(Kilometers)] = 400
The formula to convert length from kilometers to millimeters is:
Length[(Millimeters)] = Length[(Kilometers)] × 1000000
Substitute given weight Length[(Kilometers)] = 400 in the above formula.
Length[(Millimeters)] = 400 × 1000000
Length[(Millimeters)] = 400000000
Final Answer:
Therefore, 400 km is equal to 400000000 mm.
The length is 400000000 mm, in millimeters.
Consider that a private helicopter has a flight range of 150 kilometers.
Convert this range from kilometers to Millimeters.
The length in kilometers is:
Length[(Kilometers)] = 150
The formula to convert length from kilometers to millimeters is:
Length[(Millimeters)] = Length[(Kilometers)] × 1000000
Substitute given weight Length[(Kilometers)] = 150 in the above formula.
Length[(Millimeters)] = 150 × 1000000
Length[(Millimeters)] = 150000000
Final Answer:
Therefore, 150 km is equal to 150000000 mm.
The length is 150000000 mm, in millimeters.
Kilometers to Millimeters Conversion Table
The following table gives some of the most used conversions from Kilometers to Millimeters.
Kilometers (km) Millimeters (mm)
0 km 0 mm
1 km 1000000 mm
2 km 2000000 mm
3 km 3000000 mm
4 km 4000000 mm
5 km 5000000 mm
6 km 6000000 mm
7 km 7000000 mm
8 km 8000000 mm
9 km 9000000 mm
10 km 10000000 mm
20 km 20000000 mm
50 km 50000000 mm
100 km 100000000 mm
1000 km 1000000000 mm
10000 km 10000000000 mm
100000 km 100000000000 mm
A kilometer (km) is a unit of length in the International System of Units (SI), equal to 0.6214 miles. One kilometer is one thousand meters.
The prefix "kilo-" means one thousand. A kilometer is defined by 1000 times the distance light travels in 1/299,792,458 seconds. This definition may change, but a kilometer will always be one
thousand meters.
Kilometers are used to measure distances on land in most countries. However, the United States and the United Kingdom still often use miles. The UK has adopted the metric system, but miles are still
used on road signs.
A millimeter (mm) is a unit of length in the International System of Units (SI). One millimeter is equivalent to 0.001 meters or approximately 0.03937 inches.
The millimeter is defined as one-thousandth of a meter, making it a precise measurement for small distances.
Millimeters are used worldwide to measure length and distance in various fields, including engineering, manufacturing, and everyday life. Many industries, especially those requiring high precision,
have adopted the millimeter as a standard unit of measurement for small lengths.
Frequently Asked Questions (FAQs)
1. How do I convert kilometers to millimeters?
To convert kilometers to millimeters, multiply the number of kilometers by 1,000,000. This is because 1 kilometer equals 1,000,000 millimeters. For example, if you have 2 kilometers, multiplying by
1,000,000 gives you 2,000,000 millimeters. The formula is: millimeters = kilometers × 1,000,000, making conversions easy.
2. What is the formula for converting kilometers to millimeters?
The formula for converting kilometers to millimeters is: millimeters = kilometers × 1,000,000. Since there are 1,000 meters in a kilometer and 1,000 millimeters in a meter, multiplying by 1,000,000
converts the measurement directly from kilometers to millimeters, providing a quick way to handle large distances.
3. How many millimeters are there in a kilometer?
There are 1,000,000 millimeters in a kilometer. The metric system is based on powers of ten, which makes conversions straightforward. By knowing that a kilometer equals 1,000,000 millimeters, you can
easily switch between these two units for precise measurements.
4. Why do we multiply by 1,000,000 to convert kilometers to millimeters?
We multiply by 1,000,000 because 1 kilometer contains 1,000,000 millimeters. This conversion accounts for both the 1,000 meters in a kilometer and the 1,000 millimeters in each meter. It is
especially useful in scientific fields where very precise measurements are required.
5. How can I convert millimeters back to kilometers?
To convert millimeters back to kilometers, divide the number of millimeters by 1,000,000. This works because 1 kilometer equals 1,000,000 millimeters. For example, if you have 5,000,000 millimeters,
dividing by 1,000,000 gives you 5 kilometers. The formula is: kilometers = millimeters ÷ 1,000,000.
6. What is the difference between kilometers and millimeters?
Kilometers and millimeters are both units of length in the metric system, but they are used for very different scales. A kilometer is much larger, equaling 1,000,000 millimeters. Kilometers are used
for long distances, such as travel between cities, while millimeters are used for very small measurements, like the thickness of a sheet of paper. | {"url":"https://convertonline.org/unit/?convert=kilometers-millimeters","timestamp":"2024-11-13T04:58:03Z","content_type":"text/html","content_length":"101721","record_id":"<urn:uuid:85ff3488-4a67-43b1-aeec-03143dfad94f>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00886.warc.gz"} |
Guide to Essential BioStatistics XII: Selecting a statistical test – choosing Nonparametric or Parametric Statistical Tests
In the previous articles in this series, we explored the Scientific Method and Proposing Hypotheses and Type-I and Type-II errors, Designing and implementing experiments (Significance, Power, Effect,
Variance, Replication, Experimental Degrees of Freedom and Randomization), as well as Critically evaluating experimental data (Q-test; SD, SE, and 95%CI).
In the following articles, we will explore: Concluding whether to accept or reject the hypothesis (F- and T-tests, Chi-square, ANOVA and post-ANOVA testing).
In this twelfth article in the LabCoat Guide to BioStatistics series, we learn when to choose Nonparametric or Parametric Statistical Tests.
Inferential statistics
Inferential statistics enable us to test a hypothesis and draw conclusions, or inferences regarding a population through extrapolation from our experimental data sample.
Our choice of statistical method for hypothesis testing is based on whether the experimental data is normally distributed, and on the scale of the data.
Parametric and non-parametric tests
For normally distributed data, standard parametric tests such as the T-test and ANOVA tests are typically used, while nonparametric tests are appropriate if the data does not follow the normal
Parametric tests assume a Normal or Gaussian distribution of Measurement data at the Interval or Ratio scales (see previous article), while nonparametric do not – although they are subject to sample
size requirements (see below). In addition to non-Gaussian Measurement data, Nonparametric tests are used for Categorical data at the Nominal or Ordinal scales.
Parametric tests are generally more powerful than non-parametric tests and are more likely to detect a significant effect when one indeed exists.
For this reason, many biologists tend to favor parametric tests rather than nonparametric tests, as any non-conformity to the prerequisites for parametric testing can often be circumvented through
assumptions of normalcy, the identification and removal of outliers as well as data transformations.
In daily practice, it is thus usual for scientists to transform data from non-normal distributions, or to use parametric methods directly on datasets from non-normal distributions. With regard to the
latter: if each treatment comprises less than 10 data values (due to practical or economic constraints), the consensus is that any test of normal distribution will be so compromised that neither data
transformation nor the use of nonparametric tests will provide a significant benefit.
▶︎ A biological rule of thumb is that the small data sets commonly found in lab and greenhouse trials may be assumed to be normally distributed and analyzed using standard parametric tests.
NOTE: the information presented here comprises approximations and rules of thumb which are commonly used in designing and analyzing non-critical trials only. For critical experiments, nonparametric
tests should be used if the data does not follow the normal distribution (before or after transformation) or sample size requirements for parametric tests – always seek the advice of a qualified
Overview of essential parametric and non-parametric methods
For hypothesis testing, specific parametric and nonparametric tests are available to evaluate different experimental datasets. Among these, the most commonly used include tests to compare a single
group of data to a hypothetical value, to compare two paired or unpaired groups, or to compare three or more groups, as well as the prediction of values from previously measured values:
Figure 12.1: choosing statistical analysis methods.
We will obtain insight into the methods most commonly used in daily practice (t-tests, ANOVA and nonlinear regression for the estimation of ED50 values) in subsequent articles.
Microsoft Excel versus Statistics packages
This would be an excellent time to spend a minute or two considering software packages for statistical data analysis. Microsoft Excel is perhaps the most commonly used data software in biological
research and is often used (and misused) for Statistical Data Analysis.
Excel is first and foremost a spreadsheet with added data analysis modules, and it is important to understand that the software has limitations relative to professional statistic packages. Excel is
however excellent for data entry and data management and is accessible and sufficiently applicable for “quick-and-dirty” internal descriptive analyses and initial hypothesis testing carried out in
research laboratories.
For more demanding applications (e.g., external reports and scientific papers) statisticians advise that Excel should only be used for data preparation, and this data should then be transferred to
professional statistics packages for analysis. These results can then be reported directly or moved back to Excel for graphing and presentation purposes.
In Crop Protection Research, two of the most commonly used data packages for trial planning and statistical data analysis are the commercial ARM (Agricultural Research Manager) package, and the
open-source statistical package, R.
Both have relatively steep learning curves, but once they have been mastered, they become indispensable. For researchers with limited scientific knowledge, the GraphPad suite of statistical packages
provides real-time guidance during data analysis.
The first two books in the LABCOAT GUIDE TO CROP PROTECTION series are now published and available in eBook and Print formats!
Aimed at students, professionals, and others wishing to understand basic aspects of Pesticide and Biopesticide Mode Of Action & Formulation and Strategic R&D Management, this series is an easily
accessible introduction to essential principles of Crop Protection Development and Research Management.
A little about myself
I am a Plant Scientist with a background in Molecular Plant Biology and Crop Protection.
20 years ago, I worked at Copenhagen University and the University of Adelaide on plant responses to biotic and abiotic stress in crops.
At that time, biology-based crop protection strategies had not taken off commercially, so I transitioned to conventional (chemical) crop protection R&D at Cheminova, later FMC.
During this period, public opinion, as well as increasing regulatory requirements, gradually closed the door of opportunity for conventional crop protection strategies, while the biological crop
protection technology I had contributed to earlier began to reach commercial viability.
I am available to provide independent Strategic R&D Management as well as Scientific Development and Regulatory support to AgChem & BioScience organizations developing science-based products.
For more information, visit BIOSCIENCE SOLUTIONS – Strategic R&D Management Consultancy.
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Ellipsoidal figures of equilibrium: Compressible models
We present a new analytic study of ellipsoidal figures of equilibrium for compressible, self-gravitating Newtonian fluids. Using an energy variational method, we construct approximate hydrostatic
equilibrium solutions for rotating polytropes, either isolated or in binary systems. Both uniformly and nonuniformly rotating configurations are considered. Compressible generalizations are given for
most classical incompressible objects, such as Maclaurin spheroids, Jacobi, Dedekind, and Riemann ellipsoids, and Roche, Darwin, and Roche-Riemann binaries. The validity of our approximations is
established by presenting detailed comparisons of our results to those of recent three-dimensional computational studies. Although our treatment is quite different, the presentation of our results
follows closely that of Chandrasekhar in his work on the incompressible solutions using the tensor virial method. In the incompressible limit, our equilibrium solutions reduce exactly to those of
Chandrasekhar. For binary systems, however, our analysis improves on previous results even in the incompressible limit. Our energy variational method can also be used to study the stability
properties of the equilibrium solutions. Both secular and dynamical instability limits can be identified. For an isolated rotating star, we find that, when expressed in terms of the ratio T/\W\ of
kinetic energy of rotation to gravitational binding energy, the stability limits for axisymmetric configurations to nonaxisymmetric perturbations are independent of compressibility in our
approximation. We also study the effects of rotation and tidal forces on the radial stability of stars against gravitational collapse. Our most significant new results concern the stability
properties of binary configurations. Along a Roche sequence parameterized by binary separation, we demonstrate the existence of a point where the total energy and angular momentum of the system
simultaneously attain a minimum. A similar minimum exists for Darwin binaries when the polytropic index n of both components is below a critical value n[crit] ≈ 2. We show that such a turning point
along an equilibrium sequence marks the onset of secular instability. The instability occurs before the Roche limit is reached in Roche binaries, and before the surfaces of the two components come
into contact in Darwin binaries. We point out the critical importance of this instability in determining the final evolution of coalescing binary systems.
• Binaries: Close
• Hydrodynamics
• Stars: Rotation
ASJC Scopus subject areas
• Astronomy and Astrophysics
• Space and Planetary Science
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How to calculate apr for auto loan - Dollar Keg
How to calculate apr for auto loan
What is APR? What does APR mean? If you’re looking to buy a car, whether it’s new or used, you’ve probably been told to check the annual percentage rate (APR) of the loan. But what is APR? How can
you use it in your car-buying decision?
For shoppers interested in buying a new car or truck, one of the main steps that involves is determining the cost of your auto loan. Determining your monthly car payments means understanding what you
will be paying in interest over the course of your loan. The APR or annual percentage rate is a standard metric to evaluate this. This article will explain how APR works and how you can use it to
compare auto loan offers from various lenders.
We define APR as a yearly rate. But, this does not reflect the cost of borrowing money for the various time frames we use. Because there are often many longer-term loans that are broken down into
shorter periods, there’s also a calculation for APY (Annual Percentage Yield). APY takes into account the effects of compounding interest and its impact on the time value of money. For example, if
you borrow $100 from me at 10% per year, you’ll pay me back $110 in one year. That would be a 10% APR loan. If the same loan was payable in monthly installments, however, that $110 would be paid back
to me over twelve months (the equivalent of ten payments), resulting in an APY of 11%. In which case, your APY is greater than your APR.
APR stands for annual percentage rate. It is a measure of the interest rate you will pay on your auto loan, expressed as a yearly rate.
To calculate your APR, you need to know what the interest rate is (the number), and how much money you are borrowing (the dollar amount). Then, multiply the interest rate by the number of years you
plan to borrow the money for:
APR = Interest Rate x Number of Years
For example, if you have an auto loan with an interest rate of 6%, and you are borrowing $25,000 over four years, your APR is:
APR = 6% x 4 = 24%
When you apply for a car loan, you’ll receive a quote from the lender. It will include the loan amount, term, and projected monthly payment based on these figures. You’ll also notice the annual
percentage rate (APR), which should not be overlooked as it could easily add several hundred or thousands of dollars to the vehicle’s total purchase price.
What is APR (Annual Percentage Rate)?
The APR you pay on an auto loan represents the cost of borrowing funds from the lender.
Is APR the Same as the Interest Rate?
Consumers often use the terms interest rate and APR synonymously, but they aren’t quite the same. The interest rate is the percentage of the principal amount the lender charges you to borrow funds.
However, the APR includes interest and fees, like loan origination costs you’ll pay when financing the vehicle.
Why the APR Is Important for a Car Loan
The APR is important because it tacks on a sizable amount of additional costs to the loan. So, shopping around for an auto loan is essential to ensure you get the best deal possible on financing.
APR and Your Credit Score
The lowest APRs are generally reserved for borrowers with good or excellent credit. Of course, a bad credit score doesn’t necessarily mean you’ll be denied an auto loan. Still, you can expect
borrowing costs to be higher.
To illustrate, a 60-month $30,000 auto loan with a 5 percent interest rate comes with a $566 monthly payment, and you’ll pay $3,968 in interest over the loan term. But if your credit score is on the
lower end and you get a 9 percent interest rate, the monthly payment will increase to $623, and you’ll pay $7,365 in interest over the life of the loan.
Calculating APR on a Car Loan
Follow these steps to calculate the APR on an auto loan with ease.
Get Information on Your Car
You’ll need to prepare the following details to get started:
• Principal $35,000
• Interest Rate: 6 percent
• Loan Term: 48 months
• Projected interest: $2,200
• Fees: $1,500
• Taxes: $3,000
Once you have these figures, plug them into the APR equation below.
Plug Them into the APR Equation
• APR = [(Interest, taxes and fees / principal / loan term in days) * 365] * 100
• APR = [($6,700/$35,000/1,460) * 365] * 100
• APR = 4.79 percent
Or Use an Online APR Calculator
You can also input these figures into an APR calculator that will do the calculations for you.
Lower Your Interest Rate & Monthly Payments
Get a quote to refinance your auto loan in minutes with no credit check. See how much you can save with historically Low Rates! Skip Up To 3 Payments. A+ Rating From BBB.
How Lenders Calculate APR
Most lenders assess your APR based on your credit rating, assuming you meet the other eligibility criteria. As mentioned above, with a higher credit score, you’re more likely to receive a competitive
APR. But if your credit score is on the lower end, expect to be offered a higher APR to offset the risk of defaulting on the auto loan that you pose to the lender.
Other factors lenders consider when calculating APR include:
• Principal: The loan principal or the amount you’re requesting to borrow
• Down payment: You may not be required to make a down payment. However, doing so could get you a lower APR on your auto loan
• Loan term: You’ll pay more in interest if you opt for an extended loan term, which could overshadow the benefit of securing a more affordable monthly payment
What is a Good APR for a Car Loan?
It depends on your credit rating. Borrowers with good or excellent credit scores can expect to pay an APR between 3.61 percent and 5.38 percent. But if your score is lower, you could get an APR from
9.8 percent to 19.87 percent.
Compare APR and Loans to Refinance Your Car Loan
If your credit score has already improved since you took out your original car loan, you could qualify for a lower APR by refinancing. Consider Auto Approve to help you get the job done, minus the
Interest rates from lending partners in the Auto Approve network start at 2.25 percent. You can get a rate quote in minutes without impacting your credit score. Here’s how to get started:
• Step 1: Submit your details to get pre-approved. Be prepared to answer a few questions to verify your identity. You’ll also need to input data about your employment, earnings and vehicle.
• Step 2: View your loan offers. If there’s a match that works for you, upload the requested financial documents and information. You’ll need to provide a copy of your driver’s license, current
proof of insurance, a current registration, proof of income (or pay stubs) and your current auto loan contract if you have it available. The team at Auto Approve will handle all the paperwork for
you and ensure you steer clear of all the tedious tasks that often come with refinancing auto loans.
• Step 3: Sign your loan documents. Once the loan is finalized, you’ll start paying the new lender, typically within 45 days. However, you may qualify for a 90-day payment deferral that lets you
keep more of your hard-earned money in your pocket even longer.
How to calculate apr on car loan in excel
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Long time behavior of solutions of an autonomous differential equation
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Statistical Averages - Mean, Mode and Median - Data36
As I have mentioned several times, Data Science has 3 important pillars: Coding, Statistics and Business. To succeed, you have to be well-versed in all three. In this new series, I want to help you
to learn the most important parts of Statistics. This is the first step – and in this episode we are going to get to know the most basic statistical concept: statistical averages.
the 3 pillars: statistics, coding, business
Statistics. What is it for?
A few weeks ago, I ran into an excellent article about data vizualization by Nathan Yau. He writes about dataviz, but I love how he puts the importance of Statistics at the beginning of the article:
“Data is a representation of real life. It’s an abstraction, and it’s impossible to encapsulate everything in a spreadsheet, which leads to uncertainty in the numbers. How well does a sample
represent a full population? How likely is it that a dataset represents the truth? How much do you trust the numbers? Statistics is a game where you figure out these uncertainties and make estimated
judgements based on your calculations.”
Statistics is all about “compressing” a lot of information into a few numbers so our brain can process it more easily.
For example:
More than 500 million people are living in the European Union’s 28 countries. If we want to compare salaries between countries, we won’t compare each of the salaries one by one. That’s nonsense. We
will calculate and compare average salaries instead. That’s a helpful abstraction.
However, if you follow this article series about statistics, you will also see that describing a whole dataset with only one number can often yield misleading results. Taking average salaries is
cool, but it doesn’t let us see, for instance, the range of the data. In one country the difference between the poorest and richest people could be way bigger than in another, and that could be
important to know, too.
A good data scientist can always find those few numbers that describe her data in the simplest, but still the most meaningful way. Join me and I’ll show you the statistical tool-set necessary to be
the best at that!
Statistical Averages
Let’s start simple! Statistical averages. It’s an easy-to-understand concept, and very commonly used. The point of using averages is to get a central value of a dataset. Of course, there is more than
one way to decide which value is the most central… That’s why we have more than one average type.
The three most common statistical averages are:
In everyday language, the word ‘average‘ refers to the value that in statistics we call ‘arithmetic mean.‘ When calculating arithmetic mean, we take a set, add together all its elements, then divide
the received value by the number of elements. For example, the arithmetic mean of this list: [1,2,6,9] is (1+2+6+9)/4=4.5.
This is middle-school mathematics, so I assume that you are very well aware of this calculation anyway.
Note: calling the ‘arithmetic mean’ the ‘average’ is improper. The word ‘average’ can refer to any of the average types — mode and median are averages, too. If you hear anyone using ‘average’ wrong
in the coffee shop, don’t correct her… but in a data science meeting (or especially in a job interview), please stick to ‘arithmetic mean.’
Note 2: “But Tomi, in the header of this section you have written ‘Mean’ and not ‘Arithmetic Mean.’ What’s going on here?” Okay, you are right. We don’t really say ‘arithmetic mean’ during data
meetings. We just say ‘mean.’ But we think ‘arithmetic mean.’ The only reason for this is that we are lazy. And to be honest, it’s not the best practice because there are other types of means, too
(like geometric mean or harmonic mean. More info: here.). Bottom line is:
• “Mean.” Okay.
• “Arithmetic mean.” Perfect.
Here comes the most commonly used metaphor for showing the importance of the Median!
Ten workers sit in a room. Their yearly salaries are:
Worker #1 €15.000
Worker #2 €18.000
Worker #3 €18.000
Worker #4 €18.000
Worker #5 €18.000
Worker #6 €19.000
Worker #7 €20.000
Worker #8 €22.000
Worker #9 €22.000
Worker #10 €22.000
What’s the central value of their salary?
Let’s try arithmetic mean first! The result is €19.200.
Is it a good enough “compressed value” of the information in the table above? Can we say that these 10 workers are all making around €19.200 per year? I’d say: yes, $19.200 is not too far from
€15.000 nor €22.000.
But here comes the trouble: Worker #10 goes home, and the CEO of the company comes in instead. She makes €100.000 per year.
Worker #1 €15.000
Worker #2 €18.000
Worker #3 €18.000
Worker #4 €18.000
Worker #5 €18.000
Worker #6 €19.000
Worker #7 €20.000
Worker #8 €22.000
Worker #9 €22.000
CEO €100.000
Now, the arithmetic mean changes to €27.000. Is it still a good enough value to describe the average salary? Well… Not really. 9 out of the 10 people make less than €27.000 a year – so saying that
€27.000 is a good central value doesn’t sound right.
Now, the CEO goes home, and Bill Gates comes in.
Worker #1 €15.000
Worker #2 €18.000
Worker #3 €18.000
Worker #4 €18.000
Worker #5 €18.000
Worker #6 €19.000
Worker #7 €20.000
Worker #8 €22.000
Worker #9 €22.000
Bill Gates €1.000.000.000
The arithmetic mean is €100.017.000! Is everyone in the room making 100 million Euros yearly!? Not even close.
When there are extreme values – or in statistics-language: outliers – in a data set, arithmetic mean is not a good-enough representation of the data anymore.
That’s when median comes into play! You get the median of a set by simply arranging all the elements of it from smallest to greatest, then taking the middle value. Here’s an example:
What’s the median of this list? First, sort it in ascending order: [1,5,7,9,11]. Then take the middle value of the list. In this case, it’s 7. That’s the median.
Note: the mean of this sample_data list is 6.6, which is pretty close to the median.
But what happens when we have a list with an even number of elements? Let’s take a look at our workers again:
Worker #1 €15.000
Worker #2 €18.000
Worker #3 €18.000
Worker #4 €18.000
Worker #5 €18.000
Worker #6 €19.000
Worker #7 €20.000
Worker #8 €22.000
Worker #9 €22.000
Worker #10 €22.000
Luckily, this data is already sorted, so we just have to pick the middle value. Is it Worker #5 or Worker #6? The answer is: it’s the mean of the two, so €18.500. The general rule is that if you have
a list with an even number of elements, you can calculate the median by:
1. sorting the list
2. taking the mean of the middle two elements.
What’s the median when the CEO comes in? It’s the exact same: €18.500.
And when Bill Gates comes in? Same: €18.500.
Now you can see, that in these two last cases, the median was a better central value for the whole dataset than the arithmetic mean would have been.
Note: even if it was, median by itself is not a good enough number to describe a data set with outliers, mainly because only by looking at the median value, we don’t even suspect that Bill Gates is
in the room, too. And in real life data science problems, you want to know about “Bill Gates”-es in the rooms. I’ll get back to this problem in the next few articles.
The third famous average type is Mode. The definition is simple: mode is the element that occurs most often in a list. In theory, it’s useful when the numerical values in your data set are used as
categorical values. But to be honest, we rarely use it; and even when we do, we don’t really call it the mode. We just say: the most frequent element in a list.
However, I wanted to add it to the list because, you know… if you are asked during a job interview, at least now you will know what to answer. 🙂
What to use? Mean, median or mode?
There is no definite answer for this question. Based on my experience, mean and median will be used equally often, and mode almost never… But when to use which one? That really depends on the
particular case. There are some rules of thumb, but as I mentioned above, I don’t really see the point of using only one number to describe a dataset anyway… so let’s get back to this question when
we have learned more about variance, standard deviation, standard error, about different distributions, and more – in the upcoming articles!
But now…
Test yourself!
Okay, now that you know all three famous average types – mode, median and mean – it’s time to test yourself! This assignment was actually used a few years ago at junior data analyst job interviews,
Take a distribution that’s skewed to the right. Here it is on a chart:
In this data set we have elements from 1 to 20. Every element occurs more than once. (E.g. We have around one hundred 1 values, around one hundred forty 2 values, and so on… and at the end of the
list, we have around ten 19 values and one or two 20 values.)
The x-axis marks the different elements of the list: numbers from 1 to 20. The y-axis shows the number of occurrences of each of these elements.
The task is:
Draw a vertical line on the chart where you estimate the mode, the mean and the median values to be. What’s the order of the values from left to right?
The solution is this: mode < median < mean.
Since you don’t have the data, you can’t calculate the exact values to get this answer. But you can do two things:
1. You can create your own sample data that would result a similar skewed-to-the-right chart. Here’s a very simple example: [1,1,2,2,2,3,3,4,5,6]. If you calculate the mode (2), the mean (2.9) and
the median (2.5) for this sample data set, you will already know the answer to the original question: mode < median < mean.
2. You can think out the solution, too! First, try to figure out the relationship between mode and median.
Imagine that we are chopping off the right side of the x-axis. The mode would be 4, and, because this part of the chart is almost symmetrical, the median would be around 4, too.
Let’s put back the right side of the chart. Now, we have a lot of greater values than 8, and these values push up the median value – while mode doesn’t change.
So the relationship between mode and median is mode < median.
What about median vs. mean? We already know that mean is more sensitive to extreme values. So the one or two 20 values at the end of the list will have an effect on the mean, but it probably
won’t change the median value (just as in the Bill Gates example). This means that median < mean. All in all: mode < median < mean.
The 3 most common statistical averages are (arithmetic) mean, median and mode. You will use mean and median all the time, so it’s good to be confident in calculating them!
This was our first baby step in discovering the great universe of statistics for data science! The next step is to understand statistical variability.
Tomi Mester
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Proc GAMPL - what is default distribution and link function?
Hello SAS community,
I am starting to use Proc GAMPL (SAS version 9) for some time series analyses.
The response I am using is a continuous variable (e. g., sizes or abundances of an animal caught in the field).
What is the default distribution and link used if neither of these are specified on the Model statement line? I cannot find this information in the Proc GAMPL description.
Thank you.
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What do historical temperature records tell us about natural variability in global temperature?
By Patrick T Brown, 24 August 2016
Research article
Global average surface air temperature can change when it is either ‘forced’ to change by factors such as increasing greenhouse gasses, or it can change on its own through ‘unforced’ natural cycles
like El-Niño/La-Niña. In this paper we estimated the magnitude of unforced temperature variability using historical datasets rather than the more commonly used computer climate models. We used data
recorded by thermometers back to the year 1880 as well as data from “nature’s thermometers” – things like tree rings, corals, and lake sediments – that give us clues of how temperature varied
naturally from the year 1000 to 1850.
We found that unforced natural temperature variability is large enough to have been responsible for the decade-to-decade changes in the rate of global warming seen over the 20th century. However, the
total warming over the 20th century cannot be explained by unforced variability alone and it would not have been possible without the human-caused increase in greenhouse gasses. We also found that
unforced temperature variability may be the driver behind the reduced rate of global warming experienced at the beginning of the 21st century.
Global Temperature Change
The long term warming of the globe over the 20th and 21st centuries is one of the most recognized measures of human impact on the planet. However, in order to assess the human contribution to global
warming, it is critical that we understand the natural drivers of global temperature change. Our study attempts to do just that by quantifying how large natural ‘unforced’ changes in global
temperature can be. Before we delve into the methods and results, I will provide some background on global temperature change and what is meant by ‘forced’ and ‘unforced’ temperature variability.
Temperature, in essence, is a measure of energy. All changes in global average air temperature come about due to an imbalance in the atmosphere’s energy budget [P. T. Brown et al., 2014]. Think of it
this way – the atmosphere has an energy budget similar to how you may have a financial budget. In order to accumulate wealth you need to make more money than you spend. Similarly, in order for the
global temperature to increase, the atmosphere needs to accumulate more energy than is lost. The Earth receives all of its energy from the sun, but a certain amount is reflected back to space off of
things like clouds, snow and ice. The Earth also releases something called “infrared energy” to space. When the global temperature is stable, the amount of solar energy coming in equals the amount of
energy reflected and released to space, creating a balanced energy budget.
There are many factors that can change the Earth’s energy budget and thus the average surface temperature. Forced temperature change is a change in temperature that is imposed on the ocean/atmosphere
system from a source that is considered to be outside of the ocean/atmosphere system. Examples of “forcings” include changes in the brightness of the sun and changes in concentrations of greenhouse
gases due to human fossil fuel burning. However, global surface temperature can also change naturally without any outside forcing. Fittingly, this is referred to as “unforced” temperature change.
Unforced temperature changes come from natural ocean/atmosphere interactions that can cause an imbalance in the Earth’s energy budget. The El-Niño/La-Niña cycle in the Pacific Ocean is the most well
known example of unforced variability. During La-Niña years, an unusual amount of energy is taken up by the Pacific Ocean, which causes a net loss of atmospheric energy and thus short-term global
cooling. The opposite of La-Niña is El-Niño. During El-Niño years, excess energy enters the atmosphere from the Pacific Ocean causing an energy surplus and short term global warming. Natural
variability in clouds and snow/ice can also change how much solar energy is reflected back to space and thus can affect the average surface air temperature (see [P. T. Brown et al., 2015a], [P. T.
Brown et al., 2015b], [P. T. Brown et al., 2016] for more details).
The relationship between forced and unforced temperature changes can be compared to the relationship between a man and a dog out for a walk (Figure 1: A and B). In this analogy the path of the man
represents forced temperature change (such as human-caused increases in greenhouse gasses) and the path of the dog, relative to the man, represents unforced temperature change (such as El-Niño/
La-Niña cycles). Forced temperature changes are relatively deterministic and predictable; therefore imagine that the man walks his dog on the exact same route every day. On the other hand, unforced
temperature change is somewhat random and unpredictable, therefore imagine that the dog is easily distracted and continuously redirects her attention from object to object over the course of each
walk. It is important to note that the dog is on a leash so she can only wander a certain distance away from the man before the leash restricts her. This means that the path of dog will eventually
reflect both the movement of the dog and the movement of the man. In the real climate system we can only observe the path of the dog – the combined result of forced and unforced temperature change.
This means that we must figure out the extent to which the path we see is affected by the man and the extent to which it is affected by the dog, if we want to understand the causes of temperature
change over any given period of time.
Now, imagine that the man walks the dog one hundred times and follows the exact same route each time – the path of the dog would be slightly different each time. Analogously, if we could go back in
time to the beginning of the 20th century and re-run climate history with the exact same changes in greenhouse gasses (as well as other global temperature forcings that I have not mentioned), the
temperature progression would be slightly different each time because unforced variability would be different.
Notice that if the leash linking the dog to the man is short, the path of the dog will closely match the path of the man (Figure 1: A). If the man walks the dog one hundred times, the path of the dog
will look similar each time since the dog cannot stray very far from the man (Figure 1: C). However, if the leash is long (Figure 1: B), the dog can stray a reasonable distance away from the man
(Figure 1: D). Knowing the length of the leash (or the size of unforced temperature variability) in the real climate system is of critical importance for understanding what is causing the temperature
to change at any given time.
Figure 1: Man walking dog analogy for global average surface air temperature variability.
The path of the man represents forced temperature change. The path of the dog, relative to the man, represents natural unforced temperature variability. A longer leash represents the potential for
larger unforced temperature variability. The path of the dog represents the actual observed temperature change, a combination of the forced (movement of man) and the unforced (movement of dog)
influences on temperature change.
Determining the magnitude of unforced variability will also help us predict how global temperature might change in the future. As humans put more greenhouse gasses into the atmosphere every year, the
forced temperature change will continue on his upward trajectory. If the magnitude of unforced variability is small then we should expect to see global temperatures follow this upward progression
closely. However, if the magnitude of unforced variability is large, then the global temperature might deviate from the steady upward progression for decades at a time.
The most commonly used tool for determining the size of unforced natural variability is the “global climate model”. Global climate models are computer programs that use our knowledge of physics and
geography to simulate the Earth’s oceanic and atmospheric circulations. Thus, these climate models actually simulate energy imbalances in order to estimate global surface temperature changes. When a
climate model is run, it simulates a single possible trajectory of the temperature progression. The size of unforced variability is inferred from a computer climate model by running it many times
with the same forced temperature changes but with slightly different histories of natural unforced cycles – histories that could have happened. This is analogous to figuring out the length of the
leash by observing the height of the sum of the dog’s paths (Figure 1: C and D).
Our new way to estimate the size of unforced variability
Several recent studies have suggested that global climate models might underestimate the magnitude of unforced natural temperature variability ([P. T. Brown et al., 2015a]; [K. L. Swanson et al.,
2009]; [M. G. Wyatt and J. Peters, 2012]). Considering this, it is valuable to estimate the size of unforced variability from an independent source. In our study, we estimated the size of unforced
variability by examining the historical record of temperature change in two types of datasets. We used the “instrumental record” from the year 1880 to 2013, which represents the temperature record
measured directly with thermometers. This record is relatively short, so we also used “proxy reconstructions” of temperatures from the year 1000 to 1850. Proxy reconstructions represent estimates of
historical temperature that come from “natural thermometers” present in the environment. Some examples of natural thermometers include tree rings, corals, pollen, ice layers, stalactites, and lake
We used a statistical method called “Multiple Linear Regression” in order to separate forced from unforced temperature variability in these records. Our Multiple Linear Regression technique noted how
much of the temperature variability in the past was correlated with changes in forcings. Any temperature variability that was correlated with changes in forcings was not counted as part of our
estimate of unforced variability. We then used this estimate of unforced temperature variability to create our own simulations of unforced variability over the 20th and 21st centuries. We did this
with a statistical method called “noise modeling”. Our noise modeling technique used a computer’s random number generator to create thousands of hypothetical temperature trajectories over the 20th
and 21st centuries with the same amount of unforced variability as what we found in our historical datasets (Figure 2). These trajectories represented alternative histories – again the range of
temperatures that could have occurred with the same forced temperature change. They were also used to represent the range of possible outcomes that we might expect to observe under future increases
in greenhouse gasses. We used this new data to address two main questions:
1. Is unforced natural variability large enough to account for the decade-to-decade variability in the rate of global warming over the 20th century?
2. Does the reduced rate of global warming over the beginning of the 21st century indicate that forced temperature changes slowed drastically, or is unforced variability large enough to make global
warming hiatus periods inevitable in the long run?
Implications of our new estimate of the size of unforced variability
We found that unforced variability is large enough to have accounted for decade-to-decade changes in the rate of global warming over the 20th century (Figure 2: B). This means that unforced
variability is a little bit larger than most global climate models have traditionally indicated. However, our results made it clear that unforced temperature variability is not large enough to
account for the total global warming that has been observed since 1900. Therefore, our study confirms that forced temperature changes, such as those from human-caused increases in greenhouse gasses,
were necessary for the Earth to have warmed as much as it did over the past century [N. L. Bindoff et al., 2013].
Figure 2: Estimates of forced variability and the range of unforced variability for global average temperature.
The black line represents forced variability (analogous to the path of the man in Figure 1) while the gray shading represents the range of unforced variability that we found in our study (analogous
to the length of the leash in Figure 1). The yellow line is the observed temperature progression while the green, blue, and red lines represent alternative temperature progressions that could have
occurred with the same forced variability but different unforced variability. Panels A and B show two different possibilities for forced variability from 1900 to 2015 while panels C, D and E show
three different possibilities for forced variability over next several decades. Figure reproduced from [P. T. Brown et al., 2015b].
Our findings also have implications for the first decade of the 21st century. We know that well-mixed greenhouse gasses, which cause forced temperature change, increased substantially since the turn
of the century. However, global temperatures rose very little between 2002 and 2013. If unforced variability was found to be very small, the above two observations might imply that greenhouse gasses
don’t cause as much warming as previously thought. However, since we found that unforced variability is relatively large, it suggests that the temperature we observe can meander substantially away
from the underlying forced temperature changes. Therefore, we should not be surprised to see a period of a decade-or-so without global warming even as the forced temperature change continues on its
upward trajectory (Figure 2: A). This is simply a situation where the man is progressing upward while the dog is walking down. The leash may be longer than previously thought but there is still a
leash. As long as the man continues on his upward trajectory, the leash will eventually pull up on the dog (Figure 2: C, D and E) and the long term global warming trend will continue.
• N. L. Bindoff, P. A. Stott, K. M. AchutaRao, M.R. Allen, N. Gillett, D. Gutzler, K. Hansingo, G. Hegerl, Y. Hu and co-authors: Detection and Attribution of Climate Change: from Global to
Regional., in: Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge
University Press, 2013.
• P. T. Brown, W. Li, E. C. Cordero and S. A. Mauget: Comparing the model-simulated global warming signal to observations using empirical estimates of unforced noise, Sci. Rep., vol. 5, 2015b.
• P. T. Brown, W. Li, J. H. Jiang and H. Su: Unforced surface air temperature variability and its contrasting relationship with the anomalous TOA energy flux at local and global spatial scales,
Journal of Climate, 2016.
• P. T. Brown, W. Li, L. Li and Y. Ming: Top-of-Atmosphere Radiative Contribution to Unforced Decadal Global Temperature Variability in Climate Models, Geophysical Research Letters, https://doi.org
/10.1002/2014GL060625, 2014.
• P. T. Brown, W. Li and S. P. Xie: Regions of Significant Influence on Unforced Global Mean Surface Temperature Variability in Climate Models, Journal of Geophysical Research – Atmospheres, https:
//doi.org/10.1002/2014JD022576, 2015a.
• K. L. Swanson, G. Sugihara and A. A. Tsonis: Long-term natural variability and 20th century climate change, Proceedings of the National Academy of Sciences, vol. 106, 16120-16123, 2009.
• M. G. Wyatt and J. Peters: A secularly varying hemispheric climate-signal propagation previously detected in instrumental and proxy data not detected in CMIP3 data base, SpringerPlus, vol. 1, 68, | {"url":"https://www.climanosco.org/published_article/what-do-historical-temperature-records-tell-us-about-natural-variability-in-global-temperature-3/","timestamp":"2024-11-10T12:58:53Z","content_type":"text/html","content_length":"107735","record_id":"<urn:uuid:e7c2acfa-4c47-42e5-b2ec-5007dbcca6ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00343.warc.gz"} |
[2021 KMS Annual Meeting] Related talks
Invited Lecture
Speaker: Myungho Kim (Kyung Hee University)
Date: TBA
Title: Cluster algebras and monoidal categories
Cluster algebras are special commutative rings introduced by Fomin and Zelevinsky in the early 2000s. Specifically, the cluster algebra refers to a subring generated by special elements called
cluster variables in the field of rational functions, and the process of creating a new cluster variable from given cluster variables is called a mutation. Cluster algebra is being actively studied
as it is observed that the mutation operation appears in various forms in various fields of mathematics.
A monoidal categorification of a given cluster algebra means that the Grothenieck ring is isomorphic to the cluster algebra and that special elements called cluster monomials correspond to simple
objects. If there is such a monoidal categorification, then the given monoidal category and the cluster algebra are closely related and help understand each other's properties.
In this talk, I will explain that the category of finite-dimensional representations of quiver Hecke algebras and that of quantum affine algebras form monoidal categorifications of cluster algebras.
This is based on several joint works with Seok-Jin Kang, Masaki Kashiwara, Se-jin Oh, and Euiyong Park.
Special Session
(SS-03) Representation Theory and Related Topics
Speaker: Jethro van Ekeren (Universidade Federal Fluminense)
Schedule: 2021.10. 21. AM 8:40~9:10
Title: Chiral Homology and Classical Series Identities
I will discuss results of an ongoing project on the chiral homology of elliptic curves with coefficients in conformal vertex algebras. We find interesting links between this structure and classical
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Speaker: Ryo Sato (Kyoto university)
Schedule: 2021.10.21. AM 9:10~9:40
Title: Feigin-Seikhatov duality in W-superalgebras
W-superalgebras are a large class of vertex superalgebras which generalize affine Lie superalgebras and the Virasoro algebras. It has been known that princial W-algebras satisfy a certain duality
relation (Feigin-Frenkel duality) which can be regarded as a quantization of the geometric Langlands correspondence. Recently, D. Gaiotto and M. Rapčák found dualities between more general
W-superalgebras in relation to certain four-dimensional supersymmetric gauge theories. A large part of thier conjecture is proved by T. Creutzig and A. Linshaw, and a more specific subclass
(Feigin-Seikhatov duality) is done by T. Creutzig, N. Genra, and S. Nakatsuka in a different way. In this talk I will explain how to upgrade the latter case to the level of representation theory by
using relative semi-infinite cohomology. This is based on a joint work with T. Creutzig, N. Genra, and S. Nakatsuka.
Speaker: 유필상 (칭화대학교)
Schedule: 2021.10.21. AM 9:50~10:20
Title: Representation Theory via Quantum Field Theory
It is known that some subjects in mathematics may be enriched by finding their context in physics. In this talk, we argue that representation theory is no exception. After explaining the physical
context for representation theory of a finite group as the most basic example, we discuss a research program on how to use ideas of quantum field theory to study certain objects of interest in
geometric representation theory.
Speaker: George D. Nasr (University of Oregon)
Schedule: 2021.10.21. AM 10:20~10:50
Title: A Combinatorial Formula for Kazhdan-Lusztig Polynomials of Sparse Paving Matroids and its connections to Representation Theory
In 2016, Elias, Proudfoot, and Wakefield introduced Kazhdan-Lusztig polynomials for a class of combinatorial objects called matroids. Later, they presented the equivariant (representation-theoretic)
version of these polynomials. We will introduce both these topics and discuss results in the case of sparse paving matroids. For the ordinary Kazhdan-Lusztig polynomials, we present a combinatorial
formula using skew Young tableaux for the coefficients of these polynomials for sparse paving matroids. In the case of uniform matroids (a special case of sparse paving matroids), this formula
results in a nice combinatorial interpretation that arises in the equivariant version of these polynomials.
Speaker: 오재성 (고등과학원)
Schedule: 2021.10.21. AM 11:00~11:30
Title: A tugging symmetry conjecture for the modified Macdonald polynomials
In this talk, we propose a conjecture which is a symmetry relation for the modified Macdonald polynomials of stretched partitions, $\widetilde{H}_{k\mu}[X;q,q^k]=\widetilde{H}_{\mu^k}[X;q^k,q]$.
Using the LLT-expansion of the modified Macdonald polynomials and linear relations of the LLT polynomials, we prove the conjecture for one column shape partition $\mu=(1^l)$. This is based on the
joint work with Seung Jin Lee.
Speaker: 김영훈 (서울대학교, QSMS)
Schedule: 2021.10.21 AM 11:30~12:00
Title: Extensions of 0-Hecke modules for dual immaculate quasisymmetric functions by simple modules
For each composition $\alpha$, Berg {\it et al.} introduced an indecomposable $0$-Hecke module $\mathcal{V}_\alpha$ with a dual immaculate quasisymmetric function as the quasisymmetric characteristic
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(SS-01) Trends in Arithmetic Geometry
Speaker: Chang-Yeon Chough (서울대학교, QSMS)
Schedule: TBA
Title: Brauer groups in derived/spectral algebraic geometry
Toën gave an affirmative answer to Grothendieck's question of comparing the Brauer group and the cohomological Brauer group of a scheme for all quasi-compact and quasi-separated (derived) schemes by
introducing the notion of derived Azumaya algebras. I'll give a glimpse of the extension of this result to algebraic stacks in the setting of derived/spectral algebraic geometry. If time permits, my
latest work on twisted derived equivalences in the derived/spectral setting, which is based on the aforementioned extension, will be presented. | {"url":"https://qsms.math.snu.ac.kr/index.php?mid=board_sjXR83&listStyle=viewer&order_type=desc&l=en&document_srl=1889&page=4","timestamp":"2024-11-15T00:19:32Z","content_type":"text/html","content_length":"31546","record_id":"<urn:uuid:21e071f4-84d9-4adc-9a29-e116e44dd03d>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00864.warc.gz"} |
Find the capacity in litres of a conical vessel. - WorkSheets Buddy
Find the capacity in litres of a conical vessel.
Find the capacity in litres of a conical vessel with
(i) radius 7 cm, slant height 25 cm
(ii) height 12 cm, slant height 13 cm
A conical pit of top diameter 3.5 m is 12 m deep. What is its capacity in kiloliters?
More Solutions:
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Subject: Re: [geda-user] The new to do
From: Stefan Salewski <mail AT ssalewski DOT de>
To: geda-user AT delorie DOT com
Date: Wed, 15 Jul 2015 13:42:57 +0200
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On Wed, 2015-07-15 at 12:13 +0100, Chris Smith (space DOT dandy AT icloud DOT com)
[via geda-user AT delorie DOT com] wrote:
> > The problem here is that to gEDA, a net is a non-topological entity.
> You could assign attributes to nets (that wouldn't be that hard to
> implement), but then the "needs to be 20mils" attribute would apply to
> the whole of, e.g, the "GND" net which probably isn't what you are
> intending.
> >
> > Representing the internal structute of a net in a way that makes
> this kind of attribute useful is much more difficult and would require
> a major addition to what the concept of a netlist entails.
> I don’t think it’s that difficult. I think all that is needed is:
> 1. in gschem allow a net to be split into multiple segments. At the
> moment a net in gschem is a continuous line that has a number of pin
> connections. I think it should be a collection of lines between two
> pin connections.
> 2. teach pcb that a pin can be connected to multiple nets.
Yes. We discussed that about 5 years ago already. I think I made a
picture for that, can not find it currently. And John Doty told us that
gschem does support net segments, each with its own attributes indeed,
but the default is joining straight segments. One problem is moving that
information to PCB. Another problem is, that many people said we do not
need and want that. Maybe that was indeed the motivation to start with
my Peted tool.
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Cite as
Gianfranco Bilardi and Lorenzo De Stefani. The DAG Visit Approach for Pebbling and I/O Lower Bounds. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer
Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 7:1-7:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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author = {Bilardi, Gianfranco and De Stefani, Lorenzo},
title = {{The DAG Visit Approach for Pebbling and I/O Lower Bounds}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {7:1--7:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.7},
URN = {urn:nbn:de:0030-drops-173999},
doi = {10.4230/LIPIcs.FSTTCS.2022.7},
annote = {Keywords: Pebbling, Directed Acyclic Graph, Pebbling number, I/O complexity} | {"url":"https://drops.dagstuhl.de/search/documents?author=De%20Stefani,%20Lorenzo","timestamp":"2024-11-07T15:17:39Z","content_type":"text/html","content_length":"56377","record_id":"<urn:uuid:ca83c510-ee7f-4a57-8b6f-4d6f770daaaf>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00202.warc.gz"} |
Robust Stability And Stabilization Of Uncertain It(?) Stochastic Systems With Applications
Posted on:2012-05-06 Degree:Doctor Type:Dissertation
Country:China Candidate:Y J Li Full Text:PDF
GTID:1488303356492914 Subject:Control theory and control engineering
Based on Lyapunov-Krasovskii functional method, by using stochastic Lyapunov stabilitytheory, Ito? formula, linear matrix inequalities, free-weighting matrix method and so on, thisdoctoral
dissertation is devoted to the research on the robust mean square exponential stabil-ity, almost sure exponential stability, robust feedback stabilization of uncertain Ito? stochasticsystems and the
exponential stability of uncertain fuzzy stochastic neural networks. The mainresearch work and contributions of this dissertation are summarized as follows:1. The preface gives an introduction to the
related background and the latest progress inthe stability analysis, control and applications of stochastic systems, then some preliminaries,lemma and some concepts of stability for stochastic
systems are presented.2. The robust mean square exponential stability for a class of uncertain stochastic sys-tems with time-varying delays is discussed. For a class of uncertain stochastic system
withtime-varying delays, the information of both variation range and the distribution probabilityof the time delay are considered. Firstly by translating the distribution probability of the timedelay
into parameter matrices of the transferred systems,a new modeling method is proposed.In the established model, the parameter uncertainties are norm-bounded, the stochastic distur-bances are described
in term of a Brownian motion, the time-varying delay is characterizedby introducing a stochastic variable which satisfies Bernoulli random binary distribution. Byexploiting an appropriative
Lyapunov-Krasovskii functional, stochastic Lyapunov stability the-ory and some new analysis techniques, some delay-distribution-dependent roust stability forstochastic system with linear and
nonlinear stochastic disturbances are discussed. The suffi-cient conditions for the robust exponential stability are given and derived for the system underall admissible uncertainties. All results
are given in the form of Linear Matrix Inequalities,which can be easily solved by some standard numerical packages. One of the important fea-tures of the results is that the stability conditions are
dependent on the probability distributionof the delay and upper bound of the delay derivative, the upper bound of the delay derivative isallowed to be greater than or equal to 1, the limit of the
upper bound of the delay derivative mustbe less than 1 is overcome, two numerical examples are given to illustrate the effectiveness andless conservativeness of the proposed method.3. The robust mean
square exponential stability for a class of uncertain stochastic sys-tem with impulsive disturbance is studied. By constructing a simple Lyapunov-Krasovskiifunctional, applying Ito? formula and some
inequality techniques, the mean square exponential stability sufficient conditions is given and derived based on LMIs. Then, the almost sure expo-nential stability for a class of neutral stochastic
system with time-varying delay and Markovianjumps are derived by the same method. Numerical examples show the correctness of proposedstability criteria.4. The robust non-fragile feedback controller
design problem for a class of uncertainstochastic systems with nonlinear disturbance is investigated. By means of Lyapunov-Krasovskiifunctional, stochastic Lyapunov stability theory and Ito? formula,
the non-fragile feedback con-troller are designed and obtained, the sufficient conditions which making the closed-loop sys-tem robustly stable are proposed and proved. Numerical examples and
simulation show theeffectiveness of designed controller.5. The robust exponential stability for a class of uncertain neural networks system withnonlinear stochastic disturbances is studied. By
applying the stochastic Lyapunov stabilitytheory to the neural networks, using the descriptor model transformation and the free-weightingmatrix method, a sufficient condition which making the system
robust exponential stability isderived. Numerical examples have shown the effectiveness and low conservativeness.6. The robust mean square exponential stability problem for a class of uncertain
impulsivestochastic neural networks systems with mixed time-varying delays and Markovian jumps isinvestigated. By constructing Lyapunov-Krasovskii functional and using stochastic Lyapunovstability
theory, the delay-dependent robust exponential stability sufficient conditions for thesystem is derived in terms of LMIs. The numerical example has shown that whether Markovianjump occurs at the
impulsive moment or not, the stability criteria proposed is effective.7. The robust exponential stability for a class of uncertain fuzzy stochastic neural net-works systems with mixed time-varying
delay is investigated. Based on Takagi-Sugeno (T-S)fuzzy models, a class of uncertain fuzzy stochastic neural networks systems is modeled. Byconstructing Lyapunov-Krasovskii functional, combing the
stochastic stability theory with thefuzzy control theory and robust theory, the novel delay-dependent stability sufficient conditionsare derived based on LMIs and some integrated inequality
techniques. Example and simulationare provided to show less conservative and the effectiveness of the obtained results.Finally, after the summary of this dissertation, the issues of further
investigation are pro-posed.
Keywords/Search Tags: Robust stability, time-varying delay, Markovian jump, impulsive, neutral dis-tributed time-delay systems, non-fragile controller, Ito? formula, free-weight matrix, meansquare
exponential stability, linear matrix inequalities | {"url":"https://globethesis.com/?t=1488303356492914","timestamp":"2024-11-04T07:29:26Z","content_type":"application/xhtml+xml","content_length":"11750","record_id":"<urn:uuid:bba2591e-1601-48ad-af2a-91787947578f>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00749.warc.gz"} |
[Kwant] Re: Fwd: Query on Conductance Quantization in Kwant Simulations within the Integer Quantum Hall effect
18 Aug 2024 18 Aug '24
5:10 a.m.
Dear Henrique, I suspect that this is not a Kwant question but a physics one. There are three relevant length scales in this problem: the width of the sample, the magnetic length and the Fermi wave
length. You can get them from simple analytical calculations. I suspect (from the fact that the energy is very close to -4 in your plot, hence to the bottom of the band) that these three length
scales are of the same order of magnitude, hence you're in a crossover regime. To get to the "standard" QHE regime, you want the number of open channels = (width)/(Fermi wave length) to be relatively
large. Best regards, Xavier ________________________________ De : Henrique Veiga <up201805202@edu.fc.up.pt> Envoyé : jeudi 15 août 2024 17:13:11 À : kwant-discuss@python.org Objet : [Kwant] Fwd:
Query on Conductance Quantization in Kwant Simulations within the Integer Quantum Hall effect Dear Kwant community, I have been studying the integer quantum Hall effect using the Kwant software,
through which I computed the longitudinal and transverse conductance of a central sample (without disorder) in a four-terminal setup. I model the central device, which is subject to a strong magnetic
field, as a square of lateral length, L. Both the sample and the four ideal leads are described by a square-lattice tight-binding model with nearest-neighbor hoppings. By computing the transverse
conductance as a function of the Fermi energy, so that the first two Landau levels are crossed, I observe the expected behavior: the conductance appears quantized in integer steps of e²/h. However,
upon zooming in on each of the two plateaus, I notice that the conductance does not converge immediately to the expected value. Instead, it exhibits a rippling effect, which seems to be dependent on
the leads' cross-section, L_{Lead}. This phenomenon is illustrated in the attached .pdf file, and I have also included the code used to evaluate the conductance. My question is whether this behavior
is a numerical issue, or if it is expected within the context of charge transport simulations in four-terminal setups. Thank you in advance, Henrique Veiga | {"url":"https://mail.python.org/archives/list/kwant-discuss@python.org/message/MBEHQ5PEETVWKARLIPCSLUJULHVEAIKF/","timestamp":"2024-11-13T17:51:05Z","content_type":"text/html","content_length":"14914","record_id":"<urn:uuid:b9236d32-5a9e-41a9-821e-2c56acb9b8ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00667.warc.gz"} |
New Materials Frontier by Manipulating Different Quantum Degrees of Freedom in 2D Crystals, Heterostructures, Interfaces and Nano-Metamaterials
Nai-Chang Yeh
Frontier Topics Session, MRS Memorial
It’s my great honor to be here to celebrate Millie’s incredible and near-infinite amount of impact on the world and scientific society. So today, I thought, as a tribute to her, we should talk about
some of the interesting things that she might have liked to do next or some of the exciting things that are ongoing.
But, before I do that, I would just like to show you the picture I took with Millie, the last time I saw her. That was in 2016, when we were at the Kavli Prize ceremony in Oslo, Norway.
I can tell you a lot of things about Millie. She was my Ph.D. advisor, which meant she had a huge impact on my life — as you might expect. I also wrote an article — actually, I wrote two articles —
as my tributes to Millie. One was published with ACS nano, and that one gives a general overview of Millie’s life. The other article I wrote centered more around my personal interactions with
Millie, and that was published in the Proceedings of National Academy of Sciences. Those are hyperlinked, so you can click on them and read them, and perhaps gain a little more insight.
Millie and I, while I was her student. Photo courtesy of the Dresselhaus Family
Millie was a very caring and thoughtful mentor, who seemed to understand the potential of her students far better than the students themselves: You may be aware that high-temperature
superconductivity was discovered in 1986, which stimulated immense excitement worldwide in the research of superconductivity for the following decades. During the 1987 March Meeting of the American
Physical Society (APS) in New York City, which was nicknamed the Woodstock of Superconductivity at a later time, there were thousands of enthusiastic participants attending several all-night sessions
to find out the latest developments in the budding research field, and I was one of them when I was in my last year of Ph.D. research at MIT under Millie.
Boris Elman, Millie, Nobiyuki Kambe, myself, and Gene Dresselhaus. Photo courtesy of the Dresselhaus Family
I remembered that I met with Millie one day, shortly after we returned to MIT from the APS March Meeting, to discuss my research progress on graphite intercalation compounds. During our discussion,
Millie suddenly paused briefly, changing the subject and saying: “I think you will do very well in the research of high-temperature superconductivity”. I was a bit surprised by the off-subject
remark, but did not think very much about it then because I had to concentrate on completing my thesis. Several months later, Millie told me that there was a postdoctoral position available at the
IBM Watson Research Center on high-temperature superconductivity, and suggested that I went for a job interview. I did, and was given the offer on the same day of the interview. Millie then suggested
that it was a nice fit for me so that I need not go for any other job interviews. Therefore, I concentrated on finishing my Ph.D. thesis and then went to IBM for my postdoctoral studies shortly after
my graduation from MIT.
Millie shakes my hand at her 80th birthday. Photo courtesy of Paul McGrath
Although the field of high-temperature superconductivity was very different from my graduate research at MIT, it was very exciting and full of opportunities, and so I was able to be recognized for my
work in a relatively short period of time and received multiple faculty offers. When I joined the faculty of Caltech in 1989, my initial research program was primarily focused on high-temperature
superconductivity, and was tenured based on the achievements in the field. In retrospect, I was impressed by Millie’s foresight and her confidence in me, and am truly grateful for the amazing
guidance that she provided in her most unassuming manner.
Photo courtesy of the Dresselhaus Family
Now, whenever I speak, I can always picture Millie sitting in the front row and taking notes. And I always ask myself, before I speak, “What can I present to make Millie so excited that she takes
notes even more diligently than usual?”
This is what I did to come up with the topics presented today. Hopefully, you will also feel excited and inspired by them.
First, a summary of the topics I’m about to cover.
I will start by proposing that I think there is a new material frontier where quantum meets nano and low dimensionality — and I will elaborate on that in a moment. Next, I will discuss all the
different things we can do if we manipulate different quantum degrees of freedom — such as symmetry, topology, spin, valley, dimensionality, and so on. I will show you a few examples of that. Then,
I will discuss 2D atomic crystals and their increasing popularity due to new developments in making such crystals from van der Waals materials.
Photo courtesy of the Dresselhaus Family
By the way, you’ll probably notice that throughout the symposium thus far, the topics I’m discussing have come up from time to time.
Then I will talk about strain engineering and valleytronics and how we might use this to build new properties into (for example) graphene. And then, I’ll show you some of the research we are doing on
surface and interface states of topological materials, and I have some examples. There is also an exciting development in superconductivity related to interface superconductivity that you can use to
enhance the superconducting transition temperature by a factor of nearly 10. So we want to explore what might be happening.
And that is the talk. Hopefully, by the time I’m done, all these new topics will soak into your imagination, allowing you to dream of all the many things that are now possible, thanks to these
recent discoveries.
As I mentioned earlier, I believe that we’re at a new materials frontier in which quantum meets nano and low dimensionality. As we go smaller and smaller and are able to increasingly manipulate the
dimensions we’re working with — it changes things in many ways.
It allows engineers the ability to build devices that’ll solve the next generation of problems. It’ll create new applications that we haven’t even dreamed of yet. And it’ll cause a lot more people
to view materials research as a fundamental part of the R&D needed to create mass-marketed industrial products.
Emerging physics pertaining to edge, surface and interface states of novel materials and nano-metamaterials will give us the ability to almost play God! A proper understanding of this physics will
allow us to build our own custom-chosen properties into the materials we’re starting with — giving us the ability to create the perfect material for any given application.
There is a lot of interesting new physics coming from our studies of these novel states. The remainder of my talk will give you some examples.
Let’s first start by looking at some 2D atomic crystals from van der Waals materials. There are new developments that allow us to either fabricate or isolate 2D crystals from various types of van der
Waals materials, and lots of research efforts have gone into it — and for good reasons! You can look at materials like graphene and hexagonal boron nitride as examples — these materials have
honeycomb structures.
Now, transition-metal dichalcogenides, when it’s in a special phase — called the 2H phase — is actually a semiconductor. If you look at it from the top, it also looks like it has a honeycomb
structure. However, if you look at it from the side, you’ll notice there are 3 layers.
I remember that earlier today, there was a question from the audience asking if people are looking to making nanotubes out of other types of materials. I think it will be much harder to do so than it
is with carbon, because of what you see in the figure above. If you try to fold 3 layers together, it’s much harder to make the connection of different layers consisting of different elements , and
so one would have to be much cleverer because the strain for each layer will be very different. That may cause problems when you’re trying to fold this material up into a tube.
Photo of Millie courtesy of Dresselhaus Family
Anyway, back to the transition-metal dichalcogenides — the number of layers you have of it will also give you different properties. If you have one mono-layer, then it’s a direct gap material (a
direct-gap semiconductor) and if you have greater than one, it’s an indirect gap semiconductor — and even then, it depends on whether you have an even or odd number of layers; you can break the
inversion symmetry.
So those are interesting possibilities and interesting structures!
Let’s look at them a little more. Graphene is a 2D semimetal, as all of you know. Hexagonal boron nitride is a very wide bandgap insulator. Transition-metal dichalcogenides — well, if you look at the
special case of the 2H phase — it’s a 2D semiconductor.
All three of these materials have interesting “valley” degrees of freedom, in addition to the charge and spin degrees of freedom. In particular, in transition-metal dichalcogenides, if you analyze
the monolayer, you’ll find it has two interesting properties: first, it has an interesting direct bandgap, and second, the K and K' are located in different valleys, and the spin degrees of freedom
will actually flip as the result of very strong spin-valley coupling, and that makes it very exciting for a lot of potential optoelectronic, optospintronic, optovalleytronic applications.
This is an example of a material that that you can grow using chemical vapor deposition. It’s a large grain single crystal, and it looks very pretty.
WS₂ monolayer: Large domains w/ strong fluorescence
I’ll talk about hexagonal boron nitride, next. I’m not going to talk about graphene, because everybody here knows about it — although I will mention that in graphene, you have K and K' valley and
this is gapless. By contrast, hexagonal boron nitride is a parabolic material. The in-plane dielectric constants is positive. Out-of-plane is negative.
This is a natural thing in the three dimensional limit. It’s a very interesting waveguide.
Now, you can start putting all those things together and start imagining what may be happening.
There’s one interesting thing about “valley degrees of freedom.” I don’t have time to go into details, but I’ll just give you a general idea. Traditionally, solid state physics textbooks tell you
that the derivative of the dispersion relation of energy relative to momentum will give you the velocity of electrons in the solid. But the reality is that, sometimes, you have very interesting Berry
connections related to your Bloch electrons. You can have non-trivial “pseudo magnetic field”, a non-trivial Berry curvature due to the Berry connection.
So, if you have this interesting valley degrees of freedom and you have non-trivial Berry curvature, you can actually break the inversion symmetry to lift the valley degrees of freedom, which means
this would give rise to a valley Hall effect, even in the absence of an external magnetic field.
So that’s an interesting degrees of freedom thing that one can start thinking about playing with.
I’m going to show you an example of how you actually can use nano-metamaterials on something like graphene.
Supposedly, the two valleys are degenerate. Now, if you want to do valleytronics, you want to separate them. In principle, you can play with nanostructures and then try to make things work. It has
been predicted theoretically, for instance, that — graphene has these two different sublattices, but if you strain them differently, with non-trivial strain, then you would create special Berry
curvature and generate pseudo magnetic fields. So, for one sublattice, the field can be up. For the other sublattice, the field can be down. Overall, you don’t break global time reversal symmetry;
you actually have a total flux of 0. But you can break the local time reversal symmetry so the electronic states would actually show peaks in the density of states versus energy measurements. They
would show quantized Landau levels as the result of pseudo magnetic fields.
So this was a prediction — a theory — and to maximize this kind of effect, naturally, you would think you should try to build nanostructures of this nice, small nanostructure tetrahedron. In other
words, we found a very interesting kind of material and we would like to try to create these single crystals and then put graphene on top and see what happens — in order to maximize the nanoscale
strain effect. So we take a piece of silicon, we put these palladium nanocrystals on them. Then we put a monolayer of hexagonal boron nitride — which we grow, ourselves — and that monolayer exists
specifically to isolate the next layer we want to put on, which is graphene (on top of the boron nitride).
By the way, we used room temperature PECVD techniques to grow graphene without active heating, which resulted in single crystalline like large sheets without any discernible strain.
So we can put strain-free graphene on top of our deliberately created nanostructure and see what happens!
This is what happens. If you look at the bottom of the slide above, that was taken with an SEM image. And beside it (on the bottom left of the slide) is an AFM image. You can see those nanostructures
with a layer a boron nitride and then a layer of graphene on top.
Now, if we take that structure and zoom in using a scanning tunneling microscope, you can see very sharp features (the color coding indicates the distortion, the topography). Now, you can use this
topography to figure out exactly what kind of pseudo-fields are being created.
And, lo and behold, look at this! Tens of thousands of Tesla, local magnetic field. And then, of course, you see two different colors that correspond to both positive and negative magnetic fields. If
you integrate this magnetic field distribution out, the total magnetic flux is zero. But you can now imagine this kind of strong field will have dramatic effects on graphene electronic properties.
Indeed, with this kind of effect, what we have found is that you would have Landau levels appearing, and then you would have two types of local, highly strained areas. Depending on the strain value,
you will have different energy separations. The larger the strain is, the larger energy separation you will find. You do indeed get these quantized Landau levels.
Photo courtesy of the Dresselhaus Family
Also, you see symmetry breaking. Sometimes, you see a spectrum with zero bias conduction, and sometimes you don’t. That’s spontaneous local time reversal symmetry breaking and chiral symmetry
breaking. That means you separate out the two valleys. That’s a very interesting thing! It means that, with strain, you can play with the valley degrees of freedom.
The interesting thing is, if I put two of these sharp tetrahedrons very close together and only then look at the corresponding pseudo magnetic field, the situation becomes very different. You start
seeing the pseudo magnetic field displaying very interesting patterns. Also, the net field is reduced, because there are more relaxed features. Now we can start thinking about how we might want to
play with it.
This is how we can compute the pseudo magnetic fields by using the atomic structures of the strain, and we can also measure the pseudo magnetic fields directly. One quick thing I should mention is
that we can also compute these things theoretically and make comparisons. One prediction is that, if you truly break the valley degrees of freedom, you will have valley polarization, and then half of
the spectrum should show zero bias conductance peak, and the other half should not. And, indeed, we have found that, statistically, over the strain area, the histogram of those with zero bias and
those without zero bias is almost 50% each. Therefore, the theoretical prediction is true and valley polarization is almost complete.
The top left graph is the graphene topography on two tetrahedrons, and the left bottom graph is the corresponding pseudo magnetic field distribution. We can take different line cuts and color-code
the conductance as exemplified by the top and bottom graphs on the right, which allows us to start seeing these peak features that are associated with different Landau levels. Then, we can look at
the exact Landau level separation to know what the pseudo-field is, or we can make a comparison with the topography map and then see what the corresponding pseudo field may be. We did both of these
methods and found they are in a few percentage points of agreement.
So this is all great!
Theoretically, we have computed molecular dynamics. We find that one tetrahedron is a good valley splitter, whereas you can start connecting different tetrahedrons along the zigzag directions, and
that allows you to begin creating a valley propagator.
That’s the idea that we would like to pursue.
In order to do that, however, you have to go from nanoscale to mesoscopic scale. Therefore, we are building these periodic structures and trying to look at these properties in real mesoscopic scale
and so we can attempt to do — for example — valley Hall effect measurements.
So you can imagine that if you strain the graphene everywhere (this is just a representation), then you can split the valley and see different effects. You can even imagine, if you can split the
valley and send the valley polarized current into something with strong spin-orbit coupling, then you could start having spin-polarized current.
You can play with different degrees of freedom to create new effects.
You can even shine circularly polarized light to select one valley that is more favorable over the other.
The other thing I want to talk about is novel surface and interface properties related to topological materials. Before I start, I’d like to remind you all that Einstein once suggested that the
fundamental laws of physics should be expressed in terms of geometry. He exemplified this with an idea from the ancient Greeks in which he could formulate gravity within the geometrical curvature of
space and time. These days, I would say physicists try to look beyond just space and time — and instead, try to apply Einstein’s ideas to topological materials and topological field theory.
For instance, you can make a topological insulator and, on top of that, you can put a magnetic layer. If you then put a point charge there, it will induce a magnetic monopole. This becomes something
called a dyon, which is something people are always looking for in high energy physics — which is funny because condensed matter physicists actually found them!
Or you can have boundaries between superconductors and magnetic materials, and then you can create Majorana state modes, which are related to fermions but in which the antiparticles are their own
These are different possibilities, different Hamiltonians, where you start building in these interfaces.
The point is, with interesting interfaces of novel materials, you can start creating new physical states and new concepts. Prof. Wang will talk about this later, so I will just show you the picture
that illustrates how if you play with dimensionality interfaces of materials, you see edge space, which can create novel properties. You put magnetic materials on top of topological insulators, which
gives you a bulk insulator. The surface state is a conductor with spin polarization. Originally, it’s gapless, but if you apply a net magnetization, a gap can open up. And we have done experiments
to look at those things.
Also, on these surface states of topological materials, if you put magnetic spins there, you will create different spin textures. So if the spin is out-of-plane, it’s an isolated spin and would have
a strong resonance as a single peak. If you have one parallel spin in-plane, you could have topological spin textures.
We have actually found these experimentally, using STM. They are right on top of the topological materials, on the surface of the bilayer system.
Moreover, we did some spectroscopy and discovered that although this is a gapless situation, if you turn on the magnetization, it becomes gapped.
This is a cartoon. I don’t have time to show you data.
But then, the interesting thing is that around these edges of materials with magnetism and without magnetism in gapless and gapped regions, we find very sharp and spatially localized conductance
peaks. Above is an STM image of the gap map — gap means it’s magnetic, out-of-plane, and then gapless means that magnetization can be in-plane or without magnetization. Near the edges, we find sharp
resonances — sometimes it’s a double resonance, sometimes a single resonance. Why are they there? Well, they’re related to the different edge states. The edge state is actually very different
resonant magnetic impurity resonance states.
This is a very interesting situation. It’s extremely sharp, long lived, looking like a qubit (i.e., a quantum bit). I call it a “topological bit.” It survives over merely a few angstroms spatially,
extremely sharp and long lived.
We saw this and immediately thought it might be something we could manipulate. In principle, we can manipulate them almost as if they really are qubits, although whether or not that’s true has yet to
be examined. But you can shine circularly polarized light and, by coupling the light of a specific circular polarization with those different spin textures, you can favor one peak over the other.
Here, you have two equal peaks, but you don’t have circular polarized light. We are pretty sure that if you have circularly polarized light, you can actually select one peak over the other. There are
many interesting games we can play using that idea. For instance, we can gate a system to cause entanglement among different qubits that are spatially separated.
Superconductivity. This is just to show you that people have been looking for higher T[C], and the record highest T[C], 165 K, has not been broken. But then a new class of iron-based materials was
discovered in 2008, and that’s also a high T[C] material.
Now, let’s talk about Cuprate superconductors as compared to iron-based superconductors. While they’re both high T[C] materials, they do have a lot of differences. But one thing they have in common
is that they have very strange pairing symmetry. They have unconventional pairing symmetry. Cuprate superconductors have d-wave pairing symmetry that exhibits sign-changing pairing potential. In the
case of iron-based superconductors, there’s a sign change in the pairing symmetry, but in this case it is a sign-changing s-wave. That is, you have electron and hole pockets with pairing potentials
of opposite signs.
Why is it important? Because if you look at this universal gap (D) equation for superconductors — look at the energy gap (D) as a function of momentum (k,q), and you have the sum over some
interaction matrix elements (Vk,q). The quasiparticle energy Eq is always positive definite, but for this gap, if you change the sign on the two sides of the gap equation, something unusual happens.
Traditionally, if you keep the same sign for the gap on both sides of the gap equation as in conventional superconductors, you must have a negative potential Vk,q in order to satisfy the equation.
That means your interaction potential to glue electrons together must be negative — in other words, attractive. However, if you change the sign on the two sides of the gap equation, you have a
positive interaction potential. Therefore, in your high T[C] systems, you have a repulsive interaction potential for Cooper pairs, which implies “attraction from repulsion”. That’s a very novel
Okay, but if that’s the case, we want to enhance the repulsive energy. That means we must enhance the correlation — and that enhancement of the correlation should lead to a higher T[C]!
How do you do that? It’s very hard because strong electronic correlation often breaks symmetries and gives you a bad (incoherent) state where you cannot have superconductivity. But if you have found
something that has this kind of behavior (i.e., attraction from repulsion), you can try to confine the electrons to a lower dimension — for instance, force them to only one atomic layer and see what
happens — because then, the electrons have nowhere to go. They have to interact! The correlation has to increase.
This is the final topic. Recently, these iron-based superconductors have been found to have maximum T[C] up to 55 degree Kelvin, in bulk. But this particular material, iron selenide, was different.
This research was done by a Tsinghua University group, by Professor Qi-Kun Xue’s group.
These materials have these trilayer structures in one unit cell, which may be considered as a monolayer of FeSe. But if you put just one set of such a layer on top of an insulator, the T[C] went up
by almost one order of magnitude.
And this is the evidence of T[C ]from angle resolved photo emissions spectroscopy. Sorry that I don’t have time to explain it, but you see the T[C] determined showed almost 60 to 70 degrees Kelvin.
And it was an 8 degree Kelvin material in the bulk!
This has been just a brief discussion of how we can play with quantum degrees of freedom. The point I want to convey is that there are exciting developments both in physics and in potential
technology which are emerging from these edge states of novel materials and nano-metamaterials. This science is creating new capabilities for nanoelectronics.
Simply start using your imagination! Optoelectronics, optovalleytronics, optospintronics — all coming out of these different degrees of freedom that you can manipulate in these interesting low
dimensional materials and nanostructures.
Me and Millie at Kavli, 2012
One more picture of Millie.
Thank you. | {"url":"https://millie.pubpub.org/pub/oy587eu7/release/1","timestamp":"2024-11-10T01:56:53Z","content_type":"text/html","content_length":"827659","record_id":"<urn:uuid:11578646-b864-4771-be15-a10e1e53561e>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00732.warc.gz"} |
Precession resonance of Rossby wave triads and the generation of low-frequency atmospheric oscillations
The dynamics of the Earth's atmosphere is characterized by a wide spectrum of oscillations, ranging from hourly to interdecadal and beyond. The low-frequency component of the atmospheric variability
cannot be understood solely in terms of linear atmospheric waves that have shorter timescales. A newly proposed mechanism, the precession resonance mechanism, is a regime of highly efficient energy
transfer in the spectral space in turbulent systems. Here, we investigate the role of the precession resonance, and the alignment of dynamical phases, in the generation of low-frequency oscillations
and the redistribution of energy/enstrophy in the spectral space using the barotropic vorticity equation. First, the mechanism and its ability to generate low-frequency oscillations are demonstrated
in low-order models consisting of four and five nonlinearly interacting Rossby–Haurwitz waves. The precession resonance onset is also investigated in the full barotropic vorticity equation, and the
results are in agreement with the reduced models. Efficiency peaks in the energy/enstrophy transfer also correspond to regimes of strong excitation of low-frequency oscillations. The results suggest
that the organization of the dynamical phases plays a key role in the redistribution of energy in the spectral space, as well as the generation of low frequencies in the barotropic vorticity
The dynamics of a barotropic nondivergent flow in a rotating frame has been extensively studied in the fluid dynamics literature due to its applicability as a prototype model for large- and
planetary-scale motions of the atmosphere and ocean.^1,2 According to the seminal scale analysis of large-scale motions in mid-latitudes performed by Charney,^3 these large-scale atmospheric
disturbances exhibiting a high spatial coherence, and a typical timescale ranging from a few days to a week, are characterized by a quasi-horizontal and quasi-solenoidal flow. Consequently, the
atmospheric flow can be thought of as a “noisy sea” of fast propagating gravity and acoustic waves embedded in such large-scale disturbances. The linear wave theory of the so-called barotropic
nondivergent model was introduced by Rossby^4 with the β-plane approximation and extended by Haurwitz^5 for the spherical geometry context. This model was the one utilized by Charney et al.^6 in the
first known successful numerical weather prediction and still to date constitutes an essential piece of intermediate steps in the development of general circulation models.^7 Such wave motions have
been labeled as Rossby–Haurwitz waves, and the linear wave theory has been further extended some decades later with the inclusion of wave forcings^8 and/or more realistic background flows.^9–13
Apart from the linear wave theory, a significant research effort has also been made on the role of nonlinearity on the barotropic nondivergent model. Platzman^14 described the spectral representation
of the nonlinear version of the barotropic nondivergent model, with the explicit computation of the coupling coefficients among each spherical harmonic triplet, and discussed some issues regarding
the truncation influence on the integral constraints arising from the original model equations. Platzman^15 described the time evolution of the nonlinear spectral equations by analytically solving a
highly truncated version of the system for a single triad of interacting harmonics in terms of Jacobian elliptic functions. Longuet-Higgins et al.^16 demonstrated the possibility of triads of Rossby
waves to match the so-called “resonance relations,” which impose the relative linear phase of the wave triplet not to change in time. The wave triplets satisfying these resonance relations are called
resonant triads and are believed to be associated with the leading-order nonlinear effects for a weakly nonlinear regime of any dispersive wave system having quadratic nonlinearities. A background
review of the three-wave resonant interaction can be found in Craik^17 for many fluid dynamics problems, or in Pedlosky^18 and Lynch^19 for the Rossby wave context. Longuet-Higgins et al.^16 also
computed the energy transfer rate associated with the resonant Rossby wave triads as a result of the integral constraints imposed by energy and enstrophy conservations. The findings of
Longuet-Higgins et al.^16 and Platzman^15 verified the constraints of the modal energy transfer due to the so-called Fjørtoft's theorem for a two-dimensional nondivergent flow.^20 This theorem states
that, as a consequence of total energy and total enstrophy conservations, the mode with the intermediate total wavenumber either receives energy from or supplies energy to the remaining modes of an
interacting triad. For a resonant triad, this mode has also the highest absolute eigenfrequency and is usually referred to as “pump mode” from the plasma physics literature.^21
Consequently, resonant triads have been useful to understand the stability properties of a single Rossby mode to small-amplitude perturbations, for either a β-plane^22 or the spherical geometry^23,24
contexts. Due to the aforementioned energy constraints for a triad interaction resulting from energy and enstrophy invariants, a single wave mode can only be unstable to small-amplitude perturbations
^25 if (i) it is the pump mode of a triad interaction and (ii) the mismatch among the linear eigenfrequencies of the modes of this triad be smaller than a critical value that depends on the initial
amplitude of the pump wave, as will be shown in Sec. IIA. Thus, in the weakly nonlinear regime, resonant triads are crucial for triggering the wave instability. Lynch^24 performed numerical
simulations with the barotropic nondivergent model with a vorticity forcing that resonantly excites a specific Rossby–Haurwitz wave mode. He showed that, after being resonantly excited by the
forcing, this mode preferentially loses its energy to modes that constitute a nearly resonant triad with it.
Nevertheless, although resonant triads have shown to be crucial in triggering the instability of the primary wave field, the spectral broadening of the solution due to energy cascade processes that
characterize the transition to turbulence appears to rely on nonresonant triad interactions. Reznik et al.^26 studied the nonlinear interaction of Rossby–Haurwitz waves and showed that relaxing the
resonance condition for the mode eigenfrequencies to consider near resonances increases the spectral energy redistribution. In fact, in the weakly nonlinear limit, as the wave triads are assumed to
satisfy the resonance condition for the linear eigenfrequencies, the possible effect of the wave phases on the time evolution of the wave amplitudes is neglected, leading to the so-called random
phase hypothesis.^27,28 On the other hand, by relaxing the weakly nonlinear limit to consider off-resonant triads, some studies have pointed out the important role of the nonlinear dynamical phases
on the energy flow throughout the modal space.^29–31 Chian et al.^31 showed that an amplitude-phase synchronization associated with multiscale interactions in a one-dimensional model of regularized
long-wave equation is responsible for an on–off intermittency at the onset of spatiotemporal chaos. This mechanism has also been demonstrated in other turbulence contexts, such as magnetized
three-dimensional Keplerian shear flows.^32
Another mechanism that has shown to be responsible for a significant energy flow throughout the whole spectral space refers to the so-called precession resonance mechanism. The name of this mechanism
was chosen due to the precession of the nonlinear dynamical phases, and it has a loose analogy to the precession of celestial bodies, but should not be confused with it. The mechanism has been
proposed by Bustamante et al.^33 to maximize the energy transfer between different interacting wave triplets and to enhance the cascade process, in the context of drift waves. This resonance takes
place when there is a balance between the characteristic precession frequency of the combined phase of one triad and the nonlinear frequency of amplitude modulation of another wave triplet. According
to Bustamante et al.,^33 for a moderate wave amplitude regime, the characteristic precession frequency of the combined phase of an interacting triad receives significant contribution from the
mismatch among its linear eigenfrequencies. This mechanism has also been analyzed in the context of Burgers' equation in Murray and Bustamante,^34 who showed that bursts of strong energy transfer
throughout the spectral space are interchanged with periods of weak interactions; this interchange between strong and weak energy fluxes throughout the spectral space was found to be associated with
synchronized and unsynchronized regimes, respectively, between amplitudes and dynamical phases of the wave triplets. See also Arguedas-Leiva et al.,^35 where the relation between phase
synchronization across spatial scales and turbulent intermittency was demonstrated. Furthermore, the deep connection between precession resonance and the region of convergence of the wave-turbulence
normal form transformation in the barotropic vorticity equation was studied in Walsh and Bustamante.^36
Another important aspect of the precession resonance refers to its ability in generating low-frequency fluctuations in a wave system, as demonstrated by Raphaldini et al.^37 in the context of
magnetohydrodynamic (MHD) Rossby waves in the solar tachocline. According to Raphaldini et al.,^37 the inter-triad energy transfers associated with the precession resonance occur in a longer
timescale than the intra-triad energy exchanges, thus yielding longer-period modulations on the energy exchanges within a resonant triad. This aspect of the precession resonance should also be useful
in the atmospheric dynamics context. In fact, in the atmospheric flow, the low-frequency variability is defined as oscillations with periods beyond 10days and therefore is responsible for modulating
the weather systems and consequently yielding climate anomalies. To date, the mechanisms behind the generation of the atmospheric low-frequency variability still constitute a topic of current
research in the atmospheric sciences, as these timescales are longer than the characteristic periods of the atmospheric normal modes.^1 Nonlinear triad interactions may play some roles in the
dynamics of these low-frequency modes. This has been suggested by Kartashova and L'vov^38 and Raupp and Silva Dias^39 for the case of intraseasonal variations, an important component of the
atmospheric low-frequency variability.
Here, we analyze the precession resonance mechanism in the barotropic nondivergent model, with parameters relevant for the study of atmospheric flows. The main purpose is to study the role of this
mechanism in the redistribution of energy in spectral space and the generation of low-frequency oscillations. We first investigate the mechanism in the reduced spectral equations of four (two
interacting triads connected by two modes) and five (two interacting triads connected by a single mode) wave modes, selecting triads involving a prevalent atmospheric mode that has been shown to be
unstable in previous studies.
This mode is the pump wave in many triad interactions, making the excitation of neighboring modes in spectral space efficient. We vary the initial conditions and show that the maximal efficiency of
energy/enstrophy transfer between two different triads is associated with the precession resonance regime. Furthermore, numerical simulations with the full spectral equations (i.e., considering a
regular truncation), with the initial condition projected only onto a resonant triad, demonstrate that the precession resonance associated with the four-wave problem is responsible for the energy
leakage from the primary resonant triad and, therefore, the initiation of the cascade process. The results of both the reduced and the full spectral models show that the precession resonance yields
low-frequency modulations in the energy exchanges associated with the primary resonant interaction. Therefore, based on our theoretical and numerical results, we conjecture that this mechanism may be
responsible for generating considerable low-frequency variability in the atmosphere and ocean.
The dynamics of a barotropic nondivergent flow in a rotating frame is governed by the so-called barotropic nondivergent model. This model is described by the conservation of absolute vorticity, which
can be written in the conservative form as follows:
In the equation above, $v→(ϕ,θ)$ is the solenoidal velocity field, and $ζ=(∇×v→)·r→$ is the relative vorticity, with $r→$ representing a radial unit vector with respect to the sphere; $f=2ΩE sin (θ)$
is the planetary vorticity, where Ω[E] is the Earth's rotation rate and $(ϕ,θ)$ refers to the longitude–latitude coordinate system ($ϕ∈[0,2π]$ and $θ∈[−π/2,π/2]$). Using the streamfunction
where $J(·,·)$ is the Jacobian operator in spherical coordinates
$J(h,g)=1a2 cos θ(∂h∂ϕ∂g∂θ−∂h∂θ∂g∂ϕ),$
for any differentiable functions h and g, and a denotes the Earth's radius.
From (1), it follows that the mean relative vorticity is conserved. However, the conserved quantities that are relevant for the wave interactions are those having a quadratic dependence in terms of
the disturbance variables.^16,20,40 The quadratic conserved quantities of (1) refer to total energy E and total enstrophy $E$, defined, respectively, by
where S represents the surface of the sphere.
Equation (2) can be solved according to the following expansion in terms of the eigensolutions of its linearized version:
$ψ(ϕ,θ,t)=∑n=0∞∑m=−nnAn,m(t)Ynm(ϕ,θ)e−iωn,mt=∑n=0∞∑m=−nnAn,m(t)Pnm(sin θ)ei(mϕ−ωn,mt),$
where $i=−1$ refers to the imaginary unit. In (6), $An,m(t)$ denotes the complex-valued spectral amplitudes, m is the zonal wavenumber, and n is the meridional structure index; $Ynm(ϕ,θ)$ are the
spherical harmonics, which are eigenfunctions of the spherical Laplace's operator
$Pnm(sin θ)$ indicates the associated Legendre polynomials, and the eigenfrequencies $ωn,m$ satisfy the dispersion relation of Rossby–Haurwitz waves
Expansion (6) is an exact solution of the barotropic vorticity equation (2) provided the spectral amplitudes satisfy
In the equation above, we have adopted a simplified notation in which each spherical harmonic is specified by a single index, viz., $(nj,mj)→j, (nk,mk)→k$ and $(nl,ml)→l$. Thus, $δk,lj=ωk+ωl−ωj$ is
the mismatch among the mode eigenfrequencies of a wave triplet {j, k, l}, and $Ck,lj∈ℝ$ refers to their nonlinear coupling coefficients, given by
where $z=sin θ$. Notice that $Ck,lj=Cl,kj$ in general. In order for the nonlinear coupling coefficients to be nonzero, the respective mode indices must satisfy the so-called Ellsaesser conditions^41
$mk+ml=mj,(mj)2+(ml)2≠0,nknjnl≠0,nk+nj+nl is odd,(nk2−|mk|2)+(nl2−|ml|2)>0,|nk−nl|<nj<nk+nl,(mk,nk)≠(mj,nj),(ml,nl)≠(mj,nj).$
The coupling constants $Ck,lj$ contain all the information on the nonlinearity of the original equation (2). In addition, as a consequence of the integral constraints (4) and (5), for any triad of
wave modes satisfying conditions (10), the nonlinear coupling coefficients must satisfy the relations (see the Appendix A for details)
Equation (11) refers to the Fjørtoft's theorem for a two-dimensional nondivergent flow.^20 It implies that
Thus, the mode with intermediate total wavenumber will always receive energy from or supply energy to the remaining triad components. This defines the pump mode. For resonant triads ($δk,lj=0$),
the pump mode has also the highest absolute eigenfrequency. Condition (11) also implies that total energy and total enstrophy are conserved for an arbitrary truncation of the spectral equations (8).
Condition (11) has also been demonstrated by Longuet-Higgins et al.^16 for both the barotropic vorticity equation and the quasi-geostrophic potential vorticity equation in which a finite divergence
is considered in the β-plane context. Ripa^40 presented a more general framework valid for nonlinear systems having two quadratic conserved quantities and obtained a similar result, but with the
total wavenumber squared in (11) being replaced by the slowness index defined as the inverse of the wave phase speed. However, as can be noted from (7), for Rossby waves the slowness index is
proportional to the total wavenumber squared so that (11) is consistent with Ripa's theoretical results.
Before describing the precession resonance mechanism in more detail, it is suitable to review some basic concepts of a single wave triad dynamics.
A. Three-wave interaction and wave stability
Let us truncate expansion (6) to consider only a set of three spherical harmonics $(n1,m1),(n2,m2),(n3,m3)$ satisfying conditions (10). In this context, Eqs. (8) now read
. It was shown in Lynch
that the coefficients
, and
are real and have the same sign for any given triad satisfying the Elsaesser conditions
. For simplicity of presentation, we will assume in this subsection that the mode
is a pump mode, namely, that
is the coefficient with the largest size, as per Eq.
. The equations above constitute the most elementary truncation of
that exhibits nonlinearity. Following Teruya and Raupp,
to analyze the stability of a single wave mode in a triad interaction, it is suitable to eliminate the oscillatory exponential factor in the RHS of
by making the transformation
. In this context, the above system can be written as
where we have omitted the “∼” in the equations above for simplicity. Also, assuming that the pump mode holds most of the initial energy, $|A3(0)|≫|A1(2)(0)|$, the system above may be approximated by
its linearized version
Thus, instability occurs if
The first condition is always satisfied as per Lynch's result previously mentioned. For obvious reasons, this instability is called pump wave instability.^21 The second condition implies that
off-resonant triads require a higher amplitude level for pump wave instability to be possible. Consequently, for an initial amplitude level compatible with a weakly nonlinear regime, resonant and
nearly resonant triads play a crucial role to trigger wave instability.
The above theoretical result on the pump wave instability in an interacting triad is consistent with numerical studies on Rossby wave instability. The stability of a Rossby wave field has been a
topic of considerable research effort due to its role in initiating the energy cascade process responsible for the transition to a turbulent regime and therefore the breakdown of weather
predictability. Lorenz^43 studied the barotropic instability of a flow associated with a Rossby wave mode, and Hoskins^44 performed numerical simulations to analyze the stability of a Rossby–Haurwitz
wave field superimposed to a zonal flow. Gill^22 and Baines^23 were perhaps the firsts to study the Rossby wave instability on the optics of nonlinear triad interaction. Gill^22 showed that a Rossby
wave mode is always unstable irrespective of its amplitude, but for small amplitudes, the destabilizing disturbances always constitute a resonant triad with the primary wave. However, Gill^22
considered an infinite $β−$ plane in which the continuum spectrum allows a much broader possibility of resonant triads. By contrast, in the spherical geometry, Baines^23 found a narrower range of
possible unstable wave modes than Gill,^22 with some modes requiring a high amplitude to be unstable, possibly due to the more restricted set of resonant triads in the spherical geometry as a
consequence of the discrete spectrum of wave modes.
Moreover, comparing the studies of Hoskins^44 and Baines^23 in the spherical geometry context, Hoskins^44 found a more restricted set of unstable modes, showing that only modes with n>5 are
unstable; however, his spectral model considered only zonal wavenumbers that are multiples of 4, therefore excluding many possibilities of interacting triads. On the other hand, Baines^23 considered
a larger number of destabilizing disturbances (30 modes) and obtained that traveling wave modes having wavenumbers n=3 and 4 can be unstable, with required amplitudes being smaller than those
proposed by Hoskins.^44 This difference between the findings of Hoskins^44 and Baines^23 can be attributed to the higher possibilities of resonant triad interactions in the latter's spectral
equations. In fact, Lynch^24 demonstrated that the instability of the wave mode $(n,m)=(5,4)$, which had also been verified in the numerical simulations of Thuburn and Li,^45 is due to the nearly
resonant triad interaction with modes (3, 1) and (7, 3). These wave modes will be considered in our numerical simulations displayed in Secs. III and IV.
The nonlinear system (13) has the following conserved quantities: the Manley–Rowe invariants
$I12=|A1|2C2,31+|A2|2C3,12, I23=|A2|2C3,12+|A3|2C1,23,$
$H=−1C2,31C3,12C1,23(Re(A1A2A3* e−iδ t)+δC1,23|A3|2).$
With these quantities, the triad equations may be integrated by quadrature, reducing it to a single equation. The solution is periodic and may be represented in terms of Jacobian elliptic functions,
as will be shown in Sec. IIB. The single triad system with a constant forcing, which is relevant in many contexts,^39 was also shown to be integrable.^46
B. The precession resonance
The precession resonance was introduced by Bustamante et al.^33 as a highly efficient energy transfer mechanism throughout the modal space that takes place when multiple wave triads are connected. In
order to introduce this concept in more detail, it is convenient to use the polar representation of the spectral coefficients
where $Bj=|Aj|$ and $φj=arg(Aj)$. In this representation, the system of three coupled complex valued differential equations (13) is reduced to a system of four real-valued equations, three for the
moduli of the spectral amplitudes and one for the combined phases
$dB1dt=C2,31B2B3 cos (Φ),$
$dB2dt=C3,12B1B3 cos (Φ),$
$dB3dt=−C1,23B1B2 cos (Φ),$
$dΦdt=−δ−B1B2B3(C2,31B12+C3,12B22−C1,23B32) sin (Φ),$
$Φ(t)=φ1(t)+φ2(t)−φ3(t)−δ t$
is the so-called dynamical phase, and $δ:=ω1+ω2−ω3$ is the mismatch among the linear eigenfrequencies. In the case δ = 0, the solution for the moduli B[j] can be expressed as (e.g., Refs. 15 and 47)
$|B1(t)|2=−μC2C3(2K(μ)T)2sn2(2K(μ)(t−t0)T,μ)+I133C2C3(2−ρ+21−ρ+ρ2 cos (ν/3)),$
$|B2(t)|2=−μC1C3(2K(μ)T)2sn2(2K(μ)(t−t0)T,μ)+I133C1C3(2ρ−1+21−ρ+ρ2 cos (ν/3)),$
$|B3(t)|2=μC1C2(2K(μ)T)2sn2(2K(μ)(t−t0)T,μ)2+I133C1C2(ρ+1−21−ρ+ρ2 cos (ν/3)),$
while the solution for the dynamical phase is given by
with $sn, cn$, and $dn$ representing the Jacobi elliptic functions (sine, cosine, and delta amplitude, respectively) and $K(μ)$ the elliptic integral of the first kind
where the parameter μ is given by
$μ= cos (ν/3+π/6) cos (ν/3−π/6),$
with $ρ=I23/I13$ and $ν∈[0,π]$ being an angle such that
$cos (ν)=(−2+3ρ+3ρ2−2ρ3)I133−27H22(1−ρ+ρ2)3/2I133.$
In addition, in Eqs. (20), the constants C[1], C[2], and C[3] refer to the nonlinear coupling coefficients, $C1=C2,31, C2=C1,32$, and $C3=C1,23$. The solutions for the amplitudes and dynamical
phase presented above are periodic in time, with period T being defined as follows:
$T=31/42 K(μ)(1−ρ+ρ2)1/4 cos (ν/3−π/6)I13.$
In the case $δ≠0$, the solutions have a similar form. In particular, the system is still integrable, and the amplitudes $B1,B2,B3$ oscillate with a period T, satisfying a formula similar to Eq. (25)
. See Harris,^48 Chap. 3.2, for details, including the solution for the dynamical phase. Crucially, a remarkable difference is noted in the behavior of the dynamical phase $Φ$. In the case δ = 0,
the dynamical phase $Φ$ is periodic in time, with period T, but when $δ≠0$ is large enough, the dynamical phase $Φ$ has phase slips that occur periodically with period T, leading to an overall
“drift” or “precession” when considering the dynamical phase as being defined on the torus $T$. Hence, in general one can define two characteristic frequencies associated with the dynamics of an
interacting wave triad: $Γ=2π/T$, the frequency of energy exchanges among the waves, and Ω, the so-called precession frequency of triad (1, 2, 3), given by
with the dynamical phase $Φ$ being defined in (19) and $⟨·⟩$ indicating the time average. A schematic view of the meaning of the precession frequency will be shown in Sec. IIIA, Fig. 3.
As an integrable system, the triad equations are completely characterized by these two frequencies (Ω and Γ). In general, when these frequencies are incommensurable ($Ω/Γ∈ℝ$\$ℚ$), the system will
be quasi-periodic and the orbits will live in a two-dimensional torus $T2$. By contrast, for special cases in which $Ω/Γ$ is a rational number, the solution is periodic in time. Thus, by
manipulating the initial conditions, one can change the values of the constants of motion and consequently these characteristic frequencies of the solution. An easy way to do this is to re-scale the
initial amplitudes as $Bj(0)→αBj(0)$, j=1, 2, 3, with α being a nondimensional real number. From Eq. (25), it is evident that the period of energy exchange, T, scales as the inverse of the initial
amplitudes ($T∝1/I∝Bj−1$), where I is a Manley–Rowe constant. This type of manipulation will be performed throughout this study in cases of nonintegrable systems of nonlinearly interacting wave
modes as a way to explore different balances between the different frequencies of the system.
When two triads, a and b, are connected by either four- or five-wave configurations, although the system is no longer integrable, it is still useful to characterize each of the triads by these two
frequencies, $(Γa,Ωa)$ and $(Γb,Ωb)$. Assuming that the energy of triad a is initially much higher than that of triad b, the coupling terms between the two triads will make the primary triad, triad
a, to act as a forcing for the secondary one, triad b. In this context, resonance occurs when the precession frequency of triad b matches one of the Fourier harmonic frequencies of the amplitude
modulation of triad a, viz.,
where $p∈ℤ$. The equation above defines the so-called precession resonance between triads a and b.^33 In practical applications, the primary triad that contains most of the initial energy is nearly
resonant ($δa≈0$), as discussed in Sec. IIA. In addition, in the secondary wave triplet, the amplitude of one (two) of the modes is negligible for the four (five)-wave system. As we will see in
Sec. IIC, in the five-wave system, a small perturbation in the energy of the secondary modes of triad b is required for these modes to be excited; for the four-wave system (Sec. IID), however, as
the triads are connected by two modes, the secondary mode of triad b can be excited even if it has zero energy initially. Consequently, the precession frequency of the secondary triad is initially
dominated by the mismatch among the linear eigenfrequencies, $Ωb≈δb$. However, as time evolves and the secondary modes of triad b are sufficiently excited, the nonlinear terms involving the mode
amplitudes in (18d) will become important as well so that the synchronization defined by (27) will involve the fully nonlinear dynamical phases of the triads, as will be shown in Secs. III and IV.
Bustamante et al.^33 showed that, in the precession resonance regime, there is a peak in the efficiency of energy exchange between the triads. This process gets more intricate in systems involving
many triad interactions, such as in full partial differential equation models. The effects of the precession resonance are still noticeable in these full spectral models.^33,34 Here, we first
investigate the onset of the precession resonance mechanism in the reduced models of five- and four-wave modes shown, respectively, in Secs. IIC and IID. We demonstrate that, in the precession
resonance regime, the solutions display long-term modulations as a result of inter-triad energy exchanges, an aspect that was not explored in Bustamante et al.,^33 but may have important consequences
in the field of atmospheric fluid dynamics. We then proceed to investigate the effects of the precession resonance in the full barotropic vorticity equation, focusing on the redistribution of energy
and enstrophy in the large-scale circulation patterns.
Among the possible types of precession resonance, one of them, the phase alignment or synchronization, which consists of the case p=0 in Eq. (27), is verified in our numerical results presented in
Secs. III and IV and seems to be of great relevance in several other contexts, for instance, in the dynamics of the Burgers' equation^34,49,50 and in MHD waves in the solar tachocline.^51 This
mechanism fits in a more general context as one of the mechanisms involving the role of the dynamical phases' organization on the nonlinear energy transfer; other studies within this research line
include Chian et al.^31 in the context of the regularized long-wave equation and Miranda et al.^32 in the context of 3D astrophysical flows. The organization of the waves' phases in those settings is
often related to coherent structures in the physical space.^35
In the simplest setting, the precession resonance can be introduced in systems with four- or five-wave modes arranged in two triads. These reduced spectral models of (2) will be introduced in Secs.
IIC and IID and will be studied by numerical simulations in Sec. III with realistic atmospheric parameters.
C. Five-wave model
As a first step in complexity to study the precession resonance, one might consider a set of two-wave triads interacting with each other through one common mode. If one truncates expansion (6) to
consider only five spherical harmonics $(n1,m1),(n2,m2),(n3,m3),(n4,m4)$, and (n[5], m[5]) in which modes (1, 2, 5) and (3, 4, 5) satisfy (10), it follows that the corresponding spectral amplitudes
are the mismatches among the mode eigenfrequencies of triads (1, 2, 5) and (3, 4, 5), respectively.
A set of independent Manley–Rowe invariants of system (28) is given by
$J1,2=|A1|2C2,51−|A2|2C5,12, J3,4=|A3|2C5,43−|A4|2C5,34.$
Unlike the single-triad equations, system (28) does not generally have enough invariants to make it integrable so that trajectories may live in a higher dimensional phase space, with possibly chaotic
D. Four-wave system
A second way to connect two interacting wave triads is by having two modes in common. In this way, if one truncates expansion (6) to consider only four spherical harmonics $(n1,m1),(n2,m2),(n3,m3)$,
and (n[4], m[4]) in which modes (1, 2, 3) and (1, 2, 4) satisfy conditions (10), it follows that the corresponding spectral amplitudes satisfy
where $δ1=ω1+ω2−ω3$ and $δ2=ω1+ω4−ω2$ are the mismatch frequencies of each triad. The invariants of this system are
As in the five-wave model, the invariants above are not enough to make system (31) integrable in general.
Here, we analyze inter-triad energy and enstrophy fluxes in the highly truncated systems (28) and (31). As discussed in Secs. I and II, resonant and nearly resonant triads play a crucial role in the
initial pump wave instability, whereas the energy leakage from the primary (nearly) resonant triad associated with the spectral broadening of the solution can also be due to off-resonant triads.
Thus, we have integrated these two systems for some representative examples consisting of a base triad composed of Rossby–Haurwitz waves with spherical harmonics (5, 4), (3, 1), and (7, 3). As
mentioned in Sec. IIA, this triad is a nearly resonant one and, as demonstrated by the numerical simulations with the barotropic vorticity equation performed by Lynch,^24 constitutes the leading
destabilizing disturbances for the unstable mode (5, 4). Other works have investigated the role of this mode,^23,24,45 finding it to be unstable by considering a larger number of modes in the
truncation of the barotropic nondivergent model (and, therefore, a larger number of destabilizing disturbances). Lynch^24 considered in his numerical simulations a vorticity forcing resonating with
the Rossby–Haurwitz mode (5, 4). After this mode is sufficiently excited, it was found that it preferentially excites modes (3, 1) and (7, 3), with a complete cycle of energy exchanges within this
triad occurring until energy leaks to secondary modes. In addition, observational studies of the normal mode spectra based on reanalysis data sets have shown evidence to support the important role of
mode (5, 4). Gehne and Kleeman^53 showed that the main peak in the wavenumber-frequency spectrum of the zonal velocity field corresponds to barotropic Rossby–Haurwitz waves having zonal wavenumbers
m=3–4 and a meridional structure compatible with the spherical harmonic (5, 4), although they utilized a different basis function set given by the Hough vector harmonics.^54–57
In order to demonstrate the effect of the precession resonance in maximizing the inter-triad energy transfer, in the integration of the five-wave system (28), we consider a triad cluster in which the
base triad is connected with modes (7, 4) and (3, 0) via the pump mode (5, 4). For the four-wave system integration, the base triad is coupled with mode (9, 2) through modes (3, 1) and (7, 3).
A schematic view of these triad systems is presented in Fig. 1. The parameter range of the experiments is chosen so that the resulting velocity fields exhibit physically realistic values (i.e., $|v→|
<50 m/s$). While a systematic study of the efficiency landscape in a generic cluster of four or five waves is beyond the scope of this study, the reader is referenced to Murray^50 (Chap. 6) for a
more detailed study along this line.
A. Example 1: Four-wave model
Here, we show results of numerical integrations of system (31) to illustrate the precession resonance mechanism and its ability to maximize the efficiency of energy transfer between two different
wave triads and to yield low-frequency oscillations. To perform these numerical simulations, we select a representative example of four spherical harmonics obeying the four-wave system (31) given by
${(n1,m1),(n2,m2),(n3,m3),(n4,m4)}$ = ${(3,1),(7,3),(5,4),(9,2)}$ (Fig. 1). In this case, the corresponding mode eigenfrequencies (normalized by $2ΩE$) are $ω1=−0.083, ω2=−0.053, ω3=−0.133$, and
$ω4=−0.022$, corresponding to the mismatch frequencies $δ1=ω1+ω2−ω3=−0.003$ for the first triad and $δ2=ω1+ω4−ω2=−0.051$ for the second one. The corresponding nonlinear coupling coefficients are
given by $(C2,31,C1,32,C1,23)=(−0.773,−2.497,−3.270)$ (triad 1) and $(C2,41,C1,24,C1,42)=(−0.569,−5.527,−6.096)$ (triad 2).
According to Eq. (26), we further define the precession frequencies associated with the two interacting triads of the system
$Ω1,23=limτ→∞Φ1,23(τ)−Φ1,23(0)τ; Ω1,42=limτ→∞Φ1,42(τ)−Φ1,42(0)τ,$
$Φ1,23=φ3−φ2−φ1+δ1 t=φ̃3−φ̃2−φ̃1,Φ1,42=φ2−φ4−φ1+δ2 t=φ̃2−φ̃4−φ̃1,$
and $φ̃j=φj−ωjt$ are the original phases of the waves [namely, the argument of $Aj(t)e−iωjt$]. The integrations have been performed by using an explicit eighth-order Runge–Kutta numerical scheme. The
initial condition considered in our numerical simulations is given by $(A1,A2,A3,A4)=α×(1.0,1.0,1.0,1.0×10−3)/a$, where α is a free parameter to control the magnitude of the initial wave fields and,
consequently, the frequency of the energy modulation of the base triad interaction. Thus, mode 4 has very small initial energy compared with the modes of triad $({1,2,3})$ (six orders of magnitude
smaller), since it will be considered as the target mode to analyze the efficiency of the inter-triad energy transfer. The integrations have been performed by varying the parameter α from 0 to 100 in
order to find the maximum efficiency of energy transfer from the primary [(1, 2, 3)] to the secondary [(1, 2, 4)] triad that characterizes the precession resonance.^33 The initial mode amplitudes
described above constitute a simple way of investigating the precession resonance mechanism in which the characteristic frequencies of the primary triad are changed without modifying the initial
energy distribution of this triad. However, a similar analysis on the inter-triad energy exchange efficiency can also be made considering different regimes of energy exchange associated with the
primary triad by independently varying two of its initial mode amplitudes. This analysis is presented in Appendix B.
In order to measure the efficiency of the energy transfer from the primary to the secondary triad, we define the following quantity that is bounded between 0 and 1 and is zero when all the energy of
the system is in the primary triad composed of modes 1, 2, and 3:
where $κj=nj(nj+1)$, with the efficiency being defined as $ε=maxt(Q(t))$. Similarly, in the enstrophy case
with the efficiency being again defined as $ε̃=maxt(Q̃(t))$.
Figure 2 displays the inter-triad energy [Fig. 2(a)] and enstrophy [Fig. 2(b)] transfer efficiencies, the corresponding power spectrum in the low-frequency range [Fig. 2(c)] and the precession
frequencies of the two-wave triplets [Fig. 2(d)]. All these quantities are displayed as a function of the initial condition free parameter α. The spectral power in the low-frequency range is computed
from the time evolution of a particular mode energy (mode j) by
where $ω̃$ is defined such that $2π/ω̃=10$ days, and the “$̂$” indicates the time Fourier transform.
From Figs. 2(a) and 2(b), one notices that the energy/enstrophy transfer efficiencies are characterized by a broad peak centered at about $α≈15$. From this value of α, the efficiency decreases until
$α≈26$. For larger values of α, the energy/enstrophy transfer efficiencies increase with the initial condition-free parameter. It is also noticeable that the corresponding power spectrum in the
low-frequency range [Fig. 2(c)] exhibits a high power in low frequencies only in the first half of the peak of the energy/enstrophy transfer efficiency, followed by an abrupt decrease in the
low-frequency power for increasing values of α. Comparing Fig. 2(d) with Figs. 2(a)–2(c) shows that the energy/enstrophy transfer efficiencies and low-frequency spectral power are determined by a
combination of both precession frequencies. While the α range corresponding to the region with high power in the low frequencies corresponds to a case of precession resonance characterized by the
phase alignment of $Ω1,42$, the beginning and end of the broad efficiency peak are marked by sharp changes in the regime of $Ω1,23$.
In order to illustrate the meaning of the precession frequencies, Fig. 3 shows the time evolution of the original individual wave phases $φ̃j=φj−ωjt$ (left panels) and the dynamical phases of triads
(1, 2, 3) and (1, 2, 4) (middle panels), together with the corresponding estimated precession frequency of triad (1, 2, 4) (right panel), referred to some of the integrations illustrated in Fig. 2.
We recall that the phases of the individual wave modes are defined by $φj=arg(Aj),j=1,2,3,4$. Thus, the phases are unwrapped, meaning that, for each complete cycle, we sum a $2π$ increment to the
evaluated angle. Figures 3(a), 3(b), and 3(e) refer to an integration near the precession efficiency peak ($α=7.1)$, while Figs. 3(c) and 3(d) refer to an integration away from the inter-triad
energy transfer efficiency peak ($α=25.0$). The triad phases are given by $Φ123(t)=φ3(t)−φ1(t)−φ2(t)+δ1 t$ and $Φ142(t)=φ2(t)−φ1(t)−φ4(t)+δ2 t$. Graphically, the precession frequency in the time
interval $[T1,T2]$ can be estimated from the mean slope of the triad phase curves
One observes that, for $α=7.1$ [Figs. 3(a) and 3(b)], while the precession frequency $|Ω123|$ is nearly 0, the precession frequency of the second triad, $|Ω124|$, is much higher in comparison. With
an increase in the energy level of the initial condition by setting $α=25.0$ [Figs. 3(c) and 3(d)], the evolution of the individual phases changes completely [see Fig. 3(a) in comparison with Fig. 3
(c)], most notably for modes 2 and 3 (mode 3 in fact reverses its propagation from westward-anticlockwise to eastward-clockwise). The resulting triad phases change their behavior; while $Φ123$ has a
much higher slope, $Φ124$ decreases its slope, now growing slower than the former. This means that $|Ω123|>|Ω124|$ in this case.
To confirm whether the precession resonance is responsible for generating low-frequency modulations, Fig. 4 shows the spectral power of the solution in low frequencies (periods longer than 10days),
together with the corresponding time evolution of the mode energies, for different values of the parameter α. It is evident from Fig. 4 the strong correlation between the low-frequency power spectrum
of the solution and the efficiency of energy transfer between the two-wave triplets. For α = 1, one notices the target mode being excited with very weak efficiency of about 0.3%, with a primary
periodicity of about 10days and a secondary low-frequency periodicity of about 90days. Thus, although a low-frequency fluctuation in the four-wave system solution is generated in this case, its
energy/power is very weak. For $α=7.1$, which is close to the precession resonance regime, one notices a much higher inter-triad energy transfer efficiency of about 9% and a stronger spectral power
in low frequencies, predominantly with an approximate 15-day periodicity. For $α=25.0$, one notices a substantial decrease in the low-frequency power, with characteristic oscillations being
dominated by periodicities shorter than 2days.
B. Five-wave model
Here, we consider a pair of wave triads consisting of modes ${(n1,m1),(n2,m2),(n3,m3),(n4,m4),(n5,m5)}={(3,0),(7,4),(3,1),(7,3),(5,4)}$ obeying system (28) (Fig. 1). In this case, the corresponding
mode eigenfrequencies (normalized by $2ΩE$) are given by $ω1=0.0, ω2=−0.071, ω3=−0.083, ω4=−0.053$, and $ω5=−0.133$, corresponding to the mismatch frequencies $δ3=ω1+ω2−ω5=0.062$ for the first
triad and $δ4=ω3+ω4−ω5=−0.003$ for the second one. The corresponding nonlinear coupling coefficients are $(C2,51,C1,52,C1,25)=(0.655,2.116,2.771)$ [triad (1, 2, 5)] and $(C4,53,C3,54,C3,45)=
(−0.773,−2.497,−3.270)$ [triad (3, 4, 5)]. Similarly to the previous example, the initial condition adopted in the numerical integrations is given by $(A1,A2,A3,A4,A5)=α×(1.48×exp (1.530i)
×10−8,1.71×exp (0.304i)×10−8,1.03×exp (1.110i)×10−5,1.22×exp (5.060i)×10−5,1.17×exp (3.950i)×10−5)$. Thus, modes 1 and 2 are the target modes in this case, having their energy six orders of
magnitude smaller than the modes of the primary triad, (3, 4, 5). Consequently, the inter-triad energy transfer efficiency is defined by using the quantity
where $κi=ni(ni+1)$. The efficiency is then defined as the maximum in time of this quantity, $ε=maxt(Qt)$. For the enstrophy case, the analogue quantity is defined as
with again the efficiency being defined as the maximum in time of this quantity, $ε̃=maxt(Q̃t)$. Also, similarly to the four-wave example, we define the precession frequencies by
$Ω1,25=limτ→∞Φ1,25(τ)−Φ1,25(0)τ; Ω3,45=limτ→∞Φ3,45(τ)−Φ3,45(0)τ,$
$Φ1,25=φ5−φ2−φ1+δ3 t , Φ3,45=φ5−φ4−φ3+δ4 t.$
Figure 5 shows the efficiency of energy [Fig. 5(a)] and enstrophy [Fig. 5(b)] transfers to the target modes, the low-frequency power spectrum computed from (37) [Fig. 5(c)], and the precession
frequencies of the triads [Fig. 5(d)], with all these quantities being displayed as a function of the initial condition control parameter α. One can note that the maximum efficiency of energy/
enstrophy transfer occurs for α = 31. This value corresponds to the initial condition in which the nonlinear frequency of energy modulation of triad (3, 4, 5) approximately equals the mismatch among
the linear eigenfrequencies of modes 1, 2, and 5 (precession resonance regime). In addition, it is noticeable the significant positive correlation between the inter-triad energy transfer efficiency
and the low-frequency power spectrum, with the low-frequency spectral power maximum coinciding with the maximum of the energy/enstrophy transfer efficiencies. From Fig. 5(d), one notices that, while
the precession frequency $Ω3,45$ of the primary triad remains stable for all the range of α's studied here, the precession frequency of the secondary triad, $Ω1,25$, varies as a function of the
parameter α, with some abrupt changes that coincide with the interval with highly efficient energy and enstrophy transfers. In particular, the peak of the efficiency of energy and enstrophy transfers
occurs approximately when $Ω1,25$ crosses 0. $Ω1,25≈0$ means that there is an alignment of the phases of the respective triads. The alignment of the phases has been reported as a regime of highly
efficient energy transfers in other contexts.^34,35,58
To further illustrate this effect, Fig. 6 shows the time evolution of the mode energies referred to numerical solutions for two values of α outside the precession resonance regime (α = 10 and $α=60$
) as well as the numerical solution within the precession resonance regime ($α≈27$) for comparison. One clearly notices that the solution for $α≈27$ presents a long-term modulation of about
50days, while the others exhibit only faster oscillations. This is also confirmed by the respective power spectrum plots in the right panel of Fig. 6, where one notices a low-frequency spectral peak
in the solution for the precession resonance regime $α≈27$, which is absent in the other two solutions. One also notices that the inter-triad energy transfer in this kind of triad cluster
configuration is extremely inefficient outside the precession resonance regime, as can be evidenced by the low amplitude of the energy modulations associated with the solutions for α = 10 and α = 60.
We remark that low frequencies are indeed allowed outside the precession resonance regime (see, for instance, the spectral peak on Fig. 6 for α = 10). However, the integrated power spectrum over low
frequencies [Eq. (37)] tends to be weaker outside this regime. A similar analysis performed by independently varying two initial mode amplitudes of the primary triad is presented in Appendix B.
In summary, we have demonstrated in two reduced models the signatures of the precession resonance mechanism. None of these models are integrable, meaning that the nonlinear frequency Γ cannot be
calculated analytically. Rather, this frequency can only be estimated numerically through the time spectrum of the solutions for the wave amplitudes. In this sense, a proxy of this spectral content
is represented by the power at low frequencies given by Eq. (37). In both reduced models given by (28) and (31), the fact that the spectral power in the low-frequency range is clearly connected with
the precession frequency is by itself a strong evidence of the precession resonance mechanisms. This matching between the spectral power at low frequencies and the precession frequency of the
secondary triplet is also obtained by further varying the initial mode amplitudes in terms of two independent parameters, as demonstrated in Appendix B.
So far, we have presented the precession resonance mechanism in the context of highly truncated spectral models, consisting of four and five waves arranged in two triads. Although these highly
truncated models clearly demonstrate how the precession resonance mechanism operates, the full partial differential equation (2) contains a variety of other competing interacting triads that may
affect or even overshadow the solution obtained from the reduced models. Therefore, to test the robustness of the results presented in Sec. III, it is important to investigate how the precession
resonance mechanism operates in the full partial differential equation (2).
A. Numerical implementation details
Here, we use a pseudo-spectral, or collocation, method^59 to solve the barotropic vorticity equation (1). The basic structure of the method involves the following: (i) first, to calculate the
velocity $v→$ in physical space from the spectral representation of the vorticity $(ζ)$; (ii) then to calculate the product $(ζ+f)v→$ in physical space (therefore the terminology pseudo-spectral)
and to convert it to spectral space; and (iii) finally, to apply the divergence ($∇·$) in spectral space to obtain the tendency forcing $−∇·((ζ+f)v→)$. For the temporal discretization scheme, we
use the classic three-stage third-order Runge–Kutta scheme.^60 Spherical Harmonics transforms used the package provided by Schaeffer,^61 with implementation within the shallow water modeling package
of Schreiber et al.^62
In the numerical implementation, a spectral resolution of 128° (with triangular truncation), with correspondent physical resolution of 384 longitude vs 192 latitude points, ensuring no-aliasing for
quadratic terms. A time step size of 2min was used, which ensures numerical stability and almost negligible time step size errors, and simulations were performed for periods up to oneyear.
Realistic atmospheric conditions were imposed, with an Earth's rotation rate of $7.292×10−5 rad/s$ and the initial conditions defined by the imposing spherical harmonic coefficients of the vorticity
expansion given by $α/a$ on pre-selected modes, where a is the Earth radius (set to $6371.22 km$) and α will be specified within the range of 0–30, depending on the experiment. This setup ensures
that realistic atmospheric flows are generated, in terms of intensity and complexity, of what is representable with the barotropic vorticity model. No dissipation was used in these experiments, since
the experiments were performed in a transient regime (i.e., before a statistically stationary state is reached). Therefore, the time elapsed was not enough for the energy to reach small scales, which
guarantees the stability of the system in the timescale of our analysis. Additionally, the high-order spectral truncation, along with the small time step size, ensures an accurate representation of
energy and enstrophy conservations, with variations of the order of less than $10−4%−10−6%$ within an integration period of a year.
B. Numerical results
We have performed numerical experiments solving the barotropic vorticity equation initialized with equal amplitudes in the vorticity field in the modes ${(5,4),(3,1),(7,3)}$. The aim of the
experiment is to understand how the energy and enstrophy redistribute themselves among neighboring modes via nonlinear interactions, and in particular, the role played by the precession resonance
mechanism in the redistribution of energy/enstrophy.
Similarly to the study of reduced ordinary differential equation (ODE) models of Sec. III, we use a parameter α to re-scale the amplitudes (energy) of the initial condition. Figure 7 illustrates the
evolution of the vorticity field, initially (t=0) with energy only in ${(5,4),(3,1),(7,3)}$ modes, and gradually loosing coherence due to the leakage of energy to other modes for which the
amplitude was initially zero. Integrations of this model are performed for a period of oneyear. Unlike the case of the reduced models of Sec. III, in the full spectral equations (8), part of the
energy that leaves the initial triad will not return; as such, we must be aware that we might have to deal with transient behavior in the growth or decay of some of the modes. Integrations show that
the mode with wavenumber ${(n,m)=(9,2)}$ is the one to receive the largest amount of energy in the range of α's studied (see Fig. 9). For this reason, we choose to focus on mode $(n,m)=(9,2)$ in
evaluating the efficiency of the energy/enstrophy transfer as well as the spectral characteristics of the excited oscillations.
Figure 8 shows the efficiency of the energy/enstrophy transfer from the initial triad ${(5,4),(3,1),(7,3)}$ to mode $(9,2)$. Both efficiency plots have similar shape, with a peak around α = 20. The
values of the efficiency are small compared with the results from the reduced four-wave system in Sec. IIIA where the same modes were studied. The reason for this is that in the full spectral system
(8), the energy has multiple paths to leak from the initial triad ${(5,4),(3,1),(7,3)}$, exciting several modes, whereas in the reduced system (31), all the energy leaving the initial triad is
directed toward mode (9, 2). The power in the low frequencies is evaluated in Fig. 8(c), where we calculate the spectral power in frequencies $1/11 (days)−1≤ω≤1/120 (days)−1$. Unlike the reduced
system analysis, we truncate the periodicities longer than 120days to avoid the aforementioned transient behavior. The curve representing the power in the low frequencies closely follows the energy/
enstrophy transfer efficiency curves, suggesting that the regime of highly efficient energy transfer also generated low-frequency oscillations in the target mode. Finally, Fig. 8(d) shows the
precession frequency of the primary triad, $Ω1,23$, and the precession frequency $Ω1,42$ of the triad comprising modes ${(3,1)(7,3),(9,2)}$. First, the precession frequency of the primary triad
$Ω1,23$ enters a phase alignment regime at around α = 10. The precession frequency of the secondary (target) triad ($Ω1,42$) decreases toward a plateau at around α = 13, which remains up to $α≈25$
. The decrease in the precession frequency $Ω1,42$ therefore coincides with the amplification in the low frequencies of the system, an evidence that the dynamics of waves' phases and the precession
resonance mechanism do play a role here.
Figure 9 shows examples of integrations of this system for three values of the control parameter, $α=12,20,28$, displaying the time evolution of the mode energies as well as the respective spectra
for the target mode (9, 2). Modes shown in the graphs are the ones that reach a critical value of the energy of at least 0.1% of the total energy of the system. As mentioned before, (9, 2) remains as
the most efficiently excited mode. Other modes such as (5, 1), (3, 2), and (5, 4) also receive a substantial amount of energy through secondary interactions with the triad ${(3,1)(7,3),(9,2)}$.
Paths in spectral space for the energy to reach those modes are illustrated in Fig. 1. The spectral analysis of the target mode (9, 2) shows an increase in the power in the low frequencies in the
peak of the efficiency, α = 20. This corroborates with the finding described in Sec. III, for the reduced wave systems, in that precession resonance tends to excite lower frequency oscillations.
By the way that the aforementioned experiments were designed, energy transfers via the five-wave type of system are not allowed. From Eq. (28), if the energy is initialized only in modes (3, 4, 5),
the modes (1, 2) will remain zero forever. In order to test the feasibility of the five-wave type mechanism in the full barotropic vorticity equation, we initialized the energy in the base triad $
{(3,1)(7,3),(5,4)}$, plus a small perturbation in the modes (3, 0) and (7, 4). These modes are the same ones described in Sec. IIIB. Experiments were performed by controlling the amount of initial
energy in modes ${(3,1),(7,3),(5,4)}$ and keeping the energy in modes ${(3,0)(7,4)}$ constant. The results (not displayed here) show no significant energy transfer to modes ${(3,0),(7,4)}$, while
the same modes described in the previous experiment keep being excited. This suggests that the four-wave mechanism is prevalent in comparison with the five-wave mechanism in the full spectral system.
A possible reason for this is that the excited modes in four-wave arrangements are coupled to the initial triads via two modes; in contrast, in five-wave arrangements they are coupled via only one
In a third experiment, we initialize the energy in the modes ${(5,4),(3,1),(7,3)}$, but include perturbations in the initial condition to all modes such that $2≤n≤7$. All of these modes, except
those in the main interacting triad ${(5,4),(3,1),(7,3)}$, are initialized with a small initial condition defined with α = 1. The main purpose of this experiment is to test the robustness of the
results described above. Energy time-series of modes reaching at least 0.1% of the total energy are shown in Fig. 10 for α = 20. Again, we see that mode (9, 2) is among the ones to receive the
largest portion of the energy, although mode (5, 5) initially receives more energy in the beginning of the experiment and mode (5, 1) grows to be the one with most energy outside the initial triad
toward the end of one year of experiment. Unsurprisingly, more modes receive a substantial amount of energy, 0.1%, since even low levels of initial energy in other modes will allow for more possible
nonlinear interactions.
Efficiency plots for transfers toward mode (9, 2), shown in Fig. 11, qualitatively keep the pattern of the energy/enstrophy transfer efficiency without initial perturbations in Fig. 8. The same is
true for the low-frequency power spectrum, which exhibits a peak coinciding with the energy/enstrophy transfer efficiency peaks. Finally, the precession frequencies $Ω1,23$, for modes ${(3,1)(7,3),
(5,4)}$, and $Ω1,42$, for the triad ${(3,1)(7,3),(9,2)}$, are shown in Fig. 11(d). The overall behavior of the precession frequency $Ω1,23$ is very similar to the one displayed in Fig. 8, with the
system entering a phase alignment regime for $α≈9$. The precession frequency $Ω1,42$ is initially different from the one shown in Fig. 8, starting at a smaller frequency of $≈0.3×10−4 Hz$ and
subsequently increasing to a value of $≈0.5×10−4 Hz$ and plateauing at this level from $α≈10$ up to $α≈20$, after which the precession frequency increases similarly to what is observed in the
unperturbed system. Since in this case we have energy in more modes, the observed characteristics of $Ω1,42$ is possibly due to nonlinear contributions of other neighboring triads excluding the
initial triad (${(3,1)(7,3),(5,4)}$). It is important to mention that any quantitative comparison between energy transfer efficiencies in the reduced and full spectral models should be looked at
with caution. In fact, unlike the reduced spectral models in which the energy of the base triad can be transferred to only one or two other modes, the full spectral model has 8384 modes, and
therefore, the initial energy will have multiple ways to leak from the primary wave triad in this case. Therefore, the energy transfer efficiencies in the full barotropic vorticity equation should be
expected to be much lower. However, the signatures of the precession resonance mechanism are still present in the full spectral model simulations, as can be demonstrated by the matching between the
peaks of the energy/enstrophy transfer efficiency, the low-frequency power spectra, and the phase dynamics.
In the present article, we studied the precession resonance mechanism in the barotropic nondivergent model. This model describes the dynamics of a two-dimensional nondivergent flow in a rotating
frame and is well known to describe the essential features of the large- and planetary-scale motions of the atmosphere and ocean. The precession resonance was recently demonstrated by Bustamante et
al.^33 in the drift wave context as an efficient mechanism of energy transfer between different interacting triads and enhancing the cascade process responsible for the spectral energy
redistribution. It consists of a balance between the nonlinear frequency associated with the energy (amplitude) modulation of an interacting wave triad and the frequency associated with the dynamical
evolution of the relative phase among the components of an adjacent wave triplet. Another aspect of the precession resonance refers to its ability to yield low-frequency fluctuations in a wave
system. This aspect was studied by Raphaldini et al.^37 in the context of MHD Rossby waves in the solar tachocline and was proposed to be a possible mechanism for the observed long-term modulations
in the main solar cycle.
To analyze the precession resonance, we first considered two reduced spectral models of the barotropic vorticity equation consisting of five- and four-wave modes describing two interacting triads
coupled by one and two common modes, respectively, since these reduced ODE systems represent the possible ways to connect different nonlinearly interacting wave triplets. In the numerical
integrations of these reduced spectral models, we considered a primary triad containing almost the total initial energy, and we set the initial amplitudes of the primary triad to depend on a free
parameter, whose values were allowed to vary within realistic velocity magnitudes for the atmospheric flow. From these simulations, we computed the efficiency of energy and enstrophy transfers from
the primary triad to the target modes of the corresponding secondary triplet. We showed that, for both five- and four-wave systems, the precession resonance is associated with the peak of both the
energy and enstrophy transfer efficiencies. In addition, we computed the corresponding low-frequency range ($ω<1/10$$days−1$) power spectra associated with the time evolution of the mode energies
and showed that the maxima of the low-frequency power spectrum occur for the same range of initial conditions in which the maxima of the energy/enstrophy transfer efficiency occur. This range refers
to the initial amplitudes in which the energy modulation frequency of the primary triad approximately equals the mismatch among the linear eigenfrequencies of the secondary triad. By computing the
fully nonlinear precession frequencies of the two-wave triplets, it was also evident that abrupt changes in these frequencies, as a function of the initial condition free parameter, often determine
the beginning/end of regimes of high-energy transfer efficiency. This latter aspect of our results suggests the importance of the phase alignment for the energy transfers between two different wave
triplets. The phase alignment constitutes a special case of the precession resonance referring to p=0 in Eq. (27) and is characterized by a synchronization between the wave phases within an
interacting triad.
In the representative examples for the numerical integrations of the reduced spectral systems described above, we considered the modes ${(3,1),(7,3),(5,4)}$ as the primary triad that contains the
initial energy of the system. As discussed before, the pump mode of this nearly resonant triad, the Rossby–Haurwitz wave mode (5, 4), was found in the literature to be unstable, and the
Rossby–Haurwitz modes (3, 1) and (7, 3) account for the leading destabilizing disturbances, as demonstrated by the numerical simulations of Lynch.^24 This base triad was connected to modes ${(3,0),
(7,4)}$, in the five-wave system simulations, and to mode (9, 2), in the four-wave system numerical simulations.
To test the robustness of the precession resonance mechanism, we also performed numerical integrations with the full barotropic vorticity equation (1) by considering a regular triangular truncation
with a 128° spectral resolution. As in the reduced systems, we initialized the model with the total energy on the base triad ${(5,4),(3,1),(7,3)}$, with the initial condition's energy used as a
control parameter. Fluxes of energy/enstrophy to adjacent triads were evaluated as a function of the initial energy. Among the modes outside the base triad, the mode (9, 2) was found to be the most
efficiently excited, as also verified in the four-wave system integrations. The efficiency of the energy/enstrophy transfer was studied and peaks of efficiency were found, an evidence that the
precession resonance mechanism is in operation in the full spectral model as well. The spectral power in the low frequency was also evaluated, and similarly to the case of four-/five-wave systems,
the low frequencies are amplified near the regime of efficient energy/enstrophy transfer. The robustness of the results was tested by initializing the system with a small amount of energy in the
modes outside the initial triad ${(5,4),(3,1),(7,3)}$. The results previously reported were found to be robust, with mode (9, 2) remaining as the most relevant one. Quantitative differences were
found between the reduced four-wave spectral model and the inter-triad interactions in the full barotropic vorticity equation. On the reduced model, the efficiency of the energy transfer to mode (9,
2) was $≈10%$, while in the full barotropic vorticity equation the efficiency was $≈1.5%$. This difference results from the fact that in the full PDE model, the energy has multiple ways of leaving
the triad ${(5,4),(3,1),(7,3)}$, while in the reduced model it can only interact with mode (9, 2). Furthermore, the dynamics of the triad ${(9,2),(3,1),(7,3)}$ will also be affected by its
interactions with other nearby interacting triads.
Therefore, the results of both the reduced and full spectral models of the barotropic vorticity equation, with parameters relevant for the Earth's atmospheric flow, demonstrate that the precession
resonance mechanism enhances the low-frequency oscillation spectrum, along with promoting the maximum efficiency of energy transfer between different interacting wave triplets. Thus, based on the
results, we conjecture that resonant triads might be crucial for the initial pump wave instability of a Rossby mode preferentially excited by forcing mechanisms such as baroclinic instability and
thermal/orographic forcings. By contrast, the precession resonance involving two triads coupled by two common modes might play an essential role in the energy leakage from the primary triad that
initiates the cascade processes associated with the spectral energy redistribution. In addition, the energy transfers between different wave triplets yield modulations in the energy (amplitude)
oscillations associated with the energy exchanges within a primary triad, resulting in an amplification of the low-frequency power spectrum of the atmospheric flow. Although we also solved a full PDE
model of the atmosphere, the particular setting of the simulation is still idealized, since in order to evaluate the precession resonance in triad pairs within the fully spectral model, we considered
controlled initial conditions initiated in only one triad so that we could compute the efficiency of the inter-triad energy transfer. In a more realistic setting, for instance by initializing the
system with a realistic spectrum, the effect of the precession resonance would be mixed within the full overall dynamical interactions and, therefore, should be evaluated using statistical measures
of phase organizations, such as the order parameters introduced in Murray and Bustamante^34 and in Arguedas-Leiva et al.,^35 instead of tracking individual triad phases. However, this goes beyond the
scope of this work and is intended to be addressed in a future work.
In the atmospheric flow, the so-called low-frequency variability refers to phenomena whose timescales are longer than 10days and, therefore, is responsible for modulating the weather systems and
consequently yielding climate anomalies. Although much progress has been achieved in the last decades in the understanding of the atmospheric low-frequency variability, many points behind its
generation mechanisms remain as an open question in atmospheric sciences. In this direction, there has been some research effort pointing out the role of nonlinearity in the dynamics of some
phenomena within this variability timescale. For example, resonant triad interactions have been evoked to explain some features of intraseasonal oscillations;^38,39 the multiscale transfer of heat
and momentum from synoptic to planetary scales has shown to explain the morphology of the Madden–Julian oscillation (MJO),^63 the main tropical component of the intraseasonal variability; the upscale
cascade related to the so-called Rossby wave breaking^64 is believed to be important for the development of the annular modes.^65,66 In this context, the mechanism studied here with a simplified
model of the atmospheric flow was demonstrated to maximize both the cascade process that takes place after Rossby wave instability and the long-term modulation of intra-triad energy exchanges, the
two types of dynamical processes that have been evoked in the literature to explain some of the low-frequency variability modes of the atmospheric circulation.
However, it is important to mention that, as we have adopted here a simplified model of the atmospheric flow, we can only explain the essence of the processes that may potentially yield the
low-frequency variability modes described above. The modeling of these phenomena, such as the MJO, goes far beyond the scope of this study as it involves some complexities not considered in our
model, such as other wave types (e.g., gravity and Kelvin waves), as well as other vertical modes such as the baroclinic mode associated with the circulation response to deep convection heating. In
order to evaluate the relevance of the precession resonance mechanism in this context, the results presented here need to be generalized to more complex models of the atmosphere, such as the
primitive equations taking into account triads involving different wave types and vertical modes. Another important issue to be investigated refers to the formation of vortices and coherent
structures characteristic of geophysical flows.^67 Preliminary work indicates that the precession resonance mechanism favors the formation of coherent vortices. This fact can be interpreted as the
vortices being a result of the superposition of several modes with coherently interacting phases. This interplay between the nonlinear phase dynamics and coherent structures is also investigated in
other contexts, such as astrophysical flows.^32
The onset of the precession resonance and the phase alignment may also be investigated in the observational reanalysis data. Indeed, reanalysis datasets may be decomposed using normal mode function
projections that provide time-series for amplitudes and phases of different wave modes associated with low-frequency atmospheric oscillations, such as the Quasi-Biennial Oscillation (QBO) and the
MJO.^68–70 Raphaldini et al.^71 provided evidence for the interaction of atmospheric waves that contribute to the MJO with those that contribute to the QBO. In addition, observed wavenumber-frequency
spectra obtained by a normal mode projection approach^53 have shown a spectral peak associated with a barotropic Rossby mode compatible with the Rossby–Haurwitz mode (5, 4) studied here.
Consequently, this approach could, in principle, be applied for the identification of relevant interactions among barotropic Rossby modes. Events of strong energy transfers in spectral space could
also be investigated from this observational normal-mode perspective, for instance, the role of breaking Rossby waves on the disruption of the QBO,^72,73 extreme weather events,^74,75 among other
problems related to the circulation of the atmosphere.^76 The association between precession resonance/phase organization and events of strong energy transfer could, therefore, be verified in this
The work presented here was supported by FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo), Grant Nos. 2015/50686–1, 2017/23417–5, 2016/18445–7, 2020/14162–6, and 2021/06176–0, and
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brasil (CAPES)-Finance Code 001. We also thank the three anonymous reviewers for their valuable comments that helped to improve this
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Breno Raphaldini: Conceptualization (lead); Formal analysis (lead); Writing – original draft (lead); Writing – review and editing (lead). Pedro Silva Peixoto: Formal analysis (equal); Methodology
(equal); Writing – review and editing (equal). Andre S. Teruya: Formal analysis (equal); Methodology (equal). Carlos Frederico Raupp: Writing – original draft (supporting); Writing – review and
editing (equal). Miguel D. Bustamante: Conceptualization (equal); Writing – review and editing (equal).
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
In this section, we follow the study of Ripa^40 to demonstrate how the conserved quantities (4) and (5) lead to the constraints (11) for the wave triad interactions. Substituting expansion (6) into
(4) and (5) and using the orthogonality of the spherical harmonics, we obtain the following Parseval's identities:
where the summation index
indicates each spherical harmonic, viz.,
. On the other hand, multiplying
denotes the complex conjugate) and the corresponding equation for
, it follows
Now, taking the time derivative of equations (A1) and using (A2), we obtain
Therefore, in order for equations above to be satisfied, the following relations must be satisfied for each interacting wave triplet (j, k, l),
In order to further evaluate the robustness of the precession resonance mechanism in the reduced spectral models introduced in Sec. III, here we analyze the precession resonance by considering
different regimes of energy exchange among the waves of the primary triad. For this purpose, instead of a single parameter α multiplying the initial mode amplitudes of the base triad, now two of the
initial amplitudes of this triad are multiplied by different parameters α and β independently in order to yield a broader range for its characteristic frequencies.
1. Four-wave model
We proceed to evaluate the efficiency of the energy and enstrophy transfers as defined in Eqs. (35) and (36), respectively. The initial condition is now defined as a function of the dimensionless
parameters α and β, varying from 0 to 60. The initial condition considered in our numerical simulations is given by
Results are shown in Fig. 12, where we observe that the energy/enstrophy transfer efficiencies [Figs. 12(a) and 12(b)], as well as the low-frequency power [Fig. 12(c)], are strongly correlated with
the pattern of the precession frequencies of the triads displayed in Figs. 12(e) and 12(f), demonstrating the role of the precession resonance mechanism in maximizing both the inter-triad energy
transfer efficiency and the low-frequency power spectrum. Figure 12(d) illustrates the shape parameter μ of the Jacobi elliptic functions defined according to Eq. (24). From this figure, one notices
that the variation range of the parameters α and β considered here is associated with the corresponding variation range of $0<μ<1$. This variation range of the parameter μ encompasses distinct
energy modulation regimes for the primary triad. This can be noticed from Fig. 13, which illustrates the behavior of the Jacobian elliptic function $sn$ for three different values of the parameter μ.
2. Five-wave model
Similarly, we perform the 2D analysis for the energy/enstrophy transfer efficiency referred to the five-wave model (28), with the efficiencies being defined by Eqs. (38) and (39). The initial
condition is defined as $(A1,A2,A3,A4,A5)=(1.48×exp (1.530i)×10−8,1.71×exp (0.304i)×10−8,1.03×exp (1.110i)×10−5,α×1.22×exp (5.060i)×10−5,β×1.17×exp (3.950i)×10−5)$.
Similarly to the four-wave example, the results displayed in Fig. 14 referred to the integrations of system (28) show that the energy/enstrophy transfer efficiencies and the low-frequency power
[panels (a)–(c), respectively] are largely determined by the precession frequency of the target triad [panel (f)]. In addition, the variation range of the parameters α and β corresponds to a
variation range of $0<μ<1$, which encompasses distinct regimes of energy exchanges for the primary triad ${(3,1),(7,3),(5,4)}$.
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Transactions Online
Seiichiro TANI, Toshiaki MIYAZAKI, "A Design Framework for Online Algorithms Solving the Object Replacement Problem" in IEICE TRANSACTIONS on Information, vol. E84-D, no. 9, pp. 1135-1143, September
2001, doi: .
Abstract: Network caches reduce network traffic as well as user response time. When implementing network caches, the object replacement problem is one of the core problems; The problem is to
determine which objects should be evicted from a cache when there is insufficient space. This paper first formalizes the problem and gives a simple but sufficient condition for deterministic online
algorithms to be competitive. Based on the condition, a general framework to make a non-competitive algorithm competitive is constructed. As an application of the framework, an online algorithm,
called Competitive_SIZE, is proposed. Both event-driven and trace-driven simulations show that Competitive_SIZE is better than previously proposed algorithms such as LRU (Least Recently Used).
URL: https://global.ieice.org/en_transactions/information/10.1587/e84-d_9_1135/_p
author={Seiichiro TANI, Toshiaki MIYAZAKI, },
journal={IEICE TRANSACTIONS on Information},
title={A Design Framework for Online Algorithms Solving the Object Replacement Problem},
abstract={Network caches reduce network traffic as well as user response time. When implementing network caches, the object replacement problem is one of the core problems; The problem is to
determine which objects should be evicted from a cache when there is insufficient space. This paper first formalizes the problem and gives a simple but sufficient condition for deterministic online
algorithms to be competitive. Based on the condition, a general framework to make a non-competitive algorithm competitive is constructed. As an application of the framework, an online algorithm,
called Competitive_SIZE, is proposed. Both event-driven and trace-driven simulations show that Competitive_SIZE is better than previously proposed algorithms such as LRU (Least Recently Used).},
TY - JOUR
TI - A Design Framework for Online Algorithms Solving the Object Replacement Problem
T2 - IEICE TRANSACTIONS on Information
SP - 1135
EP - 1143
AU - Seiichiro TANI
AU - Toshiaki MIYAZAKI
PY - 2001
DO -
JO - IEICE TRANSACTIONS on Information
SN -
VL - E84-D
IS - 9
JA - IEICE TRANSACTIONS on Information
Y1 - September 2001
AB - Network caches reduce network traffic as well as user response time. When implementing network caches, the object replacement problem is one of the core problems; The problem is to determine
which objects should be evicted from a cache when there is insufficient space. This paper first formalizes the problem and gives a simple but sufficient condition for deterministic online algorithms
to be competitive. Based on the condition, a general framework to make a non-competitive algorithm competitive is constructed. As an application of the framework, an online algorithm, called
Competitive_SIZE, is proposed. Both event-driven and trace-driven simulations show that Competitive_SIZE is better than previously proposed algorithms such as LRU (Least Recently Used).
ER - | {"url":"https://global.ieice.org/en_transactions/information/10.1587/e84-d_9_1135/_p","timestamp":"2024-11-06T01:21:55Z","content_type":"text/html","content_length":"59585","record_id":"<urn:uuid:4e8ba9ef-ddeb-44d9-8f90-cb5f6797c6c6>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00768.warc.gz"} |
Eleven Eleven
I was born in Riga, Latvia. All my formal education is in mathematics and theoretical computer science. I got my PhD degree for thesis "Behavior of Different Types of Automata and Turing Machines on
Infinite Words" in 1990. My thesis advisor was Professor Rusins Freivalds, now well known name in quantum computing. For 20 years I was teaching in University of Latvia, where most of my students
were prospective mathematics teachers. I was also leading numerous workshops for teachers and editing math textbooks in Latvia. But one of my hobbies always was knitting that I learned in middle
school. I learned how to crochet on my own because I liked to use crochet to finish my knitting projects.
I was seeing patterns and algorithms in knitting and crochet but I was not connecting it with my professional work in mathematics until 1997 when I became a Visiting Associate Professor in Department
of Mathematics, Cornell University. Once I was participating in geometry workshop led by Professor David Henderson. He was showing a paper model of hyperbolic plane that was made using William
Thurston's idea of annuli. And then it came in my mind - if one can make it out of a paper, then I should be able to crochet it and to get a more durable model to use in my geometry class. In that
fall I was assigned to teach Math 451, using Henderson's book "Experiencing Geometry". During summer 1997 I made my first classroom set of hyperbolic planes and used it in my geometry class. It was
amazing to see how much it helped my students to understand the nature of hyperbolic plane.
Since then I have made numerous models of hyperbolic plane, including crocheting figures for 2nd and 3rd edition of "Experiencing Geometry". For the last one I am now co-author with my husband David
Henderson, who always has some idea what I could crochet next to use in his or my classes.
In 2003 I started to exchange e-mails with Margaret Wertheim about how to crochet these models (she learned about them from New Scientist), and in May 2004 The Institute for Figuring invited us to
give a public lecture "Crocheting the Hyperbolic Plane" (see www.theiff.org for more information). For classroom models I use acrylic yarn because it is more durable and easy to clean. For this
exhibit I did some of these shapes in wool to stress that we can find hyperbolic shapes in nature, we are just not used to notice them as easy as we do with flat or spherical shapes that are other
common surfaces with constant curvature.
We all know positive numbers, negative numbers and zero. For surfaces we can use a notion of curvature - zero curvature is flat surface, sphere has constant positive curvature. It is logical to ask -
but what surface has a negative constant curvature and how does it look? The answer is hyperbolic plane and looking at the models you can see how differently it can appear and how amazingly it can be
sculpted starting from the same basic shape. | {"url":"http://eleveneleven.50webs.com/taimina.html","timestamp":"2024-11-06T13:36:12Z","content_type":"text/html","content_length":"7124","record_id":"<urn:uuid:2429a2b4-6f66-4b19-a958-40d385f76fcf>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00325.warc.gz"} |
Valentin Vincendon - A "sudden strong wind": the worst excuse Ever Given?
A "sudden strong wind": the worst excuse Ever Given?
Fact-checking Evergreen's explanation for costing the world $1,000,000 per minute
The Suez Canal is an artery of world trade, connecting the Mediterranean with the Red Sea, and providing an avenue for vessels to pass between Asia and the Middle East and Europe. The main
alternative, a passage round the Cape of Good Hope at the southern tip of Africa, takes considerably longer.
On average, nearly 50 vessels per day pass along the canal, although at times the number can be much higher - accounting for some 12% of world trade. It is particularly important as an avenue for oil
and liquified natural gas, enabling shipments to get from the Middle East to Europe.
- BBC
A study by German insurer Allianz said the blockage could cost global trade $6 billion to $10 billion a week.
Evergreen Marine said the ship "was suspected of being hit by a sudden strong wind, causing the hull to deviate from waterway".
I really wondered if the orders of magnitudes in question were realistic at all, so I decided to quickly run the numbers.
This is a quick-and-dirty analysis, aimed at being understandable by anyone, with or without a background in physics.
Every time I need to take an assumption, I always pick the side that goes in favour of the wind theory. That way, the result will be an upper bound.
The Ever Given is 33m-tall, with 14.5m of this height submerged in water.
Looking at the container stack, and using the standard 2.6m height for each container, the pile should be about 26m-high.
This gives us a total above-the-water height of 44.5m.
The ship is 400m-long.
Simplifying things for the worst, we assume the side area is a rectangle. Its area is this 17,800m^2, i.e. 1.78 hectare.
This is big: ~2.5x the area of a football pitch, or ~200x your average sailing boat sails area.
Let's be extremely generous and say that the wind is blowing at 100km/h. Constantly.
Let's also assume that it is blowing perfectly perpendicular to the boat, and that it is giving away 100% of its momentum to the ship - this is a very strong assumption.
100km/h is about 27.8m/s.
At standard temperature and pressure (0°C and 100 kPa), dry air has a density of 1.2754 kg/m^3. Pretty sure it is not 0°C in Egypt right now, but let's keep a margin for some potential humidity and/
or anticyclone pressure.
It means that in 1s, 17,000*27.8*1.2754kg of air are hitting the side of the ship.
That's about 603t of air per second.
The ship's mass is 224,000t.
The wind's lateral force - again, if fully transferred to the ship - is 603t/s * 27.8 m/s, i.e. ~16.8 million Newtons.
Using Newton's second law of motion, we can deduct the ship's resulting lateral acceleration 16,800/224,000 = 0.075m/s^2
That isn't much. At all.
Gravity average acceleration at the Earth's surface is 9.81m/s^2, i.e. 131 higher.
BUT, let's unchecked, acceleration compounds.
Let's assume the ship was sailing dead in the middle of the canal.
The ship is 59m-wide.
The canal is 200m-wide.
That leaves us ~70m between the ship and the bank on each side.
Let's make one last very strong assumption: the water displaced laterally is not opposing the ship's lateral movement generated by the wind. This is wrong because the ship's lateral section is huge -
in the same order of magnitude as a football pitch, again - and absolutely not optimized aerodynamically for this type of lateral translation: a ship is supposed to sail bow-first, not laterally like
a crab.
But, this type of friction is usually proportional to the movement speed, which would remain reasonably low for most of the trajectory - as we will check retrospectively.
If the ship starts with no lateral motion, then integrating the acceleration equation twice gives us an impact after 43s.
The resulting lateral speed would be ~3.2m/s, i.e. ~11.5km/h. This is not enough to ignore water resistance, but not very high either considering this would be the absolute maximum speed for the
whole trajectory.
The wind's force is not linear to its speed: it increases with the square of its speed.
For a 70km/h constant wind speed, the impact happens after 62s at an end lateral speed of 2.3m/s (8.2km/h).
For a 50km/h constant wind speed, the impact happens after 86s at an end lateral speed of 1.6m/s (5.8km/h).
Let's not forget the angle either: a mere 30-degree (vs. a perfectly perpendicular wind) would decrease the wind force by 25%, everything else being equal. 45 degrees would decrease it by 50%.
Finally, this is in case the wind never falters and, obviously, the ship's captain does not start any avoidance manoeuvre.
The wind's force is not linear to its speed: it increases with the square of its speed.
For a 70km/h constant wind speed, the impact happens after 62s at an end lateral speed of 2.3m/s (8.2km/h).
For a 50km/h constant wind speed, the impact happens after 86s at an end lateral speed of 1.6m/s (5.8km/h).
Let's not forget the angle either: a mere 30-degree (vs. a perfectly perpendicular wind) would decrease the wind force by 25%, everything else being equal. 45 degrees would decrease it by 50%.
Such angles would increase the time-to-impact by a third, or multiply it by 2, respectively.
Finally, this is in case the wind never falters and, obviously, the ship's captain does not start any avoidance manoeuvre.
Even without strong hypotheses, it does seem like an extremely strong wind could in theory bring the ship dangerously close to the bank.
Even if the ship's captain managed to steer early enough, the bank effect - the marine version of the aeronautics ground effect - could take care of the last meters and bring us to the catastrophic
situation we all know about.
I must say that it goes against my initial intuition, so I am glad I did the math.
Hope you found it entertaining as well. | {"url":"https://www.vincendon.com/essays/evergreen-suez-canal","timestamp":"2024-11-02T20:44:39Z","content_type":"text/html","content_length":"180311","record_id":"<urn:uuid:92cce005-9fd0-4276-be32-7a2da9cbbf0e>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00437.warc.gz"} |
Engineering Mechanics: Statics & Dynamics (14th Edition) Chapter 2 - Force Vectors - Section 2.4 - Addition of a System of Coplanar Forces - Problems - Page 41 38
Work Step by Step
We know that $F_1=50(\frac{3}{5})i+50(\frac{4}{5})j$ $F_2=80 cos\space 255i+80 sin\space 255j$ and $F_3=30i+0j$ Now we find the horizontal and vertical resultant forces as $R_x=50(\frac{3}{5})+80cos
255+30=39.3N$ and $R_y=50(\frac{4}{5})+80sin\space 255+0=-37.3N$ Now the magnitude of the resultant force is $F_R=\sqrt{R_x^2+R_y^2}$ We plug in the known values to obtain: $F_R=\sqrt{(39.3)^2+
(-37.3)^2}=54.2N$ Now the direction can be determined as $\theta=tan^{-1}(\frac{R_y}{R_x})$ We plug in the known values to obtain: $\theta=tan^{-1}(\frac{37.3}{39.3})=43.5^{\circ}$ | {"url":"https://www.gradesaver.com/textbooks/engineering/mechanical-engineering/engineering-mechanics-statics-and-dynamics-14th-edition/chapter-2-force-vectors-section-2-4-addition-of-a-system-of-coplanar-forces-problems-page-41/38","timestamp":"2024-11-10T11:47:16Z","content_type":"text/html","content_length":"74823","record_id":"<urn:uuid:f16bba1e-7c79-4295-b0ef-1d6b18eeefbf>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00332.warc.gz"} |
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Rational Numbers to Linear Equations
Softcover ISBN: 978-1-4704-5675-7
Product Code: MBK/131
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Softcover ISBN: 978-1-4704-5675-7
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Rational Numbers to Linear Equations
Softcover ISBN: 978-1-4704-5675-7
Product Code: MBK/131
List Price: $50.00
MAA Member Price: $45.00
AMS Member Price: $40.00
eBook ISBN: 978-1-4704-6004-4
Product Code: MBK/131.E
List Price: $50.00
MAA Member Price: $45.00
AMS Member Price: $40.00
Softcover ISBN: 978-1-4704-5675-7
eBook ISBN: 978-1-4704-6004-4
Product Code: MBK/131.B
List Price: $100.00 $75.00
MAA Member Price: $90.00 $67.50
AMS Member Price: $80.00 $60.00
2020; 402 pp
MSC: Primary 97; 00
This is the first of three volumes that, together, give an exposition of the mathematics of grades 9–12 that is simultaneously mathematically correct and grade-level appropriate. The volumes are
consistent with CCSSM (Common Core State Standards for Mathematics) and aim at presenting the mathematics of K–12 as a totally transparent subject.
The present volume begins with fractions, then rational numbers, then introductory geometry that can make sense of the slope of a line, then an explanation of the correct use of symbols that
makes sense of “variables”, and finally a systematic treatment of linear equations that explains why the graph of a linear equation in two variables is a straight line and why the usual solution
method for simultaneous linear equations “by substitutions” is correct.
This book should be useful for current and future teachers of K–12 mathematics, as well as for some high school students and for education professionals.
Teachers of middle school mathematics; students and professionals interested in mathematical education.
□ Chapters
□ Fractions
□ Rational numbers
□ The Euclidean algorithm
□ Basic isometries and congruence
□ Dilation and similarity
□ Symbolic notation and linear equations
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Additional Material
• Requests
2020; 402 pp
MSC: Primary 97; 00
This is the first of three volumes that, together, give an exposition of the mathematics of grades 9–12 that is simultaneously mathematically correct and grade-level appropriate. The volumes are
consistent with CCSSM (Common Core State Standards for Mathematics) and aim at presenting the mathematics of K–12 as a totally transparent subject.
The present volume begins with fractions, then rational numbers, then introductory geometry that can make sense of the slope of a line, then an explanation of the correct use of symbols that makes
sense of “variables”, and finally a systematic treatment of linear equations that explains why the graph of a linear equation in two variables is a straight line and why the usual solution method for
simultaneous linear equations “by substitutions” is correct.
This book should be useful for current and future teachers of K–12 mathematics, as well as for some high school students and for education professionals.
Teachers of middle school mathematics; students and professionals interested in mathematical education.
• Chapters
• Fractions
• Rational numbers
• The Euclidean algorithm
• Basic isometries and congruence
• Dilation and similarity
• Symbolic notation and linear equations
Permission – for use of book, eBook, or Journal content
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Various computational models (method/basis) were investigated for calculation of the efg's. Each was calibrated by linear regression analysis of the calculated efg's versus the experimental nqcc's
for a chosen set of molecules. Calculations of the efg's were made on the experimental structures of these molecules. Although not independent, all three diagonal components of the efg tensor were
plotted against the corresponding components of the nqcc tensor. This assures, because the tensors are traceless, that the regression line pass through the origin, as required by equation (1). The
coefficient eQ/h in equation (1) is the slope of this line. Having thus determined eQ/h for a given model, the model may then be used for calculation of nqcc's in molecules in addition to those
chosen for calibration. The premise that underlies this procedure is that errors inherent in the computational model are systematic and can be corrected - partially, at least - by the best-fit
coefficient eQ/h.
The goal is to reproduce accurately, and efficiently, the experimental nqcc's. The best models, therefore, are those which show the best linear relationship between the calculated efg's and the
experimental nqcc's - that is, the least residual standard deviation (RSD). It is sufficient that Q[eff] - derived from the model-dependent, best-fit value of eQ/h - approximate Q to within a few
The effective moment is given by
Q[eff] = (eQ/h)/234.9647, (2)
where Q[eff] is in barns (b) when eQ/h is in MHz/a.u. (Physical constants and unit conversion factors are all contained in the numerical constant.)
Previous page Next page
Table of Contents | {"url":"http://nqcc.wcbailey.net/Intro_p2n.html","timestamp":"2024-11-13T03:04:30Z","content_type":"text/html","content_length":"11480","record_id":"<urn:uuid:7cde9a52-bf82-40ee-86fc-6e5db17ea6f4>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00236.warc.gz"} |
Introductory Exercises
This is a small list of exercises for beginners. Some of them require minimal math knowledge. I will try and make references to that knowledge, when possible.
Problems with numbers (no sequences as input)
First Degree Equation ^i
Solve the first degree equation with one variable: ax + b = 0, given a and b as input.
Second Degree Equation ^i
Solve the second degree equation with one variable: ax^2 + bx + c = 0, given a, b and c as input.
Divisibility Test ^i
Find out if n is divisible by k.
Leap Year Test ^i
Find out if year y is a leap year.
k^th digit
Extract the k^th digit of a number, counting digits from right to left
Triangle Edges ^i
Can three input numbers be the lengths of the three edges of a triangle?
Swap Variables ^i
Given variables a and b exchange their values so that in the end a will contain the old value of b and b will contain the old value of a.
Swap Variables Restricted ^i h
Given variables a and b exchange their values so that in the end a will contain the old value of b and b will contain the old value of a, without using any other variables.
Display all of number n's divisors.
Is number n a prime number?
Reverse Digits
Display in reverse order the digits of a number n
Multiple Count ^i h
How many integers divisible by n lie in interval [a, b]?
Leap Year Count ^i h
How many years between y1 and y2 are leap years?
Palindrome ^h
Is number n a [palindrome]? A palindrome is a symmetrical number, like 15351 or 12233221.
Sort numbers ^i
Display three input integers in ascending order.
Sort more numbers ^i h
Display five input integers in ascending order. Make the flow chart fit on one letter page.
GCD and LCM
Find the [greates common divisor] and the [lowest common multiple] of two numbers. Use [Euclid's algorithm]. Example: GCD of 24 and 32 is 8.
Prime Factors
Display the [prime factor decomposition] of input number n. Example: 1176 = 2^3 x 3^1 x 7^2
Numbers with two digits
Is number n formed with exactly 2 digits repeated any number of times? 23223 and 900990 are such numbers, while 593 and 44002 are not.
Decimal Fraction ^h
Display fraction m/n in decimal format, with the period between brackets. Example: 13/30 = 0.4(3)
Number Guessing
Guess a number between 1 and 1024 by asking questions of the form "is the number greater or equal to x".
• ^i means that the problem can be solved without loops, using only if-then-else structures.
• ^h means hard. The problem is difficult. | {"url":"https://www.algopedia.ro/wiki/index.php?title=Introductory_Exercises&oldid=1640","timestamp":"2024-11-06T18:15:37Z","content_type":"text/html","content_length":"23920","record_id":"<urn:uuid:64dc57ab-c3d1-4f12-964c-09073a819c76>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00428.warc.gz"} |
Corrigendum to “Mechanical wave momentum from the first principles” [Wave Motion 68 (2016) 283–290] (S0165212516301408) (10.1016/j.wavemoti.2016.11.005))
In my paper Mechanical wave momentum from the first principles, the axial momentum is represented as a product of the wave mass [Formula presented] and the wave speed [Formula presented]. The former
is introduced as the exceed of the mass density (per unit length) in the wave, [Formula presented], over that ahead of the wave, [Formula presented] It should be noted that this relation is valid not
only for mechanical waves but for electromagnetic waves too (that is, it reduces to the known expression for the axial momentum of the latter). Indeed, in an application to electromagnetic waves, we
may consider [Formula presented] as the rest mass. Since it is absent, the wave mass becomes where [Formula presented] is the energy density and [Formula presented] is the speed of light. The
momentum density follows as [Formula presented] as it should. Of course, this coincidence is not accidental. The fact is that the considerations resulting in the expression (1) in the above paper are
true for electromagnetic waves too. The author regrets that he did not note this fact at once.
Dive into the research topics of 'Corrigendum to “Mechanical wave momentum from the first principles” [Wave Motion 68 (2016) 283–290] (S0165212516301408) (10.1016/j.wavemoti.2016.11.005))'. Together
they form a unique fingerprint. | {"url":"https://cris.tau.ac.il/en/publications/corrigendum-to-mechanical-wave-momentum-from-the-first-principles","timestamp":"2024-11-04T10:15:24Z","content_type":"text/html","content_length":"48640","record_id":"<urn:uuid:6441e85f-9b45-4d50-bdf4-90be0a14e460>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00381.warc.gz"} |
Solving Cosine Equations for More Than One Solution
To solve the equation
This is not the only the solution. There are an infinite number of solutions. Typically though, we want solutions in the range 0° – 360°.
We can use the cosine curve below to find the solutions.
Draw the horizontal line y=0.4 (because the equation to solve is cos x =0.4). Where this line crosses the curve, drop a line to the x - axis. The solution is where the line crosses the x – axis.
The cosine curve has symmetry, so that
The two solutions are x=66.42° and x=293.58° . | {"url":"https://astarmathsandphysics.com/igcse-maths-notes/518-solving-cosine-equations-for-more-than-one-solution.html","timestamp":"2024-11-11T03:47:16Z","content_type":"text/html","content_length":"29359","record_id":"<urn:uuid:af8170be-9789-467f-b1c5-9e839f837ef2>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00559.warc.gz"} |
Domain theory
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Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order
theory. The field has major applications in computer science, where it is used to specify denotational semantics, especially for functional programming languages. Domain theory formalizes the
intuitive ideas of approximation and convergence in a very general way and has close relations to topology.
An alternative important approach to denotational semantics in computer science is that of metric spaces.
Motivation and intuition[edit]
The primary motivation for the study of domains, which was initiated by Dana Scott in the late 1960s, was the search for a denotational semantics of the lambda calculus. In this formalism, one
considers "functions" specified by certain terms in the language. In a purely syntactic way, one can go from simple functions to functions that take other functions as their input arguments. Using
again just the syntactic transformations available in this formalism, one can obtain so called fixed-point combinators (the best-known of which is the Y combinator); these, by definition, have the
property that f(Y(f)) = Y(f) for all functions f.
To formulate such a denotational semantics, one might first try to construct a model for the lambda calculus, in which a genuine (total) function is associated with each lambda term. Such a model
would formalize a link between the lambda calculus as a purely syntactic system and the lambda calculus as a notational system for manipulating concrete mathematical functions. The combinator
calculus is such a model. However, the elements of the combinator calculus are functions from functions to functions; in order for the elements of a model of the lambda calculus to be of arbitrary
domain and range, they could not be true functions, only partial functions.
Scott got around this difficulty by formalizing a notion of "partial" or "incomplete" information to represent computations that have not yet returned a result. This was modeled by considering, for
each domain of computation (e.g. the natural numbers), an additional element that represents an undefined output, i.e. the "result" of a computation that never ends. In addition, the domain of
computation is equipped with an ordering relation, in which the "undefined result" is the least element.
The important step to find a model for the lambda calculus is to consider only those functions (on such a partially ordered set) which are guaranteed to have least fixed points. The set of these
functions, together with an appropriate ordering, is again a "domain" in the sense of the theory. But the restriction to a subset of all available functions has another great benefit: it is possible
to obtain domains that contain their own function spaces, i.e. one gets functions that can be applied to themselves.
Beside these desirable properties, domain theory also allows for an appealing intuitive interpretation. As mentioned above, the domains of computation are always partially ordered. This ordering
represents a hierarchy of information or knowledge. The higher an element is within the order, the more specific it is and the more information it contains. Lower elements represent incomplete
knowledge or intermediate results.
Computation then is modeled by applying monotone functions repeatedly on elements of the domain in order to refine a result. Reaching a fixed point is equivalent to finishing a calculation. Domains
provide a superior setting for these ideas since fixed points of monotone functions can be guaranteed to exist and, under additional restrictions, can be approximated from below.
A guide to the formal definitions[edit]
In this section, the central concepts and definitions of domain theory will be introduced. The above intuition of domains being information orderings will be emphasized to motivate the mathematical
formalization of the theory. The precise formal definitions are to be found in the dedicated articles for each concept. A list of general order-theoretic definitions which include domain theoretic
notions as well can be found in the order theory glossary. The most important concepts of domain theory will nonetheless be introduced below.
Directed sets as converging specifications[edit]
As mentioned before, domain theory deals with partially ordered sets to model a domain of computation. The goal is to interpret the elements of such an order as pieces of information or (partial)
results of a computation, where elements that are higher in the order extend the information of the elements below them in a consistent way. From this simple intuition it is already clear that
domains often do not have a greatest element, since this would mean that there is an element that contains the information of all other elements—a rather uninteresting situation.
A concept that plays an important role in the theory is that of a directed subset of a domain; a directed subset is a non-empty subset of the order in which any two elements have an upper bound that
is an element of this subset. In view of our intuition about domains, this means that any two pieces of information within the directed subset are consistently extended by some other element in the
subset. Hence we can view directed subsets as consistent specifications, i.e. as sets of partial results in which no two elements are contradictory. This interpretation can be compared with the
notion of a convergent sequence in analysis, where each element is more specific than the preceding one. Indeed, in the theory of metric spaces, sequences play a role that is in many aspects
analogous to the role of directed sets in domain theory.
Now, as in the case of sequences, we are interested in the limit of a directed set. According to what was said above, this would be an element that is the most general piece of information that
extends the information of all elements of the directed set, i.e. the unique element that contains exactly the information that was present in the directed set, and nothing more. In the formalization
of order theory, this is just the least upper bound of the directed set. As in the case of limits of sequences, least upper bounds of directed sets do not always exist.
Naturally, one has a special interest in those domains of computations in which all consistent specifications converge, i.e. in orders in which all directed sets have a least upper bound. This
property defines the class of directed-complete partial orders, or dcpo for short. Indeed, most considerations of domain theory do only consider orders that are at least directed complete.
From the underlying idea of partially specified results as representing incomplete knowledge, one derives another desirable property: the existence of a least element. Such an element models that
state of no information—the place where most computations start. It also can be regarded as the output of a computation that does not return any result at all.
Computations and domains[edit]
Now that we have some basic formal descriptions of what a domain of computation should be, we can turn to the computations themselves. Clearly, these have to be functions, taking inputs from some
computational domain and returning outputs in some (possibly different) domain. However, one would also expect that the output of a function will contain more information when the information content
of the input is increased. Formally, this means that we want a function to be monotonic.
When dealing with dcpos, one might also want computations to be compatible with the formation of limits of a directed set. Formally, this means that, for some function f, the image f(D) of a directed
set D (i.e. the set of the images of each element of D) is again directed and has as a least upper bound the image of the least upper bound of D. One could also say that f preserves directed suprema.
Also note that, by considering directed sets of two elements, such a function also has to be monotonic. These properties give rise to the notion of a Scott-continuous function. Since this often is
not ambiguous one also may speak of continuous functions.
Approximation and finiteness[edit]
Domain theory is a purely qualitative approach to modeling the structure of information states. One can say that something contains more information, but the amount of additional information is not
specified. Yet, there are some situations in which one wants to speak about elements that are in a sense much simpler (or much more incomplete) than a given state of information. For example, in the
natural subset-inclusion ordering on some powerset, any infinite element (i.e. set) is much more "informative" than any of its finite subsets.
If one wants to model such a relationship, one may first want to consider the induced strict order < of a domain with order ≤. However, while this is a useful notion in the case of total orders, it
does not tell us much in the case of partially ordered sets. Considering again inclusion-orders of sets, a set is already strictly smaller than another, possibly infinite, set if it contains just one
less element. One would, however, hardly agree that this captures the notion of being "much simpler".
Way-below relation[edit]
A more elaborate approach leads to the definition of the so-called order of approximation, which is more suggestively also called the way-below relation. An element x is way below an element y, if,
for every directed set D with supremum such that
${\displaystyle y\sqsubseteq \sup D}$,
there is some element d in D such that
${\displaystyle x\sqsubseteq d}$.
Then one also says that x approximates y and writes
${\displaystyle x\ll y}$.
This does imply that
${\displaystyle x\sqsubseteq y}$,
since the singleton set {y} is directed. For an example, in an ordering of sets, an infinite set is way above any of its finite subsets. On the other hand, consider the directed set (in fact: the
chain) of finite sets
${\displaystyle \{0\},\{0,1\},\{0,1,2\},\ldots }$
Since the supremum of this chain is the set of all natural numbers N, this shows that no infinite set is way below N.
However, being way below some element is a relative notion and does not reveal much about an element alone. For example, one would like to characterize finite sets in an order-theoretic way, but even
infinite sets can be way below some other set. The special property of these finite elements x is that they are way below themselves, i.e.
${\displaystyle x\ll x}$.
An element with this property is also called compact. Yet, such elements do not have to be "finite" nor "compact" in any other mathematical usage of the terms. The notation is nonetheless motivated
by certain parallels to the respective notions in set theory and topology. The compact elements of a domain have the important special property that they cannot be obtained as a limit of a directed
set in which they did not already occur.
Many other important results about the way-below relation support the claim that this definition is appropriate to capture many important aspects of a domain.
Bases of domains[edit]
The previous thoughts raise another question: is it possible to guarantee that all elements of a domain can be obtained as a limit of much simpler elements? This is quite relevant in practice, since
we cannot compute infinite objects but we may still hope to approximate them arbitrarily closely.
More generally, we would like to restrict to a certain subset of elements as being sufficient for getting all other elements as least upper bounds. Hence, one defines a base of a poset P as being a
subset B of P, such that, for each x in P, the set of elements in B that are way below x contains a directed set with supremum x. The poset P is a continuous poset if it has some base. Especially, P
itself is a base in this situation. In many applications, one restricts to continuous (d)cpos as a main object of study.
Finally, an even stronger restriction on a partially ordered set is given by requiring the existence of a base of compact elements. Such a poset is called algebraic. From the viewpoint of
denotational semantics, algebraic posets are particularly well-behaved, since they allow for the approximation of all elements even when restricting to finite ones. As remarked before, not every
finite element is "finite" in a classical sense and it may well be that the finite elements constitute an uncountable set.
In some cases, however, the base for a poset is countable. In this case, one speaks of an ω-continuous poset. Accordingly, if the countable base consists entirely of finite elements, we obtain an
order that is ω-algebraic.
Special types of domains[edit]
A simple special case of a domain is known as an elementary or flat domain. This consists of a set of incomparable elements, such as the integers, along with a single "bottom" element considered
smaller than all other elements.
One can obtain a number of other interesting special classes of ordered structures that could be suitable as "domains". We already mentioned continuous posets and algebraic posets. More special
versions of both are continuous and algebraic cpos. Adding even further completeness properties one obtains continuous lattices and algebraic lattices, which are just complete lattices with the
respective properties. For the algebraic case, one finds broader classes of posets which are still worth studying: historically, the Scott domains were the first structures to be studied in domain
theory. Still wider classes of domains are constituted by SFP-domains, L-domains, and bifinite domains.
All of these classes of orders can be cast into various categories of dcpos, using functions which are monotone, Scott-continuous, or even more specialized as morphisms. Finally, note that the term
domain itself is not exact and thus is only used as an abbreviation when a formal definition has been given before or when the details are irrelevant.
Important results[edit]
A poset D is a dcpo if and only if each chain in D has a supremum.
If f is a continuous function on a domain D then it has a least fixed point, given as the least upper bound of all finite iterations of f on the least element ⊥:
${\displaystyle \operatorname {fix} (f)=\bigsqcup _{n\in \mathbb {N} }f^{n}(\bot )}$.
This is the Kleene fixed-point theorem. The ${\displaystyle \sqcup }$ symbol is the directed join.
• "Synthetic domain theory". CiteSeerX 10.1.1.55.903.
• A continuity space is a generalization of metric spaces and posets, that can be used to unify the notions of metric spaces and domains.
See also[edit]
Further reading[edit]
• G. Gierz; K. H. Hofmann; K. Keimel; J. D. Lawson; M. Mislove; D. S. Scott (2003). "Continuous Lattices and Domains". Encyclopedia of Mathematics and its Applications. 93. Cambridge University
Press. ISBN 0-521-80338-1.
• Samson Abramsky, Achim Jung (1994). "Domain theory" (PDF). In S. Abramsky; D. M. Gabbay; T. S. E. Maibaum. Handbook of Logic in Computer Science. III. Oxford University Press. pp. 1–168. ISBN
0-19-853762-X. Retrieved 2007-10-13.
• Alex Simpson (2001–2002). "Part III: Topological Spaces from a Computational Perspective". Mathematical Structures for Semantics. Retrieved 2007-10-13.
• D. S. Scott (1975). "Data types as lattices". Proceedings of the International Summer Institute and Logic Colloquium, Kiel, in Lecture Notes in Mathematics. Springer-Verlag. 499: 579–651.
• Carl A. Gunter (1992). Semantics of Programming Languages. MIT Press.
• B. A. Davey; H. A. Priestley (2002). Introduction to Lattices and Order (2nd ed.). Cambridge University Press. ISBN 0-521-78451-4.
• Carl Hewitt; Henry Baker (August 1977). "Actors and Continuous Functionals". Proceedings of IFIP Working Conference on Formal Description of Programming Concepts.
• V. Stoltenberg-Hansen; I. Lindstrom; E. R. Griffor (1994). Mathematical Theory of Domains. Cambridge University Press. ISBN 0-521-38344-7.
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High precision floats in OpenGL ES shaders
In the previous post I tried emulating higher precision by using two 10-bit precision floats, but couldn’t get it working.
As you can see in the lower part of the screenshots, the precision did increase but not by much. The upper part of the image remains blocky since the shader runs out of memory on my device and only
updates part of the image. That’s because I revved up the number of iterations to 1024, but the final app should have that ability so this is another problem that needs to be solved. I also changed
the color of the generated image to make the difference more apparent.
Generating a geometry that has one vertex per pixel
Let’s do the calculations in the vertex shader instead of the fragment shader. OpenGL ES guarantees high precision floats in vertex shaders but not in fragment shaders, so generating the fractal
there should greatly increase the resolution. Vertex shaders are executed once per polygon vertex and not per screen pixel, so we need to create a geometry that contains exactly one vertex per pixel
of the screen and fill the screen with it.
public class Geometry {
private ShortBuffer indices;
private FloatBuffer buffer;
private Geometry generatePixelGeometry(int w, int h) {
ShortBuffer indices = ShortBuffer.allocate(w*h);
FloatBuffer buffer = FloatBuffer.allocate(3*w*h);
for(int i = 0; i < h; i++) {
for(int j = 0; j < w; j++) {
buffer.put(-1 + (2*j + 1)/(float)w);
buffer.put(-1 + (2*i + 1)/(float)h);
return new Geometry(indices, buffer);
This will generate a geometry that contains one vertex in the middle of every pixel on a screen of resolution w x h. OpenGL ES 2 uses short indexing and doesn’t support int indexing for vertices.
That’s why we use a ShortBuffer for the indices buffer. This also means that we have to split up our Geometry into smaller pieces. The max value for a short is 32767. The best option is to generate
one single geometry that has fewer vertices than that and reuse it several times across the screen using multiple draw calls. Either way we have to make sure that the final scene has one vertex in
every pixel of the screen.
The vertex shader
attribute vec3 position;
uniform mat4 modelViewMatrix;
uniform float MAX_ITER;
uniform vec2 scale;
uniform vec2 offset;
varying vec4 rgba;
vec4 color(float value, float radius, float max);
vec2 iter(vec2 z, vec2 c)
// Complex number equivalent of z*z + c
return vec2(z.x*z.x - z.y*z.y, 2.0*z.x*z.y) + c;
void main(void)
vec4 posit = modelViewMatrix * vec4(position, 1.);
vec2 c = 2.*(scale*posit.xy + offset);
vec2 z = vec2(0.);
int i;
int max = int(MAX_ITER);
float radius = 0.;
for( i=0; i<max; i++ ) { radius = z.x*z.x + z.y*z.y; if( radius > 16.) break;
z = iter(z,c);
float value = (i == max ? 0.0 : float(i));
rgba = color(value, radius, MAX_ITER);
gl_Position = posit;
The vertex shader has to take the modelViewMatrix into account since we are splitting the geometry into multiple pieces and placing it at different positions in a scene. This had to be done because a
single big geometry would be too big for short.
The fragment shader
precision mediump float;
varying vec4 rgba;
void main()
gl_FragColor = rgba;
The fragment shader is as simple as it can be. It just receives the rgba vector from the vertex shader and draws it to the frame buffer.
That’s much better!
The fractal renders slightly slower than with the lower precision shader but still really quickly and does 1024 iterations without running out of memory . Let’s see how much we can zoom in now.
The pixels start filling out the screen at a scale of 0.0002, a zoom level of 5000x. Switching to the vertex mode instead shows an amazing difference. On my device highp floats in the vertex shader
are normal 32-bit floats and have a precision of 23 bits, while the mediump 16-bit floats in the fragment shader only have a 10-bit precision. That means that the precision is more then doubled.
Every pixel in fragment mode becomes in vertex mode bigger than the entire shader was in fragment mode. In fact pixels became apparent first on a scale of 2*10^(-6). See the results for your self in
the google play: GPU Mandelbrot.
In the next post I’ll give double emulation one more try.
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array definition
Page Start Prev 1 Next End
array definition
the purpose of this model is to provide the indexed elements of the "vector" array the values of the array "tester", which are increasing numbers from 1 to 12.
This shall happen every time the BooleanPulse is true. The arrays are indexed by the Integer "counter", which increases by 1 every boolean pulse.
the model simulates but gives every "vector" element the value 0. also the warning message is "Failed to solve linear system of equations (no. 21) at time 0.000000. Residual norm is 0.5."
model Array_Test_simple
Real vector[12];
Integer counter;
Real tester[12]=1:1:12;
Modelica.Blocks.Sources.BooleanPulse booleanPulse(period = 0.1) annotation(
Placement(visible = true, transformation(origin = {-62, 38}, extent = {{-10, -10}, {10, 10}}, rotation = 0)));
initial algorithm
when booleanPulse.y then
end when;
uses(Modelica(version = "4.0.0")));
end Array_Test_simple;
Page Start Prev 1 Next End
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Utilizing the Simple CLV Formula - Guide to CLV Analysis
When to Use the Simple CLV Formula
There are two customer lifetime value (CLV) formulas provided on this website. This article discusses when it is appropriate to use the simple customer lifetime value formula.
You may also want to refer to the limitations of using this formula and you should also review the article on main customer lifetime value formula. Please note there are also a free Excel spreadsheet
template available on this website that allows you to calculate customer lifetime value.
When to use the simple CLV formula
The simple CLV formula is quite appropriate to use when:
• Retention rates of customers are relatively low (say 50% or less) – which means of most of the customer profit contribution is received in the first few years
• Profit contributions from customers are relatively flat over time – that is, there is no significant changes in the average customer profitability with time
• The data inputs to the model are mainly based upon assumptions rather than historical customer data (with the customer lifetime value just being a rough estimate)
• Only a ballpark customer lifetime value estimate is needed – rather than a precise measure
• The firm is uninterested or uncomfortable in using discount rates
• Customers are highly profitable and recover their acquisition costs within the first year
How accurate is the simple CLV formula?
In many cases, the simple customer lifetime value formula is quite adequate and will provide a reasonable estimate as compared to the main CLV formula (which is available on the free Excel
spreadsheet template on this website).
There are two key differences between the simple CLV and the main CLV formulas are:
• The simple customer lifetime value formula assumes that all the data inputs (customer revenue/costs and retention rate) remains consistent over time, and
• The simple CLV formula does not use a discount rate.
Let’s compare the calculations of both CLV formulas using an example as follows:
A firm has an average acquisition cost of $500, average customer annual profit contribution is $800 – which increases by 5% per year, and the firm’s retention rate is 60% (equivalent to 2.5 years
average lifetime). In this case, a 10% discount rate is used in the main CLV formula.
Using the simple CLV formula:
• CLV = $840 (customer profit contribution – see note below) X
• 2.5 (average customer lifetime in years) –
• $500 (Customer acquisition cost)
• CLV = $2,000 – $500 = $1,600
NOTE: In this case, we have used $840 in annual customer profit contribution. While the customer profit starts at $800, it increases by 5% pa – so profit becomes $840 in year 2 and then $882 in year
3, and so on. Because the customer lifetime period s only 2.5 years, the 2nd years’ profit has been used as the approximate average customer profit.
Using the main CLV formula (using the free Excel template):
CLV (before discounting) = $1,660
CLV (after the discount rate) = $1,201
Therefore, a close approximation WITHOUT using a discount rate
As you can see, before the applicable of the discount rate, both CLV figures are very close in value – $1,600, as compared to $1,660 using the template (without using the discount rate).
However, significant differences when USING a discount rate
As you can also see, once the discount rate is applied, there is a significant difference between the two formulas – $1,600 as compared to $1,201.
Therefore, the main customer lifetime value formula (or Excel template) should be used if:
• The firm normally uses a discount rate
• The retention rate is quite high (that is, more than 75%)
• There are significant changes in the customer revenue and/or costs over time | {"url":"https://www.clv-calculator.com/customer-lifetime-value-formulas/simple-clv-formula/use-simple-clv-formula/","timestamp":"2024-11-02T11:43:51Z","content_type":"text/html","content_length":"84020","record_id":"<urn:uuid:0aabed48-cada-4675-9b57-a85a20439bbc>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00408.warc.gz"} |
- Provides surface interpolation from vector point data by Inverse Distance Squared Weighting.
v.surf.idw --help
v.surf.idw [-n] input=name [layer=string] [column=name] output=name [npoints=count] [power=float] [--overwrite] [--help] [--verbose] [--quiet] [--ui]
Don't index points by raster cell
Slower but uses less memory and includes points from outside region in the interpolation
Allow output files to overwrite existing files
Print usage summary
Verbose module output
Quiet module output
Force launching GUI dialog
input=name [required]
Name of input vector map
Or data source for direct OGR access
Layer number or name
Vector features can have category values in different layers. This number determines which layer to use. When used with direct OGR access this is the layer name.
Default: 1
Name of attribute column with values to interpolate
If not given and input is 2D vector map then category values are used. If input is 3D vector map then z-coordinates are used.
output=name [required]
Name for output raster map
Number of interpolation points
Default: 12
Power parameter
Greater values assign greater influence to closer points
Default: 2.0
v.surf.idw fills a raster matrix with interpolated values generated from a set of irregularly spaced vector data points using numerical approximation (weighted averaging) techniques. The interpolated
value of a cell is determined by values of nearby data points and the distance of the cell from those input points. In comparison with other methods, numerical approximation allows representation of
more complex surfaces (particularly those with anomalous features), restricts the spatial influence of any errors, and generates the interpolated surface from the data points.
Values to interpolate are read from column option. If this option is not given than the program uses categories as values to interpolate or z-coordinates if the input vector map is 3D.
The amount of memory used by this program is related to the number of vector points in the current region. If the vector point map is very dense (i.e., contains many data points), the program may not
be able to get all the memory it needs from the system. The time required to execute is related to the resolution of the current region, after an initial delay determined by the time taken to read
the input vector points map.
Note that vector features without category in given layer are skipped.
If the user has a mask set, then interpolation is only done for those cells that fall within the mask. The module has two separate modes of operation for selecting the vector points that are used in
the interpolation:
Simple, non-indexed mode (activated by -n flag)
When the -n flag is specified, all vector points in the input vector map are searched through in order to find the npoints closest points to the centre of each cell in the output raster map. This
mode of operation can be slow in the case of a very large number of vector points.
Default, indexed mode
By default (i.e. if -n flag is not specified), prior to the interpolation, input vector points are indexed according to which output raster cell they fall into. This means that only cells nearby
the one being interpolated need to be searched to find the npoints closest input points, and the module can run many times faster on dense input maps. It should be noted that:
□ Only vector points that lie within the current region are used in the interpolation. If there are points outside the current region, this may have an effect on the interpolated value of cells
near the edges of the region, and this effect will be more pronounced the fewer points there are. If you wish to also include points outside the region in the interpolation, then either use
the -n flag, or set the region to a larger extent (covering all input points) and use a mask to limit interpolation to a smaller area.
□ If more than npoints points fall within a given cell then, rather than interpolating, these points are aggregated by taking the mean. This avoids the situation where some vector points can be
discarded and not used in the interpolation, for very dense input maps. Again, use the -n flag if you wish to use only the npoints closest points to the cell centre under all circumstances.
The power parameter defines an exponential distance weight. Greater values assign greater influence to values closer to the point to be interpolated. The interpolation function peaks sharply over the
given data points for 0 < p < 1 and more smoothly for larger values. The default value for the power parameter is 2.
By setting npoints=1, the module can be used to calculate raster Voronoi diagrams (Thiessen polygons).
g.region, r.surf.contour, r.surf.idw, r.surf.gauss, r.surf.fractal, r.surf.random, v.surf.rst
Overview: Interpolation and Resampling in GRASS GIS
Michael Shapiro, U.S. Army Construction Engineering Research Laboratory
Improved algorithm (indexes points according to cell and ignores points outside current region) by Paul Kelly
Available at: v.surf.idw source code (history)
Latest change: Thursday Jan 26 14:10:26 2023 in commit: cdd84c130cea04b204479e2efdc75c742efc4843
Main index | Vector index | Topics index | Keywords index | Graphical index | Full index
© 2003-2024 GRASS Development Team, GRASS GIS 8.5.0dev Reference Manual | {"url":"https://grass.osgeo.org/grass85/manuals/v.surf.idw.html","timestamp":"2024-11-07T09:43:32Z","content_type":"text/html","content_length":"10064","record_id":"<urn:uuid:c9aeccf5-29da-4eab-82de-06da8e8d1c85>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00467.warc.gz"} |
Now showing items 1-10 of 17
A populated iterated greedy algorithm with inver-over operator for traveling salesman problem
(Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2013)
In this study, we propose a populated iterated greedy algorithm with an Inver-Over operator to solve the traveling salesman problem. The iterated greedy (IG) algorithm is mainly based on the central
procedures of destruction ...
A general variable neighborhood search algorithm for the no-idle permutation flowshop scheduling problem
(Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2013)
In this study, a general variable neighborhood search (GVNS) is presented to solve no-idle permutation flowshop scheduling problem (NIPFS), where idle times are not allowed on machines. GVNS is a
metaheuristic, where inner ...
Lagrangian heuristic for scheduling a steelmaking-continuous casting process
(Proceedings of the 2013 IEEE Symposium on Computational Intelligence in Scheduling, CISched 2013 - 2013 IEEE Symposium Series on Computational Intelligence, SSCI 2013, 2013)
One of the biggest bottlenecks in iron and steel production is the steelmaking-continuous casting (SCC) process, which consists of steel-making, refining and continuous casting. The production
scheduling of SCC is a complex ...
Null control in linear antenna arrays with ensemble differential evolution
(Proceedings of the 2013 IEEE Symposium on Differential Evolution, SDE 2013 - 2013 IEEE Symposium Series on Computational Intelligence, SSCI 2013, 2013)
This paper describes a synthesis method for null insertion in linear antenna array geometries by using newly proposed ensemble differential evolution (DE) algorithm. The given ensemble DE algorithm
uses the advantages of ...
Metaheuristic algorithms for the quadratic assignment problem
(Proceedings of the 2013 IEEE Symposium on Computational Intelligence in Production and Logistics Systems, CIPLS 2013 - 2013 IEEE Symposium Series on Computational Intelligence, SSCI 2013, 2013)
This paper presents two meta-heuristic algorithms to solve the quadratic assignment problem. The iterated greedy algorithm has two main components, which are destruction and construction procedures.
The algorithm starts ...
Ensemble of differential evolution algorithms for electromagnetic target recognition problem
(IET Radar, Sonar and Navigation, 2013)
In this study, an ensemble of differential evolution (DE) algorithms is presented to classify electromagnetic targets in resonance scattering region. The algorithm aims to synthesize a special
incident signal for each ... | {"url":"https://dspace.yasar.edu.tr/xmlui/handle/20.500.12742/1/discover?rpp=10&filtertype_0=has_content_in_original_bundle&filtertype_1=author&filter_relational_operator_1=equals&filter_relational_operator_0=equals&filter_1=Tasgetiren%2C+M.F.&filter_0=false&filtertype=dateIssued&filter_relational_operator=equals&filter=2013","timestamp":"2024-11-09T20:48:32Z","content_type":"text/html","content_length":"53983","record_id":"<urn:uuid:d382c65e-d16a-4335-a734-06e0c152a5a2>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00162.warc.gz"} |
quantum algorithms qiskit
Quanvolutional Neural Network s (Pennylane demo).
Resource person during the 5-day Faculty Development Programme on. There are quite a few available today, each with its own set of quirks In the 1.44 system, one can see that if f ( For older news
items published in 2021 click here, for 2020 click here, for 2019 click here, for 2018 click here, and for items published in 2015-2017, click here. Deutsch-Jozsa Algorithm is the first quantum
algorithm that demostrate Quantum algorithm faster than the best classical algorithm. Quantum computing is fast emerging as one the key disruptive technologies of our times. It is part of many
quantum algorithms, most notably Shor's Grovers algorithm has a huge advantage over classical Add Business Sign Up Sign In. Access our more advanced systems on an as-needed basis, and pay only for
the quantum compute time you use. 4.1 Applied Quantum Algorithms. This tutorial series is designed to provide readers from any background with two services: 1) A concise and thorough understanding of
some of the most popular/academically Quantum Algorithms for Applications. In this first version you can explore running simple circuits or more complex variational algorithms (based on VQE). Click
any link to open the tutorial directly in Quantum Lab. For more look at 1.44 equation (page 33) in the M. Nielsen and I. Chuang textbook, where one can find the final state before the measurement.
Here is the list of the tutorials (existing and planned). View Phone, Address, Reviews, Complaints, Compliments and Similar Businesses to Quantum Dental. Week 2 :IBM Quantum Composer and Quantum CNN
with Quantum Fully Connected Layer. The Deutsch Jozsa algorithm is a good place to start since it was the first example of a quantum algorithm that performs better than the best classical algorithm.
Portfolio optimization - This tutorial shows how to solve a mean-variance portfolio optimization problem for n This is the repository for the interactive open-source Learn Quantum Computation using
Qiskit textbook. The textbook is intended for use as a university quantum Implementation The algorithm can be implemented incredibly easily since Qiskit has a baked in function for the algorithm
called Shor(N). 4.1.3 Solving combinatorial optimization problems using QAOA.
Future quantum computers may one day tackle these problems exponentially faster than classical computers. The backend can be set as K=tc.set_backend("jax") and K is the backend with a full set of
APIs as a conventional ML framework, which can also be accessed by tc.backend. The measurement of the top qubit will appear on bit 0 of the 5-bit line and the measurement of the second qubit will
appear on bit 1 of the 5-bit line. Next we create the QSVM with the following code: svm = QSVM 4.1.4 Solving Satisfiability Problems using Grover's Algorithm. The quantum computing market is
predicted to grow by nearly $1.3 billion over the next five In "Barren Plateaus in Quantum Neural Network Training Landscapes", we focus on the training of quantum neural networks , and probe
questions related to a key difficulty in classical neural networks , which is the problem of vanishing or exploding gradients. Access our most advanced core systems: 27-qubit Falcon R5 Click any link
to open the tutorial directly in Quantum Lab. Qiskit tutorials: Finance. This is the repository for the interactive open-source Learn Quantum Computation using Qiskit textbook. 4.1.1 Solving Linear
Systems of Equations using HHL. IBMs new Qiskit primitives make it easier to develop algorithms for quantum computers Big Blue has emerged as quantum computing's clear front-runner April 12, 2022 -
1:00 pm Greetings from the Qiskit Community team! Brookshire. Finally, you'll explore quantum algorithms and understand how they differ from classical algorithms, along with learning how to use
pre-packaged algorithms in Qiskit(R) Aqua. Shors algorithm was demonstrated in 2001 by a group at IBM, which factored 15 into 3 and 5, using a quantum computer with 7 qubits. This algorithm consists
of three steps (More information about how to implement a phase oracle and a phase estimation is found in the Qiskit textbook ): First, put the node and coin As a continuation to its predecessor:
"Introduction to Coding Quantum Algorithms: A Tutorial Series Using Qiskit", this tutorial series aims to help understand several of the most promising Nvidia is accelerating its efforts to bridge
the GPU and quantum computing realms through cuQuantum, its Tensor-capable (opens in new tab) quantum simulation toolkit. Before implementing quantum algorithms on real quantum computers, it is
important to highlight the definition of a quantum circuit concretely, as we will be building quantum circuits to implement these algorithms. 4.1.2 Simulating Molecules using VQE. Access our most
powerful core and exploratory systems with shorter wait times and hands on service. This Certificate is Presented to Ms. Neha Sharma . 3.3 The Deutsch Jozsa algorithm Hybrid quantum -classical Neural
Network s with PyTorch and Qiskit ( Qiskit textbook) Gradients of parameterized quantum gates using the parameter-shift rule and gatedecomposition (arxiv) Model 2. The programming model extends the
existing interface in Qiskit with a set of new primitive programs. Click any link to open the tutorial directly in Quantum Lab. Jul 2021. Quantum Dental Dental Laboratories in Katy, TX. Qiskit
tutorials: Machine learning. The Brookshire Police Department investigates a shooting near Fourth Street and Purdy. KPConv-PyTorch. Ground state solvers - This tutorial discusses the ground state
calculation interface of Qiskit Chemistry. Chitkara University Punjab for sharing her valuable knowledge as a. They create a quantum circuit and run an approximate simulation on it 1000 times: Why
Stack Exchange Network Stack Exchange network consists of 180 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and
build their careers. Tutorials for Quantum Algorithms This is a collection of tutorials for quantum algorithms. BROOKSHIRE, Texas - Multiple people were We recommend using TensorFlow or Jax backend
since PyTorch lacks advanced jit and vmap features. Qiskit Runtime speeds up processing time by combining classical and quantum computing in a streamlined architecture. Dental Laboratories. Since
almost all quantum algorithms use probability distributions, Sampler and Estimator are likely to have broad applicability across the entire spectrum of quantum algorithms. Qiskit Pocket Guide:
Quantum Development with Qiskit by Francis Harkins, James Weaver. Recent news items published within the last 6 months on quantum computing developments are listedan below. IBM uses a quantum state
tomography algorithm to calculate how closely the resulting state matches the expected state. The textbook is intended for use as a university quantum algorithms course supplement as well as a guide
for self-learners who are interested in learning quantum programming. Now youre ready to run your. We are excited to share a solution to one of our most frequent customer requests: a Qiskit provider
for Amazon Braket.Users can now take their existing algorithms written in Qiskit, a widely used open-source quantum programming SDK and, with a few lines of code, run them directly on Amazon
Braket.The qiskit-braket-provider currently supports access to superconducting quantum processing Come up with your own original circuit and you'll be very famous! FOX 26 Houston. It is a
fundamentally new computing paradigm that has the potential to efficiently solve certain challenging problems which cannot be solved efficiently in a classical setting. gates and you can design any
quantum circuit. Week 1 :Introduction and IBM Quantum Perspective, Q Mission in India Invited talk, Quantum Computing Applications, Quantum Computing Basics. In this section, we're going to introduce
you to a very popular one used in this field: IBM's Quantum Information Science Kit, or Qiskit. Quantum Computers will offer new drugs, better AI, new encryption schemes and solve problems more
efficiently. Quantum Science and Technology A multidisciplinary, high impact journal devoted to publishing research of the highest quality and significance covering the science and application of all
quantum-enabled technologies. Qiskit Runtime is a quantum computing service and programming model that allows users to optimize workloads and efficiently execute them on quantum systems at scale. Now
suppose we want to use qiskit to construct a circuit for CNOT using |+> as the control qubit and |0> as the target qubit.We will need to create a quantum register to hold two qubits with qr =
QuantumRegister(2).We will also need to give each qubit in the register as an argument to the cx method of the QuantumCircuit class. The algorithm on a quantum computer is similar to the algorithm we
just coded, except we do not split the potential propagator into two half steps. Create and execute quantum programs at scale with Qiskit Runtime.Gain hands-on support and training as well as curated
access to the best minds in quantum computing via membership to the IBM Quantum Network.. Current Premium Plan systems: Click on the hyperlinked item to go to the press release or news article for
more details. Qiskit tutorials: Optimization. Quantum phase estimation (QPE) is the key subroutine of several quantum computing algorithms as well as a central ingredient in quantum computational
chemistry and quantum simulation. Qiskit is an open-source SDK for working with quantum computers at the level of extended quantum circuits, operators, and algorithms. And this algorithm works with
the measurement counts. Loading Quantum Protocols and Quantum Algorithms After all the work done in the previous posts, we are now ready to actually implement Shors factoring algorithm on a real
quantum computer, using once Qiskit is an open-source SDK for working with quantum computers at the level of pulses, circuits, and application modules. Pay-As-You-Go Plan. 2. Introduction to Quantum
Computing: Quantum Algorithms and Qiskit. Quantum Computing, sponsored by AICTE-ATAL and organized by IIIT. And, it doesnt even matter if you use quantum machine learning algorithms. At the heart of
QML is the ability to hybridise classical and quantum worlds. Finally, you'll explore quantum algorithms and understand how they differ from classical algorithms, along with learning how to use
pre-packaged algorithms in Qiskit(R) Aqua. | {"url":"http://www.stickycompany.com/the/pan/nintendo/92054278c435137-quantum-algorithms-qiskit","timestamp":"2024-11-03T04:24:04Z","content_type":"text/html","content_length":"18644","record_id":"<urn:uuid:2b857fe0-67de-4730-88f5-98ceba318266>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00088.warc.gz"} |
Scan and Over
Over and Scan provide Map-Reduce and Fold as single-character operators.^1
The accumulators, Scan and Over, perform the same computations. They share the same Do, Converge and While forms for unary arguments.
They differ only in that
• Scan (f\) returns the result of every iteration
• Over (f/) returns the final result
Number of iterations
For long int n and test expression t
rank of f expression i (#iterations) count[r] r[0]
1 r:f\[y] ≥1 (Converge) i y
1 r:n f\y n (Do) 1+i y
1 r:t f\y ≥0 (While) 1+i y
2 r:x f\y count y i f[x;first y]
3–8 r:f\[x;y;z;…] max count(y;z;…) i f[x;first y;first z;…]
In each of the forms above, f/ performs the same computation as f\, with the same number of iterations.
Zero iterations
The Converge, Do, and While forms all return a result in which the first item is the result of zero iterations; i.e. y.
q)0{'`ouch}\100 / zero iterations
q)(til 6;5(.1*)\3.) / (# iterations;result)
3 0.3 0.03 0.003 0.0003 3e-05
Data type change
Where f returns a result of different type than its argument/s the result of f\ is a mixed list.
q)5(.1*)\3. / argument and results are all floats
3 0.3 0.03 0.003 0.0003 3e-05
q)5(.1*)\3 / iteration 1 changes type
Above, the test 11> is applied to the 0th iteration, i.e. y, the argument 100. The test fails, and the composition 1+ is not applied.
q){'`ouch} 100 / evaluation of the lambda signals an error
[0] {'`ouch} 100
q)(11>){'`ouch}\100 / no evaluation of the lambda
For Converge, the result of the evaluation that terminates iteration is not returned.
Do as Cond
Suppose you have a test result test that is not a boolean but a long int; i.e. 0 or 1. You could use it in a conditional – or as a left argument to Do.
{$[test;f x;x]} <==> test f/x
In Jeff Borrors’ classic textbook Q for Mortals you will find frequent references to moments of “Zen meditation” leading to flashes of insight into the workings of q.
My teacher Myokyo-ni liked to quote the Third Patriarch:
The Great Way is not difficult. It avoids only picking and choosing.
The Do form of the Scan iterator has a pattern sometimes called ‘the Zen monks’.
The pattern is to apply a function and not to apply it.
Consider the trim keyword. It must find the spaces in a string, then the continuous spaces from each end. If we had to write trim in q it might be
q){b:x<>" ";(b?1b)_ neg[reverse[b]?1b] _ x}" Trim the spaces. "
"Trim the spaces."
We notice the repetitions:
• both b and reverse[b] are searched for 1b
• two uses of the Drop operator
We want to do the search/drop thing from both ends of the string.
q){x{y _ x}/1 -1*(1 reverse\" "<>x)?'1b}" Trim the spaces. "
"Trim the spaces."
Applying a series of left arguments
Notice the {y _ x}/ reduction above. Lambda {y f x} commutes an operator f by switching its arguments. The pattern R{y f x}/L successively applies a list of left arguments L to an argument R.
Here we use 1 reverse\ to get the boolean vector and its reversal. The 1 f\ pattern is known as the ‘Zen monks’. (Of course, we could write {(x;f x)} but where’s the fun in that?)
Here is another use for it, in finding the shape (rows and columns) of a matrix.
q)show m:{max[count each x]$'x}string`avoids`picking`and`choosing
"avoids "
"picking "
"and "
q)shp:{count each 1 first\x} / shape of a matrix
q)shp m
The Zen Buddhist pension plan
A day without work is a day without food. — Baizhang Huaihai
Can you find any other work for the monks?
Final iteration
For the While form of Scan, the result of the evaluation that terminates iteration is returned.
q)(11>) (1+)\1
The test expression t is applied to the result of every iteration (even the 0th); so
t last t f\y <==> 0
Test expression
The test expression t in t f/y and t f\y is any applicable value that could return a zero when applied to the result of an iteration. That is, a
• keyword
• projection
• lambda
• dictionary
• list
And the zero is any value that casts to zero.
q)show bd:`bob`ted`sue`carol`mary`tom`dick!2000.01.01+1000* -3+til 7 /birthdays
bob | 1991.10.15
ted | 1994.07.11
sue | 1997.04.06
carol| 2000.01.01
mary | 2002.09.27
tom | 2005.06.23
dick | 2008.03.19
q)show likes:.[!]reverse 1 (1 rotate)\key bd / who likes whom
ted | bob
sue | ted
carol| sue
mary | carol
tom | mary
dick | tom
bob | dick
q)likes\[`tom] / chain of likes
q)bd likes\`tom / follow chain of likes to someone born 1/1/2000
Above, both f and the test expression are dictionaries, and the zero is 2000.01.01.
Higher-rank arguments
1. In x f/y and x f\y what is the rank of f?
In x f/y and x f\y the rank of f can be only 1 or 2.
□ If 1, x is either a non-negative integer or a test expression.
□ If 2, x is the left value for iteration 1, which is f[x;y 0].
2. Predict the value of t and f:
(t;f):0 0
test fn\1
q)(t;f):0 0;test fn\1
test was evaluated 11 times and fn 10.
3. Write a binary f so that for atom x and vector y, x f\y is a general list, not a vector.
The result of x f\y will be a vector as long as f[a;b] returns an atom for atoms a and b. Any other result will have x f\y evaluate to some other kind of list.
q)3{y?x}\2 4 5 4 2
4. If test is a long 0 or 1, the lambda {$[test;f x;x]} could be replaced by test f/x. Explain why this is not a smart substitution.
The two expressions are equivalent provided test is either long 0 or 1. But if it were, say, 100 then, while the lambda would still apply f once, test f/x would apply it 100 times.
5. Above, shp returns the shape of a matrix – rows followed by columns. That is, the shape of a 2-dimensional array. Write a version that returns as a vector the shape of an \(N\)-dimensional array.
Keep applying first – Converge will stop when it reaches an atom. Ignore the count of the atom.
shp:{-1_ count each first\[x]} /lambda
shp: -1_ (count') (first\) :: /composition
6. Find simpler equivalents for:
first 1 f\x
last 1 f\x
last 5 f\x
7. Where f has rank 3, compare f\[x;y;z] to f'[x;y;z]. Compare f'[x;y;z] to f[x]'[y;z]. Is f\[x;y;z] equivalent to f[x]\[y;z]? Explain.
In f'[x;y;z] all three arguments must conform; in f\[x;y;z] only the right arguments must conform.
In f[x]'[y;z] the binary projection f[x;;] is applied to corresponding items of y and z. In each iteration the left argument is the full value of x.
Similarly in f[x]\[y;z] the value of the first argument is fixed as x; it is the value of the second argument that iterates successively. The items of result r:
r[0] <==> f[x; y; z@0]
r[1] <==> f[x; r@0; z@1]
r[2] <==> f[x; r@1; z@2]
1. Scan and Over descend from the very first implementation of Iverson Notation in the 1960s. APL had Map-Reduce when computer programs were still punched on cards. ↩ | {"url":"https://q201.org/explore-scan/","timestamp":"2024-11-02T11:24:59Z","content_type":"text/html","content_length":"63800","record_id":"<urn:uuid:5ba51292-66cf-40cc-aff5-4340fd8eec35>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00729.warc.gz"} |
Chemical Masses
Relative mass on the atomic scale
It is convenient to describe masses on a relative scale of numbers that has no units of ‘u’. Relative mass of an entity is equal to the mass of that entity (u) is equal to 1/12 mass of a carbon 12
atom (1u), which in turn is also equal to a number with no units.
Relative mass of an isotope
These are very close to whole numbers and, for most purposes, are usually quoted as those whole numbers.
H = 1 H = 2 C = 12 C= 13 Cl= 35 Cl = 37
Relative atomic mass
Naturally occurring elements exist as a mixture of different isotopes. Th e relative atomic mass of the element will be affected by the relative proportions of the different isotopes.
E.g. About 75% of naturally occurring chlorine is 75% of 35Cl and 25% of 37Cl. Thus the relative atomic mass of Cl is ;
0.75 x 35 + 0.25 x 37 = 35.5.
Relative atomic mass (R.A.M.) of an element
Th e relative atomic mass (R.A.M.) of an element is the average mass of the atoms in the naturally-occurring isotopic mixture of a carbon 12 atom (1u). It can be calculated from knowing the natural
isotope abundance.
• m1, m2, m3 are the masses of the individual isotopes (use accurate values in u if they are given, or use the mass numbers of the isotopes)
• P1, P2, P3 are the percentages of these isotopes in the naturally occurring mixture for this element.
R.A.M. values of elements are oft en expressed to the nearest whole number. E.g. H = 1, Li = 7, C = 12 and Na = 23. But sometimes more accurate values are needed, E.g. Cl = 35.45. Use the values
provided on a Periodic Table.
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onfidence interval calculator for population mean
Confidence interval calculator for population mean
Population Confidence Interval Calculator is an online statistics and probability tool for data analysis programmed to construct a confidence interval for a population proportion. The Confidence
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What is the confidence interval?
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A funny encounter with immersions of spheres
A funny encounter with immersions of spheres
• Venue:
• Date:
• Speaker:
Tam Nguyen Phan
• Time:
16:00 – 17:30
• It is a common topology homework exercise to prove that any embedding of a closed, orientable n-manifold M, such as the n-sphere, into R^{n+1}, is separating (i.e. has disconnected complement).
However, this homework problem has never been posed with “embedding”
replaced by “immersion”, probably because it seems at first glance to complicate the problem for no good reason. But there is actually an interesting reason. This talk is based on joint work with
Michael Freedman. | {"url":"https://topology.math.kit.edu/english/123_979.php","timestamp":"2024-11-04T17:21:01Z","content_type":"text/html","content_length":"31856","record_id":"<urn:uuid:af4bb65b-22fa-4f64-bb7f-5e0fa4e1f426>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00773.warc.gz"} |
Find the sum of three consecutive odd integers if the sum of the first two integers is equal to twenty-four less than four times the third integer
Get an answer to your question ✅ “Find the sum of three consecutive odd integers if the sum of the first two integers is equal to twenty-four less than four times the third ...” in 📙 Mathematics if
there is no answer or all answers are wrong, use a search bar and try to find the answer among similar questions.
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al Circuit
Half Subtractor:-
The half subtractor is also a building block for subtracting two binary numbers. It has two inputs and two outputs. This circuit is used to subtract two single bit binary numbers A and B. The 'diff'
and 'borrow' are two output states of the half subtractor.
Block diagram:-
Truth Table:-
The SOP form of the Diff and Borrow is as follows:
Diff= A'B+AB'
Borrow = A'B
*formed using simulator provided.
Full Subtractor:-
The Half Subtractor is used to subtract only two numbers. To overcome this problem, a full subtractor was designed. The full subtractor is used to subtract three numbers A, B, and C, which are
minuend, subtrahend, and borrow, respectively. The full subtractor has three input states and two output states i.e., diff and borrow.
Block diagram:-
Truth Table:-
The SOP form can be obtained with the help of K-map as:-
Diff=xy' z'+x' y' z+xyz+x'yz'
Borrow=x' z+x' y+yz
*formed using simulator provided. | {"url":"https://circuitverse.org/projects/tags/Half%20Subtractor","timestamp":"2024-11-11T18:06:45Z","content_type":"text/html","content_length":"94700","record_id":"<urn:uuid:ff4a671c-d731-401c-8252-3ca649a8d670>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00721.warc.gz"} |
An Etymological Dictionary of Astronomy and Astrophysics
least common multiplier (LCM)
کوچکترین بستاگر ِهمدار
kucektarin bastâgar-e hamdâr
Fr.: plus petit commun multiple
Of two or more → integers, the smallest positive number that is divisible by those integers without a remainder.
→ least; → common; → multiplier.
least squares
کوچکترین چاروشها
kucaktarin cârušhâ
Fr.: moindres carrés
Any statistical procedure that involves minimizing the sum of squared differences.
→ least; → square.
least-squares deconvolution (LSD)
واهماگیش ِکمترین چاروشها
vâhamâgiš-e kucaktarin cârušhâ
Fr.: déconvolution des moindres carrés
A → cross correlation technique for computing average profiles from thousands of → spectral lines simultaneously. The technique, first introduced by Donati et al. (1997, MNRAS 291,658), is based on
several assumptions: additive → line profiles, wavelength independent → limb darkening, self-similar local profile shape, and weak → magnetic fields. Thus, unpolarized/polarized stellar spectra can
indeed be seen as a line pattern → convolved with an average line profile. In this context, extracting this average line profile amounts to a linear → deconvolution problem. The method treats it as a
matrix problem and look for the → least squares solution. In practice, LSD is very similar to most other cross-correlation techniques, though slightly more sophisticated in the sense that it cleans
the cross-correlation profile from the autocorrelation profile of the line pattern. The technique is used to investigate the physical processes that take place in stellar atmospheres and that affect
all spectral line profiles in a similar way. This includes the study of line profile variations (LPV) caused by orbital motion of the star and/or stellar surface inhomogeneities, for example.
However, its widest application nowadays is the detection of weak magnetic fields in stars over the entire → H-R diagram based on → Stokes parameter V (→ circular polarization) observations (see also
Tkachenko et al., 2013, A&A 560, A37 and references therein).
→ least; → square; → deconvolution.
least-squares fit
سز ِکوچکترین چاروشها
saz-e kucaktarin cârušhâ
Fr.: ajustement moindres carrées
A fit through data points using least squares.
→ least squares; → fit.
۱) پریژیدن؛ ۲) پریژ
1) parižidan; 2) pariž
Fr.: 1) quitter; 2) congé, permission
1a) Go away from.
1b) To let remain or have remaining behind after going, disappearing, ceasing, etc.
2a) Permission to be absent, as from work or military duty.
2b) The time this permission lasts (Dictionary.com).
M.E. leven, from O.E. laefan "to allow to remain in the same state or condition" (cf. O.Saxon farlebid "left over;" Ger. bleiben "to remain") ultimately from PIE *leip- "to stick, adhere;" also
"fat," from which the cognates: Gk. lipos "fat;" O.E. lifer "liver," → life.
Parižidan, on the model of Sariqoli barēzj "leavings;" Yaghnobi piraxs- "to stay behind, remain;" ultimately from Proto-Ir. *apa-raic-, from *raic- "to abandon, leave;" cf. Av. raēc- "to leave, let"
(Cheung 2006), → heritage.
Leavitt law
قانون ِلویت
qânun-e Leavitt
Fr.: loi de Leavitt
Same as the → period-luminosity relation.
Named after Henrietta Swan Leavitt (1868-1921), American woman astronomer, who discovered the relation between the luminosity and the period of → Cepheid variables (1912); → law.
Leclanché cell
پیل ِلوکلانشه
pil-e Leclanché (#)
Fr.: pile de Leclanché
A → primary cell in which the anode is a rod of carbon and the cathode a zinc rod both immersed in an electrolyte of ammonia plus a depolarizer.
Named after the inventor Georges Leclanché (1839-1882), a French chemist, → cell.
Ledâ (#)
Fr.: Léda
1) The ninth of Jupiter's known satellites and the smallest. It is 16 km in diameter and has its orbit at 11 million km from its planet. Also called Jupiter XIII, it was discovered by Charles Kowal
(1940-), an American astronomer, in 1974.
2) An asteroid, 38 Leda, discovered by J. Chacornac in 1856.
In Gk. mythology, Leda was queen of Sparta and the mother, by Zeus in the form of a swan, of Pollux and Helen of Troy.
Ledoux's criterion
سنجیدار ِلودو
sanjidâr-e Ledoux
Fr.: critère de Ledoux
An improvement of → Schwarzschild's criterion for convective instability, which includes effects of chemical composition of the gas. In the Ledoux criterion the gradient due to different molecular
weights is added to the adiabatic temperature gradient.
After the Belgian astrophysicist Paul Ledoux (1914-1988), who studied problems of stellar stability and variable stars. He was awarded the Eddington Medal of the Royal Astronomical Society in 1972
(Ledoux et al. 1961 ApJ 133, 184); → criterion.
cap (#)
Fr.: gauche
Of, pertaining to, or located on or toward the west when somebody or something is facing north. Opposite of → right.
M.E. left, lift, luft, O.E. left, lyft- "weak, idle," cf. Ger. link, Du. linker "left," from O.H.G. slinc, M.Du. slink "left," Swed. linka "limp," slinka "dangle."
Cap "left," from unknown origin.
left-hand rule
رزن ِدست ِچپ
razan-e dast-e cap
Fr.: règle de la main gauche
See → Fleming's rules.
→ left; → hand; → rule.
چپال، چپدست
capâl (#) , capdast (#)
Fr.: gaucher
Using the left hand with greater ease than the right.
→ left; → hand + -ed.
Capâl, from cap, → left, + -al, → -al. Capdast, with dast, → hand.
۱) لنگ؛ ۲) ساق
1) leng (#); 2) sâq (#)
Fr.: jambe
1) The part of the body from the top of the → thigh down to the → foot.
2) Anatomy: The lower limb of a human being between the → knee and the → ankle.
M.E., from O.Norse leggr; cognate with Dan. læg, Swed. läg "the calf of the leg."
Leng, related to Mid.Pers. zang "shank, ankle;" Av. zanga-, zənga- "bone of the leg; ankle bone; ankle;" Skt. jánghā- "lower leg;" maybe somehow related to E. → shank.
qânuni (#)
Fr.: légal
1) Permitted by law; lawful.
2) Of or relating to law; connected with the law or its administration (Dictionary.com).
From M.Fr. légal or directly from L. legalis "legal, pertaining to the law," from lex (genitive legis) "law."
Qânuni, of or relating to qânun, → law.
Fr.: légende
1) A non-historical or unverifiable story handed down by tradition from earlier times and popularly accepted as historical.
2) The body of stories of this kind, especially as they relate to a particular people, group, or clan (Dictionary.com).
M.E. legende "written account of a saint's life," from O.Fr. legende and directly from M.L. legenda literally, "(things) to be read," noun use of feminine of L. legendus, gerund of legere "to read"
(on certain days in church).
Cirok, from Kurd. cirok "story, fable," related to Kurd. cir-, cirin "to sing, [to recite?];" Av. kar- "to celebrate, praise;" Proto-Ir. *karH- "to praise, celebrate;" cf. Skt. kar- "to celebrate,
praise;" O.Norse herma "report;" O.Prussian kirdit "to hear;" PIE *kerH[2]- "to celebrate" (Cheung 2007).
Fr.: légendaire
Of, relating to, or of the nature of a legend.
→ legend; → -ary.
Legendre equation
هموگش ِلوژاندر
hamugeš-e Legendre
Fr.: équation de Legendre
The → differential equation of the form: d/dx(1 - x^2)dy/dx) + n(n + 1)y = 0. The general solution of the Legendre equation is given by y = c[1]P[n](x) + c[2]Q[n](x), where P[n](x) are Legendre
polynomials and Q[n](x) are called Legendre functions of the second kind.
Named after Adrien-Marie Legendre (1752-1833), a French mathematician who made important contributions to statistics, number theory, abstract algebra, and mathematical analysis; → equation.
Legendre transformation
ترادیسش ِلوژاندر
tarâdiseš-e Legendre
Fr.: transformation de Legendre
A mathematical operation that transforms one function into another. Two differentiable functions f and g are said to be Legendre transforms of each other if their first derivatives are inverse
functions of each other: df(x)/dx = (dg(x)/dx)^-1. The functions f and g are said to be related by a Legendre transformation.
→ Legendre equation; → transformation.
gânungozâri (#)
Fr.: législation
1) The act of making or enacting laws.
2) A law or a body of laws enacted (Dictionary.com).
From Fr. législation, from L.L. legislationem, from legis latio, "proposing (literally 'bearing') of a law," → legislator.
Qânungoz&acric;ri "act or process followed by the qânungoz&acric;r", → legislator.
qânungozâr (#)
Fr.: législateur
1) A person who gives or makes laws.
2) A member of a legislative body (Dictionary.com).
From L. legis lator "proposer of a law," from legis, genitive of lex, → law, + lator "proposer," agent noun of latus "borne, brought, carried."
Qânungozâr, literally "he who places the law," from qânun, → law, + gozâr, present stem and agent noun of gozâštan "to place, put; perform; allow, permit," related to gozaštan "to pass, to cross," → | {"url":"https://dictionary.obspm.fr/index.php?showAll=1&&search=L&&formSearchTextfield=&&page=7","timestamp":"2024-11-12T11:50:14Z","content_type":"text/html","content_length":"35821","record_id":"<urn:uuid:d4bf147c-9832-44ee-bee7-c5d4f08de034>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00575.warc.gz"} |
Topics in Geometry
Topics in Geometry: curves, planes, forms, exceptions and their applications.
An idea of the course:
This course is at the cross-roads of algebra, analysis, arithmetic, geometry, topology and trigonometry. It is designed not as an introduction to any particular subject, but rather as a motivation
for further studies. Most of the objects will be revisited or recycled, seen from different perspectives. No knowledge beyond the undergraduate math program is assumed, and I will be happy to review
the necessary pre-requisites and to adjust the syllabus to the needs of the audience. Similarly to Conway's book [2] we aim to help the student establish a sensual contact with quadratic (and even
cubic!) forms and geometric notions. Similarly to Clemens [1] we will try to keep every lecture coherent. Asterisk (*) means that the proofs are out of scope of this course.
List of lectures (f - only formulation):
1. [3] (f) Quaternions, modular functions (theta, eta) and their applications in arithmetics.
2. Euler's 4-square identity, Lagrange's 4-square theorem, Fermat 2-square theorem and Gauss integers. Valuations.
3. Oriented area and determinant. Area of triangle in coordinates. Sperner lemma. Monsky theorem.
4. Valuation ring, valuation ideal, units, fraction field. Valuation implies Metric implies Topology. Cauchy sequences and completion. p-adic numbers (construction and homeomorphism with a Cantor
set). Teichmuller representatives. Frobenius map.
5. [Proof of Ostrowki theorem.] Convergence of exponent and logarithm in (non-)-archimidean topologies and over p-adics. Laurent and Puiseux series.
6. Functional and differential equations for Exp, Log. Convergence. De Moivre and Euler formulae, trigonomentric formulae, trigonometric polynomials of Tchebyshev and Korkin-Zolorarev.
7. Newton polygon and its additivity. Tropicalization of a polynomial over non-archimedean field. Newton's method and Hensel lemma. Solving algebraic equations in Puiseux series.
8. Coverings, liftings. Fiber products, pullbacks and fibers. Monodromy. Critical points and values. Belyi polynomials and plane trees.
9. Riemann-Hurwitz formula. Mason-Stothers theorem. Belyi rational functions. Cyclic type of local monodromy in terms of multiplicities.
10. Applications of Mason-Stothers. A proof of genus zero case of the "difficult part" of Belyi theorem. Monodromy of Belyi polynomial in terms of the tree.
11. Belyi functions and 8-color stratifications of the sphere. Normalization. Dirichlet domains, Voronoi diagrams. Quotients and invariant functions.
12. Orbits. Invariants. Quasi-Invariants and (non-)liftings.
13. Group extensions and compositions of functions. Invariants of Kleinian groups of types An, Dn, E7. Dihedral and octahedral functions as compositions. Commutative square of cyclic, Joukovsky and
Tchebyshev. Barycentric triangulations and flags. Big sums of inverses of integers.
14. [Jensen formula, Mahler measure, Lehmer conjecture.] Gluings of polygons, Harer-Zagier formula (f). Equivalence between graphs embedded in oriented surfaces, ribbon graphs, abstract graphs
equipped with cyclic orders of semi-edges in every vertex, triples of permutations, homomorphisms from a free group F2 to permutation group.
15. Polar duality on a sphere and real projective plane. Concurence of altitudes in a spherical triangle via Jacobi identity. Area of a spherical triangle as angle defect. Regular polygons on sphere
and hyperbolic plane. Geometrization of topological tesselations. Dual regular tesselation by geodesics. Normalization and a metric proof of existence of Belyi functions with given graph.
16. [Balanced necklaces (f).] Ice melting. Pick's theorem from ice melting and sum of angles. Lattice polygons and reflexive broken lines, formulation of 12-theorem.
17. Broken watches problem. Discussion of the homework: two solutions for balanced necklace problem, a quadrangle without equiareal triangulations.
18. Arithmetic and topological ways to solve ad-bc=1. Intersection of curves on the torus. First homology of a map between tori as linearization and as the best linear approximation for universal
19. Identification of Euclidean and Minkowski 3-space with the Lie algebra of motions, after some Lie generalities.
20. Dirichlet (fundamental) domain for the modular group SL(2,Z).
21. Area of the fundamental domain. Identification of the hyperbolic plane with a quadric in 2x2-matrices. Transition functions for locally trivial fibrations and vector bundles. Line bundles form a
group with respect to tensor product. Definition of modular forms. Interpretation of some modular forms as sections of line bundles. Functions on the space of lattices and Eisenstein series.
22. PSL(2,Z) is generated by two elements, of orders 2 and 3. Moreover, it is a free product (f), only mentioned ping-pong lemma. Binary quadratic forms with fixed negative discriminant: finiteness
of isomorphism classes. Modular forms as functions on lattices homogeneous with respect to homotheties. Prove that Eisenstein series are modular and find their constant terms. Sum of valuations/
ramification of a modular form is linear with respect to its weight. Formulate construction of Hilbert field for imaginary quadratic irrationality.
23. Euler's [E41] - infinite products for (sin(s)-y), infinite sum for cotangent function, Leibniz series for pi/4, values of zeta(2), zeta(4), ... Fourier expansions of Eisenstein series. Dimension
of spaces of modular forms. Ramanujan's Delta-function as parabolic form of weight 12, j-invariant. Ring of modular forms is freely generated by E4 and E6.
24. Bernoulli numbers and first two coefficients of Eisenstein series. Transformation law for E2. Quasi-modular forms (f). Transformation law for Dedekind eta-function and Ramanujan's delta-function
(via logarithmic derivative). Asymptotics of coefficients of (parabolic) modular forms.
25. Weierstrass function - definition, Taylor series, uniformization of elliptic curve. Fourier transform of Gaussian. Poisson summation formula. Modularity of Jacobi theta-function. Numbers of
representations of an integer as a sum of 2 or 4 squares. Modular forms of higher levels with characters.
26. Classification of 2x2 matrices into hyperbolic, parabolic and elliptic types; relations between types of conjugacy classes of the group with geometry of the quotient -- elliptic elements and
ramification in finite part, parabolic elements and cusps. Reminders: definition of modular forms of higher levels. Gamma(2) is a free group and H/Gamma(2) is a Riemann sphere with 3 cusps and no
elliptic points. Modular interpretations of Belyi functions. Platonic solids as Belyi functions on modular curves H/Gamma(N) for N=2,3,4,5. Klein quartic, H/Gamma(7) and PSL(2,7). Eta-function as
generalized theta-function; Serre-Stark theorem (f). Theta-functions of lattices. Applications: signature 8 theorem and asymptotics for number of vectors of given length. Theta-functions of
Korkin-Zolotarev and Leech lattices. Milnor's pair of 16-dimensionals drums that sound the same, but differ in shape. Venkov's proof of Kneser's theorem on Niemeier lattices via modular forms
27. the proof of 12-theorem via modular forms (after Bjorn Poonen and Fernando Rodriguez-Villegas).
28. Weight 2 cusp-forms are holomorphic differentials
29. Metrics of constant curvature with conical singularities. Consequences of Riemann's uniformization theorem. Moduli spaces of lattices and moduli space of elliptic curves.
Basic bibliography:
1. C. Herbert Clemens: A Scrapbook of Complex Curve Theory.
2. John (Horton) Conway. The Sensual (Quadratic) Form.
3. John Conway, Derek Smith. On Quaternions and Octonions.
4. Don Zagier. Elliptic Modular Forms and Their Applications.
5. Sergei Lando, Alexander Zvonkin. Graphs on Surfaces and Their Applications. [also note an appendix by Zagier with crash-course on finite group representations ]
6. Jean-Pierre Serre. A Course in Arithmetic. (A chapter on modular forms is self-contained.)
Software: PARI.math.u-bordeaux.fr, SageMath.org
Some short inspiring papers: Complementary bibliography:
• [E41] Leonhard Euler. De summis serierum reciprocarum. Commentarii academiae scientiarum Petropolitanae, Volume 7, pp. 123-134.
• C.F. Gauss. Disquisitiones Arithmeticae. 1801. In particular: Section V, art. 172.
• Felix Klein. Lectures on the Icosahedron and the Solution of the Fifth Degree, 1884. [ Also see an expository article On Klein's Icosahedral Solution of the Quintic by Oliver Nash. ]
• Henri PoincarĂ©. Analysis Situs, 1895-1904
• Yuri I. Manin. "Cubic forms: algebra, geometry, arithmetics."
• H.S.M. Coxeter. Virtually any book, e.g. "Regular Polytopes" or "Projective Geometry"
• Robin Hartshorne. Projective Geometry. [ Do not confuse with "Algebraic Geometry"] | {"url":"https://users.mccme.ru/galkin/tg.html","timestamp":"2024-11-08T20:55:08Z","content_type":"application/xhtml+xml","content_length":"10997","record_id":"<urn:uuid:035322d2-49e8-4fe1-90ca-5fb565bf4b21>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00158.warc.gz"} |
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers),structure,space, and change. There is a range of views among mathematicians and
philosophers as to the exact scope and definition of mathematics.
Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good
models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation,
measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required
to solve mathematical problems can take years or even centuries of sustained inquiry.
Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic
systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics
developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that
has continued to the present day. | {"url":"http://piglix.com/piglix/Mathematics","timestamp":"2024-11-13T18:06:04Z","content_type":"text/html","content_length":"13686","record_id":"<urn:uuid:6622341b-d012-4101-a7f4-b0f7e7b596d4>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00458.warc.gz"} |
Headers for the CRC8 calculation for Maxim Integrated Monitoring devices.
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2019-02-05 (date of creation)
2022-05-30 (date of last update)
This module supports the calculation of a CRC8 based on the polynomial described in the Maxim data sheets. The polynomial is 0xA6.
Definition in file mxm_crc8.h. | {"url":"https://iisb-foxbms.iisb.fraunhofer.de/foxbms/gen2/docs/html/v1.3.0/_static/doxygen/tests/html/mxm__crc8_8h.html","timestamp":"2024-11-14T23:54:05Z","content_type":"application/xhtml+xml","content_length":"19113","record_id":"<urn:uuid:128b1091-0b35-4080-aa2e-da653087f027>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00139.warc.gz"} |
What is the APF of hcp?
HCP is one of the most common structures for metals. HCP has 6 atoms per unit cell, lattice constant a = 2r and c = (4√6r)/3 (or c/a ratio = 1.633), coordination number CN = 12, and Atomic Packing
Factor APF = 74%. HCP is a close-packed structure with AB-AB stacking.
What is APF in crystal structure?
In crystallography, atomic packing factor (APF), packing efficiency, or packing fraction is the fraction of volume in a crystal structure that is occupied by constituent particles. It is a
dimensionless quantity and always less than unity.
How do I get APF from hcp?
c a = 8 3 = 1.633 2 Page 3 2. Show that the atomic packing factor for HCP is 0.74. Now, the unit cell volume is the product of the base area times the cell height, c.
Why do fcc and hcp have the same APF?
Because FCC and HCP are both composed of close-packed layers, they have the same type and number of interstitial sites–the only difference is the location of these interstitial sites. You already
know that unit cells of metals are not fully packed (74% for FCC and HCP), which means they have some empty space.
How do you find APF?
Remember, APF is just the volume of the atoms within the unit cell, divided by the total volume of the unit cell. You use this to calculate the APF of any crystal system, even if it’s non-cubic or
has multiple kinds of atoms!
How is APF calculated?
The atomic packing factor is defined as the ratio of the volume occupied by the average number of atoms in a unit cell to the volume of the unit cell. for FCC a = 2√2 r where a is side of the cube
and r is atomic radius.
What is FCC BCC hcp?
The hexagonal closest packed (hcp) has a coordination number of 12 and contains 6 atoms per unit cell. The face-centered cubic (fcc) has a coordination number of 12 and contains 4 atoms per unit
cell. The body-centered cubic (bcc) has a coordination number of 8 and contains 2 atoms per unit cell.
Why is APF important?
APF annual recognizes the Most Significant Futures Works for the purpose of identifying and rewarding the work of professional futurists and others whose work illuminates aspects of the future.
What is APF simple cubic?
The atomic packing factor [A.P.F]: It can be defined as the ratio between the volume of the basic atoms of the unit cell (which represent the volume of all atoms in one unit cell ) to the volume of
the unit cell it self.
Is nickel a FCC?
Nickel is highly crystalline, having an FCC structure as shown below. Ni has an atomic radius of 0.1246 nm and a lattice parameter of 3.52 Angstroms. (FCC also gives a coordination number of 12 and
an atomic packing factor of 0.74) The image below shows the basic unit of the FCC.
How many atoms are in FCC?
four atoms
FCC unit cells consist of four atoms, eight eighths at the corners and six halves in the faces. | {"url":"https://corfire.com/what-is-the-apf-of-hcp/","timestamp":"2024-11-13T22:35:01Z","content_type":"text/html","content_length":"37159","record_id":"<urn:uuid:51d9311b-68c3-4583-af4b-cd2ba38d5597>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00888.warc.gz"} |
Hence, the roots are real and equal.
22. Let the transformed eq... | Filo
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Hence, the roots are real and equal. 22. Let the transformed equation of so that the term containing the cubic power of is absent be . Then,
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Question Text Hence, the roots are real and equal. 22. Let the transformed equation of so that the term containing the cubic power of is absent be . Then,
Updated On Apr 21, 2024
Topic Coordinate Geometry
Subject Mathematics
Class Class 12 | {"url":"https://askfilo.com/user-question-answers-mathematics/hence-the-roots-are-real-and-equal-22-let-the-transformed-3130303532323837","timestamp":"2024-11-13T23:11:41Z","content_type":"text/html","content_length":"239093","record_id":"<urn:uuid:0155a195-f4f5-4429-af23-ac47971d10a2>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00394.warc.gz"} |
Number 80
Interesting facts about the number 80
• (80) Sappho is asteroid number 80. It was discovered by N. R. Pogson from Madras Observatory on 5/2/1864.
Areas, mountains and surfaces
• The total area of Cyprus (main island) is 3,565 square miles (9,234 square km). Country Cyprus, Akrotiri and Dhekelia, Sovereign Base Areas of the United Kingdom, and Northern Cyprus. 80th
largest island in the world.
• There is a 80 miles (128 km) direct distance between Brooklyn (USA) and Philadelphia (USA).
• There is a 80 miles (128 km) direct distance between Caracas (Venezuela) and Valencia (Venezuela).
• There is a 50 miles (80 km) direct distance between Changzhou (China) and Suzhou (China).
• There is a 80 miles (128 km) direct distance between Sūrat (India) and Vadodara (India).
• More distances ...
• Pokémon Slowbro (Yadoran) is a Pokémon number # 080 of the National Pokédex. Slowbro is a water-type and psychic Pokémon in the first generation. Group is a Pokémon Egg Monster and Water 1.
indexes ontros of Slowbro are Johto index 081 , Kalos Center index 134 - Pokédex Coastal Zone.
History and politics
• United Nations Security Council Resolution number 80, adopted 14 March 1950. Commending India, Pakistan and Kashmir for ceasefire agreements. Resolution text.
In other fields
• M80 Radio is a radio station in Portugal (and formerly also Spain) playing music from the 1970s, 1980s, 1990s and 2000s.
• 80 is the smallest cubeful integer with a cubeful successor
• 80 is the smallest number n where n and n+1 are both products of 4 or more primes.
Motorsport and cars
• The Audi 80 is a D-segment passenger car produced by the German manufacturer Audi from 1972 to 1996.
• Mercury is the chemical element in the periodic table that has the symbol Hg and atomic number 80.
• The best athletes to wear number 80
Jerry Rice, NFL; Kellen Winslow, NFL; Steve Largent, NFL; Chris Carter, NFL
• The T-80 is a main battle tank that was designed and manufactured in the former Soviet Union and manufactured in Russia.
What is 80 in other units
The decimal (Arabic) number
converted to a
Roman number
Roman and decimal number conversions
The number 80 converted to a Mayan number is
Decimal and Mayan number conversions.
Weight conversion
80 kilograms (kg) = 176.4 pounds (lbs)
80 pounds (lbs) = 36.3 kilograms (kg)
Length conversion
80 kilometers (km) equals to 49.710 miles (mi).
80 miles (mi) equals to 128.748 kilometers (km).
80 meters (m) equals to 128.748 feet (ft).
80 feet (ft) equals 24.384 meters (m).
80 centimeters (cm) equals to 31.5 inches (in).
80 inches (in) equals to 203.2 centimeters (cm).
Temperature conversion
80° Fahrenheit (°F) equals to 26.7° Celsius (°C)
80° Celsius (°C) equals to 176° Fahrenheit (°F)
Power conversion
80 Horsepower (hp) equals to 58.83 kilowatts (kW)
80 kilowatts (kW) equals to 108.78 horsepower (hp)
Time conversion
(hours, minutes, seconds, days, weeks)
80 seconds equals to 1 minute, 20 seconds
80 minutes equals to 1 hour, 20 minutes
Number 80 morse code:
---.. -----
Sign language for number 80:
Number 80 in braille:
Year 80 AD
• The Emperor Titus inaugurates the Flavian Amphitheatre with 100 days of games.
• The original Roman Pantheon was destroyed in a fire, together with many other buildings.
• The Eifel Aqueduct was constructed.
• The aeolipile, the first steam engine, was invented by Hero of Alexandria.
• Vologases II of Parthia died.
• Julius Agricola, the Roman Governor of Britain,advanced into Scotland at the head of a Roman Army.
Gregorian, Hebrew, Islamic, Persian and Buddhist Year (Calendar)
Gregorian year 80 is Buddhist year 623.
Buddhist year 80 is Gregorian year 463 a. C.
Gregorian year 80 is Islamic year -559 or -558.
Islamic year 80 is Gregorian year 699 or 700.
Gregorian year 80 is Persian year -543 or -542.
Persian year 80 is Gregorian 701 or 702.
Gregorian year 80 is Hebrew year 3840 or 3841.
Hebrew year 80 is Gregorian year 3680 a. C.
The Buddhist calendar is used in Sri Lanka, Cambodia, Laos, Thailand, and Burma. The Persian calendar is the official calendar in Iran and Afghanistan.
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Advanced math operations
Is Prime?
The number 80 is not a
prime number
. The closest prime numbers are
The 80th prime number in order is
Factorization and factors (dividers)
The prime factors of 80 are
2 * 2 * 2 * 2 * 5 The factors of
80 are
, 80.
Total factors 10.
Sum of factors 186 (106).
Prime factor tree
The second power of 80
is 6.400.
The third power of 80
is 512.000.
The square root √
is 8,944272.
The cube root of
is 4,308869.
The natural logarithm of No. ln 80 = log
80 = 4,382027.
The logarithm to base 10 of No. log
80 = 1,90309.
The Napierian logarithm of No. log
80 = -4,382027.
Trigonometric functions
The cosine of 80 is -0,110387.
The sine of 80 is -0,993889.
The tangent of 80 is 9,003655.
Number 80 in Computer Science
Code type Code value
Port 80/TCP HTTP (HyperText Transfer Protocol) - used for transferring web pages
ASCII Html P Uppercase P P
(ISO 8859-1 Characters)
Unix time Unix time 80 is equal to Thursday Jan. 1, 1970, 12:01:20 a.m. GMT
IPv4, IPv6 Number 80 internet address in dotted format v4 0.0.0.80, v6 ::50
80 Decimal = 1010000 Binary
80 Decimal = 2222 Ternary
80 Decimal = 120 Octal
80 Decimal = 50 Hexadecimal (0x50 hex)
80 BASE64 ODA=
80 MD5 f033ab37c30201f73f142449d037028d
80 SHA1 b888b29826bb53dc531437e723738383d8339b56
80 SHA224 0717d705040fdb9b4da516f41c800742b65ce06ad584154c9a6599d0
80 SHA256 48449a14a4ff7d79bb7a1b6f3d488eba397c36ef25634c111b49baf362511afc
80 SHA384 8e3f10fab019b9abbc9a5bc88c789d4715511f4cc93de2bb09c3cea3cfb3858d79e1f4bda5ba8342ea4c8f5607463606
More SHA codes related to the number 80 ...
If you know something interesting about the 80 number that you did not find on this page, do not hesitate to write us here.
Numerology 80
The meaning of the number 8 (eight), numerology 8
Character frequency 8: 1
The number eight (8) is the sign of organization, perseverance and control of energy to produce material and spiritual achievements. It represents the power of realization, abundance in the spiritual
and material world. Sometimes it denotes a tendency to sacrifice but also to be unscrupulous.
More about the the number 8 (eight), numerology 8 ...
The meaning of the number 0 (zero), numerology 0
Character frequency 0: 1
Everything begins at the zero point and at the zero point everything ends. Many times we do not know the end, but we know the beginning, it is at the zero point.
More about the the number 0 (zero), numerology 0 ...
№ 80 in other languages
How to say or write the number eighty in Spanish, German, French and other languages.
Spanish: 🔊 (número 80) ochenta
German: 🔊 (Nummer 80) achtzig
French: 🔊 (nombre 80) quatre-vingts
Portuguese: 🔊 (número 80) oitenta
Hindi: 🔊 (संख्या 80) अस्सी
Chinese: 🔊 (数 80) 八十
Arabian: 🔊 (عدد 80) ثمانون
Czech: 🔊 (číslo 80) osmdesát
Korean: 🔊 (번호 80) 팔십
Danish: 🔊 (nummer 80) firs
Hebrew: (מספר 80) שמונים
Dutch: 🔊 (nummer 80) tachtig
Japanese: 🔊 (数 80) 八十
Indonesian: 🔊 (jumlah 80) delapan puluh
Italian: 🔊 (numero 80) ottanta
Norwegian: 🔊 (nummer 80) åtti
Polish: 🔊 (liczba 80) osiemdziesiąt
Russian: 🔊 (номер 80) восемьдесят
Turkish: 🔊 (numara 80) seksen
Thai: 🔊 (จำนวน 80) แปดสิบ
Ukrainian: 🔊 (номер 80) вісімдесят
Vietnamese: 🔊 (con số 80) tám mươi
Other languages ...
News to email
If you know something interesting about the number 80 or any other natural number (positive integer), please write to us here or on Facebook.
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Frequently asked questions about the number 80
• How do you write the number 80 in words?
80 can be written as "eighty".
Number 80 in News
NyTimes: More Than 80 Nations Back Talks to Ease Path to Peace in Ukraine
Meeting in Switzerland, world leaders backed a joint statement urging more diplomacy, but were divided on how to engage Russia.
June 16, 2024
NyTimes: Rebuilding All Destroyed Gaza Homes Could Take 80 Years, U.N. Report Says
The projection didn’t take into account the time it would take to repair the homes that were damaged but not destroyed.
May 3, 2024
France24: Almost 80 farmers arrested as protests blockade key food market, close in on Paris
French police on Wednesday detained scores of people who broke into the Paris region's main wholesale market complex during nationwide protests by farmers, the Créteil public prosecutor's office told
AFP. The arrests came as convoys of tractors edged closer to Paris, Lyon and other key locations, with many ignoring police …
Jan. 31, 2024
BBC: 'A story of absolute heroism': 80 years on from WW2 sabotage mission
In 1943, Operation Jaywick sent 14 men from Australia on a mission to sink Japanese ships in Singapore.
Sept. 26, 2023
BBC: The leafy street where 80 fake firms have sham addresses
Financial crime experts say "burner companies" are "most likely part of a criminal network".
Sept. 14, 2023
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Depreciation Calculator
Calculate the depreciation of an asset using the straight line, declining balance, double declining balance, or sum-of-the-years’ digits method.
Depreciation Schedule:
Year Depreciation Amount
2024 $19,000
2025 $19,000
2026 $19,000
2027 $19,000
2028 $19,000
This calculation is based on widely-accepted formulas for educational purposes only, and it is not a recommendation for how to handle your finances. Consult with a financial professional before
making financial decisions.
How to Calculate Depreciation
Most physical goods—machinery, rental properties, equipment, and even office supplies—have a limited useful life. And if these goods and materials are essential for conducting business, they will
eventually need to be replaced.
Suppose there is a piece of equipment that costs $100 and the useful estimated life of this piece of equipment is 10 years.
While it is quite possible that the business paid for this piece of equipment all at once (i.e., they did not need to finance it), that business might still consider spreading these costs over time.
Typically, items under $2,500 are not depreciated and are instead taken as an expense all at once.
Depreciation is a broad finance and accounting term used to describe various ways of distributing these costs. As we will further illustrate throughout this article, there are several different
depreciation strategies that can make sense in certain situations.
What is Depreciation?
It is important to remember that depreciation is a cost allocation strategy, and not what a business is paying for an asset each month. Depreciation is an accounting strategy that helps spread the
total cost of acquiring an asset over that asset’s useful lifetime.
If an asset’s book value is 90% of its original value, that doesn’t necessarily mean you could sell the product for that price on the open market. Instead, this simply means that 10% of that asset’s
value has been accounted for.
Depreciation helps make it easier for businesses and other organizations to control their expenses. This can be useful in a variety of situations, such as determining the entire value of a company or
minimizing tax obligations.
How to Calculate Straight-Line Depreciation
One of the most common depreciation strategies is straight-line depreciation. This strategy will evenly spread an asset’s cost over its entire life. In other words, depreciation will be linear.
To calculate straight-line depreciation, you need to know the cost of the asset, its estimated service life, and its salvage value. The salvage value is the amount you could sell the asset or its
parts for once its useful life has been exhausted.
Straight-Line Depreciation Formula
The formula for straight-line depreciation is:^[1]
depreciation = asset cost – salvage value / service life
Using the straight-line method, depreciation is equal to the asset cost minus the salvage value, divided by the service life.
Suppose an asset costs $100, its estimated salvage value is $10, and its estimated useful life is 10 years:
depreciation = $100 – $10 / 10
depreciation = $9
In this case, the business can account for a $9 depreciation expense annually for the asset. Of course, if the asset is still useful after ten years, the business can continue using it, but they
cannot continue accounting for its depreciation.
How to Calculate Declining Balance
The declining balance depreciation method assumes that the asset will lose value at a faster rate towards the beginning of its useful life than it will at the end of its useful life (think about how
a new car loses much of its value as soon as it drives off the lot).
In order to calculate the declining balance formula, you will need to incorporate a corresponding depreciation factor.
Declining Balance Formula
The formula for declining balance depreciation is:^[2]
depreciation = value × factor × 1 / service life
value = depreciable value at the beginning of the year
service life = years of useful life
factor = depreciation factor (see below for tips to find this)
Using the declining balance method, depreciation is equal to the value times the depreciation factor times 1 divided by the service life.
In the formula above, the depreciation factor you choose will depend on how quickly you believe the asset will lose its value. The larger the factor, the more quickly the asset’s value will be
depreciated in the beginning of its life.
This strategy is useful for assets that decline in value quickly, such as computers or other technology that rapidly becomes obsolete. In some cases, there might be an industry standard available
that you can use as a baseline.
In other cases, you might simply want to depreciate at double the linear rate using the double-declining balance formula below.
Double-Declining Balance Formula
The double-declining balance formula is shown below:
depreciation = value × 2 × 1 / service life
value = depreciable value at the beginning of the year
service life = years of useful life
Using the double-declining balance method, depreciation is equal to the value times 2 times 1 divided by the service life.
Suppose that the value of the product at the beginning of the year is $50 and you believe the product has 4 more useful years remaining. In this case, all you’d need to do is:
Depreciation = $50 x 2 x (1/4)
Depreciation = $25
The double-declining balance formula calculates that the depreciation entry for that year would be $25.
How to Calculate Sum-of-the-Years’ Digits
The sum-of-the-years’ digits depreciation method is another common depreciation strategy that places a higher emphasis on the early years and less emphasis on the later years. As the years go on, the
depreciation percentage will change accordingly.
Sum-of-the-Years’ Digits Formula
The sum-of-the-years’ digits formula is:
depreciation = (asset cost – salvage value) × factor
n = service life in years
factor = years of service life remaining divided by the sum of each year’s digits (see example below)
year 1 factor = n / 1 + 2 + … + n
year 2 factor = n – 1 / 1 + 2 + … + n
final year factor = 1 / 1 + 2 + … + n
Which Depreciation Method Should I Use?
The depreciation method that makes the most sense for you will depend on several different factors. To start with, you might want to consider the value of a product that may deteriorate over time.
For example, the drop in value between a new product and a product that is one year old will almost always be greater than, say, the drop in value between years 2 and 3.
You might also want to think about the likelihood that you will sell the product before its useful life has been exhausted.
In most cases, the straight-line method will be easiest to apply and will make the most sense. Regardless of which method you use, be sure to use depreciation in order to distribute the cost of
assets over time.
For example, let’s compare these depreciation methods using the following parameters:
• Asset Cost = $100,000
• Salvage Value = $5,000
• Service Life = 5 years
• Depreciation Factor (DB method only) = 2.06
• Convention = Full-Month
• Placed in Service = January 2022
The resulting depreciation schedules look like this:
Table showing an example depreciation schedule for a $100,000 asset with a $5,000 salvage value and a
5-year service life.
Year Depreciation Amount for Method
Straight-Line Declining Balance (factor = 2.06) Double-Declining Balance Sum-of-the-Years’ Digits
2022 $19,000 $41,200 $40,000 $31,667
2023 $19,000 $24,226 $24,000 $25,333
2024 $19,000 $14,244 $14,400 $19,000
2025 $19,000 $8,376 $8,640 $12,667
2026 $19,000 $4,925 $5,184 $6,333
Total $95,000 $92,971 $92,224 $95,000
So, two methods attain the fully depreciated status:
$100,000 – $95,000 = $5,000 (salvage value)
But what about the other two methods, which haven’t depreciated enough to attain this status?
In cases where the declining balance method will not fully depreciate an asset over its service life there are a few options.
The most commonly adopted option is to switch the depreciation method to use the straight-line method later in the asset’s life.
Another option is to use a higher depreciation factor to further accelerate the depreciation ensuring the asset is fully depreciated.
If the asset is not fully depreciated then it will have a higher book value than its salvage value, ultimately resulting in a lower tax benefit.
You might also be interested in our appreciation calculator to figure out the increase in value of an asset over time.
Frequently Asked Questions
What does depreciation do?
Depreciation helps spread the total cost of acquiring an asset over that asset’s useful lifetime, thus allowing companies to spread out their costs instead of recognizing them all at once, which
could skew their true profitability.
Why is depreciation important?
Depreciation is important because it allows companies to spread the cost of an asset over that asset’s useful life. This allows companies to reduce their tax bill by recognizing the lost value of an
asset over time, as well as show investors and stakeholders an accurate depiction of company performance.
Without depreciation, company performance may be skewed with an asset’s full expense being recognized in one period, instead of the over the time period it’s actually being used.
Why is depreciation an expense?
Depreciation is an expense because it is a normal business operating expense: as an asset is used over time and has more wear and tear, the value of that asset drops. Depreciation is recognized on
the income statement, which is where normal business operating expenses are shown.
1. Chamber of Commerce, The Small Business's Guide to Straight Line Depreciation, https://www.chamberofcommerce.org/small-businesss-guide-to-straight-line-depreciation
2. Internal Revenue Service, Publication 946, How To Depreciate Property, https://www.irs.gov/publications/p946 | {"url":"https://www.inchcalculator.com/depreciation-calculator/","timestamp":"2024-11-07T12:18:46Z","content_type":"text/html","content_length":"107689","record_id":"<urn:uuid:514abe2c-cee5-4822-9871-aca9e7beb1df>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00598.warc.gz"} |
Napier's Bones Presentation Set
Napier's Bones
John Napier (1550-1617)
John Napier is perhaps best remembered for his contributions to mathematics: natural logarithms and numbered rods for multiplication, later known as "Napier's Bones".
Napier's Bones
Napier's Bones are an arrangement of multiplication tables written on rectangular rods to facilitate multiplication, division, and the extraction of square and cube roots. Ten rods are divided into
nine rows of digits corresponding to multiples of the digit at the top. An additional rod has columns of square and cube values to help calculate square roots and cube roots.
In his book Rabdologia John Napier began his description of the construction of the rods as follows:
“Construct from silver, ivory, boxwood, or other similar solid material a number of square rods. Make 10 rods for numbers less than 11,111; 20 rods for numbers lessthan 111,111,111; 30 rods for
numbers less than 1,111,111,111,111; etc.”
While sets of silver might be destined for nobility, ivory or boxwood were more common choices. This set has been created by rotary engraving numbers into aircraft grade aluminum in recognition of
the magnificence of a set made from silver. Each rod is carefully hand finished to give a special feel of history. The three dimensional engraving of the numbers and careful finish guarantee each set
will endure for centuries.
Napier specified a specific layout for the number columns:
"... the third face of each rod is always opposite to the first and the fourth to the second, and that the simples on each face are opposed not merely in the sense that one is on the upper and the
other on the lower face or that one is on the right-hand and the other on the left-hand face but also in that they are at opposite ends, one at the top and the other at the bottom of the rod, and
that these two opposed simples always add up to nine."
In contemporary times, Napier’s original layout is frequently modified in favor a larger single digit at the top of the rod. There’s no arguing that it’s easier to read, however we chose to make this
set according to Napier’s exact instructions. Napier notes that a multiplication may be validated by turning the rods over and manipulating the complement digits of the result. It’s interesting to
realize that Napier’s layout using the arrangement of opposing numbers represents one of the first user interface optimizations for automated calculation in recorded history.
Click a picture to see a larger size.
This set is of the highest presentation quality - supplied with an exquisitely finished walnut box with a clear top to showcase the rods, a tablet to hold the bones during calculations, and a booklet
that provides an introduction to John Napier, the rods, and instructions for multiplication, division, square roots, and cube roots.
It's a great gift to recognize a colleague, an heirloom present for a young person, and a compelling conversation piece for collectors of all ages.
Using Napier's Bones
The booklet supplied with Napier's Bones describes multiplication, division, square roots, and cube roots. In this example we'll illustrate the multiplication of 2489 by 7. Arrange the rods to show
the columns for 2, 4, 8, and 9 on the tablet, then look down to the 7th row.
Starting from the right and proceeding left, add up the numbers in the diagonals. When the sum exceeds 9, write the least significant digit as the answer for that diagonal and add a carry digit to
the left adjacent diagonal. In this illustration this happens twice. If the multiplier were to be 70, the procedure is the same, but with the extra step of adding a 0 to the end (think of this as
shifting the decimal point).
John Napier (1550-1617) was born at Merchiston Castle near Edinburgh, Scotland into a wealthy family. He studied briefly at St. Andrews, travelled on the Continent, and returned to Scotland. In 1572
he married Elizabeth Stirling, and they had two children. Elizabeth died in 1579, and John later married Agnes Chiholm, with whom he had 10 children.
Napier had diverse interests in mathematics, theology, and agriculture. Like his father, who had been one of the earliest to promote the Protestant cause in Scotland, he was also a fervent Protestant
and was appointed a commissioner to the Gneral Assembly on behalf of the Presbytery of Edinburgh in 1588. He later published the Plaine Discovery of the Whole Revelation of St. John, which he
considered to be his most important work.
In 1614 Naper published his famous Mirifici Logarithmorum Canonis Descriptio, describing his work on logarithms. The book contained 90 pages of tables and descriptions. The invention of logarithms
was hugely significant in that it made multiplication, division, and root extraction far less time consuming, particularly for very large numbers. His collaboration with Henry Briggs was crucial to
Brigg's construction of log base 10 tables and their widespread adoption.
In 1617, just before he died, Napier published Rabdologia, seu Numerationis per Virgulas Libri Duo: Cum Appendice de expecitissimo Multiplicationis Promptuario. Quibus accessit et Arithmeticae
Loaclis Liber Unus, commonly referred to as "The Rabdologia", or Rabdology.
This book describes three devices to facilitate calculations:
Numbered rods The technique of making and using numbered rods to facilitate multiplication.
Promptuary A large set of numbered paper strips for multiplication and division.
Location arithmetic A technique using a grid and counters to perform binary arithmetic.
Only the first was widely adopted. Rabdology was translated into several languages and variations of the numbered rods were used for several centuries.
Rabdology is credited as containing one of the earliest references to the use of a decimal point, in this case a improvement of Simon Stevin's decimal notation.
Rabdology has been translated by Wiliam Frank Richardson as volume 15 of the Charles Babbage Institute reprint series for the history of computing. This volume is difficult to find, but well worth
the effort.
For Sale
Armstrong Metalcrafts is building a limited number of these for sale. The retail price is $295. To inquire about a purchase, please use our contact form or send an email to "sales" at | {"url":"https://www.armstrongmetalcrafts.com/Products/NapiersBones.aspx","timestamp":"2024-11-02T07:44:22Z","content_type":"application/xhtml+xml","content_length":"33286","record_id":"<urn:uuid:49f75011-02c7-4566-a6cd-ba6842a80d33>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00158.warc.gz"} |
A Beginner's Guide to t-tests: Real-life Applications of t-test: One-Sample, Two Sample and Paired Sample t-test
The t-test was developed by William Sealy Gosset, an English statistician and a beer brewer.
William Sealy Gosset used this test to produce consistent and also high-quality beer. He published this test in a paper under the pen-name "
". Hence, this test is also called the
Student's t-test
Three types of t-test:
A) One-sample t-test
Suppose you represent a testing agency. The government wants to know whether the average weight of a certain species of animal is 10 kg. To test this, you take a random sample (of let us say 10
animals) and use the one-sample t-test to check whether the sample mean is 10 kg or is it statistically different from 10 kg.
B) Two-sample t-test
You own a shop. There are two major consumer groups: i) those who visit your shop by car and ii) those on a motorbike. You want to test - is there any difference in spending between these two
consumer groups? For this purpose, you use a two-sample t-test.
The two-sample t-test is also used to analyse the results from A/B testing.
What if there are more than two groups?
Then, we have to perform pair-wise two-sample t-tests, which is cumbersome.
A better approach is to perform ANOVA (Analysis of Variance).
What if the variances of the two samples are different?
In that case, we have to use
Welch's t-test, also called the unequal variances t-test.
C) Paired samples t-test
You are a trainer. You trained 10 students. You want to know if there was any improvement in students' knowledge level due to training. To compare, pre-training scores with post-training scores, you
use paired samples t-test.
This test may appear similar to a two-sample t-test but there is a vital difference. In paired samples t-test, subjects (in this example, students) are the same. In paired samples t-test, we compare
the same subject (students in this example) using some measures (e.g. scores in exams) before and after the interventions (i.e. training in this example).
Assumptions of t-test
• Normally distributed data
□ If this assumption is not valid, then we have to use non-parametric tests
• Samples are drawn randomly from the population
• Homogeneity of variance (variability of data in each group is similar)
□ If this assumption is not fulfilled, then we can use Welch's t-test, also called the unequal variances t-test. | {"url":"https://www.datasciencesmachinelearning.com/2023/03/the-t-test.html","timestamp":"2024-11-10T15:49:03Z","content_type":"application/xhtml+xml","content_length":"63938","record_id":"<urn:uuid:966908c3-35c5-4673-b675-337336f05030>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00147.warc.gz"} |
New Meta-Analysis of GWAS Highlights Missing Heritability Problem
Also debunks most of Sasha Gusev's claims about IQ not being like height
A new meta-analysis of GWAS and family-GWAS (FGWAS) just dropped yesterday.
Here’s the table with the main results:
Sasha Gusev was wrong
Recall that Sasha Gusev claimed 5 things: (read my response here)
1. IQ is much less heritable and more confounded than height
2. Genetic effects on IQ differ within families much more than for height
3. IQ estimates are much more biased by participation than height
4. The genetics of IQ is much more environmentally sensitive than height
5. Unlike height, no one knows what IQ is actually measuring
This meta-analysis provides data about 1 and 2. Sasha gave the following chart, claiming the within-family effects of IQ were much lower than the population effects:
And he also claimed that the height effects don’t differ. I pointed out his p-values were shit, and it was likely that the IQ effects did not differ.
Guess what didn’t replicate in the meta-analysis? The IQ effects being different:
Is IQ less heritable than height?
But you see above that the heritabilities for height and IQ are different. Does this mean that IQ is “more cultural”? No. The difference between GWAS heritability estimates for height and IQ can be
explained by the purely random measurement error inherent in IQ tests.
In classic psychological test theory, IQ is modeled as follows:
IQ is the sum of underlying general intelligence and an error component. The error component is totally random and doesn’t correlate with anything, including SNPs.
So, given this model holds, if you know the correlation between SNPs and IQ, you can figure out the correlation between SNPs and g, if you know the magnitude of the error component relative to g.
This magnitude is captured by test reliability. We assume each person’s g doesn’t change. This means any difference in test scores over time is due to having different error components. This lets us
estimate the variance of the error component.
\(r = \frac{Cov(IQ_{t}, IQ_{t+1})}{Var(IQ)} = \frac{Cov(g + \epsilon_{t}, g + \epsilon_{t+1})}{Var(IQ)} = \frac{Cov(g, g)}{Var(IQ)} = \frac{Var(g)}{Var(IQ)}\)
The reliability r is the how much variance g explains of IQ, similar to how heritability is how much variance genetics explains of phenotype.
So now say we want the correlation of SNPs with g instead of IQ:
\(Corr(g, SNPs) = \frac{Cov(g, SNPs)}{\sigma_g \sigma_{SNP}} = \frac{Cov(IQ, SNPs)}{\sqrt{r}\sigma_{IQ}\sigma_{SNP}} = \frac{Corr(IQ,SNP)}{\sqrt{r}}\)
\(h^{2}(g)_{SNP} = \frac{h^{2}(IQ)_{SNP}}{r}\)
Because the heritabilities are just the squared correlations.
This is known as correcting for attenuation.
It’s unclear what the average r was for the cognitive tests in this meta-analysis. But for the UK Biobank, it was a mere r=.55.
So I’ll give a number of possible r values. As you can see, the 95% CIs overlap with the height h^2. Consequently, there’s no evidence that general intelligence is less heritable than height in
Where did all the heritability go?
Sasha Gusev et al. like to claim that the heritability vanished in GWAS because of “cultural” factors, like the Equal Environments Assumption (EEA) being violated.
They take the low GWAS heritabilities of IQ as evidence that twin designs were flawed and IQ is “just a test” and is “determined by culture.”
But by that logic, height is just a mostly random measurement and is mostly determined by culture. Perhaps the EEA is violated and people bigotedly feed identical twins more similar diets than
fraternal twins. Maybe height is determined by stereotype threat and the kinds of books your parents read to you as a child.
Because they’re missing 40+% of heritable variance in height GWAS. Where did the variance go?
By Occam’s razor, the IQ and height variance are missing in the same place. And because height is missing along with IQ, it’s probably not missing because of “culture.”
Where did the variance go? As far as I know, nobody knows for sure. However, they’re missing about half of the variance. This reminds me of another molecular study that was off from quantitative
methods by a factor of 2. I hypothesized that the rest of the variance was hiding in the noncoding region of the DNA.
The coding region of the DNA is highly variable and small, about 2% of the genome. Most GWAS SNPs at this point are in this region. The non-coding region is 98% of DNA and varies far less. Most
extremely rare variants will therefore likely be in this region. Just based on the size of the two regions, if each region has half of the variance, it will take 50^21 times the power to detect the
average SNP in the non-coding region as the coding region.
So we will need sample sizes in the tens of millions, potentially the hundreds of millions.
Another further out possibility is extranuclear genetics, including the possibility of non-DNA or RNA inheritance. Did you know your whole nuclear genome is just 800 megabytes in size? Do you feel
like 800 megabytes? Maybe this is a meaningless question, since you’re not likely to be more than 1.6 GB in size informationally, given he have half the expected heritability.
Given all the height heritability that’s missing, it’s hard to think of another explanation beyond them just missing alleles. Can anyone knowledgeable think of something else?
Joseph Bronski is a reader-supported publication. To receive new posts and support my work, consider becoming a free or paid subscriber.
SE =~ 1/sqrt(n). To detect an effect t, 1.96SE < t. So sqrt(n) < 1.96/t → sqrt(min_n) = 1.96*t^-1. If t is multiplied by 1/50 then sqrt(min_n) is multiplied by 50, so min_n is multiplied by 50^2.
Truly and I wholeheartedly mean it when I say bravo to you for taking upon the most honorable duty of taking a fat dump on Sasha Gusev and his mystical pseudo-analytic voodoo bogus, only thing that
could make it better is if you were able to literally shit on his face
Expand full comment
IQ variation must be mainly caused by factors influencing gene expression.
Expand full comment
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Print for a date range | Microsoft Community Hub
Forum Discussion
I would like a feature that I can use ad hoc that will let me do a standard print of my worksheet table for a date range that I specify for each print request. Is that doable?
If you've gotten rid of that top row that had only a few formulas in it, as well as both rows at the bottom that were not true data rows, then the formula below should work, and keep up with new
rows at the top. However, I think you'll find that it works best to insert the new row just above row 3, not at the very top. At the very top, it will format itself based on the header row above,
which you don't want to have happen.
This formula goes in cell A5 of the "Prt Date Rng" tab
=FILTER(tblQ1,(tblQ1[Date]>=Start)*(tblQ1[Date]<=Finish),"No Rows")
What you'll see there is that it no longer refers to the cell addresses, but rather to the table by name and the Date column by name.
Note: I've attached a slightly updated version of the older workbook, with the table modified (and an empty row waiting for you) and the new formula.
Regarding the $ symbol in a cell reference ( $A$3, $A3, A$3 😞 What these do, and it's significant, is to render the letter or the number (the column or the row reference, respectively)
absolute rather than relative. Which means, as you copy a formula elsewhere, the parts of a cell reference that are absolute don't change, where the relative ones do, relative to the new cell.
Here's a more complete description of how that works.
I am trying to understand the formula. It starts at A24 and ends at T2401. It appears that the beginning parameter moves down as new rows are inserted and the ending parameter remains the same
(causing it to effectively move up)
The copy I have has different start and end rows, since you've been adding new rows in yours. Nevertheless, I think that I understand what's happening, and, yes, it is caused by your entering new
rows at the top. It's also complicated, I think by the fact (at least on the copy I have) you have extraneous rows at the top AND a row at the bottom, rows that contain formulas, but not the data to
which those formulas refer. You also have an extra cell at the very bottom that does an average of the Glucose column.
From a sheer data table point of view, my sense is that if you made this a cleaner table--i.e., eliminated those two incomplete rows and the Glucose average calc at the bottom--and started making all
entries at the bottom, then the formula, once adjusted to the top and bottom, would automatically track with each new entry, each new row. And I believe the formulas would automatically cascade into
the new rows as the raw data to which they apply is added.
So my recommendation would be:
• first, eliminate those "incomplete" rows at the top and bottom, eliminate also the single cell that computes the average Glucose. [That latter can be done anywhere in the workbook; it does NOT
need to be at the bottom of that column.]
• second, adjust the formula in the Print Range sheet to the new top and bottom rows, whatever they are
• third, try inserting and entering your new data at the top. IF that works, if the formula accepts that, then you're home free. If it doesn't, see if you can adapt your practices to entering new
rows at the bottom of the table.
Let me know if those suggestions make sense, and the results.
HI. I deleted the extra rows at the bottom and reset the top and bottom rows. Then I inserted a new row3 at the top and went back to the report sheet. The top row incremented to 4, but the bottom row
incremented by one staying inline with the new bottom row. I see no practical way of entering at the bottom. It is just too awkward to have to navigate there for each row I add. Then I would also
need to have the time column part of the sort, which it is not at this time. Doesn't the use of the "$" help with keeping the formula inline with the growth of the rows? I am not very familiar with
the meaning of the $ in formulas.
If you've gotten rid of that top row that had only a few formulas in it, as well as both rows at the bottom that were not true data rows, then the formula below should work, and keep up with new
rows at the top. However, I think you'll find that it works best to insert the new row just above row 3, not at the very top. At the very top, it will format itself based on the header row above,
which you don't want to have happen.
This formula goes in cell A5 of the "Prt Date Rng" tab
=FILTER(tblQ1,(tblQ1[Date]>=Start)*(tblQ1[Date]<=Finish),"No Rows")
What you'll see there is that it no longer refers to the cell addresses, but rather to the table by name and the Date column by name.
Note: I've attached a slightly updated version of the older workbook, with the table modified (and an empty row waiting for you) and the new formula.
Regarding the $ symbol in a cell reference ( $A$3, $A3, A$3 😞 What these do, and it's significant, is to render the letter or the number (the column or the row reference, respectively)
absolute rather than relative. Which means, as you copy a formula elsewhere, the parts of a cell reference that are absolute don't change, where the relative ones do, relative to the new cell.
Here's a more complete description of how that works.
• You may recall that I needed row 2 hidden to preserve formatting when I add next to the header. I copied your new formula into my current version and tried it with row 2 hidden with no data and
the report sheet seems to work OK anyway. Thank you. I have attached the current version to reference.
• You may recall that I needed row 2 hidden to preserve formatting when I add next to the header.
I did not recall that, but it made sense as soon as I read it. It was the same reason I suggested using a row lower. And I'm glad that the formula works now and adjusts as new rows are added.
Nevertheless, on the attached I'm making one more effort to convince you to use the bottom row for new entries. You may not have realized you can "Freeze" the top of a workbook or worksheet, so
that the column headings remain visible no matter how far down you are on the table. Here I've re-sorted the data so that the latest are at the bottom, so all you need to do is add a row. No
"insert" needed.
I also took a stab at revising the formulas that, if I'm reading them correctly, give the averages for the last 7 or 90 days.....
You are, of course, totally free to keep going as you have, to ignore my OCD-ness on the topic. (smiley face)
• I tried viewing and adding new rows at the bottom. I find it too awkward. When I open the file, I really need to see the most current at the top. I work with this several times a day, as you can
see from the number of entries per day. I also often view it between additions. So, I really need to use the sheet in this sort order. However, I really appreciate the new "report" sheet. It will
be very useful. Thank you.
• You are most welcome. It's been a privilege to work with you on it. Appreciate your patience as well. | {"url":"https://techcommunity.microsoft.com/discussions/excelgeneral/print-for-a-date-range/3739547/replies/3754318","timestamp":"2024-11-07T09:27:41Z","content_type":"text/html","content_length":"354439","record_id":"<urn:uuid:d0f82cdb-434c-4941-a32b-6e15ab071eda>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00727.warc.gz"} |
FA Chapter 18 Questions Mark-up and Margins
Reader Interactions
1. Question 5
Good day sir, can you please help to explain why we need to subtract the Purchases 8600. I thought purchase add to inventory, while sales reduce inventory but on your calculations is vise versa.
Thank you sir
□ It is because we are having to work backwards. We are given the inventory on 4 June but we need to calculate what the inventory would have been on 31 May.
2. can you please explain how the answer is 123000 questiion 2
□ Given that the mark up is 42% (of cost), then the cost of sales must be 100/142 x 193,200 = 136,056.
The inventory fell by 13,200 over the year, and so only the remainder of the cost of sales needed to be purchased, and 136,056 – 13,200 = 122,856 (which to the nearest 1,000 is 123,000).
(If you click on ‘review quiz’ after submitting your answers then you will see the explanations for the correct answers)
☆ i think it should be cogs + closing inventory to get purchase rather than subtract
☆ The question does not give us the closing inventory, only the decrease over the year. My previous answer is correct.
3. I don’t understand why at the question number 5, the goods returned need to be added, as I think on 31 May we did sell that 700 already, and in next days we receive back 700 so we have to minus
700 on 31 May? Thank you Sir
□ We returned the goods to the supplier. So on 31 May we had more than on 4 June because on 31 May we had not sent them back to the supplier.
4. Can you kindly break down question 4, how do you arrive at the the 27.8 % answer. its quite unclear.
□ The revenue is understated and so the correct revenue is 10,000 higher. Higher revenue also means that the profit will be 10,000 higher.
The closing inventory was overstated and lower closing inventory means that the profit will be lower by 5,000 (but this does not affect the revenue).
Therefore the correct revenue is 90,000 and the correct profit is 25,000.
5. sir, what is the meaning of inventory decreased?
□ It means that the closing inventory is lower than the opening inventory.
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Work Done in Compressing a Gas
Work done by a gas
Gases can work through enlargement or compression against relentless external pressure. Compressing of a gas is also known is the isothermal process. Work done by gases is additionally generally
known as pressure-volume or PV.
Let’s consider gas is contained in a piston. Two tubes leading to the piston can let gas in and out of the piston to control the movement of the piston. Energy is added to the gas molecules in case
the gas becomes hested. We can observe the rise in average K.E. of the molecules by measurement however the temperature of the gas will increase. As the gas molecules move quicker, they also collide
with the piston more often. These progressively more frequent collisions transfer energy to the piston and move it against external pressure, increasing the overall volume of the gas. In this
example, the gas worked on the surroundings, which includes the piston and also the rest of the universe.
To calculate how much work a gas has done or it’s been done to it against a continuous external pressure, we use a variation on the previous equation: where P external is that the external pressure
(as opposed to the pressure of the gas within the system) and ΔV is that the modification within the volume of the gas, that can be calculated from the very first and final volume of the gas:
ΔV = V final − V initial
Since work is energy, it has units of Joules (where 1J = 1 kilogram. m^2 \ s^2) you may also see other units used, such as atmospheres for pressure and liters for volume, resulting in L. atm as the
unit for work. We can convert L ⋅ atm to convert to Joules using the conversion factor of 101.325 J / 1L⋅atm.
As a matter of convention, negative work happens whenever a system does work on their surroundings.When the gas does work the volume of a gas increases and the work done is negative.When work is done
on the gas, the volume of the gas decreases and work is positive.When the gas is compressed, energy is transferred to the gas so the energy of the gas increases due to positive work.
Calculating Work Done on a Gas
To illustrate a way to use the equation for PV work, we can imagine a bicycle pump. We will assume that the air within the pump can be approximated as a perfect gas in a piston. We can work on the
air in the pump by compressing it. Initially, the gas contains a volume of 3.00 L. We apply a constant external pressure of 1.10 atm, to push down the handle of the bike pump until the gas is
compressed to a volume of 2.50 L. How much work was done on the gas?
We can use the equation from the last section to calculate that how much work has been done to compress the gas:
w = −Pexternal × ΔV
= −Pexternal × (Vfinal −Vinitial)
If we plug in the values for P external, V final and V initial for our example, we get:
w = −1.10atm × (2.50L− 3.00L)
= −1.10atm × −0.50L
= 0.55L ⋅ atm
We know that work was done on the gas since the level of the gas decreased. That means the worth of work we have calculated ought to be positive, which matches our result. We can additionally convert
our calculated work to Joules using the conversion factor. Thus, we did 56 J of work to compress the gas in the bicycle pump from 3.00 L to 2.50 L. | {"url":"https://www.w3schools.blog/work-done-in-compressing-a-gas","timestamp":"2024-11-07T06:59:34Z","content_type":"text/html","content_length":"141287","record_id":"<urn:uuid:4abe34a1-16ae-49c0-a2b9-d0c9bb6a3feb>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00846.warc.gz"} |