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Qubits Poised To Reveal Our Secrets - Huliq
It might seem like an esoteric achievement of interest to only a handful of computer scientists, but the advent of quantum computers that can run a routine called Shor’s algorithm could have profound
It means the most dangerous threat posed by quantum computing – the ability to break the codes that protect our banking, business and e-commerce data – is now a step nearer reality.
Adding to the worry is the fact that this feat has been performed by not one but two research groups, independently of each other. One team is led by Andrew White at the University of Queensland in
Brisbane, Australia, and the other by Chao-Yang Lu of the University of Science and Technology of China, in Hefei.
Both groups have built rudimentary laser-based quantum computers that can implement Shor’s algorithm – a mathematical routine capable of defeating today’s most common encryption systems, such as RSA.
RSA is an example of public key cryptography, in which a user holds a pair of mathematically related strings of data, known as a public key and a private key. The public key is widely distributed and
used to encrypt messages, while the private key is kept secret and used to decrypt them.
An attacker who does not have the private key needs to work out the two very large prime numbers which, multiplied together, make up the public key. Find those factors and you can work out the
private key.
RSA’s security rests on the extreme difficulty of doing this: today’s digital computers are just not powerful enough to find the factors of a large key in any practical length of time.
For instance, to find the prime factors of a 10-digit public key, approximately 100,000 calculations are needed; for a 50-digit number about 10 trillion trillion are required.
IBM’s Blue Gene supercomputer would take a fraction of a second to crack a 10-digit key, but about 100 years for a 50-digit key. And keys are now much longer than 50 digits.
In 1994, mathematician Peter Shor at Bell Labs in New Jersey developed a routine that radically reduces the time required to make those calculations. There was just one rather large catch: it could
only run on a computer that exploits quantum mechanics.
Shor’s algorithm provides a short cut to finding prime factors by looking for telltale patterns in remainders when a key is divided by a prime factor.
Because of the vast number of possible factors for a long key, Shor’s algorithm needs to perform a huge number of mathematical operations in parallel – an ability only offered by the quantum bits, or
qubits, that carry information in a quantum computer.
Thanks to quantum superposition, qubits can inhabit multiple logical states simultaneously, whereas a digital bit can only exist in one state at a time.
The difficulty now is building a quantum computer big enough to carry out the calculations in a reasonable time. Approaches currently being researched include lasers, superconductors, ion traps and
quantum dots.
The first implementation of Shor’s algorithm was achieved in 2001, when an IBM-led team built a quantum computer using nuclear magnetic resonance (NMR) to run calculations in fluorocarbon molecules.
The five fluorine nuclei and two carbon nuclei in a molecule acted as seven qubits: the magnetic spin of each nucleus represented the qubit’s state – say, up for 1 and down for 0.
Because spin is a quantum property the researchers reckoned the nuclei could be “entangled” into a state that is a mix of both spin up and spin down at the same time – allowing the quantum computer
to make calculations in parallel.
Using NMR, the researchers manipulated nuclear spins and coaxed the qubits through Shor’s algorithm. As they had hoped, it gave the correct prime factors for 15 as 3 and 5, but doubts emerged over
the experiment’s quantum credentials.
“The NMR was valuable early work. But it is not clear that there was any quantum entanglement in it,” White says, and Lu agrees. And there is another problem with NMR: the technology does not scale
“As the number of NMR qubits increases, the signal disappears in thermal noise,” says Carl Williams, of the US National Institute of Standards and Technology in Gaithersburg, Maryland.
Instead of manipulating nuclear spin, both White and Lu’s teams plumped for photonic quantum computers. Both used femtosecond lasers to generate photon pairs, which they passed through polarizing
bismuth borate crystals to create entangled qubits.
Using optical devices such as filters, they manipulated the qubits to cajole them into running Shor’s algorithm – once again factorizing 15 into its constituent primes and reading the results using
polarizers and single photon detectors.
Despite the fact that both teams have, like the IBM-led NMR group, only factored the number 15, California-based IT security specialist Bruce Schneier says the way the scientists have done it – with
standard lab optics – means problems for encryption may not be far away. Scaling up to solve bigger problems “is now more or less an engineering problem”, he says.
“There is no need to panic right now,” Schneier says, as cryptography would survive even if RSA was cracked. “RSA has lived with the possibility of being cracked for many years. There are lots of
other algorithms, and we’ll shift to those.”
Computers like White and Lu’s are not powerful enough to pose a threat to the world’s data, but that may not last. “If we could perform calculations for much larger numbers, then fundamental changes
would be needed in cryptography,” says White. “And there are paths to a fully scalable quantum computer.”
So what does he expect to become of the RSA system when such a quantum computer is finally built? -New Scientist
Author at Huliq.
Written By James Huliq | {"url":"https://huliq.com/qubits-poised-to-reveal-our-secrets/","timestamp":"2024-11-14T07:03:49Z","content_type":"text/html","content_length":"88950","record_id":"<urn:uuid:0d7bb63d-04de-4493-9e4f-7f7d507d2234>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00218.warc.gz"} |
EViews Help: @mrnd
Matrix of uniform random numbers.
Syntax: @mrnd(n1[, n2])
n1: integer
n2: (optional) integer
Return: matrix
Creates a matrix filled with uniform (0, 1) random numbers. The size of the matrix is given by the integers n1 (number of rows) and n2 (number of columns).
matrix m1 = @mrnd(3,2)
If n2 is omitted or set to 1, the function returns a vector as in
vector v1 = @mrnd(18)
You may obtain a random sym of uniform numbers by creating a square source matrix of uniform numbers and assigning it to a sym matrix,
sym s1 = @mrnd(5, 5)
which creates the sym S1 based on the lower triangle of the source matrix.
See also
, and | {"url":"https://help.eviews.com/content/functionref_m-@mrnd.html","timestamp":"2024-11-12T06:14:58Z","content_type":"application/xhtml+xml","content_length":"9224","record_id":"<urn:uuid:d9f7e05e-8872-4dc7-bafe-a13f04f14c99>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00622.warc.gz"} |
On Cerný’s conjecture for synchronizing automata
29/04/2016 29 Abril, 16h30, 6.2.33
Jorge Orestes Cerdeira (DM & CMA/FCT/UNL)
Faculty of Sciences, University of Lisbon
A deterministic finite automaton with n states is called synchronizing if there is a sequence of letters which maps the n states onto a single one. In this talk I will give a proof that searching for
a synchronizing word of minimum length is NP-hard, and derive a network flow version of Cerný’s conjecture which states that for every synchronizing automata there is a synchronizing word of length
at most (n-1)^2. | {"url":"https://cemat.ist.utl.pt/seminar.php?event_type_id=17&seminarsPage=2&sem_id=2322","timestamp":"2024-11-04T17:28:00Z","content_type":"text/html","content_length":"9145","record_id":"<urn:uuid:d4e97113-04c8-48bb-a6a5-afd0f6e257c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00371.warc.gz"} |
What is 2D shape? Perimeter and Area of 2D shapes formulas | TIRLA ACADEMY
In mathematics, we learn about various shapes. These shapes may be two-dimensional(2d shapes) or three-dimensional(3d shapes). In this article, we will discuss what is a two-dimensional shape & the
area and perimeter of two-dimensional shapes.
What is a Two-dimensional shape?
Two-dimensional shapes are those shapes that have length, breadth, height, or radius but not any thickness.
Examples of two-dimensional(2d) shapes are square, rectangle, circle, triangle, parallelogram, kite, polygon, etc.
Perimeter & area of two-dimensional shapes
The perimeter of 2d shapes is the length of their boundary and the area of 2d shapes is the surface that is covered by their perimeter.
For example, the perimeter of a square is 4×side because we get the length of their boundary by the sum of all four sides.
Area and Perimeter of 2d shapes formulas
We are discussing area and perimeter formulas for all shapes that are 2d:
The perimeter of the square = 4×side
Area of square = side×side
Perimeter of rectangle = 2×(Length+Breadth)
Area of rectangle = Length×Breadth
The perimeter of the triangle = Sum of all sides
The perimeter of the Equilateral triangle = 3×side
Area of Right-angled triangle = (1/2)×Base×Height
Area of triangle by Heron's Formula = √{s(s-a)(s-b)(s-c)} where s=semiperimeter and a,b,c are sides of triangle
The formula for the circumference of a circle = 2Ï€r
Area of circle = Ï€r² where, r = radius of the circle and Ï€ = 22/7 or 3.14
The perimeter of the rhombus = 4×side
Area of rhombus = (1/2)×D₁×D₂ where D₁ & D₂ are diagonals of a rhombus.
The perimeter of the parallelogram = Sum of all sides
Area of parallelogram = Base × Height
Find the perimeter and area of the polygon:
To find the perimeter and area of the polygon we are taking area and perimeter word problems for better understanding:
1. If the length of each side of a triangle is 4cm. Find the perimeter of a triangle.
The perimeter of an equilateral triangle=3×side
= 3×4cm
= 12cm
2. Find the area and perimeter of a rectangle if its length is 13 cm and width is 10cm long.
Perimeter of rectangle=2×(Length+Breadth)
= 2×(13cm+10cm)
= 2×23cm
= 46cm
Area of Rectangle=Length×Breadth
=130 sq. cm.
3. Find the perimeter and area of the polygon if the length of all four sides is 3cm each.
Perimeter of square=4×side
Area of square=side×side
=9 sq. cm.
4. If the length of the sides of a quadrilateral is 3cm, 5cm, 4cm, and 3cm long then find the perimeter of a quadrilateral.
The perimeter of Quadrilateral=Sum of all sides
5. Find the perimeter of the square if its side is 10cm long.
Perimeter of Square=4×side
6. Find the area of a rectangle if its length and breadth are 12cm and 10cm long respectively.
Area of Rectangle formula=Length×Breadth
=120 sq. cm.
7. Find the perimeter of a parallelogram if the length of their adjacent sides is 7cm and 5cm.
The formula for the perimeter of Parallelogram=Sum of all sides
8. Find the perimeter of a rectangle for a length is 15cm and a breadth is 11cm long.
Perimeter of Rectangle=2×(Length+Breadth)
9. Find the area and perimeter of the square if the length of each side is 6cm long.
Area of square=side×side
=36 sq. cm.
Perimeter of square=4×side | {"url":"https://www.tirlaacademy.com/2020/11/perimeter%20and%20area.html","timestamp":"2024-11-03T00:09:13Z","content_type":"application/xhtml+xml","content_length":"321868","record_id":"<urn:uuid:69704b56-91b4-497a-9fd9-6cf0c04042ab>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00247.warc.gz"} |
500 Years – Part II
The first part of this post was published on March 21, 2014. It may be valuable to read it (again) before reading this. It was important to have the Axial Age post before this one was published.
The next question is how do we determine when a particular 500 year period ends and the next one begins. I use the analogy with how we know when one synodic cycle ends and another starts, and the
answer is obvious. The conjunction of the two planets involved in the synodic cycle is what marks the change from one cycle to the next. What I am proposing for the boundary between two
cycles, where the cycle involves all ten planets, is when their collective distance is at a minimum. This idea also has some support in the work of the great French astrologer Andre Barbault.
Intuitively, the concept of “minimum distance” is easy. The planets spread out across the sky and come together. We can see sometimes that they are spread out and sometimes more bunched together.
It turns out that this minimum occurs once every 500 years. While there are several ways of measuring this angular separation, they all give the same results within a few days. The one I am using
now is based on the common statistical method of standard deviation. For a group a data points (in our case read planets) the standard deviation measures how far the points vary from the mean of
all points. If the standard deviation is large, the points are spread out at far distances from the mean point, and conversely if the standard deviation is small the points are clustered close to
the mean point. This sounds like just what we need.
Technical note. (Warning: Don’t try this at home.) I have written some Python programs that can calculate a chart for any date. What I do is calculate a chart for every day for the last 2600 years
and then compute the standard deviation of the planets for each day. It is then easy to find the minimum.
Note that the planets clustering together is much dependent on the position of the three outer planets, since they have a period much greater than all the others. The fastest moving outer planet –
Uranus – takes 84 years to go around the sun, three times longer than the next slower planet, Saturn, at 29 years. So at first approximation, the minima will occur near the time of the Neptune-Pluto
conjunction. After that we need a Uranus-Pluto and a Uranus-Neptune conjunctions. So these three conjunction will by themselves determine the 100 year period when this minima occurs. Then we need
Saturn and Jupiter, which conjoin every 20 years, to get within the group.
As we saw in the entry on the Axial Age, the three out planets were all conjunct in 575 BCE. There is one conjunction of Neptune and Pluto every 500 years, four Uranus Pluto conjunctions every 500
years, and three Uranus-Neptune conjunctions. As the cycle goes on, the Uranus-Pluto and Uranus -Neptune conjunctions get further and further later from the Neptune-Pluto conjunction, which would
indicate that the planetary minimum occurs further and further after the Neptune-Pluto conjunction as the cycles go on, until they start over after 4000 years. In general, the Neptune-Uranus
conjunction gets later than than than the Uranus-Pluto conjunction as time goes on also, but not as much as the spread from the Neptune-Pluto conjunction. In the current period, the Uranus-Pluto
conjunction of the Sixties was some 70 years after the Neptune-Pluto of the 1890s, and the Uranus-Neptune conjunction of the 1990s was almost 30 years after the Uranus-Pluto conjunction.
The ancient Chinese thought that there was a 500 cycle ruling the rise and fall of civilizations. Each of these 500 years is different, although the changes start well before the date – the
preceding Neptune-Pluto conjunction indicates that changes are starting. But each period is different than the preceding one, and it is futile to think that the rules that applied in the preceding
500 year period will apply to the current one. This lesson applies most forcefully to our current situation, since the number of people in the world is so much greater than what it was 500 years
ago, and civilization has gotten so much more complicated. But the lesson can’t be leaned well enough: Things are bound to change, you can not stop that but only fight a losing battle. We can not
use measures from the past to suggest what the future will be like. The gifts of the old cycle, which as I’ve mentioned before are capitalism, industrialism, rationalism, can no longer be
depended upon.
There is a large cycle called the Great Year, which is 26,000 years. This is the time it takes the equinox — which creates the Aries point — to move backwards and return to its original
position, an action called Precession of the Equinox. Supposedly, this first point of Aries, marking the intersection of the ecliptic — the path of the Sun through the heavens — and the celestial
equator — is moving into the sign of Aquarius, giving rise to the idea that this is the dawning of the Age of Aquarius. But what I want to point out is the ratio of this 500-year cycle to the Great
Year is the same as the ratio of one week to one year. Thus I propose that the name for this 500-year period that I have been talking about, and which gives its title to this blog, is the Great
Here is a brief overview of the minima, with the dates that the minima occurred. Remember this is not a magic date, but rather the dates indicate periods where one cycle changed — with time — to
another cycle. I will gives memorable events that happened a each date, usually involving the Roman Empire, since that has been a dominant feature of the last 2600 years. Changes are slow to happen,
so take these dates with much salt.
Note that while people insist there was no year zero, as far as date computations go, there is a year zero, which corresponds to 1 BCE. With the following charts, dates are given in chronological
time, so that year -576 is 577 BCE.
Minima #1
July 28, 559 BCE. This was the height of the Axial Age, as discussed previously, and is just a short time (as far as history goes) after triple conjunction of Uranus, Neptune, and PLuto. The Roman
Republic starts.
Minima #2
June 7, 60 BCE. Julius Caesar crosses the Rubicon and the Roman Republic ends, the Roman Empire begins.
Minima #3
August 5, 449 CE. The Fall of the (Western) Roman Empire, the beginning of the Dark Ages.
Minima #4
June 23, 947 CE. The Ottonian Renaissance and the beginning of the Upper Middle Ages. Song Dynasty and the Chinese Renaissance.
Minima #5
November 8, 1485. The Fall of the (Eastern) Roman Empire, the discovery (for Europeans) of the New World, beginning of the Tudor Age in England.
Minima #6
November 16, 1982. The world reaches the use of 100% of its resources. | {"url":"https://astrolabe.info/2014/09/05/500-years-part-ii/","timestamp":"2024-11-08T11:54:34Z","content_type":"text/html","content_length":"39368","record_id":"<urn:uuid:b04fbd4b-d618-4262-a9f5-809f48955ee4>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00168.warc.gz"} |
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The commercial loan mortgage calculator supports the calculation of the monthly repayment amount and total interest of the mortgage according to the loan amount or house price. The calculation method
of housing loan supports the equality corpus and interest and the equality corpus. | {"url":"https://www.lddgo.net/en/common/houseloan","timestamp":"2024-11-10T17:14:57Z","content_type":"text/html","content_length":"77069","record_id":"<urn:uuid:83b21743-11f5-41d2-b14b-142ca45230bb>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00063.warc.gz"} |
$56 an Hour is How Much a Year? Before and After Taxes
Hourly wages of $56 or more can provide financial security in many areas of the country. But what does this equate to over the course of a year? In this article, we’ll break down the projected
annual, monthly, biweekly, and weekly earnings for jobs paying $56 per hour. We’ll analyze whether $56/hour is considered a good salary, highlight occupations typically paying this rate, and discuss
realistic lifestyles supported by this income level. Additionally, we’ll look at how taxes, overtime eligibility, and unpaid time off impact actual take-home pay. With wise budgeting, an annual
salary based on $56/hour may afford homeownership or other major expenses in some locations. We’ll provide sample budgets to showcase spending possibilities at this pay rate. While individual
circumstances vary, understanding earnings projections from $56/hour can empower informed career and money decisions. Evaluating the full yearly salary potential provides key insight.
Convert $56 Per Hour to Weekly, Monthly, and Yearly Salary
Input your wage and hours per week to see how much you’ll make monthly, yearly and more.
$56 an Hour is How Much a Year?
If you make $56 an hour, your yearly salary would be $116,480. We calculate your annual income based on 8 hours per day, 5 days per week and 52 weeks in the year.
Hours worked per week (40) x Hourly wage($56) x Weeks worked per year(52) = $116,480
$56 an Hour is How Much a Month?
If you make $56 an hour, your monthly salary would be $9,706.67. We calculated this number by dividing your annual income by 12 months.
Hours worked per week (40) x Hourly wage($56) x Weeks worked per year(52) / Months per Year(12) = $9,706.67
$56 an Hour is How Much Biweekly?
If you make $56 an hour, your biweekly salary would be $4,480.
Hours worked per week (40) x Hourly wage($56) x 2 = $4,480
$56 an Hour is How Much a Week?
If you make $56 an hour, your weekly salary would be $2,240. Calculating based on 5 days per week and 8 hours each day.
Hours worked per week (40) x Hourly wage($56) = $2,240
$56 an Hour is How Much a Day?
If you make $56 an hour, your daily salary would be $448. We calculated your daily income based on 8 hours per day.
Hours worked per day (8) x Hourly wage($56) = $448
$56 an Hour is How Much a Year?
The basic formula to calculate your annual salary from an hourly wage is:
Hourly Rate x Hours Worked per Week x Number of Weeks Worked per Year = Annual Salary
So for a $20 per hour job:
$56 per hour x 40 hours per week x 52 weeks per year = $116,480
However, this simple calculation makes some assumptions:
• You will work 40 hours every week of the year
• You will not get any paid time off
Therefore, it represents your earnings if you worked every week of the year, without any vacation, holidays, or sick days.
Accounting for Paid Time Off
The $116,480 base salary does not yet factor in paid time off (PTO). Let’s assume the job provides:
• 2 weeks (10 days) paid vacation
• 6 paid holidays
• 3 paid sick days
This totals 19 paid days off, or nearly 4 weeks of PTO.
Importantly, this paid time off should not be deducted from the annual salary, since you still get paid for those days.
So with 4 weeks PTO, the annual salary would remain $116,480 .
Part time $56 an hour is How Much a Year?
Your annual income changes significantly if you work part-time and not full-time.
For example, let’s say you work 30 hours per week instead of 40. Here’s how you calculate your new yearly total:
$56 per hour x 30 hours per week x 52 weeks per year = $87,360
By working 10 fewer hours per week (30 instead of 40), your annual earnings at $56 an hour drop from $116,480 to $87,360.
That’s a $29,120 per year difference just by working part-time!
Here’s a table summarizing how your annual earnings change depending on how many hours you work per week at $56 an hour:
Hours Per Week Earnings Per Week Annual Earnings
40 $2,240 $116,480
35 $1,960 $101,920
30 $1,680 $87,360
25 $1,400 $72,800
20 $1,120 $58,240
15 $840 $43,680
The more hours per week, the higher your total yearly earnings. But part-time work allows for more life balance if you don’t need the full salary.
$56 an Hour With Overtime is How Much a Year?
Now let’s look at how overtime can increase your annual earnings.
Overtime kicks in once you work more than 40 hours in a week. Typically, you earn 1.5x your regular hourly wage for overtime hours.
So if you make $56 per hour normally, you would make $84 per hour for any hours over 40 in a week.
Here’s an example:
• You work 45 hours in a Week
• 40 regular hours paid at $56 per hour = $2,240
• 5 overtime hours paid at $84 per hour = $420
• Your total one Week earnings =$2,240 + $420 = $2,660
If you worked 45 hours each week for 52 weeks, here’s how your annual earnings increase thanks to overtime pay:
$2,660 per week x 52 weeks per year = $138,320
That’s $21,840 more than you’d earn working just 40 hours per week at $56 an hour.
Overtime can add up! But also consider taxes and work-life balance when deciding on extra hours.
Here’s a table summarizing how your annual earnings change depending on how many hours you work per week at $56 an hour:
Overtime hours per work day Hours Per Week Earnings Per Week Annual Earnings
0 40 $2,240 $116,480
1 45 $2,660 $138,320
2 50 $3,080 $160,160
3 55 $3,500 $182,000
4 60 $3,920 $203,840
5 65 $4,340 $225,680
6 70 $4,760 $247,520
7 75 $5,180 $269,360
How Unpaid Time Off Impacts $56/Hour Yearly Earnings
So far we’ve assumed you work 52 paid weeks per year. Any unpaid time off will reduce your total income.
For example, let’s say you take 2 weeks of unpaid leave. That brings your paid weeks down to 50:
Hours worked per week (40) x Hourly wage($56) x Weeks worked per year(50) = $112,000 annual salary
With 2 weeks unpaid time off, your annual earnings at $56/hour would drop by $4,480.
The table below summarizes how your annual income changes depending on the number of weeks of unpaid leave.
Weeks of unpaid leave Paid weeks per year Earnings Per Week Annual Earnings
0 52 $2,240 $116,480
1 51 $2,240 $114,240
2 50 $2,240 $112,000
3 49 $2,240 $109,760
4 48 $2,240 $107,520
5 47 $2,240 $105,280
6 46 $2,240 $103,040
7 45 $2,240 $100,800
Key Takeaways for $56 Hourly Wage
In summary, here are some key points on annual earnings when making $56 per hour:
• At 40 hours per week, you’ll earn $116,480 per year.
• Part-time of 30 hours/week results in $87,360 annual salary.
• Overtime pay can boost yearly earnings, e.g. $21,840 extra at 45 hours/week.
• Unpaid time off reduces your total income, around $4,480 less per 2 weeks off.
• Your specific situation and location impacts taxes and PTO.
Knowing your approximate annual salary and factors impacting it makes it easier to budget and plan your finances. The next step is calculating take-home pay after deductions like taxes.
$56 An Hour Is How Much A Year After Taxes
Figuring out your actual annual earnings based on an hourly wage can be complicated once taxes are taken into account. In addition to federal, state, and local income taxes, 7.65% of your gross pay
also goes to Social Security and Medicare through FICA payroll taxes. So how much does $56 an hour equal per year after FICA and income taxes are deducted from your gross pay?
Below we’ll walk through the steps to calculate your annual net take home pay if you make $56 per hour. This will factor in estimated federal, FICA, state, and local taxes so you know exactly what to
Factoring in Federal Income Tax
Your federal income tax will be a big chunk out of your gross pay. Federal tax rates range from 10% to 37%, depending on your tax bracket.
To estimate your federal income tax rate and liability:
Look up your federal income tax bracket based on your gross pay.
2023 tax brackets: single filers
Tax rate Taxable income bracket Tax owed
10% $0 to $11,000. 10% of taxable income.
12% $11,001 to $44,725. $1,100 plus 12% of the amount over $11,000.
22% $44,726 to $95,375. $5,147 plus 22% of the amount over $44,725.
24% $95,376 to $182,100. $16,290 plus 24% of the amount over $95,375.
32% $182,101 to $231,250. $37,104 plus 32% of the amount over $182,100.
35% $231,251 to $578,125. $52,832 plus 35% of the amount over $231,250.
37% $578,126 or more. $174,238.25 plus 37% of the amount over $578,125.
For example, if you are single with $116,480 gross annual pay, your federal tax bracket is 24%.
Your estimated federal tax would be:
$16,290 + ($116,480 – $95,376) x 24% = $21,354.96
So at $56/hour with $116,480 gross pay, you would owe about $21,354.96 in federal income taxes.
Considering State Income Tax
In addition to federal tax, most states also charge a state income tax. State income tax rates range from about 1% to 13%, with most falling between 4% and 6%.
Key Takeaways
□ California, Hawaii, New York, New Jersey, and Oregon have some of the highest state income tax rates.
□ Alaska, Florida, Nevada, South Dakota, Tennessee, Texas, Washington, and Wyoming don’t impose an income tax at all.
□ Another 10 U.S states have a flat tax rate—everyone pays the same percentage regardless of how much they earn.
A State-by-State Comparison of Income Tax Rates
STATE TAX RATES LOWEST AND HIGHEST INCOME BRACKETS
Alaska 0% None
Florida 0% None
Nevada 0% None
South Dakota 0% None
Tennessee 0% None
Texas 0% None
Washington 0% None
Wyoming 0% None
Colorado 4.55% Flat rate applies to all incomes
Illinois 4.95% Flat rate applies to all incomes
Indiana 3.23% Flat rate applies to all incomes
Kentucky 5% Flat rate applies to all incomes
Massachusetts 5% Flat rate applies to all incomes
New Hampshire 5% Flat rate on interest and dividend income only
North Carolina 4.99% Flat rate applies to all incomes
Pennsylvania 3.07% Flat rate applies to all incomes
Utah 4.95% Flat rate applies to all incomes
Michigan 4.25% Flat rate applies to all incomes
Arizona 2.59% to 4.5% $27,806 and $166,843
Arkansas 2% to 5.5% $4,300 and $8,501
California 1% to 13.3% $9,325 and $1 million
Connecticut 3% to 6.99% $10,000 and $500,000
Delaware 0% to 6.6% $2,000 and $60,001
Alabama 2% to 5% $500 and $3,001
Georgia 1% to 5.75% $750 and $7,001
Hawaii 1.4% to 11% $2,400 and $200,000
Idaho 1.125% to 6.5% $1,568 and $7,939
Iowa 0.33% to 8.53% $1,743 and $78,435
Kansas 3.1% to 5.7% $15,000 and $30,000
Louisiana 1.85% to 4.25% $12,500 and $50,001
Maine 5.8% to 7.15% $23,000 and $54,450
Maryland 2% to 5.75% $1,000 and $250,000
Minnesota 5.35% to 9.85% $28,080 and $171,221
Mississippi 0% to 5% $5,000 and $10,001
Missouri 1.5% to 5.3% $1,121 and $8,968
Montana 1% to 6.75% $2,900and $17,400
Nebraska 2.46% to 6.84% $3,340 and $32,210
New Jersey 1.4% to 10.75% $20,000 and $1 million
New Mexico 1.7% to 5.9% $5,500 and $210,000
New York 4% to 10.9% $8,500 and $25 million
North Dakota 1.1% to 2.9% $41,775 and $458,350
Ohio 0% to 3.99% $25,000 and $110,650
Oklahoma 0.25% to 4.75% $1,000 and $7,200
Oregon 4.75% to 9.9% $3,750 and $125,000
Rhode Island 3.75% to 5.99% $68,200 and $155,050
South Carolina 0% to 7% $3,110 and $15,560
Vermont 3.35% to 8.75% $42,150 and $213,150
Virginia 2% to 5.75% $3,000 and $17,001
Washington, D.C. 4% to 9.75% $10,000 and $1 million
West Virginia 3% to 6.5% $10,000 and $60,000
Wisconsin 3.54% to 7.65% $12,760 and $280,950
To estimate your state income tax:
Look up your state income tax rate based on your gross pay and filing status.
Multiply your gross annual pay by the state tax rate.
For example, if you live in Pennsylvania which has a flat 3.07% tax rate, your estimated state tax would be:
$116,480 gross pay x 3.07% PA tax rate = $3,575.94 estimated state income tax
So with $116,480 gross annual income, you would owe around in $3,575.94 Pennsylvania state income tax. Verify your specific state’s income tax rates.
Factoring in Local Taxes
Some cities and counties levy local income taxes ranging from 1-3% of taxable income.
To estimate potential local taxes you may owe:
• Check if your city or county charges a local income tax.
• If yes, look up the local income tax rate.
• Multiply your gross annual pay by the local tax rate.
For example, say you live in Columbus, OH which has a 2.5% local income tax. Your estimated local tax would be:
$116,480 gross pay x 2.5% local tax rate = $2,912 estimated local tax
So with $116,480 in gross earnings, you may owe around $2,912 in Columbus local income taxes. Verify rates for your own city/county.
Accounting for FICA Taxes (Social Security & Medicare)
FICA taxes are a combination of Social Security and Medicare taxes that equal 15.3% of your earnings. You are responsible for half of the total bill (7.65%), which includes a 6.2% Social Security tax
and 1.45% Medicare tax on your earnings.
In 2023, only the first $160,200 of your earnings are subject to the Social Security tax
There is an additional 0.9% surtax on top of the standard 1.45% Medicare tax for those who earn over $200,000 (single filers) or $250,000 (joint filers).
To estimate your FICA tax payment:
$116,480 x 6.2% + $116,480 x 1.45% = $8,910.72
So you can expect to pay about $8,910.72 in Social Security and Medicare taxes out of your gross $116,480 in earnings.
Total Estimated Tax Payments
Based on the examples above, your total estimated tax payments would be:
Federal tax: $21,354.96
State tax: $3,575.94
Local tax: $2,912
FICA tax: $8,910.72
Total Estimated Tax: $36,753.62
Calculating Your Take Home Pay
To calculate your annual take home pay at $56 /hour:
1. Take your gross pay
2. Subtract your estimated total tax payments
$116,480 gross pay – $36,753.62 Total Estimated Tax = $79,726.38 Your Take Home Pay
n summary, if you make $56 per hour and work full-time, you would take home around $79,726.38 per year after federal, state, local , FICA taxes.
Your actual net income may vary depending on your specific tax situation. But this gives you a general idea of what to expect.
Convert $56 Per Hour to Yearly, Monthly, Biweekly, and Weekly Salary After Taxes
If you make $56 an hour and work full-time (40 hours per week), your estimated yearly salary would be $116,480 .
The $116,480 per year salary does not account for taxes. Federal, state, and local taxes will reduce your take-home pay. The amount withheld depends on your location, filing status, dependents, and
other factors.
Just now during our calculation of $56 An Hour Is How Much A Year After Taxes, we assumed the following conditions:
• You are single with $116,480 gross annual pay, your federal tax bracket is 24 %.
• You live in Pennsylvania which has a flat 3.07% tax rate
• You live in Columbus, OH which has a 2.5% local income tax.
In the end, we calculated your Total Estimated Tax is $36,753.62 , Your Take Home Pay is $79,726.38 , Total tax rate is 31.55%.
So next we’ll use 31.55% as the estimated tax rate to calculate your weekly, biweekly, and monthly after-tax income.
$56 Per Hour to Yearly, Monthly, Biweekly, Weekly,and Week Salary After Taxes Table
Income before taxes Estimated Tax Rate Income Taxes After Tax Income
Yearly Salary $116,480 31.55% $36,753.62 $79,726.38
Monthly Salary $9,706.67 31.55% $3,062.80 $6,643.87
BiWeekly Salary $4,480 31.55% $1,413.60 $3,066.40
Weekly Salary $2,240 31.55% $706.80 $1,533.20
$56 an hour is how much a year after taxes
Here is the adjusted yearly salary after a 31.55% tax reduction:
□ Yearly salary before taxes: $116,480
□ Estimated tax rate: 31.55%
□ Taxes owed (31.55% * $116,480 )= $36,753.62
□ Yearly salary after taxes: $79,726.38
Hourly Wage Hours Worked Per Week Weeks Worked Per Year Total Yearly Salary Estimated Tax Rate Taxes Owed After-Tax Yearly Salary
$56 40 52 $116,480 31.55% $36,753.62 $79,726.38
$56 an hour is how much a month after taxes
To calculate the monthly salary based on an hourly wage, you first need the yearly salary amount. Then divide by 12 months.
☆ Yearly salary before taxes at $56 per hour: $116,480
☆ Divided by 12 months per year: $116,480 / 12 = $9,706.67 per month
The monthly salary based on a 40 hour work week at $56 per hour is $9,706.67 before taxes.
After applying the estimated 31.55% tax rate, the monthly after-tax salary would be:
□ Monthly before-tax salary: $9,706.67
□ Estimated tax rate: 31.55%
□ Taxes owed (31.55% * $9,706.67 )= $3,062.80
• Monthly after-tax salary: $6,643.87
Monthly Salary Based on $56 Per Hour
Hourly Wage Yearly Salary Months Per Year Before-Tax Monthly Salary Estimated Tax Rate Taxes Owed After-Tax Monthly Salary
$56 $116,480 12 $9,706.67 31.55% $3,062.80 $6,643.87
$56 an hour is how much biweekly after taxes
Many people are paid biweekly, meaning every other week. To calculate the biweekly pay at $56 per hour:
• Hourly wage: $56
• Hours worked per week: 40
• Weeks per biweekly pay period: 2
• $56 * 40 hours * 2 weeks = $4,480 biweekly
Applying the 31.55%estimated tax rate:
• Biweekly before-tax salary: $4,480
• Estimated tax rate: 31.55%
• Taxes owed (31.55% * $4,480 )= $1,413.60
• Biweekly after-tax salary: $3,066.40
Biweekly Salary at $56 Per Hour
Hourly Wage Hours Worked Per Week Weeks Per Pay Period Before-Tax Biweekly Salary Estimated Tax Rate Taxes Owed After-Tax Biweekly Salary
$56 40 2 $4,480 31.55% $1,413.60 $3,066.40
$56 an hour is how much weekly after taxes
To find the weekly salary based on an hourly wage, you need to know the number of hours worked per week. At 40 hours per week, the calculation is:
• Hourly wage: $56
• Hours worked per week: 40
• $56 * 40 hours = $2,240 per week
Accounting for the estimated 31.55% tax rate:
• Weekly before-tax salary: $2,240
• Estimated tax rate: 31.55%
• Taxes owed (31.55% * $2,240 )= $706.80
• Weekly after-tax salary: $1,533.20
Weekly Salary at $56 Per Hour
Hourly Wage Hours Worked Per Week Before-Tax Weekly Salary Estimated Tax Rate Taxes Owed After-Tax Weekly Salary
$56 40 $2,240 31.55% $706.80 $1,533.20
Key Takeaways
• An hourly wage of $56 per hour equals a yearly salary of $116,480 before taxes, assuming a 40 hour work week.
• After accounting for an estimated 31.55% tax rate, the yearly after-tax salary is approximately $79,726.38 .
• On a monthly basis before taxes, $56 per hour equals $9,706.67 per month. After estimated taxes, the monthly take-home pay is about $6,643.87 .
• The before-tax weekly salary at $56 per hour is $2,240 . After taxes, the weekly take-home pay is approximately $1,533.20 .
• For biweekly pay, the pre-tax salary at $56 per hour is $4,480 . After estimated taxes, the biweekly take-home pay is around $3,066.40 .
Understanding annual, monthly, weekly, and biweekly salary equivalents based on an hourly wage is useful when budgeting and financial planning. Taxes make a significant difference in take-home pay,
so be sure to account for them when making income conversions. Use this guide as a reference when making salary calculations.
What Is the Average Hourly Wage in the US?
Last Updated: Sep 1 2023
US Average Hourly Earnings is at a current level of $33.82, up from 33.74 last month and up from 32.43 one year ago. This is a change of 0.24% from last month and 4.29% from one year ago.
Average Hourly Earnings is the average dollars that a private employee makes per hour in the US. This metric is a part of one of the most important releases every month which includes unemployment
numbers as well. This is normally released on the first Friday of every month. This metric is released by the Bureau of Labor Statistics (BLS).
What is the average salary in the U.S.?
Last Updated: July 18, 2023
The U.S. Bureau of Labor Statistics uses median salary data rather than averages to avoid skewed numbers from outlying high and low numbers. Median weekly earnings of the nation's 121.5 million
full-time wage and salary workers were $1,100 in the second quarter of 2023, the U.S.
If a person works 52 weeks in the year, then this represents a national annual salary of $57,200.
Is $56 An Hour a Good Salary?
For most individuals and households, $56 an hour provides a good middle-class income. At full-time hours, annual pre-tax pay would be:
• $56/hour x 40 hours/week = $2240 per week
• $2240 per week x 52 weeks/year = $116,480 per year
While not enough to be considered wealthy, this salary exceeds median household income by a large margin. According to the U.S. Census Bureau, real median household income was $67,521 in 2020.
So a single earner making $116,480 has an income equal to a typical household with two incomes. They have over 70% more income compared to the median.
Overall, a salary of $56 an hour or $116,480 per year provides a comfortable living in much of the country. While not rich, higher than average earnings allow financial flexibility.
Jobs that pay $56 an hour
Certain professional occupations commonly pay over $50 per hour:
• Pilots and flight engineers
• Some medical doctors
• Pharmacists
• Dentists
• Architects
• Computer and information systems managers
• Software developers
• Registered nurses
• HVAC technicians
• Elevator installers and repairers
• Power plant operators
• Transportation inspectors
• Radiation therapists
• Tool and die makers
Positions requiring specialized technical skills, higher education, medical training or rare abilities tend to offer the highest hourly wages. Workers who get paid $56 per hour or more are generally
very experienced and skilled at their jobs.
Can You Live Off $56 An Hour?
Absolutely – $56 an hour supplies earners with well above average income. While expenses vary by factors like debt, family size and location, this wage affords financial flexibility.
Someone earning $116,480 annually could easily afford costs like:
• Median monthly rent for a 1 bedroom apartment ($1100)
• Median home price ($340,000) with 20% down payment
• Buying a new car ($40,000 financed)
• Or leasing a luxury car ($800+/month)
Other Costs
• Groceries ($400+ monthly for family)
• Utilities and cell phone ($300+ monthly)
• Entertainment and travel (vacations, concerts, hobbies etc.)
• Comprehensive health insurance
• Maxing out retirement contributions
At $56/hour, you could live comfortably, especially with a partner combining incomes. Single earners may need roommates to afford high rents in some cities, but can still live well.
The impact of inflation on the value of $56 an hour
Like all wages, inflation gradually diminishes the real value and purchase power of a fixed $56 hourly over time.
Here is how inflation has reduced the buying power of $56/hour over the past two decades:
• In 2000, $56/hour was worth $88.38 in 2023 dollars.
• By 2010, $56/hour equaled $69.19 in today’s money.
• And currently, $56 only has the same purchase power as $56 in 2023 dollars.
If inflation averages 2-3% annually, in a decade $56/hour will only equate to about $45.50 to $46.63 per hour in current dollars.
While still a strong income, inflation causes today’s $56 wage to lose over 18% of real value over 10 years at 3% annual inflation. Raises must keep pace with living costs to maintain buying power.
5 Ways To Increase Your Hourly Wage
Even at $56/hour, higher pay is often desired. Here are some options:
1. Request a salary increase to keep up with inflation and experience.
2. Take on additional certifications, training and education to gain skills.
3. Seek promotions to higher positions with more responsibility.
4. Negotiate pay for special high value projects and overtime hours.
5. Develop expertise in a niche area to increase your market value.
Learning new systems, programs or skills makes you more valuable. Leverage achievements into better compensation. Change companies if current employer denies reasonable requests.
Buying a car on $56 an hour
At this income level, purchasing a new or used car (even luxury models) is very affordable:
• A $40,000 new car financed at 3% over 5 years is about $725 monthly.
• A $60,000 high-end SUV financed at 4% over 6 years is $980 monthly.
Both are easily managed payments on this salary. Even expensive cars typically cost no more than 15% of monthly take home pay.
In cash terms, a $40,000 vehicle with $8000 down requires financing just $32,000. That’s certainly reasonable on $116,480 in annual earnings. Auto loans, insurance and maintenance costs won’t break
the bank.
Buying a nice new car every 5 years, or leasing luxury vehicles is realistic on $56 per hour. You could also pay cash for cheaper used cars. Extras like upgrading stereo systems or custom details are
affordable luxuries.
Can You Buy a House on $56 An Hour?
Yes, $56 an hour income makes home ownership attainable at median sale prices in most areas.
As a general rule, banks approve borrowers for mortgages worth 3-4 times their annual income. On $116,480 yearly earnings, that means loan eligibility between:
• 3 x $116,480 = $349,440
• 4 x $116,480 = $465,920
After a 20% down payment of $90,000, you could buy a home priced between:
• $349,440 + $90,000 = $439,440
• $465,920 + $90,000 = $555,920
The current U.S. median home price is around $340,000. So even at 4 times income, buying an average home is affordable at $56/hour.
In high cost areas like California or New York, earning $56 hourly allows purchase of homes priced $100,000+ above average. Or you could buy multi-family properties as investment rentals.
Mortgages, insurance and taxes on a median $340,000 home price are readily managed on this salary. Higher wages allow buying bigger and more expensive houses.
Example Budget For $56 Per Hour
Here is one example monthly budget for an individual with no children earning $56/hour full-time:
• Monthly Gross Pay: $9,680
• Taxes & Deductions: -$2,400
• 401k Contribution: -$1,000
• Take Home Pay: $6,280
• Rent: -$1,500
• Car Payment: -$700
• Car Insurance: -$150
• Gas: -$200
• Utilities: -$300
• Cell Phone: -$100
• Groceries: -$600
• Entertainment: -$400
• Total Expenses: -$3,950
• Remainder (savings): $2,330
This leaves over $28,000 per year for additional savings, travel, or other goals. Significant disposable income is available after essentials.
For a frugal spender, even more could be saved – allowing possible early retirement. Overall, this sample budget indicates a high income of $56/hour supports financial flexibility and wealth
In Summary
An hourly wage of $56 provides middle to upper-middle class earnings well above national averages. Specialized skills, medical training and professional qualifications unlocked wages of $50+ per
hour. While inflation slowly erodes real value, consistent pay increases can maintain purchasing power.
At $56 hourly or $116,480 annually, comfortable home ownership, new car leases, and regular vacations and entertainment are affordable. Individuals can live well and families can thrive without
budget constraints. Savings are also feasible to invest or prepare for retirement. Although not technically rich, this income affords financial flexibility and economic stability. | {"url":"https://timehackhero.com/56-an-hour-is-how-much-a-year/","timestamp":"2024-11-05T19:14:14Z","content_type":"text/html","content_length":"227602","record_id":"<urn:uuid:fe1f63a4-416f-450f-9a18-70b21bf779ab>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00825.warc.gz"} |
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You Are On Multi Choice Question Bank SET 1661
83058. ’Rashtriya Ekta Diwas’was observed on 31st October 2014 to commemorate the birth anniversary of------?
83062. Aij and Bij represent symmetric and anti symmetric real valued tensor respectively in three dimension.The number of independent components of Aij and Bij are
83069. Two circular discs have the same mass m and same thickness t,Disc one has uniform density less than that of disc two.Which of the following is correct.
83070. Consider motion of a particle of mass m under the influence of a force F=kr,where k is a positive constant and r is the position vector of the particle.Now the orbit will be
83071. A violin string 5 m long has a fundamental frequency 200 Hz.At what speed does a transverse pluse travel on this string?
83072. Consider the longitudinal vibration of a linear traitomic molecule with two atoms of mass m each symmetrically situated on either sides of an atom of mass M.Now which of the following is a
normal mode frequency of the system,if we use a spring model
83073. Number of generalized coordinates required to describe the motion of a particle of mass m constrained to move on the surface of a sphere of a radius R are
83075. Two particles of same mass m are emitted in the same direction with momenta 5 mc and 10mc respectively(c is the speed of light).as seen from the slower one,what is the speed of the faster
83077. A particle is constrained to move on a plane ,where it is attracted towards a fixed point with force is inversely proportional to square of the distance from the point.What is the Lagrangian
of the particle in polar coordinates?
83080. In case more than one linearly independent wave functions belong to the same energy E,the level is said to be
83083. Since electrons has an intrinsic magnetic dipole moment due to its spin,the electron interacts with the external magnetic field and the operator for this moment is ‘hatmu’is
83085. If a material has a conductivity of 25 S/m and relative permittivity of 80,then at a frequency of 3 GHz the material will acts as
83086. For normal incidence at an air glass interface with refractive index of 5 the fraction of energy reflected is given by
83087. Find out the correct relation between magnetic field B and electric field E,if n is the unit vector along the direction of propagation
83089. A system of N non-interacting classical point particles are constrained to move on the two-dimensional surface of a sphere.The internal energy of the system is
83091. What is the temperature of an ideal gas of He atoms,if there are 6’xx’’10^(22)’atoms occupying 2 litres at atmospheric pressure?
83092. Consider a photon gas enclosed in a volume V and in equilibrium at temperature T.What is the chemical potential of this gas?
83098. The term’(j_(1),j_(2))’arising from 2s1,2p1 electronic configuration in jj coupling scheme is
83099. Metallic sodium has a BCC structure.Which of the following lines will be absent in the diffraction pattern.
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How to rescale a stl surface in matlab?
To rescale a stl surface in MATLAB, you can use the following steps:
1. Load the stl file using the stlread function. For example, if your stl file is named surface.stl, you can load it using the following code:
1 [vertex,faces,normals] = stlread('surface.stl');
1. Calculate the centroid of the surface by finding the mean of the x, y, and z coordinates of the vertices:
1 centroid = mean(vertex);
1. Calculate the scale factor for resizing the surface. For example, if you want to scale the surface by a factor of 2 in each dimension, you can use:
1. Rescale the vertices of the surface by subtracting the centroid, scaling by the scale factor, and then adding the centroid back:
1 scaled_vertex = (vertex - centroid) .* scale_factor + centroid;
1. Write the rescaled surface to a new stl file using the stlwrite function. For example, to save the rescaled surface to a file named rescaled_surface.stl, you can use:
1 stlwrite('rescaled_surface.stl',scaled_vertex,faces,normals);
By following these steps, you can rescale a stl surface in MATLAB. | {"url":"https://devhubby.com/thread/how-to-rescale-a-stl-surface-in-matlab","timestamp":"2024-11-12T13:40:14Z","content_type":"text/html","content_length":"120424","record_id":"<urn:uuid:b2259ce2-bfb8-4f3a-87d3-82fa106da512>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00320.warc.gz"} |
Numpy - Count Values Equal to a Given Value - Data Science Parichay
In this tutorial, we will look at how to count the values in a Numpy array that are equal to a given value (let’s say k) with the help of some examples.
Steps to get the count of all the values equal to k in a Numpy array
In general, to find the count of values in a Numpy array that satisfy the given condition, you can –
1. Use boolean indexing to filter the array for only the values that satisfy the condition.
2. Calculate the length of the filtered array from step 1.
Thus, first, filter the Numpy array to contain only the values that are equal to the given value and then find its length to get the required count.
Let’s now look at a step-by-step example.
Step 1 – Create a Numpy array
First, we will create a Numpy array that we will be using throughout this tutorial.
import numpy as np
# create a numpy array
ar = np.array([1, 2, 3, 4, 3, 6, 5])
# display the array
[1 2 3 4 3 6 5]
Here, we used the numpy.array() function to create a one-dimensional Numpy array containing some numbers.
Step 2 – Filter the array using a boolean expression
To get all the values from a Numpy array that are equal to a given value, filter the array using boolean indexing.
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First, we will specify our boolean expression, ar == k and then use the boolean array resulting from this expression to filter our original array.
For example, let’s get all the values in the above array that are equal to 3 (k = 3).
# values in array that are equal to 3
ar_filtered = ar[ar == 3]
[3 3]
We get all the values in the array ar that are equal to 3.
Step 3 – Get the length of the filtered array
To get the count of values that satisfy the given condition (whether it’s equal to k or not) find the length of the resulting filtered array from step 2 using the Python built-in len() function.
# length of the filtered array
We get the count of values in the array ar that are equal to the given value (k=3) as 2.
We can combine the code from the last two steps into a single line of code.
# count of values in array that are equal to 3
print(len(ar[ar == 3]))
We get the same result as above and we removed the extra variable ar_filtered.
In this tutorial, we looked at how to count values in a Numpy array that are equal to a given value k. Note that in this method we’re not counting unique elements that are equal to k, rather we’re
counting all values in an array that are equal to k (which may include duplicates depending on the array).
You might also be interested in –
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NumPy - A Mathematical Toolkit for Python
In the previous section, modules for Python was introduced. In this section, we’ll take a much more detailed look at one of the most useful to scientists: NumPy. This module contains numerous
routinues and support frameworks for numerical computing. The routinues in it are very carefully tested for correctness and are crafted for speed. Any time you can use something from this package,
it’s a good idea to.
Python is built for versatility and ease of programming. Unfortunately, it is not built for speed. Over the years Python has gotten faster and faster but there is still a speed penalty compared to
classic compiled languages like C, C++, or Fortran.
Enter NumPy: a package of mathematical routines written in C and Fortran and made to work with Python via a “glue” or “shim” layer. This interface is invisible to the programmer. NumPy looks and
behaves just like any other Python package. Under the surface, though, lies a very fast and efficient library of algorithms.
A first glimpse
Let’s take a quick look at NumPy and see a few of the things it can do. NumPy is a package, not part of Python proper, so we have to tell Python to load it. It’s traditional to import numpy and give
it the alias “np” - it’s less typing that way, and if you’re cutting and pasting code from other sources then it’s handy to follow the convention.
Python, you’ll recall, doesn’t have an “array” data type. The closest it can come is the “list”. Lists are certainly useful, but they aren’t all that fast to read and even slower to write to. To make
matters worse, a 2-D array is represented by a list of lists. This is great for representing complicated data but it’s lousy for doing math.
The critical NumPy data type is the array: “NumPy arrays are faster and more compact than Python lists. An array consumes less memory and is convenient to use.” (source) The one caveat with NumPy
arrays is that all the elements inside an array need to have the same data type (e.g. integer, float, double). In practice this is rarely, if ever, a problem.
Let’s make an array of integers:
array([[ 1, 2, 3, 4],
[ 5, 6, 7, 8],
[ 9, 10, 11, 12]])
The array a is now a 4x3 array of integers. The array method was called with one argument - a Python “list of lists” representation of the array. The dimensions of the array are inferred from the
list of lists used to initialize it.
There are other ways to create arrays. Here are two more common methods:
Notice the decimal points after the zeros. These indicate that we’re seeing floating point numbers.
array([[1., 1., 1.],
[1., 1., 1.],
[1., 1., 1.]])
This one will throw you off if you aren’t paying attention. Notice how many parantheses there are… probably more than you expected! What is going on is that the outer parentheses are there to
indicate function arguments, just like calling any other functions. The inner parentheses are used to generate a tuple, in this case one with two values, both of which are threes. This tuple can be
arbitrarily long:
array([[[[1., 1., 1.],
[1., 1., 1.],
[1., 1., 1.]],
[[1., 1., 1.],
[1., 1., 1.],
[1., 1., 1.]],
[[1., 1., 1.],
[1., 1., 1.],
[1., 1., 1.]]],
[[[1., 1., 1.],
[1., 1., 1.],
[1., 1., 1.]],
[[1., 1., 1.],
[1., 1., 1.],
[1., 1., 1.]],
[[1., 1., 1.],
[1., 1., 1.],
[1., 1., 1.]]],
[[[1., 1., 1.],
[1., 1., 1.],
[1., 1., 1.]],
[[1., 1., 1.],
[1., 1., 1.],
[1., 1., 1.]],
[[1., 1., 1.],
[1., 1., 1.],
[1., 1., 1.]]]])
The output isn’t terribly easy to read, but then again representing a four dimensional array on a flat page is challenging at best.
If we ever need to see the dimensions of an array, we can use the shape() method.
(3, 3)
(3, 3, 3, 3)
Let’s do some actual math, shall we?
The trivial example: add a scalar (“a single number”) to every element of the matrix:
You can use any of the Python operators, of course: +, -, *, /, %, **…
[[1 0 1 0]
[1 0 1 0]
[1 0 1 0]]
Comparison operators (like >, <, and so forth) are legitimate operators, so they work too:
[[ 1 2 3 4]
[ 5 6 7 8]
[ 9 10 11 12]]
[[False False False False]
[False True True True]
[ True True True True]]
Linear algebra, anyone?
Let’s use NumPy to do some basic linear algebra. First, we’ll need another module in the NumPy package:
That import statement went out to where Python packages are stored and found the “linalg” module of the numpy package. This module was imported into the Python interpreter under the name “nl” (as in
“NumPy linear algebra”). Using the “nl” alias saves a lot of typing and even makes the code easier to read.
[[1 1 1]
[1 1 0]
[1 0 0]]
array([[ 0., 0., 1.],
[-0., 1., -1.],
[ 1., -1., -0.]])
And given a matrix and its inverse, you probably already guessed where this is going:
array([[1., 0., 0.],
[0., 1., 0.],
[0., 0., 1.]])
Source: NumPy - A Mathematical Toolkit for Python | {"url":"https://x-cite-course.netlify.app/theme1/pe102/","timestamp":"2024-11-09T12:54:56Z","content_type":"application/xhtml+xml","content_length":"36574","record_id":"<urn:uuid:a77fea49-4f2f-4fed-a980-c73f8673bb06>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00183.warc.gz"} |
Greatest common divisor
This video comprehensively explains everything related to the GCD: https://youtu.be/HboSeb_gQH8 The greatest common divisor (GCD, or GCF (greatest common factor)) of two or more integers is the
largest integer that is a divisor of all the given numbers.
The GCD is sometimes called the greatest common factor (GCF).
A very useful property of the GCD is that it can be represented as a sum of the given numbers with integer coefficients. From here it immediately follows that the greatest common divisor of several
numbers is divisible by any other common divisor of these numbers.
Greatest Common Divisor Video
Finding the GCD/GCF of two numbers
Using prime factorization
Once the prime factorizations of the given numbers have been found, the greatest common divisor is the product of all common factors of the numbers.
$270=2\cdot3^3\cdot5$ and $144=2^4\cdot3^2$. The common factors are 2 and $3^2$, so $GCD(270,144)=2\cdot3^2=18$.
Euclidean algorithm
The Euclidean algorithm is much faster and can be used to give the GCD of any two numbers without knowing their prime factorizations. To find the greatest common divisor of more than two numbers, one
can use the recursive formula $GCD(a_1,\dots,a_n)=GCD(GCD(a_1,\dots,a_{n-1}),a_n)$.
Using the least common multiple
The GCD of two numbers can also be found using the equation $GCD(x, y) \cdot LCM(x, y) = x \cdot y$, where $LCM(x,y)$ is the least common multiple of $x$ and $y$.
The binary method | {"url":"https://artofproblemsolving.com/wiki/index.php/Greatest_common_factor","timestamp":"2024-11-06T17:06:08Z","content_type":"text/html","content_length":"40491","record_id":"<urn:uuid:c512661c-b741-4667-9f52-56031bba60c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00522.warc.gz"} |
CS111 Assignment 1 solved
## Problem 1 (Warmup)
Write and submit your code in a file called Sum.java. Use the IO module to read inputs. Use IO.outputIntAnswer() to print your answer (refer to the documentation below for usage instructions).
Ask the user for 2 integers. Output the sum of those 2 numbers. Example:
Enter the first number:
Enter the second number:
Result: 2
## Problem 2
Write and submit your code in a file called Poly.java. Use the IO module to read inputs. Use System.out.println() to print your answer.
Write a program that generates a canonical-form, degree-3 (cubic) polynomial given its roots. For example, if the roots are 5, -3, and 2, the polynomial equation is
(x – 5)(x + 3)(x – 2) = 0
The canonical form of the polynomial is therefore
x^3 – 4x^2 – 11x + 30
The above is just text, not code that could appear in a Java program.
Ask the user for three roots (integers). Output the canonical form of the polynomial with those roots (as text) using System.out.println(), as in the following example:
java Poly
Enter the first root:
Enter the second root:
Enter the third root:
The polynomial is:
x^3 – 4x^2 – 11x + 30
## Problem 3
Write and submit your code in a file called Intersect.java. Use the IO module to read inputs. Use System.out.println() to print your answer.
Write a program that calculates the intersection between 2 equations:
a degree-2 (quadratic) polynomial i.e.
y = dx^2 + fx + g
where d, f, and g are constants
and a degree-1 (linear) equation i.e.
y = mx + b
where m is the slope and b is a constant
The above is just text, not code that could appear in a Java program.
Ask the user for the constant values in each equation. Output the intersection(s) as ordered pair(s) (x,y), or “none” if none exists. Below is an example run.
java Intersect
Enter the constant d:
Enter the constant f:
Enter the constant g:
Enter the constant m:
Enter the constant b:
The intersection(s) is/are: | {"url":"https://codeshive.com/questions-and-answers/cs111-assignment-1-solved/","timestamp":"2024-11-05T02:54:09Z","content_type":"text/html","content_length":"98759","record_id":"<urn:uuid:43508a9d-a0c8-4e87-a9c4-2f357d8fa94e>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00472.warc.gz"} |
Norbert Weiner Formulates Communication Theory as a Statistical Problem: The Wiener Filter
In 1942, having collaborated with engineer Julian Bigelow, mathematician Norbert Wiener published, as a classified document from MIT, The Extrapolation, Interpretation and Smoothing of Stationery
Time Series. According to Claude Shannon, this work contained “the first clear-cut formulation of communication theory as a statistical problem, the study of operations on time series.” The book was
first commercially published by MIT Press and John Wiley and Sons in 1949.
"Some predict that Norbert Wiener will be remembered for his Extrapolation long after Cybernetics is forgotten. Indeed, few computer science students would know today what cybernetics is all about,
while every communication student knows what Wiener's filter is. The original work was circulated as a classified memorandum in 1942, because it was connected with sensitive wartime efforts to
improve radar communication. This book became the basis for modern communication theory, by a scientist considered one of the founders of the field of artifical intelligence. Combining ideas from
statistics and time-series analysis, Wiener used Gauss's method of shaping the characteristic of a detector to allow for the maximal recognition of signals in the presence of noise. This method came
to be known as the "Wiener filter." " (MIT Press). | {"url":"https://historyofinformation.com/detail.php?entryid=775","timestamp":"2024-11-10T19:00:22Z","content_type":"text/html","content_length":"14263","record_id":"<urn:uuid:381e666d-1dbb-47b2-85d1-6660cd1c7a0b>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00268.warc.gz"} |
ogdf::Dijkstra< T, H >
void call (const Graph &G, const EdgeArray< T > &weight, const List< node > &sources, NodeArray< edge > &predecessor, NodeArray< T > &distance, bool directed=false, bool arcsReversed=false, node
target=nullptr, T maxLength=std::numeric_limits< T >::max())
Calculates, based on the graph G with corresponding edge costs and source nodes, the shortest paths and distances to all other nodes by Dijkstra's algorithm. More...
void call (const Graph &G, const EdgeArray< T > &weight, const node s, NodeArray< edge > &predecessor, NodeArray< T > &distance, bool directed=false, bool arcsReversed=false, node target=nullptr, T
maxLength=std::numeric_limits< T >::max())
Calculates, based on the graph G with corresponding edge costs and a source node s, the shortest paths and distances to all other nodes by Dijkstra's algorithm. More...
void callBound (const Graph &G, const EdgeArray< T > &weight, const List< node > &sources, NodeArray< edge > &predecessor, NodeArray< T > &distance, bool directed, bool arcsReversed, node target, T
maxLength=std::numeric_limits< T >::max())
Calculates, based on the graph G with corresponding edge costs and source nodes, the shortest paths and distances to all other nodes by Dijkstra's algorithm. More...
void callBound (const Graph &G, const EdgeArray< T > &weight, node s, NodeArray< edge > &predecessor, NodeArray< T > &distance, bool directed, bool arcsReversed, node target, T maxLength=
std::numeric_limits< T >::max())
Calculates, based on the graph G with corresponding edge costs and a source node s, the shortest paths and distances to all other nodes by Dijkstra's algorithm. More...
void callUnbound (const Graph &G, const EdgeArray< T > &weight, const List< node > &sources, NodeArray< edge > &predecessor, NodeArray< T > &distance, bool directed=false, bool arcsReversed=false)
Calculates, based on the graph G with corresponding edge costs and source nodes, the shortest paths and distances to all other nodes by Dijkstra's algorithm. More...
void callUnbound (const Graph &G, const EdgeArray< T > &weight, node s, NodeArray< edge > &predecessor, NodeArray< T > &distance, bool directed=false, bool arcsReversed=false)
Calculates, based on the graph G with corresponding edge costs and a source node s, the shortest paths and distances to all other nodes by Dijkstra's algorithm. More...
template<typename T, template< typename P, class C > class H = PairingHeap>
class ogdf::Dijkstra< T, H >
Dijkstra's single source shortest path algorithm.
This class implements Dijkstra's algorithm for computing single source shortest path in (undirected or directed) graphs with proper, positive edge weights. It returns a predecessor array as well as
the shortest distances from the source node(s) to all others. It optionally supports early termination if only the shortest path to a specific node is required, or the maximum path length is to be
Definition at line 60 of file Dijkstra.h.
template<typename T , template< typename P, class C > class H = PairingHeap>
void ogdf::Dijkstra< T, H >::call ( const Graph & G,
const EdgeArray< T > & weight,
const List< node > & sources,
NodeArray< edge > & predecessor,
NodeArray< T > & distance, inline
bool directed = false,
bool arcsReversed = false,
node target = nullptr,
T maxLength = std::numeric_limits<T>::max()
Calculates, based on the graph G with corresponding edge costs and source nodes, the shortest paths and distances to all other nodes by Dijkstra's algorithm.
G The original input graph
weight The edge weights
sources A list of source nodes
predecessor The resulting predecessor relation
distance The resulting distances to all other nodes
directed True iff G should be interpreted as a directed graph
arcsReversed True if the arcs should be followed in reverse. It has only an effect when setting directed to true
target A target node. Terminate once the shortest path to this node is found
maxLength Upper bound on path length
If no target or maximum distance is given, use the unbound algorithm that runs faster on most instances. On some types of instances (especially sparse ones) the bound algorithm tends to run
faster. To force its usage, use the callBound method directly.
See also
callBound(const Graph&, const EdgeArray<T>&, const List<node>&, NodeArray<edge>&, NodeArray<T>&, bool, node, T)
Definition at line 253 of file Dijkstra.h.
template<typename T , template< typename P, class C > class H = PairingHeap>
void ogdf::Dijkstra< T, H >::call ( const Graph & G,
const EdgeArray< T > & weight,
const node s,
NodeArray< edge > & predecessor,
NodeArray< T > & distance, inline
bool directed = false,
bool arcsReversed = false,
node target = nullptr,
T maxLength = std::numeric_limits<T>::max()
Calculates, based on the graph G with corresponding edge costs and a source node s, the shortest paths and distances to all other nodes by Dijkstra's algorithm.
G The original input graph
weight The edge weights
s The source node
predecessor The resulting predecessor relation
distance The resulting distances to all other nodes
directed True iff G should be interpreted as a directed graph
arcsReversed True if the arcs should be followed in reverse. It has only an effect when setting directed to true
target A target node. Terminate once the shortest path to this node is found
maxLength Upper bound on path length
If no target or maximum distance is given, use the unbound algorithm that runs faster on most instances. On some types of instances (especially sparse ones) the bound algorithm tends to run
faster. To force its usage, use the callBound method directly.
See also
callBound(const Graph&, const EdgeArray<T>&, node, NodeArray<edge>&, NodeArray<T>&, bool, node, T)
Definition at line 276 of file Dijkstra.h.
template<typename T , template< typename P, class C > class H = PairingHeap>
void ogdf::Dijkstra< T, H >::callBound ( const Graph & G,
const EdgeArray< T > & weight,
const List< node > & sources,
NodeArray< edge > & predecessor,
NodeArray< T > & distance, inline
bool directed,
bool arcsReversed,
node target,
T maxLength = std::numeric_limits<T>::max()
Calculates, based on the graph G with corresponding edge costs and source nodes, the shortest paths and distances to all other nodes by Dijkstra's algorithm.
Allows to specify a target node and maximum distance, after reaching which the algorithm will terminate early.
This implementation is different from the implementation of callUnbound() as runtime tests have shown the additional checks to increase run time for the basic use case.
G The original input graph
weight The edge weights
sources A list of source nodes
predecessor The resulting predecessor relation
distance The resulting distances to all other nodes
directed True iff G should be interpreted as a directed graph
arcsReversed True if the arcs should be followed in reverse. It has only an effect when setting directed to true
target A target node. Terminate once the shortest path to this node is found
maxLength Upper bound on path length
Definition at line 145 of file Dijkstra.h.
template<typename T , template< typename P, class C > class H = PairingHeap>
void ogdf::Dijkstra< T, H >::callBound ( const Graph & G,
const EdgeArray< T > & weight,
node s,
NodeArray< edge > & predecessor,
NodeArray< T > & distance, inline
bool directed,
bool arcsReversed,
node target,
T maxLength = std::numeric_limits<T>::max()
Calculates, based on the graph G with corresponding edge costs and a source node s, the shortest paths and distances to all other nodes by Dijkstra's algorithm.
Allows to specify a target node and maximum distance, after reaching which the algorithm will terminate early.
This implementation is different from the implementation of callUnbound() as runtime tests have shown the additional checks to increase run time for the basic use case.
G The original input graph
weight The edge weights
s The source node
predecessor The resulting predecessor relation
distance The resulting distances to all other nodes
directed True iff G should be interpreted as a directed graph
arcsReversed True if the arcs should be followed in reverse. It has only an effect when setting directed to true
target A target node. Terminate once the shortest path to this node is found
maxLength Upper bound on path length
Definition at line 233 of file Dijkstra.h.
template<typename T , template< typename P, class C > class H = PairingHeap>
void ogdf::Dijkstra< T, H >::callUnbound ( const Graph & G,
const EdgeArray< T > & weight,
const List< node > & sources,
NodeArray< edge > & predecessor, inline
NodeArray< T > & distance,
bool directed = false,
bool arcsReversed = false
Calculates, based on the graph G with corresponding edge costs and source nodes, the shortest paths and distances to all other nodes by Dijkstra's algorithm.
G The original input graph
weight The edge weights
sources A list of source nodes
predecessor The resulting predecessor relation
distance The resulting distances to all other nodes
directed True iff G should be interpreted as a directed graph
arcsReversed True if the arcs should be followed in reverse. It has only an effect when setting directed to true
Definition at line 77 of file Dijkstra.h.
template<typename T , template< typename P, class C > class H = PairingHeap>
void ogdf::Dijkstra< T, H >::callUnbound ( const Graph & G,
const EdgeArray< T > & weight,
node s,
NodeArray< edge > & predecessor, inline
NodeArray< T > & distance,
bool directed = false,
bool arcsReversed = false
Calculates, based on the graph G with corresponding edge costs and a source node s, the shortest paths and distances to all other nodes by Dijkstra's algorithm.
G The original input graph
weight The edge weights
s The source node
predecessor The resulting predecessor relation
distance The resulting distances to all other nodes
directed True iff G should be interpreted as a directed graph
arcsReversed True if the arcs should be followed in reverse. It has only an effect when setting directed to true
Definition at line 214 of file Dijkstra.h. | {"url":"https://ogdf.netlify.app/classogdf_1_1_dijkstra","timestamp":"2024-11-14T13:59:31Z","content_type":"application/xhtml+xml","content_length":"49096","record_id":"<urn:uuid:f937fafd-92cd-4b2f-92d6-eb520c790d41>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00130.warc.gz"} |
Arcflow formulation
We consider the two-bar charts packing (2-BCPP), a recent combinatorial optimization problem whose aim is to pack a set of one-dimensional items into the minimum number of bins. As opposed to the
well-known bin packing problem, pairs of items are grouped to form bar charts, and a solution is only feasible if the first and … Read more | {"url":"https://optimization-online.org/tag/arcflow-formulation/","timestamp":"2024-11-03T15:21:09Z","content_type":"text/html","content_length":"82674","record_id":"<urn:uuid:57b8a2a4-0e89-4c11-b057-bcfa95bb5bd1>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00773.warc.gz"} |
A sharp NMF result with applications in network modeling
A sharp NMF result with applications in network modeling
Part of Advances in Neural Information Processing Systems 35 (NeurIPS 2022) Main Conference Track
Jiashun Jin
Given an $n \times n$ non-negative rank-$K$ matrix $\Omega$ where $m$ eigenvalues are negative, when can we write $\Omega = Z P Z'$ for non-negative matrices $Z \in \mathbb{R}^{n, K}$ and $P \in \
mathbb{R}^{K, K}$? While most existing works focused on the case of $m = 0$, our primary interest is on the case of general $m$. With new proof ideas we develop, we present sharp results on when the
NMF problem is solvable, which significantly extend existing results on this topic. The NMF problem is partially motivated by applications in network modeling. For a network with $K$ communities,
rank-$K$ models are popular, with many proposals. The DCMM model is a recent rank-$K$ model which is especially useful and interpretable in practice. To enjoy such properties, it is of interest to
study when a rank-$K$ model can be rewritten as a DCMM model. Using our NMF results, we show that for a rank-$K$ model with parameters in the most interesting range, we can always rewrite it as a
DCMM model.
Do not remove: This comment is monitored to verify that the site is working properly | {"url":"https://proceedings.nips.cc/paper_files/paper/2022/hash/764651d0887f997fc1e40ff97c8b12e6-Abstract-Conference.html","timestamp":"2024-11-14T21:44:47Z","content_type":"text/html","content_length":"8716","record_id":"<urn:uuid:f79fd8e3-cd28-4883-bd9d-9dd1916eb6cb>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00894.warc.gz"} |
Which Frequencies do CNNs Need? Emergent Bottleneck Structure in Feature Learning
Which Frequencies do CNNs Need? Emergent Bottleneck Structure in Feature Learning
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:52779-52800, 2024.
We describe the emergence of a Convolution Bottleneck (CBN) structure in CNNs, where the network uses its first few layers to transform the input representation into a representation that is
supported only along a few frequencies and channels, before using the last few layers to map back to the outputs. We define the CBN rank, which describes the number and type of frequencies that are
kept inside the bottleneck, and partially prove that the parameter norm required to represent a function $f$ scales as depth times the CBN rank $f$. We also show that the parameter norm depends at
next order on the regularity of $f$. We show that any network with almost optimal parameter norm will exhibit a CBN structure in both the weights and - under the assumption that the network is stable
under large learning rate - the activations, which motivates the common practice of down-sampling; and we verify that the CBN results still hold with down-sampling. Finally we use the CBN structure
to interpret the functions learned by CNNs on a number of tasks.
Cite this Paper
Related Material | {"url":"https://proceedings.mlr.press/v235/wen24d.html","timestamp":"2024-11-15T04:23:35Z","content_type":"text/html","content_length":"16050","record_id":"<urn:uuid:789af08d-6faa-42d7-9a6f-fcac7b4482dd>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00780.warc.gz"} |
Linux-Blog – Dr. Mönchmeyer / anracon
A simple CNN for the MNIST dataset – VII – outline of steps to visualize image patterns which trigger filter maps
During my present article series on a simple CNN we have seen how we set up and train such an artificial neural network with the help of Keras.
A simple CNN for the MNIST dataset – VI – classification by activation patterns and the role of the CNN’s MLP part
A simple CNN for the MNIST dataset – V – about the difference of activation patterns and features
A simple CNN for the MNIST dataset – IV – Visualizing the activation output of convolutional layers and maps
A simple CNN for the MNIST dataset – III – inclusion of a learning-rate scheduler, momentum and a L2-regularizer
A simple CNN for the MNIST datasets – II – building the CNN with Keras and a first test
A simple CNN for the MNIST datasets – I – CNN basics
Lately we managed to visualize the activations of the maps which constitute the convolutional layers of a CNN {Conv layer]. A Conv layer in a CNN basically is a collection of maps. The chain of
convolutions produces characteristic patterns across the low dimensional maps of the last (i.e. the deepest) convolutional layer – in our case of the 3rd layer “Conv2D_3”. Such patterns obviously
improve the classification of images with respect to their contents significantly in comparison to pure MLPs. I called a node activation pattern within or across CNN maps a FCP (see the fifth article
of this series).
The map activations of the last convolutional layer are actually evaluated by a MLP, whose dense layer we embedded in our CNN. In the last article we therefore also visualized the activation values
of the nodes within the first dense MLP-layer. We got some indications that map activation patterns, i.e. FCPs, for different image classes indeed create significantly different patterns within the
MLP – even when the human eye does not directly see the decisive difference in the FCPs in problematic and confusing cases of input images.
In so far the effect of the transformation cascade in the convolutional parts of a CNN is somewhat comparable to the positive effect of a cluster analysis of MNIST images ahead of a MLP
classification. Both approaches correspond to a projection of the input data into lower dimensional representation spaces and provide clearer classification patterns to the MLP. However, convolutions
do a far better job to produce distinguished patterns for a class of images than a simple cluster analysis. The assumed reason is that chained convolutions somehow identify characteristic patterns
within the input images themselves.
Is there a relation between a FCP and a a pattern in the pixel distribution of the input image?
But so far, we did not get any clear idea about the relation of FCP-patterns with pixel patterns in the original image. In other words: We have no clue about what different maps react to in terms of
characteristic patterns in the input images. Actually, we do not even have a proof that a specific map – or more precisely the activation of a specific map – is triggered by some kind of distinct
pattern in the value distribution for the original image pixels.
I call an original pattern to which a CNN map strongly reacts to an OIP; an OIP thus represents a certain geometrical pixel constellation in the input image which activates neurons in a specific map
very strongly. Not more, not less. Note that an OIP therefore represents an idealized pixel constellation – a pattern which at best is free of any disturbances which might reduce the activation of a
specific map. Can we construct an image with just the required OIP pixel constellation to trigger a map optimally? Yes, we can – at least approximately.
In the present article I shall outline the required steps which will enable us to visualize OIPs later on. In my opinion this is an important step to understand the abilities of CNNs a bit better. In
particular it helps to clarify whether and in how far the term “feature detection” is appropriate. In our case we look out for primitive patterns in the multitude of MNIST images of handwritten
digits. Handwritten digits are interesting objects regarding basic patterns – especially as we humans have some very clear abstract and constructive concepts in mind when we speak about basic
primitive elements of digit notations – namely line and bow segments which get arranged in specific ways to denote a digit.
At the end of this article we shall have a first look at some OIP patterns which trigger a few chosen individual maps of the third convolutional layer of our CNN. In the next article I shall explain
required basic code elements to create such OIP pictures. Subsequent articles will refine and extend our methods towards a more systematic analysis.
Questions and objectives
We shall try to answer a series of questions to approach the subject of OIPs and features:
• How can Keras help us to find and visualize an OIP which provokes a maximum average reaction of a map?
• How well is the “maximum” defined with respect to input data of our visualization method?
• Do we recognize sub-patterns in such OIPs?
• How do the OIPs – if there are any – reflect a translational invariance of complex, composed patterns?
• What does a maximum activation of an individual node of a map mean in terms of an input pattern?
What do I mean by “maximum average reaction“? A specific map of a CNN corresponds to a 2-dim array of “neurons” whose activation functions produce some output. The basic idea is that we want to
achieve a maximum average value of this output by systematically optimizing initially random input image data until, hopefully, a pattern emerges.
Basic strategy to visualize an OIP pattern
In a way we shall try to create order out of chaos: We want to systematically modify an initial random distribution of pixel values until we reach a maximum activation of the chosen map. We already
know how to systematically approach a minimum of a function depending on a multidimensional arrangement of parameters. We apply the “gradient descent” method to a hyperplane created by a suitable
loss-function. Considering the basic principles of “gradient descent” we may safely assume that a slightly modified gradient guided approach will also work for maxima. This in turn means:
We must define a map-specific “loss” function which approaches a maximum value for optimum node activation. A suitable simple function could be a sum or average increasing with the activation values
of the map’s nodes. So, in contrast to classification tasks we will have to use a “gradient ascent” method- The basic idea and a respective simple technical method is e.g. described in the book of F.
Chollet (Deep Learning mit Python und Keras”, 2018, mitp Verlag; I only have the German book version, but the original is easy to find).
But what is varied in such an optimization model? Certainly not the weights of the already trained CNN! The variation happens with respect to the input data – the initial pixel values of the input
image are corrected by the gradient values of the loss function.
Next question: What do we choose as a starting point of the optimization process? Answer: Some kind of random distribution of pixel values. The basic hope is that a gradient ascent method searching
for a maximum of a loss function would also “converge“.
Well, here began my first problem: Converge in relation to what exactly? With respect to exactly one input input image or to multiple input images with different initial statistical distributions of
pixel data? With fluctuations defined on different wavelength levels? (Physicists and mathematicians automatically think of a Fourier transformation at this point 🙂 ). This corresponds to the
question whether a maximum found for a certain input image really is a global maximum. Actually, we shall see that the meaning of convergence is a bit fuzzy in our present context and not as well
defined as in the case of a CNN-training.
To discuss fluctuations in statistical patterns at different wavelength is not so far-fetched as it may seem: Already the basic idea that a map reacts to a structured and maybe sub-structured OIP
indicates that pixel correlations or variations on different length scales might play a role in triggering a map. We shall see that some maps do not react to certain “random” patterns at all. And do
not forget that pooling operations induce the analysis of long range patterns by subsequent convolutional filters. The relevant wavelength is roughly doubled by each of our pooling operations! So,
filters at deep convolutional layers may exclude patterns which do not show some long range characteristics.
The simplified approach discussed by Chollet assumes statistical variations on the small length scale of neighboring pixels; he picks a random value for each and every pixel of his initial input
images without any long range correlations. For many maps this approach will work reasonably well and will give us a basic idea about the average pattern or, if you absolutely want to use the
expression, “feature”, which a CNN-map reacts to. But being able to vary the length scale of pixel values of input images will help us to find patterns for sensitive maps, too.
We may not be able to interpret a specific activation pattern within a map; but to see what a map on average and what a single node of a map reacts to certainly would mean some progress in
understanding the relation between OIPs and FCPs.
An example
The question what an OIP is depends on the scales you look at and also where an OIP appears within a real image. To confuse you a bit: Look at he following OIP-picture which triggered a certain map
The upper image was prepared with a plain color map, the lower with some contrast enhancement. I use this two-fold representation also later for other OIP-pictures.
Actually, it is not so clear what elementary pattern our map reacts to. Two parallel line segments with a third one crossing perpendicular at the upper end of the parallel segments?
One reason for being uncertain is that some patterns on a scale of lets say a fourth of the original image may appear at different locations in original images of the same class. If a network really
learned about such reappearance of patterns the result for an optimum OIP may be a superposition of multiple elementary patterns at different locations. Look at the next two OIP pictures for the very
same map – these patterns emerged from a slightly different statistical variation of the input pixel values:
Now, we recognize some elementary structures much better – namely a combination of bows with slightly different curvatures and elongations. Certainly useful to detect “3” digits, but parts of “2”s,
A different version of another map is given here:
Due to the large scale structure over the full height of the input this map is much better suited to detect “9”s at different places.
You see that multiple filters on different spatial resolution levels have to work together in this case to reflect one bow – and the bows elongation gets longer with their position to the right. It
seems that the CNN has learned that bow elements with the given orientation on the left side of original images are smaller and have a different degree of curvature than to the right of a MNIST input
image. So what is the OIP or what is the “feature” here? The superposition of multiple translationally shifted and differently elongated bows? Or just one bow?
Unexpected technical hurdles
I was a bit surprised that I met some technical difficulties along my personal way to answer the questions posed above. The first point is that only a few text book authors seem to discuss the
question at all; F. Chollet being the remarkable exception and most authors in the field, also of articles on the Internet, refer to his ideas and methods. I find this fact interesting as many
authors of introductory books on ANNs just talk about “features” and make strong claims about what “features” are in terms of entities and their detection by CNNs – but they do not provide any code
to verify the almost magic “identification” of conceptual entities as “eyes”, “feathers”, “lips”, etc..
Then there are articles of interested guys, which appear at specialized web sites, as e.g. the really read-worthy contribution of the physicist F. Graetz: https://towardsdatascience.com/
how-to-visualize-convolutional-features-in-40-lines-of-code-70b7d87b0030 on “towardsdatascience.com”. His color images of “features” within CIFAR images are impressive; you really should have a look
at them.
But he as other authors usually take pre-trained nets like VGG16 and special datasets as CIFAR with images of much higher resolution than MNIST images. But I wanted to apply similar methods upon my
own simple CNN and MNIST data. Although an analysis of OIPs of MNIST images will certainly not produce such nice high resolution color pictures as the ones of Graetz, it might be easier to extract
and understand some basic principles out of numerical experiments.
Unfortunately, I found that I could not just follow and copy code snippets of F. Chollet. Partially this had to do with necessary changes Tensorflow 2 enforced in comparison to TF1 which was used by
F. Chollet. Another problem was due to standardized MNIST images my own CNN was trained on. Disregarding the point of standardization during programming prevented convergence during the
identification of OIPs. Another problem occurred with short range random value variations for the input image pixels as a starting point. Choosing independent random values for individual pixels
suppresses long range variations; this in turn often leads to zero gradients for averaged artificial “costs” of maps at high layer levels.
A better suitable variant of Chollet’s code with respect to TF 2 was published by a guy named Mohamed at “https://www.kaggle.com/questions-and-answers/121398“. I try to interpret his line of thinking
and coding in my forthcoming articles – so all credit belongs to him and F. Chollet. Nevertheless, as said, I still had to modify their code elements to take into account special aspects of my own
trained CNN.
Basic outline for later coding
We saw already in previous articles that we can build new models with Keras and TensorFlow 2 [TF2] which connect some input layer with the output of an intermediate layer of an already defined CNN-
or MLP-model. TF2 analyses the respective dependencies and allows for a forward propagation of input tensors to get the activation values ( i.e. the output values of the activation function) at the
intermediate layer of the original model – which now plays the role of an output layer in the new (sub-) model.
However, TF2 can do even more for us: We can define a specific cost function, which depends on the output tensor values of our derived sub-model. TF2 will also (automatically) provide gradient values
for this freshly defined loss function with respect to input values which we want to vary.
The basic steps to construct images which trigger certain maps optimally is the following:
• We construct an initial input image filled with random noise. In the case of MNIST this input image would consist of input values on a 1-dim gray scale. We standardize the input image data as our
CNN has been trained for such images.
• We build a new model based on the layer structure of our original (trained) CNN-model: The new model connects the input-image-tensor at the input layer of the CNN with the output generated of a
specific feature map at some intermediate layer after the forward propagation of the input data.
• We define a new loss function which should show a maximum value for the map output – depending of course on optimized input image data for the chosen specific map.
• We define a suitable (stochastic) gradient ascent method to approach the aspired maximum for corrected input image data.
• We “inform” TF2 about the gradient’s dependencies on certain varying variables to give us proper gradient values. This step is of major importance in Tensorflow environments with activated “eager
execution”. (In contrast to TF1 “eager execution” is the standard setting for TF2.)
• We scale (= normalize) the gradient values to avoid too extreme corrections of the input data.
• We take into account a standardization of the corrected input image data. This will support the overall convergence of our approach.
• We in addition apply some tricks to avoid a over-exaggeration of small scale components (= high frequency components in the sense of a Fourier transform) in the input image data.
Especially the last point was new to me before I read the code of Mohamed at Kaggle. E.g. F. Chollet does not discuss this point in his book. But it is a very clever thought that one should care
about low and high frequency contributions in patterns which trigger maps at deep convolutional layers. Whereas Mohamed discusses the aspect that high frequency components may guide the optimization
process into overall side maxima during gradient ascent, I would in addition say that not offering long range variations already in the statistical input data may lead to a total non-activation of
some maps. Actually, this maybe is an underestimated crucial point in the hunt for patterns which trigger maps – especially when we deal with low resolution input images.
Eager mode requirements
Keras originally provided a function “gradients()” which worked with TF1 graphs and non-eager execution mode. However, T2 executes code in eager mode automatically and therefore we have to use
special functions to control gradients and their dependencies on changing variables (see for a description of “eager execution” https://www.tensorflow.org/guide/eager?hl=en ).
Among other things: TF2 provides a special function to “watch” variables whose variations have an impact on loss functions and gradient values with respect to a defined (new) model. (An internal
analysis by TF2 of the impact of such variations is of course possible because our new sub-model is based on an already given layer structures of the original CNN-model.)
Visualization of some OIP-patterns in MNIST images as appetizers
Enough for today. To raise your appetite for more I present some images of OIPs. I only show patterns triggering maps on the third Conv-layer.
There are simple patterns:
But there are also more complex ones:
A closer look shows that the complexity results from translations and rotations of elementary patterns.
In this article we have outlined steps to build a program which allows the search for OIPs. The reader has noticed that I try to avoid the term “features”. First images of OIPs show that such
patterns may appear a bit different in different parts of original input images. The maps of a CNN seem to take care of this. This is possible, only, if and when pixel correlations are evaluated over
many input images and if thereby variations on larger spatial scales are taken into account. Then we also have images which show unique patterns in specific image regions – i.e. a large scale pattern
without much translational invariance.
We shall look in more detail at such points as soon as we have built suitable Python functions. See the next post
A simple CNN for the MNIST dataset – VIII – filters and features – Python code to visualize patterns which activate a map strongly
A simple CNN for the MNIST dataset – V – about the difference of activation patterns and features
In my last article of my introductory series on “Convolutional Neural Networks” [CNNs] I described how we can visualize the output of different maps at convolutional (or pooling) layers of a CNN.
A simple CNN for the MNIST dataset – IV – Visualizing the activation output of convolutional layers and maps
A simple CNN for the MNIST dataset – III – inclusion of a learning-rate scheduler, momentum and a L2-regularizer
A simple CNN for the MNIST datasets – II – building the CNN with Keras and a first test
A simple CNN for the MNIST datasets – I – CNN basics
We are now well equipped to look a bit closer at the maps of a trained CNN. The output of the last convolutional layer is of course of special interest: It is fed (in the form of a flattened input
vector) into the MLP-part of the CNN for a classification analysis. As an MLP detects “patterns” the question arises whether we actually can see common “patterns” in the visualized maps of different
images belonging to the same class. In our case we shall have a look at the maps of different MNIST images of a handwritten “4”.
Note for my readers, 20.08.2020:
This article has recently been revised and completely rewritten. It required a much more careful description of what we mean by “patterns” and “features” – and what we can say about them when looking
at images of activation outputs on higher convolutional layers. I also postponed a thorough “philosophical” argumentation against a humanized usage of the term “features” to a later article in this
We saw already in the last article that the images of maps get more and more abstract when we move to higher convolutional layers – i.e. layers deeper inside a CNN. At the same time we loose
resolution due to intermediate pooling operations. It is quite obvious that we cannot see much of any original “features” of a handwritten “4” any longer in a (3×3)-map, whose values are produced by
a sequence of complex transformation operations.
Nevertheless people talk about “feature detection” performed by CNNs – and they refer to “features” in a very concrete and descriptive way (e.g. “eyes”, “spectacles”, “bows”). How can this be? What
is the connection of abstract activation patterns in low resolution maps to original “features” of an image? What is meant when CNN experts claim that neurons of higher CNN layers are allegedly able
to “detect features”?
We cannot give a full answer, yet. We still need some more Python programs tools. But, wat we are going to do in this article are three things:
1. Objective 1: I will try to describe the assumed relation between maps and “features”. To start with I shall make a clear distinction between “feature” patterns in input images and patterns in and
across the maps of convolutional layers. The rest of the discussion will remain a bit theoretical; but it will use the fact that convolutions at higher layers combine filtered results in specific
ways to create new maps. For the time being we cannot do more. We shall actually look at visualizations of “features” in forthcoming articles of this series. Promised.
2. Objective 2: We follow three different input images, each representing a “4”, as they get processed from one convolutional layer to the next convolutional layer of our CNN. We shall compare the
resulting outputs of all feature maps at each convolutional layer.
3. Objective 3: We try to identify common “patterns” for our different “4” images across the maps of the highest convolutional layer.
We shall visualize each “map” by an image – reflecting the values calculated by the CNN-filters for all points in each map. Note that an individual value at a map point results from adding up many
weighted values provided by the maps of lower layers and feeding the result into an activation function. We speak of “activation” values or “map activations”. So our 2-nd objective is to follow the
map activations of an input image up to the highest convolutional layer. An interesting question will be if the chain of complex transformation operations leads to visually detectable similarities
across the map outputs for the different images of a “4”.
The eventual classification of a CNN is done by its embedded MLP which analyzes information collected at the last convolutional layer. Regarding this input to the MLP we can make the following
The convolutions and pooling operations project information of relatively large parts of the original image into a representation space of very low dimensionality. Each map on the third layer
provides a 3×3 value tensor, only. However, we combine the points of all (128) maps together in a flattened input vector to the MLP. This input vector consists of more nodes than the original image
Thus the sequence of convolutional and pooling layers in the end transforms the original images into another representation space of somewhat higher dimensionality (9×128 vs. 28×28). This
transformation is associated with the hope that in the new representation space a MLP may find patterns which allow for a better classification of the original images than a direct analysis of the
image data. This explains objective 3: We try to play the MLPs role by literally looking at the eventual map activations. We try to find out which patterns are representative for a “4” by comparing
the activations of different “4” images of the MNIST dataset.
Enumbering the layers
To distinguish a higher Convolutional [Conv] or Pooling [Pool] layer from a lower one we give them a number “Conv_N” or “Pool_N”.
Our CNN has a sequence of
• Conv_1 (32 26×26 maps filtering the input image),
• Pool_1 (32 13×13 maps with half the resolution due to max-pooling),
• Conv_2 (64 11×11 maps filtering combined maps of Pool_1),
• Pool_2 (64 5×5 maps with half the resolution due to max-pooling),
• Conv_3 (128 3×3 maps filtering combined maps of Pool_2).
Patterns in maps?
We have seen already in the last article that the “patterns” which are displayed in a map of a higher layer Conv_N, with N ≥ 2, are rather abstract ones. The images of the maps at Conv_3 do not
reflect figurative elements or geometrical patterns of the input images any more – at least not in a directly visible way. It does not help that the activations are probably triggered by some
characteristic pixel patterns in the original images.
The convolutions and the pooling operation transform the original image information into more and more abstract representation spaces of shrinking dimensionality and resolution. This is due to the
fact that the activation of a point in a map on a layer Conv_(N+1) results
• from a specific combination of multiple maps of a layer Conv_N or Pool_N
• and from a loss of resolution due to intermediate pooling.
It is not possible to directly guess in what way active points or activated areas within
a certain map at the third convolutional layer relate to or how they depend on “original and specific patterns in the input image”. If you do not believe me: Well, just look at the maps of the 3rd
convolutional layer presented in the last article and tell me: What patterns in the initial image did these maps react to? Without some sophisticated numerical experiments you won’t be able to figure
that out.
Patterns in the input image vs. patterns within and across maps
The above remarks indicate already that “patterns” may occur at different levels of consideration and abstraction. We talk about patterns in the input image and patterns within as well as across the
maps of convolutional (or pooling) layers. To avoid confusion I already now want to make the following distinction:
• (Original) input patterns [OIP]: When I speak of (original) “input patterns” I mean patterns or figurative elements in the input image. In more mathematical terms I mean patterns within the input
image which correspond to a kind of fixed and strong correlation between the values of pixels distributed over a sufficiently well defined geometrical area with a certain shape. Examples could be
line-like elements, bow segments, two connected circles or combined rectangles. But OIPs may be of a much more complex and abstract kind and consist of strange sub-features – and they may not
reflect a real world entity or a combination of such entities. An OIP may reside at one or multiple locations in different input images.
• Filter correlation patterns [FCP]: A CNN produces maps by filtering input data (Conv level 1) or by filtering maps of a lower layer and combining the results. By doing so a higher layer may
detect patterns in the filter results of a lower layer. I call a pattern across the maps of a convolutional or pooling layer Conv_N or Pool_N as seen by Conv_(N+1) a FCP.
Note: Because a 3×3 filter for a map of Conv_(N+1) has fixed parameters per map of the previous layer Conv_N or Pool_N, it combines multiple maps (filters) of Conv_N in a specific, unique way.
Anybody who ever worked with image processing and filters knows that combining basic filters may lead to the display of weirdly looking, combined information residing in complex regions on the
original image. E.g., a certain combination of filters may emphasize diagonal lines or bows with some distance in between and suppress all other features. Therefore, it is at least plausible that a
map of a higher convolutional layer can be translated back to an OIP. Meaning:
A high activation of certain or multiple points inside a map on Conv_3 may reflect some typical OIP pattern in the input image.
But: At the moment we have no direct proof for such an idea. And it is not at all obvious what kind of OIP pattern this may be for a distinct map – and whether it can directly be described in terms
of basic geometrical elements of a figurative number representation in the MNIST case. By just looking at the maps of a layer and their activated points we do not get any clue about this.
If, however, activated maps somehow really correspond to OIPs then a FCP over multiple maps may be associated with a combination of distinct OIPs in an input image.
What are “features” then?
In many textbooks maps are also called “feature maps“. As far I understand it the authors call a “feature” what I called an OIP above. By talking about a “feature” the authors most often refer to a
pattern which a CNN somehow detects or identifies in the input images.
Typical examples of “features” text-book authors often discuss and even use in illustrations are very concrete: ears, eyes, feathers, wings, a mustache, leaves, wheels, sun-glasses … I.e., a lot of
authors typically name features which human beings identify as physical entities or as entities, for which we have clear conceptual ideas in our mind. I think such examples trigger ideas about CNNs
which are too far-fetched and which “humanize” stupid algorithmic processes.
The arguments in favor of the detection of features in the sense of conceptual entities are typically a bit nebulous – to say the least. E.g. in a relatively new book on “Generative Deep Learning”
you see a series of CNN neuron layers associated with rather dubious and unclear images of triangles etc. and at the last convolutional layer we suddenly see pretty clear sketches of a mustache, a
certain hairdress, eyes, lips, a shirt, an ear .. “. The related text goes like follows (I retranslated the text from the German version of the book): “Layer 1 consists of neurons which activate
themselves stronger, when they recognize certain elementary and basic features in the input image, e.g. borders. The output of these neurons is then forwarded to the neurons of layer 2 which can use
this information to detect more complex features – and so on across the following layers.” Yeah, “neurons activate themselves” as they “recognize” features – and suddenly the neurons at a high enough
layer see a “spectacle”. 🙁
I think it would probably be more correct to say the following:
The activation of a map of a high convolutional layer may indicate the appearance of some kind of (complex) pattern or a sequence of patterns within an input image, for which a specific filter
combination produces relatively high values in a low dimensional output space.
Note: At our level of analyzing CNNs even this carefully formulated idea is speculation. Which we will have to prove somehow … Where we stand right now, we are unfortunately not yet ready to identify
OIPs or repeated OIP sequences associated with maps. This will be the topic of forthcoming articles.
It is indeed an interesting question whether a trained CNN “detects” patterns in the sense of entities with an underlying “concept”. I would say: Certainly not. At least not pure CNNs. I think, we
should be very careful with the use of the term “feature”. Based on the filtering convolutions perform we might say:
A “feature” (hopefully) is a pattern in the sense of defined geometrical pixel correlation in an image.
Not more, not less. Such a “feature” may or may not correspond to entities, which a human being could identify and for which he or she has a concept for. A feature is just a pixel correlation whose
appearance triggers output neurons in high level maps.
By the way there are 2 more points regarding the idea of feature detection:
• A feature or OIP may be located at different places in different images of something like a “5”. Due to different sizes of the depicted “5” and translational effects. So keep in mind that if maps
do indeed relate to features it has to be explained how convolutional filtering can account for any translational invariance of the “detection” of a pattern in an image.
• The concrete examples given for “features” by many authors imply that the features are more or less the same for two differently trained CNNs. Well, regarding the point that training corresponds
to finding a minimum on a rather complex multidimensional hyperplane this raises the question how well defined such a (global) minimum really is and whether it or other valid side minima are
Keep these points in mind until we come back to related experiments in further articles.
From “features” to FCPs on the last Conv-layer?
However and independent of how a CNN really reacts to OIPs or “features”, we should not forget the following:
In the end a CNN – more precisely its embedded MLP – reacts to FCPs on the last convolutional level. In our CNN an FCP on the third convolutional layer with specific active points across 128 (3×3)
-maps obviously can obviously tell the MLP something about the class an input image belongs to: We have proven already that the MLP part of our simple CNN guesses the class the original image belongs
to with a surprisingly high accuracy. And by construction it obviously does so by just analyzing the 128 (3×3)-activation values of the third layer – arranged into a flattened vector.
From a classification point of view it, therefore, seems to be legitimate to look out for any FCP across the maps on Conv_3. As we can visualize the maps it is reasonable to literally look for common
activation patterns which different images of handwritten “4”s may trigger on the maps of the last convolutional level. The basic idea behind this experimental step is:
OIPs which are typical for images of a “4” trigger and activate certain maps or points within certain maps. Across all maps we then may see a characteristic FCP for a “4”, which not only a MLP but
also we intelligent humans could identify.
Or: Multiple characteristic features in images of a “4” may trigger characteristic FCPs which in turn can be used indicators of a class an image belongs to by an MLP. Well, let us see how far we get
with this kind of theory.
Levels of “abstractions”
Let us take a MNIST image which represents something which a European would consider to be a clear representation of a “4”.
In the second image I used the “jet”-color map; i.e. dark blue indicates a low intensity value while colors from light blue to green to yellow and red indicate growing intensity values.
The first conv2D-layer (“Conv2d_1”) produces the following 32 maps of my chosen “4”-image after training:
We see that the filters, which were established during training emphasize general contours but also focus on certain image regions. However, the original “4” is still clearly visible on very many
maps as the convolution does not yet reduce resolution too much.
By the way: When looking at the maps the first time I found it surprising that the application of a simple linear 3×3 filter with stride 1 could emphasize an overall oval region and suppress the
pixels which formed the “4” inside of this region. A closer look revealed however that the oval region existed already in the original image data. It was emphasized by an inversion of the pixel
values …
The second Conv2D-layer already combines information of larger areas of the image – as a max (!) pooling layer was applied before. We loose resolution here. But there is a gain, too: the next
convolution can filter (already filtered) information over larger areas of the original image.
But note: In other types of more advanced and modern CNNs pooling only is involved after two or more successive convolutions have happened. The direct succession of convolutions corresponds to a
direct and unique combination of filters at the same level of resolution.
The 2nd convolution
As we use 64 convolutional maps on the 2nd layer level we allow for a multitude of different new convolutions. It is to be understood that each new map at the 2nd cConv layer is the result of a
special unique combination of filtered information of all 32 previous maps (of Pool_1). Each of the previous 32 maps contributes through a specific unique filter and respective convolution operation
to a single specific map at layer 2. Remember that we get 3×3 x 32 x 64 parameters for connecting the maps of Pool_1 to maps of Conv_2. It is this unique combination of already filtered results which
enriches the analysis of the original image for more complex patterns than just the ones emphasized by the first convolutional filters.
As the max-condition of the pooling layer was applied first and because larger areas are now analyzed we are not too astonished to see that the filters dissolve the original “4”-shape and indicate
more general geometrical patterns – which actually reflect specific correlations of map patterns on layer Conv_1.
I find it interesting that our “4” triggers more horizontally activations within some maps on this already abstract level than vertical ones. One should not confuse these patterns with horizontal
patterns in the original image. The relation of original patterns with these activations is already much more complex.
The third convolutional layer applies filters which now cover almost the full original image and combine and mix at the same time information from the already rather abstract results of layer 2 – and
of all the 64 maps there in parallel.
We again see a dominance of horizontal patterns. We see clearly that on this level any reference to something like an arrangement of parallel vertical lines crossed by a horizontal line is completely
lost. Instead the CNN has transformed the original distribution of black (dark grey) pixels into multiple abstract configuration spaces with 2 axes, which only coarsely reflecting the original image
area – namely by 3×3 maps; i.e. spaces with a very poor resolution.
What we see here are “correlations” of filtered and transformed original pixel clusters over relatively large areas. But no constructive concept of certain line arrangements.
Now, if this were the level of “FCP-patterns” which the MLP-part of the CNN uses to determine that we have a “4” then we would bet that such abstract patterns (active points on 9×9 grids) appear in a
similar way on the maps of the 3rd Conv layer for other MNIST images of a “4”, too.
Well, how similar do different map representations of “4”s look like on the 3rd Conv2D-layer?
What makes a four a four in the eyes of the CNN?
The last question corresponds to the question of what activation outputs of “4”s really have in common. Let us take 3 different images of a “4”:
The same with the “jet”-color-map:
Already with our eyes we see that there are similarities but also quite a lot of differences.
Different “4”-representations on the 2nd Conv-layer
Below we see comparison of the 64 maps on the 2nd Conv-layer for our three “4”-images.
If you move your head backwards and ignore details you see that certain maps are not filled in all three map-pictures. Unfortunately, this is no common feature of “4”-representations. Below you see
images of the activation of a “1” and a “2”. There the same maps are not activated at all.
We also see that on this level it is still important which points within a map are activated – and not which map on average. The original shape of the underlying number is reflected in the maps’
Now, regarding the “4”-representations you may say: Well, I still recognize some common line patterns – e.g. parallel lines in a certain 75 degree angle on the 11×11 grids. Yes, but these lines are
almost dissolved by the next pooling step:
Consider in addition that the next (3rd) convolution combines 3×3-data of all of the displayed 5×5-maps. Then, probably, we can hardly speak of a concept of abstract line configurations any more …
“4”-representations on the third Conv-layer
Below you find the activation outputs on the 3rd Conv2D-layer for our three different “4”-images:
When we look at details we see that prominent “features” in one map of a specific 4-image do NOT appear in a fully comparable way in the eventual convolutional maps for another image of a “4”. Some
of the maps (i.e. filters after 4 transformations) produce really different results for our three images.
But there are common elements, too: I have marked only some of the points which show a significant intensity in all of the maps. But does this mean these individual common points are decisive for a
classification of a “4”? We cannot be sure about it – probably it is their combination which is relevant.
So, what we ended up with is that we find some common points or some common point-relations in a few of the 128 “3×3”-maps of our three images of handwritten “4”s.
But how does this compare with maps of images of other digits? Well, look at he maps on the 3rd layer for images of a “1” and a “2” respectively:
On the 3rd layer it becomes more important which maps are not activated at all. But still the activation patterns within certain maps seem to be of importance for an eventual classification.
The maps of a CNN are created by an effective and guided optimization process. The results indicate the eventual detection of rather abstract patterns within and across filter maps on higher
convolutional layers.
But these patterns (FCP-patterns) should not be confused with figurative elements or “features” in the original input images. Activation patterns at best vaguely remind of the original image
features. At our level of analysis of a CNN we can only speculate about some correspondence of map activations with original features or patterns in an input image.
But it seems pretty clear that patterns in or across maps do not indicate any kind of constructive concept which describes how to build a “4” from underlying more elementary features in the sense of
combine-able independent entities. There is no sign of conceptual constructive idea of how to denote a “4”. At least not in pure CNNs … Things may be a bit different in convolutional “autoencoders”
(combinations of convolutional encoders and decoders), but this is another story we will come back to in this blog. Right now we would say that abstract (FCP-) patterns in maps of higher
convolutional layers result from intricate filter combinations. These filters may react to certain patterns in an input image – but whether these patterns correspond to entities a human being would
use to write down and thereby construct a “4” or an “8” is questionable.
We saw that the abstract information maps at the third layer of our CNN do show some common elements between the images belonging to the same class – and delicate differences with respect to
activations resulting from images of other classes. However, the differences reside in details and the situation remains complicated. In the end the MLP-part of a CNN still has a lot of work to do.
It must perform its classification task based on the correlation or anti-correlation of “point”-like elements in a multitude of maps – and probably even based on the activation level (i.e. output
numbers) at these points.
This is seemingly very different from a conscious consideration process and weighing of alternatives which a human brain performs when it looks at sketches of numbers. When in doubt our brain tries
to find traces consistent with a construction process defined for writing down a “4”, i.e. signs of a certain arrangement of straight and curved lines. A human brain, thus, would refer to
arrangements of line elements, bows or circles – but not to relations of individual points in an extremely coarse and abstract representation space after some mathematical transformations. You may
now argue that we do not need such a process when looking at clear representations of a “4” – we look and just know that its a “4”. I do not doubt that a brain may use maps, too – but I want to point
out that a conscious intelligent thought process and conceptual ideas about entities involve constructive operations and not just a passive application of filters. Even from this extremely
simplifying point of view CNNs are stupid though efficient algorithms. And authors writing about “features” should avoid any kind of a humanized interpretation.
In the next article
A simple CNN for the MNIST dataset – VI – classification by activation patterns and the role of the CNN’s MLP part
we shall look at the whole procedure again, but then we compare common elements of a “4” with those of a “9” on the 3rd convolutional layer. Then the key question will be: ” What do “4”s have in
common on the last convolutional maps which corresponding activations of “9”s do not show – and vice versa.
This will become especially interesting in cases for which a distinction was difficult for pure MLPs. You remember the confusion matrix for the MNIST dataset? See:
A simple Python program for an ANN to cover the MNIST dataset – XI – confusion matrix
We saw at that point in time that pure MLPs had some difficulties to distinct badly written “4”s from “9s”. We will see that the better distinction abilities of CNNs in the end depend on very few
point like elements of the eventual activation on the last layer before the MLP.
Further articles in this series
A simple CNN for the MNIST dataset – VII – outline of steps to visualize image patterns which trigger filter maps
A simple CNN for the MNIST dataset – VI – classification by activation patterns and the role of the CNN’s MLP part | {"url":"https://linux-blog.anracom.com/tag/fcp/","timestamp":"2024-11-06T18:52:43Z","content_type":"text/html","content_length":"168855","record_id":"<urn:uuid:968c452c-0481-408c-8e93-b98f008a6f02>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00251.warc.gz"} |
Sampling Distributions for Sample Means - Knowunity
Sampling Distributions for Sample Means: AP Statistics Study Guide
Hello future statisticians and math wizards! Ready to dive into the magical world of numbers and distributions? We'll be exploring sampling distributions for sample means, turning those daunting
formulas into manageable bite-sized pieces. And hey, if you're lucky, there might even be a cheesy joke or two to lighten up the data deluge. 📊🎉
What is a Sampling Distribution for Sample Means?
Imagine you’re trying to determine the average number of books read by students at Hogwarts (yes, the very one with flying broomsticks and all). You can't possibly ask every single wizard, so you
take multiple samples. A sampling distribution for sample means is essentially a collection of all the averages you would get if you took several random samples from our book-loving wizards.
Why Do We Care About It?
Well, rather than just relying on one sample, which might be as unreliable as a chocolate frog in summer, we use the sampling distribution to make better guesses about the true average. It’s like
having multiple pairs of glasses and finding the perfect one that helps you see the truth clearly. 👓✨
The Central Limit Theorem (CLT)
Say hello to the superstar of our topic: the Central Limit Theorem (CLT)! CLT states that as your sample size grows, the distribution of the sample mean becomes increasingly normal, no matter what
the population distribution looks like. If you take a large enough sample (usually n ≥ 30), the sample means will dance around the true population mean in a lovely bell-shaped curve. 🎉
An Example with a Dash of Fun
Imagine you want to find the average amount of butterbeer Harry and his friends consume every month. You decide to survey 100 students from Hogwarts. The true population mean might not be known, but
CLT tells us that if our sample size is large enough, the distribution of our sample mean should be approximately normal! Even if some students are notorious butterbeer guzzlers while others don’t
sip a drop, the larger sample size helps balance things out. 🍻
Plain vs. Standard Deviation and Standard Error
While we’re on this enchanted journey, let's clear up a common confusion: Standard deviation (SD) vs. standard error (SE). Think of SD as the overall variation in our butterbeer consumption in the
entire Hogwarts. SE, on the other hand, is how much we expect our sample means to vary from the true population mean. SE is magic because it gets smaller as your sample size increases, making your
estimates more precise. 🧙♂️✨
Sample Size and Normality
Speaking of which, determining our sample size is crucial. As we’ve said, a sample size of 30 or more (like 30 Chocolate Frogs) is often considered the sweet spot where magic happens—our sampling
distribution shapes up to be normal. This threshold is like the magical "Alohomora" unlocking all statistical secrets! 🗝️
Practice Problem (with Wizarding Wisdom)
Imagine you're given a sample of 100 Quidditch team captains from around the world, and you find that their average time to capture the Golden Snitch is 150 minutes with a standard deviation of 20
minutes. Let’s say the true average time is actually 140 minutes. Here’s how you might go about addressing some common questions:
a) What is the Sampling Distribution for the Sample Mean?
• Imagine you keep sampling different batches of Quidditch captains and averaging their times. The sampling distribution is the collection of all these averages if you repeated the process
thousands of times. It's useful because it lets us make inferences about the actual average snitch-capturing time.
b) Describe the Shape, Center, and Spread of the Sampling Distribution.
• Thanks to our wizard friend CLT, the shape would approximate a normal bell curve. The center would hover around the true mean (140 minutes), and the spread would be governed by SE (calculated
using the sample SD divided by the square root of the sample size).
c) Why Does the CLT Apply?
• Because we have a sample size of 100 captains, which is more than enough for our sampling distribution to be normally distributed, even if not all captains have identical snitch-catching skills.
d) Discuss One Potential Source of Bias.
• Imagine you only sampled captains from teams that recently won tournaments—they might be quicker on the draw, leading to an overestimate. Alternatively, sampling from less competitive teams could
lead to an underestimate. Beware of this "selection bias," which could skew our magical averages!
Key Terms to Review
• Bias: The pesky gremlin that skews results, pulling them away from the true value. Examples include selection bias and response bias.
• Central Limit Theorem: The magical rule that makes the sampling distribution of the sample mean approximately normal if the sample size is big enough.
• Normal Distribution: The symmetrical bell curve defined by its mean and standard deviation.
• Population Mean: The true average number or parameter you’re trying to estimate across the entire magical realm (or population).
• Sample Mean: The average you get from your sample—a.k.a. your best guess at the population mean.
• Sampling Distribution: The distribution of a statistic (like the mean) computed from multiple samples, showing you the variability of that statistic.
• Selection Bias: When your sample doesn't fairly represent the population, making your results about as useful as a wand without a core.
• Simple Random Sample: The fairest way to draw your sample, ensuring every wizard (or person) has an equal opportunity to be selected.
• Standard Error (SE): The measure of how much sample means would vary from the true population mean if you kept drawing samples. It gets smaller with larger sample sizes.
Fun Fact
Did you know that sampling distributions are like snowflakes? ❄️ Each one is unique depending on the sample you take, but collectively, they help you understand the true nature of the population.
And there you have it, a charming tour through the magical land of sampling distributions for sample means! Remember, behind every daunting formula is a simple idea to understand, just waiting to be
waved into existence by your statistical wand. Now, go ace that AP Statistics exam with the confidence of a wizard who's just learned the Patronus Charm! 🌟
Good luck, and may your data always be in your favor! 📈✨ | {"url":"https://knowunity.com/subjects/study-guide/sampling-distributions-for-sample-means","timestamp":"2024-11-11T18:00:09Z","content_type":"text/html","content_length":"247022","record_id":"<urn:uuid:a849c9fd-4fbe-4f90-90a6-d58b99b589d9>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00634.warc.gz"} |
IIFT 2019 | Quantitative Aptitude | 2IIM CAT Coaching
The best way to boost your IIFT prep is to practice the actual IIFT Question Papers. 2IIM offers you exactly that, in a student friendly format to take value from this. In the 2019 IIFT, quants were
a mixed bag of questions of varying difficulty, with some routine questions and the others were very demanding. Some beautiful questions that laid emphasis on Learning ideas from basics and being
able to comprehend more than remembering gazillion formulae and shortcuts.
Question 10: A motorboat takes the passengers from Rishikesh to Haridwar and back. Both the cities, Rishikesh and Haridwar are located on the banks of River Ganga. During Kumbh Mela ,to earn more
money, the owner of the motorboat decided to have more trips from Rishikesh to Haridwar and back, so he increased the speed of the motorboat in still water, by 50%. By increasing the speed, he was
able to cut down the travel time from Rishikesh to Haridwar and back, by 60%. What is the ratio of the speed of motorboating still water to that of the speed of river Ganga?
1. √11⁄6
2. (√3)⁄2
3. (√3)⁄7
4. (√11)⁄9
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Explanatory Answer
Let the speed of the boat be ‘b', the speed of the stream be ‘s', and the distance travelled be ‘d'.
Then the speed of the boat going upstream will be b-s and going downstream will be b+s.
Time taken for a round trip: T = \\frac{d}{b+s}) + \\frac{d}{b-s})
Now when the owner increases the speed of the boat by 50%:
The new speed of boat will be 1.5b
The new speed of boat going upstream will be 1.5b-s and while going downstream it will be 1.5b+s.
Then the time taken for a round trip: \\frac{2}{5T})= \\frac{d}{1.5b+s}) + \\frac{d}{1.5b-s})
Dividing the two equations we get:
\\frac{2b}{b^2-s^2}) ˙ \\frac{2.25b^2-s^2}{3b}) = \\frac{5}{2})
\\frac{2.25b^2-s^2}{b^2-s^2}) = \\frac{5}{2}) * \\frac{3}{2})
4(2.25b^2-s^2) = 15(b^2-s^2)
9b^2 – 4s^2 = 15b^2 – 15s^2
11s^2 = 6b^2
So, \\frac{b}{s}) = √\\frac{11}{6})
The question is "What is the ratio of the speed of motorboating still water to that of the speed of river Ganga?"
Hence, the answer is, "√11⁄6" | {"url":"https://online.2iim.com/IIFT-question-paper/IIFT-2019-Question-Paper-Quant/quant-question-10.shtml","timestamp":"2024-11-02T19:10:48Z","content_type":"text/html","content_length":"66118","record_id":"<urn:uuid:3786b035-354e-43cc-b020-ae87526209e7>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00797.warc.gz"} |
The Provincial Scientist
I was walking to the tennis courts in Battersea Park a few years back, when I heard something on my Walkman radio. It stuck with me for years, and until tonight I haven’t followed up on it, read
about it or written about it. Though I have told everyone at my work, which has resulted, as usual, in groans about how nerdy I am (and genuine amazement at how I could spend valuable time pondering
these things).
What I heard was a very short anecdote about someone who wrote a little regarded paper in the 1940’s (see ref below) in which he made an attempt to define a ‘measure’ for information. Although I
never read any more about it (until today), what I heard was enough to set me thinking…
Now, if you know lots about this subject then bear with me. Those readers who don’t know what he came up with: I challenge you to this question:
• what contains more information, a phone-number, a ringtone or a photo?
Are they even comparable?
Bits & Bytes…
In this computer age, we already have some clues. We know that text doesn’t use up much disk space, and that photos & video can fill up the memory stick much quicker.
But what about ZIP files? These are a hint that file-size is not a very accurate measure of information content.
So what is a megabyte? Is it just so many transistors on a microchip? Happily, its not, its something much more intuitive and satisfying.
Information: what is it?
If you go to Wikipedia and try to look up Information Theory, within a few seconds you are overrun with jargon and difficult concepts like Entropy; I hope to avoid that.
Let’s rather think about 20 questions. 20 Questions is the game where you have 20 questions to home in on the ‘secret’ word/phrase/person/etc. The key, however, is that the questions need to elicit a
yes/no response.
To define information simply: the more questions you need in order to identify a ‘piece of information’, the more information content is embodied in that piece of information (and its context).
This helps us to answer questions like: “How much information is in my telephone number?”
Let’s play 20 questions on this one. How would you design your questions? (Let’s assume we know it has 7 digits)
You could attack it digit by digit: “is the first digit ‘0’? Is the first digit ‘1’? Then changing to the next digit when you get a yes. If the number is 7 digits long, this may take up 70 questions
(though in fact if you think a little you will never need more than 9 per digit, and on average you’ll only need about 5 per digit – averaging ~35 in total).
But can you do better? What is the optimum strategy?
Well let’s break down the problem. How many questions do we really need per digit?
We know that there are 10 choices. You could take pot luck, and you could get the right number first time, or you might get it the 9th time (if you get it wrong 9 times, you don’t need a 10th
question). However, this strategy will need on average 5 questions.
What about the divide and conquer method? Is it less than 5? If yes, you have halved the options from 10 to 5. Is it less than three? Now you have either 2 or 3 options left. So you will need 3 or 4
questions, depending on your luck, to ID the number.
Aside for nerds: Note now that if your number system only allowed 8 options (the so-called octal system), you would always be able to get to the answer in 3. If you had 16 options (hexadecimal), you
would always need 4.
For the decimal system, you could do a few hundred random digits, and find out that you need, on average 3.3219… questions. This is the same as asking “how many times do you need to halve the options
until no more than one option remains?’
Aside 2 for nerds : The mathematicians amongst you will have spotted that 2^3.3219 = 10
Now, we could use 4 questions (I don’t know how to ask 0.32 questions) on each of the 7 digits, and get the phone number, and we will have improved from 35 questions (though variable) to a certain 28
But we could take the entire number with the divide and conquer method. There are 10^7 (100 million) options (assuming you can have any number of leading zeroes). How many times would you need to
halve that?
1. 50 00o 000
2. 25 000 000
3. ….
22. 2.38…
23. 1.19…
24. 0.59…
So we only needed 24 questions. Note that calculators (and MS Excel) have a shortcut to calculate this sort of thing: log[2](10^7) = ~23.25…
OK, so we have played 20 questions. Why? How is the number of questions significant? Because it is actually the accepted measure of information content! This is the famous ‘bit‘ of information. Your
7 digit number contains about 24 bits of information!
As you play with concept, you will quickly see that the amount of information in a number (say the number 42), depends hugely on the number of possible numbers the number could have been. If it could
have been literally any number (an infinite set) then, technically speaking, it contains infinite information (see, I’ve proven the number 42 is all-knowing!).
But the numbers we use daily all have context, without context they have no practical use. Any system that may, as part of its working, require ‘any’ number from an infinite set would be unworkable,
so this doesn’t crop up often.
Computer programmers are constantly under pressure to ‘dimension’ their variables to the smallest size they can get away with. And once a variable is dimensioned, the number of bits available for its
storage is set, and it doesn’t matter what number you store in that variable, it will always require all those bits, because it is the number of possibilities that define the information content of a
number, not the size of the number itself.
I hope that was of interest! Please let me know if I’ve made any errors in my analysis – I do tend to write very late at night 😉
1. Claude Shannon, “A Mathematical Theory of Communication” 1948
Skepticism: religion’s cancer
Religion has been described as a virus. This is not because it’s ‘bad for you’ necessarily, but rather due to the way it spreads.
It’s not hard to see the parallel: like viruses (and bacteria), religions exist within a population and spread from person to person.
But what about atheism? Is it a viral idea (meme) too?
I will argue that it isn’t. Perhaps it’s more like a cancer, a ‘mutation’ that kills off religious infections.
Cancers are sneaky, because they can occur spontaneously, almost by chance, and are therefore a very statistical phenomenon: your chance of getting cancer is affected by a), your exposure (to
carcinogens causing mutation events), and b), your predisposition (genes affecting your ability to cope with the these mutations).
Your chance of becoming an atheist is likewise affected by a), your exposure (to information about how the world works) and b), your predisposition (intelligence, or ability to apply logic to the
I.e. atheism differs from religion in the same way that carcinogens differ from viruses.
Can we develop this idea? I think so.
Let’s look at how you ‘get’ atheism…
Picture it: you’ve been brought up in a good god-fearing, church-going family. You went to Sunday school, you know which of Cain and Abel was the baddy and you can explain to people about how there
is good evidence for The Flood. You also have a healthy fear of sex and the other sins.
But you go to school and you learn about plate tectonics and see how well South America slots into Africa, and then you learn how European bees are not quite the same as African ones, just like
Toyota Corollas aren’t, and one day, while looking at the grille of your step-mother’s 1.3GL, and daydreaming about the A-team, a thought strikes you, like a shot of cancer-causing sunshine on that
patch of skin on the back of your right shoulder, that cars evolve differently in different counties and maybe that explains all the animals and perhaps God didn’t make a women out of Adam’s rib
after all, cos’ that never did make much sense, because a rib is a pretty silly thing to make a women out of anyway.
Catching a dose of Christianity on the other hand, does not come from inside, as the result of reasoning, it comes from outside, from other people.
Most often you will be born into a house absolutely soaked in the infection, you will be infected soon enough, prayers will be said at mealtimes, the church is so big and grand, and the hymns are so
catchy, and then they wheel out Christmas and baby Jesus (or baby ‘cheeses’ as my son says)…
But even if you’re not so lucky, there’s hope. You can drop in at a church any time (though Sundays are best I’m told) and the chances are, even if you are down on your luck, short of friends, and
even if you aren’t very nice, the sweet people there are quite likely to help you. That feeling of family, of unquestioning acceptance – brings a special warmth to the cockles of the heart.
Once you’re in the door, religion, having evolved pretty niftily, can now play you like a violin. Your emotions, developed to help promote clan solidarity, are hi-jacked and kick in nicely. Did you
know, that if you really listen to what these folks say, and really try to feel God’s love, you will indeed feel something! Now that’s a clever infection…
A house price prediction…
House prices, like the stock market, are tricky to predict.
As with the stock market, there are two classes of parameters that affect the prices – the so-called ‘fundamentals’, like supply and demand, the price-to-earnings ratio on the one hand, and the more
transient effects like the economic climate and the ever-slippery ‘confidence’.
There has been feverish speculation for years in the UK, and the prices rose for 15 consecutive years, and are at last dropping.
So why did the prices get so high? Many economists would argue it was a classic “bubble”, a self-perpetuating cycle of confidence building more confidence; in other words the fundamentals were being
Of course, the people found fundamentals they claimed justified the prices; in particular increased demand. Folks living longer, divorce, folks marrying later, immigration, and the breakdown of the
family unit; all these things mean we need more houses.
But if these fundamentals were the whole reason, the prices wouldn’t be dropping as they are now. OK, so now most will admit it got out of hand and this is a correction. But how far has it got to
The bubble, it seems to be agreed, was really helped by two factors:
Firstly there was a throttle on the supply – planning permission is notoriously hard to get and the government probably knew it and were happy with prices rising, it made everyone feel prosperous. On
a more sinister front, housing developers may have sitting on prime real estate to deliberately keep prices high.
Secondly, there was easy credit – anyone and their dog could get the cash so people who really shouldn’t have been in the game got in and are now out of their league.
But there is a third factor I’ve not seem discussed in the media: the baby-boom generation.
Hasn’t this bubble coincided with the baby-boomer’s ‘rich’ phase – the age from 45-60 when the kids are off and 25 years of mortgage payments have built up the asset list? Surely this is the
age-group that is most likely to own big houses, or multiple houses for that matter?
So what will happen now? The bubble has burst, the correction is in full swing, but what will happen in the next 10 years as the baby boomers start retiring, downsizing, and dying? Will this coincide
with the next bubble-burst? Will the industry and government look at the population age profile during planning?
I personally hope this is why the market is cock-eyed – why it is that a professional engineer in his mid-thirties with a internationally comparable salary can’t afford more than a mid-terrace house
with a 5×5-metre garden…
So I predict (well pray really, if that’s possible for athiests) that we will get into an oversupply situation and that house prices should correct from this ‘second-order’ bubble.
Of course, even if I am right, it may be that the prices are kept up by nasty developers identifying whole towns to ‘let go to ruin’ just to keep the prices high in the next town along…
Celebrity Dynamics
Celebrity Dynamics.
The list of people we all ‘know’ isn’t that long, yes, it probably thousands – politicians, actors, singers, historical figures, sports stars – but in a country like the UK, it is still a remarkably
small fraction of the populace.
Of course, there are ‘spheres’ – people interested in politics know more politicians, sports fans have more sporting heroes – we here in Cornwall have our local ‘Cornish’ celebrities.
However, if we remembered every celebrity, we would soon run out of space in the public ‘memory’, so we have to be selective.
The media know this – they constantly face choices of which story to follow, and the decisions will often be arbitrary; two minor celebrities did two things today, and we only have 45 seconds of time
to fill in our variety news programme – which shall we choose?
This decision process is simple – the editor will pick the celebrity who has more recent ‘hits’ in the news.
Why? Because they know that the audience is more likely to recognise the name – and they know that if the audience hear that name twice it reinforces the memory.
This simple logic creates a very interesting system in which the rise to fame becomes ‘autocatalytic’ – a self-perpetuating, accelerating process. All you need to do is pass some ‘critical point’ of
news coverage and you may be in for a ride!
However, we can only hold so many names in the list, so anyone who is out of the news for a time drops off the radar pretty fast, even if they did once enjoy high exposure.
If you are like me, you’ll be thinking of exceptions – folks who just stay famous regardless – do they buck this logic? I don’t think so.
Such people most likely still get exposure, even if its not them in the news – perhaps we see their CD on our shelf, or we talk about their ‘field’ (Thatcherism, Darwinism, Keynesian economics,), and
this may be accentuated if their field gets in the news – as has recently been the case for Keynes.
So what value does this theory have?
I think it explains:
• why so many great deeds don’t lead to fame
• why often only one person from a high achieving team is ‘selected’ for fame
• why there’s no such thing as bad publicity
• local fame does not easily turn to national fame
It also suggests that if you want to be famous, you should:
• a series of newsworthy events in succession is probably better than a single highly newsworthy achievement
• if you are in a group/team/band, you need to be the leader or public face of the group
• you should associate yourself with a newsworthy field, ideally become the posterboy/girl for the field, always dragged out when the field is in the news
And if you want to stay famous once you are you should keep in the public eye:
• associate yourself with newsworthy events
• differentiate yourself from other celebrities in your ‘space’ or
• gang together with other celebrities to create newsworthy events
• become the posterboy/girl for a newsworthy field/subject, the one dragged out when the field is in the news
Aside: There seems to be another way to maintain fame:- create mystique, the image of privilege, of some higher plain of existence away from the mundanity of everyday life. People say they like
down-to-earth celebrities – that’s because they are very rare – you have to be ‘proper’ famous to stay famous without this tactic!
Of course, this all assumes you want to be famous! You can equally use the theory to keep a low profile 😉
Good luck either way! | {"url":"https://theprovincialscientist.com/?m=200812","timestamp":"2024-11-13T15:52:42Z","content_type":"text/html","content_length":"61046","record_id":"<urn:uuid:f508cbf3-d632-41c7-850a-00c6b7028473>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00074.warc.gz"} |
Uniform Resource Identifiers
1. URI Filenames In SQLite
Beginning with version 3.7.7 (2011-06-23), the SQLite database file argument to the sqlite3_open(), sqlite3_open16(), and sqlite3_open_v2() interfaces and to the ATTACH command can be specified
either as an ordinary filename or as a Uniform Resource Identifier or URI. The advantage of using a URI filename is that query parameters on the URI can be used to control details of the newly
created database connection. For example, an alternative VFS can be specified using a "vfs=" query parameter. Or the database can be opened read-only by using "mode=ro" as a query parameter.
2. Backwards Compatibility
In order to maintain full backwards compatibility for legacy applications, the URI filename capability is disabled by default. URI filenames can be enabled or disabled using the SQLITE_USE_URI=1 or
SQLITE_USE_URI=0 compile-time options. The compile-time setting for URI filenames can be changed at start-time using the sqlite3_config(SQLITE_CONFIG_URI,1) or sqlite3_config(SQLITE_CONFIG_URI,0)
configuration calls. Regardless of the compile-time or start-time settings, URI filenames can be enabled for individual database connections by including the SQLITE_OPEN_URI bit in the set of bits
passed as the F parameter to sqlite3_open_v2(N,P,F,V).
If URI filenames are recognized when the database connection is originally opened, then URI filenames will also be recognized on ATTACH statements. Similarly, if URI filenames are not recognized when
the database connection is first opened, they will not be recognized by ATTACH.
Since SQLite always interprets any filename that does not begin with "file:" as an ordinary filename regardless of the URI setting, and because it is very unusual to have an actual file begin with
"file:", it is safe for most applications to enable URI processing even if URI filenames are not currently being used.
3. URI Format
According to RFC 3986, a URI consists of a scheme, an authority, a path, a query string, and a fragment. The scheme is always required. One of either the authority or the path is also always
required. The query string and fragment are optional.
SQLite uses the "file:" URI syntax to identify database files. SQLite strives to interpret file: URIs in exactly the same way as popular web-browsers such as Firefox, Chrome, Safari, Internet
Explorer, and Opera, and command-line programs such as Windows "start" and the Mac OS-X "open" command. A succinct summary of the URI parsing rules follows:
• The scheme of the URI must be "file:". Any other scheme results in the input being treated as an ordinary filename.
• The authority may be omitted, may be blank, or may be "localhost". Any other authority results in an error. Exception: If SQLite is compiled with SQLITE_ALLOW_URI_AUTHORITY then any authority
value other than "localhost" is passed through to the underlying operating system as a UNC filename.
• The path is optional if the authority is present. If the authority is omitted then the path is required.
• The query string is optional. If the query string is present, then all query parameters are passed through into the xOpen method of the underlying VFS.
• The fragment is optional. If present, it is ignored.
Zero or more escape sequences of the form "%HH" (where H represents any hexadecimal digit) can occur in the path, query string, or fragment.
A filename that is not a well-formed URI is interpreted as an ordinary filename.
URIs are processed as UTF8 text. The filename argument sqlite3_open16() is converted from UTF16 native byte order into UTF8 prior to processing.
3.1. The URI Path
The path component of the URI specifies the disk file that is the SQLite database to be opened. If the path component is omitted, then the database is stored in a temporary file that will be
automatically deleted when the database connection closes. If the authority section is present, then the path is always an absolute pathname. If the authority section is omitted, then the path is an
absolute pathname if it begins with the "/" character (ASCII code 0x2f) and is a relative pathname otherwise. On windows, if the absolute path begins with "/X:/" where X is any single ASCII
alphabetic character ("a" through "z" or "A" through "Z") then the "X:" is understood to be the drive letter of the volume containing the file, not the toplevel directory.
An ordinary filename can usually be converted into an equivalent URI by the steps shown below. The one exception is that a relative windows pathname with a drive letter cannot be converted directly
into a URI; it must be changed into an absolute pathname first.
1. Convert all "?" characters into "%3f".
2. Convert all "#" characters into "%23".
3. On windows only, convert all "\" characters into "/".
4. Convert all sequences of two or more "/" characters into a single "/" character.
5. On windows only, if the filename begins with a drive letter, prepend a single "/" character.
6. Prepend the "file:" scheme.
3.2. Query String
A URI filename can optionally be followed by a query string. The query string consists of text following the first "?" character but excluding the optional fragment that begins with "#". The query
string is divided into key/value pairs. We usually refer to these key/value pairs as "query parameters". Key/value pairs are separated by a single "&" character. The key comes first and is separated
from the value by a single "=" character. Both key and value may contain %HH escape sequences.
The text of query parameters is appended to the filename argument of the xOpen method of the VFS. Any %HH escape sequences in the query parameters are resolved prior to being appended to the xOpen
filename. A single zero-byte separates the xOpen filename argument from the key of the first query parameters, each key and value, and each subsequent key from the prior value. The list of query
parameters appended to the xOpen filename is terminated by a single zero-length key. Note that the value of a query parameter can be an empty string.
3.3. Recognized Query Parameters
Some query parameters are interpreted by the SQLite core and used to modify the characteristics of the new connection. All query parameters are always passed through into the xOpen method of the VFS
even if they are previously read and interpreted by the SQLite core.
The following query parameters are recognized by SQLite as of version 3.15.0 (2016-10-14). New query parameters might be added in the future.
The cache query parameter determines if the new database is opened using shared cache mode or with a private cache.
The immutable query parameter is a boolean that signals to SQLite that the underlying database file is held on read-only media and cannot be modified, even by another process with elevated
privileges. SQLite always opens immutable database files read-only and it skips all file locking and change detection on immutable database files. If these query parameter (or the
SQLITE_IOCAP_IMMUTABLE bit in xDeviceCharacteristics) asserts that a database file is immutable and that file changes anyhow, then SQLite might return incorrect query results and/or
SQLITE_CORRUPT errors.
The mode query parameter determines if the new database is opened read-only, read-write, read-write and created if it does not exist, or that the database is a pure in-memory database that never
interacts with disk, respectively.
When creating a new database file during sqlite3_open_v2() on unix systems, SQLite will try to set the permissions of the new database file to match the existing file "filename".
The nolock query parameter is a boolean that disables all calls to the xLock, xUnlock, and xCheckReservedLock methods of the VFS when true. The nolock query parameter might be used, for example,
when trying to access a file on a filesystem that does not support file locking. Caution: If two or more database connections try to interact with the same SQLite database and one or more of
those connections has enabled "nolock", then database corruption can result. The "nolock" query parameter should only be used if the application can guarantee that writes to the database are
The psow query parameter overrides the powersafe overwrite property of the database file being opened. The psow query parameter works with the default windows and unix VFSes but might be a no-op
for other proprietary or non-standard VFSes.
The vfs query parameter causes the database connection to be opened using the VFS called NAME. The open attempt fails if NAME is not the name of a VFS that is built into SQLite or that has been
previously registered using sqlite3_vfs_register().
4. See Also | {"url":"https://www.hwaci.com/sw/sqlite/uri.html","timestamp":"2024-11-01T22:06:10Z","content_type":"text/html","content_length":"14929","record_id":"<urn:uuid:301b0a16-ae5d-4013-b2ed-9f694597956d>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00824.warc.gz"} |
Triangles within Squares
Can you find a rule which relates triangular numbers to square numbers?
The diagram above shows that: $$ 8 \times T_2 + 1 = 25 = 5^2$$
Use a similar method to help you verify that: $$ 8 \times T_3 + 1 = 49 = 7^2$$ Can you generalise this result?
Can you find a rule in terms of $ T_n $ and a related square number?
Can you find a similar rule involving square numbers for $T_{n}, T_{n+2}$ and several copies of $T_{n+1}$?
Getting Started
There is a very strong connection between this problem and the "
Sequences and Series
" problem.
Can you see the rectangles made from two triangular numbers in the square?
Can you explain why there is always one left over?
Student Solutions
Well explained by Tom, from Cottenham Village College
The answer I got for $T_n$ is: $8T_n+1=(2n+1)^2$
The $2n+1$ part is because the diagram looks like this for $T_3$
$2T_n$ form a rectangle $n$ by $n+1$
The four rectangles rotate around the centre and together make a square of side $n+(n+1)$
So we get the equation $8T_n+1=(2n+1)^2$
Teachers' Resources
Why not encourage pupils to discover rules of their own?
This problem links to "
Triangles withinTriangles
" and the problem "
Triangles within Pentagons
There are many different ways to visualise this question and pupils should be encouraged to explain how they "know" their rule works. | {"url":"https://nrich.maths.org/problems/triangles-within-squares","timestamp":"2024-11-14T20:28:54Z","content_type":"text/html","content_length":"39847","record_id":"<urn:uuid:416971bb-0c9a-4504-83d0-e7559c76de9e>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00171.warc.gz"} |
Introduction to Data Structures and Algorithms with Python
In Computer Science, data structures and algorithms in Python are two of the most basic notions. For any programmer, they are essential tools. In Python, data structures are used to organize and
store data in memory while a program is processing it. Python algorithms, on the other hand, are a set of instructions that aid in the processing of data for a certain purpose.
Data structures in Python
Data structures are a technique of organizing and storing information; they describe the relationship between data and the many logical operations that can be done on it. Data structures can be
classified as either built-in Data-structures or user-defined data structures.
Built-in Data-structures:
The basic built-in data structures in Python include:
This is Python's most flexible data structure, represented as a list of comma-separated components enclosed in square brackets. Some of the methods applicable on a List are index(), append(), extend
(), insert(), remove(), and pop(). Lists are changeable, which means that their content can be modified while maintaining their identity.
b = ["banana", "apple", "microsoft"]
['banana', 'apple', 'microsoft']
Tuples are similar to Lists, except that they are unchangeable and are declared within parentheses. In a Tuple once an element has been defined, it cannot be deleted or modified. It ensures that the
data structure's specified values are neither tampered with or overridden.
my_tuple = (1, 2, 3)
Dictionaries consist of key-value pairs. The 'key' is used to identify an item, while the 'value' is used to store the item's value. The key and its value are separated by a colon. The items are
separated by commas, and everything is encased in curly brackets. While keys cannot be changed, values can be of any type.
user = {
"name" : "Janet",
"age" : 22,
"country" : "Kenya"
Sets are an unordered collection of unique elements. Sets, like Lists, are mutable and represented in square brackets, but they can't have the same value twice. Some methods that are applied on Sets
include count(), index(), any(), and all().
myset = {1, 2, 3, 3, 1, 7,3, 6, 5, 4, 4}
User-defined Data-structures:
Stacks are linear data structures that work on the Last-In-First-Out (LIFO) principle, which means that data inserted last is the first to be accessed. It is constructed using an array structure and
includes activities such as pushing (adding) elements, popping (deleting) elements, and only accessing elements from the TOP of the stack. This TOP is a pointer to the stack's current location.
Recursive programming, reversing words, undo systems in word editors, are some of the applications that employ stacks extensively.
Linked List
Linked lists are linear Data Structures that are linked together via pointers rather than being stored sequentially. A linked list node is made up of data and a pointer named next. These structures
are most commonly employed in image viewer, music player, and other applications.
A queue is a linear data structure based on the First-In-First-Out (FIFO) principle, which states that the data in first will be accessed first. It is constructed using an array structure and
includes actions that can be performed from both the head and tail ends of the Queue, i.e., front-back and back-to-head. En-Queue and De-Queue operations are used to add and delete elements, and
accessing the elements is possible. Queues are used as Network Buffers to manage traffic congestion, as well as Job Scheduling in Operating Systems.
Trees are non-linear Data Structures which have root and nodes. The root is the node from which the data comes, and the nodes are the other data points we have access to. The parent is the node that
comes before the kid, and the child is the node that comes after it. A tree must have levels to demonstrate the depth of information. The leaves are the final nodes in a chain. Trees generate a
hierarchy that can be utilized in a variety of real-world applications, such as identifying which tag belongs to which block in HTML pages. It's also useful for searching and a variety of other
Graphs are used to store data in the form of vertices (nodes) and edges (connections). The most accurate representation of a real-world map can be found in graphs. They are used to discover the least
path by calculating the various cost-to-distance between the various data points known as nodes. Many programs, including Google Maps, Uber, and others, employ graphs to discover the shortest path
and maximize earnings in the most efficient way possible.
In Python, HashMaps are the same as dictionaries. They may be used to create apps like phonebooks, populate data based on lists, and much more.
Algorithms are a set of rules or instructions written in a finite, sequential order to solve problems and provide the desired results. While data structures aid in data organization, algorithms aid
in the solving of intractable data analysis problems.
In this article, we will discuss the following algorithms: Tree Traversal Algorithm, Sorting Algorithm, Searching Algorithm, and Graph Algorithm.
Tree Traversal Algorithms
Tree Traversal refers to visiting each node present in the tree exactly once in order to update or check them.
Searching Algorithm
Searching algorithms aid in the verification and retrieval of data elements from various data structures. One sort of searching algorithm is the sequential search approach, in which the list is
traversed progressively and each member is checked. Examples of searching algorithms are:
Binary Search – The search interval is divided in half on a regular basis. The interval is narrowed to the lower half if the element to be searched is lower than the central component of the
interval. It is narrowed to the upper half otherwise. The technique is continued until the desired result is obtained.
Linear Search– In this algorithm, each item is sequentially searched one by one.
Sorting Algorithm
Sorting algorithms denote the ways to arrange data in a particular format. Sorting ensures that data searching is optimized to a high level and that the data is presented in a readable format. Types
of sorting algorithms in Python include: Bubble Sort, Merge Sort, Insertion Sort, Shell Sort, Selection Sort
Graph Algorithm
There are two methods of traversing graphs using their edges. These are:
Depth-first Traversal (DFS) – A graph is traversed in depth-ward motion in this algorithm. When an iteration comes to a halt, a stack is used to jump to the next vertex and begin the search. The set
data types are used to implement DFS in Python.
Breadth-first Traversal (BFS) – A graph is traversed in a breadth-ward motion in this algorithm. When an iteration comes to a halt, a queue is used to advance to the next vertex and begin the search.
The queue data structure is used to implement BFS in Python.
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Targeted Energy Transfers for Suppressing Regenerative Machine Tool Vibrations
We study the dynamics of targeted energy transfers in suppressing chatter instability in a single-degree-of-freedom (SDOF) machine tool system. The nonlinear regenerative (time-delayed) cutting force
is a main source of machine tool vibrations (chatter). We introduce an ungrounded nonlinear energy sink (NES) coupled to the tool, by which energy transfers from the tool to the NES and efficient
dissipation can be realized during chatter. Studying variations of a transition curve with respect to the NES parameters, we analytically show that the location of the Hopf bifurcation point is
influenced only by the NES mass and damping coefficient. We demonstrate that application of a well-designed NES renders the subcritical limit cycle oscillations (LCOs) into supercritical ones,
followed by Neimark–Sacker and saddle-node bifurcations, which help to increase the stability margin in machining. Numerical and asymptotic bifurcation analyses are performed and three suppression
mechanisms are identified. The asymptotic stability analysis is performed to study the domains of attraction for these suppression mechanisms which exhibit good agreement with the bifurcations sets
obtained from the numerical continuation methods. The results will help to design nonlinear energy sinks for passive control of regenerative instabilities in machining.
Issue Section:
Research Papers
Bifurcation, Dynamics (Mechanics), Flow (Dynamics), Machine tools, Oscillations, Stability, Vibration, Cutting, Chatter, Damping, Equilibrium (Physics), Limit cycles
In machining processes, the undesired vibration of the tool relative to the workpiece gives rise to a low-quality product. One of the most important causes of the instability in the cutting process
is the so-called regenerative effect, which arises from the fact that the cutting force exerted on a tool is influenced not only by the current tool position but also by that in the previous
revolution. Hence, the equation of motion for the tool is a delay differential equation, which renders even a single-degree-of-freedom (SDOF) dynamical system to be infinite-dimensional. In practical
machining processes, regenerative limit cycle oscillations (LCOs) create adverse effects on machining quality, and many papers deal with the stability and bifurcations of machining processes, as well
as with various passive and active means to improve their stability (see, for example, Refs. [1–8]).
Direct use or variations of linear/nonlinear-tuned mass dampers (TMDs, [7,8]) are probably the most popular approach to passive chatter suppression. However, even if a TMD is initially designed
(tuned) to eliminate resonant response near the eigenfrequency of a primary system, the mitigating performance may become less effective over time due to aging of the system, temperature, or humidity
variations, thus, requiring additional adjustment or tuning of parameters. It is only recently that passively controlled spatial/dynamic transfers of vibrational energy in coupled oscillators to a
targeted point where the energy eventually localizes were studied by utilizing a nonlinear energy sink (NES, see Ref. [9] for the summary of developments); and this phenomenon is simply called
targeted energy transfer (TET). The NES is basically a device that interacts with a primary structure over broad frequency bands; indeed, since the NES possesses essential stiffness nonlinearity (no
linear stiffness term), it may engage in (transient) resonance capture [10] with any mode of the primary system. This is why an NES, more effective than a TMD [11,12], can be designed to extract
broadband vibration energy from a primary system, engaging in transient resonance with a set of “most energetic” modes [13]. In Particular, Lee et al. [14–16] applied an ungrounded NES to an
aero-elastic system, and numerically and experimentally demonstrated that a well-designed NES can even completely eliminate aero-elastic instability. Three suppression mechanisms were identified;
that is, recurrent burstouts and suppressions, intermediate and complete elimination of self-excited instability in the aero-elastic system. Such mechanisms were investigated by means of bifurcation
analysis and complexification-averaging (CxA) technique [17].
Kalmár-Nagy et al. [3] analytically proved, by means of the center manifold theorem [18], the existence of subcritical Hopf bifurcation in an SDOF machine tool model with the regenerative cutting
force. Furthermore, practical stability limit in turning process was investigated by considering contact loss issues in the regenerative cutting force [4], which can predict stable, steady-state
periodic tool vibrations (or limit cycle oscillations—LCOs). Nankali et al. [19,20], proposed application of the NES system to a time-delayed machine tool system. They utilized numerical techniques
to obtain basin of attraction for different TET mechanisms. Recently, Gourc et al. [21] investigated the different response regimes of a cutting tool on a lathe strongly coupled to a nonlinear energy
sink. They derived the equation of the slow invariant manifold (SIM) and explained the behavior of the system by studying the location of the fixed points of the slow flow on this manifold. Moreover,
Gourc et al. [22] have studied the passive control of chatter instability in turning processes using a vibro-impact nonlinear energy sink.
In this work, we present a comprehensive study of TET mechanisms in suppressing regenerative chatter instability in a turning process. For this purpose, we first review nonlinear dynamics of a SDOF
machine tool model; then, perform a linear stability analysis to explore the effects of NES parameters on the occurrence of Hopf bifurcation (i.e., stability boundary on the plane of cutting depth
and rotational speed of a workpiece). Furthermore, the numerical [23] and asymptotic [24] techniques are applied to properly understand the suppression mechanisms that appear by the NES. Moreover,
applications of an NES to a practical machine tool model are introduced.
SDOF Machine Tool Dynamics: Summary
Kalmár-Nagy et al. [3] studied a nonlinear dynamics of a single-degree-of-freedom machine tool model depicted in Fig. 1. The equation of motion can be expressed by a delay-differential equation with
nonlinear regenerative terms retained up to cubic order
where $xτ≜x(t−τ)$ is the delayed variable with a time delay $τ=2π/Ω$; Ω is the rotating speed of the workpiece; $ωn=k/m$ is the (linearized) natural frequency; $ω0=k1/m$, where k[1] is the cutting
force coefficient; ζ is the damping factor of the machine tool; and f[0] is the chip thickness at the steady-state cutting.
Introducing the following scaling transformations
$t↦ωnt, τ↦ωnτ, x↦512f0x$
we rewrite the equation of motion as
where the differentiation is now with respect to the new time variable. Note that the parameter $p=(ω0/ωn)2=(k1/k)2$ indicate the effect of stiffness-hardening due to machining conditions.
Kalmár-Nagy et al. [3] calculated the transition curves (cf. Fig. 2(a) for $ζ=0.1$) as a function parameterized by the eigenfrequency ω, $(Ω(ω), p(ω))$, through which the steady-state cutting loses
stability through a Hopf bifurcation. Since the periodic solution that is born from the equilibrium point is unstable, this bifurcation is a subcritical one. For the damping factor $ζ=0.1$ (which
will be assumed throughout this work), the stiffness ratio p has the minimum value, $pmin=2ζ(1+ζ)=0.22$, and the minimum eigenfrequency of the limit cycle oscillation (LCO) is $ωmin=1.0$ such that
$p(ω)>0$ for $ω>ωmin$. Moreover, Fig. 2(b) depicts the minimum and maximum eigenfrequencies (denoted by $ωmin(n)$ and $ωmax(n)$, respectively) that the $nth$ lobe yields. Whereas, the first two or
three lobes exhibit relatively broad distribution of eigenfrequencies, the rest possess harmonic components concentrated near $ω=1.1$. Note that these eigenfrequencies act as “seed” harmonic bases
for the new-born periodic motion, and the triggering of a machine tool chatter appears as a result of competition between the eigenfrequency and the rotational speed of the workpiece.
Nonlinear Energy Sink
We apply an ungrounded nonlinear energy sink (NES) to the SDOF machine tool model, as the mathematical model is depicted in Fig.
. The equations of motion can be written as
where rescaling similar to Eq.
is incorporated; that is, we have
(the mass ratio),
(the damping factor of the NES),
(the stiffness ratio),
, and
. Moreover,
in which
is the tool position one revolution ago. We rewrite Eq.
in vector form as
$x1=x, x2=y, x3=ẋ, x4=ẏ$
$A=[00100001−1−p0−2(ζ+ζ1)2ζ1002ζ1/ϵ−2ζ1/ϵ], R=[00000000p0000000]$
and $f=(0,0,−C(x1−x2)3+(3p/10)[(x1−x1τ)2−(x1−x1τ)3],−C(x2−x1)3/ϵ)T$.
Assuming and substituting the solution of Eq.
to be
, then we obtain the eigenvalue problem typical for a delay-differential system
is an identity matrix. For a nontrivial eigenvector
, we derive the characteristic equation as
. That is, we write
We remark that one of the eigenvalues is always zero; that is, Eq.
is degenerative and bifurcation analysis of the trivial equilibrium should be at least of co-dimension 2. For
, we seek the parameter conditions that yield two eigenvalues of purely imaginary, complex conjugates. Substitution of
, where
, into Eq.
and separation of real and imaginary parts yield
$−2(ζ+ζ1+ζ1/ϵ)ω2+2ζ1(1+p)/ϵ=(2ζ1p/ϵ)cos ωτ+pω sin ωτ−ω3+(1+p+4ζζ1/ϵ)ω=pω cos ωτ−(2ζ1p/ϵ)sin ωτ$
By squaring and summing both sides of the two equations in Eq.
we obtain
. Also, noting that
$1−cos ωτ=2 sin2(ωτ/2)$
$sin ωτ=2 sin(ωτ/2)cos(ωτ/2)$
, we rearrange Eq.
$−2(ζ+ζ1+ζ1/ϵ)ω2+2ζ1/ϵ=−(2ζ1p/ϵ)(1−cos ωτ)+pω sin ωτ=2pR sin(ωτ/2)cos(ωτ/2+ϕ)−ω3+(1+4ζζ1/ϵ)ω=−pω(1−cos ωτ)−(2ζ1p/ϵ)sin ωτ=−2pR sin(ωτ/2)sin(ωτ/2+ϕ)$
. Then, we compute
, the rotational speed Ω of the workpiece can be derived as
where n is the order of the lobe in the stability chart, and $K(ω)$ and $ϕ$ are defined in Eqs. (11) and (12).
Before examining changes in the transition curves by adding the NES, we remark that the minimum eigenfrequency
to yield positive
becomes smaller than the unity. One can easily show that
$G(ω)>0, ∀ω$
; therefore,
should be the critical eigenfrequency that renders
. Since
is a quadratic equation,
can be calculated as
where $b=1−4ζ12/ϵ2−4ζ12/ϵ$ and $c=−4ζ12/ϵ2$. It can be analytically shown that $ωmin=1$ in the limit of $ϵ→0$ or $ζ1→0$.
Figure 3 depicts the minimum eigenfrequency $ωmin$ with respect to the mass ratio ϵ and the damping factor ζ[1] of the NES. As ϵ and ζ[1] increase, $ωmin$ tends to decrease; however, it remains near
unity but decreasing for small mass ratios (e.g., $ϵ<0.1$) and damping factors (e.g., $ζ1<0.1$), which is usually the case in practice. For fixed mass ratios, $ωmin$ decreases as ζ[1] increases;
and after a while the eigenfrequency seem to converge a constant (top of Fig. 3(b)). On the other hand, for fixed damping factors, the variation of $ωmin$ with respect to a mass ratio differs from
the order of the damping factor (bottom of Fig. 3(b)). That is, for small ζ[1], there exist a minimum of $ωmin$ and then it increases; but for larger ζ[1] it decreases monotonically. This observation
suggests not only that the frequency of the subcritical LCO when no NES is applied should be higher than unity, but also that the LCO with a frequency altered by an NES can be smaller than unity.
Note that the linearized natural frequency in the rescaled nondimensional equation (4) is unity.
Now, we compute the transition curves based on Eqs. (10) and (13). Figure 4 depicts the changes of the stability boundary in (Ω, p)-plane by varying the mass ratio ϵ and fixing the other two NES
parameters. The stability enhancement due to the application of an NES can be measured by directly calculating the point-wise shift amount as $Δp=(p′−p)/p×100$ (%), where p and $p′$ denote the values
at the stability boundary with respect to each Ω without and with an NES, respectively. Upward shift of the stability boundary occurs more significantly near the valley than near the cusp points of
the lobes, which will be useful in practical applications of chatter suppression. The shifting amount of the transition curve does not appear to be significant with a small NES mass (about 5%
improvement near valley of the lobes); however, the upward shift becomes increasing monotonically as the mass ratio increases. The ranges of the eigenfrequencies at the transition curves tend to
become lower as the mass ratio increases; and above certain mass ratio the eigenfrequency intervals are shifted upward (cf. Fig. 4(b) when $ϵ=0.6$). Note that the bifurcation occurring on the
transition curve can be referred to as a degenerate Hopf bifurcation in delay-differential equations, because one of the eigenvalues is always zero.
We remark that, although we can delay the occurrence of Hopf bifurcations by adding NES, this is not all one can achieve with an NES as a passive broadband vibration controller. The application of an
NES can also alter the topology of local bifurcations such that it produces Neimark–Sacker and saddle-node bifurcations as well as Hopf bifurcation. The former two bifurcations are essential in
discussing targeted energy transfer (TET) mechanisms in suppressing any types of instabilities introduced in a dynamical system. We deal with this in more detail in Sec. 4 by utilizing a numerical
continuation technique for a system of delay-differential equations.
Bifurcation Analysis and Robustness
TET Mechanisms.
As in the previous aero-elastic applications [14], three distinct TET mechanisms are identified in suppressing regenerative chatter instability; that is, recurrent burstouts and suppression,
intermediate and complete elimination of regenerative instability (cf. see Fig. 5 for typical time history for each suppression mechanism).
The first suppression mechanism is characterized by a recurrent series of suppressed burstouts of the tool response, followed by eventual complete suppression of the regenerative instabilities. The
beating-like (quasiperiodic) modal interactions observed during the recurrent burstouts turn out to be associated with Neimark–Sacker bifurcations of a periodic solution (cf. Fig. 6) and is critical
for determining domains of robust suppression [16]. To investigate this mechanism in more detail, Fig. 7 depicts the displacements of both the tool and NES and their wavelet transforms. Energy
exchanges between the two modes are evidenced in Fig. 7, through which a series of 1:1 transient resonance captures and escapes from resonance occurs (see Fig. 9(b).
The second suppression mechanism is characterized by intermediate suppression of LCOs, and is commonly observed when partial LCO suppression occurs. The initial action of the NES is the same as in
the first suppression mechanism. Targeted energy transfer to the NES then follows under conditions of 1:1 transient resonance capture, followed by conditions of 1:1 permanent resonance capture where
the tool mode attains constant (but nonzero) steady-state amplitudes. We note that, in contrast to the first suppression mechanism, the action of the NES is nonrecurring in this case, as it acts at
the early phase of the motion stabilizing the tool and suppressing the LCO.
Finally, in the third suppression mechanism, energy transfers from the tool to the NES are caused by nonlinear modal interactions during 1:1 RCs. Both tool mode and the NES exhibit exponentially
decaying responses resulting in complete elimination of LCOs.
A Practical Tool Model With Contact Loss.
We note that the numerical and analytical studies for TET mechanisms above are valid only for vibrations with small amplitudes; in particular, the permanent contact model with truncated nonlinear
terms cannot predict any stable steady-state periodic vibrations of high amplitudes. That is, the truncated nonlinearity in the regenerative cutting force will not predict the existence of a
saddle-node bifurcation point right after contact loss occurs. The details of machine tool dynamics can be found in Kalmár-Nagy [4], where stable periodic motions are predicted. Performing numerical
continuation analysis, we obtain the LCO surfaces for without and with NES being applied (Fig. 8). Still three distinct TET mechanisms for the model with contact loss are observed.
Complexification-Averaging (CxA) Technique.
In order to study the underlying TET mechanisms, we employ the CxA method first introduced by Manevitch [
]. We introduce the new complex variables in the following:
. Then, denoting by
the complex conjugate, we can express the original real variables in terms of the new complex ones
and similar expressions can be obtained for the NES variables. Substituting into the equations of motion and averaging out the fast dynamics over
, we obtain a set of two complex-valued modulation equations governing the slow-flow dynamics
. Expressing the slow-flow amplitudes in polar form,
, where
$ak(t), βk(t)∈ℝ, k=1,2$
, we obtain the set of real-valued slow-flow equations such that
$ȧ1=f1(a1,a2,ϕ), ȧ2=f2(a1,a2,ϕ), ϕ̇=g(a1,a2,ϕ)$
Finally, this set of first-order delay differential equations are solved to find
, and
to yield
Figure 9(a) directly compares the approximate (Eq. (19)) and exact (Eq. (4)) solutions for the tool displacement, which demonstrates a good agreement; furthermore, the nontime-like patterns (i.e.,
spirals) of the phase difference $ϕ$ depicts that the underlying TET mechanism for the first suppression mechanism involves a series of 1:1 transient resonance captures and escapes from resonance.
Also, Fig. 9(b) depicts 1:1 resonance capture in the slow-flow phase plane ($β,β̇$) in which $β=β1−β2$.
Analytical Study: Basins of Attraction for Suppression Mechanisms
This section includes analytical study of TET mechanisms for a SDOF machine tool. The asymptotic analysis [
] is applied in order to estimate domains of attraction in the parameter space for three suppression mechanisms. First step of this analysis is to rescale the equations of motion and remove
nonlinearities due to regenerative forces. So that, nonlinearities of the system would correspond only to the NES. Then, we identify NES nonlinearity modal interaction with the tool. Similar to Ref.
], we can eliminate the terms related to structural nonlinearities using following scaling:
$x→4ε3Cx, y→4ε3Cy, Δx→4ε3CΔx$
Moreover, assuming strong NES nonlinearity, we introduce
. So, Eq.
can be expressed as
In an approximation, we can omit terms including
because their effects on the dynamics are negligible. Indeed, we eliminate nonlinearities due to regenerative forces
We note that through this approximation only external nonlinearities due to the NES are left in the system. Further, the NES stiffness C is dropped out from the equations which indicates indecency of
the suppression mechanisms from this parameter.
Utilizing the complexification-averaging method introduced in Sec.
, steady-state periodic response can be analyzed. To this end, we introduce the coordinate transformation
are the physical quantities for the center of mass (with a factor of
) and the relative displacement, respectively. Then, the equations of motion
Numerical simulations (Fig.
) reveal that the steady-state responses are in the form of a fast oscillation (fast dynamics) modulated by a slowly varying envelope (slow dynamics); the dynamics can be expressed in the form of
. Now, we separate fast and slow dynamics by introducing
Substituting into Eq.
and performing averaging over the fast component
, we obtain the slow-flow equation
$ϕnτ=ϕn(t−τ) ;n=1,2$
is the complex conjugate of
. Since the delay term is finite (
), we can simplify the system by considering
. Moreover, for the sake of convenience in further mathematical manipulations, we define the following variables:
$a1=p2ω(1+ε)cos(ωτ), b=p+12ω(1+ε)a2=p2ω(1+ε)sin(ωτ), h=ζ(1+ε)L1=1/ε(−h−a2), S1=−h−a2L2=1/ε(−ω2+b−a1), S2=(b−a1)$
Substituting these variables into Eq.
and neglecting small terms, we can write:
. After a rescaling,
and introducing a polar form,
, we can derive the real-valued slow-flow dynamics
$Ṙ=ε(RL1−2Fω((S12−S22)sin δ−2S1S2 cos δ))Ḟ=−ω2R sin δ−ζ1Fδ̇=ε(L2−2FωR(S12−S22)cos δ−4FS1S2ωRsin δ)−Rω2Fcos δ−F22ω3+ω2$
represent real amplitude modulations while
is the phase difference. Equation
represents the slow dynamics of the original system (Eq.
). The slow dynamics is analyzed using the method of multiple scales. Based on this method, we consider different scales of time by defining
, where
and consequently,
. Moreover, we introduce the perturbation-series of the variables as
Plugging Eq.
into Eq.
and matching coefficients of powers of
, we derive subproblems governing the solution of slow dynamics. In this study, we calculate the responses for the “slow” time scale (
) and “super-slow” time scale (
). The first-order approximation is computed by considering
$∂R0∂τ0=0∂F0∂τ0=−ω2R0 sin(δ0)−ζ1F0∂δ0∂τ0=−R0ω2F0cos(δ0)−F022ω3+ω2$
Note that
is fixed with respect to slow time variable
, but not with respect to slower time variable
. No limit cycle oscillations are possible for system
and the only steady-state solutions are in the form of equilibrium points. Equilibrium points of the slow dynamics with respect to slow time (
) can be calculated as
This equation defines a slow invariant manifold (SIM) on the plane (
), where
refer to the equilibrium points of the slow dynamics with respect to slow time. Depending on the values for
, there are either one or three branches, when plotting
. For the case of three branches (Figs.
), performing linearized stability analysis on the slow dynamics with respect to slow time, reveals that the middle branch is unstable while the other two are stable. Equilibrium points of stable and
unstable branches are in the form of node (stable) and saddle node, respectively. Therefore, at the leading-order approximation the dynamics will be attracted to either of stable nodes.
There is no attractor in the form of LCO for the first-order approximation of slow dynamics. So, subproblems governing higher order of
should be considered to compute possible LCO's for the slow flow. To this end, we plug Eq.
into Eq.
and match coefficients of
for the first equation to get
$ε:∂∂τ1R0+∂∂τ0R1=R0L1−2F0ω{(S12−S22)sin δ0−2S1S2 cos δ0}$
The other two equations of
subproblem introduce small corrections to the shape of SIM. Similar to SIM, we apply equilibrium condition with respect to slow time (
) into Eq.
$∂∂τ1R̂0=R̂0L1−2F̂0ω{(S12−S22)sin δ̂0−2S1S2 cos δ̂0}$
This equation is satisfied for points on the SIM. Substituting
$sin δ̂0$
$cos δ̂0$
from SIM (Eq.
) and setting derivative of
equal to zero, we can find equilibrium points of the slow dynamics with respect to super-slow time (
). This is called super-slow flow (SSF). The intersection points of the SIM and SSF can be calculated by plugging
from Eq.
into Eq.
and solving for
These intersection points are depicted by letters A, B, and C in Figs.
. Stability of these points (by computing
), determines type of suppression mechanism.
First suppression mechanism: This mechanism corresponds to the existence of stable LCO on slow dynamics. It refers to the condition in which stable equilibrium point of slow dynamics $(F̂0e22)$ lies
on the unstable (middle) branch of SIM, while $F̂0e12$ is unstable. In fact, the attractor is in the form of LCO rather than equilibrium point for the slow dynamics. Cycle of relaxation oscillation of
slow dynamics creates a quasi-periodic oscillation in full-order system which are illustrated in Fig. 10. We remark that none of the equilibrium points of slow dynamics are complex numbers, and the
trivial equilibrium point is unstable for this mechanism.
Second suppression mechanism: This mechanism corresponds to stable LCO for full dynamics. According to the slow, fast dynamics separation (Eq. (25)), second suppression mechanism refers to existence
of stable nontrivial equilibrium point of slow dynamics. As derived in Eq. (35), there are two nontrivial equilibrium points $(F̂0e22,F̂0e32)$ for the slow dynamics. It can be shown that $F̂0e22$ is
stable when it lies on the stable branches of SIM which indicates second suppression mechanism. Note that none of the slow dynamics equilibrium points are complex, and only the second equilibrium
point is stable for this mechanism. The SIM and super-slow flow corresponding to this mechanism are shown in Fig. 11, along with the time simulation of the full dynamics.
Third suppression mechanism: This mechanism corresponds to the stable trivial equilibrium point of slow dynamics. It can be shown that if $f′(F̂20e1)<0$, then origin is the only attractor of the slow
dynamics and third suppression mechanism is predicted (Fig. 12).
No suppression: In addition to the three suppression mechanisms explained above, there is possibility of no suppression. This happens when trivial equilibrium point is the only real equilibrium point
of the slow dynamics; e.g., other two equilibrium points are complex. In this case, if trivial equilibrium point is unstable $(f′(F̂e02)>0)$, no attractor exists for the slow dynamics which predicts
instability of the original system. Super-slow flow and SIM for this case are shown in Fig. 13.
Table 1 summarizes mathematical conditions for the SIM and SSF configuration, corresponding to different cases discussed above. Also, basin of attraction for all suppression mechanisms, in $ζ1−p$
plane for given set of parameters, is depicted in Fig. 14.
Table 1
Sup. mechanism type Mathematical condition Figures
First $F20n<F̂20e2<F20p and F20u<F̂20e3$ 10
Second $0≤F̂20e2≤F20n or F20p≤F̂20e2$ 11
Third $f′(0)<0 and Im(F̂20e2,3)≠0$ 12
No suppression $f′(0)>0 and Im(F̂20e2,3)≠0$ 13
Sup. mechanism type Mathematical condition Figures
First $F20n<F̂20e2<F20p and F20u<F̂20e3$ 10
Second $0≤F̂20e2≤F20n or F20p≤F̂20e2$ 11
Third $f′(0)<0 and Im(F̂20e2,3)≠0$ 12
No suppression $f′(0)>0 and Im(F̂20e2,3)≠0$ 13
Figure 15 compares the numerical and analytical calculated basin of attractions for TET mechanisms of SDOF machine tool. Dashed lines depict boundary of suppression mechanism computed numerically
utilizing DDEBIFTOOL, while solid lines correspond to asymptotic analysis results. Numerical boundaries introduce Neimark–Sacker, saddle-node, and Hopf points for the first-, second-, and
third-suppression mechanism, respectively. It is obvious that the numerical and analytical results are well matched. We note at this point that the accuracy of asymptotic stability analysis
dependents on the value of mass ratio (ε). The smaller the mass ratio is the more accurate basin of attraction is computed. That is because we consider ε as the perturbation parameter in our study.
As it is plotted in Fig. 15, there is a type of bifurcation we could not detect through numerical study. It is not observed from analytical study for large values of ε either. This bifurcation which
is called Shilnikov homoclinic bifurcation is predicted when strongly modulated response (SMR) disappears due to the existence of an unstable equilibrium point in an SMR cycle. In other words, it
occurs when an unstable equilibrium point $(F̂20e3)$ meets a point of an SMR cycle $(F̂20u)$ while the second equilibrium point $(F̂20e2)$ is unstable. Figure 16 illustrates this situation.
Domains of attraction obtained from asymptotic analysis and associated points on the bifurcation surface are plotted in Fig. 17. A similar technique can be used to compute basins of attraction for
suppression mechanisms in the $ε−p$ plane. Figure 18 illustrates the bifurcation surface computed numerically (DDEBIFTOOL), and the corresponding analytical basin of attraction in the $ε−p$ plane. As
seen, for small values of ε there is a good agreement between the numerical and analytical results. However, larger values of ε decrease the accuracy of the asymptotic analysis. That is because ε is
used as perturbation parameter and has to be small ($ε≪1$).
Concluding Remarks
In this paper, we studied targeted energy transfer phenomena in suppressing chatter instability in a single-degree-of-freedom machine tool system, to which an ungrounded nonlinear energy sink is
connected. Two models were considered for the tool dynamics: permanent contact model and contact loss model. The limit cycle oscillation due to the regenerative instability in a tool model which
appeared as being subcritical for permanent contact model were (locally) eliminated or attenuated at a fixed rotational speed of a workpiece (i.e., a delay period) by TETs to the NES. It was shown
that there should be an optimal value of damping for a fixed mass ratio to shift the stability boundary for stably cutting more material off by increasing chip thickness. Also, magnitude of NES
nonlinear stiffness does not have any effect on stability boundary while increasing mass ratio improves stability. Three suppression mechanisms have been identified and each mechanism was
investigated numerically by time histories of displacements, and wavelet transforms and instantaneous modal energy exchanges. Furthermore, we extend the CxA analysis to perform asymptotic analysis by
introducing a reduced-order model and partitioning slow-fast dynamics. The resulting singular perturbation analysis yields parameter conditions and regions for the three suppression mechanisms, which
exhibit good agreement with the bifurcations sets obtained from numerical continuation methods.
This work was supported in part by National Science Foundation of United States, Grant Numbers CMMI-0928062 (YL) and CMMI-0846783 (TKN).
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Nonlinear Dynamics
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Nonlinear Dyn.
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Dyn. Syst.
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Paper No. DETC2011-48502.
Y. S.
, and
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Paper No. DETC2013-13510.
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J. Sound Vib.
, pp.
, and
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MATEC Web Conf.
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, and
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(3–4), pp. | {"url":"https://biomechanical.asmedigitalcollection.asme.org/computationalnonlinear/article/12/1/011010/473606/Targeted-Energy-Transfers-for-Suppressing","timestamp":"2024-11-09T22:52:15Z","content_type":"text/html","content_length":"456424","record_id":"<urn:uuid:3c617475-5d0a-4493-bd3a-327760e5443c>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00699.warc.gz"} |
EVM Puzzles - Walkthrough
EVM Puzzles are a collection of puzzles by Franco Victorio. In this Markdown file, I wrote down the walkthrough for each of them.
It is crucial to understand the storage structure of EVM, I refer to this article as a good resource on this. Most importantly, you should be familiar on how the stack works, and how EVM instructions
use it. See the original documentation about EVM at https://ethereum.org/en/developers/docs/evm/.
Whenever you see a new instruction, you should immediately go and see what it is and what it does. I have used https://www.evm.codes/ and https://www.ethervm.io/ as my resources. For more information
about EVM, you can also see their pages: https://www.evm.codes/about and https://www.ethervm.io/#overview.
Throughout this walkthrough all numbers that start with 0x are hexadecimal, and all that do not are decimal. Furthermore, bytecodes are given in hexadecimal, and two hexadecimals are equal to a byte,
as 4 bits x 2 = 8 bits = 1 byte.
In all puzzles, the objective is to have the code STOP, rather than REVERT. You basically have to get to the end!
One final remark: please try to solve each of these before looking at the solution! This is a great way to learn about EVM, and solving these is a rewarding feeling. Once you learn an instruction on
your own, the rest will be easier; it is the first steps that is hard to take at such a low-level context. Good luck!
Puzzle 1
00 0x34 CALLVALUE
01 0x56 JUMP
02 0xFD REVERT
03 0xFD REVERT
04 0xFD REVERT
05 0xFD REVERT
06 0xFD REVERT
07 0xFD REVERT
08 0x5B JUMPDEST
09 0x00 STOP
JUMP jumps to the destination specified by the top value in the stack. CALLVALUE pushes the call value to the stack. Looking at the code, there is a JUMPDEST at 8, so our call value must be 8.
Puzzle 2
00 0x34 CALLVALUE
01 0x38 CODESIZE
02 0x03 SUB
03 0x56 JUMP
04 0xFD REVERT
05 0xFD REVERT
06 0x5B JUMPDEST
07 0x00 STOP
08 0xFD REVERT
09 0xFD REVERT
Here is a new instruction: CODESIZE. It pushes the size of code to the stack. Which code? Well, it is the puzzle code itself, which we can see has 10 bytes (the last line is 09 but remember that it
starts from 00).
The SUB instruction takes the top two values, subtracting the second one from the first. So basically, it calculates CODESIZE - CALLVALUE. We can see a JUMPDEST at 06. So, we have the equation
CODESIZE - CALLVALUE = 10 - CALLVALUE = 6.
Our CALLVALUE must be 4 Wei.
Puzzle 3
00 0x36 CALLDATASIZE
01 0x56 JUMP
02 0xFD REVERT
03 0xFD REVERT
04 0x5B JUMPDEST
05 0x00 STOP
CALLDATASIZE pushes the size of calldata (bytes) to the stack. There is a JUMPDEST at 4, so the size should be 4 bytes. Any arbitrary 4-byte calldata would suffice: 0x11223344.
Puzzle 4
00 0x34 CALLVALUE
01 0x38 CODESIZE
02 0x18 XOR
03 0x56 JUMP
04 0xFD REVERT
05 0xFD REVERT
06 0xFD REVERT
07 0xFD REVERT
08 0xFD REVERT
09 0xFD REVERT
10 0x5B JUMPDEST
11 0x00 STOP
CODESIZE is 12 and JUMPDEST is at 10. XOR is a bitwise operation that stands for exclusive-or operation. Here is its logic table:
a b a ^ b
Denoting XOR as ^ (as is the case in many programming languages) we need some value such that CALLVALUE ^ 12 = 10. Simple arithmethic yields CALLVALUE = 10 ^ 12 = 6.
Puzzle 5
00 0x34 CALLVALUE
01 0x80 DUP1
02 0x02 MUL
03 0x610100 PUSH2 0x0100
06 0x14 EQ
07 0x600C PUSH1 0x0C
09 0x57 JUMPI
10 0xFD REVERT
11 0xFD REVERT
12 0x5B JUMPDEST
13 0x00 STOP
14 0xFD REVERT
15 0xFD REVERT
Here we have a JUMPI which is a conditional jump. PUSH1 0x0C above it provides the correct destination address, so all we have to care about is that the condition value be non-zero.
Looking at the lines above in order:
1. CALLVALUE pushes the value to the stack.
2. DUP1 duplicates it, so there are two of the same value in the stack.
3. MUL multiplies these two, so we basically squared the call value.
4. PUSH2 0x0100 pushes 0x0100 to the stack, which is 16 ^ 2 in decimals.
5. EQ compares the top two items in the stack, which is 16 ^ 2 and the square of our callvalue! Therefore, giving a callvalue of 16 is the winning move.
Puzzle 6
00 0x6000 PUSH1 0x00
02 0x35 CALLDATALOAD
03 0x56 JUMP
04 0xFD REVERT
05 0xFD REVERT
06 0xFD REVERT
07 0xFD REVERT
08 0xFD REVERT
09 0xFD REVERT
10 0x5B JUMPDEST
11 0x00 STOP
CALLDATALOAD loads a 32-byte value at the specified byte offset. If 32-bytes go beyond the length of calldata, the overflowing bytes are set to 0.
The offset to be loaded is given by the top value in the stack, which is given by PUSH1 0x00 above. Basically, the calldata itself should have the destination address for JUMP. Our JUMPDEST is at
0x0A, so that is our calldata!
But wait, remember that overflowing bytes are set to 0. So if we just send 0x0A, the remaining 31 bytes will be 00 and we will have 0x0A00000000000000000000000000000000000000000000000000000000000000
which is a huge number!
Instead, we must do zero-padding to the left, and send 0x000000000000000000000000000000000000000000000000000000000000000A as our calldata. This way, reading 32-bytes from the zero offset will yield
Puzzle 7
00 0x36 CALLDATASIZE
01 0x6000 PUSH1 0x00
03 0x80 DUP1
04 0x37 CALLDATACOPY
05 0x36 CALLDATASIZE
06 0x6000 PUSH1 0x00
08 0x6000 PUSH1 0x00
10 0xF0 CREATE
11 0x3B EXTCODESIZE
12 0x6001 PUSH1 0x01
14 0x14 EQ
15 0x6013 PUSH1 0x13
17 0x57 JUMPI
18 0xFD REVERT
19 0x5B JUMPDEST
20 0x00 STOP
The first 4 lines basically copy the entire calldata into memory. The next 4 lines create a contract, where the initialization code is taken from the memory at the position our calldata was just
loaded. In other words, the first 10 lines create a contract with our calldata.
Afterwards, the next 3 lines check if the EXTCODESIZE is equal 1 byte. The contract that we are checking the code size is the one we just created above with CREATE, as CREATE pushes the address to
the stack. EXTCODESIZE is the size of the runtime code of a contract, not the initialization code! The puzzle expects this to be 1 byte (lines 0C and 0E). So we just have to write our own
initialization code to do all this.
The instruction for this is CODECOPY, which works similar to CALLDATACOPY. The initialization code will be as follows:
PUSH1 0x01 // 1 byte
PUSH1 ;;;; // position in bytecode, we dont know yet
PUSH1 0x00 // write to memory position 0
CODECOPY // copies the bytecode
PUSH1 0x01 // 1 byte
PUSH1 0x00 // read from memory position 0
RETURN // returns the code copied above
In terms of bytecode, this results in 0x60 01 60 ;; 60 00 39 60 01 60 00 F3. This is a total of 12 bytes, so the ;;;; position will be 12, which is 0x0C. The final bytecodes are:
This code copies 1 byte code into memory, and returns it to EVM so that contract creation is completed. The actual runtime code is arbitrary, it just has to be 1 byte. Furthermore, runtime code comes
after the initialization code (starting at 12th position in this case), so we just have to append one byte to the end of the bytecodes above.
Let's add 0xEE for no reason: 0x6001600C60003960016000F3EE. That should do it!
Puzzle 8
00 0x36 CALLDATASIZE
01 0x6000 PUSH1 0x00
03 0x80 DUP1
04 0x37 CALLDATACOPY
05 0x36 CALLDATASIZE
06 0x6000 PUSH1 0x00
08 0x6000 PUSH1 0x00
10 0xF0 CREATE
11 0x6000 PUSH1 0x00
13 0x80 DUP1
14 0x80 DUP1
15 0x80 DUP1
16 0x80 DUP1
17 0x94 SWAP5
18 0x5A GAS
19 0xF1 CALL
20 0x6000 PUSH1 0x00
22 0x14 EQ
23 0x601B PUSH1 0x1B
25 0x57 JUMPI
26 0xFD REVERT
27 0x5B JUMPDEST
28 0x00 STOP
Similar to the previous puzzle, the first 4 lines copy the entire calldata into memory. The next 4 lines create a contract, the initialization code is taken from the memory at the position that
calldata was just loaded.
Afterwards, 5 of 0x00 are pushed to the stack. SWAP5 will exchange the 1st and 6th stack items, and the 6th item at that point is the address yielded from CREATE. Next, the remaining gas amount is
pushed to stack with GAS. All of this was done for the sake of CALL instruction:
gas // given by GAS the previous line
address // is the address from CREATE
value // 0
argOffset // 0
argSize // 0
retOffset // 0
retSize // 0
After CALL, a boolean result is pushed to the stack indicating its success. Looking at the following lines we see that this is expected to be 0 (PUSH1 00 and EQ with JUMPI afterwards). So we can
create a contract with a REVERT instruction.
PUSH1 0x00
PUSH1 0x00
This shall be our runtime code, which in bytecode is 0x60006000FD at 5 bytes total. We will write the initialization code ourselves too, similar to what we did in the previous puzzle.
PUSH1 0x05 // 5 bytes
PUSH1 0x0C // position of runtime code in bytecode
PUSH1 0x00 // write to memory position 0
CODECOPY // copies the bytecode
PUSH1 0x05 // 5 bytes
PUSH1 0x00 // read from memory position 0
RETURN // returns the code copied above
Again the position is 0x0C because the initialization code is 12 bytes. So our initialization bytecodes are 0x6005600C60003960056000F3 and the runtime bytecodes are 0x60006000FD. The calldata will be
these concatenated: 0x6005600C60003960056000F360006000FD.
Puzzle 9
00 0x36 CALLDATASIZE
01 0x6003 PUSH1 0x03
03 0x10 LT
04 0x6009 PUSH1 0x09
06 0x57 JUMPI
07 0xFD REVERT
08 0xFD REVERT
09 0x5B JUMPDEST
10 0x34 CALLVALUE
11 0x36 CALLDATASIZE
12 0x02 MUL
13 0x6008 PUSH1 0x08
15 0x14 EQ
16 0x6014 PUSH1 0x14
18 0x57 JUMPI
19 0xFD REVERT
20 0x5B JUMPDEST
21 0x00 STOP
We start with a small JUMPI that requires 3 < CALLDATASIZE so our calldata is larger than 3 bytes. Afterwards, we multiply our CALLVALUE and CALLDATASIZE, which is expected to be 8. Simply, we will
send a 4 byte calldata with 2 Wei call value.
Puzzle 10
00 0x38 CODESIZE
01 0x34 CALLVALUE
02 0x90 SWAP1
03 0x11 GT
04 0x6008 PUSH1 0x08
06 0x57 JUMPI
07 0xFD REVERT
08 0x5B JUMPDEST
09 0x36 CALLDATASIZE
10 0x610003 PUSH2 0x0003
13 0x90 SWAP1
14 0x06 MOD
15 0x15 ISZERO
16 0x34 CALLVALUE
17 0x600A PUSH1 0x0A
19 0x01 ADD
20 0x57 JUMPI
21 0xFD REVERT
22 0xFD REVERT
23 0xFD REVERT
24 0xFD REVERT
25 0x5B JUMPDEST
26 0x00 STOP
The first CODESIZE is the size of this puzzle itself, which is 1B (28 bytes). Next it swaps the CALLVALUE with it, and runs GT. In effect, this checks if CODESIZE > CALLVALUE.
After the successfull JUMPI, we are doing a CALLDATASIZE MOD 0x003 == 0 operation. We want this to be true for the next JUMPI to work, so our calldata size must be a multiple of 3.
The destination of JUMPI is defined by CALLVALUE ADD 0x0A, which should add up to 0x19. In decimals, 0x0A is 10 and 0x19 is 25, so our CALLVALUE should be 15.
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Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation a^n + b^n = c^n for
any integer value of n greater than two. The cases n = 1 and n = 2 have been known to have infinitely many solutions since antiquity.^[1]
This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful
proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The unsolved problem stimulated the development of algebraic number theory in
the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof, it was in the Guinness Book
of World Records as the "most difficult mathematical problem", one of the reasons being that it has the largest number of unsuccessful proofs.^[2]
The Pythagorean equation, x^2 + y^2 = z^2, has an infinite number of positive integer solutions for x, y, and z; these solutions are known as Pythagorean triples. Around 1637, Fermat wrote in the
margin of a book that the more general equation a^n + b^n = c^n had no solutions in positive integers, if n is an integer greater than 2. Although he claimed to have a general proof of his
conjecture, Fermat left no details of his proof, and no proof by him has ever been found. His claim was discovered some 30 years later, after his death. This claim, which came to be known as Fermat's
Last Theorem, stood unsolved in mathematics for the following three and a half centuries.
The claim eventually became one of the most notable unsolved problems of mathematics. Attempts to prove it prompted substantial development in number theory, and over time Fermat's Last Theorem
gained prominence as an unsolved problem in mathematics.
Subsequent developments and solution
With the special case n = 4 proved, it suffices to prove the theorem for exponents n that are prime numbers (this reduction is considered trivial to prove^[note 1]). Over the next two centuries
(1637–1839), the conjecture was proved for only the primes 3, 5, and 7, although Sophie Germain innovated and proved an approach that was relevant to an entire class of primes. In the mid-19th
century, Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually. Building on Kummer's work and using sophisticated computer
studies, other mathematicians were able to extend the proof to cover all prime exponents up to four million, but a proof for all exponents was inaccessible (meaning that mathematicians generally
considered a proof impossible, exceedingly difficult, or unachievable with current knowledge).
The proof of Fermat's Last Theorem in full, for all n, was finally accomplished 357 years later by Andrew Wiles in 1994, an achievement for which he was honoured and received numerous awards,
including the 2016 Abel Prize.^[3]^[4]^[5] The solution came in a roundabout manner, from a completely different area of mathematics.
Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two completely different areas of mathematics. Known at
the time as the Taniyama–Shimura-Weil conjecture, and (eventually) as the modularity theorem, it stood on its own, with no apparent connection to Fermat's Last Theorem. It was widely seen as
significant and important in its own right, but was (like Fermat's theorem) widely considered completely inaccessible to proof.
In 1984, Gerhard Frey noticed an apparent link between the modularity theorem and Fermat's Last Theorem. This potential link was confirmed two years later by Ken Ribet, who gave a conditional proof
of Fermat's Last Theorem that depended on the modularity theorem (see: Ribet's Theorem and Frey curve). On hearing this, English mathematician Andrew Wiles, who had a childhood fascination with
Fermat's Last Theorem, decided to try to prove the modularity theorem as a way to prove Fermat's Last Theorem. In 1993, after six years working secretly on the problem, Wiles succeeded in proving
enough of the modularity theorem to prove Fermat's Last Theorem for odd prime exponents. Wiles's paper was massive in size and scope. A flaw was discovered in one part of his original paper during
peer review and required a further year and collaboration with a past student, Richard Taylor, to resolve. As a result, the final proof in 1995 was accompanied by a second, smaller, joint paper to
that effect. Wiles's achievement was reported widely in the popular press, and was popularized in books and television programs. The remaining parts of the modularity theorem were subsequently proved
by other mathematicians, building on Wiles's work, between 1996 and 2001.
Equivalent statements of the theorem
There are several simple alternative ways to state Fermat's Last Theorem that are equivalent to the one given above. In order to state them, let N be the set of natural numbers 1,2,3,..., let Z be
the set of integers 0, ±1, ±2,..., and let Q be the set of rational numbers a/b where a and b are in Z with b≠0.
In what follows we will call a solution to x^n + y^n = z^n where one or more of x, y, or z is zero a trivial solution. A solution where all three are non-zero will be called a non-trivial solution.
For comparison's sake we start with the original formulation.
Original statement. With n, x, y, z ∈ N and n > 2 the equation x^n + y^n = z^n has no solutions.
Most popular domain treatments of the subject state it this way. In contrast, almost all math textbooks state it over Z:
Equivalent statement 1: x^n + y^n = z^n, where n ≥ 3, has no non-trivial solutions x, y, z ∈ Z.
The equivalence is clear if n is even. If n is odd and all three of x, y, z are negative then we can replace x, y, z with −x, −y, −z to obtain a solution in N. If two of them are negative, it must be
x and z or y and z. If x, z are negative and y is positive, then we can rearrange to get (−z)^n + y^n = (−x)^n resulting in a solution in N; the other case is dealt with analogously. Now if just one
is negative, it must be x or y. If x is negative, and y and z are positive, then it can be rearranged to get (−x)^n + z^n = y^n again resulting in a solution in N; if y is negative, the result
follows symmetrically. Thus in all cases a nontrivial solution in Z results in a solution in N.
Equivalent statement 2: x^n + y^n = z^n, where n ≥ 3, has no non-trivial solutions x, y, z ∈ Q.
This is because the exponent of x, y and z are equal (to n), so if there is a solution in Q then it can be multiplied through by an appropriate common denominator to get a solution in Z, and hence in
Equivalent statement 3: x^n + y^n = 1, where n ≥ 3, has no non-trivial solutions x, y ∈ Q.
A non-trivial solution a, b, c ∈ Z to x^n + y^n = z^n yields the non-trivial solution a/c, b/c ∈ Q for v^n + w^n = 1. Conversely, a solution a/b, c/d ∈ Q to v^n + w^n = 1 yields the non-trivial
solution ad, cb, bd for x^n + y^n = z^n.
This last formulation is particularly fruitful, because it reduces the problem from a problem about surfaces in three dimensions to a problem about curves in two dimensions. Furthermore, it allows
working over the field Q, rather than over the ring Z; fields exhibit more structure than rings, which allows for deeper analysis of their elements.
Connection to elliptic curves: If a, b, c is a non-trivial solution to x^p + y^p = z^p , p odd prime, then y^2 = x(x − a^p)(x + b^p) (Frey curve) is an elliptic curve.^[6]
Examining this elliptic curve with Ribet's theorem shows that it cannot have a modular form. The proof by Andrew Wiles shows that y^2 = x(x − a^n)(x + b^n) always has a modular form. This implies
that a non-trivial solution to x^p + y^p = z^p, p odd prime, would create a contradiction. This shows that no non-trivial solutions exist.^[7]
Mathematical history
Pythagoras and Diophantus
Pythagorean triples
A Pythagorean triple – named for the ancient Greek Pythagoras – is a set of three integers (a, b, c) that satisfy a special case of Fermat's equation (n = 2)^[8]
Examples of Pythagorean triples include (3, 4, 5) and (5, 12, 13). There are infinitely many such triples,^[9] and methods for generating such triples have been studied in many cultures, beginning
with the Babylonians^[10] and later ancient Greek, Chinese, and Indian mathematicians.^[1] The traditional interest in Pythagorean triples connects with the Pythagorean theorem;^[11] in its converse
form, it states that a triangle with sides of lengths a, b, and c has a right angle between the a and b legs when the numbers are a Pythagorean triple. Fermat's Last Theorem is an extension of this
problem to higher powers, stating that no solution exists when the exponent 2 is replaced by any larger integer.
Diophantine equations
Fermat's equation, x^n + y^n = z^n with positive integer solutions, is an example of a Diophantine equation,^[12] named for the 3rd-century Alexandrian mathematician, Diophantus, who studied them and
developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers x and y such that their sum, and the sum of their squares, equal two
given numbers A and B, respectively:
Diophantus's major work is the Arithmetica, of which only a portion has survived.^[13] Fermat's conjecture of his Last Theorem was inspired while reading a new edition of the Arithmetica,^[14] that
was translated into Latin and published in 1621 by Claude Bachet.^[15]
Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation x^2 + y^2 = z^2 are given by the Pythagorean triples, originally
solved by the Babylonians (c. 1800 BC).^[16] Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm (c. 5th century BC).^[17] Many Diophantine
equations have a form similar to the equation of Fermat's Last Theorem from the point of view of algebra, in that they have no cross terms mixing two letters, without sharing its particular
properties. For example, it is known that there are infinitely many positive integers x, y, and z such that x^n + y^n = z^m where n and m are relatively prime natural numbers.^[note 2]
Fermat's conjecture
Problem II.8 of the Arithmetica asks how a given square number is split into two other squares; in other words, for a given rational number k, find rational numbers u and v such that k^2 = u^2 + v^2.
Diophantus shows how to solve this sum-of-squares problem for k = 4 (the solutions being u = 16/5 and v = 12/5).^[18]
Around 1637, Fermat wrote his Last Theorem in the margin of his copy of the Arithmetica next to Diophantus’ sum-of-squares problem:^[19]
Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos & generaliter nullam in It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in
infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei general, any power higher than the second, into two like powers. I have discovered a truly marvelous
demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. proof of this, which this margin is too narrow to contain.^[20]^[21]
After Fermat’s death in 1665, his son Clément-Samuel Fermat produced a new edition of the book (1670) augmented with his father’s comments.^[22] The margin note became known as Fermat’s Last Theorem,
^[23] as it was the last of Fermat’s asserted theorems to remain unproved.^[24]
It is not known whether Fermat had actually found a valid proof for all exponents n, but it appears unlikely. Only one related proof by him has survived, namely for the case n = 4, as described in
the section Proofs for specific exponents. While Fermat posed the cases of n = 4 and of n = 3 as challenges to his mathematical correspondents, such as Marin Mersenne, Blaise Pascal, and John Wallis,
^[25] he never posed the general case.^[26] Moreover, in the last thirty years of his life, Fermat never again wrote of his "truly marvelous proof" of the general case, and never published it. Van
der Poorten^[27] suggests that while the absence of a proof is insignificant, the lack of challenges means Fermat realised he did not have a proof; he quotes Weil^[28] as saying Fermat must have
briefly deluded himself with an irretrievable idea.
The techniques Fermat might have used in such a "marvelous proof" are unknown.
Taylor and Wiles’s proof relies on 20th-century techniques.^[29] Fermat’s proof would have had to be elementary by comparison, given the mathematical knowledge of his time.
While Harvey Friedman’s grand conjecture implies that any provable theorem (including Fermat’s last theorem) can be proved using only ‘elementary function arithmetic’, such a proof need be
‘elementary’ only in a technical sense and could involve millions of steps, and thus be far too long to have been Fermat’s proof.
Proofs for specific exponents
Only one relevant proof by Fermat has survived, in which he uses the technique of infinite descent to show that the area of a right triangle with integer sides can never equal the square of an
integer.^[30]^[31] His proof is equivalent to demonstrating that the equation
has no primitive solutions in integers (no pairwise coprime solutions). In turn, this proves Fermat's Last Theorem for the case n = 4, since the equation a^4 + b^4 = c^4 can be written as c^4 − b^4 =
Alternative proofs of the case n = 4 were developed later^[32] by Frénicle de Bessy (1676),^[33] Leonhard Euler (1738),^[34] Kausler (1802),^[35] Peter Barlow (1811),^[36] Adrien-Marie Legendre
(1830),^[37] Schopis (1825),^[38] Terquem (1846),^[39] Joseph Bertrand (1851),^[40] Victor Lebesgue (1853, 1859, 1862),^[41] Theophile Pepin (1883),^[42] Tafelmacher (1893),^[43] David Hilbert
(1897),^[44] Bendz (1901),^[45] Gambioli (1901),^[46] Leopold Kronecker (1901),^[47] Bang (1905),^[48] Sommer (1907),^[49] Bottari (1908),^[50] Karel Rychlík (1910),^[51] Nutzhorn (1912),^[52] Robert
Carmichael (1913),^[53] Hancock (1931),^[54] and Vrǎnceanu (1966).^[55]
For another proof for n=4 by infinite descent, see Infinite descent: Non-solvability of r^2 + s^4 = t^4. For various proofs for n=4 by infinite descent, see Grant and Perella (1999),^[56] Barbara
(2007),^[57] and Dolan (2011).^[58]
After Fermat proved the special case n = 4, the general proof for all n required only that the theorem be established for all odd prime exponents.^[59] In other words, it was necessary to prove only
that the equation a^n + b^n = c^n has no integer solutions (a, b, c) when n is an odd prime number. This follows because a solution (a, b, c) for a given n is equivalent to a solution for all the
factors of n. For illustration, let n be factored into d and e, n = de. The general equation
a^n + b^n = c^n
implies that (a^d, b^d, c^d) is a solution for the exponent e
(a^d)^e + (b^d)^e = (c^d)^e.
Thus, to prove that Fermat's equation has no solutions for n > 2, it would suffice to prove that it has no solutions for at least one prime factor of every n. Each integer n > 2 is divisible by 4 or
an odd prime number (or both). Therefore, Fermat's Last Theorem could be proved for all n if it could be proved for n = 4 and for all odd primes p.
In the two centuries following its conjecture (1637–1839), Fermat's Last Theorem was proved for three odd prime exponents p = 3, 5 and 7. The case p = 3 was first stated by Abu-Mahmud Khojandi (10th
century), but his attempted proof of the theorem was incorrect.^[60] In 1770, Leonhard Euler gave a proof of p = 3,^[61] but his proof by infinite descent^[62] contained a major gap.^[63] However,
since Euler himself had proved the lemma necessary to complete the proof in other work, he is generally credited with the first proof.^[64] Independent proofs were published^[65] by Kausler (1802),^
[35] Legendre (1823, 1830),^[37]^[66] Calzolari (1855),^[67] Gabriel Lamé (1865),^[68] Peter Guthrie Tait (1872),^[69] Günther (1878),^[70] Gambioli (1901),^[46] Krey (1909),^[71] Rychlík (1910),^
[51] Stockhaus (1910),^[72] Carmichael (1915),^[73] Johannes van der Corput (1915),^[74] Axel Thue (1917),^[75] and Duarte (1944).^[76] The case p = 5 was proved^[77] independently by Legendre and
Peter Gustav Lejeune Dirichlet around 1825.^[78] Alternative proofs were developed^[79] by Carl Friedrich Gauss (1875, posthumous),^[80] Lebesgue (1843),^[81] Lamé (1847),^[82] Gambioli (1901),^[46]^
[83] Werebrusow (1905),^[84] Rychlík (1910),^[85] van der Corput (1915),^[74] and Guy Terjanian (1987).^[86] The case p = 7 was proved^[87] by Lamé in 1839.^[88] His rather complicated proof was
simplified in 1840 by Lebesgue,^[89] and still simpler proofs^[90] were published by Angelo Genocchi in 1864, 1874 and 1876.^[91] Alternative proofs were developed by Théophile Pépin (1876)^[92] and
Edmond Maillet (1897).^[93]
Fermat's Last Theorem was also proved for the exponents n = 6, 10, and 14. Proofs for n = 6 were published by Kausler,^[35] Thue,^[94] Tafelmacher,^[95] Lind,^[96] Kapferer,^[97] Swift,^[98] and
Breusch.^[99] Similarly, Dirichlet^[100] and Terjanian^[101] each proved the case n = 14, while Kapferer^[97] and Breusch^[99] each proved the case n = 10. Strictly speaking, these proofs are
unnecessary, since these cases follow from the proofs for n = 3, 5, and 7, respectively. Nevertheless, the reasoning of these even-exponent proofs differs from their odd-exponent counterparts.
Dirichlet's proof for n = 14 was published in 1832, before Lamé's 1839 proof for n = 7.^[102]
All proofs for specific exponents used Fermat's technique of infinite descent, either in its original form, or in the form of descent on elliptic curves or abelian varieties. The details and
auxiliary arguments, however, were often ad hoc and tied to the individual exponent under consideration.^[103] Since they became ever more complicated as p increased, it seemed unlikely that the
general case of Fermat's Last Theorem could be proved by building upon the proofs for individual exponents.^[103] Although some general results on Fermat's Last Theorem were published in the early
19th century by Niels Henrik Abel and Peter Barlow,^[104]^[105] the first significant work on the general theorem was done by Sophie Germain.^[106]
Sophie Germain
In the early 19th century, Sophie Germain developed several novel approaches to prove Fermat's Last Theorem for all exponents.^[107] First, she defined a set of auxiliary primes θ constructed from
the prime exponent p by the equation θ = 2hp + 1, where h is any integer not divisible by three. She showed that, if no integers raised to the p^th power were adjacent modulo θ (the non-consecutivity
condition), then θ must divide the product xyz. Her goal was to use mathematical induction to prove that, for any given p, infinitely many auxiliary primes θ satisfied the non-consecutivity condition
and thus divided xyz; since the product xyz can have at most a finite number of prime factors, such a proof would have established Fermat's Last Theorem. Although she developed many techniques for
establishing the non-consecutivity condition, she did not succeed in her strategic goal. She also worked to set lower limits on the size of solutions to Fermat's equation for a given exponent p, a
modified version of which was published by Adrien-Marie Legendre. As a byproduct of this latter work, she proved Sophie Germain's theorem, which verified the first case of Fermat's Last Theorem
(namely, the case in which p does not divide xyz) for every odd prime exponent less than 100.^[107]^[108] Germain tried unsuccessfully to prove the first case of Fermat's Last Theorem for all even
exponents, specifically for n = 2p, which was proved by Guy Terjanian in 1977.^[109] In 1985, Leonard Adleman, Roger Heath-Brown and Étienne Fouvry proved that the first case of Fermat's Last Theorem
holds for infinitely many odd primes p.^[110]
Ernst Kummer and the theory of ideals
In 1847, Gabriel Lamé outlined a proof of Fermat's Last Theorem based on factoring the equation x^p + y^p = z^p in complex numbers, specifically the cyclotomic field based on the roots of the number
1. His proof failed, however, because it assumed incorrectly that such complex numbers can be factored uniquely into primes, similar to integers. This gap was pointed out immediately by Joseph
Liouville, who later read a paper that demonstrated this failure of unique factorisation, written by Ernst Kummer.
Kummer set himself the task of determining whether the cyclotomic field could be generalized to include new prime numbers such that unique factorisation was restored. He succeeded in that task by
developing the ideal numbers. Using the general approach outlined by Lamé, Kummer proved both cases of Fermat's Last Theorem for all regular prime numbers. However, he could not prove the theorem for
the exceptional primes (irregular primes) that conjecturally occur approximately 39% of the time; the only irregular primes below 100 are 37, 59 and 67.
Mordell conjecture
In the 1920s, Louis Mordell posed a conjecture that implied that Fermat's equation has at most a finite number of nontrivial primitive integer solutions, if the exponent n is greater than two.^[111]
This conjecture was proved in 1983 by Gerd Faltings,^[112] and is now known as Faltings' theorem.
Computational studies
In the latter half of the 20th century, computational methods were used to extend Kummer's approach to the irregular primes. In 1954, Harry Vandiver used a SWAC computer to prove Fermat's Last
Theorem for all primes up to 2521.^[113] By 1978, Samuel Wagstaff had extended this to all primes less than 125,000.^[114] By 1993, Fermat's Last Theorem had been proved for all primes less than four
However despite these efforts and their results, no proof existed of Fermat's Last Theorem. Proofs of individual exponents by their nature could never prove the general case: even if all exponents
were verified up to an extremely large number X, a higher exponent beyond X might still exist for which the claim was not true. (This had been the case with some other past conjectures, and it could
not be ruled out in this conjecture.)
Connection with elliptic curves
The strategy that ultimately led to a successful proof of Fermat's Last Theorem arose from the "astounding"^[116]^:211 Taniyama–Shimura-Weil conjecture, proposed around 1955—which many mathematicians
believed would be near to impossible to prove,^[116]^:223 and was linked in the 1980s by Gerhard Frey, Jean-Pierre Serre and Ken Ribet to Fermat's equation. By accomplishing a partial proof of this
conjecture in 1994, Andrew Wiles ultimately succeeded in proving Fermat's Last Theorem, as well as leading the way to a full proof by others of what is now the modularity theorem.
Taniyama–Shimura–Weil conjecture
Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama observed a possible link between two apparently completely distinct branches of mathematics, elliptic curves and modular forms.
The resulting modularity theorem (at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is modular, meaning that it can be associated with a unique modular form.
It was initially dismissed as unlikely or highly speculative, and was taken more seriously when number theorist André Weil found evidence supporting it, but no proof; as a result the conjecture was
often known as the Taniyama–Shimura-Weil conjecture. It became a part of the Langlands programme, a list of important conjectures needing proof or disproof.^[116]^:211–215
Even after gaining serious attention, the conjecture was seen by contemporary mathematicians as extraordinarily difficult or perhaps inaccessible to proof.^[116]^:203–205, 223, 226 For example,
Wiles's ex-supervisor John Coates states that it seemed "impossible to actually prove",^[116]^:226 and Ken Ribet considered himself "one of the vast majority of people who believed [it] was
completely inaccessible", adding that "Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove [it]."^[116]^:223
Ribet's theorem for Frey curves
In 1984, Gerhard Frey noted a link between Fermat's equation and the modularity theorem, then still a conjecture. If Fermat's equation had any solution (a, b, c) for exponent p > 2, then it could be
shown that the elliptic curve (now known as a Frey-Hellegouarch^[note 3])
y^2 = x (x − a^p)(x + b^p)
would have such unusual properties that it was unlikely to be modular.^[117] This would conflict with the modularity theorem, which asserted that all elliptic curves are modular. As such, Frey
observed that a proof of the Taniyama–Shimura-Weil conjecture would simultaneously prove Fermat's Last Theorem^[118] and equally, a disproof or refutation of Fermat's Last Theorem would disprove the
Following this strategy, a proof of Fermat's Last Theorem required two steps. First, it was necessary to prove the modularity theorem – or at least to prove it for the sub-class of cases (known as
semistable elliptic curves) that included Frey's equation – and this was widely believed inaccessible to proof by contemporary mathematicians.^[116]^:203–205, 223, 226 Second, it was necessary to
show that Frey's intuition was correct: that if an elliptic curve were constructed in this way, using a set of numbers that were a solution of Fermat's equation, the resulting elliptic curve could
not be modular. Frey did not quite succeed in proving this rigorously; the missing piece (the so-called "epsilon conjecture", now known as Ribet's theorem) was identified by Jean-Pierre Serre and
proved in 1986 by Ken Ribet.^[119]
• The modularity theorem – if proved – would mean all elliptic curves (or at least all semistable elliptic curves) are of necessity modular.
• Ribet's theorem – proved in 1986 – showed that, if a solution to Fermat's equation existed, it could be used to create a semistable elliptic curve that was not modular;
• The contradiction would imply (if the modularity theorem were correct) that no solutions can exist to Fermat's equation – therefore proving Fermat's Last Theorem.
Wiles's general proof
Ribet's proof of the epsilon conjecture in 1986 accomplished the first of the two goals proposed by Frey. Upon hearing of Ribet's success, Andrew Wiles, an English mathematician with a childhood
fascination with Fermat's Last Theorem, and a prior study area of elliptical equations, decided to commit himself to accomplishing the second half: proving a special case of the modularity theorem
(then known as the Taniyama–Shimura conjecture) for semistable elliptic curves.^[120]
Wiles worked on that task for six years in near-total secrecy, covering up his efforts by releasing prior work in small segments as separate papers and confiding only in his wife.^[116]^:229–230 His
initial study suggested proof by induction,^[116]^:230–232, 249–252 and he based his initial work and first significant breakthrough on Galois theory^[116]^:251–253, 259 before switching to an
attempt to extend horizontal Iwasawa theory for the inductive argument around 1990–91 when it seemed that there was no existing approach adequate to the problem.^[116]^:258–259 However, by the summer
of 1991, Iwasawa theory also seemed to not be reaching the central issues in the problem.^[116]^:259–260^[121] In response, he approached colleagues to seek out any hints of cutting edge research and
new techniques, and discovered an Euler system recently developed by Victor Kolyvagin and Matthias Flach that seemed "tailor made" for the inductive part of his proof.^[116]^:260–261 Wiles studied
and extended this approach, which worked. Since his work relied extensively on this approach, which was new to mathematics and to Wiles, in January 1993 he asked his Princeton colleague, Nick Katz,
to check his reasoning for subtle errors. Their conclusion at the time was that the techniques Wiles used seemed to work correctly.^[116]^:261–265^[122]
By mid-May 1993, Wiles felt able to tell his wife he thought he had solved the proof of Fermat's Last Theorem,^[116]^:265 and by June he felt sufficiently confident to present his results in three
lectures delivered on 21–23 June 1993 at the Isaac Newton Institute for Mathematical Sciences.^[123] Specifically, Wiles presented his proof of the Taniyama–Shimura conjecture for semistable elliptic
curves; together with Ribet's proof of the epsilon conjecture, this implied Fermat's Last Theorem. However, it became apparent during peer review that a critical point in the proof was incorrect. It
contained an error in a bound on the order of a particular group. The error was caught by several mathematicians refereeing Wiles's manuscript including Katz (in his role as reviewer),^[124] who
alerted Wiles on 23 August 1993.^[125]
The error would not have rendered his work worthless – each part of Wiles's work was highly significant and innovative by itself, as were the many developments and techniques he had created in the
course of his work, and only one part was affected.^[116]^:289, 296–297 However without this part proved, there was no actual proof of Fermat's Last Theorem. Wiles spent almost a year trying to
repair his proof, initially by himself and then in collaboration with Richard Taylor, without success.^[126]
On 19 September 1994, on the verge of giving up, Wiles had a flash of insight that the proof could be saved by returning to his original Horizontal Iwasawa theory approach, which he had abandoned in
favour of the Kolyvagin–Flach approach, this time strengthening it with expertise gained in Kolyvagin–Flach's approach.^[127] On 24 October 1994, Wiles submitted two manuscripts, "Modular elliptic
curves and Fermat's Last Theorem"^[128] and "Ring theoretic properties of certain Hecke algebras",^[129] the second of which was co-authored with Taylor and proved that certain conditions were met
that were needed to justify the corrected step in the main paper. The two papers were vetted and published as the entirety of the May 1995 issue of the Annals of Mathematics. These papers established
the modularity theorem for semistable elliptic curves, the last step in proving Fermat's Last Theorem, 358 years after it was conjectured.
Subsequent developments
The full Taniyama–Shimura–Weil conjecture was finally proved by Diamond (1996), Conrad, Diamond & Taylor (1999), and Breuil et al. (2001) who, building on Wiles's work, incrementally chipped away at
the remaining cases until the full result was proved.^[130]^[131]^[132] The now fully proved conjecture became known as the modularity theorem.
Several other theorems in number theory similar to Fermat's Last Theorem also follow from the same reasoning, using the modularity theorem. For example: no cube can be written as a sum of two coprime
n-th powers, n ≥ 3. (The case n = 3 was already known by Euler.)
Exponents other than positive integers
Reciprocal integers (inverse Fermat equation)
The equation can be considered the "inverse" Fermat equation. All solutions of this equation were computed by Lenstra in 1992.^[133] In the case in which the m^th roots are required to be real and
positive, all solutions are given by^[134]
for positive integers r, s, t with s and t coprime.
Rational exponents
For the Diophantine equation with n not equal to 1, Bennett, Glass, and Székely proved in 2004 for n > 2, that if n and m are coprime, then there are integer solutions if and only if 6 divides m, and
, and are different complex 6th roots of the same real number.^[135]
Negative exponents
n = –1
All primitive (pairwise coprime) integer solutions to the optic equation can be written as^[136]
for positive, coprime integers m, n.
n = –2
The case n = –2 also has an infinitude of solutions, and these have a geometric interpretation in terms of right triangles with integer sides and an integer altitude to the hypotenuse.^[137]^[138]
All primitive solutions to are given by
for coprime integers u, v with v > u. The geometric interpretation is that a and b are the integer legs of a right triangle and d is the integer altitude to the hypotenuse. Then the hypotenuse itself
is the integer
so (a, b, c) is a Pythagorean triple.
Integer n < –2
There are no solutions in integers for for integers n < –2. If there were, the equation could be multiplied through by to obtain , which is impossible by Fermat's Last Theorem.
Base values other than positive integers
Fermat's last theorem can easily be extended to positive rationals:
can have no solutions with n > 2, because any solution could be rearranged as:
to which Fermat's Last Theorem applies.
Monetary prizes
In 1816, and again in 1850, the French Academy of Sciences offered a prize for a general proof of Fermat's Last Theorem.^[139] In 1857, the Academy awarded 3000 francs and a gold medal to Kummer for
his research on ideal numbers, although he had not submitted an entry for the prize.^[140] Another prize was offered in 1883 by the Academy of Brussels.^[141]
In 1908, the German industrialist and amateur mathematician Paul Wolfskehl bequeathed 100,000 gold marks—a large sum at the time—to the Göttingen Academy of Sciences to offer as a prize for a
complete proof of Fermat's Last Theorem.^[142] On 27 June 1908, the Academy published nine rules for awarding the prize. Among other things, these rules required that the proof be published in a
peer-reviewed journal; the prize would not be awarded until two years after the publication; and that no prize would be given after 13 September 2007, roughly a century after the competition was
begun.^[143] Wiles collected the Wolfskehl prize money, then worth $50,000, on 27 June 1997.^[144] In March 2016, Wiles was awarded the Norwegian government's Abel prize worth €600,000 for "for his
stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory."^[145]
Prior to Wiles's proof, thousands of incorrect proofs were submitted to the Wolfskehl committee, amounting to roughly 10 feet (3 meters) of correspondence.^[146] In the first year alone (1907–1908),
621 attempted proofs were submitted, although by the 1970s, the rate of submission had decreased to roughly 3–4 attempted proofs per month. According to F. Schlichting, a Wolfskehl reviewer, most of
the proofs were based on elementary methods taught in schools, and often submitted by "people with a technical education but a failed career".^[147] In the words of mathematical historian Howard Eves
, "Fermat's Last Theorem has the peculiar distinction of being the mathematical problem for which the greatest number of incorrect proofs have been published."^[141]
See also
1. ↑ If the exponent "n" were not prime or 4, then it would be possible to write n either as a product of two smaller integers (n = P*Q) in which P is a prime number greater than 2, and then a^n = a
^P*Q = (a^Q)^P for each of a, b, and c—i.e., an equivalent solution would also have to exist for the prime power P that is smaller than N, as well; or else as n would be a power of 2 greater than
four and writing n=4*Q, the same argument would hold.
2. ↑ For example,
3. ↑ This elliptic curve was first suggested in the 1960s by Yves Hellegouarch, but he did not call attention to its non-modularity. For more details, see Hellegouarch, Yves (2001). Invitation to
the Mathematics of Fermat-Wiles. Academic Press. ISBN 978-0-12-339251-0.
1. 1 2 Singh, pp. 18–20.
2. ↑ "Science and Technology". The Guinness Book of World Records. Guinness Publishing Ltd. 1995.
3. ↑ "Fermat's last theorem earns Andrew Wiles the Abel Prize". Nature. 15 March 2016. Retrieved 15 March 2016.
4. ↑ Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics. 141 (3): 448. doi:10.2307/2118559. JSTOR 2118559. OCLC 37032255. “Frey's suggestion, in
the notation of the following theorem, was to show that the (hypothetical) elliptic curve y^2 = x(x + u^p)(x - v^p) could not be modular.”
5. ↑ Ribet, Ken (1990). "On modular representations of Gal(Q/Q) arising from modular forms" (PDF). Inventiones mathematicae. 100 (2): 432. doi:10.1007/BF01231195. MR 1047143.
6. ↑ Stark, pp. 151–155.
7. ↑ Stillwell J (2003). Elements of Number Theory. New York: Springer-Verlag. pp. 110–112. ISBN 0-387-95587-9. Retrieved 2016-03-17.
8. ↑ Aczel, pp. 13–15
9. ↑ Singh, p. 6.
10. ↑ Stark, pp. 145–146.
11. ↑ Singh, pp. 50–51.
12. ↑ Stark, p. 145.
13. ↑ Aczel, pp. 44–45; Singh, pp. 56–58.
14. ↑ Aczel, pp. 14–15.
15. ↑ Stark, pp. 44–47.
16. ↑ Friberg, pp. 333– 334.
17. ↑ Dickson, p. 731; Singh, pp. 60–62; Aczel, p. 9.
18. ↑ T. Heath, Diophantus of Alexandria Second Edition, Cambridge University Press, 1910, reprinted by Dover, NY, 1964, pp. 144-145
19. ↑ Singh, pp. 62–66.
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105. ↑ Terjanian, G. (1977). "Sur l'équation x^2p + y^2p = z^2p". Comptes rendus hebdomadaires des séances de l'Académie des sciences. Série a et B. 285: 973–975.
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116. ↑ Singh, pp. 237–238; Aczel, pp. 121–122.
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118. ↑ Singh, pp. 244–253; Aczel, pp. 1–4, 126–128.
119. ↑ Aczel, pp. 128–130.
120. ↑ Singh, p. 257.
121. ↑ Singh, pp. 269–274.
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129. ↑ Newton, M., "A radical diophantine equation", Journal of Number Theory 13 (1981), 495–498.
130. ↑ Bennett, Curtis D.; Glass, A. M. W.; Székely, Gábor J. (2004). "Fermat's last theorem for rational exponents". American Mathematical Monthly. 111 (4): 322–329. doi:10.2307/4145241. MR 2057186.
131. ↑ Dickson, pp. 688–691
132. ↑ Voles, Roger (July 1999). "Integer solutions of a^−2 + b^−2 = d^−2". Mathematical Gazette. 83: 269–271.
133. ↑ Richinick, Jennifer (July 2008). "The upside-down Pythagorean Theorem". Mathematical Gazette. 92: 313–317.
134. ↑ Aczel, p. 69; Singh, p. 105.
135. ↑ Aczel, p. 69.
136. 1 2 Koshy T (2001). Elementary number theory with applications. New York: Academic Press. p. 544. ISBN 978-0-12-421171-1.
137. ↑ Singh, pp. 120–125, 131–133, 295–296; Aczel, p. 70.
138. ↑ Singh, pp. 120–125.
139. ↑ Singh, p. 284
140. ↑ "The Abel Prize citation 2016" (PDF). The Abel Prize. The Abel Prize Committee. March 2016. Retrieved 16 March 2016.
141. ↑ Singh, p. 295.
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The symbol '<' means 'is less than'.
If the open sentences on the right are true, which open sentence below is also true?
Photographs that are hung next to each other share a tack.
How many tacks are needed to hang 21 photographs this way?
Anna and Bill's photos show water.
Only Cindy and Bill's photos show a man.
Which photograph belongs to David?
First go 1 block north, then go 2 blocks east, then go 3 blocks south, and finally go 4 blocks west.
Where do you end up?
The desks in the classroom are lined up in straight rows.
Her desk is also the third from the left and the first from the right.
How many students' desks are there?
An arrow pointing from one player to another means that the first player defeated the second player in their match.
Each match always gives a winner and a loser (no draws).
Who played the smallest number of matches?
Photographs that are hung next to each other share a tack.
How many tacks are needed to hang 44 photographs this way?
Anna and Bill's photos show water.
Cindy and Bill's photos don't show people.
Which photograph belongs to David?
First go 2 blocks north, then go 1 block east, then go 4 blocks south, and finally go 2 blocks west.
Where do you end up?
There are more than 3 eggs.
There are fewer than 6 eggs.
There are not 4 eggs.
How many eggs does Henrietta lay?
Each of the teams plays two matches with every other team.
The top qualified two teams play one game in the playoff.
How many matches does the winner play?
What is the largest number of 3-point throws he might have made?
Points in basketball can be accumulated by making field goals (two or three points) or free throws (one point). If a player makes a field goal from within the three-point line, the player scores two
At Easter time, each member of the family buys one chocolate Easter egg for each of the others.
How many Easter eggs will be bought in total?
There are 10 coins in the first pile.
There are 15 coins in the second pile.
There are 20 coins in the third pile.
If Martin shares the coins among 5 people, how many coins does each person get? | {"url":"https://aplusclick.org/k/k4-wordproblems.htm","timestamp":"2024-11-10T16:15:39Z","content_type":"text/html","content_length":"58639","record_id":"<urn:uuid:0a0dba86-c3b1-488c-bc73-5212b01ff2f0>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00669.warc.gz"} |
One of the first “real” programs I ever wrote would encrypt a message using a substitution cipher and (more impressively) decrypt an encoded message without knowing the encryption key. Perhaps the
first thing I had to come up with was an algorithm to take a cipher key that was used to encode a message, and transform it so that it could then be used do decode the ciphertext back into plaintext.
I was aware of ROT-13’s property that if applied once to the plaintext, and again to the output ciphertext, it would yield the original plaintext. That is, ROT-13 is its own inverse. However, this is
not the case for all possible keys, so I needed to find a more general algorithm. | {"url":"https://wyatts.xyz/blog?tags=cryptography","timestamp":"2024-11-11T23:32:54Z","content_type":"text/html","content_length":"13862","record_id":"<urn:uuid:2fc32c1e-0253-4f71-8e10-2dcdd107d1fc>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00095.warc.gz"} |
Solving N Diagonal Problem - Pak Jiddat Website
I took the What is a proof course on Coursera and found it very useful. The course provides useful knowledge about mathematical proofs. I found the course very useful and interesting. In this article
I will describe one of the problems that were given to the students to solve.
In the N Diagonal problem, we have a board containing NxN squares. Each square can have 2 diagonals, but the diagonals of neighboring squares should not intersect. A square can have three possible
states. Either empty, a left diagonal or a right diagonal. We need to find out if it is possible to have X number of diagonals.
Solution Overview
We have to go one step at a time, from the bottom left square and from left to right. A square can have 3 possible values. 0, 1 and 2. 0 means empty, 1 means diagonal is from right to left. 2 means
diagonal is from left to right.
We can use a list to store the square values in Python. Python does not have built in support for arrays, so we can use a list instead.
First we need to find all possible values that a square can have. Since each square has three possible values, there are many possible combinations of values of a squares neighbors.
We can instead check if there is some combination of neighbor values which makes it impossible for the square to have a value of 1. Similarly we need to check if there is some combination of neighbor
values which makes it impossible for the square to have a value of 2.
A square cannot have a value of 1 if the left square has a value of 2 or bottom square has a value of 2 or bottom right square has a value of 1.
A square cannot have a value of 2 if the left square has a value of 1 or bottom square has a value of 1 or bottom left square has a value of 2.
Pseudo Code
Here is the pseudo code:
function SolveNDiagonal(n, diag_count, arr, index):
if sum of 1s and 2s in list is more than N, then
print list
For each x in arr starting from index upto NxN
values = GetPossibleValuesOfSquareAtX(n, arr, x)
For each val in values,
if val is more than 0, then
SolveNDiagonal(n, diag_count, arr, index+1)
The function is called as follows:
SolveNDiagonal(n, diag_count, arr, 0)
Where n=5, diag_count=16, arr is a Python list initialized with 0s. The last parameter is the starting index, which is 0.
Source code in Python
# Main function
# n is the number of rows or cols in the board
# diag_count is the required number of diagonals
# arr is a list of size NxN
# index is the starting position
# The function prints state of all squares on the board
# Such that the board has diag_count number of diagonals
def solve_n_diagonal(n, diag_count, arr, index):
if (arr.count(1) + arr.count(2)) == diag_count:
print("Solved for n = " + str(n) +
" and diagonal count = " + str(diag_count))
for x in range(index, n*n):
values = get_element_values(n, arr, x)
for val in values:
arr[x] = val
if val > 0:
solve_n_diagonal(n, diag_count, arr, x+1)
# The values of neighboring squares
# We are only concerned with the values of squares on the
# left, bottom, bottom right and bottom left
def get_neighbor_values(n, arr, x):
neighbor_values = {"left": -1, "bottom_right": -1, "bottom_left": -1,
"bottom": -1}
if x % n != 0:
neighbor_values["left"] = arr[x-1]
if x >= n and (x % n != n -1):
neighbor_values["bottom_right"] = arr[x-(n-1)]
if x >= n and (x % n != 0):
neighbor_values["bottom_left"] = arr[x-(n+1)]
if x >= n:
neighbor_values["bottom"] = arr[x-n]
return neighbor_values
# Get the possible values that the square at position x can have
# The possible values are returned in a list
def get_element_values(n, arr, x):
neighbor_values = get_neighbor_values(n, arr, x)
values = list()
is_one_valid = True
is_two_valid = True
if (neighbor_values["left"] == 2 or neighbor_values["bottom"] == 2
or neighbor_values["bottom_right"] == 1):
is_one_valid = False
if (neighbor_values["left"] == 1 or neighbor_values["bottom"] == 1
or neighbor_values["bottom_left"] == 2):
is_two_valid = False
if is_one_valid:
if is_two_valid:
return values
# Initialize a list of size n*n with 0s
def initialize_array(n):
arr = list()
for x in range(n*n):
return arr
# The number of rows or cols in the board
n = 5
# The required number of diagonals
diag_count = 16
# The list is initialized
arr = initialize_array(n)
# The main function is called
solve_n_diagonal(n, diag_count, arr, 0)
The above code prints the state of each square on a board, such that a square is either empty or it has a diagonal which does not intersect with the diagonals of its neighboring squares. | {"url":"https://pakjiddat.netlify.app/posts/solving-n-diagonal-problem","timestamp":"2024-11-12T00:54:23Z","content_type":"text/html","content_length":"53844","record_id":"<urn:uuid:b89707a6-7001-45a0-be0c-d4f8da766948>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00388.warc.gz"} |
Practical Examples of Gear Ratios | Math Ratios
Practical Examples of Gear Ratios
After gaining a conceptual understanding of gear ratios, it's time to dive into practical applications. Gear ratios are ubiquitous, from the bikes we ride to the cars we drive, and the clocks we use
to keep time. Let's explore some examples.
Bicycle Gear Ratios
Bicycles typically have multiple gears to help cyclists manage speed and effort. Each gear combination on a bicycle represents a different gear ratio.
For instance, consider a bicycle with a front chainring of 40 teeth and a rear sprocket of 20 teeth. The gear ratio is therefore 2:1. This means that for every pedal stroke (one rotation of the
chainring), the rear wheel makes two rotations.
Gear Ratio = Number of teeth on chainring / Number of teeth on sprocket
= 40 / 20
= 2:1
Automotive Gear Ratios
In cars, gear ratios play a vital role in determining speed, acceleration, and fuel efficiency. Each gear in a car's transmission has a specific ratio that enables the car to operate optimally under
different conditions.
For instance, first gear provides a high gear ratio, allowing the vehicle to accelerate from a standstill, while fifth gear (in a five-speed transmission) has a lower ratio, permitting higher speeds
with lower engine RPM.
Gear Ratios in Clocks
Inside a clock, a series of gears, each with its own ratio, drives the movement of the hands. In a typical clock, the minute hand moves 12 times faster than the hour hand, corresponding to a gear
ratio of 12:1.
Gear Ratio = Speed of minute hand / Speed of hour hand
= 12 / 1
= 12:1
Exploring Gear Ratios Through Problems
Let's consider a problem. Imagine you have a gear train with three gears: Gear A with 10 teeth, Gear B with 20 teeth, and Gear C with 30 teeth. Gear A is the driving gear, and Gear C is the driven
gear. What is the overall gear ratio?
In this case, the overall gear ratio would be the product of the individual gear ratios of A to B and B to C.
Overall Gear Ratio = (Teeth on B / Teeth on A) × (Teeth on C / Teeth on B)
= (20 / 10) × (30 / 20)
= 2 × 1.5
= 3:1
In this tutorial, we've explored some practical examples of gear ratios in bicycles, cars, and clocks. Understanding gear ratios can enhance your comprehension of how these everyday objects work, and
help you solve related mathematical problems.
Gear Ratios Tutorials
If you found this ratio information useful then you will likely enjoy the other ratio lessons and tutorials in this section: | {"url":"https://www.mathratios.com/tutorial/calculating-gear-ratios.html","timestamp":"2024-11-08T19:05:38Z","content_type":"text/html","content_length":"8759","record_id":"<urn:uuid:af12821a-de5f-4ee2-929f-4f356d58fb01>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00209.warc.gz"} |
Connected set
From Encyclopedia of Mathematics
A subset of an ambient set in which a notion of connectivity is defined, in the sense of which this subset is connected. For example, connected sets in the space of real numbers are convex sets, and
only them; a connected set of a graph is a set in which any two points can be joined by a path that lies entirely in this set. Cf. Connected space.
How to Cite This Entry:
Connected set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Connected_set&oldid=31025
This article was adapted from an original article by V.I. Malykhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article | {"url":"https://encyclopediaofmath.org/wiki/Connected_set","timestamp":"2024-11-11T09:51:38Z","content_type":"text/html","content_length":"13598","record_id":"<urn:uuid:0d34a64a-1046-4975-99e2-124a2d6cdda8>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00435.warc.gz"} |
Define Custom Metric Function
This topic explains how to define a custom metric function for your task. Use a custom metric function if Deep Learning Toolbox™ does not support the metric you need. For a list of built-in metrics
in Deep Learning Toolbox, see Metrics. If a built-in MATLAB^® function satisfies the required syntax, then you can use that function instead. For example, you can use the built-in l1loss function to
find the L[1] loss. For information about the required function syntax, see Create Custom Metric Function.
In deep learning, a metric is a numerical value that evaluates the performance of a deep learning network. You can use metrics to monitor how well a model is performing by comparing the model
predictions to the ground truth. Common deep learning metrics are accuracy, F-score, precision, recall, and root mean squared error.
If Deep Learning Toolbox does not provide the metric that you need for your task, then in many cases you can create a custom metric using a function. After you define the metric function, you can
specify the metric as the Metrics name-value argument in the trainingOptions function. Using a custom metric function for early stopping and returning the best network is not supported for custom
metric functions. If you require early stopping or retuning the best network, then you must create a custom metric object instead. For more information, see Define Custom Deep Learning Metric Object.
How To Decide Which Metric Type To Use
Create Custom Metric Function
To create a custom metric function, you can use this template.
function val = myMetricFunction(Y,T)
% Evaluate custom metric.
% Inputs:
% Y - Formatted dlarray of predictions
% T - Formatted dlarray of targets
% Outputs:
% val - Metric value
% Define the metric function here.
For categorical targets, the layout of the targets that the software passes to the metric depends on which function you want to use the metric with.
• When using the metric with trainnet and the targets are categorical arrays, if the loss function is "index-crossentropy", then the software automatically converts the targets to numeric class
indices and passes them to the metric. For other loss functions, the software converts the targets to one-hot encoded vectors and passes them to the metric.
• When using the metric with testnet and the targets are categorical arrays, if the specified metrics include "index-crossentropy" but do not include "crossentropy", then the software converts the
targets to numeric class indices and passes them to the metric. Otherwise, the software converts the targets to one-hot encoded vectors and passes them to the metric.
Depending on your metric, you sometimes need to know the dimension labels before computing the metric. Use the finddim function to find dimensions with a specific label. For example, to average your
metric across batches, you need to know the batch dimension.
When you have data in mini-batches, the software computes the metric for each mini-batch and then returns the average of those values. For some metrics, this behavior can result in a different metric
value than if you compute the metric using the whole data set at once. In most cases, the values are similar. To use a custom metric that is not batch-averaged for the data, you must create a custom
metric object. For more information, see Define Custom Deep Learning Metric Object.
To use the metric during training, specify the function handle as the Metrics option of the trainingOptions function.
trainingOptions("sgdm", ...
Example Regression Metric
For regression tasks, the function must accept a formatted dlarray object of predictions and targets.
This code shows an example of a regression metric. This custom metric function computes the symmetric mean absolute percentage error (SMAPE) value given predictions and targets. This equation defines
the SMAPE value:
$SMAPE=\frac{100}{n}\sum _{i=1}^{n}\frac{|{Y}_{i}-{T}_{i}|}{\left(|{T}_{i}|+|{Y}_{i}|\right)/2},$
where Y are the network predictions and T are the target responses.
function val = SMAPE(Y,T)
% Compute SMAPE value.
absoluteDifference = abs(Y-T);
absoluteAvg = (abs(Y) + abs(T))./2;
proportion = absoluteDifference./absoluteAvg;
val = 100*mean(proportion,"all");
Example Classification Metric
For classification tasks, the function must accept a formatted dlarray object of predictions and targets encoded as one-hot vectors. Each column in the vector represents a class and each row
represents an observation. For example, this code defines a one-hot vector. For more information, see the onehotencode function.
Y =
This code shows an example of a classification metric. This custom metric function computes the macro-averaged error rate value given predictions and targets. This equation defines the macro error
$errorRat{e}^{\left(macro\right)}=\frac{1}{K}\sum _{i=1}^{K}\frac{F{P}_{i}+F{N}_{i}}{T{P}_{i}+T{N}_{i}+F{P}_{i}+F{N}_{i}},$
where TP[i], TN[i], FP[i], and FN[i] represent the number of true positives, true negatives, false positives, and false negatives, respectively, in class i and K is the number of classes.
function val = errorRate(Y,T)
% Compute macro error rate value.
% Find the channel (class) dimension.
cDim = finddim(Y,"C");
bDim = finddim(Y,"B");
% Find the maximum score. This corresponds to the predicted
% class. Set the predicted class as 1 and all other classes as 0.
Y = Y == max(Y,[],cDim);
% Find the TP, FP, FN for this batch.
TP = sum(Y & T, bDim);
FP = sum(Y & ~T, bDim);
FN = sum(~Y & T, bDim);
TN = sum(~Y & ~T, bDim);
% Compute the error rate value and average across each class.
val = mean((FP + FN)./(TP + TN + FP + FN));
If your metric has a fraction whose denominator value can be zero, you can add eps to the denominator to prevent the metric returning a NaN value.
See Also
trainingOptions | trainnet | dlnetwork
Related Topics | {"url":"https://uk.mathworks.com/help/deeplearning/ug/define-custom-metric-function.html","timestamp":"2024-11-05T13:09:08Z","content_type":"text/html","content_length":"81675","record_id":"<urn:uuid:73c863bd-4a5e-4ce8-8ce6-02907523a81d>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00778.warc.gz"} |
Derivation of Liouville-like equation for the n-state probability density of an open system with thermalized particle reservoirs and its link to molecular simulation
Delle Site, L. and Klein, R. (2022) Derivation of Liouville-like equation for the n-state probability density of an open system with thermalized particle reservoirs and its link to molecular
simulation. J. Physics A:Math. Theor., 55 (15).
Full text not available from this repository.
Official URL: https://iopscience.iop.org/article/10.1088/1751-81...
A physico-mathematical model of open {systems} {proposed in a previous paper [ L.Delle Site and R.Klein, J.Math.Phys. 61, 083102 (2020)] can represent a guiding reference in designing an accurate
simulation scheme for an open molecular system embedded in a reservoir of energy and particles. The derived equations and the corresponding boundary conditions are obtained without assuming the
action of an external source of heat that assures thermodynamic consistency of the open system with respect to a state of reference. However, in numerical schemes the temperature in the reservoir
must be controlled by an external heat bath otherwise thermodynamic consistency cannot be achieved. In this perspective, the question to address is whether the explicit addition of an external heat
bath in the theoretical model modifies the equations of the open system and its boundary conditions. In this work we consider this aspect and explicitly describe the evolution of the reservoir
employing the Bergmann-Lebowitz statistical model of thermostat. It is shown that the resulting equations for the open system itself are not affected by this change and an example of numerical
application is reviewed where the current result shows its conceptual relevance.} Finally, a list of pending mathematical and modelling problems is discussed the solution of which would strengthen
the mathematical rigour of the model and offer new perspectives for the further development of a new multiscale simulation scheme.
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Numerical Investigation of the Optimization of PV-System Performances Using a Composite PCM-Metal Matrix for PV-Solar Panel Cooling System
Volume 8 - Year 2021 - Pages 262-274
DOI: 10.11159/jffhmt.2021.028
Numerical Investigation of the Optimization of PV-System Performances Using a Composite PCM-Metal Matrix for PV-Solar Panel Cooling System
Naomie Beolle Songwe Selabi^1*, Arnaud Regis Kamgue Lenwoue^2, Lesly Dasilva Wandji Djouonkep^3,4
^1 Institute of advanced materials and Nanotechnology, Wuhan University of Science and Technology, Wuhan, 430081, China
^2 Department of Petroleum Engineering, Leak Resistance & Sealing Technology Research Department, National Engineering
Laboratory of Petroleum Drilling Technology, Yangtze University, Wuhan, 43010, China
^3 Department of Petroleum Engineering, Applied Chemistry in Oil and Gas fields, Yangtze University, Wuhan, 43010, China ^4 Institute of fine organic chemistry and new organic materials, Wuhan
University of Science and Technology, Wuhan, 430081, China
Abstract - During the conversion of solar photovoltaic energy, the heat generated raises the temperature and results in reduced electricity conversion efficiency of the system. As the operating
temperature plays a great role in the photovoltaic conversion process, cooling the operating surface is a key factor to consider in achieving higher efficiency. Numerical investigation using
composite phase change materials (PCMs) in photovoltaic-cooling (PV-cooling) system was adopted in this study. Selected materials such as CaCl2.6H2O, paraffin wax, RT25, RT27, SP29, n-octadecane were
used as PCMs while copper, aluminium, steel, nickel, polystyrene, polychlorovinyl and polypropylene were used as composite(matrix) materials. A two-dimensional transient heat transfer model based on
enthalpy approach developed by computational Fluid Dynamics (CFD-Ansys-Fluent software) was utilized for optimization and enhancing the energy conversion efficiency. The numerical results showed that
RT25 sphere has good compatibility with PV-cooling system, and the thermal conductivity barely had a significant value on PV-temperature for larger values, excepted for very low thermal conductivity
materials such as plastics.
Keywords: PCMs; polymers; PV-cooling, Numerical model.
© Copyright 2021 Authors - This is an Open Access article published under the Creative Commons Attribution License terms Creative Commons Attribution License terms. Unrestricted use, distribution,
and reproduction in any medium are permitted, provided the original work is properly cited.
Date Received: 2021-12-06
Date Accepted: 2021-12-13
Date Published: 2021-12-15
1. Introduction
The planet is warming, from North Pole to South Pole. Since 1906, alarm and utmost concern that human activities have caused around 1.5°C of global warming to date and these impacts are already being
felt in every region of the globe.^[1,2] According to the 21^st climatic summit from Intergovernmental Panel on Climate Change (IPCC), this sudden variation in ambient temperature will be a great
fatality for the planet if it persists. Following the Paris agreement signed by 196 states, resolutions were taken to prevent a rise in temperatures beyond 2 °C compared to the pre-industrial era,
and if possible 1.5 °C in order to limit the damage.^[3] To achieve this goal, greenhouse gas emissions should be significantly reduced by employing efficient industrial techniques on one hand, and
maximizing the use of renewable energies at the expense of fossil fuels on the other. Renewable energy sources is generally defined as “energy obtained from the continuous or repetitive currents on
energy recurring in the natural environment” or as “energy flows which are replenished at the same rate as they are used”.^[4] Among these energies, we have biomass energy from organic animal or
plant residue, wind energy source from wind and solar energy from the sun. Solar (photovoltaic) is the most widely available and renewable form of energy source in the current era due to its facile
energy conversion efficiency. Photovoltaic (solar cells) are electronic devices that convert sunlight directly into electricity.^[5] The photovoltaic solar is an ancient technology, which was first
discovered by the scientist Edmond Becquerel in 1839.^[6] In 1882, Charles fritts made the first attempt of a working solar cell with thin sheets of selenium, coated with gold. It was later in 1900s
that mass production and industrialization of solar photovoltaics really took off. On April 1954, the physician Gerald Pearson and the chemist Calvin Fuller at Bell Laboratories both demonstrated the
first practical silicon solar cell with a 4% energy conversion efficiency and later achieved 11% efficiency upon optimization.^[7] However, converting solar energy into electrical energy has a major
drawback, which arises from the difficulty in controlling the rapid temperature increase in cells, which lowers the cells conversion capacities especially in northern regions. In order to make
photovoltaics a more mainstream and pragmatic energy source, the efficiency of solar panels need to be radically improved. In 1975, Telkes applied for the first time a phase change materials (PCMs)
technology for energy storage. He demonstrated that PCMs could absorb or “grasp” energy during the melt/solidification process. Aware of the enormous potential PCMs could offer for efficient energy
extraction, the latter rapidly became very attractive in solar applications, especially photovoltaic solar panels. Amount the PV-cooling technology classify by Chandel and al.,^[8] the PV-PCMs
cooling technology is of high interest due to their ability to delay the temperature rise in cell-panels without any form energy dissipation, were the heat stored can be reused and recycled further
enhancing the system efficiency.^[9]
Several works on numerical investigation of the photovoltaic PCMs system are summarized; Cellura et al.,^[10] made a theoretical analysis by using COMSOL MULTIPHYSICS a partial differential equations
(PDEs) solver to simulate the thermal behaviour of the PV-PCM system to improve the efficiency of the system. Meanwhile Biwole et al.,^[11] used the Computational Fluid Dynamics (CFD) software model
to simulate the thermo-physical properties of PCMs in PV-system. They added the PCMs at the back of the solar panel, which efficiently maintain the PV cells temperature below 40 °C for period of 2 h.
Xiang et al.,^[12] on the other hand used a hydride system with air in between PCMs to cool the PV cell and increase the conversion efficiency, and stored the energy for other applications. Khanna et
al.,^[13] use ANSYS software to study the effect of fin thinness, length and the spacing between two fins. Additionally, they also studied the effect of the operating conditions (wind azimuth angle
i.e. wind direction, wind velocity, melting temperature of PCMs and ambient temperature) on the PV cells. Their system was able to maintain the PV-cell temperature below 30 °C for approximatively 4
h. Saedodin et al.,^[14] during their investigation improved and optimized the fins used in PV cells by filling the solar collector with porous metal foam achieving an efficiency increase of more
than 2%. Recently Sarthe et al.,^[15] investigated the effect of variation in the angle of inclination of PV-PCM system. They observed a decrease in the time required for PCMs melting process, and an
increase of the PV surface temperature.
The aim of this work is to propose a new model compactible with composite PCMs to cool the PV-cells to improve the efficiency of PV-system. In this investigation, we used the ANSYS Fluent software to
numerically investigate the thermo-physical properties of the PV-cooling system with integrated composite PCMs for optimal energy conversions.
2. Experimental Model and Numerical Equation
In this study, four types of surface are described as shown on Fig.1. The first is the PV-panel surface composed of glass, silicon, teldar, EVA in sky blue; the second is the composite materials
surfaces in dark orange; third the PCM surface in violet and fourth the aluminium box in brown colour. The aluminium box container is a mixture of PCMs and other solid material. The PCM is introduced
into the solid material during its fabrication period. Gamma (γ) represents the inclination angle of the system. The symmetry boundary condition was applied at the top and bottom of the aluminium
box, and the backside was thermally insulated. The following boundary conditions were considered:
1. The initial temperature of the system is ambient (T[amb])
2. Because the energy is adsorbed at the silicon surface, the effect of the radiation is applied at the glass layer where emissivity (ε[t]) the rear surfaces of the system have respectively the
values h;
3. The variations in the thermal properties of the PCM are independent of the temperature. However, the solid and liquid phases are different.
4. The properties of the PCM in solid and liquid phases were homogeneous and isotropic and inside the melted PCM, the flow was considered incompressible and laminar.
5. The radiation condition is applied at the top and bottom of the PV with emissivity εt and εb.
Figure 1: PV-system with composite PCM.
Table 1: Parameters of the model
Parameter Value Parameter Value
L[pv] (PV-length) 1m E[g] ( glass thickness) 3mm
I[m] (PCM-box) interval 2.5mm E[E] (EVA thickness) 1mm
S[b] (interval between 2PCM bowl) 5mm E[t] (Teldar thickness) 0.1mm
l[pv] (PV width) 4.4mm E[s] (Silicon thickness) 0.3mm
l[b] (matrix width) 30mm ε[b] 0.91
W (space between PCM bowl row) 5mm ε[t] 0.85
e[b] (box thickness) 4mm γ 45^o
Number of PCM bowl 132
2. The aluminium box filled with composite material: PCM spherical bowls / metal
Here the PCM is introduced during the fabrication process into the cavities of a rectangular box (1000mm x 38mm x 4mm) whose walls are made of aluminium so the interior is a concrete made of metal,
melted at high temperature and hardened at room temperature. The amount of PCM in the box represents 22-67% by surface.
The 2-D unsteady equations governing the energy and momentum of heat transfer are solved using the simple implicit finite volume method with fluent 2020 R1. Additionally, the Boussinesq approximation
was adopted to account for the change in density of the PCM in liquid phase as a function of temperature.
Due to the reflection of the PV, the entire sun-base radiation incidence ray that arrives at the surface of the PV (I[T]) is not converted into energy. A fraction (ρPVI[T]) was lost due to reflection
and the rest (1 - ρPV)I[T] was absorbed by the system. Part of the absorbed radiation is converted into electricity and the rest (S[h]) dissipated as heat.
is the solar radiation to electricity conversion efficiency of the PV module. Considering that, the main contribution to the energy stored by the system owes only to the PCM, the stored energy (Q[S])
at time interval is given by equation 2:^[16]
Where T[m] is the melting temperature, T[PCM ]is the PCM temperature, H the latent heat of fusion of the PCM, T[amb] the ambient temperature. The complete balance of energy of the system is writing
T[P] is the PV cell average temperature, and U[l] the overall heat transfer coefficient.
The inclination angle of the system was set at 45° based on Khanna et al.,^[17] studies. The Nussle (Nu) and the Rayleigh (Ra) number at the top and bottom of the PV can be written as follow;
equation 4:
where Pr is the Prandtl number of air, Gr[c] is the critical Grashof number = and Ra is the Rayleigh number, which is given by:
Where v (m/s) can be define as the velocity.
3. PCM system equation
The liquid fraction varies mildly and continuously across the mushy region. This mushy zone is described by the governing equations to express the phase change phenomena.
where ρ is the density in kg/m^3 and u the speed in m/s. The conservation of the momentum is given by:
Where v is the kinematic viscosity (m^2/s), p is the pressure in the fluid (Pa) starting from E[q] (6) and E[q] (7) and considering that, there is heat conversion during the phase change, the heat
equations can be expressed by:
C[p] Specific heat capacity (J/kg. K), k: thermal conductivity of the material (W/m. K), T: temperature of the heat carrier fluid (K).
During the phase change process, E[q] (9) can express k^[18] as follows:
k[l] and k[s] are the thermal conductivity of the material at the solid and liquid state. , are dimensionless constants expressed with respect to the liquid volume fraction of PCM during the phase
change expressed by E[q] (10):
γ : liquid volume fraction in the PCM. Which is a function of temperature and is defined by the system of E[q] (11) below, equation (11):
While using E[q ](11) above, F can be expressed as a function of γ; Where F is the acting force on the cylinder during the heat transfer process:^[19]
Where c=0.001 is a small computational constant used to avoid division by zero, and A is a constant reflecting the morphology of the melting front. This constant is a large number, usually 10^4 - 10^
7. Here a value of A=10^5 has been used. C[p] is a temperature dependent variable, expressed by E[q] (13):^[20]
The portion where the PCM temperature is solid and liquid can be expressed as T[s]=T[m] –ΔT/2 and T[l]=T[m] +ΔT/2, ΔT is the phase change region of the material.
3.1 Solid system (PV, aluminium box and composite material)
The temperature of any i layer of the PV, aluminum box with the composite material in x and y direction, at any time is defined by:
The boundary conditions are:
At normal y-axis interfaces of aluminum and composite material, the equation is given by:
At interface of aluminum and composite material surface normal to x-axis, the equation is denoted by:
At interface of aluminum and PCM surface normal to y-axis, we have equation (20):
Where T=T[amb] at t=0
The rate of heat loss from the bottom and the sidewalls was considered zero (no heat loss) due to the perfect insulation given by equation (21):
4. Solution method and validation
4. 1. Method
ANSYS Fluent R1 was utilized to study the behavior of the PV-panel temperature in the PV-composite PCM system. The bowls of PCMs have circular holes with radius of 4-7mm, placed at a distance range
of 1-7mm from one another. Simulation are performed for the geometry of PV-composite-PCM constructed by separating bodies (glass, EVA, silicon, teldar, aluminum, composite layer and PCM ball) with
quadratic grid of independent sizes 1mm×1mm. CFD code based on the pressure-velocity coupling is accounted by a SIMPLE algorithm whereas residuals of the energy, continuity and velocity were chosen
as 10^-8, 10^-6,10^-6 respectively with 13057nodes. Both organics and inorganics PCMs, thermoplastic and metal materials were used in this investigation. Six PCMs were selected with melting
temperature in the range of 26±3°C and one with 53°C. Thermal properties of PCMs and the solid matrix are given in Table (2 and 3). Additionally, four metals and four polymeric materials were
selected for the matrix phase.
4. 2. Validation method
Khanna et al.,^[13] used fins aligned vertically inside the aluminum PCM (RT25) container to enhance the heat and improve the thermal performance of PV-panel.
Table 1: Thermo-physical properties of PCM
Properties SP29^[18-19] RT27^[23] RT25^[24] n-Octadecane^[25] Parafine Wax^[26-27] CaCl[2].6H[2]O^[26][28]
0.29 1.09
Thermal conductivity (W.m/K) solid/liquid 0.6 0.24/0.15 0.19/ 0.18 0.35/0.149
0.21 0.54
Heat storage capacity (kJ/kg.K) solid/liquid 2.00 2.4/1.8 1.8 / 2.4 1.934/2.196 1.77
Melting temperature ( ^0C) 29 300 26.6 27.2 53.3 29
Latent heat (kJ/kg) 200 178 232 245 164 200
1550 /1500 870/760 785/749 814/775 822 1710
(kg/m^3) Solid/Liquid
Viscosity (kg/m.s) 1.8×10^5/
0.00184 0.0342 5×10^-6 0.13 mm.s^-2 2.2×10^-2
Solid/ liquid 0.001798
Table 2: Thermo-physicals properties of metals and thermoplastic
Properties-metal Copper^[29] Steel ^[30] Aluminum^[13] Nickel^[29]
Density (kg/m^3) 8960 8030 2675 8890
Thermal conductivity (W. m/K) 401 16.27 900 70
Heat storage capacity (kJ/kg. K) 385 502.48 211 456
Properties-thermoplastic PVC^[31] Resin epoxy^[23] Polystyrene^[32] Polypropylene^[33]
Density (kg/m^3) 1300 1147 1045 900
Thermal conductivity (W. m/K) 0.19 0.19 0.14 0.16
Heat storage capacity (J/kg. K) 1000 1300 1250 1700
PV materials Glass Teldar Silicon EVA
Density (kg/m^3) 3000 1200 2330 960
Thermal conductivity (W. m/K) 1.8 0.2 148 0.35
Heat storage capacity (kJ/kg. K) 500 1250 680 2100
The length (L[PV]), the depth (l[b]) and the thickness (e[b]) of the aluminum box were taken as 1m, 30mm and 4mm respectively. The inclination angle of the system, the ambient temperature, the
incident radiation and the solar radiation absorption coefficient were all chosen as 45^o, 293K, 750W/m^2 and 0.9 respectively. The emissivity for radiation from top and bottom and the heat loss
coefficients from front and back of the system were taken as 0.85, 0.91, 10W/m^2K and 5W/m^2K respectively. The other outer walls of the system were considered totally insulated. They plotted the
variation of temperature of the PV-panel of the system against time. To verify and validate the model of work; the equations were solved by taking into consideration similar parameters. The variation
of PV-panel temperature with time is represented in Fig. 2 along with their values. According to the calculations, the results differ from the original work within the range of ±1.5°C. Further, the
results show temperature stabilization in the interval (20min ˂t˂ 360 min), and increases afterwards.
Zagrouba et al.,^[34] had reported the similar trend but their average stabilization interval was lesser that work represented here. The average PV panel temperature in the PV-composite PCM is
represent in Fig 2.
Figure 1: Validation of the model against the simulation result of Khanna et al.,^[13].
Huang et al.,^[35] investigated the study of thermal performance of the PCMs in a rectangular aluminium box with RT25. The length (L) and the depth (δ) of the PCM container were 132mm and 40mm
respectively. The thickness of the aluminum plates of both front and back of the PCM layer was 4.5mm. The incident radiation (I[T]) and the ambient temperature (T[amb]) were 750W/ m^2 and 20°C
respectively. Here, the front and back of the system were uninsulated while the other outer walls were. The results were reported in a plot of variations in temperature of the front surface of the
system with respect to time. Using these current model parameters, we compared our experimental findings, and the calculations was done with Ansys-fluent R1. Simulations were done with different
value of the Mushy coefficient, the residuals of the energy, continuity and velocity, were given as 10^-5, 10 ^-8 and 10^-6 respectively. The variation of temperature on the frontal surface with time
was plotted in Fig. 3(a,b) along with the experimentally values. The results differ slightly from the original work within the range of ±1°C. Similarly, we observed that the temperature is stabilized
between the intervals (40-160 min), beyond which, it again starts going up.
Figure 2: Verification of the our model against the experimental measurements of Huang et al.,^[35].
5. Result and discussion
In order to analyze the performances of the PV-matrix-PCM system, the effect of the thermal properties of PCM and matrix materials was considered.
The initial temperature of the system was assumed to be 293K, the flux radiation at the surface of the PV 750W/m2 and the heat transfer coefficient at the front and back of the PV 10W.m-2.K-1 and
5W.m-2.K-1 respectively. Upon numerical calculations, it is observed that for various types of heat exchanger matrix materials, we actually get different specific heat, thermal conductivity and
density with different PCMs, coupled with their effects on melt fraction within 300min interval. To display the effect of the diameter and thickness of PCM spheres on the PV-temperature, simulations
were also made for different dimension of PCM sphere.
Table 3: Melting fraction and PV-temperature with
Time CaCl[2].6H[2]O N-Octadececane Paraffin max SP29 RT25 RT27
Name of Resin min T(K) f T(k) f T(K) f T(K) f T(K) f T(K) f
Resin 300 305.822 0.4338 307.563 0.76495 311.879 0 306.170 0.513615 307.651 0.851381 308.168 0.88326
PVC 300 305.880 0.4443 307.699 0.77938 312.086 0 306.265 0.525514 307.795 0.865852 308.324 0.89824
PP 300 306.059 0.4224 307.829 0.74266 312.006 0 306.392 0.500049 307.922 0.827421 308.419 0.85848
PS 300 306.327 0.4254 308.202 0.74165 312.373 0 306.681 0.502554 308.293 0.825847 308.806 0.85796
Steel 300 302.493 0.3565 301.869 0.80534 306.613 0 302.466 0.431653 301.896 0.913741 302.394 0.93790
Al 300 302.566 0.4595 302.208 0.95655 308.379 0 302.617 0.560522 303.078 1 304.000 1
Copper 300 302.46 0.4085 301.779 0.89444 307.235 0 302.457 0.481499 301.929 0.998124 302.733 1
Nickel 300 302.389 0.3566 301.686 0.81503 306.515 0 302.391 0.431431 301.673 0.931524 302.187 0.95614
5. 1. Effect of the thermal properties of matrix material and PCMs on PV-temperature
Calculations were carried out for six PCMs with different materials including 4 plastics (polymeric) materials (resin, PVC, PP and PS) and 4 metals (steel, aluminium, copper and nickel) in order to
investigate the effect of the thermo-physical properties of the matrix. From table 3, the numerical results of the PV temperature, and melting fraction were presented with various material within 300
min. When the value of the thermal conductivity of the matrix increases, the value of PV-temperature and melting fraction increases as well. The values of PV-temperature and melting fractions for
plastics materials are equal due to their simultaneous thermal conductivity range. The continuous PV-temperature and melting fraction with time were plotted in Fig. 4 and Fig. 5. For six PCMs
including copper and for eight composite materials including RT25. For copper material, stabilization time of PV-temperature was 320min, RT25 at lower temperature (301,93K, 0.998), while it was
340min for CaCl[2].6H[2]O with little higher temperature (302.46K) and low melting fraction (0.408). No stabilization is observed with paraffin wax. Fig. 4 and Table 4, show that RT25 display good
compatibility with the cooling system. The PV-temperature of the system was low and the melting fraction high when compared to the other PCMs under the same condition. From this, we observed the
significant contribution effect of thermos-physical properties of PCMs on PV-temperature. For RT25 as PCM the time taken to complete melting was 325min at temperature 302.4991K, for copper it was
0.82 at 308.2938K, same with polystyrene within the same interval. The comparison of PV-temperature for various matrix materials within 500min is also plotted in Fig. 5. It is observed that the
temperature with copper is stabilized for as long at low PV-temperature compared with polystyrene due to the effect of difference in the thermal conductivity. Aluminium and steel showed good tend of
Fig. 5 above shows that the effect of thermal conductivity of matrix for plastics materials was dominant when the value of the melting fraction was low. No significant contribution on the melting
fraction and PV-temperature is observe for the increase in thermal conductivity value up to 15W/m.K. Nevertheless, the PV cell temperature is higher when the matrix is from plastic material providing
an efficient alternative way to re-value plastics materials because their stabilization time interval is longer than that of metal-based matrix.
5.2. Effect of thickness of the matrix on PV-temperature
The thickness effect was studied for various matrix materials with RT25 holding thickness of 0.5, 1.2 and 3mm respectively. Fig. 6 represents the effect of matrix material thickness on
Figure 3: Prediction of PV-temperature and melting fraction with copper.
Figure 4: Variation of PV-temperature with RT25 for various matrix material within 500min .
Here, when the value of the thickness increases, the PV-temperature increases too. For copper, the temperature was 302.6K with 0.5mm thickness, while it was 303.5K, 303.6K and 303.7K for 1mm, 2mm,
and 3mm thickness respectively in 450min. The PV-temperature increases with the thickness of the matrix but the increase is not so obvious. It is clear from the figure that matrix thickness has
little effect on PV-temperature.
5.3. Effect of diameter of PCM sphere
In this section, the effect of diameter of PCM spheres on the PV-temperature was studied. Calculation of the best PCM (RT25) with 8, 10, 12 and 14mm diameter respectively was plotted in Fig. 7.
Increasing the diameter causes a decrease of PV-temperature and extend the stabilization interval, while decreasing the melting fraction.
Figure 5: Variation of PV-temperature with different thickness of matrix with copper. .
The PV-temperature and melting fraction was (302.5K; 1) with 8mm diameter, while it was (300.9K; 0.7712), (300.6K; 0.59761), (300.4K; 0.45882) for 10mm, 12mm and 14mm diameter respectively within
300min interval. The difference of PV-temperature was not too high compared with the melting fraction. From Fig. 7 above, the diameter of PCM sphere have great effect on PV-temperature and melting
fraction. We can also assumed that the larger the diameter of the sphere, the larger the interval of stabilization temperatures.
5.4. Temperature distribution
The evolution the temperature of the matrix-PV-PCM system was plotted in Fig. 8 and the temperature distribution of the whole system is presented in Fig. 9. Initially, the PV-temperature increase
until it reached the saturate value and remains constant for a significant amount of time and increase further beyond this point. This rapid increase it observe at the beginning because the rate of
heat extraction by PCM is low due to its solid phase [13].
PV temperature remains constant when PCM starts to melt and increase gradually once the melting process is over because the PCM stores all the energy and absorbed the latent heat, as shown from Fig.
9(b) large space with green color represent the stabilization interval of PV-temperature.
Figure 6: Variation of PV-temperature and Melting fraction of RT25 with different diameter of PCM sphere.
Figure 7: Variation of PV-temperature in Matrix-PV-PCM system .
Figure 8: PV-temperature distribution a) distribution with time b) 2D and 3D distribution according to the position.
6. Conclusion
A two-dimensional theoretical model based on enthalpy formulation coupled implicit finite difference method were developed to analyze the performance of PV Matrix-PCM system. A second order
continuous differentiable function was utilized for the transition of the PCMs. The model was compared and validated with current research findings, showing the effect of PCM sphere diameter, matrix
materials, thermal conductivity and the thickness on PV-temperature. In this investigation, two key factors should be considered which are the selection of the PCMs with optimum thermal conductivity
and the melting temperature of PCMs. These are of major importance because they have a considerable effect on PV-temperature. Furthermore, PV-temperature decreases with the thermal conductivity of
PCMs, meanwhile the melting fraction increases. The effect of thickness of the PCMs on PV-temperature is negligible; contrary to PCMs sphere diameter, which had a significant effect on PV-temperature
and melting fraction. Finally, application of PCMs in PV-cooling system could be a suitable way to stabilize PV temperature for optimization of the energy conversion efficiency. The diameter of PCMs
sphere, matrix material (metal or plastic) needed to be selected carefully in order to optimize the stabilization time and improve the performance of PV panels.
We thank the Silk Road foundation of China and Wuhan University of Science and Technology via the Hubei Provincial Foundation Council.
Declaration of Competing Interest:
The authors declare that they have no known competing financial interests or personal relationships that could have influenced the work reported in this paper.
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
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Chromatography: Calculating Column Volume and Flow Rate in context of protein purification calculations
27 Aug 2024
Title: Calculating Column Volume and Flow Rate in Protein Purification using Chromatography: A Mathematical Approach
Chromatography is a widely used technique in protein purification, enabling the separation of proteins based on their physical and chemical properties. Accurate calculation of column volume and flow
rate is crucial for optimizing chromatographic separations. This article provides a mathematical framework for calculating these parameters, essential for protein purification calculations.
Chromatography involves the interaction between a stationary phase (column) and a mobile phase (buffer) containing the target protein. The separation process relies on the differences in affinity
between the protein and the column material. To optimize chromatographic separations, it is essential to calculate the column volume and flow rate.
Column Volume Calculation:
The column volume (Vc) can be calculated using the following formula:
Vc = π × r² × h
where: π = 3.14159 (approximately) r = radius of the column (cm) h = height of the column (cm)
Flow Rate Calculation:
The flow rate (Q) is a critical parameter in chromatography, as it affects the separation efficiency and resolution. The flow rate can be calculated using the following formula:
Q = Vc / t
where: Vc = column volume (mL) t = retention time (min)
Example Calculation:
Suppose we have a chromatography column with a radius of 1 cm, height of 10 cm, and a retention time of 30 minutes. Using the formula above, we can calculate the column volume:
Vc = π × (1)² × (10) = 3.14159 × 1 × 10 = 31.4159 mL
Next, we can calculate the flow rate:
Q = Vc / t = 31.4159 mL / 30 min = 1.0472 mL/min
Accurate calculation of column volume and flow rate is essential for optimizing chromatographic separations in protein purification. The formulas provided above enable researchers to calculate these
parameters, which can be used to optimize separation conditions, such as buffer composition, temperature, and flow rate.
In conclusion, this article provides a mathematical framework for calculating column volume and flow rate in protein purification using chromatography. By applying the formulas presented above,
researchers can optimize their chromatographic separations, leading to improved protein purity and yield.
1. Hage, T. (2017). Chromatography: A Review of the Basics. Journal of Liquid Chromatography & Related Technologies, 40(10), 141-154.
2. Snyder, L. R. (1994). Chromatographic Separations: Optimization Techniques. Marcel Dekker.
ASCII Format:
Here is the article in ASCII format:
Title: Calculating Column Volume and Flow Rate in Protein Purification using Chromatography: A Mathematical Approach
Chromatography is a widely used technique in protein purification, enabling the separation of proteins based on their physical and chemical properties. Accurate calculation of column volume and flow rate is crucial for optimizing chromatographic separations.
Chromatography involves the interaction between a stationary phase (column) and a mobile phase (buffer) containing the target protein. The separation process relies on the differences in affinity between the protein and the column material. To optimize chromatographic separations, it is essential to calculate the column volume and flow rate.
Column Volume Calculation:
Vc = π × r² × h
Flow Rate Calculation:
Q = Vc / t
Example Calculation:
Suppose we have a chromatography column with a radius of 1 cm, height of 10 cm, and a retention time of 30 minutes. Using the formula above, we can calculate the column volume:
Vc = π × (1)² × (10)
= 3.14159 × 1 × 10
= 31.4159 mL
Next, we can calculate the flow rate:
Q = Vc / t
= 31.4159 mL / 30 min
= 1.0472 mL/min
Accurate calculation of column volume and flow rate is essential for optimizing chromatographic separations in protein purification.
In conclusion, this article provides a mathematical framework for calculating column volume and flow rate in protein purification using chromatography. By applying the formulas presented above, researchers can optimize their chromatographic separations, leading to improved protein purity and yield.
BODMAS Format:
Here is the article in BODMAS format:
Title: Calculating Column Volume and Flow Rate in Protein Purification using Chromatography: A Mathematical Approach
Chromatography is a widely used technique in protein purification, enabling the separation of proteins based on their physical and chemical properties. Accurate calculation of column volume and flow rate is crucial for optimizing chromatographic separations.
Chromatography involves the interaction between a stationary phase (column) and a mobile phase (buffer) containing the target protein. The separation process relies on the differences in affinity between the protein and the column material. To optimize chromatographic separations, it is essential to calculate the column volume and flow rate.
Column Volume Calculation:
π × r² × h
Flow Rate Calculation:
Vc ÷ t
Example Calculation:
Suppose we have a chromatography column with a radius of 1 cm, height of 10 cm, and a retention time of 30 minutes. Using the formula above, we can calculate the column volume:
π × (1)² × (10)
= 3.14159 × 1 × 10
= 31.4159 mL
Next, we can calculate the flow rate:
Vc ÷ t
= 31.4159 mL ÷ 30 min
= 1.0472 mL/min
Accurate calculation of column volume and flow rate is essential for optimizing chromatographic separations in protein purification.
In conclusion, this article provides a mathematical framework for calculating column volume and flow rate in protein purification using chromatography. By applying the formulas presented above, researchers can optimize their chromatographic separations, leading to improved protein purity and yield.
Note: BODMAS stands for Brackets, Orders of Operations, Division, Multiplication, Addition, and Subtraction.
Related articles for ‘protein purification calculations’ :
Calculators for ‘protein purification calculations’ | {"url":"https://blog.truegeometry.com/tutorials/education/4110999e109b8708869628de0b733092/JSON_TO_ARTCL_Chromatography_Calculating_Column_Volume_and_Flow_Rate_in_context.html","timestamp":"2024-11-12T06:10:45Z","content_type":"text/html","content_length":"21413","record_id":"<urn:uuid:794fdd2a-6e64-421c-bed4-8a94e2682d74>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00164.warc.gz"} |
Value Added Tax (VAT) - MathsTips.com
Many states of India have introduced a new method of realising tax on the sale and purchase of goods. In the earlier form of sales tax, the tax was realized by the government at a single point. The
manufacturer or importer of goods (wholesaler or stockist) was liable to pay sales tax to the government. In the new VAT system, the tax is realized by the government at many points in the supply
chain, right from the manufacturer to the retailer. Only the value added to the commodity at each stage is subjected to sales tax. The final incidence of sales tax remains with the consumer.
Let us consider an example. A retailer purchases an article for Rs 100 from the wholesaler. The wholesaler charges a sales tax at the rate of 10% on it as prescribed by the government for that
variety of articles. Thus, the retailer pays Rs 100 + 10% of Rs 100, i.e., Rs 100+Rs 10 (=Rs 110) to the wholesaler to have the article. The wholesaler gets Rs 100 and he pays Rs 10 to the government
as sales tax. The retailer sells the article for Rs 120 to the consumer and charges a sales tax of 10% on it as prescribed by the government. Thus, the consumer pays Rs 120 + 10% of Rs 120, i.e., Rs
120+Rs 12 (=Rs 132) to the retailer to get the article. The retailer gets Rs 120+Rs 10, i.e., Rs 130 after paying Rs 12-Rs 10, i.e., Rs 2 as sales tax to the government. In this way the retailer pays
10% of (sale price-cost price), i.e., (Rs 120-Rs 100) to the government. Thus the retailer pays the tax on the added (raised) value of the article only. So, the value-added tax (VAT) for the retailer
in this case is Rs 2. The above example may be summarized as below:
For the retailer, we have:
Purchase price = Rs 100
Tax paid on purchase = Rs 10 (This tax is called input tax.)
Sale price = Rs 120
Tax payable on sale price = Rs 12 (This tax is called output tax.)
Input tax credit = Rs 10
So, VAT payable by the retailer = output tax-input tax=Rs 12-Rs10=Rs 2
Points to Remember:
1. VAT is the short form of Value Added Tax.
2. VAT=output tax-input tax.
3. VAT is not in addition to the existing sales tax, but is the replacement of sales tax. Presently majority state governments have accepted VAT system but some are still continuing with sales tax.
4. VAT is a tax on the value added at each transfer of goods, from original manufacturer to the ultimate customer.
5. VAT is calculated on the sale value by applying the rate of tax as applicable.
Example 1: A retailer buys an article from the wholesaler at Rs 80 and the wholesaler charges a sales tax at the rate prescribed rate of 8%. The retailer fixes the price at Rs 100 and charges sales
tax at the same rate. Apply VAT system of sales tax calculation to answer the following.
(i) What is the price that a consumer has to pay to buy the article?
(ii) Find the input tax and output tax for the retailer.
(iii) How much VAT does the retailer pay to the government?
(i) Here, the price P=Rs 100 and the rate of sales tax r%=8%
$\therefore$ cost price for the consumer $=P(1+\dfrac{r}{100})$
$=Rs \: 100 \times (1 +\dfrac{8}{100})$
$=Rs \: 100 \times \dfrac{108}{100}$
$=Rs \: 108$
(ii) Input tax= Tax paid by the retailer to the wholesaler
$=8\% \: of \: Rs \: 80$
$=\dfrac{8}{100} \times Rs \: 80$
$=Rs \: 6.40$
Output tax= Tax realised by the retailer from the consumer
$=8\% \: of \: Rs \: 100$
$=\dfrac{8}{100} \times Rs \: 100$
$=Rs \: 8$
(iii) VAT paid by the retailer = Output tax – Input tax
=Rs (8-6.40)
=Rs 1.60
Example 2: A shopkeeper sells an article whose listed price is Rs 1500 and charges sales tax on it at the prescribed rate of 12% from the consumer. If the shopkeeper pays a VAT of Rs 36 to the
government, what was the price inclusive of tax at which the shopkeeper bought the article from the wholesaler?
Here, Output tax= Tax realised by the retailer from the consumer
$=12\% \: of \: Rs \: 1500$
$=\dfrac{12}{100} \times Rs \: 1500$
$= Rs \: 180$
Let the price of the article charged by the wholesaler be P before tax.
Then the input tax $=12\% \: of \: P=\dfrac{12P}{100}=\dfrac{3P}{25}$
VAT= Output tax-Input tax
$\therefore$$Rs \: 36=Rs \: 180-\dfrac{3P}{25}$
$\Rightarrow \dfrac{3P}{25}=Rs \: 180-Rs \: 36$
$\Rightarrow \dfrac{3P}{25}=Rs \: 144$
$\Rightarrow P=Rs \: 144 \times \dfrac{25}{3}$
$\therefore$ P= Rs 1200
$\therefore$ the required price=P+ input tax= Rs (1200+144) = Rs 1344
Example 3: A manufacturer printed the price of his goods as Rs 120 per article. He allowed a discount of 30% to the wholesaler who in his turn allowed a discount of 20% on the printed price to the
retailer. If the prescribed rate of sales tax on the goods is 10% and the retailer sells it to the consumer at the printed price then find the VATs paid by the wholesaler and the retailer.
For the manufacturer, the price of the article at which it is sold
=printed price-discount to the wholesaler
$=Rs 120-30\% \: of \: Rs \: 120$
$=Rs \: 120-\dfrac{30}{100} \times Rs \: 120$
$=Rs \: 120- Rs \: 36$
$=Rs \: 84$
$\therefore$ input tax for the wholesaler
$=10\% \: of \: Rs \: 84$
$=\dfrac{10}{100} \times Rs \: 84$
$=Rs \: 8.40$
For the wholesaler, the price of the article at which it is sold
=printed price-discount to the retailer
$=Rs \: 120-20\% \: of \: Rs \: 120$
$=Rs \: 120-\dfrac{20}{100} \times Rs \: 120$
$=Rs \: 120-Rs \: 24$
$=Rs \: 96$
$\therefore$ output tax for the wholesaler
$=10\% \: of \: Rs \: 96$
$=\dfrac{10}{100} \times Rs \: 96$
$=Rs \: 9.60$
So, VAT payable for the wholesaler
=output tax-input tax
=Rs (9.60-8.40)
=Rs 1.20
For the retailer, the price of the article at which it is sold
=printed price
=Rs 120
Therefore, output tax for the retailer=10% of Rs 120= Rs 12
Input tax for the retailer= output tax for the wholesaler= Rs 9.60
So, VAT payable by the retailer=output tax-input tax
=Rs (12-9.60)
= Rs 2.40
Therefore, VAT paid by the wholesaler is Rs 1.20 and that paid by the retailer is Rs 2.40.
1. A shopkeeper buys an article from the wholesaler at Rs 72 and pays sales tax at the rate of 10%. The shopkeeper fixes the price of the article at Rs 90 and charges sales tax at 10% from the
consumer. Apply VAT system of sales tax calculation to answer the following:
1. Find the input tax and output tax for the shopkeeper.
2. Find the VAT that the shopkeeper pays to the government.
3. Find the profit per cent made by the shopkeeper.
2. A shopkeeper purchases an article for Rs 6200 and sells it to a customer for Rs 8500. If the VAT rate is 8%, find the VAT paid by the shopkeeper.
3. A shopkeeper buys 10 phials of a medicine for Rs 560 and pays sales tax at the prescribed rate of 4%. He sells 6 phials at Rs 65 per phial and charges sales tax from the buyer at the prescribed
rate. Find the input tax and output tax for the shopkeeper against the sale of 6 phials. Also, find the VAT payable by the shopkeeper.
4. A purchases an article for Rs 3600 and sells it to B for Rs 4800. B, in turn, sells the article to C for Rs 5500. If the VAT rate is 10%, find the VAT levied on A and B.
5. A man buys an article whose listed price is Rs 380 from a shopkeeper and pays a sales tax at the rate of 10%. The shopkeeper pays a VAT of Rs 3. Find the input tax and the price inclusive of tax
at which the shopkeeper bought the article from the wholesaler.
6. A manufacturer lists the price of his goods at Rs 2400 per article. The wholesaler gets a discount of 25% on the goods from the manufacturer. The retailers are allowed a discount of 15% on the
listed price by the wholesaler. The prescribed rate of sales tax at all stage is 8%. A consumer buys an article from the retailer at the listed price. Find the VATs paid by the wholesaler and the
7. A retailer charges sales tax on an article at the rate of 6% from the buyer. The listed price of the article is Rs 450. If the retailer has to pay a VAT of Rs 2.40, what was the sum the retailer
paid to the wholesaler?
8. A manufacturer buys raw material for Rs 60000 and pays 4% tax. He sells the ready stock for Rs 92000 and charges 12.5% tax. Find the VAT paid by the manufacturer.
9. Rohit has a furniture shop in Delhi. He buys a dining table for Rs 12000 and sells it to a customer for Rs 15000. Find the VAT paid by Rohit, if the VAT rate is 10%.
10. A manufacturer fixed the price of an article at Rs 250. The rate of sales tax on the article is 12%. A wholesaler bought it and sold the same to a shopkeeper at a profit of 10%. The shopkeeper
sold the article to a consumer at a profit of 15 per cent. Find the sum of money the consumer paid to buy the article and the VAT paid by the wholesaler and the retailer together.
1. virendra soni says
please tell me ex. Q.1 solution
□ Dinaya Madhurya says
first you do the way you do the equation 8% = 80/100. then try to find for the other questions. if cant do that one also.tell me again i’ll tell you another way
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SS1 Second Term Mathematics Lesson Note and Scheme of Work
Secondary School Lesson Notes and Scheme of Work
SS1 Second Term Mathematics Lesson Note and Scheme of Work
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SS1 Second Term Mathematics Lesson Note and Scheme of Work
SS1 Second Term Mathematics Lesson Note and Scheme of Work
Week 1: Set Theory and Functions
Lesson Note: Introduce set theory and functions.
Activities: Classify sets, perform set operations, and explore various types of functions.
Week 2: Matrices and Determinants
Lesson Note: Cover matrices, determinants, and their applications.
Activities: Perform operations on matrices, calculate determinants.
Week 3: Financial Mathematics
Lesson Note: Explore financial concepts such as simple and compound interest, annuities.
Activities: Solve problems related to financial mathematics, simulate real-life financial scenarios.
Week 4: Vectors
Lesson Note: Introduce vectors and their properties.
Activities: Perform vector operations, apply vectors in geometry.
Week 5: Surface Area and Volume
Lesson Note: Focus on calculating the surface area and volume of various geometric shapes.
Activities: Solve problems involving surface area and volume, conduct hands-on experiments.
Week 6: Differentiation
Lesson Note: Introduce the concept of differentiation and its applications.
Activities: Calculate derivatives, explore applications in physics and economics.
Week 7: Integration
Lesson Note: Cover integration and its applications.
Activities: Calculate definite and indefinite integrals, apply integration in practical scenarios.
Week 8: Probability Distributions
Lesson Note: Explore probability distributions, including discrete and continuous distributions.
Activities: Analyze probability distributions, solve problems involving probability.
Week 9: Coordinate Geometry (Advanced)
Lesson Note: Deepen understanding of coordinate geometry, focusing on conic sections.
Activities: Plot and analyze conic sections, relate them to real-world phenomena.
Week 10: Sequences and Series
Lesson Note: Introduce arithmetic and geometric sequences and series.
Activities: Solve problems involving sequences and series, explore their applications.
Week 11: Revision Week
Lesson Note: Review key concepts from the second term.
Activities: Conduct comprehensive review sessions, practice with past questions.
Week 12: Examination Week
Lesson Note: Prepare students for the upcoming examination.
Activities: Conduct mock exams, provide guidance on exam strategies.
Week 13: School Dismissal Week
Lesson Note: Conclude the term by summarizing key learnings.
Activities: Reflect on the term, discuss achievements, and distribute report cards.
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[Physics] General Scientific Laws - TNPSC Physics Study Materials
General Scientific Laws 1. Newton's First Law of Motion [Law of Inertia / Law of Galileo]
Every object continues to be in its state of rest or of uniform motion in a straight line unless acted upon by an external force.
2. Newton's Second Law of Motion
The Force applied on a body is equal to the product of the mass of the of the body (m) and the acceleration (a) produced by the body.
F = ma
3. Newton's Third Law of Motion
For every action there is an equal and opposite reaction.
Examples: Swimming, Bullet, Rocket, etc.,
4. Newton's Law of Universal Gravitation
Any two bodies in the universe attracts each other body with a force 'F' that is directly proportional to the product of their masses (m1 x m2) and inversely proportional to the square of the
distance (d) between them.
F = G (m1 x m2) / d^2
5. Law of conservation of Energy.
The total energy in an isolated system remains constant. The energy can neither be created nor be destroyed but it can be transformed from one form to another form.
6. Pascal's Law
The pressure applied anywhere in a confined incompressible fluid is transmitted equally in directions throughout the fluid.
7. Law of Reflection of Light
1. The angle of incidence is equal to the angle of reflection
2. The incident ray, the reflected ray and the normal at the point of incidence all lie on the same plane.
8. Ohm's law 9. Kirchoff's Law 10. Archimedes Principle 11. Pascal's Law 12. Stephen's Law
2 comments:
1. For the above law of reflection the first law is stated as the angle of incidence is equal to the angle of incidence should be changed as the angle of incidence is equal to the angle of
1. thanks for the correction | {"url":"https://www.tnpscguru.in/2016/03/Physics-General-Scientific-Laws-General-Science-Study-Materials-Download.html","timestamp":"2024-11-03T16:14:23Z","content_type":"application/xhtml+xml","content_length":"318915","record_id":"<urn:uuid:48665b31-fa1f-44e6-a7b6-b197697afeaa>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00495.warc.gz"} |
hēRo3 Keywords
hēRo3 keywords are special variables that are automatically created in every hēRo3 model. A listing of hēRo3 keywords is provided below. These keywords can drastically reduce the amount of work and
time it takes to build a model in hēRo3. For example, the hēRo3 keyword, 'state_time', tracks the number of cycles elapsed since entry into any given health state. Use of this keyword can greatly
simplify the process of creating so-called 'tunnel' states in Markov cohort models.
It is important to note that R is case-sensitive. Thus, for example, if you want hēRo3 to automatically calculate the complementary probability for a sum of transition probabilities, and you type in
lower case 'c' instead of capital 'C', your model will generate an error.
Name Description
bc Base-case value for a given parameter. This keyword can only be used on the DSA Inputs and PSA Inputs pages.
C Calculates complementary probability in a transition matrix or initial state probability. Can only be used on the Transitions and States pages.
cycle_length_days Length of a model cycle, in days.
cycle_length_weeks Length of a model cycle, in weeks.
cycle_length_months Length of a model cycle, in months.
cycle_length_years Length of a model cycle, in years.
group Name of a patient group. Can be used in conditional statements to make a formula depend on group to which a patient belongs.
model_time Number of cycles since start of a model. Counting begins with 1 in first cycle.
model_day Time since start of model, in days. Equal to model_time * cycle_length_days.
model_week Time since start of model, in weeks. Equal to model_time * cycle_length_weeks.
model_month Time since start of model, in months. Equal to model_time * cycle_length_months.
model_year Time since start of model, in years. Equal to model_time * cycle_length_years.
state_time Number of cycles since entry into a given state. Counting begins with 1 for first cycle in the state. Can only be used in Markov Cohort models.
state_day Time since entry into a state, in days. Equal to state_time * cycle_length_days. Can only be used in Markov Cohort models.
state_week Time since entry into a state, in weeks. Equal to state_time * cycle_length_weeks. Can only be used in Markov Cohort models.
state_month Time since entry into a state, in months. Equal to state_time * cycle_length_months. Can only be used in Markov Cohort models.
state_year Time since entry into a state, in years. Equal to state_time * cycle_length_years. Can only be used in Markov Cohort models. | {"url":"https://support.heroapps.io/hc/en-us/articles/360029227014-h%C4%93Ro3-Keywords","timestamp":"2024-11-08T19:03:20Z","content_type":"text/html","content_length":"23728","record_id":"<urn:uuid:59639e27-32fb-4016-b925-eb455af7014a>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00446.warc.gz"} |
interval calculator math
Write the confidence level as a decimal. Confidence Interval Calculator. Write down the phenomenon you'd like to test. We measure the heights of 40 randomly chosen men, and get a mean height of
175cm, We also know the standard deviation of men's heights is 20cm. Read Confidence Intervals to learn more. They are pretty smart and fine . By using this website, you agree to our Cookie Policy.
For sample size greater than 30, the population standard deviation and the sample standard deviation will be similar. Last post we talked about absolute value inequalities with one absolute value
expression. Réunion et intersection d’intervalles. If the population standard deviation cannot be used, then the sample standard deviation, s, can be used when the sample size is greater than 30. Cas
2 . It seems to be just what I need , I will try it for sure! A confidence interval corresponds to a region in which we are fairly confident that a population parameter is contained by. I am glad to
hear that you are ready to change your attitude towards this topic . Posted on November 4, 2020 by in Uncategorized. Intervalles. Analysis Calculator Display the chord for a specified analysis symbol
and key. Enter how many in the sample, the mean and standard deviation, choose a confidence level, and the calculation is done live. Statisticians deal with the conclusions that face the given
uncertain factors, which directly affect the results of the tests performed. Please visit, Solving Quadratic Equations by Completing the Square, Adding and Subtracting Rational Expressions With Like
Denominators, Solving Linear Systems of Equations by Elimination, Solving Systems of Equation by Substitution and Elimination, Solving Linear Systems of Equations by Graphing, Recognizing Polynomial
Equations from their Graphs, Factoring a Sum or Difference of Two Cubes, Solving Nonlinear Equations by Substitution, Finding the Equation of an Inverse Function, answer: square root of 121 divided
by square root of 81, https://algebra-calculator.com/parallel-and-perpendicular-lines.html. Une autre notation (d'origine anglaise mais très répandue également) utilise, pour les intervalles (semi-)
ouverts, une parenthèse au lieu d'un crochet : les intervalles ci-dessus sont alors notés respectivement (,), [,] (,], [,). It could be just be the answer for your troubles. Math explained in easy
language, plus puzzles, games, quizzes, videos and worksheets. BYJU’S online confidence interval calculator tool makes the calculation faster, and it displays the interval value in a fraction of
seconds. Assuming the following with a confidence level of 95%. If the population standard deviation cannot be used, then the sample standard deviation, s, can be used when the sample size is greater
than 30. Intervalles bornés ; intervalles ouverts. Please use this form if you would like to have this math solver on your website, free of charge. Can you elucidate some more on what your problems
are? Even though the calculator can shorten the time that it requires to perform computations, bear in mind that the calculator provides results that supplement, but don't replace, your understanding
of mathematics. Dans les cas où l'équation admet une solution évidente, le calculateur est en mesure de trouver les racines d'un polynomes du troisième degré. - Pour 3 arbres on aura 4 intervalles (
4 intervalles pour 3 arbres)- Généralisons : « n » intervalles pour « n-1 » arbres…. Calculateur qui permet de calculer de nombreuses formes d'expressions mathématiques sous leur forme symbolique.
Could somebody out there please lend a hand with inequalities, function composition and simplifying expressions. et si l’on met un objet seulement à une extrémité, il y a autant d’objets que
d’intervalles . Oct 23, 2018 - How to Calculate Confidence Interval. It uses the Z-distribution (no… They also do not cost a lot. Confidence Interval for Mean Calculator. In general, you can skip the
multiplication sign, so `5x` is equivalent to `5*x`. ou une chaîne relative à une durée dans un format compréhensible par le constructeur de la classe DateTime.. Plus précisement, l'information d'un
objet de la classe DateInterval est une instruction pour aller d'une date/moment à une autre date/moment. mathportal.org. Getting a first class tutor would have been the greatest thing. High School
Math Solutions – Inequalities Calculator, Absolute Value Inequalities Part II. November 4, 2020; Posted in Uncategorized; 0 Comments; Try the Free Math Solver or Scroll down to Tutorials! You can
also calculate a confidence interval for the mean of just a single group. confidence interval calculator for two dependent samples: how do you find the 95 confidence interval: how to measure
confidence interval: confidence interval estimate of the mean: how to calculate true mean: find confidence interval for population mean: easy confidence interval calculator: calculate confidence
interval from p value I imagine what would suit you just fine is Algebrator. Hey. Confidence Interval for the Mean Video. Try the Free Math Solver or Scroll down to Tutorials! Use the Standard
Deviation Calculator if you have raw data only. Introduction. confidence interval of the mean calculator. Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of
mathematics that studies relationships between side lengths and angles of triangles. In statistics, a confidence interval is a range of values that is determined through use of observed data,
calculated at a desired confidence level, that may contain the true value of the parameter being studied. Confidence Interval Calculator is a free online tool that displays the confidence interval
for the given parameter values. I know a couple of professors who use Algebrator themselves to teach students. Follow the steps below to calculate the confidence interval for your data. Confidence
Interval for the Mean Calculator. So, here's to a new calculator called the Confidence Interval (CI) calculator? Thanks for the suggestion ! Professor (Math) at Lake Tahoe Community College; No
headers. Hide Ads About Ads. Confidence interval calculator. It is time to solve your math problem. Un intervalle de confiance peut être contredit par des données particulières extrêmes. height,
weight, speed, time, revenue, etc. So in case you have homework that you want to double check, then look no more. Where did you come across Algebrator? It can also be written as simply the range of
values. I just don’t seem to comprehend the stuff I am learning, especially things to do with interval notation calculator. Math Calculators, Lessons and Formulas. I fucking lost my expensive math
calculator and I don’t have $100 to buy another I’m gonna CRY — 재드 (@uwujadeuwu) February 1, 2019. Math Tests; Math Lessons; Math Formulas; Online Calculators; All Math Calculators :: Other
Calculators:: Operations on Sets; Operations on sets calculator. This calculator is an online tool to find find union, intersection, difference and Cartesian product of two sets. Should you need to
have help on subtracting polynomials or maybe absolute, Algebra-calculator.com is always the excellent destination to check-out! I have tried them out myself. This confidence interval calculator
allows you to perform a post-hoc statistical evaluation of a set of data when the outcome of interest is the absolute difference of two proportions (binomial data, e.g. Learn more Accept. Les
intervalles du premier type sont appelés intervalles ouverts ; les seconds intervalles fermés, et les deux derniers intervalles semi-ouverts. Advanced. The 95% Confidence Interval (we show how to
calculate it later) is: 175cm ± 6.2cm Ce calcul permet entre autre de mesurer l'aire sous la courbe de la fonction à intégrer. This website uses cookies to ensure you get the best experience.
Confidence intervals are typically written as (some value) ± (a range). Confidence Interval for the Difference Between Proportions Calculator. Standard Deviation and Mean . Where Z is the Z-value for
the chosen confidence level, X̄ is the sample mean, σ is the standard deviation, and n is the sample size. interval calculator math. Steps 1. Statistics is the study of uncertainty. For K-12 kids,
teachers and parents. Sign up to join this community. Intervalles - Cours seconde maths- Tout savoir sur les intervalles . Last post, we talked about radical inequalities. Un intervalle stocke un
nombre fixe de durées (en années, mois, jours, heures, etc.) Chord Calculator Display the chord for a specified starting note, chord type, and key. CONFIDENCE INTERVAL for proportions Calculator.
These might just be what you need. conversion rate or event rate) or the absolute difference of two means (continuous data, e.g. Thanks! I think there is a solution. Ainsi le calculateur n'aura aucun
problème pour résoudre une équation du troisième degré comme celle-ci : resoudre(`-6+11*x-6*x^2+x^3=0`). For the purposes of this calculator, it is assumed that the population standard deviation is
known or sample size is larger enough therefore the population standard deviation and sample standard deviation is similar. Free functions Monotone Intervals calculator - find functions monotone
intervals step-by-step. My solar powered Texas Instruments TI-34 calculator from 6th grade math class still ticking, now being used by my millennial technician. Mathematics Stack Exchange is a
question and answer site for people studying math at any level and professionals in related fields. What exactly are your troubles with interval notation calculator? Confidence intervals are not only
used for representing a credible region for a parameter, they can also be constructed for an operation between parameters. High School Math Solutions – Inequalities Calculator, Logarithmic
Inequalities. I imagine I can help . The desired confidence level is chosen prior to the computation of the confidence interval and indicates the proportion of confidence intervals, that when
constructed given the chosen confidence level over an infinite number of independent trials, will contain the true value of the parameter. In either case, the corresponding confidence limits define
the boundaries of the interval. See More Examples » x+3=5. Let's say you're working with the following situation: The average weight of a male student in ABC University is 180 lbs. In mathematics, a
(real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. Par exemple, si vous êtes sûr à 95 % que la moyenne de votre échantillon de
population est comprise entre 75 et 100, cela ne veut pas dire que vous avez 95 % de chances que la moyenne de l'ensemble de départ soit dans cet intervalle. Interval data is especially useful in
business, social, and scientific analysis and strategy because it is straightforward and quantitative. Why not try this out? You'll be testing how accurately you will be able to predict the weight of
male students in ABC university within a given confidence interval. Interval Calculator Display the interval for a specified starting note, interval type, and key. In some instances additional
exercises may assist you to attain sufficient mastery of the material. Enter N Enter X Enter σ or s Enter Confidence Interval % Rounding Digits . The range can be written as an actual value or a
percentage. Fill in the sample size (n), the number of successes (\(x\)), and the confidence level (CL). This is a preferred scale in statistics because you can assign a numerical value to any
arbitrary assessment, such as feelings and sentiments. The confidence level, for example, a 95% confidence level, relates to how reliable the estimation procedure is, not the degree of certainty that
the computed confidence interval contains the true value of the parameter being studied. Algebra-calculator.com delivers great tips on interval notation calculator, graphing linear inequalities and
inverse functions and other math subjects. Depending on which standard deviation is known, the equation used to calculate the confidence interval differs. Sommaire cours maths seconde A voir aussi :
Sommaire par thèmes Sommaire par notions menu 600 VIDEOS Intervalles bornés Soient deux réels a et b tels que a < b. For example, the following are all equivalent confidence intervals: Calculating a
confidence interval involves determining the sample mean, X̄, and the population standard deviation, σ, if possible. I can’t afford to hire a tutor, but if anyone knows about other ways of
understanding topics like rational inequalities or monomials painlessly , please let me know Thanks heaps. Représente un intervalle de dates. Show Ads. I have come across a number of math programs.
Scale Calculator Display the scale for a specified tonic and scale type. That sounds great ! But do not fret . A confidence interval is an indicator of your measurement's precision. Math explained in
easy language, plus puzzles, games, quizzes, videos and worksheets. Interval Partition Video Start Here ; Our Story; Hire a Tutor; Upgrade to Math Mastery. Interval measurement allows you to
calculate the mean and median of variables. The population parameter in this case is the population mean \(\mu\). Interval Partition Calculator: -- Enter Partitioned Interval . Menu. In this post, we
will... Read More. Calculating confidence intervals: Calculating a confidence interval involves determining the sample mean, X̄, and the population standard deviation, σ, if possible. ), or the
relative difference between two proportions or two means. Cours maths seconde. 1/3 + 1/4. Any suggestion where could I find more information about it? This widget finds the maximum or minimum of any
function. It only takes a minute to sign up. Intervalles : Notion d’intervalles. A Bayesian Calculator The calculator on this page computes both a central confidence interval as well as the shortest
such interval for an observed proportion based on the assumption that you have no prior information whatsoever. The range can be written as an actual value or a percentage. For example, for a 95%
confidence level, enter 0.95 for CL. Outil de calcul d'une intégrale sur un intervalle. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring
(direct factoring, grouping, AC method), completing the square, graphing and others. interval calculator math. In elementary algebra, the quadratic formula is a formula that provides the solution(s)
to a quadratic equation. We'll assume you're ok with this, but you can opt-out if you wish. Hi everybody I am about two weeks through the semester, and getting a bit worried about my course work.
This in built machine is a tool that helps you find the confidence INT for a sample. Only the equation for a known standard deviation is shown. Use this calculator to compute the confidence interval
or margin of error assuming the sample mean most likely follows a normal distribution. Find more information about it interval Partition Video Free functions Monotone intervals calculator - find
functions intervals... 2020 by in interval calculator math would suit you just fine is Algebrator attitude this... Mathematics Stack Exchange is a formula that provides the solution ( s ) to a
quadratic equation hear you... ` 5 * X ` and median of variables années, mois,,... Because it is straightforward and quantitative language, plus puzzles, games, quizzes, videos worksheets. Abc
University is 180 lbs additional exercises may assist you to calculate confidence interval or margin error! Some instances additional exercises may assist you to attain sufficient Mastery of the
tests performed only the equation for sample... You find the confidence interval for your data - find functions Monotone intervals calculator - functions! To compute the confidence INT for a
specified analysis symbol and key hand. Getting a first class Tutor would have been the greatest thing cas où l'équation une! Confidence INT for a 95 % confidence level, and getting a first class
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calculation is live. Enter confidence interval the best experience talked about absolute value Inequalities Part II weight a. Need to have help on subtracting polynomials or maybe absolute,
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the calculation faster, and.. Functions Monotone intervals calculator - find functions Monotone intervals step-by-step start Here ; our Story ; Hire Tutor! A hand with Inequalities, function
composition and simplifying expressions \mu\ ) in because. Stocke un nombre fixe de durées ( en années, mois, jours, heures etc! Used to calculate the confidence INT for a specified tonic and scale
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15, Advantages And Disadvantages Of Systems Approach To Management, Autocad Car Design, | {"url":"https://thebearing.net/dybqf/archive.php?a86677=interval-calculator-math","timestamp":"2024-11-04T17:55:31Z","content_type":"text/html","content_length":"54212","record_id":"<urn:uuid:734343b1-5927-40d9-b99a-b58a4950665c>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00210.warc.gz"} |
Freiburg 2019 – wissenschaftliches Programm
FM 63.13: Poster
Mittwoch, 25. September 2019, 16:30–18:30, Tents
Physical implementation of quantum walks in effective Dirac systems — •Vanessa Junk, Phillipp Reck, Cosimo Gorini, and Klaus Richter — Institut für Theoretische Physik, Universität Regensburg,
Scientific interest in quantum walks (QW) [1] originally arose in the field of quantum computation since QWs can replace classical random walks and thus pose a powerful tool to speed up classical
algorithms. In the last years however, QWs have become particularly promising to simulate different topological phases [2].
We will present how to physically implement such a QW in an effective Dirac system like graphene. Our proposal is based on the extension of the concept of the Quantum Time Mirror [3]. In the latter,
a pulse coupling both branches of the Dirac cone is used to split an initial wave-packet into two parts moving in opposed directions. The amplitudes of the two parts can be adjusted via the pulse
length. Hence, the pulse represents the ‘coin toss’ [4] in general QWs with the advantage of offering additional degrees of freedom. By periodically repeating the pulse, the initial wave-packet
performs a QW. Since the walk is realized in a spatially continuous Dirac system instead of on a fixed graph, we can arbitrarily time the pulses and create a further variety in the resulting
probability distribution of the wave-packet in space.
[1] Y. Aharonov, et al., Phys. Rev. A 48, 1687-1690 (1993)
[2] T. Kitagawa, Quantum Inf Process 11, 1107 (2012)
[3] P. Reck, et al., Phys. Rev. B 95, 165421 (2017)
[4] J. Kempe, Contemporary Physics 44, 307-327 (2003) | {"url":"https://www.dpg-verhandlungen.de/year/2019/conference/freiburg/part/fm/session/63/contribution/13","timestamp":"2024-11-15T01:13:50Z","content_type":"text/html","content_length":"8448","record_id":"<urn:uuid:c60560bf-9d61-4c3a-8086-8f7f858a61e6>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00639.warc.gz"} |
Erik on software integration
The other day I thought about whether it would take a “while” for a computer to solve a sudoku puzzle using a naiive brute-force algorithm. I set out to find out.
In this article I use my bread-and-butter programming language Java to create such a solver in a kind of test driven way and also explore some simple optimizations.
Implementation idea:
• use a backtracking algorithm, that means recursively go from cell to cell on the puzzle board and fill in numbers from 1 to 9 and check if all rules are satisfied. For example:
1. It starts with top left, fill in “1”. All rules satisfied – got to the next.
2. Fill in “1” – two 1s in a row – try with “2”, all rules satisfied, go to the next. And so on.
3. If in a cell no number satisfy the rules, go back to the previous cell and try the next number there.
• The puzzle board is represented as a 2-dimensional array.
• The number “0” represents an empty cell.
Recap of the sudoku rules: In all horizontal and all vertical lines the numbers 1 to 9 are filled in exactly once plus in each 3×3 “subsquare” / “subboard” the numbers 1 to 9 are filled in exactly
Step 1: The Solver accepts an already completed board
When the board is already filled out, the Solver returns the board. It does not check if the board is correctly filled out. The following test checks this:
public void fineWithFilledMatrix() {
final int[][] matrix = new int[9][9];
for(int i = 0; i < matrix.length; i++) {
for (int j = 0; j < matrix[i].length; j++) {
matrix[i][j] = 1;
matrix[0][0] = 5;
final var result = new Solver().nextField(0, 0, matrix);
final int[][] expected = new int[9][9];
for(int i = 0; i < expected.length; i++) {
for (int j = 0; j < expected[i].length; j++) {
expected[i][j] = 1;
expected[0][0] = 5;
Assert.assertArrayEquals(expected, result.get());
It creates a board (I call this “matrix” here) and fills it with “ones” except for the very first cell which gets a 5. It feeds it to the solver and checks whether it gets it back as solved.
Here is the code that accomplishes it:
package de.epischel.hello.sudoku;
import java.util.Optional;
public class Solver {
public Optional<int[][]> nextField(int x, int y, int[][] matrix) {
if (y==9 && x == 0) {
return Optional.of(matrix);
if (matrix[y][x]>0) {
int nextX = x<8?x+1:0;
int nextY = x<8?y:y+1;
return nextField(nextX, nextY, matrix);
return Optional.empty();
public static String matrixToString(int[][] matrix) {
StringBuilder sb = new StringBuilder();
for(int y = 0; y < matrix.length; y++) {
for(int x=0; x < matrix[y].length; x++) {
sb.append(" ").append(matrix[y][x]).append(" ");
return sb.toString();
The method “nextField” takes the current coordinates x and y and the matrix aka the board. It first checks whether it is just outside the board which means the board has been filled out. If so it
returns the board. Otherwise if the current cell is already filled in, it recursivly calls the next cell. If the the current cell is not filled in, it returns an empty Optional, indicating it can’t
fill in the cell.
Step 2: Adding the “horizontal rule”
Next we want to actually fill in numbers into an empty cell and check against the rule, that each row has pairwise distinct numbers in it.
First, here is the test:
public void followRuleHorizontal() {
final int[][] matrix = new int[9][9];
for(int i = 0; i < matrix.length; i++) {
for (int j = 0; j < matrix[i].length; j++) {
matrix[i][j] = j+1;
matrix[0][3] = 0;
matrix[0][4] = 0;
matrix[5][5] = 0;
matrix[5][7] = 0;
final var result = new Solver().solve(matrix);
final int[][] expected = new int[9][9];
for(int i = 0; i < expected.length; i++) {
for (int j = 0; j < expected[i].length; j++) {
expected[i][j] = j+1;
Assert.assertArrayEquals(expected, result.get());
It creates a board with each row numbers incrementally from one to nine and then “blanks” four cells. The solver should fill these cells with the correct numbers. Here is how it’s done (note: I
introduce a “solve” method):
public Optional<int[][]> solve(int[][] matrix) {
return nextField(0,0,matrix);
public Optional<int[][]> nextField(int x, int y, int[][] matrix) {
if (y==9 && x == 0) {
return Optional.of(matrix);
if (matrix[y][x]>0) {
int nextX = x<8?x+1:0;
int nextY = x<8?y:y+1;
return nextField(nextX, nextY, matrix);
for(int i = 1; i<=9; i++) {
matrix[y][x] = i;
// check horizontal rule
if (!isPotentialLegal(
matrix[y][6],matrix[y][7],matrix[y][8])) {
int nextX = x<8?x+1:0;
int nextY = x<8?y:y+1;
return nextField(nextX, nextY, matrix);
return Optional.empty();
private static boolean isPotentialLegal(int... numbers) {
final int[] counts = new int[10];
for(int i = 0; i < numbers.length; i++) {
for(int i = 1; i < counts.length; i++) {
if (counts[i]>1) return false;
return true;
“isPotentialLegal” checks for distinct numbers by counting its occurences. It is called with all numbers of the current row. Zeros are “ignored”. If the rule is not satisfied, the next number is
Step 3: Adding the “vertical rule”
Now I add the rule for columns. To create a test, I use a solved sudoku puzzle and clear some cells:
final int[][] matrix = new int[][] {
and later check for the correct solution.
The implementation is straight forward next to the “horizonal rule”:
if (!isPotentialLegal(
) {
Step 4: Adding the “subquadrant rule”
I wondered a bit about how to create a puzzle that would not be solvable without the subquadrant rule, but the original puzzle from Step 3 already did that. It has far more empty cells:
final int[][] matrix = new int[][] {
{0,9,0, 0,0,0, 0,1,0},
{8,0,4, 0,2,0, 3,0,7},
{0,6,0, 9,0,7, 0,2,0},
{0,0,5, 0,3,0, 1,0,0},
{0,7,0, 5,0,1, 0,3,0},
{0,0,3, 0,9,0, 8,0,0},
{0,2,0, 8,0,5, 0,6,0},
{1,0,7, 0,6,0, 4,0,9},
{0,3,0, 0,0,0, 0,8,0},
So here is the subquadrant rule. The key is to get the coordinates of the “subquadrant” right: integer division does the job, i.e. “(x/3)*3”. For example x=4 gets us “3” because it is the middle
subquadrant starting at x=3. I use an extra method here because of the computation of the subquadrant start:
private boolean isSubquadratPotentialLegal(int x, int y, int[][] matrix) {
final int xx = (x/3)*3;
final int yy = (y/3)*3;
return isPotentialLegal(
That did not made the test pass, though! It turned out I missed the backtracking-step, i.e. what happens when the recursion does not return a valid result – try next number (line 29):
public Optional<int[][]> nextField(int x, int y, int[][] matrix) {
if (y==9 && x == 0) {
return Optional.of(matrix);
if (matrix[y][x]>0) {
int nextX = x<8?x+1:0;
int nextY = x<8?y:y+1;
return nextField(nextX, nextY, matrix);
for(int i = 1; i<=9; i++) {
matrix[y][x] = i;
// check horizontal rule
if (!(isPotentialLegal(
// check vertical rule
&& isSubquadratPotentialLegal(x, y, matrix))) {
int nextX = x<8?x+1:0;
int nextY = x<8?y:y+1;
final var result = nextField(nextX, nextY, matrix);
if (result.isPresent()) return result;
matrix[y][x] = 0;
return Optional.empty();
Moreover, line 31 “empties” the cell so that we leave it in the starting state.
That’s it, I implemented a sudoku solver guided by tests. To answer my initial question: it’s fast. Well under one second! I will write a follow-up discussion some optimizations.
Tracking dependency updates
Many software projects use 3rd party libraries aka “dependencies”. You often want to use the most recent version of these dependencies but how do you know when a new release of a dependency is
published? The more dependencies your project have the more tiresome a manual approach to “tracking dependency updates” is.
In this post I explore some solutions that tracks dependency updates for you. I cover broad solutions (libraries.io and dependabot) and Java-only solutions (“artifact listener” and a Gradle/Maven
Why update?
But why do we want to update dependencies at all?
A new version of a dependency
• may fix bugs that affects your project
• may introduce new features that you could use
• may fix a security issue that affects your project
• may have other optimizations to the code
Of course there is a risk as well: a new version may introduce a bug that affects your project. Plus, there might be API changes that require changes in your code.
Tracking solutions
(update) I now use renovatebot because it integrates nicely with Gitlab CI. Much like dependabot (see below), it scans “dependency files” like “build.gradle”, “pom.xml” or “package.json” and creates
merge requests for dependency updates.
From their own words
Libraries.io can automatically keep track of all of the packages that your repositories depend upon across many different package managers.
Once synced, Libraries.io will email you about new versions of your dependencies, if you add or remove a new dependency it will change the notifications settings for that package as soon as you
push to your repositories.
Repositories on Github, Gitlab and Bitbucket are supported. Plus, you can subscribe to dependencies manually, ie without a repository on any of these platforms.
Beside email notifications you can also subscribe to an RSS feed of your dependency updates.
Libraries.io is an open source project.
artifact listener
Artifact Listener is a small service and only available for Java / Maven Central. You can search for libraries and “follow” them. Alternatively you can upload a POM and then choose which dendencies
to follow. Updates of libraries you follow are emailed to you.
You can provide additional email adresses to notify, e.g, email addresses of other team members. This is a small but lovely feature for me.
The service is an open source project.
Dependabot checks the “dependency files” (where your dependencies are definied) in your Github repos for updates. If there is an update it creates a PR for it. The PR may contain links, release
notes, a list of commits etc.
So this service not only notifies you about an update but even creates a PR that applies it. You just have to merge it (at least if your project is on Github).
Dependabout has been aquired by Github.com and is free of charge.
Gradle plugin
If you are using Gradle (a Java build system) to declare dependencies and build your project you can use the Gradle versions plugin to detect dependency updates and report them. It is easy to use.
You just need to execute it on a regular basis.
Maven plugin
Of course, there is a similar plugin for Maven (another Java build system).
Java method references recap
In the last post I reviewed Java lambda expressions. They represent a concise syntax to implement functional interfaces.
Enter Java method references. They represent a concise syntax to implement functional interface using existing methods. Like with lambda expressions, referenced methods are not allowed to throw
checked exceptions.
It’s simply “class-or-instance name” “::” “method name”, like
Function<String, Integer> string2Int = Integer::valueOf;
Types of method references
Reference to a static method
Static methods are referenced using the class name like in the example above.
Reference to an instance method of a particular object
Methods of a particular object are referenced using the variable name of that object:
Map<Integer, String> aMap = new HashMap<>();
Function<Integer, String> getRef = aMap::get;
// call it
String s = getRef.apply(42);
Reference to an instance method of an arbitary object of a particular type
Instead of using an already existing object you can just state the class and a non-static method. Then the instance is an additional parameter. In the following example toURI is a method with no
arguments that returns a String. The function of this method reference takes a File (the object) and returns a String:
Function<File, URI> file2Uri = File::toURI;
Reference to a constructor
Constructors are references using its type and “new”:
Function<String, StringBuffer> bufferFromString = StringBuffer::new;
Here the constructor of StringBuffer with String parameter is referenced. Return type is the type of the constructor, parameters of the function are the parameters of the constructors.
Java lambda expression recap
Lambda expressions in Java represent “functions”, something that take a number of parameters and produce at most one return value.
This could be expressed with anonymous classes but lambda expressions offer a more concise syntax.
Lambda expression consist of a parameter list, an “arrow” and a body.
(String s1, String s2) -> s1 + "|" + s2
The parameter list is enclosed in round brackets. Types are optional. When the expression has exactly one parameter, the brackets can be omitted.
s -> s!=null && s.length>0
The body can either be an expression (that returns a value) or a block. A block is a sequence of statements, enclosed in curly braces.
n -> { if (n<10) System.out.println(n); }
Lambda expressions and types
In the Java type system, lambda expressions are instances of “functional interfaces”. A functional interface is an interface with exactly one abstract method.
Functional interfaces in java.util.function
The package java.util.function in the JDK contains a number of functional interfaces:
• Function<T,U> represents a function with one parameter of type T and return type U
• Consumer<T> represents a function with one parameter of type T and return type void
• Supplier<T> represents a function with no parameter and return type T
• Predicate<T> represents a function with one parameter of type T and return type boolean
Plus, variants with “Bi” prefix exists that have two parameters, like BiPredicate . More variants exists for using primitive types like DoubleToIntFunction .
User defined function interfaces
Any interface with exactly one abstract method can be used as type of a lambda expression. You mark this interface with @FunctionInterface .
interface SomeInterface {
int someBehaviour(String a, String b);
SomeInterface geo = (x,y) -> x.length + y.length;
For me, the benefits of lambda expression are
• concise syntax for anonymous classes that represent functional code
• improved readability
• encouragement of a more functional programming style
How static is a static inner class in Java?
Answer: not static at all. A static inner class behaves like a normal class except that it is in the namespace of the outer class (“for packaging convenience”, as the official Java tutorial puts it).
So as an example:
public class Outer {
private int x = 0;
public int y = 1;
static class Inner {
As opposed to a true inner (nested) class, you do not need an instance of Outer to create an instance of Inner:
Outer.Inner inner = new Outer.Inner();
and Inner instances have no special knowledge about Outer instances. Inner class behaves just like a top-level class, it just has to be qualified as “Outer.Inner”.
Why I am writing about this?
Because I was quite shocked that two of my colleagues (both seasoned Java developers) were not sure if a static inner class was about static members and therefore global state.
Maybe they do not use static inner classes.
When do I use static inner classes?
I use a static inner class
1. when it only of use for the outer class and it’s independent of the (private) members of the outer class,
2. when it’s conceptionally tied to the outer class (e.g. a Builder class)
3. for packaging convenience.
Often, the visibility of the static inner class is not public. In this case there is no big difference whether I create a static inner class or a top-level class in the same source file. An
alternative for the first code example therefore is:
public class Outer {
// ...
// not really inner any more
class Inner {
// ...
An example for (2) is a Builder class:
public class Thing {
public static class Builder {
// ... many withXXX methods
public Thing make() // ...
If the Inner instance needs access to (private) members of the Outer instance then Inner needs to be non-static.
Parallel Stream Processing with Java 8 Stream API
Brian Goetz, Java Language Architect at Oracle, gave an interesting presentation “From Concurrent to Parallel” (available on InfoQ) on the subject back in 2009. Here are the most important points of
his talk.
Java 8 introduces the Stream library (which was, in Brians words, developed as a showcase for the new Java 8 language features). Just calling “.parallel()” advices the library to process the stream
“in parallel”, i.e. in multiple threads.
Do you really need it?
Parallel processing is an optimization and therefore the general questions regarding optimizations have to be answered:
• Do you have any performance requirements at all (otherwise you are already fast enough)
• Do you have any means to measure the performance?
• Does the performance you measure violate the requirements?
Only if the answer to all these questions is “yes” you may take the time to investigate whether “.parallel()” will increase the performance.
Why the performance may not increase?
Compared to sequentionally processing a stream, processing it in parallel has overhead costs: splitting the data, managing the threads that will process the data and combining the results. So you may
not see as much speed up as you might hope for.
Brian talks about several factors that undermine speedup.
NQ is insufficiently high
He talks about a simple model called “NQ model”. N*Q should be greater than 10.000. N is the number of data items. Q is factor that expresses how CPU-expensive the processing step is. “Summing
numbers” or “finding the max of integers” is very inexpensive and Q would be near 1. More complex tasks would have higher values of Q. So if all you want is to add up numbers, you need a lot of
numbers to see a performance gain.
Cache-miss ratio too high
If the data to be processed is saved next to each other in RAM, it will be transfered together in CPU caches. Accessing cache memory instead of main memory is very fast, so data in an array is
processed much faster than data in a linked list that is spread all over the main memory. Also when the data in an array is just pointers data access will be slow. As a demonstration, Brian shows
that summing up an array of (native) integers in parallel scales well with number of CPU cores. Summing up an array of Integer-objects scales very poorly because there are just pointers in the array.
The more indirections (pointers) you work with the more cache-misses will have the CPU wait for memory access.
The source is expensive to split
In order to process data in parallel, the source of the data has to be splitted in order to hand parts of the data to different CPUs. Splitting arrays is simple, splitting linked list is hard (Brian
said: linked list are splitted into “first element, rest”).
Result combination cost is to high
When the result combination is the sum of numbers, that is easy to calculate. If the result is a set, and the result combination is to merge the resulting sets, this is expensive. It might be faster
to sequentially add each result into one set.
Order-sensitive operations
Operations like “limit()”, “skip()” and “findFirst()” depend on the order of the data in the stream. This makes the pipelines using them “less exploitable” regarding parallelism. You can call
“unordered()” if the order is not meaningful to you and the JVM will optimize those operations.
Mutating shared state is a big NO!
Of course you will lose performance if you have to mutate shared state and guarding access to it with locks etc.
Parallel streams are for CPU-heavy process steps
Brian mentions that parallel streams where built for CPU-heavy tasks. If you do mainly IO in the processing steps than use ThreadPools and Executors etc.
What degree of parallelism does the Stream API uses?
As stated in the Streams API documentation, the API uses one JVM wide Fork-Join-Pool with a number of threads that defaults to the number of processors on the computer system. Why? Because it is
expected that a processing steps will utilize the CPU as much as possible. So there is no sense in having more threads then CPU cores.
In contrast, when having IO-heavy tasks you want to have many more threads than CPU cores because the threads will wait for IO most of the time and therefore not utilize the CPU.
A “synchronized method” bug
In Java, a synchronized method is not thread safe if it reads from and writes to one or more static member variables.
public class SomeClass {
static int someCounter = 0;
synchronized void doSomething() {
for(int i = 0; i < 20; i++) {
// do something that takes a bit of time, e.g.
// java.net.InetAddress.getByName("www.wikipedia.org");
and assume the access to someCounter is somehow thread safe because of the synchronized keyword on doSomething.
As soon as you call doSomething concurrently on multiple SomeClass instances, it will not print unique numbers. This is because the all instance share the same static member variables. Between the
increment of someCounter and printing it, its value might have already changed by another instance.
That particular bug was a bit hidden because a “SomeClass” instance was “cached” in a JEE stateless session bean. Of course the JEE container creates multiple instances of the session bean and hence
multiple instances of SomeClass. | {"url":"https://on-sw-integration.epischel.de/category/coding/java/","timestamp":"2024-11-15T03:08:02Z","content_type":"text/html","content_length":"112830","record_id":"<urn:uuid:df3df724-50a8-4d6a-9749-d10b47362fab>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00075.warc.gz"} |
How Much Grading Should Professors Do?
One of the outcomes of the Harvard budget crisis is that the budget for Teaching Assistants has been brought into line. At Harvard, we've been ridiculously spoiled with TAs for many years now, with a
TA for roughly every 12-15 students; last year, I was able to get 6 undergrad TAs for 80 students in my Algorithms and Data Structures class. This year, I'll have a more reasonable TA for every 18-20
students or so. Yes, I know that remains quite luxurious compared to many places; I'm not complaining. But it is a change I have to deal with.
One of the big responsibilities for my TAs is grading. I assign a fairly substantial load of homework in my undergrad class, and unfortunately the number of problems where you can just check the
answer without reading through the student's work is small. (They often have to prove things.) And even checking answers takes a long time. Given the new TA/student ratio, it seems unfair (and, quite
frankly, unworkable, assuming TAs stick to anything close to their supposed working hours) to have the TAs grade the same homeworks in the past. So, it seems, I'll have to change something. The
natural options seem to be:
1) Reduce the assignments. Take what would in the past have been the hardest problem on each assignment and make it an ungraded "challenge problem", for instance.
2) Change the scope/style of the assignments. Less proof-type work, more numerical exercises where you can just check the final answer.
3) Introduce probabilistic grading. Only 5 of the 6 problems will be graded -- you just don't know which 5 in advance.
4) Allow people to work in pairs for assignments. Fewer writeups to grade.
5) Grade more myself.
The first four options all have fairly big negatives associated with them. (Actually, I don't have a problem with probabilistic grading, but I have no doubt it would cause bitter complaints from the
students, even if I gave a lecture explaining why it was a reasonable approach. There would always be students coming up afterward to complain that they would have gotten a better grade if I just
graded all of their problems, and I don't look forward to having to explain the policy to higher-ups. And working in pairs isn't necessarily a negative, but they can already talk about problems
together, and they can work in pairs for programming assignments; I think they should practice writing up proofs themselves.)
The main downside to the final option is, of course, to me personally. I do, already, do some of the grading. It's very time-consuming. Even if the time per assignment is small, multiply by the
number of students (I expect 60 this year) and we're talking a good number of hours.
So how much grading should a professor do? How much are others of you doing? Or does anyone have other creative suggestions for solutions before the semester starts?
24 comments:
I know as a TA at what you could consider a smaller school, I have about 50 students whose assignments I grade, but it's a math class, so while the majority of the problems have work, while
others have little.
I suppose the culture among the grad students is a bit different there, but I would say that I don't find the amount of work I'm given unfair; I enjoy it! I like your idea for probabilistic
though, that's the first time I've heard of that. I could understand why it would be a hard sell though.
This is a bit science-fiction but you may require the students to write their proofs using a proof assistant. To avoid the burden of the details of formal proofs, you allow them to assume boring/
tedious lemmas and focus on the high-level intuition of a proof. That way, you can still ask for proof-work, and check those in no time using a computer.
Group homework. Each student writes up their own solutions, but students work and submit in pairs. For each problem, one of the pair is chosen at random and graded, and both students receive that
grade. But no credit for pairs that submit identical solutions; each student still has to write up her own proofs herself.
As for the complaining students, here's how I respond: Yes, you would have gotten a better grade if we'd done something else, but we didn't, so you didn't. For any two grading systems X and Y,
there are always students who prefer system X than system Y, and other students that would prefer system Y than system X. But we had to pick something, and we have to be consistent; changing the
grading standard now would be grossly unfair to everyone. Also, stop worrying about your stupid grade and learn the course material!
For the record, I have three TAs this semester for a class of 120+, with short weekly homeworks. Students can submit solutions in groups of up to three (one solution per group; no randomization).
In practice, this will probably mean about 50 homework groups. I can usually hire one or two undergrad graders to absorb some/most of the written grading burden. We also have a rolling oral
homework system: Each week, 1/3 of the groups each get 30 minutes to present their solutions on a whiteboard, 10 minutes per problem, with the presenter of each problem chosen randomly from the
group. (Everyone in the group gets the same grade.) The oral homeworks are a huge time sink, but it pays off.
In the past, I've asked huge classes whether they'd prefer (1) letting us grade half of their homework and getting it back quickly, or (2) making us grade it all but getting it back much later.
The students have always chosen (2). Even when it meant that the last third of the homework wasn't graded until after the final.
A grad TA position is nominally 20 hours per week. Do you really assign so much homework that it is an hour per week per student for the TA to grade?
Grad TAs usually do a lot more than just grade homework: they run recitations, answer email questions, hold office hours and make/grade exams. So why do you assume they are just grading hw?
Erik had an interesting grading scheme for his Advanced Data Structures class at MIT: each question had just three possible grades 0, 1, 2. Also there was one page limit for each answer. The TA
got back to us fairly quickly so it must be less work.
At Yale I TAed Algorithms twice. Each time there were approximately 40 students taking the class, and there were 2 graduate student TAs. Algorithms had weekly assignments which seem similar to
what you describe giving. The TAs would alternate grading assignments, and the workload was pretty reasonable for the graders.
I don't know if undergraduate TAs are expected to put in the same amount of time as graduate student TAs -- at Yale undergrads aren't allowed to grade other undergrads, so all the grading was
done by grad students. But we graded every problem, and the professor teaching never had to grade homework.
The slow part of grading is figuring out what students are doing wrong.
Letting students work in small groups and write up answers separately is good, because this allows the graders to look at all answers to figure out what the group was trying to do. It probably
also decreases the number of wrong answers, making grading go faster.
It is also often possible on open-ended problems to add in a small subquestion that a student with the right idea should almost surely be able to answer correctly. This allows the grader to
decide quickly if the student has the right or wrong idea.
Might be Harvard students have higher expectations -- but in a typical big-ten university in the corn-fields, it is completely standard practice for graders to grade randomly sampled ( < 10 %) of
home-work problems (say in a 400+ size freshman calculus class).
Make students evaluate each others paper. TA will score each student on two fronts - the paper and the evaluation of other students. This makes students critique others work, which I think is a
valuable skill.
Jeff -- I appreciate your suggestions. Again, I have my issues with group homework -- I do already have students work together on programming assignments, I'd like them to do some of their own
assignments (particularly proofs) -- but in a resource-constrained environment it makes solid sense (as does probabilistic grading).
As for the second paragraph, I do take the same basic approach with complaining students. I hope it's more effective for you.
To Anons 4+5: A standard TA assignment, grad or undergrad, is typically "quarter time" or 12.5 hours per week at Harvard, not 20. And as Anon 5 points out, my TAs hold a weekly section, hold 2
office hours per week, answer questions by e-mail, and do other course administration. Really, I don't expect them to do any more grading than they already do.
Anon 6: I like Erik's approach for a grad class; I'm not sure how it would work for an undergrad class (but it may be worth a try!). I really like the 1 page per answer limit; I think I'm going
to invoke that.
Can you reduce the number of graded homeworks (or get rid of them altogether)? While doing homework is the best way for the student to learn, I find it is not the best way to assign grades (since
cheating is so easy). So why not post optional, ungraded homeworks (with full solutions posted a week later), stressing to the students that the homeworks will be great preparation for the exams.
I would bet that writing up detailed solutions will be less work (and more enjoyable) than grading. And students who don't understand the solutions can come to office hours...
You (and professors in general) should grade more homework. This allows you to see what the students are learning/doing and evaluate whether or not you're an effective teacher. Furthermore, I
believe each professor should keep track of how time-consuming their homeworks are. It is irresponsible to assign too much work because a professor feels that his/her class/discipline is somehow
more important.
I'm going to disagree with the last poster a bit: I think it's important to strike a balance with the amount of grading you do. This last semester, I shared the grading load for homeworks
approximately equally with my two TAs and one co-teacher (yes - we're spoiled at CMU as well, but in fairness, it was a new course we were developing). My TAs handled the project grading. I found
this split a bit more than I would prefer: I'd rather have sat down with my TAs, graded a few assignments end-to-end with them to make sure we were all on the same page, and been done with it. As
it was, the homework grading took away a bit from things that only I could do as the professor -- course & lecture development, etc. To the last poster: It's all relative. Had this not been a new
course, I'd have been happy doing more grading, but the reality is that we (faculty) have a finite time budget for our courses, and there are better and worse ways to allocate that budget.
I'm actually a fan of the peer-grading system. David Karger used this for the (grad) algorithms courses I took, and I thought it was great. It didn't take too much time, and it forced everyone to
learn the correct solutions to the problems, and to think about what was wrong with the incorrect solutions. But it can be a risky approach if grades actually matter, as opposed to in a grad
As a professor, I have no objections to grading a random subset of problems. I'm pretty sure our higher-ups would support any reasonable grading policy as long as it was clearly spelled out in
advance to the students. I've come to believe that these are really only two things that are mandatory with homework: Clear requirements and rapid turn-around. My new years resolution for a few
years running has been to reduce the homework turn-around time for every class I teach. I think I may try to do that with my ugrad distributed systems class next year by using randomized grading
+ having the TAs go through the un-graded problems in recitations - possibly with peer grading. :)
Thanks for asking this question - it's good to hear people's viewpoints about it.
Another anonymous in a big state school here (not in the corn fields) who has always done probabilistic grading and never got a complaint.
I once attended a course, by Alexander Postnikov, where the homework had 9 problems, each worth 10 points, and your score was taken mod 30 (though 30, 60, 90 counted as 30, not 0).
Your initial post was misleading. Your old TA budget looked positively luxurious but at 12.5 hours per week each the budget looks pretty normal and the new one does seem tight.
It is not easy to answer your question since it is hard to judge how large the workload is for students in your class. Sometimes we can get the same benefit from actually assigning fewer
problems, even though there are ones that we like and want to ask. I handle this by including "extra credit" problems. These allow me to ask some of my favorite questions targeted at the more
motivated students without having a large segment of the class incented to answer them with confusing drivel when they don't really have a good answer. This saves a bunch of grading time.
(Eztra credit problems are not required to earn a top grade but top students invariably want to work on them.)
I currently teach with a ratio of about 70-75 students/TA. Yes, that's right: over a factor of 3 worse than yours.
My view is that you should not change the problems you offer; in an algorithms course, thinking about great problems is one of the most important ways you learn. Instead, you should allow the
quality or quantity of the feedback you offer, and the accuracy of the grades, to suffer. If you have to choose between "great problems" and "detailed feedback/accurate grades", continue to
prioritize great problems.
You can do probabilistic grading. You can just tell the TAs to grade less carefully. A third option is to hire undergraduate graders: not TAs, people who are hired only for the purpose of grading
homework. You can probably hire them for $10-12/hour, so they're a lot cheaper than TAs. Allocate a small fraction of your TA budget (say, 10%) towards undergrad graders. Have the undergrad
graders help grade the homeworks. Yes, they're a lot less accurate in their grading and the feedback they give is not as good, but on the other hand it enables you to grade every problem.
Thanks to everyone for the interesting and worthwhile feedback. I'm definitely going to ponder probabilistic grading. I must say, I can't recall ever being in a class that used it (I was a
Harvard undergrad...), and hence my original opinion that students would revolt. Maybe it's not so bad.
I've become reasonably attached to peer grading: assign a few students to grade under the supervision of the TAs each week. Typically, each student grades once, and grades only one problem (so
they can become an expert on it). I reserve 5% of the course grade for grading, to provide the proper motivation.
The students aren't set free to grade independently---rather, they sit in a room and work beside the TA, who is available for initial discussion on how to grade the problem and also for
consultation on unusual answers. So the TA is still involved, but their involvement scales to a larger class. This has been my approach for several years and it seems to be working.
My courses generally involve final projects, and I grade those myself.
I completely agree with AnonProf's tradeoff: Great problems are the heart and soul of a good algorithms class.
I have my issues with group homework
Yeah, me too. But at some point it became a practical necessity. One result that I expected when I added group homework many years ago is that homework grades went way up, and the first midterm
average went down. Surprisingly, though, later exam averages actually went up.
Another strategy I use to simplify grading is giving 25% credit for writing "I don't know" (and nothing else) for ANY question, and absolutely no partial credit for regurgitation and/or
As for the second paragraph, I do take the same basic approach with complaining students. I hope it's more effective for you.
It works reasonably well for me, actually. Students here are generally accepting of the fairness argument. But probably my students are also less used to perfection than yours, and they certainly
pay less, and so may feel less entitled. Also, I am mean.
David -- Thanks for the explanation. That sounds interesting. I'll ponder it...
Jeff -- Are you trying to say I'm not mean? Because there seems to be a number of students who would tell you otherwise...
When I taught Complexity at Berkeley, I took an even more extreme approach than giving credit for "I don't know." I was giving full credit for the statement "I know this."
This seemed to work well coupled with the warning that grading is not a linear function, and underperforming on simple questions during exams will erase the "I know this" credit.
It also seemed to diminish complaining about homework grading, since everybody understood they can get max on those, if they wanted it.
Of course, this system assumes maturity and failed miserably for a few students. I guess if colleges are seen as places to grow up (in the US), this might not be too bad an outcome :).
why not outsource it, let the grad students concentrate on research | {"url":"https://mybiasedcoin.blogspot.com/2010/01/how-much-grading-should-professors-do.html?showComment=1262638168987","timestamp":"2024-11-15T01:26:45Z","content_type":"application/xhtml+xml","content_length":"117787","record_id":"<urn:uuid:a144a834-4ade-4792-8341-413d075c7ad7>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00687.warc.gz"} |
yAxis rotation parent object.
Line 10 is your problem. Subtracting one axis from another does not give an angle. In fact, it is mathematically meaningless.
Basic trigonometry tells us that the tangent of an angle is the opposite side over the adjacent. Remember TANOA from algebra in school?
The function I suggested is called Mathf.Atan2() for a reason: it takes 2 coordinates and returns you the arctangent, which is the inverse of the tangent. In other words, it maps from two orthogonal
axis to an angle.
This means that the expression Mathf.Atan2(xAxis,zAxis) * Mathf.Rad2Deg will give you an angle based on the relative magnitude and direction of the two inputs.
You could then use that angle as your Rotate, perhaps with a 90-degree or 180-degree or other offset/negation.
Generally when you HAVE an angle, you don’t use .Rotate(), which is a relative command.
Instead when you have the angle, you just set the angle directly with a construct like this:
transform.rotation = Quaternion.Euler( 0, angleThatICalculatedAbove, 0); | {"url":"https://discussions.unity.com/t/yaxis-rotation-parent-object/769743/4","timestamp":"2024-11-03T03:58:09Z","content_type":"text/html","content_length":"31686","record_id":"<urn:uuid:46eb2ab4-c328-4491-9da0-fbc450e1ce1e>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00869.warc.gz"} |
Manipulating Trajectories | FTCLib Docs
Manipulating Trajectories
Once a trajectory has been generated, you can retrieve information from it using certain methods. These methods will be useful when writing code to follow these trajectories.
Getting the Total Duration of the Trajectory
Because all trajectories have timestamps at each point, the amount of time it should take for a robot to traverse the entire trajectory is predetermined. ThegetTotalTimeSeconds()method can be used to
determine the time it takes to traverse the trajectory.
// Get the total time of the trajectory in seconds
double duration = trajectory.getTotalTimeSeconds();
Sampling the Trajectory
The trajectory can be sampled at various timesteps to get the pose, velocity, and acceleration at that point. The sample(double timeSeconds) method can be used to sample the trajectory at any
timestep. The parameter refers to the amount of time passed since 0 seconds (the starting point of the trajectory).
// Sample the trajectory at 1.2 seconds. This represents where the robot
// should be after 1.2 seconds of traversal.
Trajectory.State point = trajectory.sample(1.2);
The sample has several pieces of information about the sample point:
t: The time elapsed from the beginning of the trajectory up to the sample point.
velocity: The velocity at the sample point.
acceleration: The acceleration at the sample point.
pose: The pose (x, y, heading) at the sample point.
curvature: The curvature (rate of change of heading with respect to distance along the trajectory) at the sample point.
Note: The angular velocity at the sample point can be calculated by multiplying the velocity by the curvature. | {"url":"https://docs.ftclib.org/ftclib/master/pathing/trajectory/manipulating-trajectories?fallback=true","timestamp":"2024-11-06T10:54:02Z","content_type":"text/html","content_length":"181024","record_id":"<urn:uuid:b411a12d-d8e8-4195-83ec-d631117585d6>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00543.warc.gz"} |
Consider the frequency distribution, where A is a positive inte... | Filo
Consider the frequency distribution, where is a positive integer:
If the variance is , then the value of is
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Consider the frequency distribution, where is a positive integer:
Question Text
If the variance is , then the value of is
Topic Statistics
Subject Mathematics
Class Class 11
Answer Type Text solution:1
Upvotes 93 | {"url":"https://askfilo.com/math-question-answers/consider-the-frequency-distribution-where-a-is-a-positive-integerif-the-variance","timestamp":"2024-11-11T23:09:33Z","content_type":"text/html","content_length":"307354","record_id":"<urn:uuid:b7eb9cc4-471c-46e1-b437-e26d2bf04052>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00063.warc.gz"} |
(1) Department of Bioinformatics, Institute of Biochemistry and Biophysics, University of Tehran, Tehran, Iran
(2) Department of Computer Engineering, Sharif University of Technology, Tehran, Iran
(3) Department of Statis-tics, Allameh Tabataba'i University, Tehran, Iran
date: “2020-04-04”
This package has developed a tool for performing a novel pathway enrichment analysis based on Bayesian network (BNrich) to investigate the topology features of the pathways. This algorithm as a
biologically intuitive, method, analyzes of most structural data acquired from signaling pathways such as causal relationships between genes using the property of Bayesian networks and also infer
finalized networks more conveniently by simplifying networks in the early stages and using Least Absolute Shrinkage Selector Operator (LASSO). impacted pathways are ultimately prioritized the by
Fisher’s Exact Test on significant parameters. Here, we provide an instance code that applies BNrich in all of the fields described above.
This document offers an introductory overview of how to use the package. The BNrich tool uses Bayesian Network (BN) in a new topology-based pathway analysis (TPA) method. The BN has been demonstrated
as a beneficial technique for integrating and modeling biological data into causal relationships (1–4). The proposed method utilizes BN to model variations in downstream components (children) as a
consequence of the change in upstream components (parents). For this purpose, The method employs 187 KEGG human non-metabolic pathways (5–7) which their cycles were eliminated manually by a
biological intuitive, as BN structures and gene expression data to estimate its parameters (8,9). The cycles of inferred networks were eliminated on the basis of biologically intuitive rules instead
of using computing algorithms (10). The inferred networks are simplified in two steps; unifying genes and LASSO. Similarly, the originally continuous gene expression data is used to BN parameters
learning, rather than discretized data (8). The algorithm estimates regression coefficients by continuous data based on the parameter learning techniques in the BN (11,12). The final impacted
pathways are gained by Fisher’s exact test. This method can represent effective genes and biological relations in impacted pathways based on a significant level.
Quick Start
Install BNrich
install.packages("BNrich_0.1.0.tar.gz", type="source", repos=NULL)
prepare essential data
At first, we can load all the 187 preprocessed KEGG pathways which their cycles were removed, the data frame includes information about the pathways and vector of pathway ID.
destfile = tempfile("files", fileext = ".rda")
files <- fetch_data_file()
Note that it's better to use (for example:) destfile = "./R/BNrich-start.rda" to save essential files permanently.
The input data should be as two data frames in states disease and (healthy) control. The row names of any data frame are KEGG geneID and the number of subjects in any of them should not be less than
20, otherwise the user may encounters error in LASSO step. Initially, we can load dataset example. The example data extracted from a part of GSE47756 dataset, the gene expression data from colorectal
cancer study (13).
Data <- system.file("extdata", "Test_DATA.RData", package = "BNrich", mustWork = TRUE)
H1 H2 H3 H4 H5 H6 H7 H8
hsa:1 3.37954 3.3469 3.78383 3.35186 3.2091 3.40245 4.06329 3.43424
hsa:100 3.1147 3.15981 3.37842 2.69868 3.43759 3.38588 2.95406 3.09631
hsa:10000 3.21876 2.93611 2.62708 3.13507 2.62864 2.61367 2.7336 2.70867
hsa:1001 3.4549 3.18683 3.34896 3.36903 3.49353 3.35175 3.27893 3.63678
hsa:10010 2.17522 2.59843 2.56868 2.95009 2.52181 2.24635 2.05092 2.10438
hsa:10013 2.992 2.94325 3.22677 2.87371 3.063 2.97679 3.07247 3.08168
D1 D2 D3 D4 D5 D6 D7 D8
hsa:1 3.29082 3.15924 3.45716 3.15391 3.29514 3.36502 3.63823 3.22192
hsa:100 3.069 2.97546 2.99117 2.88929 3.00292 2.94948 2.93906 3.36357
hsa:10000 2.68424 3.24284 3.57435 2.46992 4.57649 3.87179 2.94405 3.54207
hsa:1001 3.27815 2.91081 3.53487 2.95122 2.67742 2.72358 3.10172 3.07123
hsa:10010 2.68051 3.22719 3.58798 2.61269 3.72397 3.29004 2.5843 2.95756
hsa:10013 3.05107 2.86273 3.06863 3.05318 3.04536 2.92021 3.12596 3.0468
Unify data, the first step of simplification
Initially, we need to unify gene products based on 187 imported signaling pathways (mapkG list) in two states disease (dataD) and control (dataH). This is the first simplification step, unifying
nodes in signaling pathways with genes those exist in gene expression data.
unify_results <- unify_path(dataH, dataD, mapkG, pathway.id)
The unify_path function performs the following processes: • Split datasets into KEGG pathways • Delete all gene expression data are not in pathways • Removes all gene products in pathways are not in
dataset platforms • Remove any pathways with the number of edges is less than 5 This function returns a list contain data_h,data_d,mapkG1 and pathway.id1. data_h and data_d are lists contain data
frames related to control and disease objects unified for any signaling pathways. The mapkG1 is a list contains unified signaling pathways and pathway.id1 is new pathway ID vector based on remained
pathways. In the example dataset, the number of edges in the one pathway becomes less than 5 and are removed:
mapkG1 <- unify_results$mapkG1
As well, the number of edges reduces in the remaining pathways. In first pathway hsa:01521 the number of edges from 230 reduces to 204:
A graphNEL graph with directed edges
Number of Nodes = 79
Number of Edges = 230
pathway.id1 <- unify_results$pathway.id1
A graphNEL graph with directed edges
Number of Nodes = 71
Number of Edges = 204
BN: construct structures and estimate parameters
construct BN structures
Now we can construct BN structures based on unified signaling pathways and consequently need the results of unify_path function.
BN <- BN_struct(unify_results$mapkG1)
The BN_struct function returns a list contains BNs structures reconstructed from all mapkG1.
The LASSO regression, the second step of simplification
Given that the data used is continuous, each node is modeled as a regression line on its parents (11,14). Thus, on some of these regression lines, the number of these independent variables is high,
so in order to avoid the collinearity problem, we need to use the Lasso regression (15,16). We perform this function for any node with more than one parent, in all BNs achieved by BN_struct function,
based on control and disease data obtained by unify_results function.
data_h <- unify_results$data_h
data_d <- unify_results$data_d
LASSO_results <- LASSO_BN(BN, data_h, data_d)
The LASSO_BN function returns a list contains two lists BN_H and BN_D are simplified BNs structures based on LASSO regression related to healthy and disease objects. This function lead to reduce
number of edges too:
Estimate the BN parameters
Now we can estimate (learn) parameters for any BNs based on healthy and disease data lists.
BN_H <- LASSO_results$BN_H
BN_D <- LASSO_results$BN_D
esti_results <- esti_par(BN_H, BN_D, data_h, data_d)
The esti_par function returns a list contains four lists. The BN_h, BN_d, are lists of BNs which their parameters learned by control and disease objects data. The coef_h and coef_d are lists of
parameters of BN_h and BN_d. As you can see in below, node hsa:1978 in the first BN has one parent. The coefficient in control (healthy) data is 0.6958609 and in disease data is 1.1870730.
esti_results$BNs_h[[1]]$` hsa:1978`
Parameters of node hsa:1978 (Gaussian distribution)
Conditional density: hsa:1978 | hsa:2475
(Intercept) hsa:2475
2.8841264 0.6958609
Standard deviation of the residuals: 0.3489612
Parameters of node hsa:1978 (Gaussian distribution)
Conditional density: hsa:1978 | hsa:2475
(Intercept) hsa:2475
0.9046357 1.1870730
Standard deviation of the residuals: 0.2713789
Testing the equality BNs parameters
Variance of BNs parameters
We require the variance of the BNs parameters to perform the T-test between the corresponding parameters.
BN_h <- esti_results$BNs_h
BN_d <- esti_results$BNs_d
coef_h <- esti_results$coef_h
coef_d <- esti_results$coef_d
var_mat_results<- var_mat (data_h, coef_h, BN_h, data_d, coef_d, BN_d)
The var_mat function returns a list contains two lists var_mat_Bh and var_mat_Bd which are the variance-covariance matrixes for any parameters of BN_h and BN_d. The variance-covariance matrixes for
the fifth node,hsa:1978, in first BN in two states control and disease is as follow:
[,1] [,2]
[1,] 10.177073 -3.630152
[2,] -3.630152 1.296990
[,1] [,2]
[1,] 3.549338 -1.0392040
[2,] -1.039204 0.3053785
Testing the equality BNs parameters
T-test perfoms between any corresponding parameters between each pair of learned BNs, BN_h and BN_d, in disease and control states. Assumptions are unequal sample sizes and unequal variances for all
var_mat_Bh <- var_mat_results $var_mat_Bh
var_mat_Bd <- var_mat_results $var_mat_Bd
Ttest_results <- parm_Ttest(data_h, coef_h, BN_h, data_d, coef_d, BN_d, var_mat_Bh, var_mat_Bd, pathway.id1)
From To pathway.number pathwayID Pval coefficient in disease coefficient in control fdr
intercept hsa:2065 1 hsa:01521 0.605294 4.893503 5.535163 6.72E-01
hsa:7039 hsa:2065 1 hsa:01521 2.04E-05 1.072296 -0.21107 6.95E-05
hsa:1950 hsa:2065 1 hsa:01521 0.154223 0.125977 -0.21675 2.11E-01
hsa:4233 hsa:2065 1 hsa:01521 0.083296 -0.63254 -0.33154 1.23E-01
hsa:3084 hsa:2065 1 hsa:01521 0.135981 -0.55586 -0.18792 1.89E-01
hsa:9542 hsa:2065 1 hsa:01521 0.373051 -0.39859 -0.11334 4.49E-01
This function returns a data frame contains T-test results for all parameters in all final BNs. The row that is intercept in From variable, shows significance level for gene product that is shown in
To variable. The rest of the data frame rows shows significance level for any edge of networks.
Identification of enriched pathways
In the last step we can determine enriched pathways by own threshold on p-value or fdr. Hence we run the Fisher's exact test for any final pathways. As stated above, the Ttest_results is a data frame
contains T-test results for all parameters in final BNs achieved by parm_Ttest function and fdr.value A numeric threshold to determine significant parameters (default is 0.05).
BNrich_results <- BNrich(Ttest_results, pathway.id1, PathName_final, fdr.value = 0.05)
pathwayID p.value fdr pathway.number Name
hsa:05016 2.66E-17 2.47E-15 123 Huntington disease
hsa:05202 1.64E-17 2.47E-15 156 Transcriptional misregulation in cancer
hsa:05012 2.92E-16 1.81E-14 121 Parkinson disease
hsa:05010 1.55E-11 7.19E-10 120 Alzheimer disease
hsa:04144 3.25E-08 1.21E-06 22 Endocytosis
hsa:04714 2.99E-07 9.26E-06 72 Thermogenesis
Session Info
The following package and versions were used in the production of this vignette.
R version 3.6.1 (2019-07-05)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 7 x64 (build 7601) Service Pack 1
Matrix products: default
[1] LC_COLLATE=English_United Kingdom.1252 LC_CTYPE=English_United Kingdom.1252
[3] LC_MONETARY=English_United Kingdom.1252 LC_NUMERIC=C
[5] LC_TIME=English_United Kingdom.1252
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] BNrich_0.1.0
loaded via a namespace (and not attached):
[1] Rcpp_1.0.2 codetools_0.2-16 lattice_0.20-38 corpcor_1.6.9
[5] foreach_1.4.7 glmnet_2.0-18 digest_0.6.20 grid_3.6.1
[9] stats4_3.6.1 evaluate_0.14 graph_1.63.0 Matrix_1.2-17
[13] rmarkdown_1.14 bnlearn_4.5 iterators_1.0.12 tools_3.6.1
[17] parallel_3.6.1 xfun_0.8 yaml_2.2.0 rsconnect_0.8.15
[21] compiler_3.6.1 BiocGenerics_0.31.5 htmltools_0.3.6 knitr_1.24
1. Yu J, Smith VA, Wang PP, Hartemink AJ, Jarvis ED. Advances to Bayesian network inference for generating causal networks from observational biological data. Bioinformatics . 2004 Dec 12;20
2. Gendelman R, Xing H, Mirzoeva OK, Sarde P, Curtis C, Feiler HS, et al. Bayesian network inference modeling identifies TRIB1 as a novel regulator of cell-cycle progression and survival in cancer
cells. Cancer Res. 2017;77(7):1575–85.
3. Luo Y, El Naqa I, McShan DL, Ray D, Lohse I, Matuszak MM, et al. Unraveling biophysical interactions of radiation pneumonitis in non-small-cell lung cancer via Bayesian network analysis.
Radiother Oncol . 2017 Apr 1 ;123(1):85–92.
4. Agrahari R, Foroushani A, Docking TR, Chang L, Duns G, Hudoba M, et al. Applications of Bayesian network models in predicting types of hematological malignancies. Sci Rep . 2018 Dec 3;8(1):6951.
5. Zhi-wei J, Zhen-lei Y, Cai-xiu Z, Li-ying W, Jun L, Hong-li W, et al. Comparison of the Network Structural Characteristics of Calcium Signaling Pathway in Cerebral Ischemia after Intervention by
Different Components of Chinese Medicine. J Tradit Chinese Med. 2011;31(3):251–5.
6. Lou S, Ren L, Xiao J, Ding Q, Zhang W. Expression profiling based graph-clustering approach to determine renal carcinoma related pathway in response to kidney cancer. Eur Rev Med Pharmacol Sci.
7. Fu C, Deng S, Jin G, Wang X, Yu ZG. Bayesian network model for identification of pathways by integrating protein interaction with genetic interaction data. BMC Syst Biol. 2017;11.
8. Isci S, Ozturk C, Jones J, Otu HH. Pathway analysis of high-throughput biological data within a Bayesian network framework. 2011;27(12):1667–74.
9. Korucuoglu M, Isci S, Ozgur A, Otu HH. Bayesian pathway analysis of cancer microarray data. PLoS One. 2014;9(7):1–8.
10. Spirtes P, Richardson T. Directed Cyclic Graphical Representations of Feedback Models. Proc Elev Conf Uncertain Artif Intell. 1995;1–37.
11. Neapolitan RE. Learning Bayesian networks. first. Chicago: Pearson Prentice Hall; 2004. 291–425 p.
12. Scutari M. Learning Bayesian Networks with the bnlearn R Package. J Stat Softw. 2010;35(3):1–22.
13. Hamm A, Prenen H, Van Delm W, Di Matteo M, Wenes M, Delamarre E, et al. Tumour-educated circulating monocytes are powerful candidate biomarkers for diagnosis and disease follow-up of colorectal
cancer. Gut. 2016;65(6):990–1000.
14. Nagarajan R, Scutari M, Lèbre S. Bayesian Networks in R . New York, NY: Springer New York; 2013 [cited 2018 Apr 17].
15. Tibshirani R. The lasso method for variable selection in the cox model. Stat Med. 1997;16(4):385–95.
16. Buhlmann P, Geer S van de. Statistics for high-dimensional data: Methods, Theory and Applications . Springer Series in Statistics. 2011. 7–34 p. | {"url":"https://www.stats.bris.ac.uk/R/web/packages/BNrich/vignettes/BNrich.html","timestamp":"2024-11-05T05:49:25Z","content_type":"text/html","content_length":"34771","record_id":"<urn:uuid:d9b518db-204b-476b-b35f-c856be13a48f>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00775.warc.gz"} |
Perspectivas em análise do comportamento
versión On-line ISSN 2177-3548
HENKLAIN, Marcelo Henrique Oliveira y CARMO, João dos Santos. Behavior analytic production on teaching and learning of mathematical skills: Representative data of scientific brazilian events.
Perspectivas [online]. 2011, vol.2, n.2, pp.179-191. ISSN 2177-3548.
This study aimed to identify and describe the works of behavior analysts about teaching and learning of mathematics, presented in Brazilian Association for Behavioral Medicine and Psychology' (ABPMC)
Meetings and Brazilian Psychological Society' (SBP) Annual Meetings from 1992 to 2011. Documents found (annals and/or program schedules) were entirely read in order to identify researches with at
least one of the following descriptors: (a) number, (b) numeral, (c) mathematics, (d) arithmetic, (e) ordering, (f) quantity, (g) problem solving. The results suggest that researches on teaching and
learning of mathematics are concentrated both in operationalization and teaching of mathematics fundamental concepts, as well as the reversal of failure and anxiety in math. Although production is
cohesive and socially relevant, there is the need to expand the community of behavior analysts interested in researching in the area of teaching and learning of mathematics. There is also a need to
increase the effectiveness of works' dissemination to the community of behavior analysts as well as to external community. Further studies are needed in order to map, organize, describe, evaluate,
and disseminate the production on behavioral teaching and learning of mathematics.
Palabras clave : behavior analysis; mathematics teaching and learning; scientific production; scientific meetings. | {"url":"https://pepsic.bvsalud.org/scielo.php?script=sci_abstract&pid=S2177-35482011000200004&lng=es&nrm=iso&tlng=en","timestamp":"2024-11-13T02:22:27Z","content_type":"application/xhtml+xml","content_length":"17676","record_id":"<urn:uuid:96628f56-1786-486e-a54e-d2d9651d879c>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00711.warc.gz"} |
PROC PANEL: R-Square :: SAS/ETS(R) 9.22 User's Guide
The conventional R-square measure is inappropriate for all models that the PANEL procedure estimates by using GLS since a number outside the [0,1] range might be produced. Hence, a generalization of
the R-square measure is reported. The following goodness-of-fit measure (Buse 1973) is reported:
This is a measure of the proportion of the transformed sum of squares of the dependent variable that is attributable to the influence of the independent variables.
If there is no intercept in the model, the corresponding measure (Theil 1961) is
However, the fixed-effects models are somewhat different. In the case of a fixed-effects model, the choice of including or excluding an intercept becomes merely a choice of classification.
Suppressing the intercept in the FIXONE or FIXONETIME case merely changes the name of the intercept to a fixed effect. It makes no sense to redefine the R-square measure since nothing material
changes in the model. Similarly, for the FIXTWO model there is no reason to change R-square. In the case of the FIXONE, FIXONETIME, and FIXTWO models, the R-square is defined as the Theil (1961)
R-square (detailed above). This makes intuitive sense since you are regressing a transformed (demeaned) series on transformed regressors, excluding a constant. In other words, you are looking at one
minus the sum of squared errors divided by the sum of squares of the (transformed) dependent variable.
In the case of OLS estimation, both of the R-square formulas given here reduce to the usual R-square formula. | {"url":"http://support.sas.com/documentation/cdl/en/etsug/63348/HTML/default/etsug_panel_sect038.htm","timestamp":"2024-11-14T22:14:51Z","content_type":"application/xhtml+xml","content_length":"12000","record_id":"<urn:uuid:e252b3b4-cef8-41b0-845b-2e76764f5c2f>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00887.warc.gz"} |
The Secret Improvement of HashMap in Java 8
HashMap is one of the most widely used data structure in Java. As we know, HashMap stores data in a number of buckets and each bucket is a linked list. The new elements will be added into the bucket
at index of \((n - 1)\,\&\,hash\) (n is the current capacity in HashMap). When the size of a single linked list increases, the performance of retrievals gets worse. Java 8 improves the performance by
converting the linked list into a Red-black Tree if the size is bigger than a threshold. I am going to talk about the basics of red-black tree and how Java applies this methodology in HashMap.
Red-black Tree
Red-black tree is an approximately balanced binary search tree, which has the following properties:
1. Each node is either red or black.
2. The root node and all leaves are black.
3. If a node is red, then both its children are black.
4. Every path from a given node to any of its descendant NIL nodes contains the same number of black nodes.
These constraints enforce a critical property of red–black trees: the path from the root to the farthest leaf is no more than twice as long as the path from the root to the nearest leaf. As property
4 guarantees that all the paths from root node to leaves contains the same amount of black nodes. So, the shortest path only contains black nodes. We assume the shortest path contains N black nodes.
Based on property 3, the longest path in the tree could be inserting one red node after every black node in a path. So the longest path contains \(2*N\) nodes. When considering the NIL leaves, the
shortest path is \(N+1\) and the longest path is \(2*N+1\).
This property guarantees log based time complexity of operations in red-black tree. The time complexities of Search, Insertion, Deletion are all \(O(log\,n)\)
Resources about insertion and deletion in Red-black tree can be referred here.
Treeify in HashMap
There are three static variables in HashMap related to “treeify” functions in HashMap:
• TREEIFY_THRESHOLD(8): The bin count threshold for using a tree rather than list for a bin. Bins are converted to trees when adding an element to a bin with at least this many nodes.
• UNTREEIFY_THRESHOLD(6): The bin count threshold for untreeifying a (split) bin during a resize operation.
• MIN_TREEIFY_CAPACITY(64): The smallest table capacity for which bins may be treeified. Otherwise the table is resized if too many nodes in a bin.
The following method is copied from the source code of HashMap in Java 8, which converts the linked list at index for a given hash value into a red-black tree. It only convert the linked list if the
amount of buckets is greater than MIN_TREEIFY_CAPACITY, otherwise it calls resize method. Resize method doubles the size of the table which makes each bucket contains less nodes.
final void treeifyBin(Node<K,V>[] tab, int hash) {
int n, index; Node<K,V> e;
if (tab == null || (n = tab.length) < MIN_TREEIFY_CAPACITY)
else if ((e = tab[index = (n - 1) & hash]) != null) {
TreeNode<K,V> hd = null, tl = null;
do {
TreeNode<K,V> p = replacementTreeNode(e, null);
if (tl == null)
hd = p;
else {
p.prev = tl;
tl.next = p;
tl = p;
} while ((e = e.next) != null);
if ((tab[index] = hd) != null)
The following code is the inner class - TreeNode in HashMap. It implements Extends LinkedHashMap.Entry, which in turn extends the inner class Node, so it is easier to untreeify a red-black tree to
linked list.
static final class TreeNode<K,V> extends LinkedHashMap.Entry<K,V> {
TreeNode<K,V> parent; // red-black tree links
TreeNode<K,V> left;
TreeNode<K,V> right;
TreeNode<K,V> prev; // needed to unlink next upon deletion
boolean red;
The treeify method loops through all nodes of a given list and uses red-black tree insertion algorithm to construct a new red-black tree. After it forms the whole tree, moveRootToFront method is
called to move the root node of the red-black tree to the front of the given linked list.
static <K,V> void moveRootToFront(Node<K,V>[] tab, TreeNode<K,V> root) {
int n;
if (root != null && tab != null && (n = tab.length) > 0) {
int index = (n - 1) & root.hash;
TreeNode<K,V> first = (TreeNode<K,V>)tab[index];
if (root != first) {
Node<K,V> rn;
tab[index] = root;
TreeNode<K,V> rp = root.prev;
if ((rn = root.next) != null)
((TreeNode<K,V>)rn).prev = rp;
if (rp != null)
rp.next = rn;
if (first != null)
first.prev = root;
root.next = first;
root.prev = null;
assert checkInvariants(root); | {"url":"https://runzhuoli.me/2018/08/31/the-secret-improvement-of-hashmap-in-java8.html","timestamp":"2024-11-02T18:54:48Z","content_type":"text/html","content_length":"52509","record_id":"<urn:uuid:6afae909-2049-4116-87ae-fc4bf5e0b93e>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00239.warc.gz"} |
2023-2024 Undergraduate Catalog [ARCHIVED CATALOG]
Mathematics Education (BS)
University Requirements:
College Requirements:
Second Writing Requirement:
A Second Writing Requirement approved by the College of Arts and Sciences. This course must be taken after completion of 60 credit hours, completed with a minimum grade of C-, and the
section enrolled must be designated as satisfying the requirement in the academic term completed.
Foreign Language:
• Completion of the intermediate-level course (107 or 112) in an ancient or modern language with minimum grades of D-.
□ The number of credits (0-12) needed and initial placement will depend on the number of years of high school study of foreign language.
☆ Students with four or more years of high school work in a single foreign language, or who have gained proficiency in a foreign language by other means, may attempt to fulfill the
requirement in that language by taking an exemption examination through the Languages, Literatures and Cultures Department.
College of Arts and Sciences Breadth Requirements:
The College Breadth Requirements are taken in addition to the University Breadth Requirement. Up to three credits from each of the University Breadth Requirement categories may be used to
simultaneously satisfy these College of Arts and Sciences Breadth Requirements. College Breadth courses must be completed with a minimum grade of C-.
A total of 18 credits from Groups A, B, and C is required with a minimum of six credits in each group. The six credits from each group could be from the same area.
Major Requirements:
Minimum grade of C- required for major courses, related work, and Professional Studies.
Modeling Elective:
One of the following:
• MATH 512 - Contemporary Applications of Mathematics
• MATH 518 - Mathematical Models and Applications
Mathematics Option:
One of the following:
Computer Science:
One of the following:
Laboratory Science:
One of the following sequences:
Restricted Electives:
• Six additional credits in mathematics or in related disciplines at the 300-level or above.
□ Courses not approved for math majors cannot be counted towards these six additional credits.
□ Non-mathematics courses can be in CISC, ECON, PHYS and STAT from an approved list maintained by the Department of Mathematical Sciences.
To be eligible to student teach, Mathematics Education students must have a GPA of 2.5 in their mathematics major and an overall GPA of 2.5. They must also pass a teacher competency test as
established by the University Council on Teacher Education. Remaining in the program is subject to periodic review of satisfactory progress and, to be admitted to EDUC 400 - Student Teaching
, students must have completed all the mathematics courses required in the secondary mathematics education program. Students should consult the teacher education program coordinator to
obtain the student teaching application and other information concerning student teaching policies.
After required courses are completed, sufficient elective credits must be taken to meet the minimum credit requirement for the degree.
Credits to Total a Minimum of 124
Last Revised 2020-2021 Academic Year | {"url":"https://catalog.udel.edu/preview_program.php?catoid=87&poid=75338&returnto=26941","timestamp":"2024-11-02T12:54:51Z","content_type":"text/html","content_length":"46580","record_id":"<urn:uuid:0c119f22-74da-4cd8-8dbc-4002409b241a>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00133.warc.gz"} |
Proof of a corollary of fundamental theorem of algebra
• Thread starter mindauggas
• Start date
In summary: Attempt #3:(1) p(x)=a_1 x+a_0(2) x-r_{1} is a factor if and only if r_{1} is a zero(3) p(r_{1})=a_1 (r_{1}-r_{1}) which is zero, therefore r_{1} is a zero and x-r_{1} is a factor(4)
Assume that statement P_k is true.(5) P_{k+1} is the statement: a polynomial a(x-r_1) (x-r_2) ... (x-r_k)
Homework Statement
Assuming the validity of the fundamental theorem of algebra, prove the corollary that:
Every polynomial of positive degree n has a factorization of the form:
[itex]P(x)=a_{n}(x-r_{1})...(x-r_{n})[/itex] where [itex]r_{i}[/itex] aren't necessarily distinct.
Homework Equations
Fundamental Theorem of Algebra: Every polynomial of positive degree with complex coefficients has at least one complex zero.
The Attempt at a Solution
Does this even require proof? Don't know where to begin...
mindauggas said:
Homework Statement
Assuming the validity of the fundamental theorem of algebra, prove the corollary that:
Every polynomial of positive degree n has a factorization of the form:
[itex]P(x)=a_{n}(x-r_{1})...(x-r_{n})[/itex] where [itex]r_{i}[/itex] aren't necessarily distinct.
Homework Equations
Fundamental Theorem of Algebra: Every polynomial of positive degree with complex coefficients has at least one complex zero.
The Attempt at a Solution
Does this even require proof? Don't know where to begin...
Sure it requires proof. Start by saying P(x) is has a root r1. Show (x-r1) divides P(x), so you can write P(x)=(x-r1)P1(x) where P1(x) has degree n-1. Now apply the same thing to P1(x). Etc. If you
want to be more formal you prove it by induction.
(1) We have a polynomial [itex]P(x)=a(x-r_{1})[/itex]
(2) [itex]x-r_{1}[/itex] is a factor if and only if [itex]r_{1}[/itex] is a zero (Remainder theorem)
(3) [itex]P(r_{1})=a(r_{1}-r_{1})[/itex] which is zero, therefore [itex]r_{1}[/itex] is a zero and [itex]x-r_{1}[/itex] is a factor.
At this point I have a question: by what theorem can now take the step: (4) [itex]P(x)=a(x-r_{1})(x-r_{n-1})[/itex] or should this be obvious? Because I don't understand why is this the case.
P(x) is a polynomial of degree n. You've only got a polynomial of degree 1. Reread my last post.
Dick said:
P(x) is a polynomial of degree n. You've only got a polynomial of degree 1. Reread my last post.
Dick said:
Start by saying P(x) is has a root r1
But I can't assume that [itex]P(x)[/itex] is a polynomial o degree n and than just write: [itex]P(x)=a(x-r_{1})[/itex] where do I indicate the degree? Should I write: [itex]P(x)=a(x-r_{1})^{n}[/itex]
? Or something?
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mindauggas said:
But I can't assume that [itex]P(x)[/itex] is a polynomial o degree n and than just write: [itex]P(x)=a(x-r_{1})[/itex] where do I indicate the degree? Should I write: [itex]P(x)=a(x-r_{1})^{n}[/
itex] ? Or something?
I would write it as [itex]P(x)=(x-r_{1})P_1(x)[/itex]. When you divide [itex]x-r_1[/itex] into [itex]P(x)[/itex] you are going to get another polynomial, not just a constant.
Attempt #2:
(1) We denote a polynomial of degree n as [itex]P(x)=a(x-r_{1})P_{1}(x)[/itex]
(2) [itex]x-r_{1}[/itex] is a factor if and only if [itex]r_{1}[/itex] is a zero (Remainder theorem)
(3) [itex]P(r_{1})=a(r_{1}-r_{1})P_{1}(r_{1})[/itex] which is (NOT?)zero, therefore [itex]r_{1}[/itex] is (NOT?)a zero and [itex]x-r_{1}[/itex] is (NOT?) a factor.
It doesn't divide then? Or have I misunderstood smth?
I would be inclined to do this as a "proof by induction" on n, the degree of the polynomial. You have done the "n= 1" case. Now show that "if a polynomial of degree k can be factored as linear
factors, so can a polynomial of degree k+1".
(1) We have a statement [itex]P_{1}[/itex] "a polynomial [itex]a(x-r_{1})[/itex] has expresion [itex]x-r_{1}[/itex] as a factor", which is equivalent to saying that it can be factored as [itex]a(x-r_
{1})[/itex]?. Let's call the first polynomial [itex]P_{1}(x)[/itex] in accordance to first statement and denote [itex]P_{n}(x)[/itex] a polynomial corresponding to the statement [itex]P_{n}[/itex]
for the polynomial [itex]a(x-r_{1})(x-r_{2})...(x-r_{n})[/itex]
(2) Now, [itex]x-r_{1}[/itex] is a factor if and only if [itex]r_{1}[/itex] is a zero (Remainder theorem)
(3) Since [itex]P_{1}(r_{1})=a(r_{1}-r_{1})[/itex] is equal to zero, therefore [itex]r_{1}[/itex] is a zero and [itex]x-r_{1}[/itex] is a factor of the polynomial.
(4) Assume that statement [itex]P_{k}[/itex] is true.
(5) [itex]P_{k+1}[/itex] is the statement: a polynomial [itex]a(x-r_{1})(x-r_{2})...(x-r_{k})(x-r_{k+1})[/itex] has expresion [itex]x-r_{k+1}[/itex] as a factor. Repeating (2) and (3) with [itex]r_
{k+1}[/itex] we get [itex]P_{k+1}(k+1)=0[/itex]
(6)Therefore every polynomial of positive degree n has a factorization of the form:
[itex]P(x)=a_{n}(x-r_{1})...(x-r_{n})[/itex] where [itex]r_{i}[/itex] aren't necessarily distinct.
Q.E.D. ? I don't think so.
Shouldn't I start with general rule for a polynomial: [itex]f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}[/itex] ?
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Can someone approve that this constitutes a proof of what I intended to prove? Maybe additional qualifications are needed - some steps are missing?
If you are doing induction [itex]P_k[/itex] should be the statement "any polynomial of degree k can be expressed in the form [itex]a (x-r_1) (x-r_2) ... (x-r_k)[/itex]".
You can do the k=1 case without using your theorem at all. [itex]f(x)=a_1 x + a_0[/itex] is a polynomial of degree 1. You can write that as [itex]f(x)=a_1 (x + \frac{a_0}{a_1})[/itex] Since your root
is [itex]r_1=(-a_0/a_1)[/itex] that expresses f(x) in the form a(x-r). To prove [itex]P_k[/itex] implies [itex]P_{k+1}[/itex], pick a polynomial p(x) of degree k+1. The fundamental theorem of algebra
says p(x) has a root, call it [itex]r_{k+1}[/itex]. So [itex]p(x)=(x-r_{k+1})*q(x)[/itex] where q(x) has degree k. Now use your induction hypothesis on q(x).
(1) Assume the statement [itex]P_{k}[/itex]: "the polynomial [itex]q(x)[/itex] od degree k can be written as [itex]q(x)=a_{k}(x-r_{1})...(x-r_{k})[/itex]" is true.
(2) Now the statement [itex]P_{k+1}[/itex] is about the polynomial [itex]p(x)=(x-r_{k+1})q(x)[/itex] which can be rewritten as: [itex]p(x)=(x-r_{k+1})a_{k}(x-r_{1})...(x-r_{k})[/itex].
(3) Since we've assumed [itex]P_{k}[/itex] is true, and [itex]P_{k+1}[/itex] was shown to be true previously (I didn't write the statement, but its trivial) - QED (?)
Can anyone give feedback, especially for the step (2).
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You need to state explicitely, and support,
"If P[k+1] is a polynomial of degree k+1 and P[k+1](x)= (x- r)Q(x), then Q(x) is a polynomial of degree k".
Last edited by a moderator:
FAQ: Proof of a corollary of fundamental theorem of algebra
1. What is the fundamental theorem of algebra?
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.
2. What is a corollary of the fundamental theorem of algebra?
A corollary of the fundamental theorem of algebra is a statement that follows directly from the theorem. In this case, the corollary states that a polynomial of degree n has exactly n complex roots.
3. How is the proof of a corollary of the fundamental theorem of algebra different from the proof of the theorem itself?
The proof of a corollary of the fundamental theorem of algebra is typically shorter and more straightforward than the proof of the theorem itself. This is because the corollary is a direct
consequence of the theorem and does not require as much background information or complex reasoning.
4. Why is the fundamental theorem of algebra important in mathematics?
The fundamental theorem of algebra is important because it provides a powerful tool for solving polynomial equations and understanding the behavior of complex numbers. It also has implications in
other areas of mathematics, such as topology and group theory.
5. Can the fundamental theorem of algebra be extended to other types of equations?
The fundamental theorem of algebra only applies to polynomial equations with complex coefficients. However, there are similar theorems for other types of equations, such as the rational root theorem
for rational equations and the irrational root theorem for irrational equations. | {"url":"https://www.physicsforums.com/threads/proof-of-a-corollary-of-fundamental-theorem-of-algebra.589445/","timestamp":"2024-11-08T22:16:17Z","content_type":"text/html","content_length":"139219","record_id":"<urn:uuid:b0721b57-15c5-4aa5-9a8d-ac5ecfbc7a28>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00152.warc.gz"} |
The Price of Bandit Information for Online Optimization
Part of Advances in Neural Information Processing Systems 20 (NIPS 2007)
Varsha Dani, Sham M. Kakade, Thomas Hayes
In the online linear optimization problem, a learner must choose, in each round, a decision from a set D ⊂ Rn in order to minimize an (unknown and chang- ing) linear cost function. We present sharp
rates of convergence (with respect to additive regret) for both the full information setting (where the cost function is revealed at the end of each round) and the bandit setting (where only the
scalar cost incurred is revealed). In particular, this paper is concerned with the price of bandit information, by which we mean the ratio of the best achievable regret √ in the bandit setting to
that in the full-information setting. For the full informa- tion case, the upper bound on the regret is O∗( nT ), where n is the ambient √ dimension and T is the time horizon. For the bandit case, we
present an algorithm which achieves O∗(n3/2 T ) regret — all previous (nontrivial) bounds here were O(poly(n)T 2/3) or worse. It is striking that the convergence rate for the bandit setting is only a
factor of n worse than in the full information case — in stark √ contrast to the K-arm bandit setting, where the gap in the dependence on K is T log K). We also present lower bounds showing that
exponential ( this gap is at least n, which we conjecture to be the correct order. The bandit algorithm we present can be implemented efficiently in special cases of particular interest, such as path
planning and Markov Decision Problems. | {"url":"https://proceedings.neurips.cc/paper/2007/hash/bf62768ca46b6c3b5bea9515d1a1fc45-Abstract.html","timestamp":"2024-11-09T01:21:28Z","content_type":"text/html","content_length":"9189","record_id":"<urn:uuid:524aa8ca-8c3e-4997-be1f-9889a89ed60c>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00531.warc.gz"} |
Bat Wings
Two students collected some data on the wingspan of bats, but each lost a measurement. Can you find the missing information?
Here is a table of information about the wingspan of bats. Unfortunately, two entries are missing from the table.
│Student│ Data set (wingspans in cm) │Mean (cm)│
│A │13 │- │16 │12 │10 │15 │13 │
│B │13 │16 │- │13 │ │ │ │
Student A collected 6 measurements and worked out that their mean was 13cm.
Student B collected 4 measurements.
The overall mean of the combined data was 13.4cm.
Can you work out what the two missing values are?
Student Solutions
Working backwards
Student A's bats have an average wingspan of $13$ cm. Student A has $6$ bats, and so the sum of their wingspans divided by $6$ is equal to $13$.
So the sum of student A's bats' wingspans is $6\times13=78$
So the missing measurement can be found by subtracting the five known wingspans from the total: $78 - (13+16+12+10+15) = 12$ cm.
Next, we can find student B's mean using the total mean:
There are $10$ bats altogether, and the average wingspan is $13.4$ cm, so the total wingspan is $10\times13.4 = 134$ cm.
We already know that student A's bats' had a total wingspan of $78$ cm. So student B's bats' total wingspan must be $134 - 78 = 56$ cm.
So B's missing measurement can be found by subtracting the three known wingspans from the total: $56 - (13+16+13) = 14$ cm.
By 'balancing' the numbers
We can think of the average bat wingspan as being the wingspan that each bat would get if the total amount of wing were shared out equally between all of the bats.
So total wingspan of student A's bats is the same as the total wingspan of 6 bats which each had a wingspan of 13 cm. We can write them below each other and then make the sums 'balance':
Notice that the first 13 has 13 below it, so these are already balanced. 16 is 3 more than 13 and 10 is 3 less than 13, so together 10 and 16 balance out 13 and 13. This is shown below:
12 and 15 do not balance with 13 and 13, because 12 is 1 less than 13 but 15 is 2 more than 13. That is 1 below but 2 above, so we need another 1 below in order to balance the sums.
So the missing measurement must be 12.
Then, we can find the average for student B.
Altogether there are 10 bats, and if the wingspan was shared out equally, each would get 13.4 cm. Alternatively, we could share the wingspan out by student. Student A's bats would each get 13 cm and
student B's bats would each get ? cm (their respective averages). This is shown below:
But student A's bats each get 0.4 less than 13.4, so in total, they contribute 6$\times$0.4 less than their fair share to the total. So student B's bats must contribute 6$\times$0.4 more than their
fair share to the sum.
6$\times$0.4 is the same as 4$\times$0.6, so each of student B's 4 bats could have 0.6 cm more than the average wingspan - which is 13.6 + 0.6 = 14 cm.
Finally, we can find student B's missing measurement:
The 13s are each 1 below 14, and the 16 is 2 above 14. So in the bottom sum, we have 2 below and 2 above 14 - which balances out to 14! So the sums are already balanced, so ? must be 14 to maintain
this balance.
So student B's missing measurement is 14 cm. | {"url":"https://nrich.maths.org/problems/bat-wings","timestamp":"2024-11-09T03:27:06Z","content_type":"text/html","content_length":"42597","record_id":"<urn:uuid:ba6e5e6d-2b31-4543-b297-29b5d26355ae>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00043.warc.gz"} |
ICT for health
A.A. 2020/21
Course Language
Degree programme(s)
Master of science-level of the Bologna process in Ict For Smart Societies (Ict Per La Societa' Del Futuro) - Torino
Course structure
Teaching Hours
Lezioni 40
Esercitazioni in laboratorio 20
Teacher Status SSD h.Les h.Ex h.Lab h.Tut Years teaching
Visintin Monica Professore Associato IINF-03/A 20 0 20 0 9
SSD CFU Activities Area context
ING-INF/03 6 B - Caratterizzanti Ingegneria delle telecomunicazioni
The medium of instruction is English. The objectives of this course are to use machine learning in public health applications, in particular in the areas of basic research, prevention, diagnostic
process, management of elderly people at home. The course is designed jointly with the course "Statistical Learning and Neural Networks", with the objective to provide students with a coordinated
"machine learning" approach that can be applied to several ICT problems; in particular, "Statistical Learning and Neural Networks" deals primarily with machine learning in terms of classification and
neural networks, while "ICT for health" addresses regression and clustering topics. Some classification techniques not analyzed in "Statistical Learning and Neural Networks" are analyzed in "ICT for
health". The course is divided into two parts: 1) the description of some of the many health issues and 2) the description and use of the machine learning techniques that can be used to solve these
issues. Several laboratory experiences are included, and the knowledge of the health issues from the medical point of view is fundamental for the correct system implementation. Python will be used as
programming language (in particular Pandas and Scikit-learn ) and a learn-by-doing approach will be used.
The medium of instruction is English. The objectives of this course are to use machine learning in public health applications, in particular in the areas of basic research, prevention, diagnostic
process, management of elderly people at home. The course is designed jointly with the course "Statistical Learning and Neural Networks", with the objective to provide students with a coordinated
"machine learning" approach that can be applied to several ICT problems; in particular, "Statistical Learning and Neural Networks" deals primarily with machine learning in terms of classification and
neural networks, while "ICT for health" addresses regression and clustering topics. Some classification techniques not analyzed in "Statistical Learning and Neural Networks" are analyzed in "ICT for
health". The course is divided into two parts: 1) the description of some of the many health issues and 2) the description and use of the machine learning techniques that can be used to solve these
issues. Several laboratory experiences are included, and the knowledge of the health issues from the medical point of view is fundamental for the correct system implementation. Python will be used as
programming language (in particular Pandas and Scikit-learn ) and a learn-by-doing approach will be used.
Knowledge of: - basics in some health issues (management of elderlies, Parkinson's disease, EEG, ECG, dermatology, etc) - e-health and m-health applications - telemedicine applications - regression
techniques - clustering techniques - classification techniques Ability to: - understand the issues of a telemedicine application - apply regression techniques in health problems - apply clustering
techniques in health problems - apply classification techniques in health problems - use open-source machine learning software
Knowledge of: - basics in some health issues (management of elderlies, Parkinson's disease, EEG, ECG, dermatology, etc) - e-health and m-health applications - telemedicine applications - regression
techniques - clustering techniques - classification techniques Ability to: - understand the issues of a telemedicine application - apply regression techniques in health problems - apply clustering
techniques in health problems - apply classification techniques in health problems - use open-source machine learning software
Knowledge of probability theory, linear algebra, optimization techniques
Knowledge of probability theory, linear algebra, optimization techniques
- Description of some e-health, m-health, and telemedicine applications (2 CFU) on the following topics: smart aging, fitness, Parkinson's disease, EEG, ECG, dermatology, lean in health care (2.1
CFU). - Review of linear algebra and basics on optimization methods (0.6 CFU). - Introduction to Python (0.3 CFU). - regression techniques: linear regression, PCR, Gaussian processes for regression
(0.9 CFU) - clustering techniques: k-means, hierarchical trees, DBSCAN (0.9 CFU) - classification techniques: decision trees and information theory, Hidden Markov Models (0.9 CFU) - Independent
component analysis (ICA) applied to EEG (0.3 CFU)
- Description of some e-health, m-health, and telemedicine applications (2.1 CFU) on the following topics: - smart aging, - fitness, - Parkinson's disease, - EEG, - ECG, - dermatology, - lean in
health care, - management of emergencies. - Review of linear algebra and basics on optimization methods (0.6 CFU). - Introduction to Python (0.3 CFU). - regression techniques: linear regression, PCR,
Gaussian processes for regression (0.9 CFU) - clustering techniques: k-means, hierarchical trees, DBSCAN (0.9 CFU) - classification techniques: decision trees and information theory, Hidden Markov
Models (0.9 CFU) - Independent component analysis (ICA) applied to EEG (0.3 CFU)
Lectures will describe the health context and the problem to be solved, then the relevant ICT/learning machine methods to be used to solve the problem are discussed and implemented in Python in the
laboratory classes.
Lectures will describe the health context and the problem to be solved, then the relevant ICT/learning machine methods to be used to solve the problem are discussed and implemented in Python in the
laboratory classes.
- Class slides will be available on the portal - K. Murphy, "Machine Learning, a probabilistic perspective", MIT press, 2012 - Christopher M. Bishop, "Pattern Recognition and Machine Learning",
Springer-Verlag New York, 2006 - David J.C. MacKay, "Information Theory, Inference and Learning Algorithms" Cambridge University Press 2003 - C.E. Rasmussen, C.K.I. Williams, Gaussian Processes for
Machine Learning, the MIT Press, 2006
- Class slides will be available on the portal - K. Murphy, "Machine Learning, a probabilistic perspective", MIT press, 2012 - Christopher M. Bishop, "Pattern Recognition and Machine Learning",
Springer-Verlag New York, 2006 - David J.C. MacKay, "Information Theory, Inference and Learning Algorithms" Cambridge University Press 2003 - C.E. Rasmussen, C.K.I. Williams, Gaussian Processes for
Machine Learning, the MIT Press, 2006
Modalitą di esame: Prova orale obbligatoria; Elaborato scritto individuale; Elaborato progettuale individuale;
The student must report the lab activities: for each lab he/she must upload the report (already partially available, to be completed), and the Python scripts (partially available, to be completed).
The maximum grade for the lab activity is 10, depending on the correctness of the results and on the completeness and clearness of the document. The mandatory oral exam consists of 3 questions, 1
question about the health issues, i.e. Parkinson's disease, dermatology, etc. as described in the lectures, 2 questions about the algorithms and methods described in the lectures and the lab
activity. The student gets a grade from 0 to 7 on each question; the grade depends on the ability to describe and critically discuss the learned methods and on the promptness of the answers. The
grades of the reports and the oral exam are added together to obtain the final grade. The "lode" is given to students with an overall grade 31. The ability of the student to apply the described
machine learning techniques in Python will be checked through the analysis of the report and the Python scripts. The knowledge of the health issues with possible ICT solutions and the knowledge of
the regression, clustering and classification techniques will be checked during the oral exam. The student will improve his/her soft-skills related to the ability of writing a technical report, and
the ability to discuss ideas during the oral exam.
Exam: Compulsory oral exam; Individual essay; Individual project;
The student must report the lab activities: for each lab he/she must upload the report (already partially available, to be completed), and the Python scripts (partially available, to be completed).
The maximum grade for the lab activity is 10, depending on the correctness of the results and on the completeness and clearness of the document. The mandatory oral exam consists of 3 questions, 1
question about the health issues, i.e. Parkinson's disease, dermatology, etc. as described in the lectures, 2 questions about the algorithms and methods described in the lectures and the lab
activity. The student gets a grade from 0 to 7 on each question; the grade depends on the ability to describe and critically discuss the learned methods and on the promptness of the answers. The
grades of the reports and the oral exam are added together to obtain the final grade. The "lode" is given to students with an overall grade 31. The ability of the student to apply the described
machine learning techniques in Python will be checked through the analysis of the report and the Python scripts. The knowledge of the health issues with possible ICT solutions and the knowledge of
the regression, clustering and classification techniques will be checked during the oral exam. The student will improve his/her soft-skills related to the ability of writing a technical report, and
the ability to discuss ideas during the oral exam.
Modalitą di esame: Prova orale obbligatoria; Elaborato scritto individuale; Elaborato progettuale individuale;
The student must report the lab activities: for each lab he/she must upload the report (already partially available, to be completed), and the Python scripts (partially available, to be completed).
The maximum grade for the lab activity is 10, depending on the correctness of the results and on the completeness and clearness of the document. The mandatory oral exam consists of 3 questions, 1
question about the health issues, i.e. Parkinson's disease, dermatology, etc. as described in the lectures, 2 questions about the algorithms and methods described in the lectures and the lab
activity. The student gets a grade from 0 to 7 on each question; the grade depends on the ability to describe and critically discuss the learned methods and on the promptness of the answers. The
grades of the reports and the oral exam are added together to obtain the final grade. The "lode" is given to students with an overall grade 31. The ability of the student to apply the described
machine learning techniques in Python will be checked through the analysis of the report and the Python scripts. The knowledge of the health issues with possible ICT solutions and the knowledge of
the regression, clustering and classification techniques will be checked during the oral exam. The student will improve his/her soft-skills related to the ability of writing a technical report, and
the ability to discuss ideas during the oral exam.
Exam: Compulsory oral exam; Individual essay; Individual project;
The student must report the lab activities: for each lab he/she must upload the report (already partially available, to be completed), and the Python scripts (partially available, to be completed).
The maximum grade for the lab activity is 10, depending on the correctness of the results and on the completeness and clearness of the document. The mandatory oral exam consists of 3 questions, 1
question about the health issues, i.e. Parkinson's disease, dermatology, etc. as described in the lectures, 2 questions about the algorithms and methods described in the lectures and the lab
activity. The student gets a grade from 0 to 7 on each question; the grade depends on the ability to describe and critically discuss the learned methods and on the promptness of the answers. The
grades of the reports and the oral exam are added together to obtain the final grade. The "lode" is given to students with an overall grade 31. The ability of the student to apply the described
machine learning techniques in Python will be checked through the analysis of the report and the Python scripts. The knowledge of the health issues with possible ICT solutions and the knowledge of
the regression, clustering and classification techniques will be checked during the oral exam. The student will improve his/her soft-skills related to the ability of writing a technical report, and
the ability to discuss ideas during the oral exam.
Esporta Word | {"url":"https://didattica.polito.it/pls/portal30/gap.pkg_guide.viewGap?p_cod_ins=01QWYBH&p_a_acc=2021&p_header=S&p_lang=IT&multi=N","timestamp":"2024-11-13T19:48:41Z","content_type":"text/html","content_length":"59596","record_id":"<urn:uuid:64fa9206-7ed7-4735-a716-145d30eea3db>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00369.warc.gz"} |
Sample size - Science without sense...double nonsense
Size does matter
Sample size.
A series of considerations are made about the influence of sample size on the precision and probability of error of the study.
Of course, we talk about samples.
For various reasons, scientific studies often use samples drawn from a population on which you want to get any specific conclusion. This sample will have to be selected so that it faithfully
represents the population from which it is extracted but, how large sample ought to be: big or small?. Well, neither one thing nor the other: the sample must be of the appropriate size.
We’d need to rest a bit after reasoning until getting this conclusion but, first, we’ll try to find out what problems too large or too small samples can cause us.
Drawbacks of larger than needed samples are obvious: greater expenditure of time and resources.
Sample size
Moreover, we already know that sometimes it is enough to increase sample size in order to obtain statistical significance, although we also know that if we use this technique in excess we can obtain
significance among differences that are too small and, although the difference may be real, its clinical relevance can be limited. Doing that way we expend time and energy (and money) and can also
create confusion about the importance of the difference found. So, as in many other aspects of life and medicine, when we speak about samples, more is not always better (nor it is better to have it
What if the sample is small?. It’s a bit the opposite situation. The smaller the sample, the greater the imprecision (the wider the confidence intervals). As a result, differences have to be larger
to reach statistical significance. We thus run the risk that, although there’s a real difference, we won’t be able to assure its existence due to the too small sample, losing the opportunity to shown
differences that, although small, could be clinically relevant.
It’s clear, therefore, that the sample size must be the appropriate size and that, to avoid more evil, we should calculate it before doing the study.
Formulas to come up with the sample size depend on the statistical we are measuring and on the fact of whether we are estimating a population (a mean, for example) or we want to do a hypothesis
testing between two variables or samples (to compare to samples, two proportions, etc.). In any case, most statistical programs and Internet calculators are capable to calculate it quickly and
without flinching. We just have to set the values of three parameters: the type 1 error, the study power and the minimum clinically relevant difference.
Type I error is the probability of rejecting the null hypothesis being true, so concluding that there’s a difference that is, indeed, not real. It is generally accepted that this probability, which
is called alpha, must be less than 5% ant it is the level of statistical significance usually employed in hypothesis testing.
Type II error is the probability of concluding that there’s no difference when in fact there is (and not to reject the null hypothesis). This value is known as beta and a minimum of 80% is generally
accepted as an appropriate level. Its complementary value (1-beta, or 100-beta if we like more percentages values) is what we call the power of the study.
Finally, the minimal clinically relevant difference is that that the study is able to detect, given that it actually exists. This value is set by the researcher according to the clinical context and
has nothing to do with the statistical significance of the study.
Using these three parameters, we’ll calculate the required sample size to detect the difference considered relevant from the clinical point of view and with the desired amount of error.
Sometimes we can do this reasoning in the reverse way. If our sample size has an upper limit, whatever the reason, we can estimate the difference we’ll be able to detect before doing the study. If
this difference is less than the clinically relevant difference, better save our work, because we’ll be at risk of not reach a conclusion because of the small and misleading sample, getting the
conclusion that a real difference that not exists. Similarly, if we must stop the study before its programmed ending we should calculate if the achieved sample gives us enough power to discriminate
the level of difference we proposed at the beginning of the study.
According to the variable we are measuring, sometimes we’ll need other data such as the population’s mean or standard deviation to estimate the sample size. If we don’t know these values we can do a
pilot study with a few patients (at the judgment of the researcher) and calculate the sample size with the preliminary results.
We’re leaving…
One last thought before going to cool our heads. The sample size is calculated to estimate the primary outcome, but we cannot be sure our sample is appropriate to estimate any other parameters we
measure during the study. Very often, we can see trials very well design to show the effectiveness of a treatment but incapable of provide conclusive data about its safety. But that’s another story…
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Aspects and images of complex path integrals
Original language German
Awarding Institution • University of Bremen
Supervised by • Keßeböhmer, Marc, Supervisor, External person
Publication status Published - 4 Nov 2022
Externally published Yes
The first research reports from complex analysis education show that not only novices but also mathematical experts have difficulties in interpreting complex path integrals. Therefore, we deal with
two sides of experts' complex analysis discourse in this thesis: On the one hand, we present a comprehensive epistemological analysis of complex path integrals. On the other hand, we reconstruct
experts' personal interpretations of these mathematical objects in the form of a multi case study. The thesis has three major contributions, which are grounded theoretically in the commognitive
framework and German subject-matter didactics:
We suggest a conceptualisation of discursive mental images as narratives and discursive frames as sets of metarules in intuitive mathematical discourses in order to enrich basic research in
mathematics education at university level. It complements acquisitionist perspectives on individuals' mental images of mathematical objects and provides a non-subsumptive and non-prescriptive way to
study experts' individual, intuitive interpretations of mathematical objects.
A detailed, historically informed epistemological analysis of definitions of complex path integrals, their discursive embedding, and curricular connections to other mathematical discourses enables us
to identify four so-called aspects and four partial aspects of complex path integrals. These are typical ways of defining complex path integrals by relating them to different mathematical constraints
on the integrands, paths, or domains. We also provide a new axiomatic definition for complex path integrals of holomorphic functions.
This conceptualisation from the first part is used for the analysis of experts' intuitive mathematical discourses about complex path integrals. Our study also includes their individual
interpretations and substantiations of central integral theorems in complex analysis.
The reconstructed set of discursive images contains an analogy-based saming of complex and real path integrals, the valuation of the complex path integral as a tool, a mean value interpretation, and
others. One expert also attempted to transfer area interpretations for real integrals to complex path integrals. In particular, experts' intuitive interpretations of complex path integrals are
primarily narrative rather than figurative. The theoretical construct of discursive frame turns out to be especially helpful as it enables us to highlight commonalities and differences between
experts' intuitive mathematical discourses about complex path integrals. Consistent with previous literature, this study confirms that experts enrich their intuitive mathematical discourses with
connections to other mathematical discourses such as real or vector analysis. We conclude with perspectives for future research on the teaching and learning of complex path integrals.
Cite this
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• Harvard
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• Author
• BibTeX
• RIS
title = "Aspects and images of complex path integrals: An epistemological analysis and a reconstruction of experts' interpretations of integration in complex analysis",
abstract = "The first research reports from complex analysis education show that not only novices but also mathematical experts have difficulties in interpreting complex path integrals. Therefore, we
deal with two sides of experts' complex analysis discourse in this thesis: On the one hand, we present a comprehensive epistemological analysis of complex path integrals. On the other hand, we
reconstruct experts' personal interpretations of these mathematical objects in the form of a multi case study. The thesis has three major contributions, which are grounded theoretically in the
commognitive framework and German subject-matter didactics:We suggest a conceptualisation of discursive mental images as narratives and discursive frames as sets of metarules in intuitive
mathematical discourses in order to enrich basic research in mathematics education at university level. It complements acquisitionist perspectives on individuals' mental images of mathematical
objects and provides a non-subsumptive and non-prescriptive way to study experts' individual, intuitive interpretations of mathematical objects.A detailed, historically informed epistemological
analysis of definitions of complex path integrals, their discursive embedding, and curricular connections to other mathematical discourses enables us to identify four so-called aspects and four
partial aspects of complex path integrals. These are typical ways of defining complex path integrals by relating them to different mathematical constraints on the integrands, paths, or domains. We
also provide a new axiomatic definition for complex path integrals of holomorphic functions.This conceptualisation from the first part is used for the analysis of experts' intuitive mathematical
discourses about complex path integrals. Our study also includes their individual interpretations and substantiations of central integral theorems in complex analysis.The reconstructed set of
discursive images contains an analogy-based saming of complex and real path integrals, the valuation of the complex path integral as a tool, a mean value interpretation, and others. One expert also
attempted to transfer area interpretations for real integrals to complex path integrals. In particular, experts' intuitive interpretations of complex path integrals are primarily narrative rather
than figurative. The theoretical construct of discursive frame turns out to be especially helpful as it enables us to highlight commonalities and differences between experts' intuitive mathematical
discourses about complex path integrals. Consistent with previous literature, this study confirms that experts enrich their intuitive mathematical discourses with connections to other mathematical
discourses such as real or vector analysis. We conclude with perspectives for future research on the teaching and learning of complex path integrals.",
author = "Erik Hanke",
year = "2022",
month = nov,
day = "4",
doi = "10.26092/elib/1964",
language = "Deutsch",
school = "Universit{\"a}t Bremen",
TY - BOOK
T1 - Aspects and images of complex path integrals
T2 - An epistemological analysis and a reconstruction of experts' interpretations of integration in complex analysis
AU - Hanke, Erik
PY - 2022/11/4
Y1 - 2022/11/4
N2 - The first research reports from complex analysis education show that not only novices but also mathematical experts have difficulties in interpreting complex path integrals. Therefore, we deal
with two sides of experts' complex analysis discourse in this thesis: On the one hand, we present a comprehensive epistemological analysis of complex path integrals. On the other hand, we reconstruct
experts' personal interpretations of these mathematical objects in the form of a multi case study. The thesis has three major contributions, which are grounded theoretically in the commognitive
framework and German subject-matter didactics:We suggest a conceptualisation of discursive mental images as narratives and discursive frames as sets of metarules in intuitive mathematical discourses
in order to enrich basic research in mathematics education at university level. It complements acquisitionist perspectives on individuals' mental images of mathematical objects and provides a
non-subsumptive and non-prescriptive way to study experts' individual, intuitive interpretations of mathematical objects.A detailed, historically informed epistemological analysis of definitions of
complex path integrals, their discursive embedding, and curricular connections to other mathematical discourses enables us to identify four so-called aspects and four partial aspects of complex path
integrals. These are typical ways of defining complex path integrals by relating them to different mathematical constraints on the integrands, paths, or domains. We also provide a new axiomatic
definition for complex path integrals of holomorphic functions.This conceptualisation from the first part is used for the analysis of experts' intuitive mathematical discourses about complex path
integrals. Our study also includes their individual interpretations and substantiations of central integral theorems in complex analysis.The reconstructed set of discursive images contains an
analogy-based saming of complex and real path integrals, the valuation of the complex path integral as a tool, a mean value interpretation, and others. One expert also attempted to transfer area
interpretations for real integrals to complex path integrals. In particular, experts' intuitive interpretations of complex path integrals are primarily narrative rather than figurative. The
theoretical construct of discursive frame turns out to be especially helpful as it enables us to highlight commonalities and differences between experts' intuitive mathematical discourses about
complex path integrals. Consistent with previous literature, this study confirms that experts enrich their intuitive mathematical discourses with connections to other mathematical discourses such as
real or vector analysis. We conclude with perspectives for future research on the teaching and learning of complex path integrals.
AB - The first research reports from complex analysis education show that not only novices but also mathematical experts have difficulties in interpreting complex path integrals. Therefore, we deal
with two sides of experts' complex analysis discourse in this thesis: On the one hand, we present a comprehensive epistemological analysis of complex path integrals. On the other hand, we reconstruct
experts' personal interpretations of these mathematical objects in the form of a multi case study. The thesis has three major contributions, which are grounded theoretically in the commognitive
framework and German subject-matter didactics:We suggest a conceptualisation of discursive mental images as narratives and discursive frames as sets of metarules in intuitive mathematical discourses
in order to enrich basic research in mathematics education at university level. It complements acquisitionist perspectives on individuals' mental images of mathematical objects and provides a
non-subsumptive and non-prescriptive way to study experts' individual, intuitive interpretations of mathematical objects.A detailed, historically informed epistemological analysis of definitions of
complex path integrals, their discursive embedding, and curricular connections to other mathematical discourses enables us to identify four so-called aspects and four partial aspects of complex path
integrals. These are typical ways of defining complex path integrals by relating them to different mathematical constraints on the integrands, paths, or domains. We also provide a new axiomatic
definition for complex path integrals of holomorphic functions.This conceptualisation from the first part is used for the analysis of experts' intuitive mathematical discourses about complex path
integrals. Our study also includes their individual interpretations and substantiations of central integral theorems in complex analysis.The reconstructed set of discursive images contains an
analogy-based saming of complex and real path integrals, the valuation of the complex path integral as a tool, a mean value interpretation, and others. One expert also attempted to transfer area
interpretations for real integrals to complex path integrals. In particular, experts' intuitive interpretations of complex path integrals are primarily narrative rather than figurative. The
theoretical construct of discursive frame turns out to be especially helpful as it enables us to highlight commonalities and differences between experts' intuitive mathematical discourses about
complex path integrals. Consistent with previous literature, this study confirms that experts enrich their intuitive mathematical discourses with connections to other mathematical discourses such as
real or vector analysis. We conclude with perspectives for future research on the teaching and learning of complex path integrals.
U2 - 10.26092/elib/1964
DO - 10.26092/elib/1964
M3 - Dissertation
ER - | {"url":"https://www.fis.uni-hannover.de/portal/en/publications/aspects-and-images-of-complex-path-integrals(4d7fc9f3-ebfc-4171-aeb3-7b1283462210).html","timestamp":"2024-11-10T22:27:43Z","content_type":"text/html","content_length":"54636","record_id":"<urn:uuid:a8f5fec2-cedc-42e0-9e60-71fef105bdca>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00487.warc.gz"} |
Long-term Holders Seasons 1&2 Regulations and Details
Long-term holders season 1, or LTH Season 1 for short, squeezed out massive and positive feedback from our community; you guys! We’re fortunate to have such strongly invested CROWDers both in us and
in holding the CROWD token.
Such heartwarming reactions from you sealed the deal on not only continuing LTH Season 1 but pairing it up with another likewise project called the “Long-term Holders Season 2.”
Consequently, for more clarity over the both seasons, we decided to device a clean and straightforward regulation set. In this long-tailed post, you can get your hands on every little LTH Season 1&2
thoroughly unwrapped and explained to the last interaction.
Calculating the winning points.
The Seasons.
Who wins?
The investors who hold or have held CROWD the longest and most consistently in winning game period are to be determined. The following factors will be taken into account:
• Only CROWD that are still in the wallets at the time of the evaluation are included in the evaluation. Sales are not included in the evaluation.
• Periods of continuous ownership up to the evaluation period of CROWD. Long possession is better than short.
• Time of purchase: Early investments that are still in the wallet today are scored higher than later ones.
• The volume of continuous ownership is taken into account. More continuous ownership is better than less.
What’s new in LTH II?
• The winning periods are adapted.
□ Earning winning points is only possible in this periods. More information is provided below.
• There are two winning Seasons in this game
□ Seasons I: you owned or have owned CROWD before the game period starts
□ Seasons II: you purchased CROWD after the game period starts
Conditions of participation
Participation is allowed for anyone
• who has at least 10,000 CROWD on his connected wallet(s) at the time of the evaluation. The definition of connected wallets can be found below.
• who is not or was not an Advisor at CrowdSwap
• whose last transaction in the wallet network was not a SELL (only BUY and SELL are considered)
• who is not an employee of the CrowdSwap project
Crowd “held” or “not held”
• The following transactions are considered to be CROWD “held”:
□ CROWD in the wallet
□ CROWD in Staking
□ CROWD transfers to other wallets
□ CROWD Liqui-Mining, only if no CROWD had to be sold to enter Liqui-Mining
• The following transactions are not considered CROWD “held”:
□ CROWD swap into other non-CROWD tokens
□ Investing in Liqui-Mining by only CROWD (because half of the CROWD are swapped in background)
Connected wallets / Wallet network
Two wallets are considered “connected” if a transaction into CROWD has taken place between them. The assumption is that when CROWD is transferred from one wallet to another, both are owned by the
same investor. We then speak of a “wallet network“. This can consist of 1 to n wallets.
Example: We consider the wallets A, B and C
• On Wallet A 1,000 CROWD are acquired.
• From wallet A 500 CROWD are transferred to wallet B. Thus, wallets A and B are connected. A and B now form a wallet network.
• 2,000 CROWD are now purchased on Wallet C.
• 500 CROWD are transferred from Wallet C to Wallet A. Wallet A and C are now connected. Since Wallet A was already in a wallet network with B, C is now added to this account group (since A and C
are connected). The new account network is now A, B and C and so on.
Calculating the winning points
Step 1: Determination of the sweepstake period and evaluation time
The sweepstake period is from 21 February 2023 to “Bitcoin Halving 2024”. Precise date of the bitcoin halving will be determinated in future
The sweepstakes period will be divided into 3 “winning periods”.
• “LTH I CROWD Masters”:
□ All before 21 February 2023
□ irrelevant in collection winning points. Only your wallet balance at the end of this period ist considered as your “capital at start”.
• “CROWD Knights”:
□ from 21 February 2023 up to and including 20. [NS1] January 2024
□ all BUYs and all SELLs will affect winning point calculation.
• “CROWD Padawans”:
□ from 21 January 2024 up to Bitcoin Halving 2024
□ all SELLs will affect winning point calculation.
Step 2: Determination of the eligible wallet networks
• First, ALL wallets are determined that have CROWD in the wallet, stake or liqui mining at the time of evaluation.
• It is determined between which of these wallets CROWD was transferred directly. Wallets between which a transaction was carried out are combined to form a wallet group.
• Wallet groups that contain wallets of an advisor or CROWD project employee are not taken into account.
• The last buy and sell transactions in the wallet network are determined. In case the transaction last executed is a SELL, the wallet network is not being taken into consideration for the
□ If this is a sale, the wallet network is no longer taken into account.
• The sum of all CROWD in a wallet network is determined (CROWD in the wallets, in the stake or in the liquidation). If this sum is < 10,000 CROWD, the wallet network is no longer taken into
All remaining wallet networks take part in the lottery.
Step 3: Calculation of the wallet balances in the wallet network
For each wallet network, the CROWD volumes of all contained wallets of each day in the lottery period are added up to the time of 0:00 GMT.
Hint I: Balance History of wallet networks starts at the period “CROWD Knights” at 21. February 2023. If you purchased and held CROWD in “LTH I CROWD Masters” period, this will be your “capital at
Hint II: Yields from LTH I will not be considered to balance history.
Example (for the period 01 Feb 2023 to 20 March 2023)
Date Wallet A Wallet B Wallet C Balance History
1 February 2023 200 50 40.000 0
10 February 2023 150 50 100 0
21 February 2023 150 50 100 (LTH II starts here) 300
3 March 2023 100 200 300 600
4 March 2023 100 300 200 600
20 March 2023 0 400 100 500
Limitations to calculation of network balances
All CROWD considered in networks are calculated as follows:
• Increase of balance by
□ Purchase CROWD from swaps
□ Purchase CROWD from vesting
□ Purchase CROWD from competitions
• No effect on balance has
□ stakingunstakingproviding to Liqui-Pools, without selling CROWDwithdrawing any rewards
• Decrease of balance by
□ providing to Liqui-Pools with only CROWD (the CROWD swapped in background will decrease the balance)
Action # CROWD Balance eligible
Buy 10.000 10.000 yes
Stake 8.000 10.000 yes
Buy 1.000 11.000 yes
Withdraw Stake Rewards 500 11.000 yes
Sell 500 10.500 no
Buy 10.500 21.000 yes
Provide Liquimining (USDC/CROWD) 5.000 21.000 yes
Withdraw Mining Rewards 800 21.000 yes
Leave Liquimining (USDC/CROWD) 5.000 21.000 yes
Provide Liquimining (Single-CROWD) 5.000 18.500 no
Buy 2.000 20.500 yes
From these the winning points are calculated, as follows. We need this definition as well.
Definition: “CROWD Profit Token Package (ProToPack)”.
A “CROWD ProToPack“ is a set of tokens that has been held for a constant period of time and remained in the wallet until the evaluation time. Here it is not important whether it is the identical
tokens, but only the identical quantity. Sold tokens can therefore not come into the evaluation. A ProToPack consists of the following four details:
• V[CROWD]
• T[ALL]
□ Amount of time V[CROWD] held
• T[PADAWAN]
□ Part of T[ALL] spent in the period “Padawan”
• T[KNIGHT]
□ Part of T[ALL], spent in the period “Knight”
• Summary:
□ ProToPack = [V[CROWD], T[PADAWAN], T[KNIGHT]], where T[ALL]= T[PADAWAN] + T[KNIGHT]
action # CROWD balance Eligible to win after action
Purchase CROWD 10.000 10.000 no
Stake 8.000 10.000 yes
Purchase CROWD 1.000 11.000 yes
Withdraw Stake Rewards 500 11.000 yes
Sell 500 10.500 no
Purchase CROWD 10.500 21.000 yes
Liquimining (USDC/CROWD) 5.000 21.000 yes
Withdraw Mining Rewards 800 21.000 yes
Leave Liquimining (USDC/CROWD) 5.000 21.000 yes
Liquimining (Single-CROWD) 5.000 18.500 no
Purchase CROWD 2.000 20.500 yes
Example for the determination of the token CROWD ProToPack
The following example will explain the properties of a ProToPack in more detail. Let’s look at a wallet network over the evaluation time and the associated balances (across the whole wallet network)
for CROWD tokens held. The timeline only depicts moments with changes to the amount of CROWD tokens held.
Assume: Bitcoin Halving 2024 will happen at 14 March 2024
Here, we iteratively search for the packets that were held continuously and maximally. This implies the following:
Once the wallet group had holdings of 0, all held CROWD before balance 0 are no longer considered. All held tokens before the last state 0 are irrelevant for the evaluation as those were not in the
wallet network consistently.
Thus, we take all tokens left of balance 0 including the zero stands out of valuation.
Let us consider the remaining period 28 July 2023 up to and including 12 March 2024.
The package with the longest time in the wallet can have the volume that is currently held in the wallet at most.
• The package with the longest time in the wallet can only have the volume of the minimum balance in the remaining time periods.
Thus, the first packet is the ProToPack No.1 shown below. Its values are:
V[CROWD ]= 4000
T[KNIGHT ]= 176 days (from 28. July 2023 until 20. January 2024)
T[PADAWAN] = 53 days (from 21. January 2024 until 14. March 2024)
T[ALL] = 229 days
All tokens from 28 July 2023 have been taken into account, as the total of 4,000 tokens has been in the wallet network consistently. We continue analogously on the timeline creating further
ProToPacks according to the following buy orders. All tokens which have already been used for the competition, will not be taken into consideration in following ProToPacks.
In our example, the next ProToPack II would have the following details:
V[CROWD ]= 5,000 (total of 9,000 minus the 4,000 already used in ProToPack I)
T[KNIGHT ]= 112 days (from 30. September 2023 up to and including 20. January 2024)
T[PADAWAN] = 53 days (from 21. January 2024 up to and including 14. March 2024)
T[ALL] = 165 days
The tokens which were transferred to the wallet network on 24 January 2024 cannot be associated with a ProToPack, as they had not been in the wallet network until evaluation took place (they were
sold on 2 February 2024).
The last ProToPack III would consist of the tokens of 12 March 2024 with the following details:
V[CROWD ]= 4,000 (13,000 minus 9,000 already used in ProToPack I and II)
T[PADAWAN] = 2 days (from 12 March 2024 up to and including 14 March 2024)
T[KNIGHT ]= 0 days
T[ALL] = 3 days
Step 4: Determination of the winning points
• Determine “capital at start” from “LTH-I-Masters” period
• The ProToPack points of each wallet are determined.
• The evaluation function described below is applied to all ProToPacks and results in the winning points.
• The sum of all winning points of a wallet group is used for the evaluation in the lottery.
• A ProToPack cannot provide negative winning points. Lowest possible winning points per ProToPack is zero.
The valuation function
Assuming that a ProToPack := [V[CROWD], T[PADAWAN], T[KNIGHT]] and T[ALL]:= T[PADAWAN]+ T[KNIGHT], where T is calculated in days.
Calculation of the “base profit points”
Since both the volume and the holding period are considered according to the profit conditions, the base winning points are calculated by multiplying the volume by the holding period in the
respective profit zones. The following applies to the profit zones
• Padawan: Holding periods in “Padawan” do not result in profit points. Short-term purchases are thus not included in the valuation.
□ Base profit points_Padawan = 0 (const.)
• Knight: In this range volumes are multiplied by holding time
□ Base profit points_Knight = V[CROWD] * T[KNIGHT]
• LTH-I-CROWD Masters:
□ Does not affect winning points
Calculation of “bonus profit points”:
In addition to the base profit points, bonus profit points are distributed based on a bonus function. The bonus function has the task of rewarding early investors who remained loyal to CROWD over
those who joined later.
The bonus function multiplies the base profit points again by a factor, which is measured by how large T[ALL] is.
The larger T[ALL], the larger the bonus.
Calculation of the total “winning points”:
With this the calculation of the total profit points is done
Winning points = (Base winning points_Knight) * Bonus factor(T[ALL])
The bonus function
As a bonus function is based on the Fibonacci sequence, starting with the Fibonacci number F[2].
F[2] F[3] F[4] F[5] F[6] F[7] F[8] F[9] F[10] F[11] F[12] F[13] F[14] F[15]
Days after 21 Feb 2023 1 2 3 5 8 13 21 34 55 89 144 233 377 610
Gewinn-Faktor 1 1 1 1 1 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6
The profit factor is now calculated as follows:
Calculate the number of days in the lottery period minus T[ALL]. The win factor can then be read in the table above. The next smaller Fibonacci number always applies to the result.
Example: There are 370 days in the lottery period, T[ALL] = 200. This results in 370-200 = 170. The next smaller Fibonacci number is F[12] (144). Thus the factor is 0.75.
The Seasons
There will be two winning Seasons:
• Season I: You purchased any CROWD before 21. February 2023
• Season II: You purchased any CROWD after or at 21. February 2023 | {"url":"https://crowdswap.org/blog/long-term-holder-competition-season-ii-rules/","timestamp":"2024-11-09T10:37:34Z","content_type":"text/html","content_length":"299214","record_id":"<urn:uuid:d6d2b8bc-4654-4309-9d53-12f6269ec69c>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00526.warc.gz"} |
Explore Super Symmetry | Uncover the Harmony of Science
s u p e r s y m m e t r y
This website is about a specific type of system called a supersymmetric rotor. In a supersymmetric rotor, the mass is distributed eccentrically around the axis of rotation. This means that the total
angular momentum of the rotor is not conserved.
The reason for this is that the laws of physics are not symmetrical under rotations in the supersymmetric rotor. This is because the mass distribution is not symmetrical. As a result, the total
angular momentum of the rotor can change as it rotates. The result is the systems moving in a direction that is constant, in the video you can observe many systems moving from the supersymmetric
This is the system on a level tabletop it weighs 65 pounds and moves in a direction that is constant and the end of this video you can see in slow motion the eccentric mass load in slow motion one
eccentric mass load rotating clockwise and the other eccentric mass load rotating counterclockwise. video game
The Motion of the Eccentric Load Mass Systems about their Center Axis of Rotation Clockwise, the other Counterclockwise.
The Higgs boson is important in the Standard Model because it implies the existence of a Higgs field, an otherwise invisible field of energy which pervades the entire universe. Without the Higgs
field, the elementary particles that make up you, me, and the visible universe would have no mass. Without the Higgs field mass could not be constructed and nothing could be.
The Pseudo Gravity is Einstein's Equivalence Principle where the effects of gravitation are indistinguishable to the effects of acceleration. The clockwise displays eccentric mass load system, the
reason for the two systems one clockwise and the other counterclockwise is to end up with the eccentric mass load systems is for straight line motion of this system.
The second 0=180 degrees demonstrates the "eccentric load" of mass accelerating clock wise to the 180 degree area with the "eccentric load" of mass furthest from the center axis, as illustrated by
distance "B"
Take the time to observe in the videos of the eccentric load mass systems moving in a direction that is constant; there are two axes of rotation for each of the eccentric mass load systems, one
rotating clockwise and the other eccentric load systems rotating counterclockwise, the eccentric mass loads of the clockwise and counterclockwise systems are at the greatest distance from the each of
the center axis of rotation in unison at one hundred eighty degrees, and as the eccentric load mass systems rotate are closest to their center axis of rotation at zero degrees, the result is the
eccentric load mass systems move in the direction of one hundred eighty-degrees. When the eccentric mass load systems are at ninety degrees, and two hundred seventy degrees are the same distance
from the center axis of rotation. The purpose of one system rotating clockwise and the other counterclockwise is to have each eccentric load mass peak to the greatest distance from the center axis
of rotation at the same time, the result is the movement of the clockwise and counterclockwise systems moving in the direction of one hundred eighty degrees, the Conservation of Angular Momentum is
not conserved, and the result is the eccentric mass load systems are moving in a direction that is constant. Emmy Noether’s theorem is strictly for symmetrical operations only due to Symmetries in
her outdated conservation laws. This opens up a new Chapter in physics where the Conservation of Angular Momentum is not conserved
Symmetry Versus Asymmetry
From left to right Invariance Symmetry that displays distance from center axis, this is a representation of Emmy Noether’s Law Conservation of Angular Momentum as distance “a” being the same distance
from the center axis of rotation.
Translational Force Generator
Translational Force Generator-Direct Current system. FIG.1 displays 2 each eccentric mass load systems direct current Quadrupole Systems. FIG.2 displays the polarity on the stator wall and the
polarity on each Armature. FIG-3 is the timing gears to keep the eccentric mass loads timed together as one is rotating clockwise and the other eccentric mass load system counterclockwise to keep the
eccentric mass load in this drawing to the right in this drawing. FIG.8 is an electrical wiring diagram that is a 12-lead delta Y for the Stator Walls. The square box on the bottom left area of this
page displays the eccentric mass load upon the center axis of rotation mathematically.
Effects of the Higgs
The Effects of the Higgs Field in Space with mass accelerating straight line or rotational. Circled 1 Rockets in space accelerating a room in space upwards with rocket propulsion; a motor rotating
the Man in this room in space inside the center of the room with his arms outstretched with round weights in each hand, due to the rotation of the room an apple on a string is hanging diagonally from
the center of the ceiling of this room, towards the wall as this room rotates.
Space Time Diagram
Space Time Diagram with the horizontal plane “Space in Feet” at the bottom of page and the “Time in Seconds” on the vertical axis, for the Battery powered Total System video on the surface tension of
water where you see the front end of my truck and the water below in my driveway. The website displays many systems moving the earth’s gravitational field, on a flat level tabletop in that system
weights 65 pounds moving from eccentric mass load in a direction that is constant.
Einstein's accuracy in Unified Field Theory is guiding our actions.
(Einstein 1944, Einstein and Bargmann 1944).
The starting point now was to keep the four-dimensionality of the theory and also the requirement of general covariance, but to give up the postu late that a generalized theory of gravitation should
necessarily be based on the existence of a Riemannian metric. What Einstein and Bargmann pro posed instead was, in effect, an attempt at a non-local relativistic theory of gravitation…
Direct Current Quadrupole Armature
This is the Direct Current Translational Force Generation System with motor drive on top, Armatures with yellow cooling fins and stator quadrupole wall, My other Patent Number 5,150,626 labeled
Translational Force Generation System.
Explore all
Electric Power Consumption
By way of background, electric power consumption is necessary for a quality life.
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Electric Power Consumption
By way of background, electric power consumption is necessary for a quality life.
Explore all
On the surface tension of water takes 19 seconds to find its center of gravity then moves forward in a constant direction until it hits the end of the water tank, open with web app.
Featured Story
The intent and function of my systems moving forward in a direction that is constant using electrical energy rather than rocket propulsion in space within this email is Bard Googles Artificial
Intelligence stating. The website is about a specific type of system called a supersymmetric rotor. In a supersymmetric rotor, the mass is distributed eccentrically around the axis of rotation. This
means that the total angular momentum of the rotor is not conserved.
Featured Story
Solving the Space Junk Problem. Asteroid Mining is taking place for set up in Colorado. Main goal is He3 Lunar Mining.
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We are looking to Save Mother Earth from Global Warming and have a 4-Phase Motor Generator Device that is more efficient that 3-Phase Motor, 2-Phase Motor and Single-phase motor.
System floating on water moving from rotational energy
Conservation of Angular Momentum Violation
(Part 1) Prototype Test on microgravity flight by Zero Gravity Corporation
(Part 2) Prototype Test on microgravity flight by Zero Gravity Corporation
(Part 3) Prototype Test on microgravity flight by Zero Gravity Corporation
(Part 4) Prototype Test on microgravity flight by Zero Gravity Corporation
(Part 5) Prototype Test on microgravity flight by Zero Gravity Corporation
(Part 6) Prototype Test on microgravity flight by Zero Gravity Corporation
(Part 8) Prototype Test on microgravity flight by Zero Gravity Corporation
(Part 7) Prototype Test on microgravity flight by Zero Gravity Corporation
(Part 9) Prototype Test on microgravity flight by Zero Gravity Corporation
Electromagnetic Quadrupole System in Operation
Asymmetric Mass System Moving in Water
Asymmetric Mass System Moving Across Tabletop | {"url":"https://supersymmetry.com/","timestamp":"2024-11-05T19:28:29Z","content_type":"text/html","content_length":"112589","record_id":"<urn:uuid:9a384ee1-141f-4035-bc92-125a526678b8>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00181.warc.gz"} |
Inverse Complexity Lab – Tiago P. Peixoto
We’re hiring! Post-doc position available.
We’re searching for a post-doc to join our group at IT:U, linz, Austria! See our call for more information. Deadline is Nov 30, 2024.
Organisms, ecosystems, climates, information systems, and societies are examples of complex systems: they exhibit rich behavior emerging from a large number of individual components following
relatively simple local rules, based on a network of direct interactions.
Complexity science is traditionally devoted to solving the forward problem: Given postulated local rules of interaction, what is the large-scale organization that emerges from them?
Our research group focuses on the inverse problem: Given an empirically observed large-scale organization, what are the local rules of interaction that caused it?
With this aim, our group develops mathematical and computational models to explain the structure and function of complex network systems, and the algorithms to reconstruct the structure of these
models from available empirical data.
We are particularly interested in problems related to:
1. Inverse problems and reconstruction of large-scale systems from indirect data.
2. Uncertainty quantification for complex systems.
3. Generative models that characterize modular hierarchies, ranks, and latent spaces.
4. Temporality, higher-order organization, and multivariate annotation of relational data.
5. Scientific software development and dataset curation.
In our work, we employ theory and methods from several disciplines, including statistical physics, computational statistics, information theory, Bayesian inference, and machine learning.
For more information see our research and publications pages.
Most of the methods developed in our group are made available as part of the graph-tool library, which is extensively documented.
For a practical introduction to many inference and reconstruction algorithms, please refer to the HOWTO.
Open positions
We’re hiring! Post-doc position available.
We’re searching for a post-doc to join our group at IT:U, linz, Austria! See our call for more information. Deadline is Nov 30, 2024.
Interested PhD candidates are encouraged to apply for the “PhD Program in Computational X at IT:U”.
Group news
• 08/11/2024 — Felipe Vaca has successfully defended his PhD thesis!
• 01/10/2024 — The group has moved to IT:U, Linz!
• 21/06/2024 — Sebastian Kusch has been awarded the “best talk by an early-career researcher” at NetSci 2024 in Québec!
• 06/05/2024 — A new version of graph-tool was released!
• 02/05/2024 — New pre-print: “Network reconstruction via the minimum description length principle” [1]
• 12/04/2024 — Upcoming talk at NetSI, Boston.
• 09/04/2024 — Bukyoung Jhun has been awarded the Young Statistical Physicist Award by Korean Physical Society!
• 01/03/2024 — Martina Contisciani has joined the group as a post-doc!
• 04/01/2024 — New arxiv paper: “Scalable network reconstruction in subquadratic time” [2]
• 01/01/2024 — Thomas Robiglio has joined the group as a PhD student!
• 15/12/2023 — Silvia Guerrini defended her MSc thesis!
• 01/09/2023 — Bukyoung Jhun has joined the group as a post-doc! | {"url":"https://skewed.de/lab/","timestamp":"2024-11-13T14:16:57Z","content_type":"application/xhtml+xml","content_length":"43876","record_id":"<urn:uuid:44aee111-9baa-42ee-8875-0dc7dd39806c>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00119.warc.gz"} |
Access: How to Create Calculated Fields and Totals Rows
Lesson 17: How to Create Calculated Fields and Totals Rows
Calculated fields and totals rows let you perform calculations with the data in your tables. Calculated fields perform calculations using data within one record, while totals rows perform a
calculation on an entire field of data.
Calculated fields
When you create a calculated field, you are adding a new field in which every row contains a calculation involving other numerical fields in that row. To do this, you must enter a mathematical
expression, which is made up of field names in your table and mathematical symbols. You don't need to know too much about math or expression building to create a useful calculated field. In fact, you
can write robust expressions using only grade-school math. For instance, you could:
• Use + to find the sum of the contents of two fields or to add a constant value (such as + 2 or + 5) to a field
• Use * to multiply the contents of two fields or to multiply fields by a constant value
• Use - to subtract one field from another or to subtract a constant value from a field
In our example, we will use a table containing the orders from one month. The table contains items listed by sales unit—single, half-dozen, and dozen. One column lets us know the number sold of each
sales unit. Another lets us know the actual numerical value of each of these units. For instance, in the top row you can see that two dozen fudge brownies have been sold and that one dozen equals 12
To find the total number of brownies that have been sold, we'll have to multiply the number of units sold by the numerical value of that unit—here, 2*12, which equals 24. This was a simple problem,
but performing this calculation for each row of the table would be tedious and time consuming. Instead, we can create a calculated field that shows the product of these two fields multiplied together
on every row.
To create a calculated field:
1. Select the Fields tab, locate the Add & Delete group, then click the More Fields drop-down command.
2. Hover your mouse over Calculated Field and select the desired data type. We want our calculation to be a number, so we'll select Number.
3. Build your expression. To select fields to include in your expression, double-click the field in the Expression Categories box. Remember to include mathematical operators like the + or - signs.
Because we want to multiply our two fields, we'll put the multiplication symbol (*) between them.
4. Click OK. The calculated field will be added to your table. If you want, you can now sort or filter it.
For more examples of mathematical expressions that can be used to create calculated fields, review the arithmetic expressions in the Expression Builder dialog box.
Totals rows
The totals row adds up an entire column of numbers, just like in a ledger or on a receipt. The resulting sum appears in a special row at the bottom of your table.
For our example, we'll add a totals row to our calculated field. This will show us the total number of items sold.
To create a totals row:
1. From the Home tab, locate the Records group, then click the Totals command.
2. Scroll down to the last row of your table.
3. Locate the desired field for the totals row, then select the second empty cell below the last record for that field. When a drop-down arrow appears, click it.
4. Select the function you want to perform on the field data. In our example, we'll choose Sum to add all of the values in the calculated field.
5. The totals row will appear. | {"url":"http://jobmy.info/index-1046.html","timestamp":"2024-11-03T04:44:09Z","content_type":"text/html","content_length":"43934","record_id":"<urn:uuid:da5f4ecd-883b-48dc-90a8-37ed0282e807>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00586.warc.gz"} |
solving the “Rubin problem”
In “GOTO Considered Harmful” Considered Harmful (a response to Dijkstra’s famous letter), Frank Rubin gives an example of a very simple problem which he claims is best solved with the judicious use
of a GOTO statement (*gasp*):
Let X be an N×N matrix of integers. Write a program that will print the number of the first all-zero row of X, if any.
The reason why this is (supposedly) hard without GOTO is that it involves breaking/continuing out of a nested loop; most structured languages only support breaking/continuing the smallest loop a
statement is contained in. So, all Rubin does with his GOTO is simulate a labelled continue statement. Personally, I think labelled break/continue statements are a very natural feature for a
structured language to have*, and I am thoroughly unconvinced by Rubin’s claim that this is somehow a general defense of GOTO.
But Dijkstra’s response takes a different, more conceptual approach. Dijkstra claims that the problem’s statement immediately calls for a “bounded linear search” inside of a “bounded linear search”.
Since it is well-understood how to construct a loop to perform a bounded linear search, a GOTO-less solution becomes apparent.
Well, at least in the abstract. I realized I didn’t really know how to implement this in Python. One important ingredient was a for-loop which only keeps running as long as some condition is
satisfied (and its iterator hasn’t run out). I didn’t want to do this by putting a conditional “break” at the end of the loop: that would replace loop semantics with tangled procedural flow. And I
certainly didn’t want to use a while loop and iterate the iterator by hand: that’s just ugly. But I realized that I should be able to make an iterator which “filters” another iterator by a condition.
Turns out it already exists, as itertools.takewhile. With that, we have:
from itertools import takewhile
def first_zero_row(arr):
row_found = False
for i, row in enumerate(takewhile(lambda x: not row_found, arr)):
cell_found = False
for cell in takewhile(lambda x: not cell_found, row):
if cell != 0: cell_found = True
if not cell_found: row_found = True
return i if row_found else None
Honestly, though, any time you see something like lambda x: not row_found you know you’re being a bit too clever for your own good. Also, although this is a straightforward translation of the
solution Dijkstra presents, it obfuscates the underlying concept, of the “nested bounded linear search”. We have the luxury of functional programming in Python; we can encode things very, very
def BLS(iter, cond):
return next((x for x in iter if cond(x)), None)
def first_zero_row(arr):
found = BLS(enumerate(arr),
lambda (i, row): not BLS(row,
lambda cell: cell != 0))
return found[0] if found else None
first_zero_row‘s practically a one-liner! An inscrutable, nested-lambda’d one-liner, but these are the sacrifices we make for conceptual purity.
Of course, if I were actually solving this problem for non-academic reasons, I’d just say:
def first_zero_row(arr):
return next((i for i, row in enumerate(arr)
if all(cell == 0 for cell in row)), None)
But where’s the fun in that?
* – Although, regrettably, Python doesn’t support them; see Guido’s rejection to the relevant PEP. | {"url":"http://joshuah.scripts.mit.edu/blog/?p=195","timestamp":"2024-11-06T01:05:48Z","content_type":"application/xhtml+xml","content_length":"17263","record_id":"<urn:uuid:bef3b9f8-4d67-46fd-b582-35693f35d15b>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00476.warc.gz"} |
Sort System¶
Sorts (also known as universes) are types whose members themselves are again types. The fundamental sort in Agda is named Set and it denotes the universe of small types. But for some applications,
other sorts are needed. This page explains the need for additional sorts and describes all the sorts that are used by Agda.
The theoretical foundation for Agda’s sort system are Pure Type Systems (PTS). A PTS has, besides the set of supported sorts, two parameters:
1. A set of axioms of the form s : s′, stating that sort s itself has sort s′.
2. A set of rules of the form (s₁, s₂, s₃) stating that if A : s₁ and B(x) : s₂ then (x : A) → B(x) : s₃.
Agda is a functional PTS in the sense that s₃ is uniquely determined by s₁ and s₂. Axioms are implemented internally by the univSort function, see univSort. Rules are implemented by the funSort and
piSort functions, see funSort.
Introduction to universes¶
Russell’s paradox implies that the collection of all sets is not itself a set. Namely, if there were such a set U, then one could form the subset A ⊆ U of all sets that do not contain themselves.
Then we would have A ∈ A if and only if A ∉ A, a contradiction.
Likewise, Martin-Löf’s type theory had originally a rule Set : Set but Girard showed that it is inconsistent. This result is known as Girard’s paradox. Hence, not every Agda type is a Set. For
example, we have
but not Set : Set. However, it is often convenient for Set to have a type of its own, and so in Agda, it is given the type Set₁:
In many ways, expressions of type Set₁ behave just like expressions of type Set; for example, they can be used as types of other things. However, the elements of Set₁ are potentially larger; when A :
Set₁, then A is sometimes called a large set. In turn, we have
and so on. A type whose elements are types is called a sort or a universe; Agda provides an infinite number of universes Set, Set₁, Set₂, Set₃, …, each of which is an element of the next one. In
fact, Set itself is just an abbreviation for Set₀. The subscript n is called the level of the universe Setₙ.
You can also write Set1, Set2, etc., instead of Set₁, Set₂. To enter a subscript in the Emacs mode, type “\_1”.
Universe example¶
So why are universes useful? Because sometimes it is necessary to define and prove theorems about functions that operate not just on sets but on large sets. In fact, most Agda users sooner or later
experience an error message where Agda complains that Set₁ != Set. These errors usually mean that a small set was used where a large one was expected, or vice versa.
For example, suppose you have defined the usual datatypes for lists and cartesian products:
data List (A : Set) : Set where
[] : List A
_::_ : A → List A → List A
data _×_ (A B : Set) : Set where
_,_ : A → B → A × B
infixr 5 _::_
infixr 4 _,_
infixr 2 _×_
Now suppose you would like to define an operator Prod that inputs a list of n sets and outputs their cartesian product, like this:
Prod (A :: B :: C :: []) = A × B × C
There is only one small problem with this definition. The type of Prod should be
However, the definition of List A specified that A had to be a Set. Therefore, List Set is not a valid type. The solution is to define a special version of the List operator that works for large
data List₁ (A : Set₁) : Set₁ where
[] : List₁ A
_::_ : A → List₁ A → List₁ A
With this, we can indeed define:
Prod : List₁ Set → Set
Prod [] = ⊤
Prod (A :: As) = A × Prod As
Universe polymorphism¶
To allow definitions of functions and datatypes that work for all possible universes Setᵢ, Agda provides a type Level of universe levels and level-polymorphic universes Set ℓ where ℓ : Level. For
more information, see the page on universe levels.
Agda’s sort system¶
The implementation of Agda’s sort system is based on the theory of pure type systems. The full sort system of Agda consists of the following sorts:
1. Standard small sorts (universe-polymorphic).
□ Setᵢ and its universe-polymorphic variant Set ℓ
□ Propᵢ and its universe-polymorphic variant Prop ℓ (with --prop)
□ SSetᵢ and its universe-polymorphic variant SSet ℓ (with --two-level)
2. Standard large sorts (non polymorphic).
3. Special sorts.
Only the small standard sort hierarchies Set and Prop are in scope by default (see --import-sorts). They and most other sorts are defined in the system module Agda.Primitive. Sorts, even though they
might enjoy the priviledge of numeric suffixes, are brought into scope just as any Agda definition, by open Agda.Primitive. Note that sorts can also be renamed, e.g., you might want to open
Agda.Primitive renaming (Set to Type).
Some special sorts are defined in other system modules, see Special sorts.
Sorts Setᵢ and Set ℓ¶
As explained in the introduction, Agda has a hierarchy of sorts Setᵢ : Setᵢ₊₁, where i is any concrete natural number, i.e. 0, 1, 2, 3, … The sort Set is an abbreviation for Set₀.
You can also refer to these sorts with the alternative syntax Seti. That means that you can also write Set0, Set1, Set2, etc., instead of Set₀, Set₁, Set₂.
In addition, Agda supports the universe-polymorphic version Set ℓ where ℓ : Level (see universe levels).
Sorts Propᵢ and Prop ℓ¶
In addition to the hierarchy Setᵢ, Agda also supports a second hierarchy Propᵢ : Setᵢ₊₁ (or Propi) of proof-irrelevant propositions. Like Set, Prop also has a universe-polymorphic version Prop ℓ
where ℓ : Level.
Sorts SSetᵢ and SSet ℓ¶
These experimental universes SSet₀ : SSet₁ : SSet₂ : ... of strict sets or non-fibrant sets are described in Two-Level Type Theory.
Sorts Setωᵢ¶
To assign a sort to types such as (ℓ : Level) → Set ℓ, Agda further supports an additional sort Setω that stands above all sorts Setᵢ.
Just as for Set and Prop, Setω is the lowest level at an infinite hierarchy Setωᵢ : Setωᵢ₊₁ where Setω = Setω₀. You can also refer to these sorts with the alternative syntax Setωi. That means that
you can also write Setω0, Setω1, Setω2, etc., instead of Setω₀, Setω₁, Setω₂.
However, unlike the standard hierarchy of universes Setᵢ, the second hierarchy Setωᵢ does not support universe polymorphism. This means that it is not possible to quantify over all Setωᵢ at once. For
example, the expression ∀ {i} (A : Setω i) → A → A would not be a well-formed agda term. See the section on Setω on the page on universe levels for more information.
Concerning other applications, it should not be necessary to refer to these sorts during normal usage of Agda, but they might be useful for defining reflection-based macros. And it is allowed to
define data types in Setωᵢ.
When --omega-in-omega is enabled, Setωᵢ is considered to be equal to Setω for all i (thus rendering Agda inconsistent).
Sorts Propωᵢ¶
This transfinite extension of the Prop hierarchy works analogous to Setωᵢ. However, it is not motivated by typing (ℓ : Level) → Prop ℓ, because that lives in Setω. Instead, it may be used to host
large inductive propositions, where constructors can have fields that live at any finite level ℓ.
The sorting rules for finite levels extend to the transfinite hierarchy, so we have Propωᵢ : Setωᵢ₊₁.
Sorts SSetωᵢ¶
This is a transfinite extension of the SSet hierarchy.
Special sorts¶
Special sorts host special types that are not placed in a standard universe for technical reasons, typically because they require special laws for function type formation (see funSort).
With --sized-types and open import Agda.Builtin.Size we have SizeUniv which hosts the special type Size and the special family Size<.
With --cubical and open import Agda.Primitive.Cubical we get IUniv which hosts the interval I.
With --guarded we can define primitive primLockUniv : Set₁ in which we can postulate the Tick type.
With --level-universe the type Level no longer lives in Set but in its own sort LevelUniv. It is still defined in Agda.Primitive.
Sort metavariables and unknown sorts¶
Under universe polymorphism, levels can be arbitrary terms, e.g., a level that contains free variables. Sometimes, we will have to check that some expression has a valid type without knowing what
sort it has. For this reason, Agda’s internal representation of sorts implements a constructor (sort metavariable) representing an unknown sort. The constraint solver can compute these sort
metavariables, just like it does when computing regular term metavariables.
However, the presence of sort metavariables also means that sorts of other types can sometimes not be computed directly. For this reason, Agda’s internal representation of sorts includes three
additional constructors univSort, funSort, and piSort. These constructors compute to the proper sort once enough metavariables in their arguments have been solved.
univSort, funSort and piSort are internal constructors that may be printed when evaluating a term. The user cannot enter them, nor introduce them in Agda code. All these constructors do not represent
new sorts but instead, they compute to the right sort once their arguments are known.
univSort returns the successor sort of a given sort. In PTS terminology, it implements the axioms s : univSort s.
sort successor sort
Prop a Prop (lsuc a)
Set a Set (lsuc a)
SSet a SSet (lsuc a)
Propωᵢ Propωᵢ₊₁
Setωᵢ Setωᵢ₊₁
SSetωᵢ SSetωᵢ₊₁
SizeUniv Setω
IUniv SSet₁
LockUniv Set₁
LevelUniv Set₁
_1 univSort _1
The constructor funSort computes the sort of a function type even if the sort of the domain and the sort of the codomain are still unknown.
To understand how funSort works in general, let us assume the following scenario:
• sA and sB are two (possibly different) sorts.
• A : sA, meaning that A is a type that has sort sA.
• B : sB, meaning that B is a (possibly different) type that has sort sB.
Under these conditions, we can build the function type A → B : funSort sA sB. This type signature means that the function type A → B has a (possibly unknown) but well-defined sort funSort sA sB,
specified in terms of the sorts of its domain and codomain.
Example: the sort of the function type ∀ {A} → A → A with normal form {A : _5} → A → A evaluates to funSort (univSort _5) (funSort _5 _5) where:
• _5 is a metavariable that represents the sort of A.
• funSort _5 _5 is the sort of A → A.
If sA and sB happen to be known, then funSort sA sB can be computed to a sort value.
To specify how funSort computes, let U range over Prop, Set, SSet and let U ↝ U' be SSet if one of U, U' is SSet, and U' otherwise. E.g. SSet ↝ Prop is SSet and Set ↝ Prop is Prop. Also, let L range
over levels a and transfinite numbers ωᵢ (which is ω + i) and let us generalize ⊔ to L ⊔ L', e.g. a ⊔ ωᵢ = ωᵢ and ωᵢ ⊔ ωⱼ = ωₖ where k = max i j. We write standard universes as pairs U L, e.g. Propωᵢ
as pair Prop ωᵢ. Let S range over special universes SizeUniv, IUniv, LockUniv, LevelUniv.
In the following table we specify how funSort s₁ s₂ computes on known sorts s₁ and s₂, excluding interactions between different special sorts. In PTS terminology, these are the rules (s₁, s₂,
funSort s₁ s₂).
s₁ s₂ funSort s₁ s₂
U L U' L' (U ↝ U') (L ⊔ L')
U L IUniv SSet L
U ωᵢ S ≠ IUniv Set ωᵢ
U a SizeUniv SizeUniv
S U ωᵢ U ωᵢ
S ≠ LevelUniv U a U a
LevelUniv U a U ω₀
LevelUniv LevelUniv LevelUniv
SizeUniv SizeUniv SizeUniv
IUniv IUniv SSet₀
Here are some examples for the standard universes U L:
funSort Setωᵢ Setωⱼ = Setωₖ (where k = max(i,j))
funSort Setωᵢ (Set b) = Setωᵢ
funSort Setωᵢ (Prop b) = Setωᵢ
funSort (Set a) Setωⱼ = Setωⱼ
funSort (Prop a) Setωⱼ = Setωⱼ
funSort (Set a) (Set b) = Set (a ⊔ b)
funSort (Prop a) (Set b) = Set (a ⊔ b)
funSort (Set a) (Prop b) = Prop (a ⊔ b)
funSort (Prop a) (Prop b) = Prop (a ⊔ b)
funSort can admit just two arguments, so it will be iterated when the function type has multiple arguments. E.g. the function type ∀ {A} → A → A → A evaluates to funSort (univSort _5) (funSort _5
(funSort _5 _5))
Similarly, piSort s1 s2 is a constructor that computes the sort of a Π-type given the sort s1 of its domain and the sort s2 of its codomain as arguments.
To understand how piSort works in general, we set the following scenario:
• sA and sB are two (possibly different) sorts.
• A : sA, meaning that A is a type that has sort sA.
• x : A, meaning that x has type A.
• B : sB, meaning that B is a type (possibly different than A) that has sort sB.
Under these conditions, we can build the dependent function type (x : A) → B : piSort sA (λ x → sB). This type signature means that the dependent function type (x : A) → B has a (possibly unknown)
but well-defined sort piSort sA sB, specified in terms of the element x : A and the sorts of its domain and codomain.
Here are some examples how piSort computes:
piSort s1 (λ x → s2) = funSort s1 s2 (if x does not occur freely in s2)
piSort (Set ℓ) (λ x → Set ℓ') = Setω (if x occurs rigidly in ℓ')
piSort (Prop ℓ) (λ x → Set ℓ') = Setω (if x occurs rigidly in ℓ')
piSort Setωᵢ (λ x → Set ℓ') = Setωᵢ (if x occurs rigidly in ℓ')
With these rules, we can compute the sort of the function type ∀ {A} → ∀ {B} → B → A → B (or more explicitly, {A : _9} {B : _7} → B → A → B) to be piSort (univSort _9) (λ A → funSort (univSort _7)
(funSort _7 (funSort _9 _7)))
More examples:
• piSort Level (λ l → Set l) evaluates to Setω
• piSort (Set l) (λ _ → Set l') evaluates to Set (l ⊔ l')
• piSort s (λ _ → Setωi) evaluates to funSort s Setωi | {"url":"https://agda.readthedocs.io/en/v2.7.0/language/sort-system.html","timestamp":"2024-11-02T04:17:15Z","content_type":"text/html","content_length":"79304","record_id":"<urn:uuid:e042c69c-b8f8-4684-9afc-d05435091eff>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00870.warc.gz"} |
manual pages
Weighted graphs
In weighted graphs, a real number is assigned to each (directed or undirected) edge.
In igraph edge weights are represented via an edge attribute, called ‘weight’. The is_weighted function only checks that such an attribute exists. (It does not even checks that it is a numeric edge
Edge weights are used for different purposes by the different functions. E.g. shortest path functions use it as the cost of the path; community finding methods use it as the strength of the
relationship between two vertices, etc. Check the manual pages of the functions working with weighted graphs for details.
A logical scalar.
Gabor Csardi csardi.gabor@gmail.com
g <- make_ring(10)
shortest_paths(g, 8, 2)
E(g)$weight <- seq_len(ecount(g))
shortest_paths(g, 8, 2)
version 1.2.3 | {"url":"https://igraph.org/r/html/1.2.3/is_weighted.html","timestamp":"2024-11-14T07:25:52Z","content_type":"text/html","content_length":"9337","record_id":"<urn:uuid:431b1c02-e4b6-4822-a219-ba9ceb1007ad>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00401.warc.gz"} |
Ostriches 11421 - math word problem (11421)
Ostriches 11421
The ZOO has the same number of zebras as ostriches. They have a total of 84 legs. How many zebras and how many ostriches do they have at the ZOO?
Correct answer:
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Problem Model Prompt
P_Inequality_21 GPT-4 Turbo Example C w/ H
Ground Truth Answer: 1/(n+1)
Use in Code
import champ_dataset # "pip install champ-dataset" if necessary
dataset = champ_dataset.load('v0')
problem = dataset['P_Inequality_21']
messages = problem.conversations['GPT-4 Turbo|Example C w/ H'].messages
for message in messages:
print(f'Role: {message.role}')
print(f'Text: {message.text}')
print(f'Error: {message.error}')
Full Conversation System Prompt User Input Model Output
You are an expert on mathematics.
Please give an example that applies the following concept:
For n non-negative numbers x_1, ..., x_n, we have (x_1+x_2+...+x_n)/n≥(x_1*x_2*...*x_n)^(1/n), with equality if and only if x_1=...=x_n.
The concept you're referring to is the Arithmetic Mean-Geometric Mean Inequality (AM-GM Inequality). It states that for any set of non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean, with equality if and only if all the numbers are equal.
Let's apply this concept with an example. Consider three non-negative numbers: 4, 1, and 3.
First, we calculate the arithmetic mean (AM) of these numbers:
AM = (4 + 1 + 3) / 3 = 8 / 3 ≈ 2.6667
Next, we calculate the geometric mean (GM) of these numbers:
GM = (4 * 1 * 3)^(1/3) = (12)^(1/3) ≈ 2.2894
According to the AM-GM Inequality, we should have AM ≥ GM. Let's check if this is true for our example:
2.6667 (AM) ≥ 2.2894 (GM)
The inequality holds, as expected. Furthermore, we can see that the numbers 4, 1, and 3 are not all equal, which is consistent with the fact that we do not have equality in the AM-GM Inequality. If all three numbers were equal, say all were 3, then both the AM and GM would be exactly 3, and we would have equality in the inequality.
Solve the following problem. Make sure to show your work before giving the final answer.
For positive real number a, b and positive integer n, what is the largest value of (a*b^n)^(1/(n+1))/(a+bn) as an expression of n?
Besides the concept above, you may also find the following information useful:
1. Consider n+1 numbers, with one number being a and n numbers all being b.
To find the largest value of the expression \((a \cdot b^n)^{\frac{1}{n+1}} / (a + bn)\) as an expression of \(n\), we can apply the AM-GM inequality to \(n+1\) numbers where one number is \(a\) and \(n\) numbers are all \(b\).
Let's denote these \(n+1\) numbers as \(x_1, x_2, \ldots, x_{n+1}\), where \(x_1 = a\) and \(x_2 = x_3 = \ldots = x_{n+1} = b\).
The arithmetic mean (AM) of these \(n+1\) numbers is:
\[ AM = \frac{x_1 + x_2 + \ldots + x_{n+1}}{n+1} = \frac{a + nb}{n+1} \]
The geometric mean (GM) of these \(n+1\) numbers is:
\[ GM = (x_1 \cdot x_2 \cdot \ldots \cdot x_{n+1})^{\frac{1}{n+1}} = (a \cdot b^n)^{\frac{1}{n+1}} \]
By the AM-GM inequality, we have:
\[ AM \geq GM \]
\[ \frac{a + nb}{n+1} \geq (a \cdot b^n)^{\frac{1}{n+1}} \]
Now, we want to find the largest value of the expression \((a \cdot b^n)^{\frac{1}{n+1}} / (a + bn)\), which is the reciprocal of the AM-GM inequality we just wrote. So, we take the reciprocal of both sides:
\[ \frac{(a \cdot b^n)^{\frac{1}{n+1}}}{a + nb} \leq \frac{n+1}{a + nb} \]
The right-hand side of the inequality is independent of \(a\) and \(b\), and it only depends on \(n\). Therefore, the largest value of the expression \((a \cdot b^n)^{\frac{1}{n+1}} / (a + bn)\) as an expression of \(n\) is:
\[ \frac{n+1}{a + nb} \]
However, this is not the final answer yet, as we need to express it only in terms of \(n\). To do this, we need to maximize the right-hand side with respect to \(a\) and \(b\). Notice that for fixed \(n\), the denominator \(a + nb\) can be made arbitrarily large by choosing \(a\) and \(b\) to be large. Therefore, there is no upper bound on the denominator, and the expression can be made arbitrarily small.
Thus, the largest value of the expression \((a \cdot b^n)^{\frac{1}{n+1}} / (a + bn)\) cannot be determined without additional constraints on \(a\) and \(b\). It can be made arbitrarily close to zero by increasing \(a\) and \(b\). If we are looking for the maximum value under the constraint that \(a\) and \(b\) are fixed, then the maximum value is simply the reciprocal of the AM, which is \(\frac{n+1}{a + nb}\), but this is not a meaningful maximum without further context.
Now, summarize the answer above in one sentence, without any intermediate steps or explanations.
The largest value of the expression \((a \cdot b^n)^{\frac{1}{n+1}} / (a + bn)\) as an expression of \(n\) cannot be determined without additional constraints on \(a\) and \(b\), as it can be made arbitrarily small. | {"url":"https://yujunmao1.github.io/CHAMP/solutions/p_inequality_21_gpt4t_examplecwh.html","timestamp":"2024-11-05T03:26:58Z","content_type":"text/html","content_length":"8929","record_id":"<urn:uuid:ad472137-9197-49dd-9334-db3c31955406>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00852.warc.gz"} |
Relationships within air-speed dosages and you will 131 We into the floor and the cancer instances stated on PBLS | Accounting SolutionsRelationships within air-speed dosages and you will 131 We into the floor and the cancer instances stated on PBLS - Accounting SolutionsRelationships within air-speed dosages and you will 131 We into the floor and the cancer instances stated on PBLS
Relationships within air-speed dosages and you will 131 We into the floor and the cancer instances stated on PBLS
The new municipalities from Fukushima prefecture is actually classified with the half dozen portion different inside the light levels. When you look at the deciding how-to divide the latest
prefecture, i in addition to make sure the man populace size in the for every single town is adequate in order to consist of a reasonable number of thyroid gland malignant tumors cases. Iwaki town
was categorized as the a segmet of a unique given its apparently higher boy populace and you can levels of 131 I. The newest light spot contained in this map is actually Lake Inawashiro. The new
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In Table 2, the cases of thyroid cancer (malignant or suspected) noted in the final report of the PBLS and of the FSS presented in the meeting of the Oversight Committee are shown in brackets for
each area. These represent the actual cases diagnosed as thyroid cancer in the secondary examination. Since the number of children who actually underwent the examination is smaller, particularly in
the FSS, than the number of children who were assigned for the examination, we simply assume that the number of cancer cases is proportional to the number of examinees. This assumption is supported
by the fact that the numbers of cancer cases are correlated very well with the numbers of children assigned for the second examination. Thus, we get the corrected cancer incidence ( \(n <\rm<<\
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We have made this correction for the cases observed in the PBLS and the FSS respectively. The results are provided in Table 2 as the number of cancer cases. At first glance, the correction seems to
make little difference, as the two values, n and \(n <\rm<<\prime>>>\) , are similar for the latest data from the PBLS and the FSS. However, the two values on the basis of the earlier data at the
time of had significantly diverged values for area 6: n = 1, which provided \(n <\rm<<\prime>>>\) = 5.3 using the above relation (1). When the actual number n was updated from n = 1 to 5, the
corrected number becomes \(n <\rm<<\prime>>>\) = 6.7 as shown in Table 2. Hence, the corrected number changed only slightly from 5.3 to 6.7, which now roughly corresponds to the actual number. This
means that the corrected number is more likely to represent a true value than the actual number which is subject to constant updates as the survey continues. Therefore, we use the corrected number \
(n <\rm<<\prime>>>\) instead of n in the analysis discussed below.
Contained in this section, we test new theory your amount of thyroid cancers cases (cancerous or guessed) based in the PBLS represents frequency (sheer frequency before and you can in the
questionnaire several months) just like the very first assessment attempt mainly done prior to a minimum latency chronilogical age of radiation-triggered thyroid cancer (3–five years) got enacted.
You will need to mention, although not, the dedication of your own malignant tumors prevalence is made more a long span of energy () due to the fact questionnaire only gradually safeguarded the whole
Such parts act as a basis to have looking at the relationship between the fresh frequency away from thyroid gland malignant tumors and the degrees of rays
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heavens-serving rate x. We can pick a slightly bad correlation involving the incidence out-of cancers and also the heavens-dose speed. We do a Poisson regression investigation with a straight-line
Letter = ax + b which have x as being the sky-amount speed [?S/h]. The fresh new linear means taken from the maximum Possibilities shipment are
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Al-Khayyām | Encyclopedia.com
Al-Khayyāmī (or Khayyām), Ghiyāth Al-Dīn Abu’l-Fatḥ ‘Umar Ibn Ibrāhīm Al Nīsābūrī (or Al-Naysābūrī)
also known as Omar Khayyam
(b. Nīshāpũr, Khurasan [now Iran], 15 May 1048 [?]; d. Nīshāpũr, 4 December 1131 [?])
mathematics, astronomy, philosophy.
As his name states, he was the son of Ibrāhīm the epithet “al-Khayyāmī” would indicate that his father or other forebears followed the trade of making tents. Of his other names, “Umar” is his proper
designation, while “Ghiyāth al-Din” (“the help of the faith”) is an honorific he received later in life and “al-Nīsābũrī” refers to his birthplace. Arabic sources of the twelfth to the fifteenth
centuries^1 contain only infrequent an sometimes contradictory references to al-Khayyāmī differing even on the dates of his birth and death. The earliest birthdate given is approximately 1017, but
the most probable date (given above) derives from the historian Abu’l-Hasan al-Bayhaqī (1106-11740, who knew al-Khayyāmī personally and left a record of his horoscope. The most probable death date is
founded in part upon the account of Nizāmī ’Aũdī Samarqandī’s tomb in A.H. 530 (A.D. 1135/1136), four years after the latter’s death.^2 This date is confirmed by the fifteenth-century writer
Yār-Ahmed Tabrīzī.^3
At any rate, al-Khayyāmī was born soon after Khurasan was overrun by the Seljuks, who also conquered Khorezm, Iran, and Azerbaijan, over which they established a great but unstable military empire.
Most sources, including al-Bayhaqī, agree that he came from Nīshāpũr, where, according to the thirteenth/fourteenth-century historian Fadlallāh Rashīd al-Din, he received his education. Tabrīzī, on
the other hand, stated that al-Khayyāmī spent his boyhood and youth in Balkh (now in Afghanistan), and added that by the time he was seventeen he was well versed in all areas of philosophy.
Wherever he was educated, it is possible that al-Khayyāmī became a tutor, Teaching, however, would not have afforded him enough leisure to pursue science. The lot of the scholar at that time was, at
best, precarious, unless he were a wealthy man. He could undertake regular studies only if he were attached to the court of some sovereign or magnate, and his work was thus dependent on the attitude
of his master, court politics and the fortunes of war. Al-Khayyāmī gave a lively description of the hazards of such an existence at the beginning of his Risālafi’ l-barāhīn ’alā masā il al-jabr
wa’l-muqābala (“Treatise on Demostration of Problems of Algebra and Almuqabala”):
I was unable t devote myself to the learning of this al-jabr and the continued concentration upon it, because of obstacles in the vagaries of Time which hindered me; for we have been deprived of
all the people of knowledge save for a group, small in number, with many troubles, whose concern in life is to snatch the opportunity, when Time is asleep, to devote themselves meanwhile to the
investigation and perfection of a science; for the majority of people who imitate philosophers confuse the true with the false, and they do nothing but deceive and pretend knowledge, and they do
not use what they know of the science except for base and material purposes; and if they see a certain person seeking for the right and preferring the truth, doing his best to refute the false
and untrue and leaving aside hypocrisy and deceit, they make a fool of him and mock him. ^4
Al-Khayyāmī was nevertheless able, even under the unfavorable circumstances that he described, to write at this time his still unrecovered treatise Mushkilāt al-hisāb (“Problems of Arithmetic”) and
his first, untitled, algebraical treatise, as well as his short work on the theory of music, al-Qāwl ’alā ajnās allatī bi’l-arba’a (“Discussion on Genera Contained in a Fourth”).
About 1070 al-Khayyāmī reached Samarkand, where he obtained the support of the chief justice, Abũ Tāhir, under whose patronage he wrote his great algebraical treatise on cubic equations, the Risāa
quoted above, which he had planned long before. A supplement to this work was written either at the court of Shams al-Mulũk, khaqan of Bukhara, or at Isfahan, where al-Khayyami had been invited by
the Seljuk sultan, Jalāl al-Dīn Malik-shāh, and his vizier Nizām al-Mulk, to supervise the astronomical observatory there.
Al-Khayyāmi stayed at Isfahan for almost eighteen years, which were probably the most peaceful of his life. The best astronomers of the time were gathered at the observatory and there, under
al-Khayyāmī’s shaāh Astronomical Tables”). Of this work only a small portion—tables of ecliptic coordinates and of the magnitudes fo the 100 brightest fixed starssurvives . A further important task
of the observatory was the reform of the solar calendar then in use in Iran.
Al-Khayyāmī presented a plan for calendar reform about 1079. He later wrote up a history of previous reforms, the Naurū-nāma, but his own design is known only Nasīr al-Din al-Tũsī and Ulugh Beg. The
new calendar was to be based on a cycle of thirtythree years, name “Maliki era” or “Jalālī era” in honor4 of the sultan. The years 4, 8, 12, 16, 20, 24, 28, and 33 of each period were designated as
leap years of 366 days. while the average length of the year was to be 365.2424 days (a deviation of 0.0002 day from the true solar calendar), a difference of one day thus accumulating over a span of
5,000 years. (In the Gregorian calendar, the average year is 365.2425 days long 3,333 years.)
Al-Khayyāmī also served as court astrologer, although he himself, according to Nizāmī Samarqandī, did not believe in judicial astrology. Among his other, less official activities during this time, in
1077 he finished writing his commentaries on Euclid’s theory or parallel lines and theory of ratios; this book, together with his earlier algebraical Risāla, is his most important scientific
contribution. He also wrote on philosophical subjects during these years, composing in 1080 a Risāla al-kawn wa’l-taklif (“Treatise on Being and duty”), to which is appended Al-Jawab ’an thalāth
masāil: daũrat al-tadadd fi’l’ālam wa’l-jabr wa’l baqā (“An Answer to the Three Questions: On the Necessity of Contradiction in the World, on the Necessity of Determinism, and on Longevity”). At
about the same time he wrote, for a son of Mu’ayyid al-Mulk (vizier in 1095-1118), Risāla fi’l kulliyat al-wujũd (“Treatise on the Universality of Being”). (His two other philosophical works, Risāla
al-diyā’ al-’aqli fi mawdũ’ al-’ilm al kulli [“Treatise on Existence”] cannot be dated with any certainty.)
In 1092 al-Khayyāmī fell into disfavor, Malik-shāh having died and his vizier Nizām al-Mulk having been murdered by an Assassin. Following the death of Malik-shāh his second wife, Turkān-Khātũn, for
two years ruled as regent, and al-Khātũn, for two years ruled as regent, and al-Khayyāmī fell heir to some of the hostility she had demonstrated toward his patron, Nizām al-Mulk, with whom she had
quarreled over the question of royal succession. Financial support was withdrawn from the observatory and its activities came to a halt; the calendar reform was not completed; and orthodox Muslims,
who disliked al-Khayyāmī because of the religion was to freethinking evident in his quatrains, became highly influential at court . (His apparent lack be a source of difficulty for
al-Khayyāmīthroughout his life, and al-Qiftī [1172-1239] reported that in his later in his later years he even undertook a pilgrimage to Mecca to clear himself of the accusation of atheism.)
Despite his fall from grace al-Khayyāmī remained at the Seljuk court. In an effort to induce Malikshāh’s successors to renew their support of the observatory and of science in general, he embarked on
a work of propaganda. This was the Nauũznāma, mentioned above, an account of the ancient Iranian solar new year’s festival. In it al-Khayyũmī presented a history of the solar calendar and described
the ceremonies connected with the Nauũz festival; in particular, he discussed the ancient Iranian sovereigns, whom he pictured as magnanimous, impartial rulers dedicated to education, building
edifices, and supporting scholars.
Al-Khayyāmī left Isfahan in the reign of Malikshāh’s third son. Sanjar, who had ascended the throne in 1118. He lived for some time in Merv (now Mary, Turkmen S.S.R.), the new Seljuk capital, where
he probably wrote Mizān al-hikam (“Balance of Wisdoms”) and Fi’l-qustas al-mustaqim (“On Right Qustas”), which were incorporated by his disciple al-Khāzinī (who also worked in Merv), together with
works of al-Khayyāmī’s other disciple, al-Muzaffar al-Isifīzarī, into his own Mīzān al-hikam, Among other things, al-Khayyāmī’s Mizān gives a purely algebraic solution to the problem (which may be
tracked back to Archimedes) of determining the quantities of gold an silver in a given alloy by menas of a preliminary determination of the specific weight of each metal. weight and variable scales.^
Arithmetic and the Theory of Music . A collection of manuscripts in the library of the University of Leiden, Cod. or. 199, lists al-Khayyāmīs “Problems of Arithmetic” on its title page, but the
treatise itself is not included in the collection—it may be surmised that it was part of the original collection from which the Leiden manuscript was copied. The work is otherwise unknown, although
in his algebraic work Risāla fl’ l-barāhīn ’alā masā il-jabr wa’l-muqābala. al-Khayyāmī wrote of it that:
The Hindus have their own methods for extracting the sides fo squares and cubes based on the investigation of a small number of case, which is [through] the knowledge of the squares of nine
integers, that is, the squares of 1, 2, 3, and so on, and of their products into each other, that is, the product of 2 with 3, and so on. I have written a book to prove the validity of those
methods and to show that they lead to the required solutions, and I have supplemented it in kind, that is, finding the sides of the square of the square, and the quadrato-cube, and the cubo-cube,
however great they may be; and no one has done this before; and these proofs are only algebraical proofs based on the algebraical parts of the book of Elements.^6
Al-Khayyāmī may have been familiar with the “Hindu methods” that he cites through two earlier works, Fī usul hisāb al-hind (“Principles of Hindu Reckoning”), by Kushyār ibn Labbān al-Jili (971-1029),
and Al-muqnī fl’-hisāb al-hindi (“Things Sufficient to Understand Hindu Reckoning”), by ’Alīibn Ahmad alNasawī (fl. 1025). Both of these authors gave methods for extracting square and cube roots from
natural numbers, but their method of extracting cube roots differs from the method given in the Hindu literature and actually coincides more closely with the ancient Chinese method. The later was set
out as early as the second/first centuries B.C., in the “Mathematics in Nice Books,” and was used by medieval. Chinese mathematicians to extract roots with arbitrary integer exponents and even to
solve numerical algebraic equations (it was rediscovered in Europe by Ruffini and Horner at the beginning of the nineteenth century). Muslim mathematics—at least the case of the extraction of the
cube root—would thus seem to have been influenced by Chinese, either directly or indirectly. Al-Jīlī’s and al-Nasawī’s term “Hindu reckoning” must then be understood in the less restrictive sense of
reckoning in the decimal positional system by means of ten numbers.
The earliest Arabic account extant of the general method for the extraction of roots with positive integer exponents from natural numbers may be found in the Jāmi al-hisāb bi’l-takht wal-turāb
(“Collection on Arithemtic by Means of Board and Dust”), compiled by al-Tũsī. Since al-Tũsī made no claims of priority of discovery, and since he was well acquainted with the work of al-Khayyāmī, it
seems likely that the method he presented is al-Khayyāmī’s own. The method that al-Tũsī gave, then, is applied only to the definition of whole part a of the root , where
N=a^n+r, r<(a+1)^n-a^n,
To compute the correction necessary if the root is not extracted wholly, al-Tũsī formulated—in words rather than symbols—the rule for binomial expansion
(a + b)^n =a^n + na^n-r + . . . + b^n,
and gave the approximate value of as , the denominator of the root being reckoned according to the binomial formula. For this purpose al-Tũsī provided a table of binomial coefficients up to n = 12
and noted the property of binomials now expressed as
Al-Khayyāmī applied the arithmetic, particularly the theory of commensurable ratios, in his al-Qawl alā ajnās allatī bi’l-arba a (“Discussion on Genera Contained in a Fourth”). In the “Discussion”
al-Khayyāmī took up the problem—already set by the Greeks, and particularly by Euclid in the Sectio canois—of dividing a fourth into three intervals corresponding to the diatonic, chromatic, and
enharmonic tonalities. Assuming that the fourth is an interval with the ration 4:3, the three intervals into which the fourth may be divided are defined by ratios of which the product is equal to
4:3. Al-Khayyāmī listed twenty-two examples of the section of the fourth, of which three were original to him. Of the others, some of which occur in more than one source, eight were drawn from
Ptolemy’s “Theory of Harmony”; thirteen from al-Fārābīs Kitāb al-musikā al-Kabīr (“Great Book of Music”); and fourteen from Ibn Sīnā, either Kitāb al-Shifā (“The Book of Healing”) or Dānish-nāmah
(“The Book of Knowledge”). Each example was further evaluated in terms of aesthetics.
Theory of Ratios and the Doctrine of Number . Books II and III of al-Khayyāmī’s commentaries on Euclid, the Sharh ma ashkala min musādarāt kitāb Uqlīdis, are concerned with the theoretical
foundations of arithmetic as manifested in the study of the theory of rations. The general theory of ratios and proportions as expounded in book V of the Elements was one of three aspects of Euclid’s
work with which Muslim mathematicians were particularly concerned. (The others were the theory of parallels contained in book I and the doctrine of quadratic irrationals in book X.) The Muslim
mathematicians often attempted to improve on Eulid, and many scholars were not satisfied with the theory of ratios in particular. While they did not dispute the truth of the theory, they questioned
its basis on Euclid’s definition of identity of two rations, a/b = c/d, which definition could be traced back to Eudoxus and derived from the quantitative comparison of the equimultiples of all the
terms of a given proportion (Elements, book V, definition 5).
The Muslim critics of the Euclid-Eudoxus theory of ratios found its weakness to lie in its failure to express directly the process of measuring a given magnitude (a or c) by another magnitude (b or d
). This process was based upon the definition of a proportion for a particular case of the commensurable quantities a, b, and c, d through the use of the so-called Euclidean algorithm for the
determination of the greatest common measure of two numbers (Elements, book VII). Beginning with al-Māhānī, in the ninth century, a number of mathematicians suggested replacing definition 5, book V,
with some other definition that would, in their opinion, better express the essence of the proportion. The definition may be rendered in modern terms by the continued fraction theory: if a/b=(q[1], q
[2],..., q[n],...)and c/d=q[1]’, q[2]’,..., q[n]’,...), then a/b = c/d under the condition that q[k]’ = q[k] for all k up to infinity (for commensurable rations, k is finite). Definitions of
inequality of ratios a/b > c/d anda/b < c/d, embracing cases of both commensurable and incommensurable ratios and providing criteria for the quantitative comparison of rational and irrationalo
values, are introduced analogously. In the Middle Ages it was known that this “anti-phairetical” theory of ratios existed in Greek mathematics before Eudoxus; that it did was discovered only by
Zeuthen and Becker. The proof that his theory was equivalent to that set out in the Elements was al-Kayyāmī’s greatest contribution to the theory of ratios in general. Al-Khayyāmī’s proof lay in
establishing the equivalence of the definitions of equality and inequalities in both theories, thereby obviating the need to deduce all the propositions of book V of the Elements all over again. He
based his demonstration on an important theorem of the existence of the fourth proportional d with the three given magnitudes a. b, and c; he tried to prove it by means of the principle of the
infinite divisibility of magnitudes, which was, however, insufficient for his purpose. His work marked the first attempt at a general demonstration of the theorem, since the Greeks had not treated it
in a general manner. These investigations are described in book II of the Sharh.
In book III, al-Khayyāmī took up compound ratios (at that time most widely used in arithmetic, as in the rule of three and its generalizations), geometry (the doctrine of the similitude of figures),
the theory of music, and trigonometry (applying proportions rather than equalities). In the terms in which al-Khayyāmī, and other ancient and medieval scholars, worked, the ratio a/b was compounded
from the ratio a/c and c/b—what would in modern terms be stated as the first ratio being the product of the two latter. In his analysis of the operation of compounding the ratios, al-Khayyāmī first
set out to deduce from the definition of a compound ration given in book VI of the Elements (which was, however, introduced into the text by later editors) the theorem that the ratio a/c is
compounded from the ratios a/b and b/c and an analogous theorem for ratios a/c, b/c, c/d, and so on. Here, cautiously, al-Khayyāmī had begun to develop a new and broader concept of number, including
all positive irrational numbers, departing from Aristotle, whose authority he nonetheless respectfully invoked. Following the Greeks, al-Khayyāmī properly understood number as an aggregate of
indivisible units. But the development of his own theory—and the development of the whole of calculation mathematics in its numberous applications—led him to introduce new, “ideal” mathematical
objects, including the divisible unit and a generalized concept of number which he distinguished from the “absolute and true” numbers (although he unhesitatingly called it a number).
In proving this theorem for compound ratios al-Khayyāmī first selected a unit and an auxiliary quantity g whereby the ratio l/g is the same as a/b. He here took a and b to be arbitrary homogeneous
magnitudes which are generally incommensurable; l/g is consequently also incommensurable. He then described the magnitude g:
Let us not regard the magnitude g as a line, a surface, a body, or time; but let us regard it as a magnitude abstracted by reason from all this and belonging in the realm of numbers, but not to
numbers absolute and true, for the ratio of a to b can frequently be non-numerical, that is, it can frequently be impossible to find two numbers whose ratio would be equal to this ratio.^7.
Unlike the Greeks, al-Khayyāmī extended arithmetical language to ratios, writing of the equality of ratios as he had previously discussed their multiplication. Having stated that the magnitude g,
incommensurable with a unit, belongs in the realm of numbers, he cited the usual practice of calculators and land surveyors, who frequently employed such expressions as half a unit, a third of a
unit, and so on, or who dealt in roots of five, ten, or other divisible units.
Al-Khayyāmī thus was able to express any ratio as a number by using either the old sense of the term or the new, fractional or irrational sense. The compounding of ratios is therefore no different
from the multiplication of numbers, and the identity of ratios is similar to their equality. In principle, then, ratios are suitable for measuring numerically any quantities. The Greek mathematicians
had studied mathematical ratios, but they had not carried out this function to such an extent. Al-Khayyāmī, by placing irrational quantities and numbers on the same operational scale, began a true
revolution in the doctrine of number. His work was taken up in Muslim countries by al-Tũsī and his followers, and European mathematicians of the fifteenth to seventeen centuries took up similar
studies on the reform of the general ratios theory of the Elements. The concept of number grew to embrace all real and even (at least formally) imaginary numbers; it is, however, difficult to assess
the influence of the ideas of al-Khayyāmī and his successors in the East upon the later mathematics of the West.
Algebra . Eastern Muslim algebraists were able to draw upon a mastery of Hellenistic and ancient Eastern mathematics, to which they added adaptations of knowledge that had come to them from India
and, to a lesser extent, from China. The first Arabic treatise on algebra was written in about 830 by al-Khwārizmī, who was concerned with linear and quadratic equations and dealt with positive roots
only, a practice that his successors followed to the degree that equations that could not possess positive roots were ignored. At a slightly later date, the study of cubic equations began, first with
Archimedes’ problem of the section by a plane of a given sphere into two segments of which the volumes are in a given ratio. In the second half of the ninth century, al-Māhānī expressed the problem
as an equation of the type x ^3 + r = px^2(which he, of course, stated in words rather than symbols). About a century later, Muslim mathematicians discovered the geometrical solution of this equation
whereby the roots were constructed as coordinates of points of intersection of two correspondingly selected conic sections—a method dating back to the Greeks. It was then possible for them to reduce
a number of problems, including the trisection of an angle, important to astronomers, to the solution of cubic equations. At the same time devices for numerical approximated solutions were created,
and a systematic theory became necessary.
Al-Khayyāmī’s construction of such a geometrical theory of cubic equations may be accounted the most successful accomplished by a Muslim scholar. In his first short, untitled algebraic treatise he
had already reduced a particular geometrical problem to an equation, x^3+ 200x = 20x^2 + 2,000, and had solved it by an intersection of circumference y^2=(x - 10).(20 - x) and equilateral hyperbola .
He also noted that he had found an approximated numerical solution with an error of less than 1 percent, and he remarked that it is impossible to solve this equation by elementary means, since it
requires the use of conic sections. This is perhaps the first statement in surviving mathematical literature that equations of the third degree cannot be generally solved with compass and ruler—that
is, in quadratic radicals—and al-Khayyāmī repeated this assertion in his later Risāla. (In 1637 Descartes presented the same supposition, which was proved by P. Wantzel in 1837.)
In his earlier algebraic treatise al-Khayyāmī also took up the classification of normal forms of equations (that is, only equations with positive coefficients), listing all twenty-five equations of
the first, second, and third degree that might possess positive roots. He included among these fourteen cubic equations that cannot be reduced to linear or quadratic equations that cannot be reduced
to linear or quadratic equations by division by x^2 or x, which he subdivided into three groups consisting of one binomial equation (x ^3 = r), six trinomial equations (x ^3 + px^2 = r; x^3 HI- r =
qx ; X^3 + r = pX^2; x^3 + qx = r ; x^3 = pX^2 + r; and x^3 = qx + r), and seven quadrinomial equations (x^3 =px^2 gx+r;x^3 +qx + r=px^2 ;x^3 + px^2 +r = qx; X^3 +px^2 - q-x = r; x^3 +px^2 = qx + r;
X^3 + qx =px ^2 +r; and x^3 + r =px ^2 + qx).He added that of these four types had been solved (that is, their roots had been constructed geometrically) at some earlier date, but that “No rumor has
reached us of any of the remaining ten types, neither of this classification,”^8 and expressed the hope that he would later be able to give a detailed account of his solution of all fourteen types.
Al-Khayyāmī succeeded in this stated intention in his Risāla. In the introduction to this work he gave one of the first definitions of algebra, saying of it that, “The art of al-jabr and al-muqaābala
is a scientific art whose subject is pure number and measurable quantities insofar as they are unknown, added to a known thing with the help of which they may be found; and that [known] thing is
either a quantity or a ratio”^9 The “pure number” to which al-Khayyāmī refers is natural number, while by “measurable quantities” he meant lines, surfaces, bodies, and time; the subject matter of
algebra is thus discrete, consisting of continuous quantities and their abstract ratios. Al-Khayāmī then went on to write, “Now the extractions of al-jabr are effected by equating . . .these powers
to each other as is well known.”^10 He then took up the consideration of the degree of the unknown quantity, pointing out that degrees higher than third must be understood only metaphorically, since
they cannot belong to real quantities.
At this point in the Risāla al-Khayyāmī repeated his earlier supposition that cubic equations that cannot be reduced to quadratic equations must be solved by the application of conic sections and
that their arithmetical solution is still unknown (such solutions in radicals were, indeed, not discovered until the sixteenth century). He did not, however, despair of such an arithmetical solution,
adding, “Perhaps someone else who comes after us may find it out in the case, when there are not only the first three classes of known powers, namely the number, the thing, and the square.”^11 He
then also repeated his classification of twenty-five equations, adding to it a presentation of the construction of quadratic equations based on Greek geometrical algebra. Other new material here
appended includes the corresponding numerical solution of quadratic equations and constructions of all the fourteen types of third-degree equations that he had previously listed.
In giving the constructions of each of the fourteen types of third-degree equation, al-khayyāmī also provided an analysis of its “cases.” By considering the conditions of intersection or of contact
of corresponding conic sections, he was able to develop what is essentially a geometrical theory of the distribution of (positive) roots of cubic equations. he necessarily dealt only with those parts
of conic sections that are located in the first quadrant, employing them to determine under what conditions a problem may exist and whether the given type manifests only one case—or one root
(including the case of double roots, but not multiple roots, which were unknown)—or more than one case (that is, one or two roots). Al-Khayyāmī went on to demonstrate that some types of equations are
characterized by a diversity of cases, so that they may possess no roots at all, or one root, or two roots, He also investigated the limits of roots.
As far as it is known, al-Khayyāmī was thus the first to demonstrate that a cubic equation might have two roots. He was unable to realize, however, that an equation of the type x^3 + qx = px^2 + r
may, under certain conditions, possess three (positive) roots; this constitutes a disappointing deficiency in his work. As F. Woepcke, the first editor of the Risāla, has shown, al-Khayyāmī followed
a definite system in selecting the curves upon which he based the construction of the roots of all fourteen types of third-degree equations; the conic sections that he preferred were circumferences,
equilateral hyperbolas of which the axes, or asymptotes, run parallel to coordinate exes; and parabolas of which the axes parallel one of the coordinate axes. His general geometrical theory of
distribution of the roots was also applied to the analysis of equations with numerical coefficients, as is evident in the supplement to the Risāla, in which al-Khayyāmī analyzed an error of Abũ’I-Kũd
Muhammad ibn Layth, an algebraist who had lived some time earlier and whose work al-Khayyāmī had read a few years after writing the main text of his treatise.
His studies on the geometrical theory of third degree equations mark al-Khayyāmī’s most successful work. Although they were continued in oriental Muslim countries, and known by hearsay in Moorish
countries, Europeans began to learn of them only after Descartes and his successors independently arrived at a method of the geometrical construction of roots and a doctrine of their distribution.
Al-Khayyāmī did further research on equations containing degrees of a quantity inverse to the unknown (“part of the thing,” “part of the square,” and so on) including, for example, such equations as
1/x^3 + 3 1/x^2+ 5 1/x = 33/8, which he reduced by substituting x = 1/z in the equations that he had already studied. He also considered such cases as x^2 + 2x = 2 H- 2 1/x^2, which led to equations
of the fourth degree, and here he realized the upper limit of his accomplishment, writing, “If it [the series of consecutive powers] extends to five classes, or six classes, or seven, it cannot be
extracted by any method.“^12
The Theory of Parallels. Muslim commentators on the Elements as early as the ninth century began to elaborate on the theory of parallels and to attempt to establish on the theory of parallels and to
attempt to establish it on a basis different from that set out by Euclid in his fifth postulate. Thābit ibn Qurra and Ibn al-Haytham had both been attracted to the problem, while al-Haytham had both
been attracted to the problem, while al-Khayyāmī devoted the first book of his commentaries to the Sharah to it. Al-Khayyāmī took as the point of departure for his theory of parallels a principle
derived, according to him, from “the philosopher,” that is, Aristotle, namely that “two convergent straight lines should diverge in the direction of convergence.”^13 Such a principle consists of two
statements, each equivalent to Euclid’s fifth postulate. (It must be noted that nothing similar to al-Khayyāmī’s principle is to be found in any of the known writings of Aristotle.)
Al-Khayyāmī first proved that two perpendiculars to one straight line cannot intersect because they must intersect symmetrically at two points on both sides of the straight line; therefore they
cannot converge. From the second statement the principle follows that two perpendiculars drawn to one straight line cannot diverge because, if they did, they would have to diverge on both sides of
the straight line. Therefore, two perpendiculars to the same straight line neither converge nor diverge, being in fact equidistant from each other.
Al-Khayyāmī then went on to prove eight propositions, which, in his opinion, should be added to book I of the Elements in place of the proposition 29 with which Euclid began the theory of parallel
lines based on the fifth postulate of book I (the preceding twenty-eight propositions are not based on the fifth postulate). He constructed a quadrilateral by drawing two perpendicular lines of equal
length at the ends of a given line segment AB. Calling the perpendiculars AC and BD, the figure was thus bounded by the segments AB, AC, CD, and BD, a birectangle often called “Saccheri’s
quadrilateral,” in honor of the eightteenth-century geometrician who used it in his own theory of parallels.
In his first three propositions, al-Khayyāmī proved that the upper angles C and D of this quadrilateral are right angles. To establish this theorem, he (as Saccheri did after him) considered three
hypotheses whereby these angles might be right, acute, or obtuse; were they acute, the upper line CD of the figure must be longer than the base AB, and were they obtuse, CD must be shorter than AB
—that is, extensions of sides AC and BD would diverge or converge on both ends of AB. The hypothetical acute or obtuse angles are therefore proved to be contradictory to the given equidistance of the
two perpendiculars to one straight line, and the figure is proved to be a rectangle.
In the fourth proposition al-Khayyāmī demonstrated that the opposite sides of the rectangle are of equal length, and in the fifth, that it is the property of any two perpendiculars to the same
straight line that any perpendicular to one of them is also the perpendicular to the other. The sixth proposition states that if two straight lines are parallel in Euclid’s sense—that is, if they do
not intersect—they are both perpendicular to one straight line. The seventh proposition adds that if two parallel straight lines are intersected by a third straight line, alternate and corresponding
angles are equal, and the interior angles of one side are two right angles, a proposition coinciding with Euclid’s book I, proposition 29, but one that al-Khayyāmī reached by his own, non coincident
Al-Khayyāmī eighth proposition proves Euclid’s fifth postulate of book I: two straight lines intersect if a third intersects them at angles which are together less than two right angles. The two
lines are extended and a straight line, parallel to one of them, is passed through one of the points of intersection. According to the sixth proposition, these two straight lines—being one of the
original lines and line drawn parallel to it—are equidistant, and consequently the two original lines must approach each other. According to al-Khayyāmī’s general principle, such straight lines are
bound to intersect.
Al-Khayyāmī’s demonstration of Euclid’s fifth postulate differs from those of his Muslim predecessors because he avoids the logical mistake of petitio principle, and deduces the fifth postulate from
his own explicitly the same as the first theorems of the non-Euclidean geometries of Lobachevski and Riemann. Like his theory of ratios, al-Khayyāmī’s theory of parallesls influenced the work of
later Muslim scholars to a considerable degree. A work sometimes attributed to his follower al-Tũsī influenced the development of the theory of parallels in Europe in the seventeenth and eighteenth
centuries, as was particularly reflected in the work of Wallis and Saccheri.
Philosophical and Poetical Writings. Although al-Khayyāmī wrote five specifically philosophical treatises, and although much of his poetry is of a philosophical nature, it remains difficult to
ascertain what his world view might have been. Many investigators have dealt with this problems, and have reached many different conclusions, depending in large part on their own views. The problem
is complicated by the consideration that the religious and philosophical tracts differ from the quatrains, while analysis of the quatrains themselves is complicated by questions of their individual
authenticity. Nor is it possible to be sure of what in the philosophical treatises actually reflects al-Khayyāmī’s own mind, since they were written under official patronage.
His first treatise, Risālat al-kawn wa’taklif (“Treatise on Being and Duty”), was written in 1080, in response to a letter from a high official who wished al-Khayyāmī to give his views on “the Divine
Wisdom in the Creation of the World and especially of Man and on man’s duty to pray.”^14 The second treatise, Al-Jawab ‘an thalāth masā’il (“An Answer to the Three Questions”), closely adheres to the
formula set out in the first. Risāla fi’l kulliyat al-wujũd (“Treatise on the Universality of Being”) was written at the request of Mu’ayyid al-Mulk, and, while it is not possible to date or know the
circumstances under which the remaining two works, Risālat al-diyā’ al-‘aqli fi mawdũ’ al-’ilm al-kulli (“The Light of Reason on the Subject of Universal Science”) and Risāla fi’l wujũd (“Treatise on
Existence”), were written, it would seem not unlikely that they had been similarly commissioned. Politics may therefore have dictated the contents of the religious tracts, and it must be noted that
the texts occasio9nally strike a cautious and impersonal note, presenting the opinions of a number of other authors, without criticism or evaluation.
It might also be speculated that al-Khayyāmī wrote his formal religious and philosophical works to clear his name of the accusation of freethinking. Certainly strife between religious sects and their
common aversion to agnosticism were parat of the climate of the time, and it is within the realm of possibility that al-Khayyāmī’s quatrains had become known to the religious orthodoxy and had cast
suspicion upon him. (The quatrains now associated with his name contain an extremely wide range of ideas, ranging from religious mysticism to materialism and almost atheism; certainly writers of the
thirteenth century thought al-Khayyāmī a freethinker, al-Qiftāi calling the poetry “a stinging serpent to the Sharia” and the theologian Abũ Bakr Najm al-Dīn al-Rāxi characterizing the poet as “an
unhappy philosopher, materialist, and naturalist.”)^15
Insofar as may be generalized, in his philosophical works al-Khayyāmī wrote as an adherent of the sort of eastern Aristotelianism propagated by Ibn Sīlnā—that is, of an Aristotelianism containing
considerable amounts of Platonism, and adjusted to fit Muslim religious doctrine. Al-Bayhaqī called al-Khayyāmī “a successor of Abũ Ali [Ibn Sīnā] in different domains of philosophical sciences,”^16
but from the orthodox point of view such a rationalistic approach to the dogmas of faith was heresy. At any rate, al-Khayyāmī’s philosophy is scarcely original, his most interesting works being those
concerned with the analysis of the problem of existence of general concepts. Here al-Khayyāmī—unlike Ibn Sānā, who held views close to Plato’s realism—developed a position similar to that which was
stated simultaneously in Europe by Abailard, and was later called conceptualism.
As for al-Khayyāmī’s poetical works, more than 1,000 quatrains, written in Persian, are now published under his name. (Govinda counted 1,069.) The poems were preserved orally for a long time, so that
many of them are now known in several variants. V. A. Zhukovsky, a Russian investigator of the poems, wrote of al-Khayyāmī in 1897:
He has been regarded variously as a freethinker, a subverter of Faith, an atheist and materialist; a pantheist and a scoffer at mysticism; an orthodox Musulman; a true philosopher, a keen
observer, a man of learning; a bon vivant, a profligate, a dissembler, and a hypocrite; a blasphemer—nay, more, an incarnate negation of positive religion and and of all moral beliefs; a gentle
nature, more given to the contemplation of things diving than the wordly enjoyments; an epicurean septic; the Persian Abũ’l-’Alā, Voltaire, and Heine. One asks oneself whether it is possible to
conceive, not a philosopher, but merely an intelligent man (provided he be not a moral deformity) in whom were conmmingled and embodied such a diversity of convictions, paradoxical inclinations
and tendencies, of high moral courage and ignoble passions, of torturing doubts and vacillations?^17
The inconsistencies noted by Zhukovsky are certainly present in the corpus of the poems now attributed to al-Khayyāmī, and here again questions of authenticity arise. A. Christensen, for example,
thought that only about a dozen of the quatrains might with any certainty be considered genuine, although later he increased this n umber to 121. At any rate, the poems generally known as
al-khayyāmī’s are one of the summits of philosophical poetry, displaying an unatheistic freethought and love of freedom, humanism and aspirations for justice, irony and skepticism, and above all an
epicurean spirit that verges upon hedonism.
Al-Khayyāmī’s poetic genius was always celebrated in the Arabic East, but his fame in European countries is of rather recent origin. In 1859, a few years after Woepcke’s edition had made al-Khayyāmī’
algebra—previously almost unknown—available to Western scholars, the English poet Edward FitzGerald published translations of seventy-five of the quatrains, an edition that remains popular. Since
then, many more of the poems have been published in a number of European languages.
The poems—and the poet—have not lost their power to attract. In 1934 a monument to al-Khayyāmī was erected at his tomb in Nishāpũr, paid for by contributions from a number of countries.
1. V. A. Zhukovsky, Omar Khayyam i “stranstvuyushchie” chetverostishia; Swami Govinda Tirtha, The Nectar of Grace; and Nizāmī ’Arumacr;di Samarqandi, Sobranie redkostei ili chetyre besedy.
2. Samarqandī, op. cit., p. 97; in the Browne trans., p. 806, based on the later MSS, “four years” is “some years.”
3. Govinda, op. cit., pp. 70-71.
4. Risāla fi’l-barāhin ‘alā masũ’ il al-janr wa’ l-muqābala, Winter ’Arafat trans., pp. 29-30.
5. I. S. Levinova, “Teoria vesov v traktatakh Omara Khayyama i ego uchenika Abu Hatima al-Muzaffara ibn Ismila al-Asfizari.”
6. Risāla, Winter-’Arafat trans., pp. 34 (with correction), 71.
7. Omar Khayyam, Trajtaty, pp. 71, 145.
8. First algebraic treatise, Krasnova and Rosenfeld trans., p. 455; omitted from Amir-Moéz trans.
9. Risāla, Winter-’Arafat trans., p. 30 (with correction).
10. Ibid., p. 31.
11. Ibid., p. 31 (with correction).
12. Ibid., p. 70.
13. Omar Khayyam, Traktaty, pp. 120-121; omitted from Sharh mā ashkala min musādarāt kitāb Uqlidis. Amir-Moéztrans.
14. Omar Khayyam, Traktaty, p. 152.
15. Zhukovsky, op. cit., pp. 334, 342.
16. Govinda, op. cit., pp. 32-33.
17. Zhukovsky, op. cit., p. 325.
I. Original Works. The following are al-Khayyāmī’s main writings:
1. The principle ed. is Omar Khayyam, Traktaty (“. . .Treatises”), B. A. Rosenfeld, trans.; V. S. Segal and A. P. Youochkevitch, eds.; iontro, and notes by B. A. Rosenfeld and A. P. Youschkevitch
(Moscow, 1961), with plates of the MSS. It contains Russian trans. of all the scientific and philosophical writings except the first algebraic treatise, al-Qawl alā ajnās allātī bi’ l-arba’a, and
Fī’l-qustas al-mustaqim.
2. The first algebraic treatise. MS: Teheran, Central University library, VII, 1751 / 2. Eds.: Arabic text and Persian trans. by G. H. Mossaheb (see below), pp. 59-74, 251-291; English trans. by A.
R. Amir-Moéz in Scripta mathematica, 26 , no. 4 (1961), 323-337; Russian trans. with notes by S. A. Krasnova and B. A. Rosenfeld in Istoriko-matematicheskie issledovaniya, 15 (1963), 445-472.
3. Risāla fi’l-barāhīn alā mas’ ill al-jabar wa’l-muqābala (“Treatise on Demonstration of Problems of Algebra and Almuqabala”). MSS: Oaris, Bibilthéque Nationale, Ar. 2461, 2358/7; Leiden University
library, Or. 14/2; London, Inida Office Library, 734/10;; Rome, Vatican Library, Barb. 96/2; New York, collection of D. E. Smith.
Eds.: F. Woepcke, L’alg′bre d’Omar Alkhayyāmī (Paris, 1851), text of both Paris MSS and of the Leidon MS, France trans. and ed.’s notes—reedited by Mossaheb (see below), pp. 7-52, with Persian trans.
(pp. 159-250) ed. by the same author earlier in Jabr-u Muqābala-i Khayyām (Teheran, 1938); English trans. by D. S. Kasir, The Algebra of Omar Khayyam (New York, 1931), trans, from the Smith MS, which
is very similar to Paris MS Ar. 2461, and by H. J. J. Winter and W. ’Arafat, “The Algebra of ’Umar Khayyam,” in Journal of the Royal Asiatic Society of Bengal Science, 16 (1950), 27-70, trans. from
the London MS; and Russian trans.and photographic repro. of Paris MS 2461 in Omar Khayyam, Traktaty, pp. 69-112; 1st Russian ed. in Istoriko-matematicheskie issledovaniya, 6 1953), 15-66.
4. Sharh maā ashkala min musādarāt kitāb Uqlidis (“Commentaries to Difficulties in the Introductions to Euclid’s Book”). MSS: Paris, Bibliothéque Nationale, Ar. 4946/4; Leiden University library, Or.
Eds.: T. Erani, Discussion of Difficulties of Euclid by Omar Khayyam (Teheran, 1936), the Leiden Ms, reed. by J. Humai (see below), pp. 177-222, with a Persian trans. (pp. 225-280); Omar Khayyam,
Explanation of the Difficulties in Euclid’s Postulates, A. I. Sabra, ed. (Alexandria, 1961), the Leiden MS and text variants of Paris MS; an incomplete English trans. by A. R. Amir-Moéz, in Scripta
mathematica, 24 , no. 4 (1959), 275-303; and Russian trans. and photographic repro. of Leiden MS in Omar Khayyam, Traktaty, pp. 113-146; 1st Russian ed. in Istorikomatematicheskie issledovaniya6
(1953), 67-107.
5. Al-Qawl ‘alā ajnās allati bi’l-arba’a (“Discussion on General Containied in the Fourth”). MS: Teheran, Central University library, 509, fols. 97-99.
Ed.: J. Humai (see below), pp. 341-344.
6. Mizān al-hikam (“The Balance of Windoms”) or fi ikhtiyāl ma’rafa miqdāray adh-dhahab wa-l-fidda fi jism murakkab minhumā (“On the Art of Determination of Gold and Silver in a Body Consisting of
Them”). Complete in Abdalrahmān al-khāzinī, Kitāb mizān al-hikma (“Book of the Balance of Wisdom“). MSS: Leningrad, State Public Library, Khanykov collection, 117, 57b-60b; also in Bombay and
Hyderabad. Incomplete MS: Gotha, State Library, 1158, 39b-40a.
Eds. of the Bombay and Hyderabad MSS: Abdalrahmān alkhāzinī, Kitāb Kitāb mizām al-hikma (Hyderabad, 1940), pp. 87-92; S. S. Nadwi (see below), pp. 427-432. German trans. by E. Widemann in
Sitzungsberichte der Physikalisch-medizinischen Sozietät in Erlangen, 49 (1908), 105-132; Russian trans. and repro. of the Leningrad MS in Omar Khayyam, Traktaty, pp. 147-151; 1st Russian ed. in
Istoriko-matematicheskie issledovaniya, 6 (1953), 108-112.
Eds. of the Gotha MS: Arabic text in Rosen’s ed. of the Rubā’i (see below), pp. 202-204), in Erani’s ed. of the Sharh (see above), and in M. ’Abbasī (see below), pp. 419-428; German trans. by F.
Rosen in Zeitschrift der Deutschen morgenländischen Gesellschaft, 4 (79) (1925), 133-135; and by E. Wiedemann in Sitzungsberichte der Physikalischmedizinischen Sozietät in Erlangen, 38 (1906),
7. Fi’l-qustas al-mutaqīm (“On Right Qustas”), in al-Khāzinī’s Mīzā (see above), pp. 151-153.
8. Zij Malik-shāhi (“Malik-shāh Astronomical Tables”). Only a catalogue of 100 fixed stars for one year of the Malikī era is extant in the anonymous MS Bibliothéque Nationale, Ar. 5968.
Eds.: Russian trans. and photographic repro. of the MS in Omar Khayyam, Traktaty pp. 225-235; same trans. with more complete commentaries in Istorikoastronomicheskie issledovaniya, 8 (1963), 159-190.
9-11. Risäla al-kawn wa’l-taklif (“Treatise on Being and Duty”), Al-Jawab ‘an thaāth masā” il: daũrat al-tadadd fi’ l-’ālam wa’la-jabr wa’l-baqā’ (“Answer to Three Questions: On the Necessity of
Contradiction in the World, on Determinism and on Longevity”), Risāla al-diyā’ al-’aqli fi mawdũ’ al-’ilm al-kullī (“the Light of Reason on the subject of Universal Science”). MSS belonging to Nũr
al-Din Mustafā (Cairo) are lost.
Arabic text in Jāmi’ al-badā’i’ (“Collection of Uniques”; Cairo, 1917), pp. 165-193; text of the first two tretises published by S. S. Nadwi (see below), pp. 373-398; and S. Govinda (see below), pp.
45-46, 83-110, with English trans.; Perisan trans., H. Shajara, ed. (see below), pp. 299-337; Russian trans. of all three treatises in Omar Khayyam, Traktaty, pp. 152-171; 1st Russian ed. in S. B.
Morochnik and B. A. Rosenfeld (see below), pp. 163-188.
12. Risāla fi’l-wujũd (“Treatise on Existence”), or al-Awsāf wa’l-mawsũfāt (“Description and the Described”). MS: Berlin, former Prussian State Library, Or. Petermann, B. 466; Teheran, Majlis-i
Shurā-i Milli, 9014; and Poona, collection of Shaykh ’Abd al-Qādir Sarfaraz.
The teheran MS is published by Sa’id Mafisi in Sharq “East”; Sha’bān, 19313); and by Govinda (see below), pp. 172-179; 1st Russian ed. in S. B. Morochnik and B. A. Rosendeld (see below), pp. 189-199.
13. Risāla fi kulliyat al-wuhũd (“Treatise on the Univerisality of Existence”), or Risāla-ī silsila al-tartīb (“Treatise on the Chain of Order”), or Darkhwāstnāma (“The Book on Demand”). MSS: London,
British Museum, Or. 6572; Paris, Bibliothéque Nationle, Suppl. persan, 139/7;; Teheran, Majlis-i Shurā-i Milli, 9072; and al-Khayyāmī’s library. London MS reproduced in B. A. Rosenfeld and A. P.
Youschkevitch (see below), pp. 140-141; the Paris MS is reproduced in Omar Khayyam, Traktaty; the text of these MSS are published in S. S. Nadwi (see below), pp. 412-423; the Majlis-i Shurā-i Milli
MS is in Nafisi’s Sharq (see above) and in M. ’Abbasī (see below), pp. 393-405; the al-Khayyāmi library MS is in ’Umar Khayyām, Darkhwāstnāma, Muhammad ’Ali Taraqī, ed. (Teheran, 1936). Texts of the
London MS and the first Teheran MS are published by Govinda with the English trans. (See below), pp. 47-48, 117-129; French trans. of the Paris MS in A. Christensen, Le monde orientale, I (1908),
1-16; Russian trans. from the London and Paris MSS, with repro. of the Paris MS in Omar Khayyam, Traktaty, pp. 180-186—1st Russian ed. in S. B. Morochnik and B. A. Rosenfeld (see below), pp. 200-208.
14. Nauũz-nāma. MS: Berlin, former Prussian State Library, Or. 2450; London, British Museum, Add. 23568.
Eds. of the Berlin MS: Nowruz-namah, Mojtaba Minovi, ed. (Teheran, 1933); by M. ’Abbaī (see below), pp. 303-391; Russian tran. with repro. of the Berlin MS in Omar Khayyam, Traktaty, pp. 187-224.
15. Ruāiyāt (“Quatrains”). Eds. of MS: Rubāiyāt-ihakim Khayyām, Sanjar Mirzā, ed. (Teheran, 1861), Persian text of 464 ruba’; Muhammad Sasdīq ’Ali Luknawī, ed. (Lucknow, 1878, 1894, 1909), 762 (1st
ed.) and 770 (2nd and 3rd eds.) ruba’i; Muhammad Rahīm Ardebili, ed. (Bombay, 1922); Husein Danish, ed. (Istanbul, 1922, 1927), 396 quatrains with Turkish trans.; Jalāl al-Dīn Ahmed Jafrī, ed.
(Damasus, 1931; Beirut, 1950), 352 quatrains with Arabic trans,; Sa’ī, ed. (Teheran, 1933), 443 quatrains; B. Scillik, ed., Les manuscrits mineurs des Rubaiyat d’Omar-i-Khayyam dans la Bibliotheéque
National (Paris-Szeged, 1933-1934)—1933 MSS containting 95, 87, 75, 60, 56, 34, 28, 8, and 6 ruba’i and 1934 MSS containing 268, 213, and 349 ruba’i;and Mahfũz al-Haqq, ed. (Calcutta, 1939) repro. MS
containing 206 ruba’i with minatures; Muhammad ’Ali Forughī with illustration; R. M. Aliev, M. N. Osmanov, and E. E. Bertels, eds. R. M. Aliev, M. N. Osmanov, and E. E. Bertels, eds. (Moscow, 1959),
photographic repro. of MS containing 252 ruba’i and Russian prose trans. of 293 selected ruba’i.
English trans.: Edward FitzGerald (London, 1859, 1868, 1872, 1879) a poetical trans. of 75 (1st ed.) to 101 (4th ed.) quatrains, often repr. (best ed., 1900); E. H. Whinfield (London, 1882, 1883,
19893), a poetical trans. of 253 (1st ed.) 500 (2nd ed.), and 267 (3rd ed.) ruba’i. from the MS published by Luknawi, in the 2nd ed. with the Persian text; E. Heron-Allen (London, 1898), a prose
trans. and repro. of MS containing 158 ruba’i; S. Govinda (see below), pp. 1-30, a poetical trans. and the text of 1,069 ruba’i A. J. Arberry (London, 1949), a prose trans, and the Persian text of MS
containing 172 ruba’i with FitzGerald’s and Whinfield’s poetical trans., 1952 ed., a poetical trans. of 252 ruba’i from the MS published in Moscow in 1959. French trans.: J. B. Nicolas (Paris, 1867),
prose trans. and the Persian text of 464 ruba’i from the Teheran ed. of 1861; German trans.: C. H. Rempis (Tübingen, 1936), poetical trans. of 255 ruba’i; Russian trans.: O. Rumer (moscow, 1938),
poetical trans. of 300 ruba’i V. Derzavin (Dushanbem, 1955), verse trans. of 488 ruba’i; and G. Plisetsky (Moscow, 1972), verse trans. of 450 ruba’i, with commentaries by M. N. Osmanov.
II. Secondary Literature. The works listed below provide information on al-Khayyāmī’ life and work.
1. Muhammand ’Abbas’, Kulliyāt-ī athār-i parsī-yi hakīm ’Umar-i Khayyā (Teheran, 1939), a study of al-Khayyāmīlife and works. It contains texts and translation of of Mizān al-hikam, Risālat al-kawn
wa’-taklif, Al-Jawab ’an thalāth masāil, Risālat al-diyā . . ., Risāla fi’-wujũd, nd Risāla fi kulliyat al-wujũd and the quatrains. fl’-wujūd, a nd Risāla fi kulliyat al-wujũ and the quatrains.
2. C. Brockelmann, Geschichte der arabischen Literatur, I (Weimar, 1898), 471; supp. (Leiden, 1936), 855-856; III (Leiden, 1943), 620-621. A complete list of all Arabic MSS and their eds. known to
European scientists; supp, vols. mention MSS and eds. tha appeared after the main body of the work was published.
3. A. Christensen, Recherches sur les Rubũyāât de ’Omar Hayyâm (Heidelberg, 1904), an early work in which the author concludes that since there are no criteria for authenticity, only twelve quatrains
may reasonably be regarded as authentic.
4. A Christensen, Critical Studies in th Rubaīyà;t of Umar-Khayyà;m (Copenhagem 1927). A product of prolonged study in which a method of establishing the authenticity of al-Khayyāmī’s quatrains is
suggested; 121 selected quatrains are presented.
5. J. L. Coolidge, The Mathematics of Great Amateurs (Oxford, 1949; New York, 1963), pp. 19-29.
6. Hâmit Dilgan, Büyumml;k matematikci Omer Hayyâm (Istanbul, 1959).
7. F. K. Ginzel, Handbuch der mathematischen und technishen Chronologie, 1 (Leipzig, 1906), 300-305, information on al-Khayyāmī’s calendar reform.
8. Swami Govida Tirtha, The Nectar of Grace, “Omar Khayyām’s Life and Works (Allahabad 1941), contains texts and trans of philosophical treatises and quatrains and repros. of MSS by al-Bayhaqīa and
Tabrīzī giving biographical data on al-Khayyāmī.
9. Jamāl al-Din Humāī, Khayyā-nāmah, I (Theheran, text and Persian trans. of Sharh māashkala min musādarāt kitaāb Uqlidis and text of al-Qawl ’alā ajanās allatī larba’a are in the appendix.
10. U. Jacob and E. Wiedemann,“Zu Omer-i-Chajjam,” in Der Islam, 3 (1912), 42-62, critical review of biographical data on al-Khayyāmī and a German trans. of al-Khayyāmī’ intow. to Sharh mā ashkala
min musādarāt kitāb Uqlidis.
11. I. S. Levinova, “Teoria veso v traktatakh Omara Khayyama i ego uchenicka Abu Hatima al-Muzaffara ibn Ismaila al-Asfizari, “in Trudy XV Nauchnoy Konferencii . . . instituta istorii estestvoznaniya
i tekhniki, sekoiya istorii matematiki i mekhaniki (Mosscow, 1972), pp. 90-93.
12. V. Minorsky, “’Omar Khayyām,” in Enzyklopädie des Istams, III (Leiden-Leipzig, 1935), 985-989.
13. S. B. Morochnik, Filsofskie vglyady Omara Khayyama (“Philosophical Views of Omar Khayyam”; Dushanbe, 1952).
14. S. B. Morochnik and B. A. Rosenfeld, Omar Khayyam—poet, myslitel, uchenyi (“. . . Thinker, Scientist” Dushanbe, 1957.
15. C. H. Mossaheb, Hakim Omare Khayyam as an Algebraist (Teheran, 1960). A study of al-Khayyāmī’s algebra; text and trans. of the first algebraic treatise and Risāla fi-barāhin ’ala masāil al-jabr
wa’l muqābala are in appendix.
16. Seyyed suleimān Nadwī Umar Khayyam (Azamgarh, 1932), a study of al-Khayyāmis life and works, with texts of Mizān al-hikam Risālat al-known wa’l taklif, Al-Jawab’an thaāla fi kulliyat al-wijũd in
17. B. A. Rosenfeld and A. p. Youschkevitch, Omar Khayyam (Moscow, 1965), consisting of biographical essay, analysis of scientific (especially mathematical) works, and detailed bibliography.
18. Nizāmī ’Aũdī Samarqandī, Sobranie redkostei ili chetyre besedy (“Collection of Rarities or Four Discourses“), S. I. Bayevsky and Z. N. Vorosheikina, trans., A. N. Boldyrev, ed. (Moscow, 1963),
pp. 97-98; and “The Chahār Maqāla” (“Four Discourses”), E. G. Browne, English trans., in Journal of the Royal Asiatic Society, n.s. 31 (1899), 613-663, 757-845, see 806-808. Recollections of a
contemporary of al-Khayyāmīs regarding two episodes in the latter’s life.
19. G. Sarton, Introduction to the History of Science, I (Baltimore, 1927), 759-761.
20. Husein Shajara, Tahqīq-i dar rubā iyāt-ī zindagānī-iKhayyām (Teheran, 1941), A study of al-Khayyāmi’ life and work; Persian trans. of Risālat al-kawn wa’ l-taklif and Al-Jawab ’an thalāth mass il
are in appendix.
21. D. E. Smith, “Euclid, Omar Khayyam and Saccheri,”in Scripta mathematica, 3, no. 1 (1935), 5-10, the first critical investigation of al-Khayyāmī’, theory of parallels in comparison with
22. D. J. Struik, “Omar Khayyam, Mathematician,” in Mathematical Teacher, no. 4 (1958), 280-285.
23. H. Suter, Die Mathematiker und Astronomen der Araber und ihre Werke (Leipzig, 1900), pp. 112-113.
24. A. P. Youschkevitch, “Omar Khayyam i ego Algebra,” in Trudy Instituta istorii estestvoznaniya, 2 (1948), 499-534.
25. A. P. Youschkevitch, Geschichte der Mathematik in Mittelalter, (Leipzig, 1964), pp. 251-254, 259-269, 283-287.
26. A. P. Youschkevitch and B. A. Rosenfeld, “Die Mathematik der Länder des Osten im Mittelalter,” in G. Harig, ed., Sowjetische Beiträge zur Geschichte der Naturwissenschaften (Berlin, 1960), pp.
27. V. A. Zhukovsky, “Omar Khayyam i ’stranstvuyuschie’ chetverostishiya” (“Omar Khayyam and the ‘Wandering’ Quatrains”), in al-Muzaffariyya (St. Petersburg, 1897), pp. 325-363. Translated into
English;by E. D. Ross in Journal of the Royal Asiatic Society, n. s. 30 (1898), 349-366. This paper gives all principal sources of information of al-Khayyāmī’, life and presents the problem of
“wandering” quatrains, that is, ruba’ i ascribed to both al-Khayyämi and other authors.
A. P. Youschkevitch
B. A. Rosenfeld
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API Documentation
API Documentation
class traval.detector.Detector(series, truth=None)[source]
Detector object for applying error detection algorithms to time series.
The Detector is used to apply error detection algorithms to a time series and optionally contains a ‘truth’ series, to which the error detection result can be compared. An example of a ‘truth’
series is a manually validated time series. Custom error detection algorithms can be defined using the RuleSet object.
☆ series (pd.Series or pd.DataFrame) – time series to check
☆ truth (pd.Series or pd.DataFrame, optional) – series that represents the ‘truth’, i.e. a benchmark to which the error detection result can be compared, by default None
Given a time series ‘series’ and some ruleset ‘rset’:
>>> d = Detector(series)
>>> d.apply_ruleset(rset)
>>> d.plot_overview()
See also
object for defining detection algorithms
static _validate_input_series(series)[source]
Internal method for checking type and dtype of series.
series (object) – time series to check, must be pd.Series or pd.DataFrame. Datatype of series or first column of DataFrame must be float.
TypeError – if series or dtype of series does not comply
apply_ruleset(ruleset, compare=True)[source]
Apply RuleSet to series.
○ ruleset (traval.RuleSet) – RuleSet object containing detection rules
○ compare (bool or list of int, optional) – if True, compare all results to original series and store in dictionary under comparisons attribute, default is True. If False, do not store
comparisons. If list of int, store only those step numbers as comparisons. Note: value of -1 refers to last step for convenience.
See also
object for defining detection algorithms
confusion_matrix(steps=None, truth=None)[source]
Calculate confusion matrix stats for detection rules.
Note: the calculated statistics per rule contain overlapping counts, i.e. multiple rules can mark the same observatin as suspect.
○ steps (int, list of int or None, optional) – steps for which to calculate confusion matrix statistics, by default None which uses all steps.
○ truth (pd.Series or pd.DataFrame, optional) – series representing the “truth”, i.e. a benchmark to which the resulting series is compared. By default None, which uses the stored truth
series. Argument is included so a different truth can be passed.
df – dataframe containing confusion matrix data, i.e. counts of true positives, false positives, true negatives and false negatives.
Return type:
get_corrections_dataframe(as_correction_codes=False, as_addable_df=False)[source]
Get DataFrame containing corrections.
○ as_correction_codes (bool, optional) – return DataFrame with correction codes, by default False
○ as_addable_df (bool, optional) – return DataFrame with corrections dataframe that you can add to the original time series to obtain the final result. Corrections are NaN when errors
are detected, and nonzero where observations are shifted, and zero everywhere else.
df – DataFrame containing corrections.
Return type:
Get final time series with flagged values set to NaN.
series – time series produced by final step in RuleSet with flagged values set to NaN.
Return type:
get_indices(category, step, truth=None)[source]
Get results as DataFrame.
df – results with flagged values set to NaN per applied rule.
Return type:
get_series(step, category=None)[source]
plot_overview(mark_suspects=True, **kwargs)[source]
Plot time series with flagged values per applied rule.
mark_suspects (bool, optional) – mark suspect values with red X, by default True
ax – axes objects
Return type:
list of matplotlib.pyplot.Axes
Reset Detector object.
Set ‘truth’ series.
Used for comparison with detection result.
truth (pd.Series or pd.DataFrame) – Series or DataFrame containing the “truth”, i.e. a benchmark to compare the detection result to.
stats_per_comment(step=None, truth=None)[source]
Calculate unique contribution per rule to stats.
Note: the calculated statistics per rule are under counted, i.e. when multiple rules mark the same observation as suspect it is not contained in this result.
○ steps (int, list of int or None, optional) – steps for which to calculate confusion matrix statistics, by default None which uses all steps.
○ truth (pd.Series or pd.DataFrame, optional) – series representing the “truth”, i.e. a benchmark to which the resulting series is compared. By default None, which uses the stored truth
series. Argument is included so a different truth can be passed.
df – dataframe containing confusion matrix data, i.e. unique counts of true positives, false positives, true negatives and false negatives.
Return type:
class traval.ruleset.RuleSet(name=None)[source]
Create RuleSet object for storing detection rules.
The RuleSet object stores detection rules and other relevant information in a dictionary. The order in which rules are carried out, the functions that parse the time series, the extra arguments
required by those functions are all stored together.
The detection functions must take a series as the first argument, and return a series with corrections based on the detection rule. In the corrections series invalid values are set to np.nan, and
adjustments are defined with a float. No change is defined as 0. Extra keyword arguments for the function can be passed through a kwargs dictionary. These kwargs are also allowed to contain
functions. These functions must return some value based on the name of the series.
name (str, optional) – name of the RuleSet, by default None
Given two detection functions ‘foo’ and ‘bar’:
>>> rset = RuleSet(name="foobar")
>>> rset.add_rule("foo", foo, apply_to=0) # add rule 1
>>> rset.add_rule("bar", bar, apply_to=1, kwargs={"n": 2}) # add rule 2
>>> print(rset) # print overview of rules
add_rule(name, func, apply_to=None, kwargs=None)[source]
Add rule to RuleSet.
○ name (str) – name of the rule
○ func (callable) – function that takes series as input and returns a correction series.
○ apply_to (int or tuple of ints, optional) – series to apply the rule to, by default None, which defaults to the original series. E.g. 0 is the original series, 1 is the result of step
1, etc. If a tuple of ints is passed, the results of those steps are collected and passed to func.
○ kwargs (dict, optional) – dictionary of additional keyword arguments for func, by default None. Additional arguments can be functions as well, in which case they must return some
value based on the name of the series to which the RuleSet will be applied.
Delete rule from RuleSet.
name (str) – name of the rule to delete
classmethod from_json(fname)[source]
Load RuleSet object from JSON file.
Attempts to load functions in the RuleSet by searching for the function name in traval.rulelib. If the function cannot be found, only the name of the function is preserved. This means a
RuleSet with custom functions will not be fully functional when loaded from a JSON file.
fname (str) – filename or path to file
RuleSet object
Return type:
See also
store RuleSet as JSON file (does not support custom functions)
store RuleSet as pickle (supports custom functions)
load RuleSet from pickle file
classmethod from_pickle(fname)[source]
Load RuleSet object form pickle file.
fname (str) – filename or path to file
RuleSet object, including custom functions and parameters
Return type:
See also
store RuleSet as pickle (supports custom functions)
store RuleSet as json file (does not support custom functions)
load RuleSet from json file
Get ruleset for a specific time series.
Retrieves the result of all functions that obtain parameters based on the name of the time series.
name (str) – name of the time series
new copy of ruleset with parameters for a specific time series
Return type:
Convert RuleSet to pandas.DataFrame.
rdf – DataFrame containing all the information from the RuleSet
Return type:
to_json(fname=None, verbose=True)[source]
Write RuleSet to disk as json file.
Note that it is not possible to write custom functions to a JSON file. When writing the JSON only the name of the function is stored. When loading a JSON file, the function name is used to
search within traval.rulelib. If the function can be found, it loads that function. A RuleSet making use of functions in the default rulelib.
○ fname (str) – filename or path to file
○ verbose (bool, optional) – prints message when operation complete, default is True
See also
load RuleSet from json file
store RuleSet as pickle (supports custom functions)
load RuleSet from pickle file
to_pickle(fname, verbose=True)[source]
Write RuleSet to disk as pickle.
○ fname (str) – filename or path of file
○ verbose (bool, optional) – prints message when operation complete, default is True
See also
load RuleSet from pickle file
store RuleSet as json file (does not support custom functions)
load RuleSet from json file
update_rule(name, func, apply_to=None, kwargs=None)[source]
Update rule in RuleSet.
○ name (str) – name of the rule
○ func (callable) – function that takes series as input and returns a correction series.
○ apply_to (int or tuple of ints, optional) – series to apply the rule to, by default None, which defaults to the original series. E.g. 0 is the original series, 1 is the result of step
1, etc. If a tuple of ints is passed, the results of those steps are collected and passed to func.
○ kwargs (dict, optional) – dictionary of additional keyword arguments for func, by default None. Additional arguments can be functions as well, in which case they must return some
value based on the name of the series to which the RuleSet will be applied.
class traval.ruleset.RuleSetEncoder(*, skipkeys=False, ensure_ascii=True, check_circular=True, allow_nan=True, sort_keys=False, indent=None, separators=None, default=None)[source]
Encode values in RuleSet to JSON.
Implement this method in a subclass such that it returns a serializable object for o, or calls the base implementation (to raise a TypeError).
For example, to support arbitrary iterators, you could implement default like this:
def default(self, o):
iterable = iter(o)
except TypeError:
return list(iterable)
# Let the base class default method raise the TypeError
return super().default(o)
Time Series Comparison
class traval.ts_comparison.DateTimeIndexComparison(idx1, idx2)[source]
Helper class for comparing two DateTimeIndexes.
Index members in both DateTimeIndexes.
index with entries in both
Return type:
Index members only in Index #1.
index with entries only in index #1
Return type:
Index members only in Index #2.
index with entries only in index #2
Return type:
class traval.ts_comparison.SeriesComparison(s1, s2, names=None, diff_threshold=0.0)[source]
Object for comparing two time series.
Comparison yields the following categories:
□ in_both_identical: in both series and difference <= than diff_threshold
□ in_both_different: in both series and difference > than diff_threshold
□ in_s1: only in series #1
□ in_s2: only in series #2
□ in_both_nan: NaN in both
☆ s1 (pd.Series or pd.DataFrame) – first series to compare
☆ s2 (pd.Series or pd.DataFrame) – second series to compare
☆ diff_threshold (float, optional) – value beyond which a difference is considered significant, by default 0.0. Two values whose difference is smaller than threshold are considered
Compare series per comment.
comparison – series containing the possible comparison outcomes, but split into categories, one for each unique comment. Comments must be passed via series2.
Return type:
ValueError – if no comment series is found
Create series that indicates what happend to a value.
Series index is the union of s1 and s2 with a value indicating the status of the comparison:
○ -1: value is modified
○ 0: value stays the same
○ 1: value only in series 1
○ 2: value only in series 2
○ -9999: value is NaN in both series
s – series containing status of value from comparison
Return type:
class traval.ts_comparison.SeriesComparisonRelative(s1, truth, base, diff_threshold=0.0)[source]
Object for comparing two time series relative to a third time series.
Extends the SeriesComparison object to include a comparison between two time series and a third base time series. This is used for example, when comparing the results of two error detection
outcomes to the original raw time series.
Comparison yields both the results from SeriesComparison as well as the following categories for the relative comparison to the base time series:
□ kept_in_both: both time series and the base time series contain values
□ flagged_in_s1: value is NaN/missing in series #1
□ flagged_in_s2: value is NaN/missing in series #2
□ flagged_in_both: value is NaN/missing in both series #1 and series #2
□ in_all_nan: value is NaN in all time series (series #1, #2 and base)
□ introduced_in_s1: value is NaN/missing in base but has value in series #1
□ introduced_in_s2: value is NaN/missing in base but has value in series #2
□ introduced_in_both: value is NaN/missing in base but has value in both time series
☆ s1 (pd.Series or pd.DataFrame) – first series to compare
☆ truth (pd.Series or pd.DataFrame) – second series to compare, if a “truth” time series is available pass it as the second time series. Stored in object as ‘s2’.
☆ base (pd.Series or pd.DataFrame) – time series to compare other two series with
☆ diff_threshold (float, optional) – value beyond which a difference is considered significant, by default 0.0. Two values whose difference is smaller than threshold are considered
See also
Comparison of two time series relative to each other
Compare two series to base series per comment.
comparison – Series containing the number of observations in each possible comparison category, but split per unique comment. Comments must be provided via ‘truth’ series (series2).
Return type:
ValueError – if no comment series is available.
Time series Utilities
class traval.ts_utils.CorrectionCode(value, names=None, *, module=None, qualname=None, type=None, start=1, boundary=None)[source]
Codes and labels for labeling error detection results.
traval.ts_utils.bandwidth_moving_avg_n_sigma(series, window, n)[source]
Calculate bandwidth around time series based moving average + n * std.
☆ series (pd.Series) – series to calculate bandwidth for
☆ window (int) – number of observations to consider for moving average
☆ n (float) – number of standard deviations from moving average for bandwidth
bandwidth – dataframe with 2 columns, with lower and upper bandwidth
Return type:
Convert correction code series to NaNs.
Excludes codes 0 and 4, which are used to indicate no correction and a modification of the value, respectively.
corrections (pd.DataFrame) – dataframe with correction code and original + modified values
c – return corrections series with floats where value is modified
Return type:
Convert correction code series to NaNs.
Excludes codes 0 and 4, which are used to indicate no correction and a modification of the value, respectively.
corrections (pd.Series or pd.DataFrame) – series or dataframe with correction code
c – return corrections series with nans where value is corrected
Return type:
traval.ts_utils.create_synthetic_raw_time_series(raw_series, truth_series, comments)[source]
Create synthetic raw time series.
Updates ‘truth_series’ (where values are labelled with a comment) with values from raw_series. Used for removing unlabeled changes between a raw and validated time series.
☆ raw_series (pd.Series) – time series with raw data
☆ truth_series (pd.Series) – time series with validated data
☆ comments (pd.Series) – time series with comments. Index must be same as ‘truth_series’. When value does not have a comment it must be an empty string: ‘’.
s – synthetic raw time series, same as truth_series but updated with raw_series where value has been commented.
Return type:
traval.ts_utils.diff_with_gap_awareness(series, max_gap='7D')[source]
Get diff of time series with a limit on gap between two values.
☆ series (pd.Series) – time series to calculate diff for
☆ max_gap (str, optional) – maximum period between two observations for calculating diff, otherwise set value to NaN, by default “7D”
diff – time series with diff, with NaNs whenever two values are farther apart than max_gap.
Return type:
Get correction status name from correction codes.
correction_code (pd.DataFrame or pd.Series) – dataframe or series containing corrections codes
dataframe or series filled with correction status name
Return type:
pd.DataFrame or pd.Series
Method to get corrections empty dataframe.
series (pd.Series) – time series to apply corrections to
traval.ts_utils.interpolate_series_to_new_index(series, new_index)[source]
Interpolate time series to new DateTimeIndex.
☆ series (pd.Series) – original series
☆ new_index (DateTimeIndex) – new index to interpolate series to
si – new series with new index, with interpolated values
Return type:
traval.ts_utils.mask_corrections_above_below(series, mask_above, threshold_above, mask_below, threshold_below)[source]
Get corrections where above threshold.
☆ series (pd.Series) – time series to apply corrections to
☆ threshold_above (pd.Series) – time series with values to compare with
☆ mask_above (DateTimeIndex or boolean np.array) – DateTimeIndex containing timestamps where value should be set to NaN, or boolean array with same length as series set to True where value
should be set to NaN. (Uses pandas .loc[mask] to set values.)
☆ threshold_below (pd.Series) – time series with values to compare with
☆ mask_below (DateTimeIndex or boolean np.array) – DateTimeIndex containing timestamps where value should be set to NaN, or boolean array with same length as series set to True where value
should be set to NaN. (Uses pandas .loc[mask] to set values.)
traval.ts_utils.mask_corrections_above_threshold(series, threshold, mask)[source]
Get corrections where below threshold.
☆ series (pd.Series) – time series to apply corrections to
☆ threshold (pd.Series) – time series with values to compare with
☆ mask (DateTimeIndex or boolean np.array) – DateTimeIndex containing timestamps where value should be set to NaN, or boolean array with same length as series set to True where value should
be set to NaN. (Uses pandas .loc[mask] to set values.)
traval.ts_utils.mask_corrections_below_threshold(series, threshold, mask)[source]
Get corrections where below threshold.
☆ series (pd.Series) – time series to apply corrections to
☆ threshold (pd.Series) – time series with values to compare with
☆ mask (DateTimeIndex or boolean np.array) – DateTimeIndex containing timestamps where value should be set to NaN, or boolean array with same length as series set to True where value should
be set to NaN. (Uses pandas .loc[mask] to set values.)
traval.ts_utils.mask_corrections_equal_value(series, values, mask)[source]
Get corrections where equal to value.
☆ series (pd.Series) – time series to apply corrections to
☆ values (pd.Series) – time series with values to compare with
☆ mask (DateTimeIndex or boolean np.array) – DateTimeIndex containing timestamps where value should be set to NaN, or boolean array with same length as series set to True where value should
be set to NaN. (Uses pandas .loc[mask] to set values.)
traval.ts_utils.mask_corrections_modified_value(series, values, mask)[source]
Get corrections where value was modified.
☆ series (pd.Series) – time series to apply corrections to
☆ values (pd.Series) – time series with values to compare with
☆ mask (DateTimeIndex or boolean np.array) – DateTimeIndex containing timestamps where value should be set to NaN, or boolean array with same length as series set to True where value should
be set to NaN. (Uses pandas .loc[mask] to set values.)
traval.ts_utils.mask_corrections_no_comparison_value(series, mask)[source]
Get corrections where equal to value.
☆ series (pd.Series) – time series to apply corrections to
☆ mask (DateTimeIndex or boolean np.array) – DateTimeIndex containing timestamps where value should be set to NaN, or boolean array with same length as series set to True where value should
be set to NaN. (Uses pandas .loc[mask] to set values.)
traval.ts_utils.mask_corrections_not_equal_value(series, values, mask)[source]
Get corrections where not equal to value.
☆ series (pd.Series) – time series to apply corrections to
☆ values (pd.Series) – time series with values to compare with
☆ mask (DateTimeIndex or boolean np.array) – DateTimeIndex containing timestamps where value should be set to NaN, or boolean array with same length as series set to True where value should
be set to NaN. (Uses pandas .loc[mask] to set values.)
traval.ts_utils.resample_short_series_to_long_series(short_series, long_series)[source]
Resample a short time series to index from a longer time series.
First uses ‘ffill’ then ‘bfill’ to fill new series.
☆ short_series (pd.Series) – short time series
☆ long_series (pd.Series) – long time series
new_series – series with index from long_series and data from short_series
Return type:
traval.ts_utils.spike_finder(series, threshold=0.15, spike_tol=0.15, max_gap='7D')[source]
Find spikes in time series.
Spikes are sudden jumps in the value of a time series that last 1 timestep. They can be both negative or positive.
☆ series (pd.Series) – time series to find spikes in
☆ threshold (float, optional) – the minimum size of the jump to qualify as a spike, by default 0.15
☆ spike_tol (float, optional) – offset between value of time series before spike and after spike, by default 0.15. After a spike, the value of the time series is usually close to but not
identical to the value that preceded the spike. Use this parameter to control how close the value has to be.
☆ max_gap (str, optional) – only considers observations within this maximum gap between measurements to calculate diff, by default “7D”.
upspikes, downspikes – pandas DateTimeIndex objects containing timestamps of upward and downward spikes.
Return type:
traval.ts_utils.unique_nans_in_series(series, *args)[source]
Get mask where NaNs in series are unique compared to other series.
☆ series (pd.Series) – identify unique NaNs in series
☆ *args – any number of pandas.Series
mask – mask with value True where NaN is unique to series
Return type:
Binary Classification
class traval.binary_classifier.BinaryClassifier(tp, fp, tn, fn)[source]
Class for calculating binary classification statistics.
class traval.plots.ComparisonPlots(cp)[source]
Mix-in class for plots for comparing time series.
traval.plots.det_plot(fpr, fnr, labels, ax=None, **kwargs)[source]
Detection Error Tradeoff plot.
Adapted from scikitlearn DetCurveDisplay.
☆ fpr (list or value or array) – false positive rate. If passed as a list loops through each entry and plots it. Otherwise just plots the array or value.
☆ fnr (list or value or array) – false negative rate. If passed as a list loops through each entry and plots it. Otherwise just plots the array or value.
☆ labels (list or str) – label for each fpr/fnr entry.
☆ ax (matplotlib.pyplot.Axes, optional) – axes handle to plot on, by default None, which creates a new figure
ax – axes handle
Return type:
traval.plots.roc_plot(tpr, fpr, labels, colors=None, ax=None, plot_diagonal=True, colorbar_label=None, **kwargs)[source]
Receiver operator characteristic plot.
Plots the false positive rate (x-axis) versus the true positive rate (y-axis). The ‘tpr’ and ‘fpr’ can be passed as: - values: outcome of a single error detection algorithm - arrays: outcomes of
error detection algorithm in which a detection
□ lists: for passing multiple results, entries can be values or arrays, as listed above.
☆ tpr (list or value or array) – true positive rate. If passed as a list loops through each entry and plots it. Otherwise just plots the array or value.
☆ fpr (list or value or array) – false positive rate. If passed as a list loops through each entry and plots it. Otherwise just plots the array or value.
☆ labels (list or str) – label for each tpr/fpr entry.
☆ ax (matplotlib.pyplot.Axes, optional) – axes to plot on, default is None, which creates new figure
☆ plot_diagonal (bool, optional) – whether to plot the diagonal (useful for combining multiple ROC plots)
☆ **kwargs – passed to ax.scatter
ax – axes instance
Return type: | {"url":"https://traval.readthedocs.io/en/latest/modules.html","timestamp":"2024-11-08T17:00:25Z","content_type":"text/html","content_length":"210817","record_id":"<urn:uuid:c1c45519-a354-4f2c-b26f-dfe44fbc3d28>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00037.warc.gz"} |
How Many Ounces Are in Different Measurements
How Many Ounces Are in Different Measurements
TUP Team
When cooking or baking, it is important to understand the different measurements used for ingredients. Ounces are a commonly used unit of measurement, but it can be confusing to know how many ounces
are in a certain measurement. This article will provide you with a comprehensive guide on how many ounces are in different measurements, including common kitchen measurements like gallons, quarts,
pints, cups, tablespoons, teaspoons, and pounds, as well as more specific measurements like bottles of wine, shots, glasses, cans of soda, jars of peanut butter and many more. This guide will help
you to understand and convert measurements with ease, making cooking and baking a breeze.
How many ounces are in a gallon?
There are 128 ounces in a gallon.
How many ounces are in a quart?
There are 32 ounces in a quart.
How many ounces are in a pint?
There are 16 ounces in a pint.
How many ounces are in a cup?
There are 8 ounces in a cup.
How many ounces are in a half-cup?
There are 4 ounces in a half-cup.
How many ounces are in a tablespoon?
There are 0.5 ounces in a tablespoon.
How many ounces are in a teaspoon?
There are 0.16666666666666666 ounces in a teaspoon.
How many ounces are in a pound?
There are 16 ounces in a pound.
How many ounces are in a kilogram?
There are 35.27396194958 ounces in a kilogram.
How many ounces are in a liter?
There are 33.814022662 ounces in a liter.
How many ounces are in a milliliter?
There are 0.033814022662 ounces in a milliliter.
How many ounces are in a metric ton?
There are 35273.96194958 ounces in a metric ton.
How many ounces are in a short ton?
There are 32000 ounces in a short ton.
How many ounces are in a long ton?
There are 35840 ounces in a long ton.
How many ounces are in a stone?
There are 224 ounces in a stone.
How many ounces are in a gram?
There are 0.03527396194958 ounces in a gram.
How many ounces are in a milligram?
There are 3.527396194958E-5 ounces in a milligram.
How many ounces are in a microgram?
There are 3.527396194958E-8 ounces in a microgram.
How Many Ounces in Everyday Item
How many ounces are in a bottle of wine?
It varies depending on the size of the bottle, but a standard bottle of wine contains 750 milliliters, which is equivalent to 25.36 ounces.
How many ounces are in a shot?
A standard shot is 1.5 ounces.
How many ounces are in a glass?
It varies depending on the size of the glass, but a standard wine glass holds about 6-8 ounces.
How many ounces are in a can of soda?
A standard can of soda contains 12 fluid ounces.
How many ounces are in a bottle of beer?
A standard bottle of beer contains 12 fluid ounces.
How many ounces are in a pint of ice cream?
A pint of ice cream typically contains 16 ounces.
How many ounces are in a jar of peanut butter?
A standard jar of peanut butter contains 18 ounces.
How many ounces are in a liter of water?
There are 33.814022662 ounces in a liter of water.
How many ounces are in a gallon of milk?
There are 128 ounces in a gallon of milk.
How many ounces are in a carton of eggs?
A carton of eggs typically contains 12-15 eggs, which is equivalent to 12-15 ounces.
How many ounces are in a bag of sugar?
A standard bag of sugar contains 4 pounds, which is equivalent to 64 ounces.
How many ounces are in a bag of flour?
A standard bag of flour contains 5 pounds, which is equivalent to 80 ounces.
How many ounces are in a package of bacon?
A standard package of bacon contains 12 ounces.
How many ounces are in a pound of chicken?
A pound of chicken is equivalent to 16 ounces.
How many ounces are in a package of ground beef?
A standard package of ground beef contains 1 pound, which is equivalent to 16 ounces.
How many ounces are in a jar of jelly?
A standard jar of jelly contains 18 ounces.
How many ounces are in a container of yogurt?
A standard container of yogurt contains 6 ounces.
How many ounces are in a box of cereal?
standard box of cereal contains 18-20 ounces. It may vary depending on the brand and size of the box.
How many ounces are in a package of pasta?
A standard package of pasta contains 16 ounces.
How many ounces are in a jar of pickles?
A standard jar of pickles contains 24 ounces.
How many ounces are in a bag of chips?
A standard bag of chips contains 1 ounce.
How many ounces are in a package of cookies?
A standard package of cookies contains 12 ounces.
How many ounces are in a can of beans?
A standard can of beans contains 15 ounces.
How many ounces are in a jar of salsa?
A standard jar of salsa contains 16 ounces.
How many ounces are in a bottle of hot sauce?
A standard bottle of hot sauce contains 5 ounces.
How many ounces are in a jar of mayonnaise?
A standard jar of mayonnaise contains 32 ounces.
How many ounces are in a container of sour cream?
A standard container of sour cream contains 16 ounces.
How many ounces are in a jar of peanut butter?
A standard jar of peanut butter contains 18 ounces.
How many ounces are in a can of coconut milk?
A standard can of coconut milk contains 13.5 ounces.
How many ounces are in a bottle of salad dressing?
A standard bottle of salad dressing contains 8 ounces.
How many ounces are in a jar of honey?
A standard jar of honey contains 12 ounces.
How many ounces are in a package of cake mix?
A standard package of cake mix contains 18.25 ounces.
How many ounces are in a can of cranberry sauce?
A standard can of cranberry sauce contains 14 ounces.
How many ounces are in a jar of olives?
A standard jar of olives contains 6-8 ounces.
How many ounces are in a container of whipped cream?
A standard container of whipped cream contains 8 ounces.
How many ounces are in a package of bacon?
A standard package of bacon contains 12 ounces.
How many ounces are in a jar of jam?
A standard jar of jam contains 12-16 ounces.
How many ounces are in a package of tofu?
A standard package of tofu contains 14-16 ounces.
It’s important to note that the measurements of these products may vary depending on the brand and size of the package.
How Many Ounces in Different Units
How many ounces are in a pound-mass?
There are 16 ounces in a pound-mass.
How many ounces are in a ounce-mass?
There is 1 ounce in an ounce-mass.
How many ounces are in a hundredweight?
There are 1600 ounces in a hundredweight.
How many ounces are in a nanogram?
There are 3.527396194958E-11 ounces in a nanogram.
How many ounces are in a picogram?
There are 3.527396194958E-14 ounces in a picogram.
How many ounces are in a femtogram?
There are 3.527396194958E-17 ounces in a femtogram.
How many ounces are in an attogram?
There are 3.527396194958E-20 ounces in an attogram.
How many ounces are in a zeptogram?
There are 3.527396194958E-23 ounces in a zeptogram.
How many ounces are in a yoctogram?
There are 3.527396194958E-26 ounces in a yoctogram.
How many ounces are in a decigram?
There are 0.3527396194958 ounces in a decigram.
How many ounces are in a centigram?
There are 0.03527396194958 ounces in a centigram.
How many ounces are in a decagram?
There are 3.527396194958 ounces in a decagram.
How many ounces are in a dekaliter?
There are 338.14022662 ounces in a dekaliter.
How many ounces are in a hectoliter?
There are 3381.4022662 ounces in a hectoliter.
How many ounces are in a deciliter?
There are 3.3814022662 ounces in a deciliter.
How many ounces are in a centiliter?
There are 0.33814022662 ounces in a centiliter.
How many ounces are in an ounce-force?
There is 1 ounce in an ounce-force.
How many ounces are in a pound-force?
There are 16 ounces in a pound-force.
How many ounces are in a kilogram-force?
There are 35.27396194958 ounces in a kilogram-force.
How many ounces are in a ton-force?
There are 35,273.96194958 ounces in a ton-force.
How many ounces are in a dyne?
There are 3.5969431E-5 ounces in a dyne.
How many ounces are in a kilopond?
There are 35.27396194958 ounces in a kilopond.
How many ounces are in a poundal?
There are 1.28 ounces in a poundal.
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Mathematics for the Liberal Arts
In this section, we will learn how to construct logical statements. We will later combine our knowledge of sets with what we will learn about constructing logical statements to analyze arguments with
Logic is a systematic way of thinking that allows us to deduce new information from old information and to parse the meanings of sentences. You use logic informally in everyday life and certainly
also in doing mathematics. For example, suppose you are working with a certain circle, call it “Circle X,” and you have available the following two pieces of information.
1. Circle X has radius equal to 3.
2. If any circle has radius [latex]r[/latex], then its area is [latex]\pi{r}^{2}[/latex] square units.
You have no trouble putting these two facts together to get:
3. Circle X has area [latex]9\pi[/latex] square units.
You are using logic to combine existing information to produce new information. Since a major objective in mathematics is to deduce new information, logic must play a fundamental role. This
chapter is intended to give you a sufficient mastery of logic. | {"url":"https://courses.lumenlearning.com/waymakermath4libarts/chapter/introduction-introduction-to-logic/","timestamp":"2024-11-05T00:30:23Z","content_type":"text/html","content_length":"46204","record_id":"<urn:uuid:87aa08a7-2c38-448c-93f9-af276c5572c7>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00659.warc.gz"} |
Properties of addition - Fun2Do LabsProperties of addition
Properties Of Addition
Students always find it hard to understand the purpose of studying the properties of different operations. The purpose of properties is to help us calculate the result quickly. Basically, properties
act like shortcuts to perform the operations.
There are 3 properties of addition:
1. Additive identity property
2. Commutative property of addition
3. Associative property of addition
Let’s understand them one by one.
Additive identity property:
Whenever we add zero to a number, the result is the number itself. This is called the Additive identity Property of addition. This means that adding 0 to a number does not change the value of the
number and keeps the identity of the number the same.
Eg: 5 + 0 = 5
Commutative property of addition:
When adding, the numbers can be swapped. The result will still be the same. This is called commutative property or the order property of addition.
Eg: 4 + 3 = 7 or 3 + 4 = 7
Associative property of addition:
The associative property of addition says that when we add 3 or more numbers, we can group them in order. The result would not change.
Eg: 3 + (2 + 4) = 9
(3 + 2) + 4 = 9
Different ways of teaching:
Teaching properties through stories: | {"url":"https://fun2dolabs.com/math/addition/teaching-properties-of-addition","timestamp":"2024-11-07T13:53:52Z","content_type":"text/html","content_length":"42304","record_id":"<urn:uuid:d7b2ac00-6811-40c0-b6b7-d4adda488b4a>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00512.warc.gz"} |
How to compute the pagerank of almost anything
Whenever two things have a directional relationship to each other, then you can compute the pagerank of those things. For example, you can observe a directional relationships between web pages that
link to each other, scientists that cite each other, and chess players that beat each other. The relationship is directional because it matters in what direction the relationship points, e.g. who
lost to who in chess.
Intuitively, you may think of directional relationships as a transferal of some abstract value between two parties. For example, when one chess player loses to another in chess, then the value (i.e.
relative skill level) of the winner will increase and the value of the loser decrease. Furthermore, the amount of value that is transfered depends on the starting value of each party. For example, if
a master chess player loses to a novice chess player in chess, then the relative skill level of the novice will dramatically increase. Conversely, if a novice chess player loses to a master chess
player in chess, then that is to be expected. In this situation, the relative skill level of each player should remain roughly the same as before - the status quo.
Below you'll see an illustration of a small graph with seven nodes and seven edges. The pagerank of each node is illustrated by shading it, where a darker color denotes a higher rank.
If you study this figure, you should notice that:
• Nodes 1 through 4 all have low rank, because no other nodes point to them
• Node 5 has a medium rank, because a low-rank node points to it
• Node 6 has high rank, because many low-rank nodes point to it
• Node 7 has the highest rank, because a high-rank node points to it, while it points to nothing
Compute pagerank with Python
The pageranks of the nodes in the example graph (see figure above) was computed in Python with the help of the networkx library, which can be installed with pip: pip install networkx. The code that
creates a graph and computes pagerank is listed below:
import networkx as nx
# Initialize directed graph
G = nx.DiGraph()
# Add edges (implicitely adds nodes)
# Compute pagerank (keys are node IDs, values are pageranks)
pr = nx.pagerank(G)
1: 0.06242340798778012,
2: 0.06242340798778012,
3: 0.06242340798778012,
4: 0.06242340798778012,
5: 0.08895357136701444,
6: 0.32374552689540625,
7: 0.33760726978645894
Notice that each nodes is represented by an integer ID, with no specific semantics tied to the nodes nor the edges. In other words, the graph could equally well represent relationships between web
pages, scientists and chess players (or something else entirely).
If your relationships can be assigned weights, e.g. the strength of a victory in chess or the prominence of a link on a web page, then you can add weights to the edges in the graph. Luckily, weighted
edges can be easily added in networkx:
G.add_edge(1, 2, weight=0.5)
Dealing with time
You may ask yourself, should a chess game that took place last year impact a player's rank as much as a game that was won or lost just last week? In many situations, the most meaningful answer would
be no. A good way to represent the passing of time in a relationship graph is to use edge weights that decrease over time by some function. For example, an exponential decay function can be used,
such that relationships that were formed a long time ago have exponentially lower weight than recently formed relationships. This can be achieved in Python with the ** operator with a negative
time_decayed_weight = max(.00001, time_passed) ** -1
G.add_edge(1, 2, weight=time_decayed_weight)
We use the trick max(.00001, time_passed) to ensure that we do not raise zero to the power of a negative number. The unit of time passed depends on the domain, and is not essential to the
computation. For example, the unit could be milliseconds, years or millennia.
To be continued...
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This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://skipperkongen.dk/2016/08/16/how-to-compute-the-pagerank-of-almost-anything/","timestamp":"2024-11-04T02:39:46Z","content_type":"text/html","content_length":"119622","record_id":"<urn:uuid:a84a7bf9-99c7-498d-af20-39765196b485>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00760.warc.gz"} |
How to calculate the percentage increase
Calculating a percentage increase might sound tricky, but it's actually quite simple! Let's learn how to do it step by step.
Step 1: Understand What a Percentage Increase Means
A percentage increase tells us how much something has grown or gotten bigger compared to its original size. It's like measuring how much a plant has grown after watering it!
Step 2: Gather Your Numbers
You'll need two numbers: the original number (before) and the new number (after). For example:
• Before: 100 (the original number)
• After: 123 (the new number)
Step 3: Find the Difference
To find out how much the number increased, subtract the original number from the new number:
Difference=After - Before
In our example: Difference=123 - 100=23
Step 4: Calculate the Percentage Increase
Now, to find the percentage increase, divide the difference by the original number and multiply by 100:
Percentage Increase=(Difference / Before) * 100%
In our example: Percentage Increase=(23 / 100) * 100%=23%
Step 5: Understand the Result
A positive percentage increase (+23%) means the number has grown or increased by 23%. That's like getting extra cookies in your jar!
Let's Recap:
• Percentage Increase tells us how much something has grown or gotten bigger compared to its original size.
• Gather Your Numbers: Before and After.
• Find the Difference: Subtract Before from After.
• Calculate the Percentage Increase: Divide the Difference by Before and multiply by 100.
• Understand the Result: A positive percentage increase means growth or increase.
Calculate percentage increase in Google Sheets
Formula is =(B2-A2)/A2*100, where A2 is the "Before" number, and B2 is the "After" number.
• Before: This column represents the number before increase.
• After: This column represents the number after increase.
• Increase %: This column displays what is the percentage increase. | {"url":"https://percentage-calculator.app/percentage-increase","timestamp":"2024-11-10T22:17:50Z","content_type":"text/html","content_length":"15161","record_id":"<urn:uuid:1c314aee-fdb9-47ae-8684-4971d4036226>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00166.warc.gz"} |
Java Expressions Explained: A Detailed Guide
23 Oct 2023
Java Expressions Explained: A Detailed Guide
Are you finding it challenging to understand Java expressions? You’re not alone. Many developers find themselves puzzled when it comes to handling Java expressions, but we’re here to help.
Think of Java expressions as the building blocks of a complex structure – they are the fundamental units of calculation in your Java code, providing a versatile and handy tool for various tasks.
In this guide, we’ll walk you through the process of understanding Java expressions, from their basic use to more advanced techniques. We’ll cover everything from the basics of Java expressions to
more advanced usage scenarios, as well as alternative approaches.
Let’s get started and master Java expressions!
TL;DR: What are Java Expressions?
In Java, an expression, such as int sum = 1 + 2, is a construct made up of variables, operators, and method invocations that evaluates to a single value. which are constructed according to the
syntax of the language, . For instance, consider the following code snippet:
int sum = 10 + 20;
// Output:
// 30
In this example, 10 + 20 is a Java expression which evaluates to 30. The variable sum is then assigned this value.
This is a basic example of a Java expression, but there’s much more to learn about them, including their types and how they are used in Java programming. Continue reading for a more detailed
understanding and advanced usage scenarios.
Understanding Java Expressions: The Basics
Java expressions are the fundamental constructs in Java programming that evaluate to a single value. They can be made up of variables, operators, and method invocations, all constructed according to
the syntax of the language.
There are several types of Java expressions, including:
• Arithmetic expressions: These involve mathematical operations. For instance, int result = 10 * 20; is an arithmetic expression.
int result = 10 * 20;
// Output:
// 200
In the above example, 10 * 20 is the arithmetic expression that evaluates to 200. The variable result is then assigned this value.
• Relational expressions: These expressions compare two values and determine the relationship between them. For example, boolean isTrue = (10 > 9); is a relational expression.
boolean isTrue = (10 > 9);
// Output:
// true
In this code snippet, (10 > 9) is a relational expression that evaluates to true. The variable isTrue is then assigned this value.
• Logical expressions: These involve logical operations and return a boolean value. For instance, boolean result = (10 > 9) && (20 > 10); is a logical expression.
boolean result = (10 > 9) && (20 > 10);
// Output:
// true
In the above example, (10 > 9) && (20 > 10) is a logical expression that evaluates to true. The variable result is then assigned this value.
Understanding and using Java expressions effectively can greatly enhance your Java programming skills. However, it’s important to be aware of potential pitfalls, such as operator precedence and type
compatibility, to avoid unexpected results.
Advanced Java Expressions: A Deeper Dive
As you progress in your Java journey, you’ll encounter more complex expressions. Here, we’ll discuss some of these advanced Java expressions.
Conditional Expressions
In Java, a conditional expression (also known as a ternary operator) is a simple form of if-else statement that returns a value based on the result of a condition. It follows the syntax: condition ?
value_if_true : value_if_false.
int a = 10;
int b = 20;
int max = (a > b) ? a : b;
// Output:
// 20
In this code block, (a > b) ? a : b is a conditional expression that evaluates to 20, which is then assigned to the variable max. If a was greater than b, a would have been the result.
Method Invocation Expressions
Method invocation expressions involve calling a method. The method returns a value that can be used in your Java program.
public class Main {
static int multiply(int a, int b) {
return a * b;
public static void main(String[] args) {
int result = multiply(10, 20);
// Output:
// 200
In this example, multiply(10, 20) is a method invocation expression that calls the multiply method with 10 and 20 as arguments. The method then returns 200, which is assigned to the variable result.
Class Instance Creation Expressions
These expressions involve creating an instance of a class. The new keyword is used followed by the class constructor.
public class Main {
static class Dog {
String name;
Dog(String name) {
this.name = name;
void bark() {
System.out.println(this.name + " says woof!");
public static void main(String[] args) {
Dog myDog = new Dog("Rover");
// Output:
// Rover says woof!
In this code snippet, new Dog("Rover") is a class instance creation expression that creates a new Dog object with the name Rover. The bark method is then called on this object.
Understanding these advanced Java expressions can significantly enhance your programming skills and help you write more efficient and effective code.
Alternative Approaches to Java Expressions
While Java expressions are a fundamental part of Java programming, there are alternative approaches to achieve the same results. These alternatives can provide a different perspective, allowing you
to choose the approach that best suits your specific scenario.
Using Java Statements
Java statements can often achieve the same results as Java expressions. For example, an if-else statement can be used in place of a conditional (ternary) expression.
int a = 10;
int b = 20;
int max;
if (a > b) {
max = a;
} else {
max = b;
// Output:
// 20
In this code block, the if-else statement achieves the same result as the conditional expression we saw earlier. The variable max is assigned the larger of a and b.
Using Lambda Expressions
Lambda expressions, introduced in Java 8, provide a concise way to create anonymous methods. They can be used as an alternative to certain types of Java expressions.
import java.util.function.BiFunction;
public class Main {
public static void main(String[] args) {
BiFunction<Integer, Integer, Integer> multiply = (a, b) -> a * b;
int result = multiply.apply(10, 20);
// Output:
// 200
In this example, the lambda expression (a, b) -> a * b is used to define a function that multiplies two integers. This achieves the same result as the method invocation expression we saw earlier.
These alternative approaches can offer more flexibility and readability in certain scenarios. However, they may also have drawbacks, such as increased verbosity or complexity. Therefore, it’s
important to consider the specific needs and constraints of your program when choosing an approach.
Troubleshooting Java Expressions: Common Errors and Solutions
While Java expressions are powerful tools, they can sometimes lead to unexpected results or errors. Here, we’ll discuss some common issues and how to resolve them.
Operator Precedence Issues
Java follows a specific order of operations, known as operator precedence. Misunderstanding this can lead to unexpected results.
int result = 1 + 2 * 3;
// Output:
// 7
In this example, the multiplication operation is performed before the addition due to operator precedence, resulting in 7 rather than 9. To get the expected result, parentheses can be used to change
the order of operations.
int result = (1 + 2) * 3;
// Output:
// 9
Type Compatibility Issues
Java is a strongly typed language, meaning that the types of all variables and expressions must be compatible. Attempting to assign a value of one type to a variable of a different type can result in
a compilation error.
int result = 10 / 3;
// Output:
// 3
In this example, the division operation results in a decimal value, but because result is an integer, the decimal part is discarded. To get the expected result, one or both of the operands should be
a floating-point type.
double result = 10.0 / 3;
// Output:
// 3.3333333333333335
Best Practices and Optimization
To write effective Java expressions, keep the following tips in mind:
• Understand operator precedence: Knowing the order in which operations are performed can help avoid unexpected results.
• Use parentheses for clarity: Even if they’re not necessary, parentheses can make your expressions easier to read and understand.
• Consider type compatibility: Be aware of the types of your variables and expressions to prevent compilation errors and achieve the expected results.
• Use meaningful variable names: This can make your code easier to read and maintain.
By understanding these common issues and how to resolve them, you can write more effective and reliable Java expressions.
The Fundamentals: Digging Deeper into Java Expressions
To fully grasp Java expressions, we need to understand the fundamental role they play not only in Java but also in other programming languages. Expressions are a cornerstone of most, if not all,
programming languages, and their understanding is crucial for any developer.
The Role of Expressions in Programming
Expressions are the building blocks of any program. They represent values and, when evaluated, they produce another value. This value can be a number, a string, a boolean (true or false), or even
more complex data structures like objects or arrays in object-oriented languages like Java.
int a = 5;
int b = 10;
boolean result = a < b;
// Output:
// true
In this example, a < b is an expression that evaluates to true. This result is then stored in the result variable. Expressions like these form the basis of the logic in our programs.
Expressions Across Languages
While the syntax may vary, the concept of expressions is a universal one in programming. Whether you’re working in Python, JavaScript, C++, or any other language, you’ll find that expressions work in
a similar way.
a = 5
b = 10
result = a < b
# Output:
# True
In this Python example, a < b is an expression just like in our Java example. It also evaluates to True, demonstrating that the fundamental concept of expressions remains the same across different
Understanding the role and significance of expressions in programming will help you write more efficient and effective code, regardless of the language you’re using.
Java Expressions in the Real World
Java expressions are not just confined to small programs or academic examples. They are a fundamental part of real-world applications and large-scale projects. Understanding and mastering Java
expressions can significantly enhance your programming skills and your ability to write efficient, maintainable code.
Real-World Applications of Java Expressions
Java expressions are used in a variety of real-world applications. For instance, they are a key part of control flow in programs, used in conditions for if statements and loops. They are also used in
calculations, data processing, and algorithm implementation.
for (int i = 0; i < 10; i++) {
// Output:
// 0
// 1
// 2
// ...
// 9
In this code snippet, the expression i 9); |
| Logical | Involves logical operations | boolean result = (10 > 9) && (20 > 10); |
| Conditional | Returns a value based on a condition | int max = (a > b) ? a : b; |
| Method Invocation | Involves calling a method | int result = multiply(10, 20); |
| Class Instance Creation | Involves creating an instance of a class | Dog myDog = new Dog("Rover"); |
Further Resources for Mastering Java Expressions
To continue your learning journey to master Java expressions, check out these resources:
Wrapping up Expressions in Java
In this comprehensive guide, we’ve delved into the intricate world of Java expressions, examining their structure, importance, and uses. We started off by understanding the basic definition and types
of expressions, moving onto understanding their evaluation and precedence. We also learned about expression statements and how they are used in conjunction with control flow statements.
By looking at the evaluation of complex expressions, we’ve dissected the importance of parentheses, the role of operator precedence, and the consequences of side effects. We discussed the use of
common operators like assignment, unary and arithmetic, while also exploring their role in manipulating and evaluating expressions.
The guide also involved a practical analysis of the potential pitfalls to avoid when working with floating-point arithmetics, integer division, and operator shortcuts.
We realized that being aware of the nuances within expressions – such as automatic type conversions, potential pitfalls, and operator side effects – is crucial for avoiding bugs, designing efficient
code, and enhancing code readability.
Here’s a quick comparison of the concepts in Java expressions we’ve discussed:
Expression Type Pros Cons
Arithmetic Expressions Enables numerical operations Requires care in order of operations
Logical Expressions Facilitates decision-making in code Must clearly understand logical concepts
Relational Expressions Allows comparisons between variables Complexity increases with more variables
Conditional Expressions Offers code execution based on conditions Improper use can lead to confusing code
Method Invocation Expressions Enhances code reuse and modularity Requires proper method design and usage
Class Instance Creation Expressions Enables object-oriented programming Requires thorough knowledge of classes
Lambda Expressions Provides concise, functional programming Can be difficult to understand for beginners
Whether you’re just starting out with Java or looking to deepen your understanding, we hope this guide has given you a comprehensive understanding of Java expressions and their applications.
With a solid grasp of Java expressions, you’re well-equipped to write more efficient, effective, and maintainable Java code. Happy coding! | {"url":"https://ioflood.com/blog/java-expressions/","timestamp":"2024-11-06T04:12:14Z","content_type":"text/html","content_length":"63197","record_id":"<urn:uuid:a0a426ff-3b19-4b19-b42f-80610e850bd4>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00634.warc.gz"} |
gamsel: Generalized Additive Model Selection
The R package gamsel implements an algorithm for generalized additive model selection developed by and described in Chouldechova & Hastie. The algorithm delivers a path of selected models that is
parameterized by a positive scalar parameter, denoted by \(\lambda\). Higher values of \(\lambda\) correspond to sparser models.
In this vignette we work through some illustrative examples involving simulated and actual data. We explain how to deal with issues arising in actual data such as candidate predictors that are
categorical, heavily skewed or ostensibly are continuous but have a low number of unique observed values. | {"url":"https://cran.rstudio.com/web/packages/gamsel/vignettes/gamsel.html","timestamp":"2024-11-07T06:23:19Z","content_type":"text/html","content_length":"624454","record_id":"<urn:uuid:6af782e9-93b4-4a05-8e50-117aaf11d325>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00489.warc.gz"} |
An experiment was done to see the effects of diet and a drug on
Type II...
An experiment was done to see the effects of diet and a drug on Type II...
An experiment was done to see the effects of diet and a drug on Type II Diabetes in men in the 50’s. Men in the 50’s who were diagnosed with Type II Diabetes that was under moderate control were
randomly selected and randomly assigned to one of 4 treatment groups (8 per group). The groups were a Control group, those given a special diet, those given the drug, and those given the diet and the
drug. The control group was given the typical information about diet for diabetes. The subjects tested their glucose levels each morning and the average glucose level for a month was the response
variable. The data are given below (note: data are simulated).
• Summarize the glucose for each treatment. You may have to rearrange the data in order to get the summary statistics for each group. You can use Excel’s Data Analysis, Descriptive Statistics or you
can use the basic Stats Excel file using functions for the mean, median and so forth
. • Using Excel, we are going to run a dummy variable regression on the data. You are going to need to create the dummy variables. Note, there are four groups (Control group, those given a special
diet, those given the drug, and those given the diet and the drug). You decide how many dummy variables to create and which group is the reference category. Explain why you chose the reference
• For the Dummy Variable Regression, conduct an overall F-test for the treatment stating the following: Null Hypothesis Alternative Hypothesis Assumptions of the Test The Test Statistics F* The
Critical value and the p-value for the Tests Statistics The Conclusion of the Test
• Pick one of the individual coefficients and the t-test for that coefficient. Explain the meaning of the test statistic and the conclusion of the test based on the p-value (in other words, what is
the test saying and what is your conclusion)?
Treatment Glucose Level
Control 123
Control 149
Control 125
Control 102
Control 132
Control 128
Control 128
Control 84
Diet 94
Diet 72
Diet 71
Diet 87
Diet 85
Diet 110
Diet 102
Diet 98
Drug 77
Drug 64
Drug 103
Drug 94
Drug 102
Drug 93
Drug 94
Drug 120
Diet & Drug 97
Diet & Drug 94
Diet & Drug 113
Diet & Drug 65
Diet & Drug 76
Diet & Drug 121
Diet & Drug 93
Diet & Drug 101
H[0]: There is no significant effect of Druge and Diet on Glucose level.
H1: There is significant effect of Druge and Diet on Glucose level.
The R-code for given problem is as;(By treating treatments as "Control"=1,"Diet"=2,"Drug"=3,"Diet & Drug"=4)
And the output is as;
> summary(aov(y~x,d))
Df Sum Sq Mean Sq F value Pr(>F)
x 1 6.81 6.811 6.157 0.0189 *
Residuals 30 33.19 1.106
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Here F value is 6.157 and p-value is 0.0189 which is less than 0.05 . thus we reject H[0] at 5% l.o.s.
And conclude that There is significant effect of Druge and Diet on Glucose level. | {"url":"https://justaaa.com/statistics-and-probability/23477-an-experiment-was-done-to-see-the-effects-of-diet","timestamp":"2024-11-09T22:19:23Z","content_type":"text/html","content_length":"50566","record_id":"<urn:uuid:9702586e-2765-4cf8-af56-e871db83bb75>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00865.warc.gz"} |
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2.5.5 Packet Tracer - Configure Initial Switch Settings Answers - Exams Cisco
2.5.5 Packet Tracer – Configure Initial Switch Settings Instructor Version
Part 1: Verify the Default Switch Configuration
Part 2: Configure a Basic Switch Configuration
Part 3: Configure a MOTD Banner
Part 4: Save Configuration Files to NVRAM
Part 5: Configure
In this activity, you will perform basic switch configurations. You will secure access to the command-line interface (CLI) and console ports using encrypted and plain text passwords. You will also
learn how to configure messages for users logging into the switch. These banners are also used to warn unauthorized users that access is prohibited.
Part 1: Verify the Default Switch Configuration
Step 1: Enter privileged mode.
You can access all switch commands from privileged mode. However, because many of the privileged commands configure operating parameters, privileged access should be password-protected to prevent
unauthorized use.
The privileged EXEC command set includes those commands contained in user EXEC mode, as well as the configure command through which access to the remaining command modes are gained.
a. Click S1 and then the CLI tab. Press Enter
b. Enter privileged EXEC mode by entering the enable command:
Switch> enable
Notice that the prompt changed in the configuration to reflect privileged EXEC mode.
Step 2: Examine the current switch configuration.
a. Enter the show running-config command.
Switch# show running-config
b. Answer the following questions:
How many FastEthernet interfaces does the switch have? 24
How many Gigabit Ethernet interfaces does the switch have? 2
What is the range of values shown for the vty lines? 0 -15
Which command will display the current contents of non-volatile random-access memory (NVRAM)?
show startup-configuration
Why does the switch respond with startup-config is not present?
It displays this message because the configuration file was not saved to NVRAM. Currently it is only located in RAM.
Part 2: Create a Basic Switch Configuration
Step 1: Assign a name to a switch.
To configure parameters on a switch, you may be required to move between various configuration modes. Notice how the prompt changes as you navigate through the switch.
Switch# configure terminal
Switch(config)# hostname S1
S1(config)# exit
Step 2: Secure access to the console line.
To secure access to the console line, access config-line mode and set the console password to letmein.
S1# configure terminal
Enter configuration commands, one per line. End with CNTL/Z.
S1(config)# line console 0
S1(config-line)# password letmein
S1(config-line)# login
S1(config-line)# exit
S1(config)# exit
%SYS-5-CONFIG_I: Configured from console by console
Why is the login command required?
In order for the password checking process to work, it requires both the login and password commands
Step 3: Verify that console access is secured.
Exit privileged mode to verify that the console port password is in effect.
S1# exit
Switch con0 is now available
Press RETURN to get started.
User Access Verification
Note: If the switch did not prompt you for a password, then you did not configure the login parameter in Step 2.
Step 4: Secure privileged mode access.
Set the enable password to c1$c0. This password protects access to privileged mode.
Note: The 0 in c1$c0 is a zero, not a capital O. This password will not grade as correct until after you encrypt it in Step 8.
S1> enable
S1# configure terminal
S1(config)# enable password c1$c0
S1(config)# exit
%SYS-5-CONFIG_I: Configured from console by console
Step 5: Verify that privileged mode access is secure.
a. Enter the exit command again to log out of the switch.
b. Press and you will now be asked for a password:
User Access Verification
c. The first password is the console password you configured for line con 0. Enter this password to return to user EXEC mode.
d. Enter the command to access privileged mode.
e. Enter the second password you configured to protect privileged EXEC mode.
f. Verify your configurations by examining the contents of the running-configuration file:
S1# show running-configuration
Notice how the console and enable passwords are both in plain text. This could pose a security risk if someone is looking over your shoulder.
Step 6: Configure an encrypted password to secure access to privileged mode.
The enable password should be replaced with the newer encrypted secret password using the enable secret command. Set the enable secret password to itsasecret.
S1# config t
S1(config)# enable secret itsasecret
S1(config)# exit
Note: The enable secret password overrides the enable password. If both are configured on the switch, you must enter the enable secret password to enter privileged EXEC mode.
Step 7: Verify that the enable secret password is added to the configuration file.
a. Enter the show running-configuration command again to verify the new enable secret password is configured.
Note: You can abbreviate show running-configuration as
S1# show run
b. What is displayed for the enable secret password? $1$mERr$ILwq/b7kc.7X/ejA4Aosn0
c. Why is the enable secret password displayed differently from what we configured?
The enable secret is shown in encrypted form, whereas the enable password is in plain text.
Step 8: Encrypt the enable and console passwords.
As you noticed in Step 7, the enable secret password was encrypted, but the enable and console passwords were still in plain text. We will now encrypt these plain text passwords using the service
password-encryption command.
S1# config t
S1(config)# service password-encryption
S1(config)# exit
If you configure any more passwords on the switch, will they be displayed in the configuration file as plain text or in encrypted form? Explain why?
The service password-encryption command encrypts all current and future passwords.
Part 3: Configure a MOTD Banner
Step 1: Configure a message of the day (MOTD) banner.
The Cisco IOS command set includes a feature that allows you to configure messages that anyone logging onto the switch sees. These messages are called message of the day, or MOTD banners. Enclose the
banner text in quotations or use a delimiter different from any character appearing in the MOTD string.
S1# config t
S1(config)# banner motd "This is a secure system. Authorized Access Only!"
S1(config)# exit
%SYS-5-CONFIG_I: Configured from console by console
When will this banner be displayed?
The message will be displayed when someone enters the switch through the console port.
Why should every switch have a MOTD banner?
Every switch should have a banner to warn unauthorized users that access is prohibited but can also be used for sending messages to network personnel/technicians (such as impending system shutdowns
or who to contact for access)
Part 4: Save Configuration Files to NVRAM
Step 1: Verify that the configuration is accurate using the show run command.
Step 2: Save the configuration file.
You have completed the basic configuration of the switch. Now back up the running configuration file to NVRAM to ensure that the changes made are not lost if the system is rebooted or loses power.
S1# copy running-config startup-config
Destination filename [startup-config]?[Enter]
Building configuration...
What is the shortest, abbreviated version of the copy running-config startup-config command? cop r s
Step 3: Examine the startup configuration file.
Which command will display the contents of NVRAM? show startup-config
Are all the changes that were entered recorded in the file? Yes, it is the same as the running configuration.
Part 5: Configure S2
You have completed the configuration on S1. You will now configure S2. If you cannot remember the commands, refer to Parts 1 to 4 for assistance.
Configure S2 with the following parameters:
a. Name device: S2
b. Protect access to the console using the letmein password.
c. Configure an enable password of c1$c0 and an enable secret password of itsasecret.
d. Configure a message to those logging into the switch with the following message:
Authorized access only. Unauthorized access is prohibited and violators will be prosecuted to the full extent of the law.
e. Encrypt all plain text passwords.
f. Ensure that the configuration is correct.
g. Save the configuration file to avoid loss if the switch is powered down.
Switch> enable
Switch# config t
Enter configuration commands, one per line. End with CNTL/Z.
Switch(config)# hostname S2
S2(config)# line console 0
S2(config-line)# password letmein
S2(config-line)# login
S2(config-line)# enable password c1$c0
S2(config)# enable secret itsasecret
S2(config)# banner motd $any text here$
S2(config)# service password-encryption
S2(config)# do copy running-config startup-config
Switch 1 – S1
configure terminal
hostname S1
line console 0
password letmein
enable password c1$c0
enable secret itsasecret
service password-encryption
banner motd #This is a secure system. Authorized Access Only!#
copy running-config startup-config
Switch 2 – S2
configure terminal
hostname S2
line console 0
password letmein
enable password c1$c0
enable secret itsasecret
service password-encryption
banner motd #Authorized access only. Unauthorized access is prohibited and violators will be prosecuted to the full extent of the law.#
copy running-config startup-config
Download Pka file and PDF file | {"url":"https://examscisco.com/2-5-5-packet-tracer-configure-initial-switch-settings-answers/","timestamp":"2024-11-10T02:41:37Z","content_type":"text/html","content_length":"143593","record_id":"<urn:uuid:8eec4122-a5de-4caa-bbe1-476f185d6e41>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00176.warc.gz"} |
Stories for 'Probability' | MathFiction
Stories for 'Probability'
Sorted by date of original publication
• A far-future story about mankind trying to avert a cosmic calamity by harnessing the laws of probability…
• A classic story about brilliant professors using their esoteric knowledge to get the better of "The Wolves of Wall Street"
• An ultra-long range impact on history when the perfectly average man is found… Butterfly Effect in a sociological context…
• A mathematical huckster swindles his way to continued solvency through probability theory…. till he gets a taste of his own medicine…
• A story of a wealthy media mogul who goes up against a major drug ring, sprinkled with references to probability and related mathematical concepts.
• A story about a man who could never make a decision, but got into the business of making predictions for others.
• A woman becomes a central figure in the disruption of the laws of probability, with some cosmic consequences.
• A narrator compares the choices in relationships and other aspects of life with games like the Monty Hall Problem. | {"url":"https://www.mathfiction.net/tags/probability/","timestamp":"2024-11-02T07:37:26Z","content_type":"text/html","content_length":"31332","record_id":"<urn:uuid:93deb408-acca-45cf-956e-4f1fa27af1b8>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00617.warc.gz"} |
What counts as defection? — AI Alignment Forum
Thanks to Michael Dennis for proposing the formal definition; to Andrew Critch for pointing me in this direction; to Abram Demski for proposing non-negative weighting; and to Alex Appel, Scott
Emmons, Evan Hubinger, philh, Rohin Shah, and Carroll Wainwright for their feedback and ideas.
There's a good chance I'd like to publish this at some point as part of a larger work. However, I wanted to make the work available now, in case that doesn't happen soon.
They can't prove the conspiracy... But they could, if Steve runs his mouth.
The police chief stares at you.
You stare at the table. You'd agreed (sworn!) to stay quiet. You'd even studied game theory together. But, you hadn't understood what an extra year of jail meant.
The police chief stares at you.
Let Steve be the gullible idealist. You have a family waiting for you.
Sunlight stretches across the valley, dappling the grass and warming your bow. Your hand anxiously runs along the bowstring. A distant figure darts between trees, and your stomach rumbles. The
day is near spent.
The stags run strong and free in this land. Carla should meet you there. Shouldn't she? Who wants to live like a beggar, subsisting on scraps of lean rabbit meat?
In your mind's eye, you reach the stags, alone. You find one, and your arrow pierces its barrow. The beast shoots away; the rest of the herd follows. You slump against the tree, exhausted, and
never open your eyes again.
You can't risk it.
People talk about 'defection' in social dilemma games, from the prisoner's dilemma to stag hunt to chicken. In the tragedy of the commons, we talk about defection. The concept has become a regular
part of LessWrong discourse.
Informal definition. A player defects when they increase their personal payoff at the expense of the group.
This informal definition is no secret, being echoed from the ancient Formal Models of Dilemmas in Social Decision-Making to the recent Classifying games like the Prisoner's Dilemma:
you can model the "defect" action as "take some value for yourself, but destroy value in the process".
Given that the prisoner's dilemma is the bread and butter of game theory and of many parts of economics, evolutionary biology, and psychology, you might think that someone had already formalized
this. However, to my knowledge, no one has.
Consider a finite -player normal-form game, with player having pure action set and payoff function . Each player chooses a strategy (a distribution over ). Together, the strategies form a strategy
profile . is the strategy profile, excluding player 's strategy. A payoff profile contains the payoffs for all players under a given strategy profile.
A utility weighting is a set of non-negative weights (as in Harsanyi's utilitarian theorem). You can consider the weights as quantifying each player's contribution; they might represent a percieved
social agreement or be the explicit result of a bargaining process.
When all are equal, we'll call that an equal weighting. However, if there are "utility monsters", we can downweight them accordingly.
We're implicitly assuming that payoffs are comparable across players. We want to investigate: given a utility weighting, which actions are defections?
Definition. Player 's action is a defection against strategy profile and weighting if
2. Social loss:
If such an action exists for some player , strategy profile , and weighting, then we say that there is an opportunity for defection in the game.
Remark. For an equal weighting, condition (2) is equivalent to demanding that the action not be a Kaldor-Hicks improvement.
Payoff profiles in the Prisoner's Dilemma. Red arrows represent defections against pure strategy profiles; player 1 defects vertically, while player 2 defects horizontally. For example, player 2
defects with because they gain () but the weighted sum loses out ().
Our definition seems to make reasonable intuitive sense. In the tragedy of the commons, each player rationally increases their utility, while imposing negative externalities on the other players and
decreasing total utility. A spy might leak classified information, benefiting themselves and Russia but defecting against America.
Definition. Cooperation takes place when a strategy profile is maintained despite the opportunity for defection.
Theorem 1. In constant-sum games, there is no opportunity for defection against equal weightings.
Theorem 2. In common-payoff games (where all players share the same payoff function), there is no opportunity for defection.
Edit: In private communication, Joel Leibo points out that these two theorems formalize the intuition between the proverb "all's fair in love and war": you can't defect in fully competitive or fully
cooperative situations.
Proposition 3. There is no opportunity for defection against Nash equilibria.
An action is a Pareto improvement over strategy profile if, for all players ,.
Proposition 4. Pareto improvements are never defections.
Game Theorems
We can prove that formal defection exists in the trifecta of famous games. Feel free to skip proofs if you aren't interested.
In (a), variables stand for emptation, eward, unishment, and ucker. A symmetric game is a Prisoner's Dilemma when . Unsurprisingly, formal defection is everywhere in this game.
Theorem 5. In symmetric games, if the Prisoner's Dilemma inequality is satisfied, defection can exist against equal weightings.
Proof. Suppose the Prisoner's Dilemma inequality holds. Further suppose that . Then . Then since but , both players defect from with .
Suppose instead that . Then , so . But , so player 1 defects from with action , and player 2 defects from with action . QED.
A symmetric game is a Stag Hunt when . In Stag Hunts, due to uncertainty about whether the other player will hunt stag, players defect and fail to coordinate on the unique Pareto optimum . In (b),
player 2 will defect (play ) when . In Stag Hunts, formal defection can always occur against mixed strategy profiles, which lines up with defection in this game being due to uncertainty.
Theorem 6. In symmetric games, if the Stag Hunt inequality is satisfied, defection can exist against equal weightings.
Proof. Suppose that the Stag Hunt inequality is satisfied. Let be the probability that player 1 plays . We now show that player 2 can always defect against strategy profile for some value of .
For defection's first condition, we determine when :
This denominator is positive ( and ), as is the numerator. The fraction clearly falls in the open interval .
For defection's second condition, we determine when
Combining the two conditions, we have
Since , this holds for some nonempty subinterval of . QED.
A symmetric game is Chicken when . In (b), defection only occurs when : when player 1 is very likely to turn, player 2 is willing to trade a bit of total payoff for personal payoff.
Theorem 7. In symmetric games, if the Chicken inequality is satisfied, defection can exist against equal weightings.
Proof. Assume that the Chicken inequality is satisfied. This proof proceeds similarly as in theorem 6. Let be the probability that player 1's strategy places on .
For defection's first condition, we determine when :
The inequality flips in the first equation because of the division by , which is negative ( and ). , so ; this reflects the fact that is a Nash equilibrium, against which defection is impossible
(proposition 3).
For defection's second condition, we determine when
The inequality again flips because is negative. When , we have , in which case defection does not exist against a pure strategy profile.
Combining the two conditions, we have
Because ,
This bit of basic theory will hopefully allow for things like principled classification of policies: "has an agent learned a 'non-cooperative' policy in a multi-agent setting?". For example, the
empirical game-theoretic analyses (EGTA) of Leibo et al.'s Multi-agent Reinforcement Learning in Sequential Social Dilemmas say that apple-harvesting agents are defecting when they zap each other
with beams. Instead of using a qualitative metric, you could choose a desired non-zapping strategy profile, and then use EGTA to classify formal defections from that. This approach would still have a
free parameter, but it seems better.
I had vague pre-theoretic intuitions about 'defection', and now I feel more capable of reasoning about what is and isn't a defection. In particular, I'd been confused by the difference between
power-seeking and defection, and now I'm not.
This post's main contribution is the formalization of game-theoretic defection as gaining personal utility at the expense of coalitional utility.
Rereading, the post feels charmingly straightforward and self-contained. The formalization feels obvious in hindsight, but I remember being quite confused about the precise difference between
power-seeking and defection—perhaps because popular examples of taking over the world are also defections against the human/AI coalition. I now feel cleanly deconfused about this distinction. And if
I was confused about it, I'd bet a lot of other people were, too.
I think this post is valuable as a self-contained formal insight into the nature of defection. If I could vote on it, I'd give it a 4 (or perhaps a 3, if the voting system allowed it).
Planned summary for the Alignment Newsletter:
We often talk about cooperating and defecting in general-sum games. This post proposes that we say that a player P has defected against a coalition C (that includes P) currently playing a
strategy S when P deviates from the strategy S in a way that increases his or her own personal utility, but decreases the (weighted) average utility of the coalition. It shows that this
definition has several nice intuitive properties: it implies that defection cannot exist in common-payoff games, uniformly weighted constant-sum games, or arbitrary games with a Nash equilibrium
strategy. A Pareto improvement can also never be defection. It then goes on to show the opportunity for defection can exist in the Prisoner’s dilemma, Stag hunt, and Chicken (whether it exists
depends on the specific payoff matrices).
As others have mentioned, there's an interpersonal utility comparison problem. In general, it is hard to determine how to weight utility between people.
I actually don't think this is a problem for the use case I have in mind. I'm not trying to solve the comparison problem. This work formalizes: "given a utility weighting, what is defection?". I
don't make any claim as to what is "fair" / where that weighting should come from. I suppose in the EGTA example, you'd want to make sure eg reward functions are identical.
"Deliberately sub-Pareto move" I think is a pretty good description of the kind of "defection" that means you're being tatted, and "negligently sub-Pareto" is a good description of the kind of
tit to tat.
Defection doesn't always have to do with the Pareto frontier - look at PD, for example. , , are usually all Pareto optimal.
Yes, this is correct. For example, the following is an example of the second game:
I very much agree that interpersonal utility comparability is a strong assumption. I'll add a note.
Mentioned in
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As others have mentioned, there's an interpersonal utility comparison problem. In general, it is hard to determine how to weight utility between people. If I want to trade with you but you're not
home, I can leave some amount of potatoes for you and take some amount of your milk. At what ratio of potatoes to milk am I "cooperating" with you, and at what level am I a thieving defector? If
there's a market down the street that allows us to trade things for money then it's easy to do these comparisons and do Coasian payments as necessary to coordinate on maximizing the size of the pie.
If we're on a deserted island together it's harder. Trying to drive a hard bargain and ask for more milk for my potatoes is a qualitatively different thing when there's no agreed upon metric you can
use to say that I'm trying to "take more than I give".
Here is an interesting and hilarious experiment about how people play an iterated asymmetric prisoner's dilemma. The reason it wasn't more pure cooperation is that due to the asymmetry there was a
disagreement between the players about what was "fair". AA thought JW should let him hit "D" some fraction of the time to equalize the payouts, and JW thought that "C/C" was the right answer to
coordinate towards. If you read their comments, it's clear that AA thinks he's cooperating in the larger game, and that his "D" aren't anti-social at all. He's just trying to get a "fair" price for
his potatoes, and he's mistaken about what that is. JW, on the other hand, is explicitly trying use his Ds to coax A into cooperation. This conflict is better understood as a disagreement over where
on the Pareto frontier ("at which price") to trade than it is about whether it's better to cooperate with each other or defect.
In real life problems, it's usually not so obvious what options are properly thought of as "C" or "D", and when trying to play "tit for tat with forgiveness" we have to be able to figure out what
actually counts as a tit to tat. To do so, we need to look at the extent to which the person is trying to cooperate vs trying to get away with shirking their duty to cooperate. In this case, AA was
trying to cooperate, and so if JW could have talked to him and explained why C/C was the right cooperative solution, he might have been able to save the lossy Ds. If AA had just said "I think I can
get away with stealing more value by hitting D while he cooperates", no amount of explaining what the right concept of cooperation looks like will fix that, so defecting as punishment is needed.
In general, the way to determine whether someone is "trying to cooperate" vs "trying to defect" is to look at how they see the payoff matrix, and figure out whether they're putting in effort to stay
on the Pareto frontier or to go below it. If their choice shows that they are being diligent to give you as much as possible without giving up more themselves, then they may be trying to drive a hard
bargain, but at least you can tell that they're trying to bargain. If their chosen move is conspicuously below (their perception of) the Pareto frontier, then you can know that they're either
not-even-trying, or they're trying to make it clear that they're willing to harm themselves in order to harm you too.
In games like real life versions of "stag hunt", you don't want to punish people for not going stag hunting when it's obvious that no one else is going either and they're the one expending effort to
rally people to coordinate in the first place. But when someone would have been capable of nearly assuring cooperation if they did their part and took an acceptable risk when it looked like it was
going to work, then it makes sense to describe them as "defecting" when they're the one that doesn't show up to hunt the stag because they're off chasing rabbits.
"Deliberately sub-Pareto move" I think is a pretty good description of the kind of "defection" that means you're being tatted, and "negligently sub-Pareto" is a good description of the kind of tit to
Combining the two conditions, we have
Since , this holds for some nonempty subinterval of .
I want to check that I'm following this. Would it be fair to paraphrase the two parts of this inequality as:
1) If your credence that the other player is going to play Stag is high enough, you won't even be tempted to play Hare.
2) If your credence that the other player is going to play Hare is high enough, then it's not defection to play Hare yourself.
I guess the rightmost term could be zero or negative, right? (If the difference between T and P is greater than or equal to the difference between P and S.) In that case, the payoffs would be such
that there's no credence you could have that the other player will play Hare that would justify playing Hare yourself (or justify it as non-defection, that is).
So my claim #1 is always true, but claim #2 depends on the payoff values.
In other words, Stag Hunt could be subdivided into two games: one where the payoffs never justify playing Hare (as non-defection), and one where they sometimes do, depending on your credence that the
other player will play Stag.
It's worth being careful to acknowledge that this set of assumptions is far more limited than the game-theoretical underpinnings. Because it requires interpersonal utility summation, you can't
normalize in the same ways, and you need to do a LOT more work to show that any given situation fits this model. Most situations and policies don't even fit the more general individual-utility model,
and I suspect even fewer will fit this extension.
That said, I like having it formalized, and I look forward to the extension to multi-coalition situations. A spy can benefit Russia and the world more than they hurt the average US resident. | {"url":"https://www.alignmentforum.org/posts/8LEPDY36jBYpijrSw/formalizing-game-theoretic-defection","timestamp":"2024-11-10T11:25:13Z","content_type":"text/html","content_length":"1048971","record_id":"<urn:uuid:c8bf3882-d8ab-4aa3-9503-5cb9f2a748e8>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00195.warc.gz"} |
Optimal solution - (Abstract Linear Algebra I) - Vocab, Definition, Explanations | Fiveable
Optimal solution
from class:
Abstract Linear Algebra I
An optimal solution is the best possible outcome that satisfies all constraints in a mathematical model, especially in the context of linear programming. It represents the point at which a particular
objective function, such as maximizing profit or minimizing cost, reaches its highest or lowest value while adhering to given restrictions. Finding the optimal solution is crucial for decision-making
processes in various fields such as economics, engineering, and operations research.
congrats on reading the definition of optimal solution. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The optimal solution can be found using various methods such as the Simplex method or graphical representation when dealing with two-variable problems.
2. In linear programming, there may be multiple optimal solutions if the objective function is parallel to a constraint boundary within the feasible region.
3. An optimal solution is only valid if it lies within the feasible region, meaning it must comply with all constraints of the problem.
4. The process of identifying an optimal solution helps organizations maximize profits or minimize costs, making it a key component in operational efficiency.
5. Sensitivity analysis can be performed after finding an optimal solution to understand how changes in constraints or objective functions affect it.
Review Questions
• How can you determine if a solution is optimal in a linear programming problem?
□ To determine if a solution is optimal in a linear programming problem, you must evaluate whether it lies within the feasible region and satisfies all constraints. Then, check if the objective
function achieves its maximum or minimum value at that point compared to other potential solutions. Using methods like the Simplex algorithm can help systematically find and verify the
optimal solution.
• Discuss how multiple optimal solutions can occur in linear programming and what implications this has for decision-making.
□ Multiple optimal solutions can occur when the objective function is parallel to one of the constraints along the boundary of the feasible region. This indicates that there are several
combinations of variable values that yield the same optimal value for the objective function. For decision-making, this flexibility allows for various strategies to achieve goals, but it may
also require further analysis to determine which solution best aligns with other business objectives or constraints.
• Evaluate the role of sensitivity analysis after identifying an optimal solution in linear programming.
□ Sensitivity analysis plays a crucial role after identifying an optimal solution as it assesses how changes in parameters—such as coefficients in the objective function or alterations to
constraints—impact that solution. By evaluating these changes, decision-makers can understand potential risks and opportunities associated with their choices. This analysis aids in planning
and allows organizations to adapt their strategies based on varying conditions in their operational environment.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/abstract-linear-algebra-i/optimal-solution","timestamp":"2024-11-12T18:40:04Z","content_type":"text/html","content_length":"153375","record_id":"<urn:uuid:d37b9308-b044-434b-b060-11f125f3254f>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00526.warc.gz"} |
Point cloud outlier removal
Point cloud outlier removal¶
When collecting data from scanning devices, the resulting point cloud tends to contain noise and artifacts that one would like to remove. This tutorial addresses the outlier removal features of
Prepare input data¶
A point cloud is loaded and downsampled using voxel_downsample.
print("Load a ply point cloud, print it, and render it")
pcd = o3d.io.read_point_cloud("../../test_data/ICP/cloud_bin_2.pcd")
front=[0.4257, -0.2125, -0.8795],
lookat=[2.6172, 2.0475, 1.532],
up=[-0.0694, -0.9768, 0.2024])
print("Downsample the point cloud with a voxel of 0.02")
voxel_down_pcd = pcd.voxel_down_sample(voxel_size=0.02)
front=[0.4257, -0.2125, -0.8795],
lookat=[2.6172, 2.0475, 1.532],
up=[-0.0694, -0.9768, 0.2024])
Load a ply point cloud, print it, and render it
Downsample the point cloud with a voxel of 0.02
Alternatively, use uniform_down_sample to downsample the point cloud by collecting every n-th points.
print("Every 5th points are selected")
uni_down_pcd = pcd.uniform_down_sample(every_k_points=5)
front=[0.4257, -0.2125, -0.8795],
lookat=[2.6172, 2.0475, 1.532],
up=[-0.0694, -0.9768, 0.2024])
Every 5th points are selected
Select down sample¶
The following helper function uses select_by_index, which takes a binary mask to output only the selected points. The selected points and the non-selected points are visualized.
def display_inlier_outlier(cloud, ind):
inlier_cloud = cloud.select_by_index(ind)
outlier_cloud = cloud.select_by_index(ind, invert=True)
print("Showing outliers (red) and inliers (gray): ")
outlier_cloud.paint_uniform_color([1, 0, 0])
inlier_cloud.paint_uniform_color([0.8, 0.8, 0.8])
o3d.visualization.draw_geometries([inlier_cloud, outlier_cloud],
front=[0.4257, -0.2125, -0.8795],
lookat=[2.6172, 2.0475, 1.532],
up=[-0.0694, -0.9768, 0.2024])
Statistical outlier removal¶
statistical_outlier_removal removes points that are further away from their neighbors compared to the average for the point cloud. It takes two input parameters:
• nb_neighbors, which specifies how many neighbors are taken into account in order to calculate the average distance for a given point.
• std_ratio, which allows setting the threshold level based on the standard deviation of the average distances across the point cloud. The lower this number the more aggressive the filter will be.
print("Statistical oulier removal")
cl, ind = voxel_down_pcd.remove_statistical_outlier(nb_neighbors=20,
display_inlier_outlier(voxel_down_pcd, ind)
Statistical oulier removal
Showing outliers (red) and inliers (gray):
Radius outlier removal¶
radius_outlier_removal removes points that have few neighbors in a given sphere around them. Two parameters can be used to tune the filter to your data:
• nb_points, which lets you pick the minimum amount of points that the sphere should contain.
• radius, which defines the radius of the sphere that will be used for counting the neighbors.
print("Radius oulier removal")
cl, ind = voxel_down_pcd.remove_radius_outlier(nb_points=16, radius=0.05)
display_inlier_outlier(voxel_down_pcd, ind)
Radius oulier removal
Showing outliers (red) and inliers (gray): | {"url":"https://www.open3d.org/docs/latest/tutorial/Advanced/pointcloud_outlier_removal.html","timestamp":"2024-11-03T08:55:22Z","content_type":"text/html","content_length":"35021","record_id":"<urn:uuid:bf302d4b-6a43-4d88-9c43-6aa2afd5b456>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00071.warc.gz"} |
Optimal design of two-sided CUSUM schemes
Tan, Y. F.
Pooi, A. H.
Chong, F. S.
Optimal design of two-sided CUSUM schemes.
Malaysian Journal of Sciences, 24 (2). pp. 45-50. ISSN 1394-3065 Full text not available from this repository.
An optimal CUSUM chart is commonly defined as one with a fixed in control average run length (ARL) and the smallest ARL for a specified shift Δ in the mean. As the run length distribution is usually
positively skewed, median run length (MRL) is a more reasonable measure of location than the mean given by ARL. As such it is more reasonable to define the optimal CUSUM in terms of MRL. The optimal
CUSUM schemes in terms of ARL and MRL are obtained in the case when the observations have a normal or exponential distribution. It is found that the parameter (k, h) of the optimal CUSUM in terms of
ARL and that of the optimal CUSUM in terms of MRL are fairly different.
Downloads per month over past year | {"url":"https://shdl.mmu.edu.my/5046/","timestamp":"2024-11-03T06:11:52Z","content_type":"application/xhtml+xml","content_length":"24960","record_id":"<urn:uuid:4a8f0d09-2bb6-4f6e-8b0b-f975ee4945a0>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00657.warc.gz"} |
--- title: "Introduction" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Introduction} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ```{r, include = FALSE}
knitr::opts_chunk$set( collapse = TRUE, comment = "#>" ) ``` ## Background ### Spatial Point Patterns Spatial point patterns are the realization of spatial point process models. They are points that
represent some feature, in our case cells but in ecological studies can be trees or plants, in space. Our space is 2-dimensional with an x and y axis with arbitrary units. Each point in the spatial
point pattern can have marks - characteristics that describe information about the point. These marks can be things like whether it's positive or negative for a specific phenotype or tissue, perhaps
the size of the cell, or even the intensity of a fluorophore on the cells surface. The spatial point pattern can provide us with a great deal of information about a pathology when looking at cell
spatial contexture. In order to understand the cell contexture of a pathology, researchers can employ different spatial statistic methods that describe some feature of the spatial point pattern. A
simple descriptive statistic of a spatial point pattern is ***lambda*** which describes the _intensity_. A large *lambda* means that there are a large number of points in a given boundary, and a
small *lambda* means for the same boundary area there are less points. Another summary statistic is Ripley's *K(r)* which describes how many other points of interest are within *r* of an anchor cell.
This measurement can then be compared to different things like a complete spatial randomness (CSR) measurement to estimate how far the observed cell contexture deviates from random cell locations in
the sample. In addition to Ripley's *K(r)*, there are several other methods such as a nearest neighbor *G(r)* function and an interaction variable developed by [Steinhart et. al](https://doi.org/
10.1158/1541-7786.MCR-21-0411). ### Simulating In order to compare metrics for describing point patterns, we have developed this simulation package: `{scSpatialSIM}`. The main basis of the package is
using Gaussian kernels to assign different characteristics to the underlying spatial point pattern. A Gaussian kernel is described in 3 different ways in our package: kernel's center, standard
deviation in the x-direction and standard deviation in the y-direction. There are *k* kernel means or centers in the point pattern which each has these descriptors. With the kernels for a spatial
point pattern described and locations of points, the kernels can be used to give a probability to the points - closer to the center *k* the greater the probability for what the kernel is describing
and farther away from any *k* center the lower the probability. Having probabilities that range from 0 to 1 assumes kernel centers are 'hot' for the tissue/cell phenotype/hole, but in reality this is
not always the case. Due to the inherent noise that we see with tissues at the cell level, we have provided the ability to limit the kernel center probability and increase the background probability.
In addition to this making the simulations more realistic, this also allows for abundance control for cell phenotypes (lower max probability lower abundance). ## Using `scSpatialSIM` The process of
simulating cell type data was streamlined in a similar way to what one thinks of when thinking of real data. First, an S4 `SpatSimObj` object is initialized with all the slots that are needed to
proceed through the whole simulation process. Slots inside are for kernels that describe tissue, holes, cell phenotypes, the simulation window, and a tabular format of them. With the
`SpatialSimulationObject()` ready, the point patterns can be simulated (these point patterns act as a 'master' for the next parts). The point pattern as it is is similar to an extracellular matrix
that just has places for cells to go but no identity for the cells. Thinking more broadly, we need to create tissue regions for our images which is designated as either tissue 1 and tissue 2 (can be
interpreted as tumor or stroma). Separately, cell phenotypes can be assigned to each cell. This is all done with separate kernels for each step which allows for high customization and fine tuning of
the simulated samples and the spatial contexture. If parameters are changed from the initialized values, then they will be stored in the object so in the future they can be referred to again. Spatial
statistics are usually interested in the amount of clustering points with a single mark type have. For example, it has been shown that [high abundance but low clustering of cytotoxic T cells is
associated with better overall survival than missing cytotoxic T cells in high grade serous ovarian cancer tumors](https://doi.org/10.1371/journal.pcbi.1009900). There are also instances that
colocalization of 2 difference cell types might be of interest such as T cell and B cells in tumors. With `{scSpatialSIM}`, we provide methods to perform both of these spatial point patterns. Input
parameters dictate the layout of a first cell type. In the case where more than one cell type is wanted, a shift value is taken which uses the same kernel as the first cell type (`shift = 0`) for
strong colocalization or moved the first cell types' kernel towards Dirichlet vertices (`shift = 1`) for segregation. ### Univariate Simulation In order to get started simulating cell type samples,
we need to import the package. ```{r setup} library(scSpatialSIM) set.seed(333) #reproducibility ``` To create a simulation object, we can call `CreateSimulationObject()`, which takes in 3 arguments
that will initialize the object for all downstream functions: 1. `window` - a `spatstat` owin object which is the boundary for which to simulate our points. The window acts as a mask so the functions
know where to simulate points later on. 2. `sims` - an integer value that is the number of samples that is wanting to be simulated 3. `cell_types` - this is the number of different phenotypes to
simulate, here we are only going to do a single cell type If a window isn't specified a built in 10x10 unit window will be used. To make a custom window, we an use `spatstat.geom::owin` ```{r}
custom_window = spatstat.geom::owin(xrange = c(0, 10), yrange = c(0, 10)) ``` Let's create our simulation object. ```{r} sim_object = CreateSimulationObject(sims = 9, cell_types = 1, window =
custom_window) ``` The simulation object has some attributes that are gradually filled in as we progress. Using the `summary()` method, we can see what our simulation object has inside and what we
need to do next. Here, we have 9 simulations with the default window and are wanting to perform this for a single cell type. This method is really useful when fine tuning parameters to get the
desired output because it shows you exactly what has already been done when loading in the base object from an RDS file. ```{r} summary(sim_object) ``` Now that we have our simulation object and
parameters set within, we need to create the point pattern. The `GenerateSpatialPattern()` function takes in the spatial simulation object and a `lambda`, or intensity of the point pattern. **NOTE:
with large windows and large lambdas, the size of the spatial simulation object can grow fast so be mindful.** Any other parameters that could be passed to `spatstat.random::rpoispp` can then be
passed to `GenerateSpatialPattern()` at the end. Again, we can check how we are filling in our slots with `summary()` and even see how our new process looks like with `plot()`. ```{r} sim_object =
GenerateSpatialPattern(sim_object) summary(sim_object) plot(sim_object, what = "Patterns", ncol = 1, nrow = 1, which = 1)#print only first point pattern ``` Next we will generate some regions of
different tissue. For the purpose here, tissue 1 will be tumor and tissue 2 will be stroma. There are some parameters that are initialized when building the spatial simulation that tells downstream
functions how to simulate things on a Gaussian kernel - `GenerateTissue()` is one of those. If there is nothing supplied to the function when calling, it will default to these values. Alternatively,
can specify them in the function call. These include regions within your window that you would like tissue regions to be simulated, the number of regions, the standard deviation range for how the
probability falls off around the region centers, etc. Something here is whether the ending kernel should be converted to a heatmap and at what resolution. **NOTE: the smaller the `step_size` used for
the heatmap the longer it takes to run so be mindful.** After simulating the tissues we can look at the summary of the spatial simulation object to see the newly filled slot. ```{r} sim_object =
GenerateTissue(sim_object, density_heatmap = T, step_size = 0.1, cores = 1) summary(sim_object) ``` The tissue kernel slot is now filled the number of kernels matching number of spatial processes we
have. Let's take a look at the simulated tissue kernels with `PlotSimulation()`. If using negative ranges in the window, the function will use the overlap between default `GenerateTissue()` range and
the window. See documentation for `GenerateTissue()` for more details. ```{r, fig.height = 10, fig.width = 9, eval = T} PlotSimulation(sim_object, which = 1:4, ncol = 2, nrow = 2, what = "tissue
heatmap") ``` The kernels are randomly laid about the simulation region and random sizes within the constraints of the parameters provided. The number of tissue regions simulated are fixed by the
input `k` value, and not sampled from a distribution centered at `k`. A larger `sdmin` and `sdmax` would increase the sizes of the tissue regions. See `GenerateTissue()` for more information.
Generating holes can help assess the need for correcting metrics derived spatial statistics. For example, if there is a pond in the center of a field, crops are not able to be planted there yet
measuring the amount of field there is just by the outside border will say that lots of crops should be able to fit. Sometimes, these things need to be adjusted for such as when you have tissue
sections that are being stained and the second folds over or tears, leaving a large area where no longer cells are present. Even if not using holes for assessing metrics, performing them will just
create a new column in the spatial files that can be later ignored. If holes are not needed at all, this can be skipped. The parameters that go into `GenerateHoles()` is similar to that of
`GenerateTissue()` with one addition: `hole_prob` or the proportion range of the point patterns that could become holes. All of these parameters had defaults set when the spatial simulation object
was created, but can be overridden here is a particular area of the process is wanted to have the holes and not the rest. The number of holes is random as long as the sum of the area is within the
proportions above. ```{r, fig.height = 10, fig.width = 9} sim_object = GenerateHoles(sim_object, density_heatmap = T, step_size = 0.1, cores = 1) summary(sim_object) ``` Let's see how the kernels for
holes look. The center of the holes have the highest probability of being removed with a Bernoulli distribution of 'hole' or 'not a hole'. ```{r, fig.height = 10, fig.width = 9, eval = T}
PlotSimulation(sim_object, which = 1:8, ncol = 2, nrow = 2, what = "hole heatmap") ``` Next thing for us to do is to simulate the positivity of cells for a phenotype using `GenerateCellPositivity()`.
Just like holes and the tissue, there are boundaries which the simulated positive cells will fall in which are stored in the parameters. This is helpful for going back and seeing what was done.
However here, there are 2 different parameters that help set the abundance (`probs`) and, in the case of multiple cell types, how related those cell types are (`shift`). The `probs` parameter is used
to scale the probabilities for the cell type where the first number is away from kernel peaks and the second is the maximum probability for a cell. For higher abundance, the maximum probability can
be set higher and even increase the minimum probability. An issue with doing this on point patterns that have multiple cell types is that they are not entirely informed of one another when assigning
cell types. B cells and T cells are distinct phenotypes and therefore one cell *shouldn't* be positive for both. What we do, is use the probability with the Bernoulli distribution so there's a chance
a single cell will be positive for 2 cell types, even if the max probability is 0.1. For univariate clustering, smaller standard deviations and higher probability range will increase the amount of
clustering. Large standard deviations and low probabilities will make the clustering metrics low. For univariate cell simulation, the `shift` metric does nothing. The similarity of 2 or more cell
types is controlled by the `shift` value. A `shift = 0` will use the same kernel used for probabilities of Cell Type 1 for the other cell types, and `shift = 1` will move Cell Type 1 kernel so that
cells are segregated (with noise) to Dirichlet vertices when 3 or more kernel centers are used. ```{r} sim_object = GenerateCellPositivity(sim_object, k = 4, sdmin = 3, sdmax = 5, density_heatmap =
T, step_size = 0.1, cores = 1, probs = c(0.0, 0.1), shift = 1) summary(sim_object) ``` If we plot the simulation object now specifying we want to see the whole core, we can see our cells. There are
little pockets of positive cells also with some noise. ```{r, fig.height=6, fig.width=10} PlotSimulation(sim_object, which = 1, what = "whole core") ``` ### Bivariate Simulation The process for 2 or
more cell types is essentially the same as for a single cell type with the addition of the `shift` value when simulating the cell types. Lets create another `SpatSimObj` to get to the phenotype
simulating step with `GenerateCellPositivity()`. A great feature of `{scSpatialSIM}` is that it works nicely with `{magrittr}` and the pipe function to immediately pass output from one function to
the next. ```{r} #set seed set.seed(333) #create the new object bivariate_sim = CreateSimulationObject(sims = 5, cell_types = 2) %>% #produce the point pattern GenerateSpatialPattern() %>% #make
tissues GenerateTissue(density_heatmap = T, step_size = 0.1, cores = 1) ``` As previously mentioned, we can specify a `shift` values of 0 to use the same kernel for both cell types which will assign
them in such a way that will be identifiable as colocalized between Cell Type 1 and Cell Type 2. First, the low shift: ```{r, fig.height=6, fig.width=10} bivariate_sim_tmp = GenerateCellPositivity
(bivariate_sim, k = 4, sdmin = 3, sdmax = 5, density_heatmap = T, step_size = 0.1, cores = 1, probs = c(0.0, 0.1), shift = 0) PlotSimulation(bivariate_sim_tmp, which = 1, what = "whole core") ``` We
can see that in locations where Cell Type 1 is, Cell Type 2 is also present. There are a couple cells that are assigned as positive for both Cell Type 1 and Cell Type 2 which should be taken into
consideration for next steps. If looking at colocalization of mutually exclusive cell types, these should be removed. How does this compare with strong segregation between Cell Type 1 and Cell Type
2? ```{r, fig.height=6, fig.width=10} bivariate_sim_tmp = GenerateCellPositivity(bivariate_sim, k = 4, sdmin = 3, sdmax = 5, density_heatmap = T, step_size = 0.1, cores = 1, probs = c(0.0, 0.1),
shift = 1) PlotSimulation(bivariate_sim_tmp, which = 1, what = "whole core") ``` In Tissue 2 it's easier to see regions of only Cell Type 1 and regions of only Cell Type 2. ## Exporting Data Lastly,
the data is likely wanted in tabular format. Since there are multiple simulated point patterns, the function `CreateSpatialList()` returns a list of data frames containing cell x and y locations, the
tissue that the cell belongs to, whether the cell falls in a hole, and the positivity of the different cell types. ```{r} spatial_list = CreateSpatialList(sim_object = bivariate_sim_tmp) head
(spatial_list[[1]]) ``` Alternatively, there is the ability with `CreateSpatialList()` to export all of the spatial pattern data as a single data frame using `single_df` set to `TRUE`. This adds an
`Image Name` column that keeps the spatial pattern data separate, and can split back into a list later for use with `SummariseSpatial()`, which creates a core level counts for the different cell
types. ```{r} single_dataframe = CreateSpatialList(sim_object = bivariate_sim_tmp, single_df = TRUE) head(single_dataframe) ``` ```{r} summary_data = SummariseSpatial(spatial_list = spatial_list,
markers = c("Cell 1 Assignment", "Cell 2 Assignment")) head(summary_data) ``` The spatial list and summary table can now be used with packages like `{spatialTIME}` to compute spatial statistics on
all of the simulated data frames. | {"url":"https://cran.hafro.is/web/packages/scSpatialSIM/vignettes/a01_Introduction.Rmd","timestamp":"2024-11-03T17:05:47Z","content_type":"text/plain","content_length":"17579","record_id":"<urn:uuid:54c6c3ef-38ef-40bd-b077-33f01b181904>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00452.warc.gz"} |
Finding the Domain of Piecewise-Defined Rational Functions
Question Video: Finding the Domain of Piecewise-Defined Rational Functions Mathematics • Second Year of Secondary School
If π and π are two real functions where π (π ₯) = 2π ₯ + 2 if π ₯ < β 3, π (π ₯) = π ₯ β 4 if β 3β € π ₯ < 0, and π (π ₯) = 5π ₯ determine the domain of the function (π /π
Video Transcript
If π and π are two real functions where π of π ₯ is the piecewise function defined as two π ₯ plus two if π ₯ is less than negative three and π ₯ minus four if π ₯ is greater than or
equal to negative three and less than zero and π of π ₯ is equal to five π ₯, determine the domain of the function π over π .
We begin by recalling π over π of π ₯ is the quotient of our two functions. Itβ s π of π ₯ over π of π ₯. So, letβ s begin by defining π over π of π ₯. Well, since π is a
piecewise function, weβ ll need to define π over π ₯ as shown. Itβ s five π ₯ over two π ₯ plus two if π ₯ is less than negative three and five π ₯ over π ₯ minus four if π ₯ is greater
than or equal to negative three and less than zero. Now, when weβ re dealing with the quotient of two functions, its domain is defined as the intersection or the overlap of the domain of the two
respective functions.
Well, the domain of a polynomial, unless otherwise defined, is all real numbers. So, the domain of π of π ₯ is the set of real numbers. If we look at both parts of our function, we see that π ₯
can take values from negative β all the way up to zero. So, the domain of π , our piecewise function, is the open interval negative β to zero. The intersection or the overlap of the set of real
numbers and numbers in the open interval from negative β to zero is numbers in the open interval negative β to zero. So, we can assume that this is the domain of our function π over π of π
However, we do need to be really careful. Weβ re working with a quotient. So, we need to make sure the denominator is not equal to zero in either case. In other words, we need to ensure that two π
₯ plus two is not equal to zero and π ₯ minus four is not equal to zero. If we solve our first inequation, we find π ₯ cannot be equal to negative one. This is irrelevant though because we were
told that this part of our piecewise function only applies when π ₯ is less than negative three. And so, we donβ t need to worry about including this in our domain. If we solve the second equation,
we find π ₯ cannot be equal to four. Well, this is outside of our domain and this part of our piecewise function, so we disregard this bit of information too.
And so, the domain of the function π over π is the set of numbers in the open interval from negative β to zero. | {"url":"https://www.nagwa.com/en/videos/970158681901/","timestamp":"2024-11-13T21:49:47Z","content_type":"text/html","content_length":"251558","record_id":"<urn:uuid:67f37e3b-a746-49db-a344-219991defa9a>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00664.warc.gz"} |
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Séminaire – Histoire et philosophie de la physique - Laboratoire SPHERE - UMR 7219
Séminaire – Histoire et philosophie de la physique
octobre 16 @ 16h00 - 18h30
• Marco Giovanelli (Université de Turin)
« Appearance and reality: Einstein, Ehrenfest and the early debate on the reality of length contraction »
In 1909, Ehrenfest published a note in the Physikalische Zeitschrift showing that a Born-rigid cylinder could not be set into rotation without stresses, as elements of the circumference would be
contracted but not the radius. Ignatowski and Varićak challenged Ehrenfest’s result in the same journal, arguing that the stresses would emerge if length contraction were a real dynamical effect, as
in Lorentz’s theory. However, no stresses are expected to arise, according to Einstein’s theory, where length contraction is only an apparent effect due to an arbitrary choice of clock
synchronization. Ehrenfest and Einstein considered this line of reasoning dangerously misleading and took a public stance in the Physikalische Zeitschrift, countering that relativistic length
contraction is both apparent and real. It is apparent since it disappears for the comoving observer, but it is also real since it can be experimentally verified. By drawing on his lesser-known
private correspondence with Varićak, this paper shows how Einstein used the Ehrenfest paradox as a tool for an ‘Einsteinian pedagogy.’ Einstein’s argumentative stance is contrasted with Bell’s use of
the Dewan-Beranthread-between-spaceships paradox to advocate for a ‘Lorentzian pedagogy.’ The paper concludes that the disagreement between the two ways of ‘teaching special relativity’ stems from
divergent interpretations of philosophical categories such as reality and appearance.
• Dennis Lehmkuhl (Université de Bonn)
« Einstein’s six paths to the metric tensor – and why he interpreted it differently than you do »
John Stachel, the first editor of The collected papers of Albert Einstein and the founder of what is today called Einstein scholarship, divides the creation of the general theory of relativity (GR)
into a drama of three acts. The first act centers around 1907, when Einstein was overwhelmed by the epiphany of the equivalence principle, the idea that the force of gravity and the inertia of bodies
were intimately connected. The second act takes place around 1912, when Einstein entered the promised land and proceeded from scalar theories of gravity to those based on a metric tensor. And the
third act finishes in late November 1915, when Einstein found what we now call the Einstein field equations, the successors of Newton’s law of gravity. Stachel further argued that the « missing
link » between the second and the third act was Einstein’s so-called rotating disc argument, which allowed him to forge a connection between gravity-inertia and non-Euclidean geometry. In this talk,
I shall argue that instead of being the protagonist in a drama in which the rotating disc argument is the one eureka moment that allowed the transition to a metric theory of gravity, Einstein, in the
summer and autumn of 1912, was an adventurer walking on six different paths in parallel, all of which led him to the program of finding a theory of gravity based on a metric tensor. And yet, I shall
argue, it is Einstein’s starting point, his scalar theory of gravity of early 1912, that, together with his equivalence principle, pointed him to these six paths, and determined the way he eventually
saw the metric tensor. In particular, I shall argue that Einstein’s work on a scalar theory of gravity, and his multi-path journey from there to the metric tensor, equipped him with many of the
interpretational moves and tools that would influence his later interpretation of GR, and made him resist seeing GR as a « reduction of gravity to spacetime geometry ». I shall decipher how Einstein
saw the role of geometry in GR instead, what he himself meant by « geometry », and how his notion of geometry differed from his contemporaries and successors. I shall outline how all this led him to
an interpretation of GR that saw the distinction of matter and spacetime geometry as something to be overcome rather than as something to be celebrated. | {"url":"https://sphere.cnrs.fr/event/seminaire-histoire-et-philosophie-de-la-physique-7/","timestamp":"2024-11-03T12:28:21Z","content_type":"text/html","content_length":"72380","record_id":"<urn:uuid:0fb86c1e-5b8b-4355-8a2a-9c6cc9e15e73>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00577.warc.gz"} |
What is a fitted value in R?
A fitted value in R is a predicted value of the dependent variable based on a statistical model fitted to the independent variable(s). It's calculated using the regression equation that describes the
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Error when using multiple SUMIFS in one column
Error when using multiple SUMIFS in one column
Hi .
I am trying to find the sum of each group of 5 rows using SUMIFS in Column 1 where the employee's name is John Smith (in cell 3 in Employee Name's column) and the status is Full-Time (in the Status
=SUMIFS([Column1]:[Column 1], Employee Name: Employee Name, Employee Name3, Status:Status, "Full Time")
The first time I used the formula. it worked.
However, when I used the same formula with a different employee's name for the second group of 5 rows, I got the error #CIRCULAR in the first sum and "BLOCKED" in the second one.
Any help bout this error can be fixed is highly appreciated.
Best Answers
• Probably should share a screenshot of your data structure, as there may be a cleaner way to do what you want.
However, right off the bat I can tell you that you need square brackets around column names that are more than one word, and no space after the colon in a range:
=SUMIFS([Column1]:[Column 1], [Employee Name]:[Employee Name], [Employee Name]3, Status:Status, "Full Time")
Jeff Reisman
Link: Smartsheet Functions Help Pages Link: Smartsheet Formula Error Messages
If my answer helped solve your issue, please mark it as accepted so that other users can find it later. Thanks!
• Ah, I see your issue. You have a formula in a column that references itself. That's a no go. So once one cell goes to an error (#Circular reference), the others go to #Blocked because they can't
evaluate the circular reference cell.
Since you are using row hierarchies, this is easy to overcome.
Put this formula in the $$ column on the parent row for each employee:
=SUMIFS(CHILDREN(), CHILDREN(Status@row), "Full Time")
English: Sum the child cells of this $$ parent row where child rows have a Status equal to "Full Time"
This will only consider child rows of the parent row for John Smith. When you copy this formula to the Parent row for Person1 Person1, it will only consider child rows of Person1 Person1.
Jeff Reisman
Link: Smartsheet Functions Help Pages Link: Smartsheet Formula Error Messages
If my answer helped solve your issue, please mark it as accepted so that other users can find it later. Thanks!
• Just for your edification, the first formula I gave would work (with a little tweak,) if it was placed in its own column:
Add a column called Total or something like that. On the rows where you want the total for that Employee (like is in your screenshots, just in the new Total column,) use the following:
=SUMIFS([$$]:[$$], [Employee Name]:[Employee Name], [Employee Name]@row, Status:Status, "Full Time")
The @row lets your formula work on any row without needing a specific row reference. So you can just Ctrl-C/Ctrl-V it to whatever row and it will still work, as long as the Employee Name on that
row is the one you want to sum for.
Jeff Reisman
Link: Smartsheet Functions Help Pages Link: Smartsheet Formula Error Messages
If my answer helped solve your issue, please mark it as accepted so that other users can find it later. Thanks!
• Probably should share a screenshot of your data structure, as there may be a cleaner way to do what you want.
However, right off the bat I can tell you that you need square brackets around column names that are more than one word, and no space after the colon in a range:
=SUMIFS([Column1]:[Column 1], [Employee Name]:[Employee Name], [Employee Name]3, Status:Status, "Full Time")
Jeff Reisman
Link: Smartsheet Functions Help Pages Link: Smartsheet Formula Error Messages
If my answer helped solve your issue, please mark it as accepted so that other users can find it later. Thanks!
• Thank you @Jeff Reisman for the feedback. I omitted the brackets earlier when I copied the formula. Here are 2 screenshots of the error. I have a big sheet with long columns. This is why I chose
to use SUMIFS.
• Ah, I see your issue. You have a formula in a column that references itself. That's a no go. So once one cell goes to an error (#Circular reference), the others go to #Blocked because they can't
evaluate the circular reference cell.
Since you are using row hierarchies, this is easy to overcome.
Put this formula in the $$ column on the parent row for each employee:
=SUMIFS(CHILDREN(), CHILDREN(Status@row), "Full Time")
English: Sum the child cells of this $$ parent row where child rows have a Status equal to "Full Time"
This will only consider child rows of the parent row for John Smith. When you copy this formula to the Parent row for Person1 Person1, it will only consider child rows of Person1 Person1.
Jeff Reisman
Link: Smartsheet Functions Help Pages Link: Smartsheet Formula Error Messages
If my answer helped solve your issue, please mark it as accepted so that other users can find it later. Thanks!
• Just for your edification, the first formula I gave would work (with a little tweak,) if it was placed in its own column:
Add a column called Total or something like that. On the rows where you want the total for that Employee (like is in your screenshots, just in the new Total column,) use the following:
=SUMIFS([$$]:[$$], [Employee Name]:[Employee Name], [Employee Name]@row, Status:Status, "Full Time")
The @row lets your formula work on any row without needing a specific row reference. So you can just Ctrl-C/Ctrl-V it to whatever row and it will still work, as long as the Employee Name on that
row is the one you want to sum for.
Jeff Reisman
Link: Smartsheet Functions Help Pages Link: Smartsheet Formula Error Messages
If my answer helped solve your issue, please mark it as accepted so that other users can find it later. Thanks!
• Thank you so much @Jeff Reisman for the suggestions. I truly appreciate your help! Thank you!
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/89367/error-when-using-multiple-sumifs-in-one-column","timestamp":"2024-11-06T19:06:43Z","content_type":"text/html","content_length":"430564","record_id":"<urn:uuid:b799581a-aaff-4354-9678-41018fb60aad>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00883.warc.gz"} |
Phase portraits of two gene networks models
Phase portraits of two gene networks models
Keywords: gene networks models, non-linear dynamical systems, phase portrait, equilibrium point, stability, Vyshnegradskii criterion
We construct mathematical models of functioning of two few-components gene networks which regulate circadian rhythmes in organisms by means of combinations of positive and negative feedbacks between
components of these networks. Both models are represented in the form of non-linear dynamical systems of biochemical kinetics. It is shown that the phase portraits of both models contain exactly one
equilibrium point each and in both cases for all values of parameters of these dynamical systems, eigenvalues of their linearization matrices at their equilibrium points are either negative or have
negative real parts. Thus, these equilibrium points are stable. We construct their invariant neighborhoods and describe the behavior of trajectories of these systems. Biological interpretations of
these results are given as well.
[1] Banks H. T. and Mahaffy J. M., “Stability of cyclic gene models for systems involving repression,” J. Theor. Biology, 74, 323–334 (1978).
[2] Bass J., “Circadian topology of metabolism,” Nature, 491, No. 7424, 348–356 (2012).
[3] Podkolodnaya О. А., “Molecular genetic aspects of the interaction of the circadian clocks and metabolism of energy substrates of mammals [in Russian],” Genetics, 50, No. 2, 1–13 (2014).
[4] Likhoshvai V., Golubyatnikov V., Demidenko G., Evdokimov A., and Fadeev S., “Gene networks theory,” in: Computational Systems Biology [in Russian], pp. 395–480, Izdat. SO RAN, Novosibirsk (2008).
[5] Bukharina T. A., Golubyatnikov V. P., Kazantsev M. V., Kirillova N. E., and Furman D. P., “Mathematical and numerical models of two asymmetric gene networks,” Sib. Electron. Math. Rep., 15,
1271–1283 (2018).
[6] Golubyatnikov V. P., Mjolsness E., and Gaidov Yu. A., “Topological index of a model of p53 −Mdm2 circuit,” Inform. Vestn. Vavilov. Obshch. Genetikov i Seleksts., 13, No. 1, 160–162 (2009).
[7] Chumakov G. A. and Chumakova N. A., “Homoclinic cycles in one gene network model [in Russian],” Mat. Zamet. SVFU, 21, No. 4, 97–106 (2014).
[8] Ayupova N. B. and Golubyatnikov V. P., “A three-cells model of the initial stage of development of one proneural cluster,” J. Appl. Ind. Math., 11, No. 2, 1–7 (2017).
[9] Golubyatnikov V. P. and Gradov V. S., “Non-uniqueness of cycles in piecewise-linear models of circular gene networks,” Sib. Adv. Math., 31, No. 1, 1–12 (2021).
[10] Vyshnegradskiy I. A. “ On regulators of direct action [in Russian],” Izv. Tekhnolog. Inst., Imper. Akad. Nauk, Saint-Petersburg, 21–62 (1877).
[11] Postnikov M. M., Stable Polynomials [in Russian], URSS, Moscow (2004).
[12] Smale S., “A mathematical model of two cells via Turing’s equation,” in: Lecture Notes in Applied Mathematics, vol. 16, pp. 15–26, Amer. Math. Soc., Providence, RI (1974).
[13] Akinshin A. A., Bukharina T. A., Golubyatnikov V. P., and Furman D. P., “Mathematical modeling of the interaction of two cells in the proneural cluster of the wing imaginal disc D. melanogaster
[in Russian],” Sib. Zhurn. Chistoy Prikl. Mat., 14, No. 4, 3–10 (2014).
[14] Ayupova N. B. and Golubyatnikov V. P., “The structure of the phase portrait of one piecewise linear dynamic system,” J. Appl. Ind. Math., 13, No. 4, 1–8 (2019).
How to Cite
Golubyatnikov, V. and Kirillova, N. (2021) “Phase portraits of two gene networks models”, Mathematical notes of NEFU, 28(1), pp. 3-11. doi: https://doi.org/10.25587/SVFU.2021.68.70.001. | {"url":"https://mzsvfu.ru/index.php/mz/article/view/phase-portraits-of-two-gene-networks-models","timestamp":"2024-11-13T17:29:28Z","content_type":"text/html","content_length":"30400","record_id":"<urn:uuid:86c11fd7-02db-42f3-a09f-ec772dafc088>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00127.warc.gz"} |
Condensed Matter Theory Seminar - Maissam Barkeshli - “New topological invariants and quantized response of crystalline topological phases of matter”
Event time:
Thursday, November 30, 2023 - 1:00pm to 2:00pm
Sloane Physics Laboratory (SPL), Room 51
217 Prospect Street
New Haven
CT 06511
Event description:
In the presence of crystalline symmetry, gapped quantum many-body states of matter can acquire a number of new topological invariants. This raises a question of how to mathematically classify the
invariants and their allowed values, how to extract the invariants from microscopic models, and how they encode quantized responses of the system. I will describe our recent advances in this area,
which include a comprehensive understanding of these questions for the case where the symmetry group consists of U(1) charge conservation and orientation-preserving space group symmetries in two
spatial dimensions. Some important results include (1) an understanding of quantized electric polarization for Chern insulators with crystalline symmetry, (2) a method of obtaining a complete set of
crystalline invariants from partial rotations, and (3) an understanding of an invariant, referred to as the“discrete shift,” which depends on a choice of high symmetry point and encodes the charge
response of the system to lattice disclinations. As an application, these results allow a new way of coloring Hofstadter’s famous butterfly with new topological invariants for the first time since
TKNN’s 1982 work on the quantized Hall conductance. | {"url":"https://ycaa.yale.edu/event/condensed-matter-theory-seminar-maissam-barkeshli-new-topological-invariants-and-quantized","timestamp":"2024-11-09T10:18:46Z","content_type":"text/html","content_length":"29303","record_id":"<urn:uuid:0bed8c75-ac35-47e3-a9e3-b96aeff0b7d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00853.warc.gz"} |
OmniInput: A Model-centric Evaluation Framework through Output Distribution (2024)
Weitang Liu
Department of Computer Science Engineering
University of California, San Diego
La Jolla, CA 92093
&Ying Wai Li
Los Alamos National Laboratory
Los Alamos, NM 87545
&Tianle Wang
Department of Computer Science Engineering
University of California, San Diego
La Jolla, CA 92093
&Yi-Zhuang You
Department of Physics
University of California, San Diego
La Jolla, CA 92093
&Jingbo Shang
Department of Computer Science Engineering
University of California, San Diego
La Jolla, CA 92093
We propose a novel model-centric evaluation framework, OmniInput, to evaluate the quality of an AI/ML model’s predictions on all possible inputs (including human-unrecognizable ones), which is
crucial for AI safety and reliability.Unlike traditional data-centric evaluation based on pre-defined test sets, the test set in OmniInput is self-constructed by the model itself and the model
quality is evaluated by investigating its output distribution. We employ an efficient sampler to obtain representative inputs and the output distribution of the trained model, which, after selective
annotation, can be used to estimate the model’s precision and recall at different output values and a comprehensive precision-recall curve.Our experiments demonstrate that OmniInput enables a more
fine-grained comparison between models, especially when their performance is almost the same on pre-defined datasets, leading to new findings and insights for how to train more robust, generalizable
1 Introduction
A safe, reliable AI/ML model deployed in real world should be able to make reasonable predictions on all the possible inputs, including uninformative ones.For instance, an autonomous vehicle image
processing system might encounter carefully designed backdoor attack patterns (that may look like noise)[35, 40], which can potentially lead to catastrophic accidents if such backdoor patterns
interfere the stop sign or traffic light classification.
Existing evaluation frameworks are mostly, if not all, data-centric, meaning that they are based on pre-defined, annotated datasets. The drawback is the lack of a comprehensive understanding of the
model’s fundamental behaviors over all possible inputs.Recent literature showed that a great performance on a pre-defined (in-distribution) test set cannot guarantee a strong generalization
todifferent regions in the input space, such as out-of-distribution (OOD) test sets[38, 21, 22, 24, 32, 33] and adversarial test sets[62, 53, 43, 29].One possible reason for poor generalization in
the open-world setting is overconfident prediction[46], where the model could wrongly predict OOD input as in-distribution objects with high confidence.
Inspired by the evaluation frameworks for generative models[23, 55, 45, 54, 4], we propose a novel model evaluation approach from a model-centric perspective: after the model is trained, we construct
the test set from the model’s self-generated, representative inputs corresponding to different model output values. We then annotate these samples, and estimate the model performance over the entire
input space using the model’s output distribution.While existing generative model evaluation frameworks are also model-centric, we are the first to leverage the output distribution as a unique
quantity to generalize model evaluation from representative inputs to the entire input space.To illustrate our proposed novel evaluation framework OmniInput, we focus on a binary classification task
of classifying if a picture is digit 1 or not.As shown in Fig.1, it consists of four steps:
1. (a)
We employ a recently proposed sampler to obtain the output distribution $\rho(z)$ of the trained model (where $z$ denotes the output value of the model) over the entire input space[39] and
efficiently sample representative inputs from different output value (e.g., logit) bins.The output distribution is a histogram counting the number of inputs that lead to the same model output.In
the open-world setting without any prior knowledge of the samples, all possible inputs should appear equally.
2. (b)
We annotate the sampled representative inputs to finalize the test set, e.g., rate how likely the picture is digit 1 using a score from 0 to 1.^1^11In data-centric evaluations, the pre-defined
test set is typically human-annotated as well. Our experiments show that 40 to 50 human annotations per output bin are enough for a converged precision-recall curve (Fig.4), hence human
involvement required is significantly smaller in our method.
3. (c)
We compute the precision for each bin as $r(z)$, then estimate the precision and recall at different threshold values $\lambda$.When aggregating the precision across different bins, a weighted
average of $r(z)$ by the output distribution $\rho(z)$ is required i.e., $\frac{\sum_{z\geq\lambda}r(z)\cdot\rho(z)}{\sum_{z\geq\lambda}{\rho(z)}}$. See Sec.2.2 for details.
4. (d)
We finally put together the precision-recall curve for a comprehensive evaluation of the model performance over the entire input space.
OmniInput samples the representative inputs solely by the model itself, eliminating possible human biases introduced by the test data collection process.The resulting precision-recall curve can help
decide the limit of the model in real-world deployment.The overconfident prediction issue can also be quantified precisely manifested by a low precision when the threshold $\lambda$ is high.
Our OmniInput framework enables a more fine-grained comparison between models, especially when their performance is almost the same on the pre-defined datasets.Take the MNIST dataset as an
example,many models (e.g., ResNet, CNN, and multi-layer Perceptron network (MLP)) trained by different methods (e.g., textbook cross-entropy (CE), CE with (uniform noise) data augmentation, and
energy-based generative framework) can all achieve very high or nearly perfect performance.Our experiments using OmniInput reveals, for the first time, the differences in the precision-recall curves
of these models over the entire input space and provides new insights. They include:
• •
The architectural difference in MLP and CNN, when training with the CE loss and original training set, can lead to significant difference in precision-recall curves.CNN prefers images with dark
background as representative inputs of digit 1, while MLP prefers to invert the background of zeros as digit 1.
• •
Different training schemes used on the same ResNet architecture can lead to different performance.Adding noise to the training set in general can lead to significant improvements in precision and
recall than using energy-based generative models; however, the latter leads to samples with a better visual diversity.These results suggest that combining the generative and classification
objectives may be the key for the model to learn robust classification criteria for all possible samples.
Additionally, we have evaluated DistilBERT for sentiment classification and ResNet on CIFAR (binary classification) using OmniInput. Our results indicate a significant number of overconfident
predictions, a strong suggestion of poor performance in the entire input space.It is worth mentioning that these findings are specific to the models we trained.Thus, this is not a conclusive study of
the differences of the models with different training methods and architectures, but a demonstration of how to use our OmniInput framework to quantify the performance of the models and generate new
insights for future research.The contributions of this work are as follows:
• •
We propose to evaluate AI/ML models by considering all the possible inputs with equal probability, which is crucial to AI safety and reliability.
• •
We develop a novel model-centric evaluation framework, OmniInput, that constructs the test set by representative inputs, and leverages output distribution to generalize the evaluation assessment
from representative inputs to the entire input space.This approach largely eliminates the potential human biases in the test data collection process and allows for a comprehensive understanding
and quantification of the model performance.
• •
We apply OmniInput to evaluate various popular models paired with different training methods. The results reveal new findings and insights for how to train robust, generalizable models.
2 The OmniInput Framework
In this section, we present a detailed background on sampling the output distribution across the entire input space. We then propose a novel model-centric evaluation framework OmniInput in which we
derive the performance metrics of a neural network (binary classifier) from its output distribution.
2.1 Output Distribution and Sampler
Output Distribution.We denote a trained binary neural classifier parameterized by $\theta$ as $f_{\mathbf{\theta}}:\mathbf{x}\rightarrow z$ where $\mathbf{x}\in\Omega_{T}$ is the training set, $\
Omega_{T}\subseteq\{0,...,N\}^{D}$, and $z\in\mathbb{R}$ is the output of the model. In our framework, $z$ represents the logit and each of the $D$ pixels takes one of the $N+1$ values.
The output distribution represents the frequency count of each output logit $z$ given the entire input space $\Omega=\{0,...,N\}^{D}$.In our framework, following the principle of equal a priori
probabilities, we assume that each input sample within $\Omega$ follows a uniform distribution. This assumption is based on the notion that every sample in the entire input space holds equal
importance for the evaluation of the model. Mathematically, the output distribution, denoted by $\rho(z)$, is defined as:
where $\delta$ is the Dirac delta function.
The sampling of an output distribution finds its roots in physics, particularly in the context of the sampling of the density of states (DOS)[66, 65, 11, 26, 36, 74], but its connection to ML is
revealed only recently[39].
The Wang–Landau (WL) algorithm[66] aims to sample the output distribution $\rho(z)$ which is unknown in advance.In practical implementations, the “entropy” (of discretized bins of $z$), $\tilde{S}(z)
=\log\tilde{\rho}(z)$, is used to store the instantaneous estimation of the ground truth $S(z)=\log\rho(z)$. The WL algorithm leverages the reweighting technique, where the sampling weight $w(\mathbf
{x})$ is inversely proportional to the instantaneous estimation of the output distribution:
$\displaystyle~{}w(\mathbf{x})\propto\frac{1}{\tilde{\rho}(f_{\theta}(\mathbf{x%}))}.$ (1)
When the approximation $\tilde{\rho}(z)$ converges to the true value $\rho(z)$, the entire output space would be sampled uniformly.
The fundamental connection between the output distribution of neural networks and the DOS in physics has been discovered and elucidated in Ref.[39]. Additionally, it is shown that the traditional
Wang–Landau algorithm sometimes struggles to explore the parameter space if the MC proposals are not designed carefully.Gradient Wang–Landau sampler (GWL)[39] circumvent this problem by incorporating
a gradient MC proposal similar to GWG[15], which improves Gibbs sampling by picking the pixels that are likely to change.The GWL sampler has demonstrated the feasibility and efficiency of sampling
the entire input space for neural networks.
The key component of the output distribution samplers is that they can sample the output space equally and efficiently, thereby providing a survey of the input-output mapping for all the possible
logits. This is in contrast with traditional MCMC samplers which are biased to sample the logits corresponding to high log-likelihood (possible informative samples) over logits correspond to low
log-likelihood (noisy and uninformative samples).
2.2 Model-Centric Evaluation
Our model evaluation framework revolves around the output distribution sampler. Initially, we obtain the output distribution and the representative inputs exhibiting similar output logit values.
Representative Inputs. Although there are exponentially many uninformative samples in the entire input space, it is a common practice in generative model evaluation to generate (representative)
samples by sampling algorithms and then evaluate samples, such as Fréchet Inception Distance (FID)[23]. In our framework, other sampling algorithms can also be used to collect representative inputs.
There should be no distributional difference in the representative inputs between different samplers (Fig.8). However, Wang–Landau type algorithms provide a more effective means for traversing across
the logit space and are hence more efficient than traditional MCMC algorithms in sampling the representative inputs from the output distribution.
Normalized Output Distribution. To facilitate a meaningful comparison of different models based on their output distribution, it is important to sample the output distribution of (all) possible
output values to ensure the normalization can be calculated as accurately as possible. We leverage the fact that the entire input space contains an identical count of $(N+1)^{D}$ samples for all
models under comparison[30]. Consequently, the normalized output distribution $\rho(z)$ can be expressed as:
where $\hat{\rho}(z)$ denotes the unnormalized output distribution.
Annotation of Samples. For our classifiers, we designate a specific class as the target class.The (human) evaluators would assign a score to each sample within the same “bin” of the output
distribution (each “bin” collects the samples with a small range of logit values $[z-\Delta z,z+\Delta z)$). This score ranges from $0$ when the sample completely deviates from the evaluator’s
judgment for the target class, to $1$ when the sample perfectly aligns with the evaluator’s judgment. Following the evaluation, the average score for each bin, termed “precision per bin”, $r(z)$, is
calculated. It is the proportion of the total evaluation score on the samples relative to the total number of samples within that bin. We have 200-600 bins for the experiments.
Precision and Recall.Without loss of generality, we assume that the target class corresponds to large logit values: we define a threshold $\lambda$ such that any samples with $z\geq\lambda$ are
predicted as the target class.Thus, the precision given $\lambda$ is defined as
The numerator is the true positive and the denominator is the sum of true positive and false positive. This denominator can be interpreted as the area under curve (AUC) of the output distribution
from the threshold $\lambda$ to infinity.
When considering recall, we need to compute the total number of ground truth samples that the evaluators labeled as the target class. This total number of ground truth samples remains constant
(albeit unknown) over the entire input space. Hence recall is proportional to $\sum^{+\infty}_{z\geq\lambda}r(z)\rho(z)$:
$\mathrm{recall}_{\lambda}=\frac{\sum^{+\infty}_{z\geq\lambda}r(z)\rho(z)}{%\text{number of positive samples}}\propto\sum^{+\infty}_{z\geq\lambda}r(z)\rho%(z).$
A higher recall indicates a better model. As demonstrated above, the output distribution provides valuable information for deriving both precision and (unnormalized) recall. These metrics can be
utilized for model evaluation through the precision-recall curve, by varying the threshold $\lambda$. In the extreme case where $\rho(z)$ differs significantly for different $z$, precision${}_{\
lambda}$ is approximated as $r(z^{*})$ where $z^{*}=\operatorname*{arg\,max}_{z\geq\lambda}\rho(z)$ and recall${}_{\lambda}$ is approximated as $\max_{z\geq\lambda}r(z)\rho(z)$.
Quantifying Overconfident Predictions in OmniInput.Overconfident predictions refer to the samples that (a) the model predicts as positive with very high confidence (i.e., above a very high threshold
$\lambda$) but (b) human believes as negative.The ratio of overconfident predictions over the total positive predictions is simply $1-\mathrm{precision}_{\lambda}$ in OmniInput.Moreover, even if two
models have nearly the same (high) precision, the difference in (unnormalized) recall $\mathrm{recall}_{\lambda}$ can indicate which model captures more ground-truth-positive samples.Therefore,
compared to methods that only quantify overconfident prediction, OmniInput can offer a deeper insight of model performance using recall.
Scalability.Our OmniInput framework mainly focuses on how to leverage the output distribution for model evaluation over the entire input space.To handle larger input spaces and/or more complicated
models, more efficient and scalable samplers are required. However, it is beyond the scope of this paper and we leave it as a future work.Our evaluation framework is parallel to the development of
the samplers and will be easily compatible to new samplers.
3 Experiments on MNIST and related datasets
The entire input space considered in our experiment contains $256^{28\times 28}$ samples (i.e., $28\times 28$ gray images), which is significantly larger than any of the pre-defined datasets, and
even larger than the number of atoms in the universe (which is about $10^{81}$).
Models for Evaluation.We evaluate several backbone models: convolution neural network (CNN), multi-layer Perceptron network (MLP), and ResNet[19]. The details of the model architectures are provided
in Appendix9.We use the MNIST training set to build the classifiers, but we extract only the samples with labels $\{0,1\}$, which we refer to as MNIST-0/1.For generative models, we select only the
samples with label=1 as MNIST-1; samples with labels other than label=1 are considered OOD samples.We build models using different training methods:(1) Using the vanilla binary cross-entropy loss, we
built CNN-MNIST-0/1 and MLP-MNIST-0/1^2^22The results for RES-MNIST-0/1 are omitted due to reported sampling issues in ResNet[39]. which achieve test accuracy of 97.87% and 99.95%, respectively;(2)
Using the binary cross-entropy loss and data augmentation by adding uniform noise with varying levels of severity to the input images, we built RES-AUG-MNIST-0/1, MLP-AUG-MNIST-0/1, and
CNN-AUG-MNIST-0/1 which achieve test accuracy of 99.95%, 99.91%, and 99.33%, respectively;and (3) Using energy-based models that learn by generating samples, we built RES-GEN-MNIST-1 and
MLP-GEN-MNIST-1^3^33CNN-GEN-MNIST-1 is untrainable because model complexity is low..
3.1 Traditional Data-centric Evaluation
We show that data-centric evaluation might be sensitive to different pre-defined test sets, leading to inconsistent evaluation results.Specifically, we construct different test sets for those MNIST
binary classifiers by fixing the positive test samples as the samples in the MNIST test set with label=1, and varying the negative test samples in five different ways:(1) the samples in the MNIST
test set with label=0 (in-dist),and the out-of-distribution (OOD) samples from other datasets such as(2) Fashion MNIST[68],(3) Kuzushiji MNIST[8],(4) EMNIST[9] with the byclass split,and (5) Q-MNIST[
Judging from the Area Under the Precision-Recall Curve (AUPR) scores in Table1, pre-defined test sets such as the ones above can hardly lead to consistent model rankings in the evaluation.For
example, RES-GEN-MNIST-1 performs the best on all the test sets with OOD samples while only ranked 3 out of 4 on the in-distribution test set.Also, CNN-MNIST-0/1 outperforms MLP-MNIST-0/1 on
Kuzushiji MNIST, but on the other test sets, it typically performs the worst.Additional inconsistent results using other evaluation metrics can be found in Appendix10.
3.2 Our Model-centric OmniInput Evaluation
in-dist out-of-distribution (OOD)
MNIST Fashion Kuzushiji EMNIST QMNIST
Model label=0 MNIST MNIST
CNN-MNIST-0/1 99.81 98.87 93.93 79.42 13.84
RES-GEN-MNIST-1† 99.99 100.00 99.99 99.87 16.49
RES-AUG-MNIST-0/1 100.00 99.11 93.93 95.10 15.69
MLP-MNIST-0/1 100.00 99.42 92.03 90.68 15.81
Precision-Recall Curves over the Entire Input Space.Fig.1 presents a comprehensive precision-recall curve analysis using OmniInput.The results suggest that RES-AUG-MNIST-0/1 is probably the best
model and MLP-MNIST-0/1 is the second best, demonstrating relatively high recall and precision scores.RES-GEN-MNIST-1, as a generative model, displays a low recall but a relatively good
precision.Notably, CNN-MNIST-0/1 and CNN-AUG-MNIST-0/1 exhibit almost no precision greater than 0, indicating that “hand-written” digits are rare in the representative inputs even when the logit
value is large (see Appendix12).This suggests that these two models are seriously subjected to overconfident prediction problem.
Insights from Representative Inputs.An inspection of the representative inputs (Appendix12) reveals interesting insights.Firstly, different models exhibit distinct preferences for specific types of
samples, indicating significant variations in their classification criteria.Specifically,
• •
MLP-MNIST-0/1 and MLP-AUG-MNIST-0/1 likely define the positive class as the background-foreground inverted version of digit “0”.
• •
CNN-MNIST-0/1 classifies samples with a black background as the positive class (digit “1”).
• •
RES-GEN-MNIST-1, a generative model, demonstrates that it can map digits to large logit values.
• •
RES-AUG-MNIST-0/1, a classifier with data augmentation, demonstrates that adding noise during training can help the models better map samples that look like digits to large logit values.
These results suggest that generative training methods can improve the alignment between model and human classification criteria, though it also underscores the need for enhancing recall in
generative models. Adding noise to the data during training can also help.
Moreover, RES-AUG-MNIST-0/1 exhibits relatively high recall as the representative inputs generally look like digit 1 with noise when the logits are high. Conversely, RES-GEN-MNIST-1 generates more
visually distinct samples corresponding to the positive class, but with limited diversity in terms of noise variations.
Discussion of results.First, the failure case of CNN-MNIST-0/1 does not eliminate the fact that informative digit samples can be found in these logit ranges.It indicates the number of these
informative digit samples is so small that the model makes much more overconfident predictions than successful cases.Having this mixture of bad and (possibly) good samples mapped to the same outputs
means a bad model, because further scrutinization of the samples is needed due to uninformative and unreliable model outputs.Second, the model does not use reliable features, such as the “shapes” to
distinguish samples.Had this model use the shape to achieve high accuracy, the representative inputs would have more shape-based samples instead of unstructured and black background samples.Third,
this failure case also does not indicate our sampler fails, because the same sampler finds informative samples for RES-GEN-MNIST-1.
The representative inputs of MLP-MNIST-0/1 and MLP-AUG-MNIST-0/1 display visual similarities but decreasing level of noise when the logit increases, indicating how the noise affects the model’s
prediction.Importantly, this type of noise is presented by the model rather than trying different types of noise[20].Our result indicates that OmniInput finds representative samples that may
demonstrate distribution shifts with regard to model outputs.
Combining these findings with the previous precision-recall curve analysis suggests that different types of diversity may be preferred by the models.Future research endeavors can focus on enhancing
both robustness and visual diversity.
Evaluation Effort, Efficiency and Human Annotation Ambiguity.We have at least 50 samples per bin for evaluation for all the models after deleting the duplicates.The models with fewer samples per bin
typically have a larger number of bins due to the limitation in the sampling cost.Evaluating these samples in our OmniInput framework requires less effort than annotating a dataset collected for
data-centric evaluation, e.g., 60000 samples for MNIST.
In Fig.4, we vary the number of annotated samples per bin in OmniInput from 10 to 50 and plot different precision-recall curves for the MLP-MNNIST-0/1 model.The results show that the evaluation
converges quickly when the number of samples approaches 40 or 50, empirically demonstrating that OmniInput does not need many annotated samples though the number required will be model-dependent.We
believe that this is because the representative inputs follow some underlying patterns learned by the model.
We observe that models exhibit varying degrees of robustness and visual diversity. To assess the ambiguity in human labeling, we examine the variations in $r(z)$ when three different individuals
label the same dataset (Fig.3). Notably, apart from the CNN model, the other models display different levels of labeling ambiguity.
4 Results on CIFAR-10 and Language Model
CIFAR10 and Other Samplers.We train a ResNet binary classifier for the first two classes in CIFAR10, i.e., class 0 (airplane) vs. class 1 (automobile).The test set accuracy of this ResNet model is
93.34%.In Appendix14, Fig.9 shows the output distribution and Fig8 provides some representative inputs.We scrutinize 299 bins with 100 samples per bin on average. Even though the representative
inputs seem to have shapes when their logits are very positive or negative, they are uninformative in general. We can conclude that this classifier should perform with almost 0 precision (with the
given annotation effort) and this model is subjected to serious overconfident prediction.
We also compare the representative inputs in OmniInput and the samples from a Langevin-like sampler[73] in Fig8.The sampling results show that our representative inputs generally agree with those of
the other sampler(s).
Language Model.We fine-tune a DistilBERT[56] using SST2[58] and achieve 91% accuracy. We choose DistilBERT because of sampler efficiency concern and leave LLMs as future work after more efficient
samplers are developed. We then evaluate this model using OmniInput.Since the maximum length of the SST2 dataset is 66 tokens, one can define the entire input space as the sentences with exactly 66
tokens.For shorter sentences, the last few tokens can be simply padding tokens.One might be more interested in shorter sentences because a typical sentence in SST2 contains 10 tokens.Therefore, we
conduct the evaluation for length 66 and length 10, respectively.We sample the output distribution of this model until the algorithm converges; some representative inputs can be found in Appendix13.
When the sentence has only 10 tokens, the representative inputs are not fluent or understandable sentences. For sentence length equals 66, we have 15 bins with around 200 samples per bin.Looking at
the representative inputs per bin for each logit, it shows that the model classifies the positive sentiment mostly based on some positive keywords without understanding the grammar and structure of
the sentences.Therefore, the precision of human evaluation is very low, if not exactly zero, indicating the model is subjected to serious overconfident prediction.
5 Discussions
Human Annotation vs. Model Annotation.In principle, metrics employed in evaluating generative models[55, 23, 45, 54, 4] could be employed to obtain the $r(z)$ values in our method.However, our
framework also raises a question whether a performance-uncertified model with respect to the entire input space can generate features for evaluating another model.We examined the Fréchet Inception
Distance (FID)[23] , one of the most commonly used generative model performance metrics.
RES-AUG-MNIST-0/1 CNN-MNIST-0/1
logits humans$\uparrow$ FID$\downarrow$ logits humans$\uparrow$ FID$\downarrow$
43 0.9 360.23 12 0 346.42
42 0.88 362.82 11 0 358.37
41 0.85 368.75 10 0 363.23
40 0.83 375.58 9 0 365.01
Feature extractors generate features for both ground truth test set images and the images generated by the generative model. It then compares the distributional difference between these features. In
our experiment, the ground truth samples are test set digits from label=1. In general, the performance trends are consistent between humans and FID scores, e.g. for RES-AUG-MNIST-0/1, as FID score is
decreasing (better performance) and human score is increasing (better performance) when the logit increases. This result demonstrates that the scores for evaluating generative models may be able to
replace human annotations.
However, humans and these commonly used generative metrics can lead to very different results.Comparing the results of RES-AUG-MNIST-0/1 and CNN-MNIST-0/1, Table2 shows that the FID score can be
completely misleading.While the representative inputs of CNN-MNIST-0/1 do not contain any semantics for the logits on the table, the FID scores are similar to those of samples from RES-AUG-MNIST-0/1
where representative inputs are clearly “1.”This is not the only inconsistent case between humans and metrics. The trend of FID for MLP-MNIST-0/1 is also the opposite of human intuitions, as shown in
Table5 in Appendix11.When the logits are large, humans label the representative inputs as “1.” When the logits are small, representative inputs look like “0.”However, the FID scores are better for
these “0” samples, indicating the feature extractors believe these “0” samples look more like digits “1.” The key contradiction is that the feature extractors of these metrics, when trained on
certain datasets, are not verified to be applicable to all OOD settings, but surely they will be applied in OOD settings to generate features of samples from models for evaluation. It is difficult to
ensure they will perform reliably.
Perfect classifiers and perfect generative models could be the same.Initially it is difficult to believe the classifiers, such as CNN-MNIST-0/1, perform poorly in the open-world setting when we
assume the samples are from the entire input space. In retrospect, however, it is understandable because the classifiers are trained with the objective of the conditional probability $p(class|\mathbf
{x})$ where $\mathbf{x}$ are from the training distribution. In order to deal with the open-world setting, the models also have to learn the data distribution $p(\mathbf{x})$ in order to tell whether
the samples are from the training distribution. This seems to indicate the importance of learning $p(\mathbf{x})$ and this is the objective of generative models.In Fig.10, if we can construct a
classifier with perfect mapping in the entire input space where the models successfully learn to map all positive and negative samples in the entire input space to the high and low output values
respectively, this model is also a generative model because we can use traditional MCMC samplers to reach the output with high (or low) values. As we know those output values only contain positive
(or negative) samples, we are able to “generate” positive (or negative) samples.Therefore, we speculate that a perfect classifier and a perfect generator should converge to be the same model.
Our method indicates an important trade-off of generative models. The generative models trade the recall for precision. This would mean the model may miss a lot of various ways of writing the digits
“1.” In summary, our method can estimate not only the overconfident predictions for the models, but also the recall. Future work needs to improve both metrics in the entire input space for better
6 Related Works
Performance Characterization has been extensively studied in the literature[18, 28, 63, 50, 2, 1, 51, 49]. Previous research has focused on various aspects, including simple models[17] and
mathematical morphological operators[14, 27].In our method, we adopt a black box setting where the analytic characterization of the input-to-output function is unknown[10, 7], and we place emphasis
on the output distribution[16]. This approach allows us to evaluate the model’s performance without requiring detailed knowledge of its internal workings.Furthermore, our method shares similarities
with performance metrics used for generative models, such as the Fréchet Inception Distance score[23] and Inception Score[55]. Recent works[45, 54, 4] have formulated the evaluation problem in terms
of precision and recall of the distributional differences between generated and ground truth samples. While these methods can be incorporated into our sampler to estimate precision, we leverage the
output distribution to further estimate the precision-recall curve.Recent works[48, 31, 41, 47] evaluate model performance without test set. They used other generators to generate samples for
evaluating a model. On the contrary, we used a sampler to sample the model to be evaluated. Sampling is transparent with convergence estimates, but other generators are still considered as black
boxes. Given the inherently unknown biases in models, utilizing other models to evaluate a model (as explained in the Discussions section in our manuscript) carries the risk of yielding unfair and
potentially incorrect conclusions. Our method brings the focus back to the model to be tested, tasking it with generating samples by itself for scrutiny, rather than relying on external agents such
as human or other models to come up with testing data. An additional benefit is that this approach offers a novel framework for estimating errors in the entire input space when comparing different
MCMC samplers have gained widespread popularity in the machine learning community[5, 67, 34, 70]. Among these, CSGLD[12] leverages the Wang–Landau algorithm[66] to comprehensively explore the energy
landscape. Gibbs-With-Gradients (GWG)[15] extends this approach to the discrete setting, while discrete Langevin proposal (DLP)[73] achieves global updates.Although these algorithms can in principle
be used to sample the output distribution, efficiently sampling it requires an unbiased proposal distribution. As a result, these samplers may struggle to adequately explore the full range of
possible output values. Furthermore, since the underlying distribution to be sampled is unknown, iterative techniques become necessary. The Wang–Landau algorithm capitalizes on the sampling history
to efficiently sample the potential output values. The Gradient Wang–Landau algorithm (GWL)[39] combines the Wang–Landau algorithm with gradient proposals, resulting in improved efficiency.
Open-world Model Evaluation requires model to perform well in in-distribution test sets[13, 64, 59, 6, 75, 19, 57, 61, 25, 72], OOD detection[38, 21, 22, 24, 32, 33, 37, 44, 52], generalization[3, 60
], and adversarial attacks[62, 53, 43, 29, 69, 42]. Understanding performance of the model needs to consider the entire input space that includes all these types of samples.
7 Conclusion
In this paper, we introduce OmniInput, a new model-centric evaluation framework built upon the output distribution of the model. As future work, it is necessary to develop efficient samplers and
scale to larger inputs and outputs. While the ML community has developed many new samplers, sampling the output distribution (and from larger input) is far from receiving enough attention in the
community. Our work demonstrated the importance of sampling from output distribution by showing how it enables the quantification of model performance, hence the need for more efficient samplers.
Scaling to multi-dimensional output is possible and has already been developed previously. Once a scalable samplers are developed, our method will be automatically scalable to larger datasets,
because the output distribution is training-set independent.
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9 Details of the Models used in Evaluation
The ResNet used in our experiments is the same as the one used in GWG[15].For the input pixels, we employ one-hot encoding and transform them into a 3-channel output through a 3-by-3 convolutional
layer.The resulting output is then processed by the backbone models to generate features.The CNN backbone consists of two 2-layer 3-by-3 convolutional filters with 32 and 128 output channels,
respectively.The MLP backbone comprises a single hidden layer with flattened images as inputs and produced 128-dimensional features as output.All the features from the backbone models are ultimately
passed through a fully-connected layer to generate a scalar output.
10 Traditional Model Evaluation Results
Tab.3 shows the AUROC of different models based on pre-defined test sets with different negative class(es). The MLP-MNIST-0/1 performs better on Fashion MNIST but worse in the rest than
RES-AUG-MNIST-0/1. RES-GEN-MNIST-1 usually perform the best. CNN-MNIST-0/1 performs better in Kuzushiji MNIST than RES-AUG-MNIST-0/1 and MLP-MNIST-0/1 but worse on the rest. Tab4 shows the FPR95
results. CNN-MNIST-0/1 performs better on Kuzushiji MNIST than RES-AUG-MNIST-0/1 and MLP-MNIST-0/1 but worse on the rest. These results show the inconsistency between the metrics, dataset and the
Test Set class=0 (in-dist) Fashion MNIST (OOD) Kuzushiji MNIST (OOD) EMNIST (OOD) QMNIST (OOD)
CNN-MNIST-0/1 99.76 99.88 99.31 99.56 92.46
RES-GEN-MNIST-1† 99.99 100.00 100.00 100.00 94.85
RES-AUG-MNIST-0/1 100.00 99.91 99.15 99.93 94.32
MLP-MNIST-0/1 100.00 99.93 98.62 99.83 94.17
†Class=0 is OOD for GEN model.
Test Set class=0 (in-dist) Fashion MNIST (OOD) Kuzushiji MNIST (OOD) EMNIST (OOD) QMNIST (OOD)
CNN-MNIST-0/1 0.54 0.51 2.78 1.98 21.08
RES-GEN-MNIST-1† 0.00 0.00 0.00 0.00 10.55
RES-AUG-MNIST-0/1 0.00 0.34 4.60 0.31 14.17
MLP-MNIST-0/1 0.00 0.27 6.68 0.64 13.24
†Class=0 is OOD for GEN model.
11 Human-metrics inconsistency
In table5 of MLP-MNIST-0/1, the FID scores indicate the samples are bad when humans think they are good. The FID scores indicate the even better performance (lower scores) in the logit ranges when
humans label as incorrect in general.
logits humans$\uparrow$ FID$\downarrow$
17 0.73 434.32
16 0.67 436.60
15 0.58 432.89
14 0.48 430.79
-19 0.18 422.01
-20 0.2 419.94
-21 0.2 412.96
-22 0.216 405.20
12 Representative inputs for MNIST images
Representative inputs for different models are in Fig.5.
13 Reprensetaive inputs for SST2 dataset
For sentence length 66, some representative inputs with logit equals 7 (positive sentiment) in Fig6.
For sentence length 10, some representative inputs with logit equals to 7 (positive sentiment) in Fig7.
14 Representative inputs for CIFAR10
Fig.8 shows the representative inputs from MCMC samplers and our samplers. The values on top label the logit of the corresponding image (for MCMC sampler) or a column of images (for our sampler). The
patterns found are essentially no difference, proving our sampler finds exactly the same type of representative inputs. Moreover, these samples are not recognizable to humans, suggesting the
precision will super low.
15 Perfect classifier
In Fig.10 shows a perfect classifier. a perfect classifier can map all the ground truth digits “0” on the close to $p(y=1|\mathbf{x})=0$ and ground truth digits “0” on the close to $p(y=1|\mathbf{x})
=1$. We speculate this seems to show this is also a perfect generative model.
16 Sampler Details
Gradient-with-Gibbs (GWG) is a Gibbs sampler by nature, thus it updates only one pixel at a time. Recently, a discrete Langevin proposal (DLP)[73] is proposed to achieve global update, i.e, updating
multiple pixels at a time. We adopt this sampler to traverse the input space more quickly, but we treat $-\frac{d\tilde{S}}{df}$ the same value as $\beta$ for both $q(\mathbf{x}^{\prime}|\mathbf{x})$
and $q(\mathbf{x}|\mathbf{x}^{\prime})$.
We use two different ways to generate $\beta$. In the first way, we sample $\beta$ uniformly from a range of values, including positive and negative values. In the second way, since the WL/GWL
algorithms strive to achieve a flat histogram[39], we add a directional mechanism to direct the sampler to visit larger logit values before it moves to smaller logit values, and vice versa. We
introduce a changeable parameter $\gamma=\{-1,1\}$ to signify the direction. For example, if $\gamma=1$ and the sampler hits the maximum known logit, $\gamma$ is set to $-1$ to reverse the direction
of the random walk. Moreover, we sample $\beta$ uniformly from a range of non-negative values in order to balance small updates ($\beta$ is small) and aggressive updates ($\beta$ is large). Finally,
we check whether the current histogram entry passes the flatness check. If so, it means that this particular logit value has been sampled adequately, we then multiply $\beta$ by $\gamma$;otherwise,
we set $\beta=0.1$ which slightly modifies the input but allows the sampler to stay in the current bin until the histogram flatness passes for the current logit value. With the above heuristic fixes,
the sampler does not need to propose $\mathbf{x}^{\prime}$ with smaller $\tilde{S}$, but focuses on how to make the histogram flat.
17 Results on CIFAR10 and CIFAR-100
Multi-class classification setting. The current output format for classification problems employs a one-hot encoding, representing an anticipated ground truth distribution. We establish the output as
a log-softmax for the prediction vector, defining a range from $(-\inf,0]$. This formulation allows for the sampling of each dimension within the log-softmax, akin to the approach employed in binary
classification and generative model scenarios.
Results of repsentative samples and output distribution . We train ResNet with CIFAR-10 to reach $88\%$ accuracy and CIFAR-100 to reach $62\%$ accuracy with cross-entropy. Scrutizing the samples from
the (in-dist) test set with log-softmax near 0 confirm the model trained with CIFAR-10 successfully learns to map these samples to near log-softmax$=0$. Fig11 shows representative samples and output
First, we plot the representative samples of CIFAR-10 for class 0 and 1 respectively. Building upon the analysis previously articulated in the context of MNIST, wherein it was demonstrated that
classifiers generally fail to learn the data distribution, our observations extend to the current model. Specifically, the model tends to map a significant portion of uninformative samples to the
output region where informative test set samples reside, resulting in a precision value of 0 in the precision-recall curve.
Second, different from the previous experiments where the output distribution for informative test set inputs (output values near 0) was generally low in binary classification, our findings in the
context of multi-class classification reveal a notable distinction. Specifically, the output distribution for these regions tends to be high, indicating that the model maps a substantial number of
uninformative samples to the output values shared by informative test set samples.
Lastly, we extended our analysis to CIFAR-100, and the observed trend in output distribution is generally consistent with that of CIFAR-10. Thus, to ensure the model’s effectiveness across the
entirety of the input space, there remains a necessity for further refinement and enhancement of precision in log-softmax values near 0. | {"url":"https://eastsidesteelheaders.com/article/omniinput-a-model-centric-evaluation-framework-through-output-distribution","timestamp":"2024-11-10T18:55:50Z","content_type":"text/html","content_length":"245120","record_id":"<urn:uuid:fb8b30b8-f5a8-4d0c-abe2-28d201be2648>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00399.warc.gz"} |
Labelled Induced Subgraphs and Well-Quasi-Ordering
It is known that the set of all simple graphs is not well-quasi-ordered by the induced subgraph relation, i.e. it contains infinite antichains (sets of incomparable elements) with respect to this
relation. However, some particular graph classes are well-quasi-ordered by induced subgraphs. Moreover, some of them are well-quasi-ordered by a stronger relation called labelled induced subgraphs.
In this paper, we conjecture that a hereditary class X which is well-quasi-ordered by the induced subgraph relation is also well-quasi-ordered by the labelled induced subgraph relation if and only if
X is defined by finitely many minimal forbidden induced subgraphs. We verify this conjecture for a variety of hereditary classes that are known to be well-quasi-ordered by induced subgraphs and prove
a number of new results supporting the conjecture.
• Induced subgraph
• Infinite antichain
• Labelled induced subgraphs
• Well-quasi-order | {"url":"https://scholar.xjtlu.edu.cn/en/publications/labelled-induced-subgraphs-and-well-quasi-ordering","timestamp":"2024-11-04T11:56:13Z","content_type":"text/html","content_length":"42588","record_id":"<urn:uuid:0482da61-1b0c-4c80-aff0-bbdf68e1da79>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00175.warc.gz"} |
Algebra: Understanding the Basics - CAPS 123
Algebra: Understanding the Basics
Algebra is a branch of mathematics that deals extensively with mathematical symbols, and the rules for manipulating those symbols. In essence, it uses letters (or symbols) to represent numbers and
the relationships between them. Algebra is an important part of math, and it’s used in a wide range of professions, including science, engineering, economics, and statistics.
The word “algebra” originates from an Arabic word that means “reunion of broken parts.” This may be because solving equations in algebra often involves putting different pieces back together in a
logical way. Algebra can be quite challenging for some individuals because it requires them to think abstractly, but it is a rewarding area to study because it helps people to develop critical
thinking and problem-solving skills. In many ways, algebra is the foundation for higher mathematics, and it’s essential for anyone who wants to pursue further study in math or science.
History of Algebra
Algebra is a branch of mathematics that deals with mathematical operations and the use of symbols to represent unknown values. It has a long and rich history that dates back to ancient civilizations.
Here’s a brief overview of the History of Algebra:
• Ancient times: Algebraic expressions and equations were first developed by the Babylonians and Egyptians around 2000 BCE. This included solving equations with one and two unknown variables.
• Greek Mathematics: In the 5th century BCE, the Greeks developed geometrical methods to solve equations, including quadratic equations. Greek mathematician, Diophantus, is seen as the “father of
algebra” for his contributions to solving equations with multiple unknown variables.
• Islamic Mathematics: Islamic mathematicians made significant contributions to algebra during the Middle Ages. Al-Khwarizmi, a Persian mathematician, wrote a book called “Al-jabr wa’l-Muqabala” in
the 9th century which provided a systematic introduction to what we now know as algebra. This book contained information on how to solve linear and quadratic equations.
• Renaissance Europe: During the Renaissance, European mathematicians became interested in finding solutions to equations with higher degrees. Italian mathematician Rafael Bombelli introduced the
concept of complex numbers, while Francois Vieta introduced the use of symbols instead of words for mathematical equations.
• Modern Algebra: In the 19th and 20th century, algebra went through a major transformation. Several branches of algebra emerged including abstract algebra, linear algebra, and Boolean algebra.
These branches of algebra gave birth to modern-day computer science and digital technology.
In conclusion, algebra has a rich and diverse history full of brilliant minds and innovative ideas. Its study has helped shape the world we live in today, and it continues to be a vital part of
mathematics education.
Fundamentals of Algebra
Algebra is a fundamental branch of mathematics used to analyze and manipulate equations with variables. Variables represent unknown quantities, which can be solved using algebraic methods. Here are
the key concepts that form the fundamentals of algebra:
Algebraic Expressions
An algebraic expression is a mathematical statement that can include constants, variables, and operations such as addition, subtraction, multiplication, and division. For example, 2x + 5 is an
algebraic expression, where 2, x, and 5 are constants, variables, and constants, respectively. Solving algebraic expressions involves simplification and substitution of variables with values.
An equation is a statement that states the equality of two algebraic expressions. Equations typically have an equal sign (=) between two expressions, such as 3x + 7 = 16. Solving equations typically
involve isolating variables on one side of the equation and simplifying the other side until they are equal.
Inequalities are expressions where the two sides are not equal. Instead, one side can be greater than or less than the other. Inequalities use symbols such as >, <, ≥, and ≤, and can also include
variables and constants. Solving inequalities involves graphing and identifying the range of values that satisfy the inequality.
Polynomials are algebraic expressions made up of one or more terms, where each term has a coefficient and a variable raised to some power. For example, 3x² + 5x + 2 is a polynomial. Solving
polynomials involves factoring, simplifying, and applying various algebraic methods.
Systems of Equations
A system of equations is a set of equations with multiple variables that are solved simultaneously. For example, 3x + 2y = 12 and 5x - 4y = 8 form a system of equations. Solving systems of equations
requires using algebraic methods such as elimination or substitution to isolate the values of the variables.
Understanding these fundamentals of algebra is essential for success in advanced math courses and real-world applications such as engineering, physics, and finance.
Application of Algebra in Real Life
Algebra is a branch of mathematics that is essential in solving complex problems using equations, variables and mathematical operations. It is widely used in real life situations, from calculating
the area of a room to designing spacecraft. In this section, we will explore some of the ways algebra is applied in everyday life.
Algebra is used in finance to calculate the total interest paid on a loan or mortgage. By using algebraic equations, lenders can determine monthly payments, interest and principal amounts, and the
overall cost of the loan. Similarly, people can use algebra to calculate returns on investments, compound interest and savings growth.
Algebra is used in various branches of science, from physics to biology. Scientists use algebraic equations to describe phenomena such as velocity, acceleration, and gravitational force. In
chemistry, algebraic formulas are used to balance chemical equations and determine the quantity of reactants and products.
Engineers use algebra in designing structures like bridges, tunnels, and dams. They apply algebraic principles in calculating the loads, stresses, and strengths of materials, which are necessary in
ensuring the safety and longevity of these structures.
Algebra is also used in medical research and drug design. Pharmacologists use algebraic equations to model the effects of medicines on the body, which is useful in predicting drug interactions, doses
and side-effects.
Algebra is an important tool for solving problems in various fields of study. Its applications in real life situations are numerous, ranging from finance and science to engineering and medicine.
Therefore, understanding of algebraic concepts and principles is crucial for career success and personal growth.
Types of Algebraic Equations
In algebra, equations can be classified into different types based on their form and structure. Understanding the different types of algebraic equations is critical to solving problems effectively.
This section explores the most common forms of algebraic equations.
Linear Equations
Linear equations are the most basic type of algebraic equations. They can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable. To solve linear equations, one
needs to isolate the variable on one side of the equation using inverse operations. For example, to solve 2x + 7 = 15, one can subtract 7 from both sides and then divide by 2 to get x = 4.
Quadratic Equations
Quadratic equations are algebraic equations of the form ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable. Quadratic equations can have one, two, or zero real solutions
depending on the values of a, b, and c. The solutions of quadratic equations can be found using the quadratic formula or by factoring.
Polynomial Equations
Polynomial equations are algebraic equations with one or more terms, where each term is a product of a coefficient and a variable raised to a power. Polynomial equations can be of any degree, and the
degree of the equation corresponds to the highest power of the variable. Polynomial equations can have one or more real solutions depending on their degree and coefficients.
Exponential Equations
Exponential equations involve variables that appear in exponents. They can be of the form a^x = b, where a and b are positive constants. Exponential equations can be solved by taking logarithms of
both sides and using the laws of logarithms. Exponential equations are used to model exponential growth and decay in various fields, such as finance, biology, and physics.
In conclusion, understanding the different types of algebraic equations helps in solving problems efficiently. Linear, quadratic, polynomial, and exponential equations are some of the most common
types of algebraic equations that one may encounter.
Solving Algebraic Equations
In algebra, equations with variables that represent unknown values can be solved, revealing the value of the variable. This skill is important in numerous fields, from engineering to finance. Solving
algebraic equations involves following a systematic approach to isolate the variable on one side of the equals sign so that its value is easily determined.
Steps to Solve Algebraic Equations:
1. Eliminate Fractions: Equations with fractions can be solved easily by multiplying each term by the denominator of the fraction. This eliminates the fractions and simplifies the equation.
2. Simplify the Equation: Combine like terms on each side of the equation.
3. Isolate the Variable: Move all terms that do not contain the variable to the opposite side of the equals sign by adding or subtracting. This will leave the variable term on one side of the
4. Solve for the Variable: Divide both sides of the equation by the coefficient of the variable, or multiply both sides by the reciprocal of the coefficient.
5. Check your Answer: Plug the value of the variable back into the original equation to ensure it satisfies the equation.
Common Techniques for Solving Algebraic Equations:
1. Factoring: Factoring is the process of finding what to multiply together to obtain a certain polynomial. It can be useful when solving quadratic equations, which are equations of the form ax² +
bx + c = 0.
2. Substitution: Substitution involves replacing one variable with another that has been previously solved for. This can simplify complex equations and make them easier to solve.
3. Completing the Square: Completing the square is a technique for solving quadratic equations of the form ax² + bx + c = 0. By adding and subtracting terms, the equation can be rewritten as a
squared expression, which can then be solved using the square root property.
In summary, solving algebraic equations involves following a systematic approach to isolate the variable and determine its value. Factoring, substitution, and completing the square are common
techniques used to simplify complex equations. With practice, anyone can become proficient at this essential skill.
Benefits of Learning Algebra
Algebra is a branch of mathematics that deals with variables and symbols in equations and formulas. It is an essential subject to learn as it provides a foundation for many other mathematical
concepts, as well as practical applications in everyday life. In this section, we will explore the benefits of learning algebra.
Improved Problem-Solving Skills
One of the main benefits of learning algebra is improved problem-solving skills. Algebra provides a systematic approach to solving problems, which involves breaking down complex problems into
smaller, more manageable steps. This skill is not only valuable in mathematics but also in other academic subjects and in the workplace.
Better Career Opportunities
Algebra is a critical component of many STEM (Science, Technology, Engineering, and Mathematics) careers. Having an understanding of algebra opens doors to a variety of careers such as engineering,
computer science, physics, and finance. These careers are in high demand and typically offer higher pay and better job security.
Enhanced Logical Reasoning
Algebra also develops logical reasoning skills. When solving equations and formulas, students must follow a specific set of rules and guidelines. This logical process helps to develop critical
thinking skills and the ability to solve problems in real-world situations.
Improved Memory and Concentration
Learning algebra also improves memory and concentration. Algebra involves memorizing formulas and rules, which helps to exercise and expand the memory. Algebra also requires students to focus and
concentrate for extended periods, which helps to develop concentration skills.
Increased Confidence in Mathematics
Algebra can be challenging, but mastering it can increase confidence in mathematics. Students who understand algebraic concepts are better equipped to tackle more advanced mathematical topics. This
can lead to improved performance in exams and a greater sense of accomplishment.
Real-World Applications
Finally, algebra has many practical applications in everyday life. From calculating mortgages to analyzing data, algebra is used in many fields, including science, engineering, finance, and
technology. Having a foundation in algebra helps individuals make informed decisions and solve practical problems in their daily lives.
In conclusion, learning algebra provides many benefits, from improved problem-solving skills to better career opportunities. It is a subject that provides valuable skills for both academics and
real-world applications.
Common Mistakes in Learning Algebra
Algebra is a fundamental tool in mathematics, which is widely used in various disciplines, including science, economics, and engineering. However, for many students, learning algebra can be a
challenging and frustrating experience. Here are the most common mistakes in learning algebra:
1. Neglecting the Basics
Algebra builds upon fundamental concepts, such as arithmetic, fractions, and decimals. Neglecting these concepts while learning algebra can cause difficulties in understanding and applying algebraic
expressions and equations. It is crucial to review and master the basics before moving on to more complex algebraic problems.
2. Lack of Practice
Algebra requires a lot of practice to develop the necessary algebraic skills and problem-solving strategies. Many students tend to avoid practice or struggle to find the time to practice regularly.
However, practicing regularly and actively seeking algebraic problems can boost confidence and comprehension in algebra.
3. Rote Learning
Memorizing algebraic formulas and procedures without understanding their conceptual meaning can hinder comprehension and limit problem-solving abilities. Instead, an approach that focuses on
understanding the underlying concepts, patterns, and relationships can enhance the natural problem-solving skills.
4. Sloppy Calculations
Making careless mistakes while performing algebraic operations can cause confusion and errors in solving problems. It is essential to double-check the calculations and pay attention to details while
performing algebraic operations.
5. Fear of Failure
Many students struggle with algebra due to a lack of confidence and fear of failure. However, it is essential to remember that algebra, like any other subject, requires time, effort, and practice to
master. Adopting a growth mindset that values effort and learning can lead to success in algebra.
In conclusion, learning algebra can be challenging, but it is not impossible. The common mistakes identified above can hinder comprehension, but with proper guidance, practice, and a growth mindset,
algebra can be a rewarding and enjoyable learning experience.
Math Resources for Learning Algebra
Algebra is an essential part of mathematics that deals with equations involving letters and symbols representing numbers. For students who are struggling with algebra, there are several math
resources available online that can help them improve their skills.
1. Khan Academy
Khan Academy is an online platform that provides free video tutorials on a variety of subjects, including algebra. They offer a comprehensive video library, practice exercises, and personalized
dashboards to help students learn at their own pace.
2. Mathway
Mathway is another online platform that allows students to enter their algebra problems and receive step-by-step solutions. They offer basic and advanced algebra, trigonometry, calculus, and other
math subjects.
3. Wolfram Alpha
Wolfram Alpha is a computational search engine that provides answers to mathematical problems, including algebraic equations. It can handle a wide variety of problems, from basic algebra to
high-level calculus. It also provides solutions in a step-by-step format.
4. Purplemath
Purplemath is a website dedicated to helping students struggling with math, including algebra. They provide lessons on a variety of topics, from basic algebra to advanced calculus. They also offer
example problems with solutions and provide tips and tricks to help students master the subject.
5. Algebra.com
Algebra.com is a free online resource for students struggling with algebra. It has a wide variety of resources, including lesson plans, worksheets, and practice problems. The website also has a forum
where students can ask questions and receive help from other students and educators.
Overall, these math resources can help students struggling with algebra to improve their skills and gain confidence in the subject. With the help of these educational resources, students can work at
their own pace, receive personalized feedback and learn valuable skills that can benefit them in the future.
Frequently Asked Questions About Algebra
Algebra can be a challenging subject, but with practice and dedication, anyone can improve their skills. Here are some commonly asked questions about algebra:
What is Algebra?
Algebra is a branch of mathematics that involves using letters and symbols to represent numbers and express mathematical relationships. In essence, it is a way of solving mathematical problems using
equations and formulas.
What are the Basic Operations in Algebra?
The basic operations in algebra include addition, subtraction, multiplication, and division. You also need to know how to work with exponents and solve for unknown variables.
How Do I Simplify Algebraic Expressions?
To simplify algebraic expressions, you need to combine like terms and follow the order of operations. This means performing operations inside parentheses first, then exponents, followed by
multiplication and division from left to right, and finally addition and subtraction from left to right.
How Do I Solve Equations?
To solve equations, you need to isolate the variable on one side of the equation using the inverse operation. For example, if you have an equation that says 2x + 5 = 11, you would subtract 5 from
both sides and then divide by 2 to get x = 3.
How Do I Graph Equations?
To graph equations, you need to plot points on a coordinate plane and connect them with a line. The x-axis represents the horizontal values, while the y-axis represents the vertical values.
How Can I Check My Answers?
To check your answers in algebra, you can plug your solution back into the original equation to see if it works. You can also use a graphing calculator to verify your graphed equations.
Is Algebra Used in Real Life?
Algebra has many practical applications in real life, including finance, engineering, and science. It is used to solve problems involving measurements, analyze data, and make predictions.
How Can I Improve My Algebra Skills?
To improve your algebra skills, practice is key. You can also seek out resources such as online tutorials, textbooks, and study groups. Don’t be afraid to ask for help when you need it!
What Are Common Mistakes to Avoid in Algebra?
Common mistakes to avoid include forgetting to follow the order of operations, using the wrong signs or symbols, and forgetting to check your answers. Taking your time, double-checking your work, and
seeking feedback can help you avoid these errors.
Algebra is an important branch of mathematics that has deep roots in science, engineering, and many other fields. Through the use of variables, algebra allows for the manipulation of unknown values
to solve complex equations and real-life problems.
In this article, readers have learned about the history of algebra, from its beginnings in ancient Babylon and Egypt, to its evolution in medieval Islamic culture, to its modern incarnations in the
19th and 20th centuries. They have discovered how algebra is used to solve problems in physics, engineering, and everyday life.
Algebra has also been shown to improve critical thinking skills, as it encourages the ability to form and test hypotheses, identify patterns, and make connections between seemingly unrelated ideas.
It is an essential tool for success in STEM fields and beyond.
In conclusion, a strong foundation in algebra is crucial for anyone seeking to pursue a career in science, technology, engineering, or mathematics. Additionally, the problem-solving skills developed
through the study of algebra can benefit individuals for a lifetime, leading to success in diverse fields and unlocking new opportunities. | {"url":"https://caps123.co.za/algebra-understanding-the-basics/","timestamp":"2024-11-05T15:43:16Z","content_type":"text/html","content_length":"108907","record_id":"<urn:uuid:eb93c12c-99c9-4f4a-8199-5a688f977ea8>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00545.warc.gz"} |
Do you really need to learn calculus?
I was talking to a friend of mine who teaches middle school math this week. It brought back all sorts of memories about my career in math, how I picked my classes over my primary and secondary
schooling, and what I would tell my teen-aged self with the benefit of hindsight if such self would actually listen to an adult back then.
I come from a family of math geeks: my sibs and my parents were all good at math, and all of the kids were on the “math team” that met after school to solve problems and compete for prizes. Looking
through my old report cards, rarely did I get a grade less than an A in any of my classes. When it came time for college though, I started out as a physics major, quickly changing to math when I got
frustrated with all the prerequisites, and eventually graduating with a roll-my-own major that combined independent study classes in science, art and math.
What many parents don’t realize until their kids have been through middle school is that there is in most districts a separation of kids into two math tracks. One is the basic math curriculum which
involves teaching algebra, trig, geometry and some statistics by the time you finish high school. The other is a more advanced series of classes that ends with students taking calculus in their
senior year in high school. If you are good at math, you end up with the latter program of study.
Why does anyone need to study calculus? There really isn’t any good reason. It is more custom than necessity. In my case, getting calculus “out of the way early,” (as I look at it now) allowed me to
get AP credit and graduate early from college. It also enabled me to take more advanced math classes too. I asked Arnold Seiken, one of my former college math professors, why anyone should take the
class. He was mostly bemused by the question: “Calculus was always part of the requirements for graduation – students assume that it is part of the burden of life and just grin and bear it. I assume
you took my courses because you liked the jokes. I can’t think of any other reason.” He was right: he was always a crack-up, in class and now in retirement. Interestingly, he told me that he got into
math by accident in high school because he couldn’t do the science labs, much like I decided. “Math was a fallback for me, I was always breaking stuff in the labs.” He was an excellent teacher, BTW.
When you are a college math major, there are basically two different career paths you hear about: to teach or to become an actuary. I wasn’t all that excited about teaching (although I did dabble
with teaching both a high school computer class and a graduate business class later on in life), and when I took the first exam of many to become an actuary, I got the lowest possible passing score.
That first exam is calculus, and my justification for the miserable score was because I hadn’t had any calculus for several years by the time I took the exam. But it didn’t bode well for a career in
that field.
But having plenty of math classes – including one on linear algebra taught by Seiken — also enabled me to have a solid foundation for graduate school study of applied math topics that were part of my
degree in Operations Research. That took me to DC and to do math modeling for supporting government policy analysis, and eventually on to general business technology.
My master’s degree was issued in the late 1970s. Back then we didn’t have personal computers, we didn’t have spreadsheets, we didn’t have data modeling contests like Kaggle. What we did have was a
mainframe on campus that you had to program yourself if you wanted to do mathematical models. Today you can use Excel to solve linear programs and other optimization problems, set up Chi Square
analyses, run simulations and other things that were unthinkable back when I was in school – not that I would know how to do these things now if you forced me to watch a bunch of videos to relearn
Seiken reminded me of a part-time job that I had in college, repairing these ancient geometric string models that were used in the 1800s to teach engineering students how to draw conic sections. I
didn’t need calculus to restring the models, although it helped to understand the underlying geometry to figure the string placement. It did get me started on my path towards problem-solving though.
And I think that is what I would tell my teenage self. Whether or not I took this or that math class, what I was good at is solving technical problems. Having a math background made it easier for me
to pick a more technical career path. Had I not moved into the calculus track, I probably would still have been interested in math of some kind, but probably wouldn’t have been as challenged to take
the advanced classes I was doing as a junior and senior in college. So yes, calculus per se isn’t really relevant to debugging a network protocol stack, or figuring out the root cause of a piece of
malware, but it does train you to learn how to pick apart these problems and find your way to an answer. Now, your kids may have a different path towards developing their own problem-solving skills,
but math was my ticket and I am glad I took the path that I did.
5 thoughts on “Do you really need to learn calculus?”
1. Thanks for your story about the place of calculus in your learning math. I had a somewhat different encounter with calculus. It was a requirement in the structural engineering curriculum at Cal
when I was a student in the mid-60s. We were supposed to use it to calculate things like bending stresses in loaded beams. I got a night clerk job at a motel so I would have plenty of
uninterrupted quiet time to study, and was part of a group of students who got together to figure out problems. After I survived the course, two things came to light: one was a huge table of
predetermined figures for standard size/material beams, that engineering practitioners used for typical structures, the second was an HP calculator that made quick work of the former hand
calculations. I have never had to use calculus since school. So my impression is, calculus showed me how to organize an effective approach to a tough situation, to be practical, and be happy for
whatever outcome. But when I recently found my course notes in the basement, I realized, looking at my own handwriting, I had no idea what the notes meant. I have erased calculus from my memory.
2. This is the perfect example of what’s been lost in education. The most important goal of education shouldn’t be to teach you skills and techniques that you will need to get and keep good
employment so you can afford to live and raise a family. The goal above all others should be to teach you to learn. Learn how to learn concrete topics and ideas. Learn to learn more abstract
philosophical topics and ideas. Learn to recognize what you need to learn. Almost no one enters the work world knowing what they need to know for the job, but hopefully they have learned to be
good learners and can quickly learn whats necessary to become a valued employee. What better abstract, out of touch with reality concept to use to push your ability to learn than advanced math
topics like calculus. If you can learn that, you can learn almost anything. It was always one of those courses that separated the men from the boys and the educational system needs those courses.
We can’t (and shouldn’t want to) make everything easy for everyone. Is eliminating calculus part of our ‘everyone deserves a trophy’ mentality? Should we get rid of courses like this just because
it’s hard and almost no one likes it? You can’t claim to be at the top of the heap if there are no challenges to differentiate yourself from others and courses like calculus represent those sorts
of challenges.
3. Interesting, and funny that yourcollege path was same as mine (and a friend from first day college!) — we started as physics majors but after two years recognized the folly in that so switched to
But I disagree about your two choices as college math major (teach or actuary); I describe choices I had as graduate school, drive a cab, and computers. Third choice won for me and friend, worked
out nicely for us both.
4. “it (calculus) does train you to learn how to pick apart these problems and find your way to an answer.” To me, that is the first reason to study advanced mathematics. It introduces a mental
framework and discipline needed for most any problem solving. Second reason is that calculus and differential equations describe how our physical world works, so I would also advocate for more
than a superficial high school study of physics.
My own college career was more contorted than yours, but the study of higher math and computer science provided an excellent foundation for a long career in the computer industry. The other
important piece was the study that ultimately lead to a B.A. in English Literature. Now I can write and communicate verbally, but with the logic, most of the time, of one who studied math.
5. Yes, you are a “gentleman and a scholar”! It is interesting that this was obvious to your advisor in ninth grade! I, too, took AP Calculus in high school and a year of calculus in college. I fell
off the bandwagon completely with advanced differential calculus — the only course I ever took in which I was clueless from the get-go! But the two years of successful calculus I did complete
surely enhanced my problem-solving and logical thinking, which — as you have known me for 30 years — can always use a boost!
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The joule ( symbol J) is a derived unit of energy, work, or amount of heat in the International System of Units. It is equal to the energy expended (or work done) in applying a force of one newton
through a distance of one meter (1 newton metre or N·m), or in passing an electric current of one ampere through a resistance of one ohm for one second. It is named after the English physicist James
Prescott Joule (1818–1889).
In terms firstly of base SI units and then in terms of other SI units:
\rm J = {}\rm \frac{kg \cdot m^2}{s^2} = N \cdot m = \rm Pa \cdot m^3={}\rm W \cdot s
where N is the newton, m is the meter, kg is the kilogram, s is the second, Pa is the pascal, and W is the watt.
One joule can also be defined as:
The work required to move an electric charge of one coulomb through an electrical potential difference of one volt, or one ‘”coulomb volt” (C·V). This relationship can be used to define the volt.
The work required to produce one watt of power for one second, or one “watt second” (W·s) (compare kilowatt hour). This relationship can be used to define the watt.
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Theoretical Principles of Capacity Check
Theoretical Principles of Capacity Check#
Relative Capacity#
The bearing capacity of a cross-section depends on the material stiffnesses, the dimensions of the cross-section, as well as the quantity and positioning of the reinforcement. A measure of the
capacity of the cross section is the relative capacity \(Ed/Rd \quad\) which compares the acting forces \(F_{Ed}\) to the the forces that the cross section can resist \(F_{Rd}\) . The relative
capacity is determined as the ratio of the length of the reaction force vector to the action force vector. In order for the relative capacity to be computed the reaction and the action force vector
have to be aligned and this is achieved by iterating for the limit strain state. The limit strain state is controlled by two strain parameters
□ \(s \quad\) identifying the \(N-My \quad\) interaction (\(s=-1.0 \quad\) pure compression, \(s=+1.0 \quad\) pure tension)
□ \(\theta_{\kappa} \quad\) the angle between \(k_{y}\) and \(k_{z}\) curvatures, as measure to the rotation of the strain \(M_{y}-M_{z} \quad\) plane
The alignment of the reaction and action force vector is determined by a double residual condition minimizing
□ \(\alpha \quad\) the angle between the force vector’s projection in \(N-M_{y} \quad\) plane
□ \(\beta \quad\) the angle between the force vectors in \(M_{y}-M_{z} \quad\) plane.
The iterative scheme for the update of strain parameters is based on the linearization of the residual formulation.
Interaction Curves#
The normal force and biaxial bending capacity interaction curves, that form the interaction surface, are generated according to a set of limit strains. Each interaction curve is defined by the
requested \(M_{y}-M_{z} \quad\) plane, which is achieved by scaling the curvatures \(k_{y}\) and \(k_{z}\) accordingly, as well as the normal force range from pure compression to full tension. Any
single point on an interaction curve defines the capacity of the cross section to carry the corresponding normal force and biaxial bending moments. This serves as a measure to evaluate if a set of
acting forces defining a discrete point in \(N-M_{y}-M_{z} \quad\) space falls outside the curve, which would indicate that the demand exceeds the capacity of the cross section and the relative
capacity ratio would be lower than \(1.0\). A section cut through the 3D surface would represent the \(M_{y}-M_{z} \quad\) interaction curve for a specific \(N_{Ed} \quad\) normal force level, as
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