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Lesson 9
Subtraction Algorithms (Part 2)
Warm-up: True or False: Does It Commute? (10 minutes)
The purpose of this True or False is to elicit insights students have about how the commutative property applies to addition and multiplication, but not subtraction. The reasoning students do here
helps to deepen their understanding of the properties of operations and how they apply to subtracting within 1,000. It will also be helpful later when students need to recognize the need to decompose
hundreds or tens to get more tens or ones.
• Display one equation.
• “Give me a signal when you know whether the equation is true and can explain how you know.”
• 1 minute: quiet think time
• Share and record answers and strategy.
• Repeat with each equation.
Student Facing
Decide if each statement is true or false. Be prepared to explain your reasoning.
• \(4 \times 5 = 5 \times 4\)
• \(125 + 200 = 200 + 125\)
• \(300 - 100 = 100 - 300\)
Activity Synthesis
• “What is different about the last equation?” (If we switch the order in subtraction, then both sides of the equal side aren't the same. If we switch the order when we subtract, we don't get the
same number.)
• Consider asking:
□ “Who can restate _____'s reasoning in a different way?”
□ “Does anyone want to add on to _____'s reasoning?”
□ “Can we make any generalizations based on the statements?”
Activity 1: Revise Subtraction Work (15 minutes)
The purpose of this activity is for students to examine an error in an algorithm in which a larger digit is subtracted from a smaller digit in the same place value position. In such a case, it is
common for students to subtract the smaller digit from the larger digit instead, not realizing that subtraction is not commutative. The given algorithm here shows the numbers in expanded form to help
students see that it is necessary to first decompose a hundred into tens before the 50 can be subtracted from 20.
When students make sense of and correct Lin’s mistake, they construct viable arguments and critique the reasoning of others (MP3).
• Groups of 2
• Give students access to base-ten blocks.
• Display the image of Lin’s work.
• “Now let’s look at how Lin subtracted 156 from 428. Take a minute to examine what she did.”
• 1–2 minutes: quiet think time
• “Work with your partner to describe the mistake and what you would tell or show Lin so she can revise her work.”
• 5 minutes: partner work time
• Monitor for students who:
□ use base-ten blocks or an algorithm to make sense of Lin’s mistake
□ decompose a hundred into 10 tens before subtracting 50 from 20, and showing this process by exchanging base-ten blocks or rewriting 400 as \(300 + 100\) and combining the 100 with 20
• Identify students who used these strategies and select them to share during synthesis.
Student Facing
Lin’s work for finding the value of \(428 - 156\) is shown.
1. What error do you see in Lin's work?
2. What would you tell or show Lin so she can revise her work?
Advancing Student Thinking
If students don’t mention the error in Lin's work, consider asking:
• “What mistake did Lin make when subtracting?”
• “How could we use base-ten blocks to help Lin revise her work?”
Activity Synthesis
• “How would you describe Lin’s mistake?” (She tried to subtract 20 from 50 when you’re subtracting 50 from 20. She needed to decompose a hundred to get more tens.)
• Select previously identified students to share what they would tell or show Lin so she can revise her work.
• If no students suggest the following revision to Lin's work, display the algorithm and ask students to explain the revision:
• “Keep Lin’s mistake in mind as we practice using this subtraction algorithm in the next activity.”
Activity 2: Try the Algorithm (20 minutes)
The purpose of this activity is for students to practice using the subtraction algorithm introduced in a previous lesson. Provide base-ten blocks for students who choose to use them to support their
reasoning about the algorithm.
MLR8 Discussion Supports: Synthesis: Before students share their reasoning, remind them to use words such as decompose, ones, tens, and hundreds.
Advances: Speaking, Representing
Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select at least 3 of 5 problems to complete.
Supports accessibility for: Organization, Attention, Social-emotional skills
• Groups of 2
• Display Kiran’s algorithm from the previous lesson.
• “Here’s a subtraction algorithm you saw in an earlier lesson. What might be the first thing you’d do if you are to use this algorithm to find the value of the subtraction expressions in the
activity?” (Write the numbers in expanded form and stack them.)
• 1 minute: quiet think time
• Share responses.
• Give students access to base-ten blocks.
• “Take some quiet time to try this algorithm. Check in with your partner if you have questions.”
• 5–7 minutes: independent work time
• If students have questions about the notation used to record the decomposition of a hundred or ten into more tens or ones, consider asking:
□ “Is there any place in the problem where you don’t have enough tens or ones?”
□ “How could you get more tens (or ones)?”
□ “How could you record a hundred being decomposed into 10 tens (or a ten decomposed into 10 ones)?”
Student Facing
Here is a subtraction algorithm you saw in an earlier lesson:
Try using this algorithm to find the value of each difference. Show your reasoning. Organize it so it can be followed by others.
1. \(283 - 159\)
2. \(425 - 192\)
3. \(639 - 465\)
4. \(591 - 128\)
5. \(832 - 575\)
Advancing Student Thinking
If students do not record the multiple decompositions in the last problem, consider asking:
• “What units did you need to decompose in this problem?”
• “Where would it make the most sense to you to record how you decomposed a hundred into more tens? A ten into more ones?”
Activity Synthesis
• Select students to share their reasoning for 2–3 problems. Choose problems to focus on based on common questions that came up. Be sure to discuss the last problem, which requires decompositions
of both a ten and a hundred.
• For the last problem, ask: “How did you decide how to record the ten and hundred that needed to be decomposed?” (I started subtracting the ones and decomposed a ten into 10 ones so I had already
crossed off the 30 and written 20. When I decomposed the hundred into tens, I just decided to write the 120 on top of the 20 like I wrote the 20 on top of the 30.)
• Record the completed algorithm, showing the decompositions of tens and hundreds.
• Consider asking:
□ “Where were there not enough tens or ones to subtract?”
□ “What was decomposed and how was it recorded?”
□ “Did you notice any places where you might have made the error we saw in Lin's work?"
Lesson Synthesis
Display student work from a problem in the second activity, such as:
“Suppose a classmate says this problem has been changed into a completely different problem because the 832 has been crossed out. How would you explain the crossed-out numbers to them?” (The 832 is
still there. It’s just been reorganized as 700 plus 120, which is 820, and then 820 plus 12 is 832. So, it’s still 832. It’s been grouped differently so we can subtract in every place value.)
Cool-down: How Did Andre Subtract? (5 minutes) | {"url":"https://curriculum.illustrativemathematics.org/k5/teachers/grade-3/unit-3/lesson-9/lesson.html","timestamp":"2024-11-11T08:14:38Z","content_type":"text/html","content_length":"94268","record_id":"<urn:uuid:8782eb0d-1b84-40ef-9aa6-eaeba8f83b9c>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00630.warc.gz"} |
Millimeters to Feet Converter | Pocket Pence
Millimeters to Feet Converter
What is millimeter?
The base unit of length in the metric system is the millimetre, which is equal to one thousandth of a meter. There is one thousand millimetres in ametre. There is a tenth of a centimetre. One
millimetre is equal to 1000 micrometres. Since an inch is officially defined as 25.4 millimetres, a millimetre is equal to 5127 of an inch.
What is foot?
The foot is an important part of the body. The terminal portion of a limb allows movement. The terminal part of the leg made up of one or more bones is known as the foot.
How to use the mm ft calculator
Simple steps to use this converter:
- Use the top drop down menu under Unit Converter to choose the category of the type of calculator ranging from length, area, math, volume to voltage, power, and many more.
- Choose the desired unit to convert from in the left black drop down bar and type in the number to convert.
- Then choose the unit to convert to in the right black drop down bar and type in the number to convert.
How to convert millimeters feet
It is easier to understand the conversion of mm to ft by looking at a step by step example.
- Multiply by the conversion factor of 1 mm = 0.0033 ft.
- So 10 mm in ft would be 10 mm x 0.0033 = 0.033 ft
Popular FAQs
When was the millimeter invented?
The first practical realization of the metric system came during the French Revolution, after the existing system of measures had become impractical for trade and was replaced by a decimal system.
What is millimeter used for?
There are meters used to measure distances and lengths. A millimeter is roughly the size of the wire used in a standard paper clip. There are 10mm in a centimeter and 1000mm in a meter in the metric
How to convert mm mm ft calculator
5.94 ft. Follow these steps to obtain the similar value:
Multiply 1800 millimeters by the base conversion rate of 0.0033 to get feet
1800 mm x 0.0033 = 5.94 ft
How to convert mm mm ft calculator
3.564 ft. Follow these steps to obtain the similar value:
Multiply 1080 millimeters by the base conversion rate of 0.0033 to get feet
1080 mm x 0.0033 = 3.564 ft
How to convert ft ft mm calculator
1212.1212 mm. Follow these steps to obtain the similar value:
Multiply 4 feet by the base conversion rate of 303.0303 to get millimeters
4 ft x 303.0303 = 1212.1212 mm
How to convert ft ft mm calculator
1515.1515 mm. Follow these steps to obtain the similar value:
Multiply 5 feet by the base conversion rate of 303.0303 to get millimeters
5 ft x 303.0303 = 1515.1515 mm
How to convert ft ft mm calculator
909.0909 mm. Follow these steps to obtain the similar value:
Multiply 3 feet by the base conversion rate of 303.0303 to get millimeters
3 ft x 303.0303 = 909.0909 mm
How to convert ft ft mm calculator
1818.1818 mm. Follow these steps to obtain the similar value:
Multiply 6 feet by the base conversion rate of 303.0303 to get millimeters
6 ft x 303.0303 = 1818.1818 mm
How many feet Millimeters?
0.0033 ft. You can calculate by:
1 millimeter is equivalent to 0.0033 feet
1 mm x 0.0033 = 0.0033 ft
How many millimeters foot
303.0303 mm. You can calculate by:
1 foot is equivalent to 303.0303 millimeters
1 ft x 303.0303 = 303.0303 mm
Examples of Millimeters to Feet rounded to 4th decimal place
Millimeters (mm) Feet (ft)
0.05 mm 0.0002 ft
0.5 mm 0.0016 ft
1 mm 0.0033 ft
2 mm 0.0066 ft
3 mm 0.0099 ft
4 mm 0.0132 ft
5 mm 0.0165 ft
6 mm 0.0198 ft
7 mm 0.0231 ft
8 mm 0.0264 ft
9 mm 0.0297 ft
10 mm 0.033 ft
20 mm 0.066 ft
30 mm 0.099 ft
40 mm 0.132 ft
50 mm 0.165 ft
60 mm 0.198 ft
70 mm 0.231 ft
80 mm 0.264 ft
90 mm 0.297 ft
100 mm 0.33 ft
Some of Millimeters to units in same category
Base Unit To Unit To Unit Full Name
1 mm 0.0 nMi Nautical Miles
1 mm 0.0 mi Miles
1 mm 0.001 fathom Fathoms
1 mm 0.003 ft Feet
1 mm 0.003 ft-us US Survey Feet
1 mm 0.001 yd Yards
1 mm 0.039 in Inches
1 mm 0.0 km Kilometers
1 mm 0.001 m Meters
1 mm 0.1 cm Centimeters
1 mm 1.0 mm Millimeters
1 mm 1000.0 μm Micrometers
1 mm 1000000.0 nm Nanometers | {"url":"https://www.pocketpence.co.uk/mm-to-ft.html","timestamp":"2024-11-14T07:56:25Z","content_type":"text/html","content_length":"136964","record_id":"<urn:uuid:885b7a15-9dd4-4301-bf67-e3f65d4af562>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00027.warc.gz"} |
Rhythm: The Music of the Spheres
"Music is a hidden arithmatic exercise of the soul, which does not know that it is dealing with numbers, because it does many things by way of unnoticed conceptions which with clear
conception it could not do. Those who believe that nothing can happen in the soul of which the soul is not conscious are wrong. For this reason the soul, although not realizing that it is
involved in mathematical computation, still senses the effect of this unnoticeable forming of numbers either as a resultant feeling of well-being in the case of harmonies or as discomfort in
the case of disharmonies."
Gottfried Wilhelm Leibniz
Rhythm is the basis of harmonic proportions and intervals, fundamental physical mechanisms. This harmonious equilibration was expounded upon by Pythagoras, father of Philosophy. Pythagorean thought
formed the basis of the philosophy of Plato, and later Neo-Pythagoreans and Platonists, and greatly influenced the development of western science.
Pythagoras laid the foundation for a holistic science which integrated spiritual, ethical, and metaphysical, as well as practical techniques. Pythagoras is famous for his axiomatic viewpoint that
"there is geometry in the humming of the strings. There is music in the spacing of the spheres." From Plotinus we hear, "All music based upon melody and rhythm, is the earthly representative of
heavenly music."
And from Sufi Hazrat Inayat Khan,
"When one looks at the cosmos, the movements of the stars and planets, the law of vibration and rhythm, all perfect and unchanging, it shows that the cosmic system is working by the law of music, the
law of harmony; and whenever that harmony in the cosmic system is lacking in any way, then in proportion disaster comes to the world, and its influence is seen in the many destructive forces which
are manifest there. The whole of astrological law and the science of magic and mysticism behind it, are based upon music."
Pythagoras systematized the laws which allow the creation of stringed instruments: musical scale intervals (octaves, fifths, fourths, thirds). He recognized that these fundamentally abstract
relationships pervade all creation--even matter irself. In music, as in nature, a wave is a shape in motion. Each note has a wave-shape.
The octave comes from exactly doubling, or halving the string length, that is in 1:2 proportion, while the harmonious fifth has a 2:3 ratio and the fourth 3:4. There is also the less obvious 4:5
interval of the third, and even less obvious consonances.
Any tone in the overtone scale is higher than the preceding tone by precisely one whole number. These are the so-called harmonics. The lower the proportions of the numbers, the stronger the
consonance, the more harmonious the sound of the two tones together. The primal polarity ratio of 1:2 is the most harmonious to our ears which are biologically geared to seven basic laws of harmony
based on the primal law of whole-number quanta (which prevails in physics as well as music):
THE SEVEN LAWS OF HARMONY
1. the overtone scale
2. the interval proportions
3. the division of the octave into twelve semitones
4. the difference between consonance and dissonance, the consonance growing as the proportion of the numbers gets smaller
5. the difference between major and minor, the major proportion being the most frequent by far
6. the predominance of the 1:2 polarity, the octave
7. the law of the Lambdoma (a column of numbers written in the form of the Greek letter lambda, whose right leg consists of whole numbers going from one to infinity while t he left leg contains
the fractions of these same whole numbers, so that the coordinates of the open isosceles triangle follow the scale of overtones or undertones). There are correspondences in physics, acoustics,
arithmatic, geometry, crystallography, cybernetics, theology and philosophy, the genetic code and I Ching.
The Mysticism of Sound: Music, The Power of the Word, and Cosmic Language (Sufi Message of Hazrat Inayat Khan Ser: Vol. 2)
3 comments:
Hmmm.... #3 is a little out of place. 12 comes from the pythagorean tradition, but pythagoras used chains of fifths to produce all the other intervals, thus his major third was (3^4)/(2^4) or 81/
64 instead of the shimmeringly pure 5/4. A tradition that limits itself to 12 semitones will always run into problems harmonically. This stems fromt the fact that no chain of a single interval
(i.e. 3/2) ever maps onto a perfect octave. There's always a comma. Limiting notes to 12 usually makes the false assumption that there is only one major second and it works for all situations
(ignoring distinction between 10/9 and 9/8). It leaves out all septimal intervals (including the blue third 7/6), except for an approximation of one of the tritones. And if there are only twelve
notes, consonance will be compromised at some point and so will dissonance - all of the exotic, rational dissonances that give variety and flavor to music are ignored or at best, rudely
"approximated." To truly respect the overtone series, harmony must be more flexible than 12 notes allow. (Or the 12 notes themselves must be flexible).
Hucbald said...
Actually David, Anon. is wrong.
Twelve Tone Equal Temperament (TTET from now on) is actually exactly what the harmonic overtone series implies is the most logical solution to the issue of temperament for fixed pitch
If you place a stack of seven perfect 2:1 octaves next to a stack of twelve 3:2 perfect fifths, you get the well known Pythagorean comma, which is slightly less than 24 cents (Less than 1/4 of a
semitone after all of those "just" iterations).
Obviously, the pitch identity of all musical systems will require octave equivalance, so you can't stretch the octaves (Philosophically: Piano tuners do it all the time, which relates to the
human perception of pitch, which is beyond the scope of this philosophical discussion), but contracting the fifths presents no problem at all, and 1/12 of the comma is less than two cents (This
gives TTET).
The average limit of human perception of pitch differences is seven cents (Most trained musicians can discern three, exceptional ones can discern two), so contracting the fifths causes almost NO
auditory difference. IOW, people who say that TTET destroyed music or that the harmonic series does not apply to TTET... ARE IDIOTS!
No human ear is decieved by the irrational numbers that represent intervals in TTET: They are perceived in virtually exactly the same way as the just ratios are, so there is little difference
between TTET and 7-limit just. None at all in most practical senses... but not all.
Sure, in so-called well-tempered systems each key has a "character" and Baroque keyboard music probably ought to use those systems (I have a Davitt Moroney recording of Bach's Art of Fugue which
uses them and is sublime), since Baroque keyboard music was composed within those systems, but the series itself implies that TTET is the best solution.
Look, the series implies that there are twenty-four possible tonics - twelve major and twelve minor - and that all of these tonics are available WITHIN A SINGLE COMPOSITION. So, in order for all
of them to SOUND THE SAME relative to one another (TO BE EQUALS), TTET is the only solution: The most logical one which is perfectly equitable according to the implications of the series itself.
I wrote a book on this subject - The Musical Implications of the Harmonic Overtone Series - which I researched for over twenty years, so these on-the-verge mindless musings which often pollute
contemporary musical thinking irritate me... to say the least. TTET did not ruin music: It is the long sought solution to the problem of equal pitch instruments... and has been since frets were
first tied to a lute amost a milennium ago.
Oh... Happy New Year, David! ;^)
Listen -- ultrasound ionizes chemicals that then generate electromagnetic fields, light and even spacetime travel.
The Law of Pythagoras -- the Tetrad -- is based on asymmetry with C to G as 2:3 (YANG) and G to C as 3:4 (YIN).
The square root of two is approximated as the Tritone -- 9/8 cubed (professor Ernest McClain) -- and this is why it's the "devil's interval." The secret of free energy alchemy was lost.
I have a "book" of information online about the real meaning behind the overtone series -- infinite transduction.
drew hempel, MA | {"url":"https://davidvaldez.blogspot.com/2007/01/rhythm-music-of-spheres.html","timestamp":"2024-11-15T03:04:45Z","content_type":"application/xhtml+xml","content_length":"97472","record_id":"<urn:uuid:e0867484-ca0e-4dd0-8377-fbb089f3401c>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00896.warc.gz"} |
About the supposed factoring of a 4096 bit RSA key
News about a broken 4096 bit RSA key are not true. It is just a faulty copy of a valid key.
Earlier today a
blog post claiming the factoring of a 4096 bit RSA key was published
and quickly made it
to the top of Hacker News
. The key in question was the PGP key of a well-known Linux kernel developer. I already
commented on Hacker News
why this is most likely wrong, but I thought I'd write up some more details. To understand what is going on I have to explain some background both on RSA and on PGP keyservers. This by itself is
pretty interesting.
RSA public keys consist of two values called N and e. The N value, called the modulus, is the interesting one here. It is the product of two very large prime numbers. The security of RSA relies on
the fact that these two numbers are secret. If an attacker would be able to gain knowledge of these numbers he could use them to calculate the private key. That's the reason why RSA depends on the
hardness of the factoring problem. If someone can factor N he can break RSA. For all we know today factoring is hard enough to make RSA secure (at least as long as there are no large quantum
Now imagine you have two RSA keys, but they have been generated with bad random numbers. They are different, but one of their primes is the same. That means we have N1=p*q1 and N2=p*q2. In this case
RSA is no longer secure, because calculating the greatest common divisor (GCD) of two large numbers can be done very fast with the euclidean algorithm, therefore one can calculate the shared prime
It is not only possible to break RSA keys if you have two keys with one shared factors, it is also possible to take a large set of keys and find shared factors between them. In 2012 Arjen Lenstra and
his team
published a paper using this attack on large scale key sets
and at the same time
Nadia Heninger and a team at the University of Michigan
independently also performed this attack. This uncovered a lot of vulnerable keys on embedded devices, but these were mostly SSH and TLS keys. Lenstra's team however also found two vulnerable PGP
keys. For more background you can watch
this 29C3 talk by Nadia Heninger, Dan Bernstein and Tanja Lange
PGP keyservers have been around since quite some time and they have a property that makes them especially interesting for this kind of research: They usually never delete anything. You can add a key
to a keyserver, but you cannot remove it, you can only mark it as invalid by revoking it. Therefore using the data from the keyservers gives you a large set of cryptographic keys.
Okay, so back to the news about the supposedly broken 4096 bit key: There is a service called
where you can upload a key and it'll check it against a set of known vulnerable moduli. This service identified the supposedly vulnerable key.
The key in question has the key id e99ef4b451221121 and belongs to the master key bda06085493bace4. Here is the vulnerable modulus:
c844a98e3372d67f 562bd881da8ea66c a71df16deab1541c e7d68f2243a37665 c3f07d3dd6e651cc d17a822db5794c54 ef31305699a6c77c 043ac87cafc022a3 0a2a717a4aa6b026 b0c1c818cfc16adb aae33c47b0803152
f7e424b784df2861 6d828561a41bdd66 bd220cb46cd288ce 65ccaf9682b20c62 5a84ef28c63e38e9 630daa872270fa15 80cb170bfc492b80 6c017661dab0e0c9 0a12f68a98a98271 82913ff626efddfb f8ae8f1d40da8d13
a90138686884bad1 9db776bb4812f7e3 b288b47114e486fa 2de43011e1d5d7ca 8daf474cb210ce96 2aafee552f192ca0 32ba2b51cfe18322 6eb21ced3b4b3c09 362b61f152d7c7e6 51e12651e915fc9f 67f39338d6d21f55
fb4e79f0b2be4d49 00d442d567bacf7b 6defcd5818b050a4 0db6eab9ad76a7f3 49196dcc5d15cc33 69e1181e03d3b24d a9cf120aa7403f40 0e7e4eca532eac24 49ea7fecc41979d0 35a8e4accea38e1b 9a33d733bea2f430
362bd36f68440ccc 4dc3a7f07b7a7c8f cdd02231f69ce357 4568f303d6eb2916 874d09f2d69e15e6 33c80b8ff4e9baa5 6ed3ace0f65afb43 60c372a6fd0d5629 fdb6e3d832ad3d33 d610b243ea22fe66 f21941071a83b252
201705ebc8e8f2a5 cc01112ac8e43428 50a637bb03e511b2 06599b9d4e8e1ebc eb1e820d569e31c5 0d9fccb16c41315f 652615a02603c69f e9ba03e78c64fecc 034aa783adea213b
In fact this modulus is easily factorable, because it can be divided by 3. However if you look at the master key bda06085493bace4 you'll find another subkey with this modulus:
c844a98e3372d67f 562bd881da8ea66c a71df16deab1541c e7d68f2243a37665 c3f07d3dd6e651cc d17a822db5794c54 ef31305699a6c77c 043ac87cafc022a3 0a2a717a4aa6b026 b0c1c818cfc16adb aae33c47b0803152
f7e424b784df2861 6d828561a41bdd66 bd220cb46cd288ce 65ccaf9682b20c62 5a84ef28c63e38e9 630daa872270fa15 80cb170bfc492b80 6c017661dab0e0c9 0a12f68a98a98271 82c37b8cca2eb4ac 1e889d1027bc1ed6
664f3877cd7052c6 db5567a3365cf7e2 c688b47114e486fa 2de43011e1d5d7ca 8daf474cb210ce96 2aafee552f192ca0 32ba2b51cfe18322 6eb21ced3b4b3c09 362b61f152d7c7e6 51e12651e915fc9f 67f39338d6d21f55
fb4e79f0b2be4d49 00d442d567bacf7b 6defcd5818b050a4 0db6eab9ad76a7f3 49196dcc5d15cc33 69e1181e03d3b24d a9cf120aa7403f40 0e7e4eca532eac24 49ea7fecc41979d0 35a8e4accea38e1b 9a33d733bea2f430
362bd36f68440ccc 4dc3a7f07b7a7c8f cdd02231f69ce357 4568f303d6eb2916 874d09f2d69e15e6 33c80b8ff4e9baa5 6ed3ace0f65afb43 60c372a6fd0d5629 fdb6e3d832ad3d33 d610b243ea22fe66 f21941071a83b252
201705ebc8e8f2a5 cc01112ac8e43428 50a637bb03e511b2 06599b9d4e8e1ebc eb1e820d569e31c5 0d9fccb16c41315f 652615a02603c69f e9ba03e78c64fecc 034aa783adea213b
You may notice that these look pretty similar. But they are not the same. The second one is the real subkey, the first one is just a copy of it with errors.
If you run a batch GCD analysis on the full PGP key server data you will find a number of such keys (Nadia Heninger has published
code to do a batch GCD attack
). I don't know how they appear on the key servers, I assume they are produced by network errors, harddisk failures or software bugs. It may also be that someone just created them in some experiment.
The important thing is: Everyone can generate a subkey to any PGP key and upload it to a key server. That's just the way the key servers work. They don't check keys in any way. However these keys
should pose no threat to anyone. The only case where this could matter would be a broken implementation of the OpenPGP key protocol that does not check if subkeys really belong to a master key.
However you won't be able to easily import such a key into your local GnuPG installation. If you try to fetch this faulty sub key from a key server GnuPG will just refuse to import it. The reason is
that every sub key has a signature that proves that it belongs to a certain master key. For those faulty keys this signature is obviously wrong.
Now here's my personal tie in to this story: Last year I started a project to analyze the data on the PGP key servers. And at some point I thought I had found a large number of vulnerable PGP keys –
including the key in question here. In a rush I wrote a mail to all people affected. Only later I found out that something was not right and I wrote to all affected people again apologizing. Most of
the keys I thought I had found were just faulty keys on the key servers.
The code I used
to parse the PGP key server data is public
, I also
wrote a background paper
and did a talk at the BsidesHN conference.
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Defining Security | EasyCrypt
Now that we have a scheme, we can define its security. For the time being, EasyCrypt exclusively allows exact—rather than asymptotic—security definitions. As such, we will only consider exact
security definitions. Furthermore, we will be a bit more generic than necessary, starting with a definition of security for any symmetric nonce-based encryption scheme, not just for the particular
scheme we are currently interested in. As alluded to before, in general, we aim to be rather generic when formalizing and proving security in EasyCrypt. The reason for this is that, if done properly,
this can significantly increase (and cannot decrease) the strength and reusability of the result. Nevertheless, at this point, we will keep things somewhat more concrete than usual for clarity
purposes. Later on, we will cover writing and reusing fully generic proof components.
IND$-NRCPA Security for Symmetric Nonce-Based Encryption Schemes, Pen-and-Paper
For the security property, we consider INDistinguishability from RANDOM ciphertexts under Nonce-Respecting Chosen-Plaintext Attacks (IND$-NRCPA): An adversary with access to a chosen-plaintext
oracle—which takes a nonce and plaintext as input, outputs a ciphertext, and does not allow for repeating nonces—should not be able to distinguish (except with a "small" probability) the actual
encryption scheme with a fixed and properly generated secret key from an oracle that returns freshly and uniformly sampled ciphertexts. Intuitively, this property captures the extent to which the
ciphertexts of a (symmetric nonce-based) encryption scheme can be distinguished from uniformly random ciphertexts: The smaller this extent, the better the security of the scheme. For defining this
indistinguishability property, it’s crucial to ensure that nonces are not used more than once, as this could trivially break security; indeed, if we would not impose this restriction, the adversary
could distinguish by simply reusing a nonce.
More formally, consider the two oracles shown below, $\mathcal{O}^{CPA\textrm{-}real}_{\Sigma}$ (which is defined with respect to an abstract symmetric nonce-based encryption scheme $\Sigma$,
instantiable by any concrete such scheme) and $\mathcal{O}^{CPA\textrm{-}ideal}$.
\begin{align*} \begin{align*} & \underline{\smash{\mathcal{O}^{CPA\textrm{-}real}_{\Sigma}}}\\ & \begin{align*} & \underline{\smash{\mathsf{init}()}}\\ & \left\lfloor~ \begin{align*} & k \
operatorname{\smash{\overset{\}{\leftarrow}}} \Sigma.\mathsf{KGen}()\\ & \mathrm{log} \leftarrow [\ ] \end{align*} \right. \end{align*} \\ & \begin{align*} & \underline{\smash{\mathsf{enc}(n, m)}}\\
& \left\lfloor~ \begin{align*} & \textsf{if}\ n otin \mathrm{log}\\ & \left\lfloor~ \begin{align*} & \mathrm{log} \leftarrow n\ ||\ \mathrm{log}\\ & c \leftarrow \Sigma.\mathsf{Enc}(k, n, m)\\ & \
textsf{return}\ c \end{align*} \right.\\ & \textsf{else}\\ & \left\lfloor~ \begin{align*} & \textsf{return}\ \bot \end{align*} \right. \end{align*} \right. \end{align*} \end{align*} &&&&&&&& \begin
{align*} & \underline{\smash{\mathcal{O}^{CPA\textrm{-}ideal}}}\\ & \begin{align*} & \underline{\smash{\mathsf{init}()}}\\ & \left\lfloor~ \begin{align*} & \\ & \mathrm{log} \leftarrow [\ ] \end
{align*} \right. \end{align*} \\ & \begin{align*} & \underline{\smash{\mathsf{enc}(n, m)}}\\ & \left\lfloor~ \begin{align*} & \textsf{if}\ n otin \mathrm{log}\\ & \left\lfloor~ \begin{align*} & \
mathrm{log} \leftarrow n\ ||\ \mathrm{log}\\ & c \operatorname{\smash{\overset{\}{\leftarrow}}} \mathcal{U}_{\mathcal{C}}\\ & \textsf{return}\ c \end{align*} \right.\\ & \textsf{else}\\ & \left\
lfloor~ \begin{align*} & \textsf{return}\ \bot \end{align*} \right. \end{align*} \right. \end{align*} \end{align*} \end{align*}
Evidently, these oracles are extremely similar: The only difference between the two concerns the creation of ciphertexts (and the corresponding initialization differences). Then, we capture the
"insecurity" of a (symmetric nonce-based) encryption scheme $\Sigma$ against a nonce-respecting chosen-plaintext distinguisher $\mathcal{D}$—i.e., a (potentially probabilistic) algorithm $\mathcal{D}
$ with the appropriate interface—as the (absolute) difference between the likelihood of (1) $\mathcal{D}$ outputting $\texttt{true}$ when given access to (the $\mathsf{enc}$ procedure of) $\mathcal
{O}^{CPA\textrm{-}real}_{\Sigma}$ and (2) $\mathcal{D}$ outputting $\texttt{true}$ when given access to (the $\mathsf{enc}$ procedure of) $\mathcal{O}^{CPA\textrm{-}ideal}$. Conceptually, the truth
value output by the distinguisher can be interpreted as the distinguisher's "guess" of which oracle it was given. In general terms, this (absolute) difference is often called the advantage of $\
mathcal{D}$ in distinguishing $\Sigma$ from random; however, since we have already established a slick name for the exact security property, we will be more precise and refer to it as the advantage
of $\mathcal{D}$ against IND$-NRCPA of $\Sigma$. Mathematically, this advantage is expressed as follows.
$\mathsf{Adv}^{\mathrm{IND\\textrm{-}NRCPA}}_{\Sigma}(\mathcal{D}) = \left|\mathsf{Pr}\left[\mathsf{Exp}^{\mathrm{IND\\textrm{-}NRCPA}}_{\mathcal{D}, \mathcal{O}^{CPA\textrm{-}real}_{\Sigma}} = 1\
right] - \mathsf{Pr}\left[\mathsf{Exp}^{\mathrm{IND\\textrm{-}NRCPA}}_{\mathcal{D}, \mathcal{O}^{CPA\textrm{-}ideal}} = 1\right]\right|$
Here, the experiment $\mathsf{Exp}^{\mathrm{IND\\textrm{-}NRCPA}}_{\mathcal{D}, \mathcal{O}}$ is defined below. This experiment is rather straightforward: It initializes the given oracle, runs the
distinguisher while providing access to the $\mathsf{enc}$ function of the oracle, and directly outputs the return value received from the distinguisher.
\begin{align*} & \underline{\smash{\mathsf{Exp}^{\mathrm{IND\\textrm{-}NRCPA}}_{\mathcal{D}, \mathcal{O}}}}\\ & \left\lfloor \begin{align*} & \mathcal{O}.\mathsf{init}()\\ & b \operatorname{\smash{\
overset{\}{\leftarrow}}} \mathcal{D}^{\mathcal{O}.\mathsf{enc}}.\mathsf{distinguish}()\\ & \textsf{return}\ b \end{align*} \right. \end{align*}
Surely, with these oracle and game definitions, $\mathsf{Adv}^{\mathrm{IND\\textrm{-}NRCPA}}_{\Sigma}(\mathcal{D})$ matches the intuitive description of insecurity we gave earlier. Namely, because $\
mathsf{Exp}^{\mathrm{IND\\textrm{-}NRCPA}}_{\mathcal{D}, \mathcal{O}^{CPA\textrm{-}real}_{\Sigma}}$ and $\mathsf{Exp}^{\mathrm{IND\\textrm{-}NRCPA}}_{\mathcal{D}, \mathcal{O}^{CPA\textrm{-}ideal}}$
merely differ in the oracle they provide to $\mathcal{D}$, a difference in probability of $\mathcal{D}$ outputting a certain truth value (arbitrarily chosen to be 1, or $\texttt{true}$, here) between
the experiments must be a consequence of distinguishing the oracles somehow. Certainly, this is precisely what $\left|\mathsf{Pr}\left[\mathsf{Exp}^{\mathrm{IND\\textrm{-}NRCPA}}_{\mathcal{D}, \
mathcal{O}^{CPA\textrm{-}real}_{\Sigma}} = 1\right] - \mathsf{Pr}\left[\mathsf{Exp}^{\mathrm{IND\\textrm{-}NRCPA}}_{\mathcal{D}, \mathcal{O}^{CPA\textrm{-}ideal}} = 1\right]\right|$ captures.
Finally, we say that a (symmetric nonce-based encryption) scheme $\Sigma$ is IND$-NRCPA secure if, for any nonce-respecting chosen-plaintext adversary $\mathcal{D}$, $\mathsf{Adv}^{\mathrm{IND\\
textrm{-}NRCPA}}_{\Sigma}(\mathcal{D})$ is "small". Because we exclusively consider exact security, "small" essentially means "bounded by other concrete values/probabilities that we believe are small
in practice".
To directly read further on the security definitions of PRFs for the pen and paper setting, jump here.
IND$-NRCPA Security for Symmetric Nonce-Based Encryption Schemes, EasyCrypt
To formalize the above-discussed oracles, adversary class, and experiment in EasyCrypt, we will make use of module types and modules, as well as several libraries from the standard library.
Specifically, we will make use of the AllCore.ec, List.ec, and Distr.ec libraries for the definitions and properties of basic concepts, lists, and distributions, respectively. To have those theories
available, we issue the following command at the beginning of the file.
require import AllCore List Distr.
The keyword require loads the content of a library and the import keyword makes all the symbols available without qualification. For more information regarding loading and importing, click here.
NRCPA Oracle Type
Before formalizing the oracles, we will formalize their type. From the pen-and-paper definition of the oracles, we can see that they implement two algorithms: $\mathsf{init()}$ and $\mathsf{enc}(n,
m)$, where $n$ is a nonce, $m$ is a plaintext, and $\mathsf{enc}(n, m)$ outputs either a valid ciphertext or an indication of failure ($\bot$). The module type that captures this is shown below.
module type NRCPA_Oraclei = {
proc init() : unit
proc enc(n : nonce, m : ptxt) : ctxt option
Here, notice that the procedures' output types. First, the output type of init is unit, which is a built-in type that only contains a single value. Among others, this type is used as the type of the
return value of procedures that do not return an actual value. Certainly, we do not expect the initialization procedure of a NRCPA oracle to return anything; as such, we use unit to denote its output
type. Second, the output type of enc is ctxt option (instead of the ctxt type you might have expected). option is an example of a type constructor (as is distr, which we briefly mentioned earlier).
Such constructors can be used to construct types by instantiating them (using prefix notation) with already-existing types. In the specific case of option, any type—say t—can be used to create a
corresponding option type that is denoted by t option. This option type contains a value Some x for each value x of type t, and an additional value None. In the formalization of the oracles, we use
the ctxt option type as a convenient way to let enc return either a valid ciphertext (as Some c where c is a ciphertext) or an indication of failure (as None).
Real NRCPA Oracle
Next, we formalize $\mathcal{O}^{CPA\textrm{-}real}_{\Sigma}$, the real oracle. Particularly, we do so using a module of type NRCPA_Oraclei. However, in contrast to the encryption scheme we
formalized as a module before, $\mathcal{O}^{CPA\textrm{-}real}_{\Sigma}$ is defined with respect to some other entity $\Sigma$ from a certain class; in this case, this is the class of symmetric
nonce-based encryption schemes. In EasyCrypt, we formalize this using a so-called functor or higher-order module—i.e., a module parameterized on other module(s)—that takes a module of the type that
represents this class; indeed, here this is the previously-defined NBEncScheme module type. So, we can formalize $\mathcal{O}^{CPA\textrm{-}real}_{\Sigma}$ as follows.
module O_NRCPA_real (S : NBEncScheme) : NRCPA_Oraclei = {
var k : key
var log : nonce list
proc init() : unit = {
k <@ S.kgen();
log <- [];
proc enc(n : nonce, m : ptxt) : ctxt option = {
var c : ctxt;
var r : ctxt option;
if (! (n \in log)) {
log <- n :: log;
c <@ S.enc(k, n, m);
r <- Some c;
} else {
r <- None;
return r;
Intuitively, the definition of O_NRCPA_real can be interpreted as follows: Given any scheme S that implements the expected procedures (or minimal syntax) of a symmetric nonce-based encryption scheme,
we can construct a NRCPA oracle by implementing the expected procedures (or minimal syntax) in the following way, using (the expected procedures of) S. Certainly, this means that the procedures of
O_NRCPA_real can formally only be given a meaning (and, hence, be referred/called from other code) if the module parameter is instantiated. In other words, O_NRCPA_real.init and O_NRCPA_real.enc are
not well-defined procedures, but O_NRCPA_real(M).init and O_NRCPA_real(M).enc are (where M is a module of type NBEncScheme). Contrarily, the module variables of O_NRCPA_real (i.e., k and log) are
independent of the instantiation of its module parameter: there is only ever one O_NRCPA_real.k and one O_NRCPA_real.log, even if O_NRCPA_real is instantiated multiple times with different modules.
Therefore, it is possible to refer to O_NRCPA_real.k and O_NRCPA_real.log, but not to O_NRCPA_real(M).k and O_NRCPA_real(M).log (again, where M is a module of type NBEncScheme).
From pen-and-paper to computer
Looking at the actual code of O_NRCPA_real, we can see that it is a relatively straightforward translation from the pen-and-paper definition of which the novel concepts may seem rather
self-explanatory. Regardless, we briefly go over these concepts for clarity and completeness.
• list is a type constructor (defined in List.ec) that can be used to construct the type of lists over a certain type. In other words, for any type t, t list is the type of lists containing
elements of type t.
• [] is a value (defined in List.ec) that denotes the empty list.
• \in is an abbreviation (defined in List.ec) that can be used an infix operator, and checks whether the left-hand side operand (of some type, say t) is an element of the right-hand side operand
(of type t list).
• :: is an infix operator (defined in List.ec) that prepends the left-hand side operand (of some type, say t) to the right-hand side operand (of type t list).
• In EasyCrypt, procedures can only have a single return statement. The main reason for this is to keep the complexity of the program logics (used for proofs) somewhat in check. For this reason, to
formalize procedures that contain multiple return statements, we accordingly replace all return statements by assignments to a return variable and a single return statement at the end.
Ideal NRCPA Oracle
Having dealt with the formalization of $\mathcal{O}^{CPA\textrm{-}real}_{\Sigma}$, we continue with the formalization of $\mathcal{O}^{CPA\textrm{-}ideal}$. Certainly, as one would expect from the
pen-and-paper definitions, the latter essentially only differs from the former in that it samples ciphertexts uniformly at random instead of legitimately generating them (i.e., via an actual
encryption scheme). As a result, the module formalizing the ideal oracle does not need to be parameterized on another module (representing an actual encryption scheme), nor does it need to maintain a
secret key. However, it does require a uniform distribution over the complete ciphertext space to sample from. So, before formalizing the oracle, we need to define this distribution; we do so as
op [lossless full uniform] dctxt : ctxt distr.
In this code, we define a (sub)distribution dctxt over the ctxt type, similar to how we defined the (sub)distribution dkey over the key type before. However, in contrast to dkey, dctxt is assumed to
have several properties, each of which is denoted by one of the keywords in the square brackets. First, we assume dctxt to be lossless; that is, we assume dctxt is proper distribution (the sum of its
probabilities exactly equals 1). Second, we assume dctxt to be full; that is, we assume dctxt assigns a non-zero probability to each value of type ctxt. Lastly, we assume dctxt to be uniform; that
is, we assume dctxt assigns either a probability of 0 or a constant probability to each value of type ctxt (this constant probability is the same for all values). Note that the combination of the
full and uniform properties mean that dctxt assigns the same non-zero probability to each value of type ctxt. Adding the lossless property on top of this results in dctxt assigning each value of type
ctxt a probability of $1 / \left| \mathcal{C} \right|$ (where $\left| \mathcal{C} \right|$ denotes the cardinality of the ciphertext space). In other words, combining the lossless, full, and uniform
properties results in the distribution that is usually referred to as the "uniform distribution". For more information about distributions in EasyCrypt, click here.
Having the required ciphertext distribution at our disposal, we can go ahead and formalize the ideal oracle. As mentioned before, this formalization is essentially identical to the formalization of
the real oracle, merely replacing the legitimate generation of ciphertexts by the appropriate sampling operation (and, of course, removing anything related to the legitimate generation of ciphertext,
as this is redundant). More precisely, we formalize $\mathcal{O}^{CPA\textrm{-}ideal}$ as the module given below.
module O_NRCPA_ideal : NRCPA_Oraclei = {
var log : nonce list
proc init() : unit = {
log <- [];
proc enc(n : nonce, m : ptxt) : ctxt option = {
var c : ctxt;
var r : ctxt option;
if (! (n \in log)) {
log <- n :: log;
c <$ dctxt;
r <- Some c;
} else {
r <- None;
return r;
IND$-NRCPA Experiment and Security
At this point, it remains to formalize the considered class of adversaries and security experiment before finally being able to formalize the security property of a (symmetric nonce-based) encryption
scheme $\Sigma$.
Starting off, recall that a class of adversaries contains any (possibly probabilistic) algorithm that implements a certain interface, potentially requiring access to a number of oracles. Notice that
if we would be able to specify access to modules of other types in a module type, adversary classes (with potential oracle access) could easily be formalized using module types. Namely, the only
requirement for a module A to be of a certain module type AT is that A implements the procedures defined by AT; otherwise, there are no restrictions on A. Indeed, this precisely matches our concept
of (a class of) adversaries: All algorithms that satisfy the expected interface constitute valid adversaries and, hence, belong to the considered class of adversaries. So, to formalize a class of
adversaries that do not expect access to any oracle, it suffices to use module types as we have already gone over before. However, to formalize a class of adversaries that does expect access to an
oracle(s), we require some additional functionality. Fortunately, EasyCrypt allows for module types that, in addition to specifying a set of procedures that is to be implemented, also indicate a
sequence of module parameter(s), so-called functor types or higher-order module types.
Applying the above to our currently case, we first create a module type formalizing the interface of the NRCPA oracles provided to the adversaries. This is necessary because NRCPA_Oraclei, the module
type formalizing the interface of the NRCPA oracles given to the experiment, include the init procedure, which we do not want to expose to the adversaries. Therefore, we create a separate module type
for the oracles actually given to the adversary; indeed, this type should only specify the enc procedure of the NRCPA_Oraclei type. Luckily, EasyCrypt allows for the direct inclusion of procedure
signatures from other module types, so we do not have to copy-paste; see the snippet below.
module type NRCPA_Oracle = {
include NRCPA_Oraclei [enc]
Using this newly defined module type, we can formalize the class of adversaries with a module type as follows.
module type Adv_IND_NRCPA (O : NRCPA_Oracle) = {
proc distinguish() : bool
Intuitively, the Adv_IND_NRCPA module type encompasses all modules that expect a module of type NRCPA_Oracle and, after being given such a module, implement distinguish procedure that takes no input
and outputs a value of type bool (i.e., a boolean).
Based on the above, we can formalize the IND$-NRCPA experiment/game, which is nothing more than a (probabilistic) program that is defined with respect to a NRCPA oracle and an IND$-NRCPA adversary.
In EasyCrypt terms, the IND$-NRCPA experiment/game is a module that takes two module parameters: one of type NRCPA_Oraclei and one of type Adv_IND_NRCPA. Here, a technicality is that—because
EasyCrypt allows code to be written exclusively inside of procedures—we must encapsulate the actual code of the experiment/game in a procedure (which we arbitrarily name run), even though there is no
corresponding (explicit) procedure/algorithm signature in the pen-and-paper definition. Otherwise, the formalization, provided in the following snippet, is a verbatim translation of the pen-and-paper
module Exp_IND_NRCPA (O : NRCPA_Oraclei) (D : Adv_IND_NRCPA) = {
proc run() : bool = {
var b : bool;
b <@ D(O).distinguish();
return b;
At last, we can express the advantage of an (IND$-NRCPA) adversary $\mathcal{D}$ against IND$-NRCPA of a symmetric nonce-based encryption scheme $\Sigma$. Recall that the pen-and-paper definition of
this advantage is the following.
$\mathsf{Adv}^{\mathrm{IND\\textrm{-}NRCPA}}_{\Sigma}(\mathcal{D}) = \left|\mathsf{Pr}\left[\mathsf{Exp}^{\mathrm{IND\\textrm{-}NRCPA}}_{\mathcal{D}, \mathcal{O}^{CPA\textrm{-}real}_{\Sigma}} = 1\
right] - \mathsf{Pr}\left[\mathsf{Exp}^{\mathrm{IND\\textrm{-}NRCPA}}_{\mathcal{D}, \mathcal{O}^{CPA\textrm{-}ideal}} = 1\right]\right|$
Formally, these probability expressions are not fully defined unless the initial memory/context is fixed. More precisely, the experiment only defines a distribution over its (state and) output
value—which allows us to talk about things such as the probability of the output of the experiment being 1—if the initial memory is fixed; otherwise, it would define a set of distributions (one
distribution for each possible initial memory). To keep flexibility in reasoning, EasyCrypt makes the choice of letting programs run in arbitrary initial memories, and those need to be specified as
part of the probability statements/advantages. At present, it is impossible to define operators that are parameterized by memories in EasyCrypt, and we must always explicitly write out the advantage
expressions when they are present. This means that the advantage expressions will only be formalized within the formalization of the corresponding security statements. For this reason, we postpone
the discussion concerning the formalization of the advantage expressions to when we start formalizing the relevant security statements.
"Nonce-Respecting" Pseudo-Random Function Family Property, Pen-and-Paper
In cryptography, it is common to base security of a scheme on computational hardness assumptions that can somehow be linked to (parts of) the scheme; we also do this here. Particularly, we base the
IND$-NRCPA security of $\mathcal{E}$ on the (assumed) "Nonce-Respecting" Pseudo-Random Function family (NRPRF) property^1 of the function family used to map nonces to plaintexts (i.e., $\left(f_{k} :
\mathcal{N} \rightarrow \mathcal{P}\right)_{k \in \mathcal{K}}$). Intuitively, this property captures the extent to which an (unknown) random function from the function family is indistinguishable
from a truly random function of the same type (i.e., from nonces to plaintexts) by observing the outputs corresponding to unique/non-repeating inputs.
More formally, consider the two oracles given below, $\mathcal{O}^{PRF\textrm{-}real}$ and $\mathcal{O}^{PRF\textrm{-}ideal}$.^2
\begin{align*} \begin{align*} & \underline{\smash{\mathcal{O}^{PRF\textrm{-}real}}}\\ & \begin{align*} & \underline{\smash{\mathsf{init}()}}\\ & \left\lfloor~ \begin{align*} & k \operatorname{\smash
{\overset{\}{\leftarrow}}} \mathcal{D}_{\mathcal{K}}\\ & \mathrm{log} \leftarrow [\ ] \end{align*} \right. \end{align*} \\ & \begin{align*} & \underline{\smash{\mathsf{get}(n)}}\\ & \left\lfloor~ \
begin{align*} & \textsf{if}\ n otin \mathrm{log}\\ & \left\lfloor~ \begin{align*} & \mathrm{log} \leftarrow n\ ||\ \mathrm{log}\\ & m \leftarrow f_k(n)\\ & \textsf{return}\ m \end{align*} \right.\\ &
\textsf{else}\\ & \left\lfloor~ \begin{align*} & \textsf{return}\ \bot \end{align*} \right. \end{align*} \right. \end{align*} \end{align*} &&&&&&&& \begin{align*} & \underline{\smash{\mathcal{O}^{PRF
\textrm{-}ideal}}}\\ & \begin{align*} & \underline{\smash{\mathsf{init}()}}\\ & \left\lfloor~ \begin{align*} & \\ & \mathrm{log} \leftarrow [\ ] \end{align*} \right. \end{align*} \\ & \begin{align*}
& \underline{\smash{\mathsf{get}(n)}}\\ & \left\lfloor~ \begin{align*} & \textsf{if}\ n otin \mathrm{log}\\ & \left\lfloor~ \begin{align*} & \mathrm{log} \leftarrow n\ ||\ \mathrm{log}\\ & m \
operatorname{\smash{\overset{\}{\leftarrow}}} \mathcal{U}_{\mathcal{P}}\\ & \textsf{return}\ m \end{align*} \right.\\ & \textsf{else}\\ & \left\lfloor~ \begin{align*} & \textsf{return}\ \bot \end
{align*} \right. \end{align*} \right. \end{align*} \end{align*} \end{align*}
As we can see, $\mathcal{O}^{PRF\textrm{-}real}$ and $\mathcal{O}^{PRF\textrm{-}ideal}$ effectively only differ in the way they create plaintexts: The former creates plaintexts by mapping the
provided nonces with a random function (that is fixed during initialization) from $\left(f_{k} : \mathcal{N} \rightarrow \mathcal{P}\right)_{k \in \mathcal{K}}$; the latter creates plaintexts by
sampling them uniformly at random, independent of the provided nonces. Then, akin to what we did for IND$-NRCPA security, we define the advantage of a nonce-respecting pseudo-random function
distinguisher $\mathcal{D}$ against $\left(f_{k} : \mathcal{N} \rightarrow \mathcal{P}\right)_{k \in \mathcal{K}}$ as the following (absolute) difference.
$\mathsf{Adv}^{\mathrm{NRPRF}}(\mathcal{D}) = \left|\mathsf{Pr}\left[\mathsf{Exp}^{\mathrm{NRPRF}}_{\mathcal{D}, \mathcal{O}^{PRF\textrm{-}real}} = 1\right] - \mathsf{Pr}\left[\mathsf{Exp}^{\mathrm
{NRPRF}}_{\mathcal{D}, \mathcal{O}^{PRF\textrm{-}ideal}} = 1\right]\right|$
In this equation, $\mathsf{Exp}^{\mathrm{NRPRF}}_{\mathcal{D}, \mathcal{O}}$ refers to the experiment defined below.
\begin{align*} & \underline{\smash{\mathsf{Exp}^{\mathrm{NRPRF}}_{\mathcal{O}}(\mathcal{D})}}\\ & \left\lfloor \begin{align*} & \mathcal{O}.\mathsf{init}()\\ & b \operatorname{\smash{\overset{\}{\
leftarrow}}} \mathcal{D}^{\mathcal{O}.\mathsf{get}}.\mathsf{distinguish}()\\ & \textsf{return}\ b \end{align*} \right. \end{align*}
Notice that this experiment takes the exact same approach as the one we defined for IND$-NRCPA security: The only difference between these experiments concerns the class of adversaries and the class
of oracles they consider.
Lastly, we say that $\left(f_{k} : \mathcal{N} \rightarrow \mathcal{P}\right)_{k \in \mathcal{K}}$ is a NRPRF if, for any nonce-respecting pseudo-random function family adversary $\mathcal{D}$, $\
mathsf{Adv}^{\mathrm{NRPRF}}(\mathcal{D}$) is "small". Again, because we only consider exact security, "small" basically means "bounded by other concrete values/probabilities that we believe are
small in practice".
Again, if you are interested in seeing the paper-based security proof before diving into the EasyCrypt formalization, you can read further on the proving security part.
"Nonce-Respecting" Pseudo-Random Function Family Property, EasyCrypt
Evidently, on a conceptual level, the definitions for IND$-NRCPA and NRPRF experiment and oracles are almost identical. Accordingly, the corresponding EasyCrypt formalization are also going to be
almost identical. As such, we go over the formalization of NRPRF at a faster pace, primarily highlighting the differences with the formalization of IND$-NRCPA and reiterating important aspects. The
preceding material discussing IND$-NRCPA most likely contains explanations of subjects/concepts left untouched here.
NRPRF Oracle Type
Once again, we start by defining the type for the oracles; as before, we do so using a module type that specifies a procedure signature for each algorithm defined expected to be implemented by NRPRF
oracles. Looking at the definitions of $\mathcal{O}^{PRF\textrm{-}real}$ and $\mathcal{O}^{PRF\textrm{-}ideal}$, we can see what (type of) algorithms these are. The following snippet presents the
corresponding formalization.
module type NRPRF_Oraclei = {
proc init() : unit
proc get(n : nonce) : ptxt option
Real NRPRF Oracle
Using the newly defined module type, we formalize the real NRPRF oracle using a (by now) straightforward translation from the pen-and-paper definition; see the snippet below. Recall that an EasyCrypt
procedure can only have one return statement, which is why we employ a return variable instead of having multiple return statements. Furthermore, remember that, since we are using an option type to
represent successes and failures, the outputs are of the form Some p (where p is of type ptxt; this represents a success) or None (this represents a failure).
module O_NRPRF_real : NRPRF_Oraclei = {
var k : key
var log : nonce list
proc init() : unit = {
k <$ dkey;
log <- [];
proc get(n : nonce) : ptxt option = {
var r : ptxt option;
if (! (n \in log)) {
log <- n :: log;
r <- Some (f k n);
} else {
r <- None;
return r;
Ideal NRPRF Oracle
For the ideal NRPRF oracle, the formalization is similar to that of the real NRPRF oracle, only differing in (analogous) ways the pen-and-paper definitions also differ.
The only novelty here is that we define and use an alias for the dctxt distribution, called dptxt. In its definition, this alias is explicitly indicated to be a distribution over the type of
plaintexts. Naturally, this is only possible because ptxt and ctxt are actually the same type. Semantically, this makes no difference at all; the only reason we do this is to increase readability by
matching the notation with the conceptual meaning. (Recall that NRPRF oracles conceptually produce plaintexts, not ciphertexts.) The specific alias definition is provided in the snippet below.
op dptxt : ptxt distr = dctxt.
Then, we formalize the ideal NRPRF oracle as follows.
module O_NRPRF_ideal : NRPRF_Oraclei = {
var log : nonce list
proc init() : unit = {
log <- [];
proc get(n : nonce) : ptxt option= {
var y : ptxt;
var r : ptxt option;
if (! (n \in log)) {
log <- n :: log;
y <$ dptxt;
r <- Some y;
} else {
r <- None;
return r;
NRPRF Experiment and Security
Having formalized the relevant oracles, we continue by formalizing the (NRPRF) adversary class. Once again, we formalize this adversary class using a module type with a single module parameter
modeling the expected NRPRF oracle. Notice that, akin to before, the current module type we have for NRPRF oracles—NRPRF_Oraclei—specifies (the signature of) an initialization procedure, which we do
not want to expose to adversaries. As such, we create a separate module type for NRPRF oracles given to adversaries, which only expose the get procedure.
module type NRPRF_Oracle = {
include NRPRF_Oraclei [get]
module type Adv_NRPRF (O : NRPRF_Oracle) = {
proc distinguish() : bool
At this point, we have everything required to formalize the NRPRF experiment, shown. Recall that, even though the adversary is given an oracle of type NRPRF_Oraclei, the init procedure of this module
is not actually exposed to the adversary due to the way we specified its module type.
module Exp_NRPRF (O : NRPRF_Oraclei) (D : Adv_NRPRF) = {
proc run() : bool = {
var b : bool;
b <@ D(O).distinguish();
return b;
1. NRPRF is not a conventional property; rather, it is a variant of the more customary Pseudo-Random Function family (PRF) property. For educational purposes, we have specifically devised this
variant to simplify the current proof. ↩
2. Typically, the NRPRF experiment and oracles would first be defined with respect to an abstract function family of the correct type before being instantiated with the actual relevant function
family for the proof. This is analogous to how we first defined IND$-NRCPA with respect to an abstract (symmetric) nonce-respecting encryption scheme before instantiating it with the actual
relevant encryption scheme for the proof. However, for educational purposes, we decide to keep it simple and stick with the description that matches the EasyCrypt formalization the best; for this
reason, we immediately consider the concrete NRPRF property of $\left(f_{k} : \mathcal{N} \rightarrow \mathcal{P}\right)_{k \in \mathcal{K}}$. ↩ | {"url":"https://easycrypt.gitlab.io/easycrypt-web/docs/simple-tutorial/security/","timestamp":"2024-11-11T19:55:51Z","content_type":"text/html","content_length":"318693","record_id":"<urn:uuid:aebaeba3-4522-41e4-87e3-18ce9e2fdb6e>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00725.warc.gz"} |
Simple fitting of single datasets
Simple fitting of single datasets
Classes used:
Making informed guesses for the initial values of the variable parameters of a model and fit the model to the data is the most straight-forward strategy. Still, different optimisation algorithms can
be chosen. The recipe presented here uses fitpy.analysis.SimpleFit with standard values and shows as well how to use the dedicated plotters and reporters.
2 type: ASpecD recipe
3 version: '0.2'
6 autosave_plots: false
9 # Create "dataset" to fit model to
10 - kind: model
11 type: Zeros
12 properties:
13 parameters:
14 shape: 1001
15 range: [-10, 10]
16 result: dummy
17 comment: >
18 Create a dummy model.
19 - kind: model
20 type: Gaussian
21 from_dataset: dummy
22 properties:
23 parameters:
24 position: 2
25 label: Random spectral line
26 comment: >
27 Create a simple Gaussian line.
28 result: dataset
29 - kind: processing
30 type: Noise
31 properties:
32 parameters:
33 amplitude: 0.2
34 apply_to: dataset
35 comment: >
36 Add a bit of noise.
37 - kind: singleplot
38 type: SinglePlotter1D
39 properties:
40 filename: dataset2fit.pdf
41 apply_to: dataset
42 comment: >
43 Just to be on the safe side, plot data of created "dataset"
45 # Now for the actual fitting: (i) create model, (ii) fit to data
46 - kind: model
47 type: Gaussian
48 from_dataset: dataset
49 output: model
50 result: gaussian_model
52 - kind: fitpy.singleanalysis
53 type: SimpleFit
54 properties:
55 model: gaussian_model
56 parameters:
57 fit:
58 position:
59 start: 1
60 range: [0, 5]
61 algorithm:
62 method: least_squares
63 parameters:
64 ftol: 1e-6
65 xtol: 1e-6
66 result: fitted_gaussian
67 apply_to: dataset
69 # Plot result
70 - kind: fitpy.singleplot
71 type: SinglePlotter1D
72 properties:
73 filename: fit_result.pdf
74 parameters:
75 show_legend: true
76 apply_to: fitted_gaussian
78 # Create report
79 - kind: fitpy.report
80 type: LaTeXFitReporter
81 properties:
82 template: simplefit.tex
83 filename: report.tex
84 compile: true
85 apply_to: fitted_gaussian
• The purpose of the first block of four tasks is solely to create some data a model can be fitted to. The actual fitting starts only afterwards.
• Usually, you will have set another ASpecD-derived package as default package in your recipe for processing and analysing your data. Hence, you need to provide the package name (fitpy) in the kind
property, as shown in the examples.
• Fitting is always a two-step process: (i) define the model, and (ii) define the fitting task.
• To get a quick overview of the fit results, use the dedicated plotter from the FitPy framework: fitpy.plotting.SinglePlotter1D.
• For a more detailed overview, particularly in case of several fits with different settings on one and the same dataset or for a series of similar fits on different datasets, use reports, as shown
here using fitpy.report.LaTeXFitReporter. This reporter will automatically create the figure showing both, fitted model and original data.
Examples for the two figures created in the recipe are given below. While in the recipe, the output format has been set to PDF, for rendering them here they have been converted to PNG. Due to the
noise added to the model having an inherently random component, your data will look slightly different. Therefore, the fit results will be slightly different as well. Nevertheless, overall it should
be fairly identical. | {"url":"https://docs.fitpy.de/v0.1/examples/simplefit.html","timestamp":"2024-11-07T11:50:02Z","content_type":"text/html","content_length":"27348","record_id":"<urn:uuid:3b31d77a-8950-4b2f-896f-a570f393e2bd>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00153.warc.gz"} |
To determine if an atomic constraint is
, the parameter mapping and template arguments are first substituted into its expression
If substitution results in an invalid type or expression, the constraint is not satisfied
The constraint is satisfied if and only if evaluation of
results in
If, at different points in the program, the satisfaction result is different for identical atomic constraints and template arguments, the program is ill-formed, no diagnostic required
Example 3
template<typename T> concept C = sizeof(T) == 4 && !true; // requires atomic constraints sizeof(T) == 4 and !true template<typename T> struct S { constexpr operator bool() const { return true; } };
template<typename T> requires (S<T>{}) void f(T); // #1 void f(int); // #2 void g() { f(0); // error: expression S<int>{} does not have type bool } // while checking satisfaction of deduced arguments
of #1; // call is ill-formed even though #2 is a better match
end example | {"url":"https://timsong-cpp.github.io/cppwp/n4868/temp.constr.constr","timestamp":"2024-11-14T07:07:52Z","content_type":"text/html","content_length":"44908","record_id":"<urn:uuid:ba66e391-0f55-45b0-ab02-ffdf7ba4d971>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00615.warc.gz"} |
Inside every child is a mathematician.
As I spend time examining the Common Core State Standards for math, I am struck by how powerful the mathematics is in there. What concerns me the most is that the CCSS was written by mathematicians
who have a fantastic handle on what mathematicians need to know, but I think the documents lack a bit of down-to-earth verbiage for teachers and students. When I read the 8 mathematical practices, I
was impressed at how the practices discussed the habits of mind that it takes to be a mathematician. The more I read it, though, the more I realized how difficult it would be in its current state to
use in the K-8 classroom. For the past two years, I have used the 8 practices in my classroom (grades 6-8) to not only drive my instruction, but to guide the methods that I use to help students
become mathematicians. I also have a version of the practices for the lower grade levels (K-3). Over the course of two years, I feel that I created a student-friendly version of the practices and
engaging lessons to help students use them every day in the classroom. I have a narrated power-point to explain my lessons and there are documents to use in the classroom at my website
Welcome to my blog!
My name is Holly Young and I am a mathematician, math teacher, and math trainer. I just recently left the classroom and decided to go out on my own to create resources for math teachers. By and
large, math teachers have the raw deal in teaching, especially in secondary schools. The onus falls on the shoulders of the math teacher to lead every day, every minute of every lesson. We don’t
have scores of amazing National Geographic videos or Discovery websites that further our curriculum. If we have to be out of the classroom for a day, then we usually lose a whole day of
instruction. It is my goal, therefore, to create useful media that math teachers can access and help forward student learning. I have lessons available for multiple grades and topics at
www.makingmathematicians.com and I will continue to add more lessons and useable media. This is a journey for me, so I will be constantly improving and creating new resources. If you have
suggestions of what resources you would like to see on my website (or blog), please feel free to let me know.
This first file that I am posting came to me as a sudden inspiration while training teachers on creating essential understandings (questions) in the classroom. As I was asking teachers to write an
essential understanding that encompassed multiple grade levels along the same CCSS strand, I kept asking them, “What is the essence of this strand, and why do we need to learn it?” The result of
that exercise came to me in a flash when I asked myself, “Why do we study the main topics in mathematics?” Math_poster | {"url":"http://www.makingmathematicians.com/hollysblog/?paged=2","timestamp":"2024-11-06T15:33:49Z","content_type":"text/html","content_length":"33813","record_id":"<urn:uuid:bedd7b07-aa8b-4132-9238-1ffabe57a623>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00236.warc.gz"} |
Applications of LASSO regression to sparse data sets
Twelve Days 2013: LASSO Regression
Day Eight: LASSO Regression
LASSO regression (least absolute shrinkage and selection operator) is a modified form of least squares regression that penalizes model complexity via a regularization parameter. It does so by
including a term proportional to $||\beta||_{l_1}$ in the objective function which shrinks coefficients towards zero, and can even eliminate them entirely. In that light, LASSO is a form of feature
selection/dimensionality reduction. Unlike other forms of regularization such as ridge regression, LASSO will actually eliminate predictors. It’s a simple, useful technique that performs quite well
on many data sets.
Regularization refers to the process of adding additional constraints to a problem to avoid over fitting. ML techniques such as neural networks can generate models of arbitrary complexity that will
fit in-sample data one-for-one. As we recently saw in the post on Reed-Solomon FEC codes, the same applies to regression. We definitely have a problem anytime there are more regressors than data
points, but any excessively complex model will generalize horribly and do you zero good out of sample.
Why LASSO?
There’s a litany of regularization techniques for regression, ranging from heuristic, hands-on ones like stepwise regression to full blown dimensionality reduction. They all have their place, but I
like LASSO because it works very well, and it’s simpler than most dimensionality reduction/ML techniques. And, despite being a non-linear method, as of 2008 it has a relatively efficient solution via
coordinate descent. We can solve the optimization in $\O(n\cdot p)$ time, where $n$ is the length of the data set and $p$ is the number of regressors.
An Example
Our objective function has the form:
$\frac{1}{2} \sum_i(y_i - \mathbb{x}^T\beta)^2 + \lambda\sum_{j = 1}^p|\beta_j|$
where $\lambda \geq 0$. The first half of the equation is just the standard objective function for least squares regression. The second half penalizes regression coefficients under the $l_1$ norm.
The parameter $\lambda$ determines how important the penalty on coefficient weights is.
There are two R packages that I know of for LASSO: lars (short for least angle regression—a super set of LASSO) and glmnet. Glmnet includes solvers for more general models (including elastic net—a
hybrid of LASSO and ridge that can handle catagorical variables). Lars is simpler to work with but the documentation isn’t great. As such, here are a few points worth noting:
1. The primary lars function generates an object that’s subsequently used to generate the fit that you actually want. There’s a computational motivation behind this approach. The LARS technique
works by solving for a series of “knot points” with associated, monotonically decreasing values of $\lambda$. The knot points are subsequently used to compute the LASSO regression for any value
of $\lambda$ using only matrix math. This makes procedures such as cross validation where we need to try lots of different values of $\lambda$ computationally tractable. Without it, we would have
to recompute an expensive non-linear optimization each time $\lambda$ changed.
2. There’s a saturation point at which $\lambda$ is high enough that the null model is optimal. On the other end of the spectrum, when $\lambda = 0$, we’re left with least squares. The final value
of $\lambda$ on the path, right before we end up with least squares, will correspond to the largest coefficient norm. Let’s call these coefficients $\beta_\text{thresh}$, and denote $\Delta = ||
\beta_\text{thresh} ||_{l_1}$. When the lars package does cross validation, it does so by computing the MSE for models where the second term in the objective function is fixed at $x \cdot \Delta,
\ x \in [0, 1]$. This works from a calculation standpoint (and computationally it makes things pretty), but it’s counter intuitive if you’re interested in the actual value of $\lambda$ and not
just trying to get the regression coefficients. You could easily write your own cross validation routine to use $\lambda$ directly.
3. The residual sum of squared errors will increase monotonically with $\lambda$. This makes sense as we’re trading off between minimizing RSS and the model’s complexity. As such, the smallest RSS
will always correspond to the smallest value of $\lambda$, and not necessarily the optimal one.
Here’s a simple example using data from the lars package. We’ll follow a common heuristic that recommends choosing $\lambda$ one SD of MSE away from the minimum. Personally I prefer to examine the CV
L-curve and pick a value right on the elbow, but this works.
# Compute MSEs for a range of coefficient penalties expressed as a fraction
# of the final L1 norm on the interval [0, 1].
cv.res <- cv.lars(diabetes$x, diabetes$y, type = "lasso",
mode = "fraction", plot = FALSE)
# Choose an "optimal" value one standard deviation away from the
# minimum MSE.
opt.frac <- min(cv.res$cv) + sd(cv.res$cv)
opt.frac <- cv.res$index[which(cv.res$cv < opt.frac)[1]]
# Compute the LARS path
lasso.path <- lars(diabetes$x, diabetes$y, type = "lasso")
# Compute a fit given the LARS path that we precomputed, and our optimal
# fraction of the final L1 norm
lasso.fit <- predict.lars(lasso.path, type = "coefficients",
mode = "fraction", s = opt.frac)
# Extract the final vector of regression coefficients
Final Notes
LASSO is a biased, linear estimator whose bias increases with $\lambda$. It’s not meant to provide the “best” fit as Gauss-Markov defines it—LASSO aims to find models that generalize well. Feature
selection is hard problem and the best that we can do is a combination of common sense and model inference. However, no technique will save you from the worst case scenario: two very highly
correlated variables, one of which is a good predictor, the other of which is spurious. It’s a crap shoot as to which predictor a feature selection algorithm would penalize in that case. LASSO has a
few technical issues as well. Omitted variable bias is still an issue as it is in other forms of regression, and because of its non-linear solution, LASSO isn’t invariant under transformations of
original data matrix. | {"url":"https://www.klittlepage.com/articles/twelve-days-2013-lasso-regression/","timestamp":"2024-11-13T08:32:26Z","content_type":"text/html","content_length":"242653","record_id":"<urn:uuid:67637388-3d67-4dd4-8ea6-6e74e22e466a>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00475.warc.gz"} |
The Transformation to the third Quantum Dimension (Bohr-distance)
When you are interested in physics you must read “Unbelievable“!
We demonstrated that with /2 point-volumes we can create a perfect “circle” with 864 triangles OAB. Point O (figure 4) is the center of the created Compton quantum circle (Rc). Between O and Rc the
quantum space is imperfect. The “bulbs” with radius QD cannot fill up spherical space homogeneously.
The “perfect” geometry is created at Rc. For the observer in O it is not possible to observe perfect circles all around (no perfect bulb possible when space is filled with point-volumes).
The orientation of the two possible circles Rc is not fixed.
With the “perfect” two-dimensional circle Rc we are able to create a perfect bulb shell tunnel with diameter Rc at the distance *Rc; the Bohr-distance.
Figure 6. Compton circles creating the 3^rd quantum dimension.
With /2 circles Rc (two circles each) we are able, in conjunction with the creation of the Compton-circle, to create two “perfect” bulb shell tunnels with radius Rc at the Bohr-distance; the
beginning of the 3^rd quantum dimension.
The ratio between the distances Rb and Rc should be .
We have observed that the ratio between the Bohr-radius and the Compton-radius is:
The Bohr/Compton distance ratio appears to be 10.8674 times larger than the ratio Rc/QD or the Rydberg/Bohr ratio.
We must however not forget that the situation at 2-QD is not the same as the 1-QD level or the 3-QD level. We have seen that a mathematical correction from 1-QD to 2-QD with the factor 2pi was
necessary. We now compare the 3rd dimension of the Bohr-bulb with 2^nd dimension of the Compton-plate. A mathematical correction is necessary.
The factor 10.8674 is the correction factor from the 2-QD to the 3-QD. From the 1^st to the 2^nd QD the correction factor is 2π. This transformation explains the origin of the mathematical natural
constant π. Is it a coincidence that the other mathematical natural constant e can be found in 10.8674, because and the difference is therefore only 0.05%? **
So both mathematical natural constants may well originate from the dimension transfer from the point-volume to three-dimensional space.
After the dimension correction we have also the ratio Rb/Rc= .
The total mathematical correction from the point-volume (1^st dimension) to our 3^rd dimension is: .
(** When we correct for neglecting the factor (1+Me/Mp)=1.0005446, when equation 5a was derived from 5, the deviation factor with 4*e is less than 2.10^-5)
Next chapter: The 12 Ionization Levels of the Atom | {"url":"https://paradox-paradigm.nl/preface/quantum-mechanics-and-the-ether-the-derivation-of-plancks-constant/the-transformation-to-the-third-quantum-dimension-bohr-distance/","timestamp":"2024-11-04T13:26:47Z","content_type":"application/xhtml+xml","content_length":"45819","record_id":"<urn:uuid:dd80e03f-2417-4cbb-b60c-bfb0adfbf158>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00683.warc.gz"} |
he length of a rectangular garden is 3 m greater than the width. The area of the garden is 70m squared. find the dimensions of the garden
1. Home
2. General
3. The length of a rectangular garden is 3 m greater than the width. The area of the garden is 70m squa... | {"url":"http://thibaultlanxade.com/general/the-length-of-a-rectangular-garden-is-3-m-greater-than-the-width-the-area-of-the-garden-is-70m-squared-find-the-dimensions-of-the-garden","timestamp":"2024-11-08T11:32:16Z","content_type":"text/html","content_length":"30686","record_id":"<urn:uuid:8a343c1f-3d03-43cb-8848-30a292a36339>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00880.warc.gz"} |
Subset, Seasonal, and Factored ARMA Models
The simplest way to specify an ARMA model is to give the order of the AR and MA parts with the P= and Q= options. When you do this, the model has parameters for the AR and MA parts for all lags
through the order specified. However, you can control the form of the ARIMA model exactly as shown in the following section.
You can control which lags have parameters by specifying the P= or Q= option as a list of lags in parentheses. A model that includes parameters for only some lags is sometimes called a subset or
additive model. For example, consider the following two ESTIMATE statements:
identify var=sales;
estimate p=4;
estimate p=(1 4);
Both specify AR(4) models, but the first has parameters for lags 1, 2, 3, and 4, while the second has parameters for lags 1 and 4, with the coefficients for lags 2 and 3 constrained to 0. The
mathematical form of the autoregressive models produced by these two specifications is shown in Table 7.1.
Table 7.1: Saturated versus Subset Models
Option Autoregressive Operator
P=(1 4)
One particularly useful kind of subset model is a seasonal model. When the response series has a seasonal pattern, the values of the series at the same time of year in previous years can be important
for modeling the series. For example, if the series SALES is observed monthly, the statements
identify var=sales;
estimate p=(12);
model SALES as an average value plus some fraction of its deviation from this average value a year ago, plus a random error. Although this is an AR(12) model, it has only one autoregressive
A factored model (also referred to as a multiplicative model) represents the ARIMA model as a product of simpler ARIMA models. For example, you might model SALES as a combination of an AR(1) process
that reflects short term dependencies and an AR(12) model that reflects the seasonal pattern.
It might seem that the way to do this is with the option P=(1 12), but the AR(1) process also operates in past years; you really need autoregressive parameters at lags 1, 12, and 13. You can specify
a subset model with separate parameters at these lags, or you can specify a factored model that represents the model as the product of an AR(1) model and an AR(12) model. Consider the following two
ESTIMATE statements:
identify var=sales;
estimate p=(1 12 13);
estimate p=(1)(12);
The mathematical form of the autoregressive models produced by these two specifications are shown in Table 7.2.
Table 7.2: Subset versus Factored Models
Option Autoregressive Operator
P=(1 12 13)
Both models fit by these two ESTIMATE statements predict SALES from its values 1, 12, and 13 periods ago, but they use different parameterizations. The first model has three parameters, whose
meanings may be hard to interpret.
The factored specification P=(1)(12) represents the model as the product of two different AR models. It has only two parameters: one that corresponds to recent effects and one that represents
seasonal effects. Thus the factored model is more parsimonious, and its parameter estimates are more clearly interpretable. | {"url":"http://support.sas.com/documentation/cdl/en/etsug/65545/HTML/default/etsug_arima_gettingstarted21.htm","timestamp":"2024-11-11T10:00:11Z","content_type":"application/xhtml+xml","content_length":"24935","record_id":"<urn:uuid:c86ed98d-f8bd-48fc-a7ab-21732d8e4bb5>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00434.warc.gz"} |
Department of Mathematics
Event time:
Friday, November 1, 2024 - 10:00am
Event description:
Combinatorial billiards is a new topic that concerns rigid and discretized billiard systems that can be modeled combinatorially or algebraically. I will introduce
a random combinatorial billiard trajectory depending on some fixed probability p; when p tends to 0, it essentially recovers Thomas Lam’s reduced random walk. This random billiard trajectory can also
be interpreted as a random growth process on core partitions. The analysis of the random billiard trajectory relies on new finite Markov chains called stoned exclusion processes, which are variants
of certain interacting particle systems. These processes have remarkable stationary distributions determined by well-studied polynomials such as ASEP polynomials, inhomogeneous TASEP polynomials, and
open boundary ASEP polynomials; in many cases, it was previously not known how to construct Markov chains with these stationary distributions. | {"url":"https://math.yale.edu/event/random-combinatorial-billiards","timestamp":"2024-11-11T16:34:18Z","content_type":"text/html","content_length":"35774","record_id":"<urn:uuid:ca7b8608-7020-4d1d-a08b-db90b2c73fef>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00640.warc.gz"} |
Multicollinearity is not correlation
Introduction: The problem with tutorials
This huge amount of data science tutorials on the internet has its pros and cons. The biggest problem, in my humble opinion, is that everyone wants to talk about data nowadays in order to get people
— specially recruiters — attention. That being said, several mistakes are being made and no one is correcting them. This leads to mistakes being spread all over beginners.
If you take 90% of tutorials about Linear Regression in this platform, you'll see a common mistake being made: saying that multicollinearity is the same as correlation. Even worse, people think that
multicollinearity invalidates their model and that you should use Pearson correlation coefficient to deal with it. All these statements are wrong. Multicollinearity is not correlation, you should not
deal with it by using Pearson coefficient and it’s not a problem depending on your situation.
Fig. 1: You can see by this search that people often confuse the terms
What’s multicollinearity?
Multicollinearity appears when one of your predictors can be predicted by one or several others. Let’s say you want to predict the salary of a young student. In order to do that, you have some
variables, like the high school price, parents' earnings, neighborhood, amount of time spent in school and amount of subjects studied in high school. If the high school price can be predicted by the
parents' earnings, which is very likely, then you have multicollinearity.
You might be thinking that this looks exactly like linear correlation, but it’s not always like this. In the example above, correlation will indicate multicollinearity, but this doesn’t happen every
time. Let’s say the parents' earnings is not a very good predictor of the school price by itself. This variable, however, can be part of a model that predicts the price of the school. We might have a
great model if we take not only the earnings, but also the neighborhood and the amount of subjects. Now, we have 3 independent variables — parents' earnings, neighborhood and amount of subjects —
that predicts another one independent variable, the school price. This is a situation in which there is multicollinearity, but you won’t be able to capture it only by using the Pearson coefficient.
So, how can we detect multicollinearity?
In order to assess if your model has multicollinearity you should use a metric called Variance Inflation Factor, also known as VIF. Mathematically, this metric is equal to the ration of the model
variance to the variance of a model that includes only that single independent variable.
Fig. 2: VIF formula
As mentioned on Wikipedia:
The square root of the variance inflation factor indicates how much larger the standard error increases compared to if that variable had 0 correlation to other predictor variables in the model.
Usually, we use this rule of thumb:
• VIF equals to 1: no multicollinearity.
• VIF higher than 1 and lower (or equal) than 5: mild multicollinearity. It’s not a big problem to leave the variable.
• VIF higher than 5: there is multicollinearity and you should leave the variable out of your model.
If you want to see the step-by-step demonstration of the formula, I highly recommend you to start with Wikipedia.
Wait, multicollinearity does affect your model!
Yes and no. There is definitely a problem when our model has independent variables which are predicted by other independent variables, but that doen’t mean your model doesn’t work anymore.
Multicollinearity inflates standard errors, which means that our coefficients are not trustworthy anymore. However, our predictions are still safe. That’s why I mentioned before that it’s not a
problem depending on your situation.
One last thought, this is only one of the assumptions people believe invalidate their model, when it doesn’t do that. If you are working with a inference problem and want to learn more about these
assumptions, please read some econometrics book. My favorite one is Introductory Econometrics, by Wooldridge.
Hope you enjoyed this article. | {"url":"https://universidadedosdados.medium.com/multicollinearity-is-not-correlation-38014cbfc710?source=user_profile---------9----------------------------","timestamp":"2024-11-06T08:49:37Z","content_type":"text/html","content_length":"105648","record_id":"<urn:uuid:714f3c7b-1a5f-45cf-8162-239f66c6ab6f>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00826.warc.gz"} |
How to use NPER function in Excel? | Easy Excel Tips | Excel Tutorial | Free Excel Help | Excel IF | Easy Excel No 1 Excel tutorial on the internet
Home Excel Functions How to use NPER function in Excel?
How to use NPER function in Excel?
How To Use NPER Function In Excel
In this lesson you can learn how to use NPER function in Excel.
The NPER function is similar to both the Future Value and Present Value functions. But the difference is that the NPER formula will be returning the quantity of an investment, using the periodic,
consistent payment and interest rate. NPER function calculates number of periods for an investment or a loan with the assumption of constant payments at regular intervals and a fixed interest rate.
Syntax is:
• Rate – the rate of interest for the period
• PMT – fixed payment payed in each period,
• PV – present value of the investment or loan,
• FV – future value of an investment or loan (the value you want to achieve at the end of all periods)
• Type – specifies the time to make a payment during the period: 0 – payments are made at the ends of periods, 1 – payments are made at the beginning of each period.
The rate parameter should be provided for the payment period, ie if the interest rate is given annually and monthly payments are then divide the interest rate for 12 months.
For investment parameters PMT, PV and FV are negative positive, while for the credit PMT – negative PV – positive, FV – negative (if calculations are made assuming a partial repayment of a loan.
Basic calculation of NPER
How many payments you must pay to reach a value of 1 000 000 $ for investment with an annual interest rate of 10%? You pay monthly payments in the amount of 1 000 $, having already set aside 150 000
$. You pay at the end of each period (month).
Formula is =NPER(0.10/12,-1000,-150000,1000000,0)
Answer: Result is 171,43 what means, that you need 172 payments to reach 1 000 000$
• 0.10/12 – rate is 10% what must be divided by count of months
• -1000 – monthly payment (with minus because you have to pay)
• -150000 – your current cash (with minus because you pay)
• 1000000 – your future value (without minus because it’s your cash)
• 0 – because you pay at the end of the period
How much do you need to pay off your loan?
How many payments you need to pay off 100 000 $ loan repaid at the end of the month the installment of 700 $ (principal + interest). Annual interest rate is 8%.
Formula is
Answer: Result is 458,2 what means, that you need 459 payments to pay off this loan.
• 0.08/12 – rate is 8% what must be divided by count of months
• -700 – monthly payment (with minus because you have to pay)
• 150000 – your current cash (without minus because it’s your cash)
• 0 – your future value because you want to pay off this debt
• 0 – because you pay at the end of the period
NPER when you pay at the beginning of the month
How many months you have to pay to the bank at $300 to save the amount of $18 000. Annual interest rate is 3,5%. You pay at the beginning of each month.
Formula is
Answer: Result is 55,22 what means, that you need 56 months to reach $18 000.
• 0.035/12 – rate is 3.5% what must be divided by count of months
• -300 – monthly payment (with minus because you have to pay)
• 0 – your current cash
• 18 000 – your future value (without minus because it’s your cash)
• 1 – because you pay at the beginning of the period
Simple NPER Formula
The example is based on finding a solid NPER of the investment that we are about to make, so we can fully comprehend how the whole situation would work. This is something that would work with a full
understanding of the formula, and we are getting the answer that we strive after with the formula.
Complete NPER Formula
The investment is going quite good, and we needs to know about the future value. This is a follow up to the previous example, and requires having a full formula in comparison to the previous example,
and we are going to use the same data as previous one, so you could see the differences between the results. The example is about knowing the benefits of our investments, and how it would affect the
ways that we are planning the executions of additional plans.
NPER Formula with Bigger than Symbol
The formula in this example includes using a mathematic symbol that will be beneficial in the long run. It is possible to make it possible to know how it will go when the rate is bigger than 10
percent. This example clarifies the possibilities of having a solid information about the number of investments when the rate is including such mathematical symbol. One great thing about this formula
is that it provides the possibilities of knowing exactly how the mathematic symbol affects the outcome.
Usage of Formula with Future Value Increment
The formula is going to have a very specific formula that also include the increment of a future value. In this example, we would like to know how many times we needs to make an investment when we
expect the future value to increase by 50%, and how it would affect the business deal and its overall performance.
A Formula with both NPER & AND
This example is a complete understanding of the formula, and we are going to use both NPER & AND for successfulness of acknowledging the information about how many times we will be making an
investment when using two formulas simultaneously.
Additional Percentage with AND
In this example, we want to add some percentages to the current value, under circumstances that the business needs to add 12% to the current value, because we have accepted a customized deal with a
company that is asking for 12 percent interest rate of the current value. This is where the NPER formula would be an excellent formula for acknowledging how many times the business needs to add the
investment into its overall performance.
NPER with Texts
The example is about clarifying that under the circumstances that we have already labeled the cells with texts, to make it more understanding for us to understand which part of the formula the cell
is talking about, even after some time has passed by, and we might have forgotten about what part of parameters belong to where in the NPER formula.
Simultaneously Using NPER with SUM
This example is using both the NPER and OR formulas simultaneously to find out how many times the company needs to perform the investments for it to successfully achieve a goal that has been set for
the business. We have set a flexible rates for the business, and we would like to know how we would be making the investments.
Numeric NPER Formula
This is a more complicated formula, and we need to have a numeric formula of the NPER formula to get the answer that we are looking for with the formula. We want to make some evaluations, which is
why we are rather having this kind of formula.
Understanding Usage of NPER Formula with another Spreadsheet
There are many different reasons that you’d might needs to have the NPER formula on another spreadsheet. But in this example, we are using a different spreadsheet, because we would like to have a
full understanding of how many times we need to perform the investment for it to yield us the estimated future value, and we are using a different spreadsheet for it. | {"url":"https://www.excelif.com/nper-function/","timestamp":"2024-11-11T21:17:33Z","content_type":"text/html","content_length":"207139","record_id":"<urn:uuid:9cb29a26-fec2-4ab7-835f-df1535827c2d>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00716.warc.gz"} |
John S. Lowengrub
Chancellor’s Professor, Mathematics
School of Physical Sciences
Professor, Biomedical Engineering
The Henry Samueli School of Engineering
Professor, Chemical Engineering & Materials Science
The Henry Samueli School of Engineering
B.A., Cornell University, 1985, Mathematics
Ph.D., Courant Institute of Mathematical Sciences, 1988, Applied Mathematics
University of California, Irvine
540H Rowland Hall
Mail Code: 3875
Irvine, CA 92697
Research Interests
mathematical materials science, mathematical fluid dynamics, mathematical biology, computational mathematics, cancer modeling, nanomaterials, quantum dots, complex fluids
Academic Distinctions
7/09–Present Chancellor’s Professor
Sept 1991–June 1992 Member
School of Mathematics, Institute for Advanced Study
June 1990–June 1993 NSF Postdoctoral Fellow
Stanford Advisor: Professor Joseph Keller
Minnesota Advisor: Professor Mitchell Luskin
July 1989–Sept 1992 Szego Assistant Professor
Department of Mathematics, Stanford University
Sept 1988–June 1989 Visiting Member
Courant Institute of Mathematical Sciences, New York University
1. The Convergence of the Vortex Method for Vortex Sheets, with R.C. Caflisch, SIAM J. Num. Anal., 26, pp. 1060-1080, 1989.
2. The Convergence of the Vortex Method for Vortex Sheets, Mathematical Aspects of Vortex Dynamics, ed. R.C. Caflisch, SIAM, pp. 120-127, 1989.
3. The Convergence of the Point Vortex Method for the 2-D Euler Equations, with J. Goodman and T.Y. Hou, Comm. Pure Appl. Math, XLIII, pp. 415-430, 1990.
4. The Convergence of the Point Vortex Method for the 3-D Euler Equations, with T.Y. Hou, Comm. Pure Appl. Math, XLIII, pp. 965-981, 1990.
5. Smooth Grid Methods for the Vorticity Formulation of the Euler Equations, with M.J. Shelley, in Vortex Dynamics and Methods, ed. C. Anderson and C. Greengard, Lectures in Applied Mathematics, 28,
AMS, pp. 423-432, 1991.
6. The Convergence of an Exact Desingularization and Local Regridding for Vortex Methods, with T.Y. Hou and M.J. Shelley, in Vortex Dynamics and Methods, ed. C. Anderson and C. Greengard, Lectures in
Applied Mathematics, v. 28, AMS, pp. 341-362, 1991.
7. The Convergence of a Point Vortex Method for Vortex Sheets, with T.Y. Hou and R. Krasny, SIAM J. Num. Anal., 28, pp. 308-320, 1991.
8. On the Well-Posedness of Two Fluid Interfacial Flows with Surface Tension, with J.T. Beale and T.Y. Hou, in Singularities in Fluids, Plasmas and Optics, ed. R. Caflisch and G. Papanicolaou, NATO
ASI Series, Kluwer Publishers, pp. 11-38, 1992.
9. Asymptotic and Numerical Results for Blowing-up Solutions to Semi-Linear Heat Equations, with J.B. Keller, in Singularities in Fluids, Plasmas and Optics, ed. R.C. Caflisch and G. Papanicolaou,
NATO ASI Series, Kluwer Publishers, pp. 111-130, 1992.
10. The Convergence of an Exact Desingularization for Vortex Methods, with T.Y. Hou and M.J. Shelley, SIAM J. Sci. Comp., 14, pp. 1-18, 1993.
11. High Order and Efficient Methods for the Vorticity Formulation of the Euler Equations, with M.J. Shelley and B. Merriman, SIAM J. Sci. Comp., 14, pp. 1107-1142, 1993.
12. Growth Rates for the Linearized Motion of Fluid Interfaces away from Equilibrium, with J.T. Beale and T.Y. Hou, Comm. Pure Appl. Math., XLVI, pp. 1269-1301, 1993.
13. Removing the Stiffness from Interfacial Flows with Surface Tension, with T.Y. Hou and M.J. Shelley, J. Comp. Phys., 114, No. 2, pp. 312-338, 1994.
14. Spatial and Temporal Stability Issues for Interfacial Flows with Surface Tension, with J.T. Beale, T.Y. Hou and M.J. Shelley, J. Math. and Comp. Modelling, 20, No. 10/11, pp. 1-27, 1994.
15. Numerical Calculations of Precipitate Shape Evolution in Elastic Media, with H.-J. Jou and P. Leo, to appear in Proceedings of an International Conference on Solid-Solid Phase Transformations,
ed. W.C. Johnson, J.M. Howe, D.E. Laughlin, W.A. Soffa, The Minerals, Metals and Materials Society, Warrendale, PA, pp. 635-640, 1994.
16. Convergence of Boundary Integral Methods for Water Waves, with J.T. Beale and T.Y. Hou, SIAM J. Num. Anal., 33, 1797, 1996.
17. The long time motion of vortex sheets with surface tension, with T.Y. Hou and M.S. Shelley, Phys. Fluids, 9, pp. 1933-1954, 1997.
18. Microstructural Evolution in Inhomogeneous Elastic Media, with H.J.-Jou and P.H. Leo, J. Comp. Phys., 131, pp. 109-148, 1997.
19. Stability of Boundary Integral Methods for Water Waves, with J.T. Beale and T.Y. Hou, AMS/IP Stud. Adv. Math, 3 (Nonlinear Evolutionary Partial Differential Equations, Beijing 1993), pp. 107-127,
20. A Diffuse Interface Model for Microstructural Evolution in Elastically Stressed Solids, with P.H. Leo and H.-J. Jou, Acta Materialia, 46, pp. 2113-2130, 1998.
21. Quasi-incompressible Cahn-Hilliard Fluids and Topological Transitions, with L. Truskinovsky, Proc. Roy. Soc. London A 454, pp. 2617-2654, 1998.
22. Almost Optimal Convergence of the Point Vortex Method for Vortex Sheets using Numerical Filtering, with R.C. Caflish and T.Y. Hou, Math. Comp., 68, pp. 1465-1496, 1999.
23. Topological Transitions in Liquid/Liquid Interfaces, with J. Goodman, H. Lee, E. Longmire, M.J. Shelley and L. Truskinovsky, Chapman & Hall/CRC Res. Notes Math, 409 (Free Boundary Problems:
Theory and Applications, Crete 1997), pp. 221-236, 1999.
24. Microstructural Evolution in Orthotropic Elastic Media, with P.H. Leo and Q. Nie, J. Comp. Phys., 157, pp. 44-88, 2000.
25. A Comparison of Experiments and Simulations on Pinch-Off in Round Jets, with E.K. Longmire and D.L. Gefroh, in Proceedings of the 1999 ASME/JSME Meeting, San Francisco.
26. Measurement and modeling of latent heat release during freezing in a small container, with R. V. Devireddy, J.C. Bischof, P.H. Leo, ASME IMECE HTD-368/BED-47 (2000), 23-31.
27. Boundary Integral Methods for Multicomponent Fluids and Multicomponent Materials, with T.Y. Hou and M.J. Shelley, J. Comp. Phys. 169 (2001), 302-362.
28. Focusing of an elongated hole in porous medium flow, with S.B. Angenent, D.G. Aronson and S.I. Betelu, Physica D 151 (2001), 228-252.
29. Modeling multiphase flows using a novel 3D adaptive remeshing algorithm, with R. Hooper V. Cristini, S. Shakya, C. W. Macosko and J. J. Derby, In Computational Methods in Multiphase Flow, Eds.:
C.A. Brebbia and H. Power, Series: Advances in Fluid Mechanics, Vol. 29, Wessex Institute of Technology Press, UK, 2001.
30. On an Elastically Induced Splitting Instability, with P.H. Leo and Q. Nie, Acta Mater. 49 (2001), 2761-2772.
31. Modelling Pinchoff and Reconnection in a Hele-Shaw Cell Part I: The Models and their Calibration, with H. Lee and J. Goodman, Phys. Fluids 14 (2002), 492-513.
32. Modelling Pinchoff and Reconnection in a Hele-Shaw Cell Part II: Analysis and Simulation in the Nonlinear Regime, with H. Lee and J. Goodman, Phys. Fluids 14 (2002), 514-545.
33. Measurement and numerical analysis of freezing in solutions enclosed in a small container, R. Devireddy, P. Leo and J. Bischof, Int. J. Heat Mass Transfer 45 (2002), 1915-1931.
34. Three dimensional crystal growth. I. Linear analysis and self-similar evolution, with V. Cristini, J. Crystal Growth, 240 (2002) 267.
35. Nonlinear simulation of tumor growth, with V. Cristini and Q. Nie, J. Math. Biol. 46 (2003) 191.
36. Microstructure evolution in three-dimensional inhomogeneous elastic media, with X. Li, Q. Nie, P.H. Leo and V. Cristini, Met. Mater. Trans. A 34A (2003) 1421.
37. Conservative multigrid methods for Cahn-Hilliard fluids, with J.-S. Kim, K. Kang, J. Comp. Phys. 193 (2004) 511-543.
38. Three dimensional crystal growth II. Nonlinear simulation and control of the Mullins-Sekerka instability, with V. Cristini, J. Crystal Growth 266 (2004) 552.
39. Conservative multigrid methods for ternary Cahn-Hilliard systems, with J.-S. Kim and K. Kang, Comm. Math. Sci. 2 (2004) 53.
40. Nonlinear theory of self-similar growth and melting, with S. Li, P.H. Leo and V. Cristini, J. Crystal Growth 267 (2004) 703.
41. A surfactant conserving volume-of-fluid method for interfacial flows with insoluble surfactant, with A. James, J. Comp. Phys. 201 (2004) 685-722.
42. Two- and three dimensional equilibrium morphology of a misfitting particle and the Gibbs-Thomson effect , with X. Li, K. Thornton, Q. Nie and P.W Voorhees, Acta Metall. 52 (2004) 5829-5843.
43. Efficient phase-field simulation of quantum dot formation in a strained heteroepitaxial film, with S.M. Wise, J.S. Kim and W.C. Johnson, J. Superlattices and Microstructures 36 (2004) 293-304.
44. Experiments and computations on drop impact at a liquid/liquid interface, with Z. Mohamed Kassim, E.K. Longmire, J.-S. Kim, X. Zheng , Proc. 5th Int. Conf. Multiphase Flow, paper no. 122 (2004)
in press.
45. Phase-field modeling of step dynamics, with Z. Hu, S.M. Wise, J.S. Kim and A. Voigt, MRS Proceedings 859E (JJ8.6), J. Evans, C. Orme, M. Asta and Z. Zhang eds., 2004.
46. Evolving interfaces via gradients of geometry dependent interior Poisson problems: Application to tumor growth, with P. Macklin, J. Comp. Phys. 203 (2005) 191-220.
47. Nonlinear stability analysis of self-similar crystal growth: Control of the Mullins-Sekerka Instability, with S. Li, P.H. Leo and V. Cristini, J. Crystal Growth 277 (2005) 578-592.
48. Modeling coarsening dynamics using interface tracking methods, invited review, Handbook of Materials Modeling, vol 1., S. Yip ed., Kluwer Acad. Press (2005) in press.
49. Interfaces and multicomponent fluids, with J.-S. Kim, Encyclopedia of Math. Phys., J.-P. Francoise, G. Naber and T.-S. Tsun eds., Elsevier (2005), invited review, to appear.
50. Adaptive unstructured volume remeshing algorithms II: Application to two- and three- dimensional level-set simulations of multiphase flows, with X. Zheng, A. Anderson and V. Cristini, J. Comp.
Phys., 208 (2005) 626-650.
51. Phase field modeling and simulation of three phase flows, with J.-S. Kim, Int. Free Bound. 7 (2005) 435-466.
52. Nonlinear morphological control of growing crystals, with S.W. Li and P.H. Leo, Physica D 208 (2005) 209-219.
53. Quantum dot formation on a strain-patterned epitaxial thin film, with S.M. Wise, J.S. Kim, K. Thornton, P.W. Voorhees and W.C. Johnson, Appl. Phys. Lett. 87 (2005) 133102.
54. A level-set method for interfacial flows with surfactant, with J.J. Xu, Z.L. Li and H.-K. Zhao, J. Comp. Phys. 212 (2006) 590-616.
55. Phase reconstruction by the weighted least action principle, with C.M. Lee J. Rubinstein and X.M. Zheng, J. Optics A- Pure Appl. Optics 8 (2006) 279-289.
56. An improved geometry-aware curvature discretization for level set methods: Application to tumor growth, with P. Macklin, J. Comp. Phys. 215 (2006) 392-401.
57. Analysis of cell growth in three-dimensional scaffolds, with J.C.Y. Dunn, W.Y. Chan, V. Cristini, J.S. Kim, S. Singh, B.M. Wu, Tissue Eng. 12 (2006) 705-716.
58. Numerical evidence of nonuniqueness in the evolution of vortex sheets, with M.C. Lopes, H.J.N. Lopes and Y. Zheng, ESAIM-Math. Model. Numer. Anal. 40 (2006) 225-237.
59. An adaptive coupled level-set/volume of fluid interface tracking method for unstructured triangular grids, with X. Yang, A. James, X. Zheng and V. Cristini, J. Comp. Phys. 217 (2006) 364-394.
60. Non-monotone temperature boundary conditions in dendritic growth, with M.E. Glicksman and S. Li, Proc. Modelling of Casting, Welding and Adv. Solid Processes XI, ed. C.A. Gandin, M. Bellet,
(2006) 512-528.
61. A deterministic mechanism for dendritic solidification kinetics, with M.E. Glicksman and S. Li, JOM 59 (2007) 27-34.
62. Nonlinear three-dimensional simulation of solid tumor growth, with X. Li, V. Cristini and Q. Nie, Discrete Contin. Dyn. System B 7 (2007) 581-604.
63. Nonlinear simulation of the effect of the microenvironment on tumor growth, with P. Macklin, J. Theor. Biol. 245 (2007) 677-704.
64. A rescaling scheme with application to the long time simulation of viscous fingering in a Hele-Shaw cell, with S. Li and P.H. Leo, J. Comp. Phys. 225 (2007) 554-567.
65. Surface phase separation and flow in a simple model of multicomponent drops and vesicles, with J.-J. Xu and A. Voigt, Fluid Dyn. Mater. Proc., 3 (2007) 1-19.
66. Computer simulation of glioma growth and morphology, with H.B. Frieboes, S. Wise, X. Zheng, P. Macklin, E. Bearer, V. Cristini, NeuroImage 37 (2007) S59-S70.
67. Morphological stability analysis of the epitaxial growth of a circular island: Application to nanoscale shape control, with Z. Hu and S. Li, Physica D 233 (2007) 151-166.
68. Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method, with S.M. Wise and J.-S. Kim, J. Comp. Phys. 226 (2007) 414-446.
69. A linear stability analysis for step meandering instabilities including the effects of elastic interactions and ES Barriers, with D.-H. Yeon, P.-R. Cha, A. Voigt and K. Thornton, Phys. Rev. E 76
(2007) 011601.
70. A deterministic mechanism for side-branching in dendritic growth, with S. Li, X. Li and M. Glicksman, Fluid Dyn. Mater. Proc. 2 (2007) 1-8.
71. Nonlinear modeling and simulation of tumor growth , with V. Cristini, H.B. Frieboes, X. Li, P. Macklin, S. Sanga, S.M. Wise and X. Zheng, in Modeling and Simulation in Science, Engineering and
Technology, ed. N. Bellomo, M. Chaplain and E. DeAngelis, Birkhauser, Boston, (2008) 113-181.
72. A ghost-cell/level-set method for nonlinear moving boundary problems, with P. Macklin, J. Sci. Comput. 35 (2008) 266-299.
73. Three-dimensional multispecies nonlinear tumor growth-I. Model and numerical method, with S.M. Wise, H.B. Frieboes and V. Cristini, J. Theor. Biol. 253 (2008) 524-543.
74. A new method for simulating strongly anisotropic Cahn-Hilliard equations , with S. Torabi, S. Wise, A. Ratz and A. Voigt, Proc. Mater. Sci. Tech. 2007.
75. Phase-field modeling of nanoscale island dynamics , with Z. Hu, S. Li, S. Wise and A. Voigt, Proc. TMS 2008 (137th meeting) Supplemental Proceedings: Materials Processing and Properties (2008)
76. Nonlinear simulations of solid tumor growth using a mixture model: Invasion and branching, with V. Cristini, X. Li and S.M. Wise, J. Math. Biol. 58 (2009) 723-763.
77. Multiscale modeling and nonlinear simulation of vascular tumour growth , with P. Macklin, S. McDougall, A.R.A. Anderson, M.A.J. Chaplain and V. Cristini, J. Math. Biol. 58 (2009) 765-798.
78. Multiparamter computational modeling of tumor invasion, with E.L. Bearer, Y.-L. Chuang, H.B. Frieboes, F. Jin, S.M. Wise, M. Ferrari, D.B. Agus, V. Cristini, Cancer Res. 69 (2009) 4493-4501.
79. Phase-field modeling of the dynamics of multicomponent vesicles: Spinodal decomposition, coarsening, budding and fission, with A. Rätz and A. Voigt, Phys. Rev. E 79 (2009) 031926.
80. A new phase-field model for strongly anisotropic systems, with S. Torabi, A. Voigt and S.M. Wise, Proc. R. Soc. London A 465 (2009) 1337-1359
81. Geometric evolution laws for thin crystal line films: Modeling and numerics, with B. Li, A. Rätz and A. Voigt, Comm. Comp. Phys. 6 (2009) 433-482. (invited review)
82. Stable and efficient finite-difference nonlinear multigrid schemes for the phase-field crystal equation, with Z. Hu, S.M. Wise, C. Wang, J. Comput. Phys. 228 (2009) 5323-5339.
83. An energy stable and convergent finite-difference scheme for the phase-field crystal equation, with S.M. Wise and C. Wang, SIAM J. Numer. Anal. 47 (2009) 2269-2288.
84. Solving PDES in complex geometries: A diffuse domain approach, with X. Li, A. Ratz and A. Voigt, Commun. Math. Sci. 7 (2009) 81-107.
85. Control of viscous fingering patterns in a radial Hele-Shaw cell , with S. Li, J. Fontana and P. Palffy-Muhoray, Phys. Rev. Lett. 102 (2009) 174501.
86. Coarsening of 3D thin films under the influence of strong surface anisotropy, elastic stresses, with P. Zhou and S.M. Wise, TMS 2009 (138th annual meeting), Supplemental Proceedings: Materials
Characterization, Computation and Modeling (2009) 39-46.
87. Dynamics of multicomponent vesicles in a viscous fluid, with J.-S. Sohn, Y.-H. Tseng, S. Li, A. Voigt, J. Comput. Phys. 229 (2010) 119-144.
88. A diffuse-interface approach for modeling transport, diffusion and adsorption/desorption of material quantities on a deformable interface, with K.E. Teigen, X. Li, F. Wang and A. Voigt, Comm.
Math. Sci., to appear.
89. Selection in spatial stochastic models of cancer: Migration as a key modulator of fitness, with C.J. Thalhauser, D. Stupack and N.L. Komarova, Biology Direct (2009), to appear.
90. Nonlinear modeling of cancer: Bridging the gap between cells and tumors, with H.B. Frieboes, F. Jin, Y.-L. Chuang, X. Li, P. Macklin, S.M. Wise and V. Cristini, Nonlinearity (2010) to appear.
(invited review).
Oct 2009- Sept 2011 Grand Opportunity Award Feedback, lineages and cancer: A multidisciplinary approach National Institutes of Health (co-PI Arthur Lander, UCI)
June 2008 Chancellor’s award for Excellence in Fostering Undergraduate Research UC Irvine
April 2008 Distinguished Mid-Career Faculty Award for Research UC Irvine
June 2006 - May 2009 NSF Grant in the Division of Materials Research New epitaxial nanostructures in the limited adatom mobility regime (co-PI, with Professor Robert Hull, U. Va.(PI))
August 2005 - July 2009 NSF Grant in the Division of Materials Research NSF-EC Cooperative Activity in Computational materials research: Bridging the atomistic to the continuum– Multiscale
investigation of self-assembling magnetic dots during epitaxial growth (co-PI, with Professors M. Asta (co-PI), P.W. Voorhees (co-PI) and K. Thornton (PI))
Jul 1994-Aug 2012 NSF Grants in the Division of Mathematical Sciences Current Grants: (1) Multiscale modeling of solid tumor growth (2) Computational problems in heterogeneous nanomaterials (3)
Morphological control of material microstructures
Jul 2006-June 2007 University of California, Irvine Research Experience for Undergraduates Agent-based models of tumor growth (awarded for Aaron Abajian). Also awarded Henry Samueli Engineering
School Undergraduate Fellowship.
June 2006-Sept 2006 University of California, Irvine Research Experience for Undergraduates Turing instability for irregular domains (awarded for Katiya Pavlova)
Jul 2004-June 2005 University of California, Irvine Research Experience for Undergraduates Nonlinear 3D modeling of tumor growth (awarded for Genevieve Brown)
Jan 2002-July 2003 University of Minnesota Research Experience for Undergraduates The development of a three dimensional adaptive tetrahedal mesh (awarded for Tony Anderson)
July 2001-June 2002 Minnesota Supercomputer Institute Research Scholarship Numerical Simulation of Microstructured Materials (awarded for Dr. Vittorio Cristini)
Jan 1998-Dec 2004 DOE Grants in the Basic Energy Sciences Division Fundamental Studies of Topological Transitions in Liquid/Liquid Flows (PI, with Professor E.K. Longmire (co-PI)).
Nov 1998 Francois Frenkiel Award American Physical Society, Fluid Dynamics Division
Sept 1996-2001 NSF Group Infrastructure Grant Infrastructural Needs for Preparing Students for the Industrial and Business Workforce (co-PI, with Professors B.Cockburn (co-PI), A. Friedman (co-PI),
and F. Santosa (PI))
Sept 1995-1997 Sloan Foundation Fellowship
July 1994-1996 McKnight Foundation Professorship University of Minnesota
June 1990-1993 NSF Postdoctoral Fellowship
Other Experience
Assistant Professor
University of Minnesota 1992—1995
Associate Professor
University of Minnesota 1995—1999
University of Minnesota 1999—2003
University of North Carolina 1999—2001
Graduate Programs
Mathematical and Computational Biology
Research Centers
Center for Complex Biological Systems (CCBS)
Mathematical, Computational and Systems Biology (MCSB) | {"url":"https://faculty.uci.edu/profile/?facultyId=5697","timestamp":"2024-11-09T17:16:57Z","content_type":"text/html","content_length":"40341","record_id":"<urn:uuid:f65eb408-2f0e-4b88-95b3-a5df8e945a15>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00640.warc.gz"} |
Public Key vs. Private Key Encryption: Key Differences Explained
Public Key vs. Private Key Encryption: Understanding the Key Differences
Public keys and Private keys are the two general encryption techniques in cryptography. Cryptography presents public key encryption as asymmetric encryption, and private key encryption is stated as
symmetric encryption.
Both play a pretty significant role in cybersecurity. However, both show differences mainly in the way data is encrypted and decrypted.
We shall in this blog look at the differences between public key and private key encryption methods and how they work-principles and use of both methods.
What is Encryption?
Encryption is the process of transforming plain-text completely into some unreadable format for security. This kind of text is sometimes known as ciphertext. On the contrary, decryption is the
reverse process whereby ciphertext is decrypted so that it may reverse back to its original form.
What is Public Key Encryption?
This form of encryption makes use of a public key as well as a private key. It is the public key that encrypts the data whereas it is the private key, that decrypts the data. That’s why this method
of encryption is referred to as “asymmetric encryption”. Why? Because the two different keys are utilized.
How Does Public Key Encryption Work?
An algorithm produces a pair of public and private keys. The public key is transmitted to the sender, and the private key is retained by the owner who generated them.
1. Once someone wishes to send the encrypted data or crypto, it uses the recipient’s public key to encrypt the message.
2. The recipient uses his private key to retrieve and decrypt the data to its original form.
3. As the public key is available in public, public key encryption shall be employed in any scenarios where it is mandatory to establish secure communication between entities who have never crossed
some encryption keys before.
A third party cannot decrypt the message intercepted without the private key.
This is the sort of encryption applied in blockchains.
What is Private Key Encryption (Symmetric Encryption)?
The key for both encryption and decryption is the same, or symmetric, in private key encryption, and hence it is commonly called symmetric encryption. Obviously, communication cannot begin until both
sender and recipient have agreed on the shared key. Because the key must be used both to encrypt and to decrypt, this is often called a symmetric approach, in contrast to the public key encryption
How Private Key Encryption Works?
1. Communication cannot occur unless the sender and receiver possess this encrypted private key, securely.
2. In this scenario, the sender encrypts his information using the common key.
3. The receiver uses the same key to decrypt the message.
4. Because both end parties have the same key, private key encryption is faster and more efficient than public key encryption. However, the necessity of having to share the secret key securely in
advance beforehand frequently introduces vulnerabilities, especially if the network size is large or that the parties involved are unknown.
Differences Between Public Key and Private Key Encryption
Having now outlined some basics of each, let’s proceed to compare and contrast them on the basis of several key criteria:
1. Use of Keys
Public Key Encryption: Uses two unique keys-public and private. It encrypts using the public key and decrypts using the private key.
Private Key Encryption: The same key is used for encryption and decryption. In the private key encryption process, the sender also shares it with the recipient.
2. Speed and Efficiency
Public Key Encryption: This is the slower approach as it involves lengthy mathematical algorithms for generating a key pair and encrypting data. It is less effective to encrypt huge data.
Private Key Encryption: Better for efficiency and speed. The same key is encrypting and decrypting. It is better for the usage of bulk encryption.
3. Security
Public Key Encryption: Stronger security due to a separate pair of keys. Even if they know the public key, the message cannot be decrypted without the private key.
Private Key Encryption: This is based on how good the protection and sharing of the key are. If somehow the key is compromised at transmission or storage, then the data is easily decrypted by
unauthorized individuals.
4. Key Distribution
Public Key Encryption: More feasible for distribution because the public key can be distributed without restrictions. There is no need for a secure channel to exchange keys since only the private key
needs to remain secret.
Private Key Encryption: It requires a secure way to transfer the private key between the sender and receiver. This can be viewed as a weakness in case the interceptor somehow gets its hands on the
5. Applications
Public Key Encryption: It is suitable for secure communication when parties have not had previous key exchanges. Mostly used when a digital signature by the sender is used for encryption of messages,
SSL/TLS certificates, and email encryption.
Private Key Encryption: Best for those scenarios in which both parties have already exchanged keys confidentially. Most people use this to encrypt data transfer in bulk, for instance VPNs, database
encryption, and file encryption.
Public Key Encryption Use Cases
Public key encryption is often used between two parties that wish to communicate safely but do not share a common encryption key in advance. Some of its popular use cases include:
SSL Certificate: SSL certificates uses this kind of cryptography to create secure communication between the website and its users.
Email Encryption: Services like PGP uses public key encryption to make sure that only the recipient can read the email.
Private Key Encryption in Practice
Private key encryption is usually applied where efficiency and speed are more important. Some examples include:
File Encryption: This type of encryption is used for files stored in a device or in the cloud. This way, only authorized users with the key can access the data.
Database Encryption: Private key encryption can be used here, especially in cases where encrypting large volumes of data is required.
Combines Use of Public and Private Keys
In many number of systems, both the methods, public key encryption and private key encryption are used together to utilize both the strengths. Hybrid encryption is one commonly used in secure
communication protocols. In hybrid encryption:
Public key encryption is used for exchanging symmetric key across sender and recipient.
Once the symmetric key has been shared, private key encryption will then encrypt the actual data, since this mode is fast and efficient.
This method ensures both security and efficiency of encrypting data.
Frequently Ask Questions
What Is the Difference Between Private and Public Key Encryption?
Private and public key encryption are the two main types of asymmetric encryption. In public key encryption there are two keys: one public and one private. The public key is used to encrypt the data
so anyone can see it but only the private key can decrypt it so only the intended recipient can read the message. Private key encryption (also known as symmetric encryption) uses one key to encrypt
and decrypt the data.
The main difference is the number of keys: public key encryption is asymmetric (two keys) and private key encryption is symmetric (one key).
Do I Encrypt with Public or Private Key?
In public key encryption the public key is used to encrypt the data. When someone wants to send you a secure message they encrypt it with your public key. Once encrypted only your private key can
decrypt it so the data is secure and private.
But in digital signatures the process is reversed: you encrypt with your private key to create a signature which can then be verified by others with your public key to ensure authenticity.
What Is the Difference Between Public Key and Private Key in PGP Encryption?
PGP (Pretty Good Privacy) uses both public and private keys for secure communication. The public key is shared with anyone who wants to send an encrypted message to you. The message is encrypted with
your public key and can only be decrypted with your private key which you keep secret.
Public and private keys in PGP are public and secret. Public is shared to enable communication, private is kept secret.
Is Private Key More Secure Than Public Key?
Both public and private keys have different uses and security depends on how they are used. Private keys are more sensitive as they are used to decrypt data. If a private key is compromised,
encrypted messages can be decrypted and confidential info can be exposed.
Public keys are meant to be public and not a direct security risk if leaked. But private keys must be kept secure and not accessible to others.
What Is an Example of Private Key Encryption?
Private key encryption example is AES. In AES, one key (private key) is used for both encryption and decryption. This is symmetric encryption used for securing sensitive data like file encryption or
device to device secure communication.
Another example is DES (Data Encryption Standard) which also uses one key for both encryption and decryption.
Public key encryption is stronger in security and ideal for establishing secure communication between parties who haven’t exchanged keys prior to this. However, it is slower and less efficient for
the encryption of large amounts of data.
Private key encryption, on the other hand is faster and more efficient but requires secure key exchange.
Key differences between public and private key encryption provide an understanding for the purposes of selecting both public and private key methods, depending upon the particular requirements and
use cases, in order to provide informed decisions. Quite often, a combination of both ways will provide an ideal balance between security and efficiency. | {"url":"https://cryptolandoff.com/public-key-vs-private-key/","timestamp":"2024-11-10T01:14:13Z","content_type":"text/html","content_length":"511267","record_id":"<urn:uuid:79a36683-53de-41e9-bc56-26d3a6ab82d6>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00837.warc.gz"} |
An operator splitting-radial basis function method for the solution of transient nonlinear Poisson problems
This paper presents an operator splitting-radial basis function (OS-RBF) method as a generic solution procedure for transient nonlinear Poisson problems by combining the concepts of operator
splitting, radial basis function interpolation, particular solutions, and the method of fundamental solutions. The application of the operator splitting permits the isolation of the nonlinear part of
the equation that is solved by explicit Adams-Bashforth time marching for half the time step. This leaves a nonhomogeneous, modified Helmholtz type of differential equation for the elliptic part of
the operator to be solved at each time step. The resulting equation is solved by an approximate particular solution and by using the method of fundamental solution for the fitting of the boundary
conditions. Radial basis functions are used to construct approximate particular solutions, and a grid-free, dimension-independent method with high computational efficiency is obtained. This method is
demonstrated for some prototypical nonlinear Poisson problems in heat and mass transfer and for a problem of transient convection with diffusion. The results obtained by the OS-RBF method compare
very well with those obtained by other traditional techniques that are computationally more expensive. The new OS-RBF method is useful for both general (irregular) two- and three-dimensional geometry
and provides a mesh-free technique with many mathematical flexibilities, and can be used in a variety of engineering applications.
• Convection-diffusion-reaction equation
• Helmholtz equation
• Method of fundamental solutions
• Nonlinear Poisson problem
• Operator splitting
• Particular solution method
• Radial basis functions
ASJC Scopus subject areas
• Modeling and Simulation
• Computational Theory and Mathematics
• Computational Mathematics
Dive into the research topics of 'An operator splitting-radial basis function method for the solution of transient nonlinear Poisson problems'. Together they form a unique fingerprint. | {"url":"https://experts.syr.edu/en/publications/an-operator-splitting-radial-basis-function-method-for-the-soluti","timestamp":"2024-11-12T13:14:55Z","content_type":"text/html","content_length":"54159","record_id":"<urn:uuid:dc5d9394-54c3-474f-bc79-17cc433b7728>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00254.warc.gz"} |
class chainer.datasets.SubDataset(dataset, start, finish, order=None)[source]¶
Subset of a base dataset.
SubDataset defines a subset of a given base dataset. The subset is defined as an interval of indexes, optionally with a given permutation.
If order is given, then the i-th example of this dataset is the order[start + i]-th example of the base dataset, where i is a non-negative integer. If order is not given, then the i-th example of
this dataset is the start + i-th example of the base dataset. Negative indexing is also allowed: in this case, the term start + i is replaced by finish + i.
SubDataset is often used to split a dataset into training and validation subsets. The training set is used for training, while the validation set is used to track the generalization performance,
i.e. how the learned model works well on unseen data. We can tune hyperparameters (e.g. number of hidden units, weight initializers, learning rate, etc.) by comparing the validation performance.
Note that we often use another set called test set to measure the quality of the tuned hyperparameter, which can be made by nesting multiple SubDatasets.
There are two ways to make training-validation splits. One is a single split, where the dataset is split just into two subsets. It can be done by split_dataset() or split_dataset_random(). The
other one is a \(k\)-fold cross validation, in which the dataset is divided into \(k\) subsets, and \(k\) different splits are generated using each of the \(k\) subsets as a validation set and
the rest as a training set. It can be done by get_cross_validation_datasets().
☆ dataset – Base dataset.
☆ start (int) – The first index in the interval.
☆ finish (int) – The next-to-the-last index in the interval.
☆ order (sequence of ints) – Permutation of indexes in the base dataset. If this is None, then the ascending order of indexes is used.
Returns an example or a sequence of examples.
It implements the standard Python indexing and one-dimensional integer array indexing. It uses the get_example() method by default, but it may be overridden by the implementation to, for
example, improve the slicing performance.
index (int, slice, list or numpy.ndarray) – An index of an example or indexes of examples.
If index is int, returns an example created by get_example. If index is either slice or one-dimensional list or numpy.ndarray, returns a list of examples created by get_example.
>>> import numpy
>>> from chainer import dataset
>>> class SimpleDataset(dataset.DatasetMixin):
... def __init__(self, values):
... self.values = values
... def __len__(self):
... return len(self.values)
... def get_example(self, i):
... return self.values[i]
>>> ds = SimpleDataset([0, 1, 2, 3, 4, 5])
>>> ds[1] # Access by int
>>> ds[1:3] # Access by slice
[1, 2]
>>> ds[[4, 0]] # Access by one-dimensional integer list
[4, 0]
>>> index = numpy.arange(3)
>>> ds[index] # Access by one-dimensional integer numpy.ndarray
[0, 1, 2]
Returns the number of data points.
Returns the i-th example.
Implementations should override it. It should raise IndexError if the index is invalid.
i (int) – The index of the example.
The i-th example.
__eq__(value, /)¶
Return self==value.
__ne__(value, /)¶
Return self!=value.
__lt__(value, /)¶
Return self<value.
__le__(value, /)¶
Return self<=value.
__gt__(value, /)¶
Return self>value.
__ge__(value, /)¶
Return self>=value. | {"url":"https://docs.chainer.org/en/stable/reference/generated/chainer.datasets.SubDataset.html","timestamp":"2024-11-12T09:34:07Z","content_type":"text/html","content_length":"31681","record_id":"<urn:uuid:f847321a-3d7c-40ac-9b4e-1d34d982463e>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00666.warc.gz"} |
Probability Sampling - Research-Methodology
Sampling techniques can be divided into two categories: probability and non-probability. In probability sampling, each population member has a known, non-zero chance of participating in the study.
Randomization or chance is the core of probability sampling technique. Probability sampling methods use some form of random selection. Therefore, application of this method offers the highest chance
of creating a sample that is truly representative of the population.
In non-probability sampling, on the other hand, sample group members are selected non-randomly; therefore, in non-probability sampling only certain members of the population has a chance to
participate in the study.
The table below illustrates the main differences between probability and non-probability sampling methods.
Probability sampling Non-probability sampling
Randomly selected samples Subjective judgement of researchers are used to select samples
Equal chance for each member of population to get selected Not everyone has equal chance to get selected
Used to reduce a sampling bias Researcher is not overly concerned with sampling bias
Effective to collect data from diverse population Useful in specific environment with sampling group members sharing similar characteristics
Useful in obtaining accurate representation of population Does not help in representing the population accurately
Finding correct audience is difficult Finding correct audience is simple
Probability sampling comprises the following sampling techniques:
Application of Probability Sampling: an Example
Let’s suppose, your dissertation topic is ‘A study into employee motivation of ABC Company and the ways of increasing it’. You chose survey primary data collection method to achieve research
Sampling process comprises the following four stages:
1. Identifying an appropriate sampling frame based on your research question(s) and objectives. ABC Company has 400 employees and accordingly, your sampling frame would be 400.
2. Determining a suitable sample size. You may decide that the sample size of 60 employees should be sufficient for the purposes of this research.
3. Choosing the most appropriate sampling technique and selecting the samples. In this case, simple random sampling, the most basic form of probability sampling technique can be applied through using
a table of randomly generated numbers. Websites such asGenerate Data, Graph Pad,Mockaroo and many others can be used to do this task easily and quickly.
Now, all you have to do is to choose a starting point in the table (a row and column number) and look at the random numbers that appear there. In this case, since the data run into three digits, the
random numbers would need to contain three digits as well. You need to ignore all the random numbers after 400, since your target population has only 400 members. Also, choose a specific number only
once and if a number recurs, simply skip it and move to the next number. In this way, the first 60 different numbers between 001 and 400 that represent 60 employees of ABC Company constitute your
sample group.
4. Checking if the sample is representative of the population.
Advantages of Probability Sampling
• The absence of systematic error and sampling bias
• Higher level of reliability of research findings
• Increased accuracy of sampling error estimation
• The possibility to make inferences about the population. Effective to collect choose samples from broad population base
• Cost-effectiveness
• Simple and straightforward in application
Disadvantages of Probability Sampling
• Higher complexity compared to non-probability sampling
• More time-consuming
• Usually more expensive than non-probability sampling
My e-book, The Ultimate Guide to Writing a Dissertation in Business Studies: a step by step approach contains a detailed, yet simple explanation of sampling methods. The e-book explains all stages of
the research process starting from the selection of the research area to writing personal reflection. Important elements of dissertations such as research philosophy, research approach, research
design, methods of data collection and data analysis are explained in this e-book in simple words.
John Dudovskiy | {"url":"https://research-methodology.net/sampling-in-primary-data-collection/probability-sampling/","timestamp":"2024-11-06T04:12:39Z","content_type":"text/html","content_length":"141274","record_id":"<urn:uuid:a484a574-ea9d-42c9-bf39-4c3d227ec380>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00082.warc.gz"} |
Christopher Alexander
Diagrams as Information Structures
Background and Influences:
Christopher Alexander, educated both as an architect and mathematician, was widely influential during 60’s -and still is- on discussions around design methodology and the use of computers in design,
architecture and urbanism. Despite being a quite controversial and “lone” scholar, he was engaged in an active conversation around these issues, due to his involvement in multiple academic
collaborations, conferences and publications. In particular, during his doctoral studies at Harvard (1958+) he worked with the Center for Cognitive Research, the Joint Center for Urban Studies of MIT
and Harvard, and the MIT Civil Engineering Systems Laboratory. In 1962 he participated in the first of a series of Conferences on Design Methods in London, attended also by personalities like Gordon
Pask and representatives from HfG Ulm. In 1967, he founded the Center for Environmental Structure in University of California-Berkeley, where he continued his teaching career for almost 40 years.
Alexander’s work argues for the relationship between science and design methodologies: ‘Scientists try to identify the components of existing structures, designers try to shape the components of new
structures.’ (Alexander, 1964) For Alexander structure means form that manifests information structures. He expresses design problems as systems. In order to synthesize a form, he first decomposes
the system into subsystems and tries to define their interrelationship.
“This is what lies behind D’ Arcy Thompson’s remark that the form is a diagram of forces. Once we have the diagram of forces in the literal sense, this will in essence also describe the form as a
complementary diagram of forces.”(Alexander, 1964)
Illustrations from “Notes on the Synthesis of Form”.
This systemic approach is directly influenced by Cybernetics and specifically Norbert Wiener and Ross Ashby’s homeostasis theory. In Alexander’s case, systems are used to describe an environmental
approach to design, which conceives our habitat as an open system of animate and inanimate agents (Maldonado, 1969) and design knowledge is approached holistically as the man-environment
relationship. Due to his continuous effort to describe design as a diagram of forces that and his participation in Design Methods Conference, his name is associated to the nascent field of Design
Methods. However, later in his life, he denounces the whole field:
“Indeed, since the book was published (Notes on the Synthesis of form, 1964), a whole academic field has grown up around the idea of “design methods”-and I have been hailed as one of the leading
exponents of these so-called design methods. I am very sorry that this has happened, and want to state, publicly, that I reject the whole idea of design methods as a subject of study, since I
think it is absurd to separate the study of designing from the practice of design. […] Study of method by itself is always barren, and people who have treated this book as if it were a book about
“design method” have almost always missed the point of the diagrams, and their great importance, because they have been obsessed with the details of the method I propose for getting at the
diagrams.” (Alexander, 1971)
Relation of problem decomposition to computational means: Diagrams and Computer Programs
Alexander’s constant redefinition of theories and visualization of information structures is even a more insightful as a narrative, than the criticism he received for being too deterministic. It is
interesting to see the relationship between his information structures (diagrams) and the capacities of computer programing. It is argued here that the units of work had a significant impact on how
he perceived and described design problems. The decomposition of the problem should be processed in a standardized manner, following rules, as if a computer performed it.
Tree Structure and HIDECS 2 (Hierarchical Decomposition of Systems): In his dissertation Notes on the synthesis of Form, Alexander describes design problem as sets that can be in a hierarchical tree.
The computer program HIDECS 2 that corresponds to this theory uses a binary stochastic algorithm that “at each level of the tree each set of variables is broken into those two subsets with minimum
information transfer between them”. Alexander realizes that this method does not take into account the holistic relatedness of system and subsystems. Usually subsystems overlap in a system.
illustrations from “Notes on the Synthesis of Form” Worked example by Alexander, taken from a recent paper of Christopher Jones ,ed. , Conference on Design Method (Oxford : Pergamon , 1 963) ., “The
Determination of Components for an Indian Village.” The problem treated is : An agricultural village of six hundred people is to be reorganized to make it fit present and future conditions developing
in rural India .
Lattice Structures and HIDECS 3: Since his realization that structural interrelations cannot always be described as trees, he writes the City is not a Tree and HIDECS 3 that aims to tackle weaknesses
of HIDECS 2, while using the same “machine representation”. He therefore introduces the structure of the semi lattice. In this program the decomposition into subsystems is not defined by a binary
condition in each step, but all at once.
Semi-lattice vs Tree, Alexander’s Diagram from “The City is not a Tree”
Alexander’s diagrams and Reference painting by Simon Nicholson (top right) from “The City is not a Tree”
Later on, Alexander will write collaboratively Pattern Language, his most famous work, where he looses his control on information structures and describes systems as open networks. Patterns contain
information of the forces that made them but they are at the same time instances of an open-ended approach to design. “Each pattern is a field — not fixed, but a bundle of relationships, capable of
being different each time that it occurs, yet deep enough to bestow life wherever it occurs.”
Alexander employed mathematics and in particular set theory to analyze design problems and used graphs (diagrams) to bridge the gap between architectural and software representation. This relation of
computational means and techniques for problem solving is probably one of the reasons that made Alexander so popular among object-oriented programming.
Bibliography & References:
- Alexander Christopher, “Notes on the Synthesis of Form”, Harvard University Press, 1964
- Alexander Christopher, “The City is not a Tree”, Ekistics Vol.23, 1967
- Alexander Christopher, “HIDECS 3: Four Computer Programs for the Hierarchical Decomposition of Systems which have an associated linear graph”, MIT, Department of Civil Engineering, 1963
-Maldonado Tomas, “How to fight complacency in design education”, Bit 4 International, 1969
- Cross Nigel, “A History of Design Methodology”, 1993, available here: https://monoskop.org/images/6/66/Cross_Nigel_1993_A_History_of_Design_Methodology.pdf
- Wright Steenson Molly, Architectures of Information: Christopher Alexander, Cedric Price and Nicholas Negroponte & MIT’s Architecture Machine Group, PhD Thesis, Princeton, April 2014 | {"url":"https://eliza-pert.medium.com/1960s-32969cc82d03","timestamp":"2024-11-02T06:56:08Z","content_type":"text/html","content_length":"125224","record_id":"<urn:uuid:01bafa7e-9214-47b4-95bd-7e91af47cf15>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00476.warc.gz"} |
Solve 0.2 mg : 1.3 ml = 0.35 mg : x ml
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Inches to Feet
Feet and inches are two of the most commonly used units of measurement in the United States. A foot is a unit of length equal to 12 inches, or one-third of a yard. An inch is a unit of length equal
to 1/12th of a foot, or approximately 2.54 centimeters. Both feet and inches are used for measuring length, height, distance, and other dimensions in many everyday applications. Converting between
these two units can be done using simple formulas or charts which provide easy reference points for making accurate conversions quickly. In this article we will look at how to convert from feet to
inches as well as from inches to feet using various methods including formulas and conversion charts.
How to Convert Inches to Feet?
The feet conversion formula is:
Feet = Inches / 12
For example, if you want to convert 24 inches into feet measurement, the calculation would be as follows:
Feet = 24 inches / 12= 2 Feet
How to Convert Feet to Inches?
The formula for converting feet to inches is:
Inches = Feet x 12
For example, if you want to convert 2 feet into inch measurement, the calculation would be as follows:
Inches = 2 Feet x 12= 24 Inches
Inches to Feet and Inches
At times, instead of converting a quantity into decimals, it is preferable to express it in feet and inches. For instance, 71 inches is equivalent to 5 feet and 11 inches. This method is known as how
to convert inches to feet and inches, wherein the quotient and remainder are used for representing the value in feet and inches, respectively. Here are a few examples:
• To convert inches to feet, for example, we have 67 inches, we divide it by 12, which gives us 5 as the quotient and 7 as the remainder. Therefore, 67 inches is equal to 5 feet and 7 inches.
• When we convert 80 inches to feet, the result is 6 feet and 8 inches, since 80 divided by 12 is equal to 6 with a remainder of 8.
• Similarly, 54 inches in feet becomes 4 feet and 6 inches after dividing 54 by 12, which yields 4 as the quotient and 6 as the remainder.
Inches to Feet Conversion Chart
If you need a quick reference, the following chart can be used as a guide to convert inch measurement to feet measurement quickly and easily.
Inches to Feet Conversion Tips and Tricks:
The word 'feet' is the correct plural form of 'foot' and is used when referring to more than one unit. It is important to note that 'feets' is not a valid word, as 'feet' already represents the
plural form.
Additionally, 1 foot is equal to 12 inches, while 1 inch is equal to 0.0833 feet. When it comes to area, 1 square inch is equivalent to 1/144 square feet, and for volume, 1 cubic inch is equal to 1/
1728 cubic feet.
In conclusion, it is important to understand how to convert inches into feet and vice versa. With the help of this article, you now have a better understanding of how conversion works between feet
and inches measurement. Furthermore, we provided an easy-to-use chart which can be used as a reference when making conversions from inches to feet or vice versa. Finally, remember that 'feet' is the
correct plural form for 'foot', so make sure not to use 'feets'. Now go ahead and start converting! | {"url":"https://toolsfor.ninja/inches-to-feet/","timestamp":"2024-11-03T23:10:02Z","content_type":"text/html","content_length":"56699","record_id":"<urn:uuid:dd26c1d7-e44c-4eab-8338-aa47de9f5032>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00171.warc.gz"} |
Should have never went to metric
Ancient civilizations primarily used the base 12 number system. It was perfect because it has factors of 2,3,4,6, and you can even combine together like 8 being 2/3rds of 12. This was the base of
mathematics for a long time, and gave rise to concepts such as angles, measurements of time, and was heavily used in tradecrafts that had to rely heavily on division: cooking, woodworking, etc.
This lasted basically until the Arabic number system which was base 10 was taken on by the Romans who then spread it over the world, but the trades people created measurements that bridged the gap
between the two, so you have names for groups of measurements that work off of the base 12 system.
Now we have further entrenched in base 10, and thanks to technology, we can use digital measurement devices to kinda get by, but it sucks when you decide to just work with a ruler and calculations in
your head, because you end up with some long decimals, and even some infinitely repeating ones. Hell even computers have a hell of a time with it because it doesn't match up with their base 2 number
system leading to strange behaviors where numbers that should be equal are not.
The fun thing is, if we were using base 12, we wouldn't need metric. All of the imperial measurement names were just a bridge between two bases, and like metric, we might have special names for
various scales, but really it would all just be moving the duodecimal point to the left and right.
We should have went back to base 12 as that was the best, but I am sure that would have been even harder for the world to have embraced. | {"url":"https://robkohr.com/articles/should-have-never-went-to-metric","timestamp":"2024-11-03T23:07:57Z","content_type":"text/html","content_length":"3321","record_id":"<urn:uuid:620964c0-2535-4d1e-81fc-59b426b9f842>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00886.warc.gz"} |
Fast parametric maximum flow algorithm and applications
The classical maximum flow problem sometimes occurs in settings in which the arc capacities are not fixed but are functions of a single parameter, and the goal is to find the value of the parameter
such that the corresponding maximum flow or minimum cut satisfies some side condition. Finding the desired parameter value requires solving a sequence of related maximum flow problems. In this paper
it is shown that the recent maximum flow algorithm of Goldberg and Tarjan can be extended to solve an important class of such parametric maximum flow problems, at the cost of only a constant factor
in its worst-case time bound. Faster algorithms for a variety of combinatorial optimization problems follow from the result.
All Science Journal Classification (ASJC) codes
• General Computer Science
• General Mathematics
Dive into the research topics of 'Fast parametric maximum flow algorithm and applications'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/fast-parametric-maximum-flow-algorithm-and-applications","timestamp":"2024-11-05T20:39:12Z","content_type":"text/html","content_length":"47936","record_id":"<urn:uuid:0ba32418-ea06-4e77-8fd8-22c52e040bac>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00090.warc.gz"} |
The Worksheet for playing Entropy Mastermind, completed following
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Neutral flow interaction with a magnetic dipole plasma. I. Theory and scaling
The interaction between a high-speed neutral gas flow and dipole magnetized plasma is investigated theoretically to examine how mass, momentum, and energy are transferred to the plasma.
Single-particle trajectory analysis reveals the existence of an ion-trapping region within which ions are born into closed orbits around the magnet, thus adding mass and energy to the plasma. Ion
deceleration through trapping or deflection is analyzed to quantify momentum transfer to the magnetic field. The drag force and transfer rate of mass and energy from the flow to the plasma are found
to scale with two dimensionless parameters: (1) a characteristic ion Larmor radius normalized by the magnet radius and (2) a characteristic neutral reaction rate normalized by the neutral gas rate of
transit. The energy transfer rate is maximized at a specific reaction rate, above which increased reactivity rapidly decreases energy capture as the interaction moves away from the ion-trapping
region. As the electron energy confinement time increases, there is a transition from a mode in which seed plasma is required to sustain the interaction to a high-density mode that is sustained
primarily by mass and energy from the neutral flow. Two distinct flow-sustained regimes are identified that depend on the ratio of the effective ionization energy to kinetic energy of the neutral gas
particles. One of the two regimes corresponds to the well-known critical ionization velocity phenomenon. The other, in which charge exchange collisions are the dominant energy transfer mechanism, has
not been previously identified as a separate physical regime.
The interaction between a high-speed neutral gas flow and magnetized plasma is a problem of fundamental interest to space plasma physics,^1–4 plasma propulsion,^5,6 atmospheric reentry,^7,8 and
plasma aerocapture concepts.^9,10 Theoretical understanding of this problem has primarily focused on the critical ionization velocity (CIV) phenomenon.^11–13 CIV theory hypothesizes that rapid
ionization of a neutral flow will occur when the kinetic energy of the flow relative to a background plasma exceeds the ionization energy of the neutral particles. The corresponding velocity is
commonly referred to as the critical ionization velocity. Evidence of this effect has been found in a number of laboratory and space experiments. Laboratory experiments have focused mainly on
studying the interaction of a high-speed plasma flow with stationary neutral gas.^14 Homopolar plasma devices, which possess a radial electric (E) field and an axial magnetic (B) field, found the E×
B velocity of the ions to be limited to the critical ionization velocity following rapid ionization of the neutral gas.^11,15 Experiments with plasma guns observed that high-speed plasmas rapidly
decelerate to the critical ionization velocity upon merging with a neutral gas.^16–19 Similar experimental support for CIV has come from studying the interaction of high-speed neutral gas flows in
the Earth's ionosphere using rockets that possess shaped-charge explosives^20 and controlled gas releases from orbiting spacecraft.^21–23
CIV theory generally considers relative motion between a neutral gas and isotropic plasma of infinite extent. As incoming neutral particles are ionized via electron impact ionization, the resulting
fast-moving ions transfer a fraction of their kinetic energy to the electron population. This increased electron energy results in a corresponding ionization rate increase—a feedback process that
rapidly ionizes the neutral flow. Theoretical research^3,12,24 has focused on understanding energy transfer between ions and electrons in the collisionless regime relevant to space and laboratory CIV
experiments. Less emphasis has been placed on the influence of plasma anisotropies, finite length scales, and collisional energy transfer. These effects can be expected to impact CIV through a
combination of finite Larmor radius effects, spatial variations in the ionization rate, and the exchange and diffusion of mass and energy. Furthermore, they are critical to understanding momentum
transfer between the neutral flow and plasma and determining specific requirements of CIV on the electron energy confinement time.
In this paper, we address these effects in a theoretical model describing the interaction between a high-speed neutral flow and dipole plasma of finite extent. In a finite system, the magnetic field
geometry is critical because the location at which a neutral is ionized determines the nature of its resulting trajectory. Therefore, we focus on understanding the spatial distribution of these
trajectories to account for finite Larmor radius effects and the capture of mass and energy from trapped particles. This analysis enables derivation of general scaling laws for the transfer of mass
and energy from the neutral flow to the plasma and for momentum coupling to the magnetic field. These laws are used to describe the processes governing distinct regimes of plasma–flow interaction and
the critical ionization-like transitions between them. In Paper II of this series,^25 we derive a higher-fidelity global model that combines the scaling laws found in this current paper with physical
models for diffusion, charge exchange (CX), ionization, and ion–electron energy transfer.
The comprehensive approach presented here is unique in addressing critical ionization in a finite, inhomogeneous magnetic topology. It is important to note that we primarily consider collisional
interaction between species, such as charge exchange and electron-impact ionization, as a departure from many prior CIV models. These effects are most readily invoked for low-temperature ($T<102$
eV), high density ($n>1016$ m^−3) plasmas. We assume a relatively high magnetic field strength (B>100 G) so that the timescale for thermal equilibration is generally faster than that for particle
diffusion from the system. Therefore, ions that become trapped in regions of high magnetic field strength provide a source of mass and energy for the plasma. Furthermore, we assume that magnetic
pressure dominates plasma pressure (low-beta), so the magnetic field is static and unaffected by the plasma.
To formulate this model, we consider an axisymmetric magnetized plasma whose axis of symmetry is oriented parallel to an incident neutral flow (Fig. 1). This alignment allows the use of 2D
cylindrical coordinates (r, z), where symmetry dictates that solutions are uniform in θ. The density and velocity of the neutral flow are denoted as $n∞$ and $u∞$, respectively. We assume that the
thermal energy of the neutral flow is negligible compared to its kinetic energy, $ε∞≈msnn∞u∞2/2$, where $ε∞$ is the freestream energy density and m[sn] represents the neutral particle mass.
A. Magnetic field model
For simplicity, we consider the field topology of a magnetic dipole formed by a closed loop of electric current. The magnetic field can be described by the flux function^26
Here, $ψ̂=ψ/(B0rc2)$ is the normalized flux function, B[0] is the magnetic field strength at the magnet center, and r[c] is the magnet radius. K and E are complete elliptic integrals of the first and
second kind, respectively. Cylindrical coordinates are normalized by the magnet radius: $r̂=r/rc$ and $ẑ=z/rc$. The normalized magnetic field vector, $B̂=B/B0$, can be found from
where $êθ$ is the unit vector in the θ direction.
A contour plot of $ψ̂$ is shown in Fig. 1 for normalized cylindrical coordinates. Here, lines of constant $ψ̂$ represent the intersection of magnetic flux surfaces with the $r̂$–$ẑ$ plane. We note that
the magnetic topology presented in this coordinate system is independent of both r[c] and B[0]. This property will be leveraged in Secs. IIC and IID when performing surface and volume integrals in
$(r̂,ẑ)$ coordinates. Also note that this formulation assumes an infinitesimal current loop. We therefore neglect particle loss to the magnet surface, which is a fair approximation provided that
diffusion scales inversely with B (or B^2). This allows us to exploit the theoretical literature on plasma dipole equilibria for simplification of the pressure profile, the validity of which is
discussed in Sec. IIC.
B. Particle orbits and deflection
New ions with charge q[i] and mass m[i] are generated through charge exchange and electron impact ionization as the neutral flow encounters the dipole plasma. The net behavior of the plasma and
neutral flow depends on the trajectories of newly created ions in the vicinity of the magnetic dipole. Understanding the impact of non-uniform magnetic fields on the resulting trajectories will
inform not only momentum transfer to the magnet, but also the definition of a control volume for energy and mass accounting. Here, we examine ion trajectories as a function of the location at which
they are created to understand their behavior in the presence of the dipole magnetic field.
The equation of motion for an ion in a magnetic field without collisions can be expressed in the dimensionless form as
represents a characteristic ion Larmor radius normalized by the magnet radius. Here, we have introduced the dimensionless variables: velocity vector $û=u/u∞$, position vector $r̂=r/rc$, and time $t̂=
tu∞/rc$. From Eqs. (4) and (5), the ion trajectory as a function of time, $r̂(t̂)$, is uniquely determined by ρ[L] and the initial ion location and velocity, $r̂(0)$ and $û(0)$, respectively. Charge
exchange (CX) and electron-impact ionization are the two mechanisms by which streaming neutrals become ions. In both cases, the newly formed ion will initially maintain the neutral flow velocity in
the axial direction, $û(0)=êz$, where $êz$ is the unit vector in the axial direction. Note that while the ion is created by a collision, we consider its resulting trajectory neglecting any further
collisions. This is a fair approximation in the limit of an ion Hall parameter $Ωi>1$ applied only for the purpose of determining whether a given ion is trapped by the field.
We examine the influence of the non-uniform magnetic field by considering the trajectory of ions formed at various locations within the $r̂$–$ẑ$ plane. Equations (4) and (5) are propagated many times
longer than the transit time of an ion across the magnet ($t̂max=200$) using the LSODA differential equation solver.^27 A few sample trajectories are depicted in Fig. 2. We observe that ions
originating upstream of the magnet are either deflected or reflected by an amount that depends largely on their initial radial location, $r̂(0)$. Ions formed away from the magnet (i.e., $r̂(0)>1$)
deflect by an amount that increases as $r̂(0)$ decreases [Figs. 2(a)–2(c)]. Ions formed on a collision course with the magnet [i.e., $r̂(0)∼1$] tend to reflect because they experience stronger magnetic
fields throughout their trajectory [Fig. 2(d)]. The behavior of ions created near the axis of the magnet ($ẑ∼0$) is distinct from those created prior to reaching the magnet. Notably, a region of
space exists where ions enter closed (or trapped) trajectories about the magnet [Figs. 2(e) and 2(f)]. A second trapped-orbit region exists at slightly larger initial radii whereby the ion
trajectories exhibit orbits similar to the well-known banana orbit in tokamak plasmas^28 [Fig. 2(g)]. Finally, ions formed beyond this second trapped region exhibit deflected orbits [Fig. 2(h)].
The nature of the different ion orbits is best demonstrated using a heat map of the final axial velocity of each ion, $ûz,f$, as a function of the location at which the ion was formed. Here, we
consider ions formed on a uniform grid with spacing $Δr̂=Δẑ=0.1$ and calculate $ûz,f$ as the average of $ûz(t̂)$ for $t̂∈[100,200]$. This averaging reduces the noise associated with trapped orbits for
which $ûz$ is oscillatory with time. Note that $ûz,f=1$ for undeflected ions, $ûz,f=0$ for trapped ions, and $ûz,f=−1$ for fully reflected ions. The heat map shown in Fig. 2(i) clearly shows the
spatial distribution of reflected, deflected, and trapped ions. Notably, a region of pass-through orbits exists on-axis because ions formed here do not encounter significant radial magnetic fields.
We will ultimately use maps of $ûz,f$ to calculate the force of the reacting neutral flow on the magnet by taking the summation of the axial momentum change per ion multiplied by the rate at which
ions are formed.
Ions with trapped orbits are particularly important because they generally reside near the magnet for timescales long compared with the collisional timescale, allowing them to reach thermal
equilibrium with the plasma. As a consequence, trapped ions are a source of plasma mass and energy. Using distributions of $ûz,f$ for different ρ[L], we observe a characteristic boundary inside which
all ions are trapped. This boundary is defined by a specific magnetic flux surface,
The resulting magnetic flux contour, depicted in Fig. 2, marks the transition between trapped ions and reflected/deflected ions. In other words, any ion formed from the flow in a location at which
$ψ̂>ψ̂*$ will become trapped by the magnetic field. In 3D space, the $ψ̂*$-contour forms a toroidal control volume that we will use to examine mass and energy transfer between the neutral flow and
plasma. We conservatively neglect the trapped-ion volume formed by the banana orbit region [Fig. 2(g)] due to the complexity associated with modeling this irregular shape. This region may be
addressed by future higher-order modeling.
C. Mass and energy capture by the dipole plasma
The results of the single particle model provide us with a crucial link between the high-speed neutral gas flow and confined plasma. As neutral particles interact with plasma ions and electrons, new
ions can be formed through ionization or charge exchange collisions. Ions formed within the boundary defined by Eq. (7) provide a source of mass and energy to the plasma. Here, we use continuity to
derive an equation for the neutral particle density spatial distribution as a function of key dimensionless parameters. The neutral density distribution is then used along with our expression for the
trapped-particle boundary to determine the rate of mass and energy transfer from the neutral flow to the plasma.
We assume the ion and electron densities are equal everywhere (n[i]=n[e]) to preserve charge neutrality. In the presence of charge exchange reactions and electron impact ionization, the
steady-state continuity equation then takes the following form:
Here, $Rtot=Rion+Rcx$ is the sum of the ionization ($Rion$) and charge exchange ($Rcx$) reaction rates. Note that we have introduced the subscript sn to distinguish between incident neutral
particles (viz. stream neutrals) and neutral particles that are formed via charge exchange reactions (viz. secondary neutrals). Because the cross section for collisions between stream neutrals and
secondary neutrals is much smaller than ionization and charge exchange cross sections for the stream energies of interest (>1eV), the secondary neutrals do not play a significant role in mass and
energy transfer from the neutral flow to the plasma. However, as we will discover in Paper II,^25 secondary neutrals play a key role in the global balance of mass and energy within the plasma.
Two simplifying assumptions enable us to reduce Eq. (8) into a form amenable to analytical modeling. First, we assume that $usn=u∞êz$ everywhere, which is equivalent to saying that the stream neutral
flow is directed along $êz$ with a Mach number $Msn≫1$. Second, following Kesner et al.^29 we assume the dipole is driven to a stationary state and any fluctuations are damped out on timescales
faster than the characteristic evolution time of the system. This second assumption allows us to approximate the electron density as $ne/ne,r=(ψ̂/ψ̂r)α$. Here, $ne,r$ is the electron density along an
arbitrary reference surface of constant scalar flux $ψ̂r$. The parameter α is a constant that represents the steepness of the density profile. In our subsequent analysis, we take α=4 to be
consistent with theoretical predictions for plasma confined by a strong magnetic dipole.^29,30 Note that the density profile considered here neglects the effects of anisotropy and the interaction
between the plasma and magnet surfaces.^30,31 Diffusion across the $ψ̂*$ surface is expected to far exceed any possible cross field loss. Despite their limitations, these simplifications enable us to
examine the general scaling of mass and energy transfer over a wide parameter space. The inclusion of higher order effects is required for more accurate predictions within a narrower parameter space;
however, this is outside the scope of the present analysis and is left for future work.
With the above simplifications, the solution to Eq. (8) is
Here, I[n] is a function of $r̂$ and $ẑ$ that depends only on the specific magnetic field topology. According to Eq. (9), the stream neutral density distribution forms a wake whose size depends on the
dimensionless scalar quantity
We note that $ζtot$ scales with the ratio of the characteristic ion formation rate and the characteristic rate at which stream neutrals transit the magnet.
The stream neutral density distribution is shown for four different values of $ζtot$ in Fig. 3. It is clear that the wake formed by the interaction of the neutral flow with the dipole plasma
increases in size as $ζtot$ increases. This result is due to an increased probability that a stream ion undergoes an ionization or charge exchange reaction with the plasma prior to transiting past
the magnet. Also shown in Fig. 3 are magnetic flux contours that define the trapped-ion volume for three different values of ρ[L], as determined from Eq. (7). Notably, for each ρ[L], there exists a
value of $ζtot$ above which the trapped-ion volume is shadowed by the wake (i.e., $n̂sn≈0$ for $ψ>ψ*$). This shadowing effect will ultimately play an important role in the transfer of stream neutral
flow mass and energy to the plasma.
The total capture rate of stream neutral particles within the trapped-ion control volume ($Ṅcap$) can be determined by integrating the volumetric stream neutral reaction rate over the volume $ψ>ψ*$
. This can be written symbolically as
Recognizing that each stream neutral brings with it $msnu∞2/2$ worth of kinetic energy, the capture rate of stream neutral energy or captured power ($Pcap$), is given by
It is important to note that the mass captured by the plasma is divided among the ion and secondary neutral populations by an amount that depends on the ratio of $Rion/Rcx$. This division will be
examined in more depth in Sec. III.
The equations for particle and energy capture can be simplified considerably by introducing the following normalization: $Ṅ̂≡Ṅ/(n∞u∞rc2)$ and $P̂≡2P/(msnn∞u∞3rc2)$. Here, $Ṅ$ and P are normalized by
the flux of neutral particles and kinetic energy on an area equal to $rc2$, respectively. The dimensionless form of Eqs. (12) and (13) then becomes
For a given magnetic field topology, the integral I[sn] is a function of only two dimensionless parameters: the bounds of the integral are governed by ρ[L] via Eq. (7) and the spatial distribution of
$n̂sn$ is governed by $ζtot$ via Eq. (9). This result suggests that variations in the mass and energy capture rates with the properties of the neutral flow (e.g., density, particle mass, and velocity)
or magnetized plasma (e.g., magnet strength, size, plasma density, and temperature) are similar with respect to ρ[L] and $ζtot$.
The dimensionless particle (and power) capture rate is shown as a function of ρ[L] and $ζtot$ in Fig. 4(a). For fixed ρ[L], $Ṅ̂cap$ increases linearly with $ζtot$ at small values of $ζtot$. This
linear regime results from a weak $ζtot$-dependence within the integral I[sn], which occurs when $n̂sn≈1$ for $ψ̂>ψ̂*$. Physically speaking, the stream neutral wake does not shadow the ion-trapping
volume at a low $ζtot$ (un-shadowed $ψ*$). As $ζtot$ increases, a maximum capture rate eventually occurs. Just beyond this maximum value, further increases in $ζtot$ yield slight reductions in
$Ṅ̂cap$ (partially shadowed $ψ*$). Eventually, a regime is reached wherein $Ṅ̂cap$ decreases rapidly with increasing $ζtot$ (fully shadowed $ψ*$).
The solid lines in Fig. 4(a) represent a simple analytical approximation for $Ṅ̂cap$, described by Eqs. (A1)–(A6) in the Appendix. Equations (A1)–(A6) are important because they enable the mass and
energy capture rates to be quickly calculated without needing to solve the full integral in Eq. (15). The value of $ζtot$ that corresponds to the maximum capture rate can be estimated from this
approximate form of $Ṅ̂cap$, which to first order is $ζtot(m)≈0.3/ρL0.6$ for $ρL≪1$. We find that full shadowing of $ψ*$ occurs for $ζtot>10ζtot(m)$. Beyond this limit, the vast majority of neutral
flow particles ionize and deflect prior to reaching the ion-trapping volume. In Paper II,^25 we incorporate the analytical approximation for $Ṅ̂cap$ within a self-consistent global model to examine
the time-dependent neutral flow–plasma interaction.
D. Momentum transfer to the dipole magnet
A force is produced on the dipole magnet when ionized stream neutral particles either deflect or are captured by the magnetic field. In this case, momentum is transferred to the magnet via the
interaction between the magnet current and the current density that results from summing over individual ion orbits. Understanding the scaling of this force is of fundamental interest to space plasma
physics.^32–34 For plasma aerocapture concepts,^9,10 this force represents the drag generated when the magnetized plasma encounters a planetary atmosphere at high velocities—a quantity critical to
assessing the feasibility of this technology. Here, we use the particle trajectory maps (Fig. 2) and stream neutral density distribution (Fig. 3) to derive a scaling law for the drag force generated
by a high-speed neutral flow interacting with a dipole plasma.
We define the drag force, F[D], as the axial force imparted by the neutral flow to the magnet. The force density can be written as a product of the volumetric reaction rate of stream neutrals (
$Rtotnsnne$) and the change in momentum of the newly formed ion ($msnΔuz$), where $Δuz=uz,f−u∞$ is a function of r and z that represents the total change in axial velocity of an ion formed at
location (r, z). The volume integral of the force density over all space yields the drag force
We define the normalized drag force as
Here, $F̂D$ effectively represents the drag of the system relative to the drag on a solid disk of radius r[c], and can be thought of as a characteristic drag coefficient that includes the effects of
magnetic deflection. With this definition, Eq. (16) can be written as
We thus arrive at a second integral function that depends on only two dimensionless parameters for a given field topology: $n̂sn$ is governed by $ζtot$ via Eq. (9) and $Δûz$ depends on ρ[L] as shown
in Fig. 2.
The force density distribution is shown in Fig. 5 for a wide range of ρ[L] and $ζtot$. Here, the force density is defined as $f̂D=(2/π)ζtotn̂snΔûz(ψ̂/ψ̂r)α$. Values of $f̂D$ are calculated on an
adaptive grid that focuses gridpoints in regions of high-reactivity. This technique helps to increase both the resolution of the resulting spatial distribution and numerical precision of the
integrated drag force. The general features of the force density distribution can be traced back to the three variables upon which it relies: the dark region near and behind the magnet results from
wake formation in the function $n̂sn$; the streaks in front of the magnet, the dark band on the magnet axis, and the lobes on the magnet periphery are all features associated with the ion orbit
mapping (Fig. 2) contained in the function $Δûz$; and the region over which the previous two effects are important is localized near a particular flux surface via the plasma density distribution,
$n̂e∼ψ̂α$. We observe that the radial extent of $f̂D$ generally increases with decreasing ρ[L]. For a given ρ[L], the radial extent of $f̂D$ increases with $ζtot$ before eventually reaching a saturation
point at which it ceases to expand. Finally, we note that the value of $ζtot$ corresponding to the saturation point increases as ρ[L] decreases.
Numerical calculations for $F̂D$ are shown as a function of $ζtot$ and ρ[L] in Fig. 4(b). As the $f̂D$-distributions in Fig. 5 suggest, $F̂D$ generally increases as ρ[L] decreases. Furthermore, $F̂D$
increases with $ζtot$ for fixed ρ[L] up to a saturation point, with saturation occurring at higher values of $ζtot$ as ρ[L] decreases. Even with the saturation effect, drag forces up to three orders
of magnitude greater than the purely aerodynamic drag are observed. The solid lines in Fig. 4(b) represent a simple analytical approximation to $F̂D$, described by Eqs. (A7)–(A10) in the Appendix.
The scaling of the saturated drag value is revealed by taking the limit $ζtot→∞$ of the approximate form of $F̂D$ [Eq. (A7)], yielding $F̂D(s)≈1.2/ρL$ to first order. Interestingly, this is roughly 70%
of the drag that would be generated if the neutral flow were to be replaced by a fully ionized flow with equivalent velocity and density. We find that the reactivity at which saturation occurs is
roughly 1–5 times higher than $ζtot(m)$. Therefore, it is possible to approach drag saturation while maintaining significant mass and energy transfer to the plasma. This point has important
consequences for plasma aerocapture applications because it suggests the possibility of optimizing in situ flow utilization without sacrificing drag performance. Such applications are discussed in
depth in Paper II.^25
Our analysis thus far has taken the plasma density ($ne,r$) and temperature (through $Rion$) as independent variables. However, these parameters will inevitably evolve as mass and energy is
transferred from the neutral flow to the plasma. Before advancing toward greater complexity with a time-resolved model, we seek physical insight into the steady-state behavior of the system by
combining the scaling laws derived in Sec. II with a simple model for plasma mass and energy balance.
We consider the conservation of mass and energy inside a control volume defined by the ion-confining magnetic flux surface, $ψ*$ (Fig. 1). Upon reacting with the plasma, flowing (stream) neutral
particles provide a source of mass and energy to the control volume, as described by Eq. (14). This mass and energy is distributed among ions, electrons, and post charge-exchange (secondary) neutrals
prior to leaving the control volume. Mass conservation for these species can be written in the dimensionless form as
where source terms are on the left side of each equation and loss terms are on the right side. Similarly, energy conservation for ions, electrons, and secondary neutrals can be expressed as
Here, $N̂j$ and $T̂j$ are the total number of particles and temperature of species j, respectively; $τ̂j$ represents the timescale for particle diffusion from the control volume; $τ̂ie$ represents the
thermal equilibration time between ions and electrons; $Ṅ̂2n→i$ is the ionization rate of secondary neutrals averaged over the control volume; and $ε̂ion$ represents the effective ionization energy,
which also accounts for additional losses due to electron scattering and electron impact excitation. The fraction of stream neutrals captured via ionizing and charge exchange collisions is given by
$ξion=Rion/Rtot$ and $ξcx=Rcx/Rtot$, respectively. Finally, we consider the injection of external power into the plasma, where ξ[i] and ξ[e] represent the fraction of injected power ($P̂inj$)
deposited into the ions and electrons, respectively. We note that Eqs. (20)–(25) utilize the following normalizations: $Nĵ=Nj/(n∞rc3), T̂j=3Tj/(msnu∞2), τ̂j=τj/(rc/u∞)$, and $ε̂ion=2εion/(msnu∞2)$.
Equations (20)–(25) describe the volume-averaged, steady-state balance of mass and energy for a high-speed neutral flow interacting with a plasma. Before proceeding, we take a moment to briefly
summarize the physics associated with each term within these equations. In Eqs. (20) and (21), the first and second source terms account for ion/electron creation from electron impact ionization of
stream and secondary neutrals, respectively. The loss term in the ion and electron continuity equations represents diffusion from the control volume. Charge exchange collisions between stream
neutrals and ions provides the only source of secondary neutrals, as described in Eq. (22). The two secondary neutral loss terms describe losses due to diffusion and electron impact ionization. In
Eq. (23), the three ion energy source terms correspond to power injected into the ion population (e.g., via resonant wave heating), the kinetic power captured from the neutral flow as a result of
both ionizing and charge exchange collisions, and thermal power addition from ionized secondary neutrals. The three ion energy loss terms represent the rate of ion energy lost to secondary neutrals
during charge exchange collisions, ion thermal energy diffusion from the control volume, and the power transferred from the ions to electrons (e.g., via Coulomb collisions). The source terms in the
electron energy equation, Eq. (24), represent power injected into the electron population and power transferred from the ions. Electron energy losses include diffusion of thermal energy and the rate
of energy lost to inelastic collisions, including ionization of both secondary and stream neutrals. Finally, Eq. (25) shows that charge exchange collisions between the ions and stream neutrals
provide a source of power for the secondary neutral population. This power is balanced by losses due to diffusion and electron impact ionization.
Finally, we note that the steady-state mass and energy balance model described above makes two key assumptions: (1) the thermal energy is much greater than the kinetic energy for ions, electrons, and
secondary neutrals and (2) the temperature of each species is uniform. Two useful simplifications result from these assumptions. First, substitution of the plasma quasineutrality requirement ($N̂e=
N̂i$) into Eqs. (20) and (21) yields $τ̂i=τ̂e$. The diffusive loss rate of ions must therefore equal that of electrons to maintain zero net charge, which is equivalent to saying that charged particle
transport is governed by ambipolar diffusion. Second, Eqs. (22) and (25) combine to give $T̂2n=T̂i$. In other words, secondary neutrals remain in thermal equilibrium with the ions.
A. First mode transition
The overall power balance within the control volume can be obtained by summing over the power balance equations for each species, Eqs. (23)–(25). Noting that $ξion+ξcx=1$ and $ξe+ξi=1$, the total
steady-state power balance is described by
Here, the sources of power are the kinetic energy capture rate from the neutral flow and injected power. The first term on the right hand side represents the loss rate of electron thermal and frozen
flow energy associated with charged particle diffusion. The second loss term is equivalent to the sum of thermal energy lost via secondary neutral and ion diffusion. In this analysis, we will refer
to the left hand and right hand side of Eq. (26) as $P̂in$ and $P̂out$, respectively.
An important transition can occur if the maximum rate of energy capture is much greater than the injected power or $P̂cap(ρL,ζtot(m))≫P̂inj$. According to Fig. 4(a), captured power scales linearly
with plasma density, $P̂cap≈N̂i/τ̂cap$, for $ζtot≪ζtot(m)$. Here, the characteristic particle capture time
is a dimensionless quantity that compares the particle capture timescale to the neutral particle transit time. If the dominant contribution to $τ̂e$ is independent of density (e.g., Bohm diffusion),
the first power loss term in Eq. (26) also scales linearly with $N̂i$. Equation (26) therefore suggests that a critical value of $τ̂e$ exists, defined by
The importance of $τ̂e*$ to steady-state power balance is demonstrated in Fig. 6, which shows the curves of $P̂out$ and $P̂in$ as a function of $N̂i$ for different values of $τ̂e$. Here, steady-state
equilibrium occurs when $P̂out=P̂in$, satisfying Eq. (26). For $τ̂e<τ̂e*$, the power lost exceeds the power captured irrespective of $N̂i$. As a consequence, $P̂inj$ dominates the equilibrium power
balance. For $τ̂e≈τ̂e*$, the linear portion of the $P̂out$-curve approximately equals that of the $P̂in$-curve, indicating that small changes in $τ̂e$ yield large changes in the equilibrium state.
Finally, for $τ̂e>τ̂e*$, the equilibrium density increases to the point where the non-linear portion of $P̂cap$ balances $P̂out$. In other words, increases in the charged particle diffusion time
require large corresponding increases in plasma density to maintain power balance via the wake shadowing effect described in Sec. IIC. The cumulation of these effects is a mode transition from a
regime where power balance is dominated by injected power to a regime where it is dominated by power captured from the neutral flow.
Similar to the overall power balance in Eq. (26), a requirement on $τ̂e*$ also exists to satisfy the overall mass balance. The addition of Eqs. (20) and (22) yields
This equation is similar in form to Eq. (26) with the exception that there is an additional loss term on the right hand side due to secondary neutral diffusion. Conservation of mass therefore
This equation states that, for $N̂2n/τ̂2n>0$, the timescale for charged particle diffusion must exceed the characteristic particle capture timescale if the plasma is to be sustained solely by mass and
energy from the neutral flow.
The mode transition described above has significant consequences on the fundamental interaction between a high-speed neutral flow and dipole-confined plasma. Taking the case shown in Fig. 6 as an
example, a ninefold increase in $τ̂e$ yields a nearly four order of magnitude increase in both $N̂i$ and $P̂cap$. We take a moment to examine the physical meaning of $τ̂e*$ to determine how the mode
transition depends on energy transfer processes between the flow and plasma. Rearranging Eq. (28) yields $(1−T̂i)/τ̂cap=(T̂e+ε̂ion)/τ̂e*$. The left hand side of this equation represents the net power
deposited into the ions assuming neutral capture occurs only through charge exchange collisions ($ξcx=1$). This quantity can be thought of as the minimum amount of power the ions can absorb from
flow interaction. The right hand side of the equation represents the net power lost from the electron population due to diffusion and inelastic collisions. If $τ̂e<τ̂e*$, the electron energy loss per
captured particle outweighs the ion energy gain, thus the plasma cannot be sustained without an additional power source. $τ̂e*$ therefore represents the minimum diffusion time required to balance
power captured into the ion population with power lost from the electron population. This highlights an important consideration for the occurrence of critical ionization. Because flow energy deposits
directly into the ion population, an energy channel must exist between the ions and electrons (e.g., Coulomb collisions) to sustain the electron temperature and balance frozen-flow losses. Thus,
$τ̂e*$ also places bounds on the efficiency of this energy channel, as ion–electron energy transfer must be generally faster than the critical diffusion time.
The solution to Eq. (26) can be used to examine how the total input power and drag force on the dipole scale with ρ[L] and $τ̂e/τ̂e*$ (Fig. 7). As anticipated, we observe the plasma to transition from
a self-sustained (via $P̂inj$) to a neutral-flow-sustained regime at $τ̂e/τ̂e*=1$. Following the mode transition, the input power into the dipole plasma reaches a maximum that increases with ρ[L]
before eventually decreasing with $τ̂e/τ̂e*$ due to dipole shadowing. A similar trend exists for the drag force with the exception that $F̂D$ increases monotonically with $τ̂e/τ̂e*$ beyond the mode
transition. This difference is due to the fact that the integral for $P̂cap$ is over $ψ>ψ*$, while the integral for $F̂D$ is over all of space. Notably for $ρL=10−3$, the plasma is observed to
capture nearly four orders of magnitude more power than the injected power and upwards of three orders of magnitude more drag compared to the aerodynamic drag on a solid object with size equivalent
to the magnet.
The preceding analysis considered $τ̂e$ as an independent variable to simplify the steady-state analysis and obtain intuitive physical understanding of the observed mode transition. In a real system,
$τ̂e$ will depend on the properties of both the plasma and magnet. For example, increasing the magnetic field strength for a given flow will increase $τ̂e$ and decrease ρ[L]. A critical magnetic field
strength will therefore exist that corresponds to $τ̂e/τ̂e*=1$, after which the captured power and drag will continue to increase as the systems transitions between curves of constant ρ[L].
B. Second mode transition
A second mode transition occurs when $Ṅ̂2n→i≫N̂2n/τ̂2n$. In this limit, nearly all of the secondary neutral particles are ionized prior to diffusing from the control volume, resulting in
$ξcxṄ̂cap≈Ṅ̂2n→i$. Insertion of this equation into the ion mass balance [Eq. (20)] yields
Simply stating, every neutral flow particle captured by the plasma ultimately diffuses from the control volume as an ion. We note that Eq. (31) is similar in nature to Eq. (26). Following the same
logic used for the derivation of Eq. (28), a critical value of $τ̂e$ exists for this mode transition defined by
Physically, for $τ̂e<τ̂e**$, the loss rate of electrons and ions via diffusion is larger than the capture rate from the neutral flow. When this occurs, the plasma is unsustainable without an
additional source of charged particles. The requirement that $τ̂e>τ̂e**$ for plasma sustainment is equivalent to the Townsend criterion.^34
Elimination of secondary neutral diffusion has a significant impact on energy balance within the control volume. Substitution of Eq. (31) into Eq. (26) yields the following total energy balance
Here, the sum of captured kinetic power and injected power balances the diffusion of thermal and frozen-flow power from the control volume. Considering again the limit $Ṅ̂cap(ρL,ζtot(m))≫P̂inj$,
energy balance can be written as
which states that the kinetic energy of each captured particle must equal the sum of the electron and ion thermal energies plus the effective ionization energy. To exist in this regime, the neutral
flow must possess enough kinetic power to ionize and heat itself to the plasma temperature. The relationship between the second mode transition and Alfvén's critical ionization velocity phenomenon
will be discussed in Sec. IIIE.
C. Electron heating requirement
The fact that kinetic power from the flow deposits directly into the ion population places an additional requirement on $τ̂e$ with respect to the timescale for ion–electron energy transfer, $τ̂ie$.
The electron energy equation [Eq. (24)] can be rewritten as
where the simplification on the right hand side results from inserting the ion mass conservation equation with $N̂e=N̂i$ and $τ̂e=τ̂i$. This equation states that the power lost from the electron
population must be balanced by a combination of injected power and power transferred from the ions. However, the mode transitions are characterized by a transformation of the plasma into a state in
which $P̂inj≪N̂i(T̂e+ε̂ion)/τ̂e$. The electron energy equation therefore yields a critical value of $τ̂ie$,
For $τ̂ie>τ̂ie*$, the plasma cannot be sustained solely on power deposited by the neutral flow. As $τ̂ie$ decreases eventually $τ̂ie=τ̂ie*$, at which point power transfer from the ions exactly balances
power lost to diffusion and inelastic collisions. Once this mode transition occurs, Eq. (36) suggests that the temperature of the plasma species will adjust to maintain $τ̂ie=τ̂ie*$.
D. Flow interaction regimes
The previous analysis suggests that the interaction between a high speed neutral flow and magnetized plasma exhibits three distinct physical regimes. In the first regime, the plasma is unsustainable
without a source of mass and energy that is independent of the neutral flow. Due to its dependence on injected mass/power, we refer to this as the injection (I) regime. The first mode transition
(Sec. IIIA) describes the transition of the plasma to a state where it is sustained entirely by the neutral flow. In this regime, secondary neutrals resulting from charge exchange are the dominant
source for ionization. For this reason, we refer to it as the charge exchange (CX) regime. The CX regime is possible because diffusion of charge exchange neutrals enables stream neutral particle
kinetic energy to be absorbed by the plasma without incurring the additional energy cost of ionization. As the stream neutral kinetic energy increases, eventually a point is reached where the kinetic
energy per captured particle is large enough to overcome the ionization and thermal energy loss per diffused particle. The second mode transition (Sec. IIIB) occurs at this point, which is marked by
an abrupt increase in stream neutral ionization. We refer to this regime as the critical ionization velocity (CIV) regime due to the well-known, eponymous velocity associated with the kinetic energy
requirement of the mode transition.
The particular regime in which a flow/plasma system will exist depends on the timescales for electron confinement, stream neutral particle capture, and ion–electron energy transfer. Equations (28),
(30), (32), (34), and (36) provide equations that describe the timescale requirements of a particular regime as a function of species temperatures and the ratio of the effective ionization energy to
the kinetic energy of flowing neutrals. Determination of the steady-state species temperatures for a given system requires the full solution to Eqs. (20)–(25) with appropriate physical models for the
various diffusion, ion–electron energy transfer, and collisional reaction processes. However, it is possible to generate an approximate mapping of the different regimes by recognizing that the
ionization energy represents a minimum energy requirement for plasma sustainment.
First, we consider the limit where ion–electron energy transfer occurs much faster than particle capture, or $τ̂ie≪τ̂cap$. The minimum requirement on $τ̂e$ for the CX regime can be found by letting
both $T̂i→0$ and $Tê→0$ in Eqs. (28) and (30), which gives
$τ̂eτ̂cap>{ε̂ionfor ε̂ion≥11for ε̂ion<1.$
Taking the same limit for Eqs. (32) and (34) gives the following requirements on $ε̂ion$ and $τ̂e$ for the CIV regime:
Equations (37)–(39) are plotted in Fig. 8(a), which provides a map of the different regime requirements as a function of the quantities $τ̂e/τ̂cap$ and $ε̂ion$. For $τ̂e/τ̂cap>1$, Fig. 8(a) indicates
that decreasing $ε̂ion$ (e.g., by increasing $u∞$) for a system in the I regime can induce the I–CX mode transition. Further decreases in $ε̂ion$ eventually lead to the CX–CIV mode transition. For $τ̂e
/τ̂cap<1$, the system exists in the I regime irrespective of $ε̂ion$. At a fixed value of $ε̂ion$, increasing $τ̂e/τ̂cap$ (e.g., by increasing B[0]) leads to either an I–CIV mode transition or an I–CX
mode transition. Notably, increasing $τ̂e/τ̂cap$ cannot induce a CX–CIV mode transition unless it is accompanied by a corresponding decrease in $ε̂ion$.
The regime boundaries are modified when the ion–electron energy transfer timescale becomes on the order of or greater than the neutral particle capture timescale. From Eq. (26), it is obvious that
$T̂i<1$ for the CX regime. Equation (36) therefore provides the following requirement for the CX regime:
Similarly, power balance for the CIV regime [Eq. (34)] requires $T̂i−Tê=1−2T̂e−ε̂ion<1−ε̂ion$. According to Eq. (36), the CIV regime therefore requires
Figure 8(b) illustrates how the electron energy requirements influence the regime boundaries for a fixed ratio of $τ̂ie/τ̂cap$ greater than unity. The main consequence is a shift in both the CX and CIV
regime boundaries toward smaller $ε̂ion$. This shift occurs because a smaller percentage of the ion energy is transferred to the electron population. As a consequence, more ion energy is required per
captured neutral particle in order to sustain the plasma. At a large $τ̂e/τ̂cap$, the CIV boundary approaches that of Fig. 8(a) or $ε̂ion=1$. In other words, the CIV boundary does not depend on the
ion–electron energy transfer timescale so long as the energy is transferred before it escapes the volume via diffusion.
The boundaries presented in Fig. 8 represent the minimum requirements on $τ̂e$ as a function of $τ̂cap, τ̂ie$, and $ε̂ion$. Using appropriate physical models for the electron diffusion coefficient,
charge exchange cross section, ionization reaction rates, and effective ionization energy, Eqs. (37)–(41) provide an easy method to determine if a system can physically exist in a given regime.
Identification of the precise mode transition boundary requires knowledge of $T̂i$ and $Tê$, necessitating a full solution to the mass and energy balance equations, as presented in Paper II.^25
E. Relationship to Alfvén's critical ionization velocity
We end our analysis with a brief discussion of the previous results in the context of Alfvén's theory for the critical ionization velocity. CIV theory predicts that rapid ionization of the neutral
gas will occur when the flow kinetic energy exceeds the ionization energy of the neutral gas. The velocity corresponding to this threshold or critical ionization velocity, is traditionally defined as
where U[iz] is the ionization energy of a neutral particle with mass m[sn].
The actual energy cost of creating an ion is greater than the ionization energy due to the cumulative effects of electron scattering and electron impact excitation of neutral particles. Including
these effects into an effective ionization energy, $εion$, the following parameter can be defined:
Here, C[i] is a function of T[e] due to the dependence of electron impact ionization and excitation cross sections on electron energy. In general, $Ci>1$ and decreases with increasing T[e]. The
dimensionless effective ionization energy used to determine the flow regime boundaries in Sec. IIID can therefore be written as
For a given neutral flow species, it is possible to decrease $ε̂ion$ by increasing either T[e] or $u∞$.
Referencing Fig. 8(a), for $τ̂e≥τ̂cap$ and $τ̂ie≪τ̂cap$, an I–CX mode transition will occur when $u∞$ exceeds a threshold velocity. Inserting Eq. (44) into Eq. (37), we find that the threshold velocity
for the CX regime is bounded by
Similarly, using Eq. (38) we find that the threshold velocity for the CX–CIV mode transition must satisfy
From Eq. (46), it is apparent that the CIV mode transition occurs when $u∞>ucr$. The fact that excitation losses increase the threshold velocity for critical ionization beyond u[cr] was examined
theoretically by Möbius et al.^12 and is supported by numerous CIV laboratory experiments.^14 Equation (45) predicts the existence of another threshold velocity that is less than the CIV threshold
velocity and can also be less than u[cr] if $τ̂e/τ̂cap>Ci$. We will refer to this second threshold velocity as the critical charge exchange velocity (CXV) due to its forcing of the I–CX mode
transition. The appearance of a second threshold velocity resulting from charge-exchange collisions is distinct from the theoretical results of Möbius et al.^12 and McNeil et al.,^24 who conclude
that CX collisions tend to decrease the CIV threshold velocity. This distinction is due to the fact that their analysis neglects: (1) the loss of ion energy during a CX collision [first term on the
RHS of Eq. (23)] and (2) the recycling of charge-exchange neutrals into plasma ions (i.e., terms related to $Ṅ̂2n→i$). Notably, we have found no experimental evidence of a distinct CXV threshold in
the open literature. One possible reason for this is that traditional CIV experiments are performed using open magnetic field configurations in which $τ̂e≈τ̂cap$. According to Eqs. (45) and (46), the
two threshold velocities become indistinguishable in this limit.
For $τ̂ie≥τ̂cap$, the shift of the mode transition boundary toward lower $ε̂ion$ brings with it an increase in the CIV and CXV threshold velocities. Again restricting our analysis to $τ̂e≥τ̂cap$, Eqs.
(40) and (41) indicate that the CXV and CIV threshold velocities are bounded by
respectively. In both cases, the threshold velocity increases with the ratio $τ̂ie/τ̂e$ because of a corresponding decrease in the fraction of ion energy transferred to the electron population. This
result agrees with Formisano et al.,^3 who found that ion–electron energy transfer inefficiencies increased the CIV threshold velocity in cometary plasmas.
In this paper, we theoretically investigated the interaction between a high-speed neutral gas flow and dipole magnetized plasma. This problem represents a departure from classical CIV theories, which
generally consider an isotropic plasma of infinite extent flowing against neutral gas within a uniform, open magnetic field configuration. Single particle trajectories were examined as a function of
the location at which neutral particles undergo ionizing reactions, either via charge exchange with plasma ions or electron impact ionization. This trajectory analysis provided a method to
incorporate non-uniform magnetic field effects within the fluid theory typically used to analyze CIV. Specifically, by identifying trapped and deflected ion orbits and integrating spatially, we were
able to derive equations for the transfer of mass, momentum, and energy from the neutral gas flow to the dipole plasma. The resulting equations were incorporated into a simple model for the
conservation of mass and energy in steady-state, from which we identified mode transitions indicative of CIV-like effects.
The main contributions of our analysis can be summarized as follows:
1. There exists a critical magnetic flux surface, $ψ*$, within which newly formed ions enter into closed trajectories around the magnetic dipole. In general, ions formed outside of $ψ*$ either
bypass or deflect from the dipole magnetic field. We found that $ψ*∼ρL$, where ρ[L] is the characteristic Larmor radius of newly formed ions normalized by the magnet radius. Furthermore, the
volume bounded by a particular flux surface increases as ψ decreases. Therefore, the scaling of $ψ*$ with ρ[L] describes how the spatial domain of trapped-ion orbits changes with the magnet
radius, magnetic field strength, and neutral flow mass and velocity.
2. Equations for the transfer of mass, momentum, and energy from the neutral flow to the dipole plasma were derived as a function of two dimensionless quantities: ρ[L] and $ζtot$, as defined in
Eqs. (6) and (11), respectively. Here, ρ[L] dictates the spatial distribution of captured and deflected ion orbits (described above), while $ζtot$ determines the probability that a streaming
neutral particle will undergo an ionizing reaction at a particular location. It was found that mass and energy transfer increase monotonically as ρ[L] decreases. A critical value of $ζtot$ was
discovered to maximize mass and energy transfer to the plasma for a given ρ[L]. The appearance of a critical value is explained by the fact that in a highly reactive plasma (large $ζtot$) a
majority of neutral flow particles are ionized and deflected prior to reaching the ion-trapping volume (i.e., wake shadowing). We determined that momentum transfer increases monotonically with
$ζtot$ and approaches an asymptotic value (i.e., drag saturation), which depends only on ρ[L] and is roughly equal to 70% of the drag that would be produced if the neutral flow was instead fully
3. The dependence of the particle capture rate on plasma density drives a mode transition from a regime where injected mass and power are required to sustain the plasma (I regime) to a regime where
the plasma can be sustained entirely by the neutral flow. The flow-sustained regime can be further divided into two separate modes. In the CX regime, diffusion of charge-exchange neutrals enables
flow energy to sustain the plasma without overwhelming the electron population with ionization energy losses. In the CIV regime, flow energy is sufficiently large to ionize and heat the entire
population of captured neutral particles. Transitions between these regimes were derived as a function of the electron confinement timescale, ion–electron energy transfer timescale, neutral flow
particle capture timescale, and the ratio of the effective ionization energy to the kinetic energy of neutral flow particles. Threshold velocities were derived for each mode transition that were
found to be similar to Alfvén's critical ionization velocity, but with additional dependencies on the timescales listed above.
The results presented here are general in the sense that we do not specify physical models for diffusion, charge exchange and ionization reactions, ionization energy losses, and ion–electron energy
transfer. One limitation that results from this generality is that the time-evolution of the density and temperature of the different species cannot be determined self-consistently. This limitation
is addressed in Paper II,^25 where we use the results of this current paper to derive a global model for the plasma/flow interaction that includes physical models for the various diffusion, reaction,
and energy transfer processes for each species. This self-consistent model will ultimately provide a framework for us to examine the characteristics of, and transitions between, the different flow
interaction regimes identified in this work.
We thank Dr. Robert Moses for the insightful conversations related to this research and its applications to plasma aerocapture. C. L. Kelly's effort was supported by the NASA Space Technology
Research Fellowship No. 80NSSC18K1191.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Simple analytical approximations for the normalized particle capture rate [Eq. (14)] and normalized drag [Eq. (18)] are included here as a function of the dimensionless quantities $ζtot$ and ρ[L].
The functions were obtained for α=4 and $ψ̂r=1$.
For $ρL≲1$ and $ζtot>0.1$, the particle capture integral is well-approximated by the function
$s0(ρL)=0.66 exp(−3.27ρL0.73),$
A comparison between the exact numerical solution for $Ṅ̂cap$ and above approximation is shown in Fig. 4(a).
For $ρL≲1$ and $ζtot>0.1$, the normalized drag can be approximated as
A comparison between the exact numerical solution for $F̂D$ and above approximation is shown in Fig. 4(b).
Astrophys. J.
P. T.
Rev. Geophys.
, (
, and
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J. M.
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© 2020 Author(s). | {"url":"https://pubs.aip.org/aip/pop/article/27/11/113512/107901/Neutral-flow-interaction-with-a-magnetic-dipole","timestamp":"2024-11-09T23:07:40Z","content_type":"text/html","content_length":"476559","record_id":"<urn:uuid:8f291aa6-fcf0-445c-b716-97731293e707>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00242.warc.gz"} |
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Difference Between IRR & XIRR in Commercial Real Estate | FNRP
When evaluating the performance of a commercial real estate investment, there are a number of metrics that are traditionally used to measure returns. One of the most common is called the internal
rate of return.
By definition, the Internal Rate of Return (IRR) is the discount rate that sets the Net Present Value of a series of cash flows equal to zero. But, just because the NPV is zero does not mean it is a
bad investment, it just means that an investor will earn the IRR as their rate of return.
In order to calculate the IRR, it is first necessary to estimate a series of cash flows that an investor expects to earn. This is done by creating a pro forma projection of a property’s income,
expenses, and debt service. If the property is profitable, there is money left after the loan payments have been made, which is available to be distributed to investors. These are the cash flows
that are used as inputs in the IRR calculation.
In this article we’ll discuss how to calculate IRR and the differences between XIRR vs. IRR.
How to Calculate The Internal Rate of Return (IRR)
The formula used to calculate IRR is complex. Fortunately, the calculation is made easier by using a function in Microsoft Excel or similar spreadsheet program. The required inputs for the IRR
function are simply the estimated pro forma cash flows, including a “year 0” cash flow that represents the initial investment in the property (a negative value). For example, assume that the
following series of cash flows are projected for a real estate investment:
Using these cash flows, the IRR function inputs would be “=IRR(-$100,000, $20,000, $25,000, $30,000, $35,000, $40,000). The resulting IRR is calculated as 13.45%, which is a solid annual return.
However, the IRR function can be misleading when calculating commercial real estate investment returns. This function is designed to assume that the time period between all cash flows is equal.
This is unrealistic because it is unlikely that the initial outflow for a property will be on the first day of the year and that each successive cash flow will be at the exact same interval in the
In reality, it is much more likely that a property will be purchased on any given day in the year and that the cash flows will occur at irregular intervals. For these reasons, the “XIRR” function is
the better measure of return for irregular cash flows.
Calculating Internal Rate of Return using the XIRR Function
The major difference between the IRR formula and the XIRR formula is that XIRR uses the dates of the future cash flows and incorporates them into the return value by calculating the exact number of
days between each period. A few days here and there may seem insignificant, but the difference can be significant.
To illustrate this point about XIRR, consider the same series of periodic cash flows. Now, assume that the initial outlay for the purchase (year 0) occurs on June 30. The first cash inflow occurs
on 12/31 of the same year and the successive annual cash flows also occur on 12/31 of the following years. They are displayed in the following table:
Again, the XIRR function incorporates both the dates and the amount of the cash flows. In Excel, the input would be like this: =XIRR(Dates, Cash Flows). The result of this calculation is 16.25%,
which is a significant difference from the result using the IRR function.
XIRR vs. IRR – Why The Difference Matters
Many commercial real estate investments offer a “waterfall” return distribution, meaning the cash flow split between the transaction sponsor and the investors is based on a series of return
“hurdles.” In many cases, the hurdle rate is calculated using the Internal Rate of Return. For example, a typical 3-tier waterfall could look something like this:
• Tier 1: If the property earns an IRR of 0% – 8%, the investors get 90% of the cash flow and the sponsor gets 10%.
• Tier 2: If the property earns an IRR of 8% – 14%, the investors get 80% of the cash flow and the sponsor gets 20%.
• Tier 3: If the property earns an IRR of 14% or above, the investors get 70% of the cash flow and the sponsor gets 30%.
In such a structure, the investors get 80% of the cash flow when the Internal Rate of Return is calculated using the IRR function, which assumes regular intervals between cash flow iterations.
However, the result of the XIRR calculation bumps the cash flow split to the next tier where the investors get 70% of the cash flow.
The point is, the difference between the results of the XIRR vs. IRR calculations can impact the cash flow split in a waterfall return structure. As such, it is critically important that investors
read all of the offering materials to determine how IRR is calculated and which function is used.
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pulsed module¶
Pulse-based experiment design and measurement.
Detailed information is provided in the documentation for each class.
Pulsed Use the hardware in pulsed mode.
FlatPulse Container for a (long) flat pulse.
Event Container for hardware events.
Template An output pulse event.
Match A template-matching event.
MeasurementHandle Handle to a running measurement.
MeasurementStatus Enumeration for current status of the measurement
Module constants:
Pulsed class¶
class presto.pulsed.Pulsed(ext_ref_clk=False, force_reload=False, dry_run=False, address=None, port=None, adc_mode=AdcMode.Direct, adc_fsample=None, dac_mode=DacMode.Direct, dac_fsample=None,
force_config=False, downsampling=1)¶
Use the hardware in pulsed mode.
Create only one instance at a time. Use each instance for only one measurement. This class is designed to be instantiated with Python’s with statement, see Examples section.
ValueError – if adc_mode or dac_mode are not valid Converter modes; if adc_fsample or dac_fsample are not valid Converter rates.
In all the Examples, it is assumed that the imports import numpy as np and from presto import pulsed have been performed, and that this class has been instantiated in the form pls = pulsed.Pulsed
(), or, much much better, using the with pulsed.Pulsed() as pls construct as below.
For an introduction to adc_mode and dac_mode, see Direct and Mixed mode. For valid values of adc_mode and dac_mode, see Converter modes. For valid values of adc_fsample and dac_fsample, see
Converter rates. For advanced configuration, e.g. different converter rates on different ports, see Advanced tile configuration.
Output a 2 ns pulse on output port 1 every 100 μs, and sample 1 μs of data from input port 1. Repeat 10 times, average 10k times.
>>> from presto import pulsed
>>> with pulsed.Pulsed() as pls:
>>> pls.setup_store(1, 1e-6)
>>> pulse = pls.setup_template(
>>> output_port=1,
>>> group=0,
>>> template=np.ones(4),
>>> )
>>> pls.output_pulse(0.0, pulse)
>>> pls.store(0.0)
>>> pls.run(
>>> period=100e-6,
>>> repeat_count=10,
>>> num_averages=10_000,
>>> print_time=True,
>>> )
>>> t_arr, data = pls.get_store_data()
connecting to 192.168.42.50 port 7878
Expected runtime: 10.0s
Total time: 10.2s
Transfering data: 3.0ms
Gracely disconnect from the hardware.
Call this method if the class was instantiated without a with statement. This method is called automatically when exiting a with block and before the object is destructed.
Period of the programmable logic clock in seconds.
This is also the granularity of the time grid for defining events in the experiment sequence.
Return type:
Frequency of the programmable logic clock in Hz.
Return type:
Get sampling frequency for which converter in Hz.
which (str) – "adc" for input and "dac" for output.
ValueError – if which is unknown.
Return type:
Obtain from hardware the result of the sparse real-time histogram.
Only valid after the measurement is performed.
Return type:
a dictionary with the following keys defined.
For a 1D histogram:
"bins" (ndarray, dtype=np.float64): left-hand edges of the
histogram bins
"counts" (ndarray, dtype=np.int64): the values (counts) of the
For a 2D histogram:
"bins_x" (ndarray, dtype=np.float64): left-hand edges of the
histogram bins along the x axis
"bins_y" (ndarray, dtype=np.float64): left-hand edges of the
histogram bins along the y axis
"counts" (ndarray, dtype=np.int64): the values (counts) of the
histogram bins are not sorted and sparse, i.e. bins with no counts are not returned
○ RuntimeError – If no template-matching data is available.
○ TypeError – If input_pairs contains unrecognized objects.
Get and plot histogram data:
>>> hist = pls.get_histogram_data()
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> ax.plot(hist["bins"], hist["counts"], '.')
>>> fig.show()
Sort histogram data:
>>> hist = pls.get_histogram_data()
>>> idx = np.argsort(hist["bins"])
>>> bins_sorted = hist["bins"][idx]
>>> counts_sorted = hist["counts"][idx]
Maximum number of data points in a single template slot.
Equal to MAX_TEMPLATE_LEN when in direct mode (dac_mode is DacMode.Direct), and half of it when using digital upconversion (dac_mode is one of DacMode.Mixedxx).
Return type:
The number of available input/output ports in the hardware.
Return type:
Obtain from hardware the result of the (averaged) stores.
Only valid after the measurement is performed.
use_compression (bool) – if True, use lossless compression when downloading measurement data. See Notes section below.
Return type:
Tuple[ndarray, ndarray]
A tuple of (t_arr, data) where
■ t_arr (ndarray): The time axis in seconds during one acquisition. dtype is float.
■ data (ndarray): The acquired data with get_fs("adc") sample rate and scaled to +/-1.0 being full-scale input. The shape of the array is (num_stores * repeat_count, num_ports,
smpls_per_store), where num_stores is number of store events programmed in a sequence with the store() method, and num_ports and smpls_per_store are the number of inputs ports and
the number of samples in one acquisition set up with setup_store().
When using digital downconversion (adc_mode=AdcMode.Mixed), the returned data is complex (dtype=np.complex128); in direct mode (adc_mode=AdcMode.Direct) data is real (dtype=np.float64).
Using lossless compression when downloading measurement data (compression=True) requires the third-party Python package lz4 which can be installed from PyPI with pip install lz4 or from
Anaconda with conda install lz4.
○ RuntimeError – If no store data is available.
○ ImportError – if the Python package lz4 is not installed when using compression=True, see Notes section above.
get_template_matching_data(input_pairs, use_compression=False)¶
Obtain from hardware the result of the template matching.
Only valid after the measurement is performed.
Return type:
hardware: Hardware¶
interface to hardware features common to all modes of operation
match(at_time, match_info)¶
Program hardware to perform template matching at given time.
○ at_time (float) – At what time in seconds the template(s) should be matched. Must be a multiple of get_clk_T()
○ template_info – The information about one or more template-matching pairs as obtained from setup_template_matching_pair().
next_frequency(at_time, output_ports, group=None)¶
Program the hardware to move to the next frequency/phase at given time.
○ at_time (float) – At what time in seconds the change should happen. Must be a multiple of get_clk_T()
○ output_ports (int or array_like) – What output ports should be affected by the change. Valid ports are in the interval [1, get_nr_ports()].
○ group (int or array_like) – Valid groups are in the interval [0, 1]. If None, move the carrier for both groups to the next entry.
ValueError – if at_time is not a multiple of get_clk_T(); if any of output_ports is out of [1, get_nr_ports()]; if group is out of [0, 1].
next_scale(at_time, output_ports, group=None)¶
Program the hardware to move to the next output scale at given time.
○ at_time (float) – At what time in seconds the change should happen. Must be a multiple of get_clk_T()
○ output_ports (int or array_like) – What output ports should be affected by the change. Valid ports are in the interval [1, get_nr_ports()].
○ group (int or array_like) – Valid groups are in the interval [0, 1]. If None, move both groups to the next entry.
ValueError – if at_time is not a multiple of get_clk_T(); if any of output_ports is out of [1, get_nr_ports()]; if group is out of [0, 1].
output_dc_bias(at_time, bias, port)¶
Program hardware to output a DC bias at given time.
○ at_time (float) – at what time in seconds the bias should be output. Must be a multiple of get_clk_T()
○ bias (float) – DC bias value in volts.
○ port (int or array_like) – output port(s) the DC bias should be output from. Valid values are in [1, 16]. See Notes for behavior when more than one port is addressed.
It is not possible to change the DC range during an experimental sequence. Only the bias value can be changed. It is required to use Hardware.set_dc_bias() to configure the range and the
initial DC-bias value for port before programming the experimental sequence. It is also recommended to use Hardware.set_dc_bias() to reset the value to zero after the experiment terminated,
i.e. after a call to run().
Updating the DC bias on one port takes 1 μs, during which no other DC bias event should be scheduled. Updating the bias on n ports takes n μs, and all ports will switch to the new value
○ ValueError – if port is out of range; if at_time is not a multiple of get_clk_T(); if bias is larger than the current setting for DC-bias range on port.
○ RuntimeError – if the DC range for port was not set.
output_digital_marker(at_time, duration, ports)¶
Program hardware to output a digital marker at given time.
○ at_time (float) – at what time in seconds the marker should be output. Must be a multiple of get_clk_T()
○ duration (float) – for how long in seconds the marker should be output. Must be a multiple of get_clk_T()
○ port (int or array_like) – digital output port(s) the marker should be output from. Valid values are in [1, 4]
ValueError – if ports is out of range; if at_time is not a multiple of get_clk_T()
output_pulse(at_time, pulse_info)¶
Program hardware to output pulse(s) at given time.
perform_measurement(*args, **kwargs)¶
Deprecated since version 2.0.0: Use run() instead.
reset_phase(at_time, output_ports, group=None)¶
Program the hardware to reset the phase of group at given time.
The phase will be reset to the current entry of the frequency/phase look-up table.
○ at_time (float) – at what time in seconds the pulse(s) should be output. Must be a multiple of get_clk_T()
○ output_ports (int or array_like) – what output ports should be affected by the reset. Valid ports are in the interval [1, get_nr_ports()].
○ group (int or array_like) – valid groups are in the interval [0, 1]. If None, reset the phase of both groups.
ValueError – if at_time is not a multiple of get_clk_T(); if any of output_ports is out of [1, get_nr_ports()]; if index is out of [0, MAX_LUT_ENTRIES - 1]; if group is out of [0, 1].
Round t to a multiple of the clock period.
t (float) – time in seconds
Return type:
the multiple of get_clk_T() that is closest to t
run(period, repeat_count, num_averages, print_time=True, verbose=False, trigger=TriggerSource.Internal, prefetch_matches=False)¶
Execute the experiment planned so far. Can be time consuming!
This method will block until the measurement and the data transfer are completed. See run_async() for a non-blocking alternative.
○ period (float) – Measurement time in seconds for one repetition. Should be long enough to include the last event programmed. If longer, there will be some waiting time between
repetitions. Must be a multiple of get_clk_T()
○ repeat_count (int or tuple of ints) – Number of times to repeat the experiment and stack the acquired data (no averaging). For multi-dimensional parameter sweeps, repeat_count is a
tuple where each element specifies the number of repetitions along that axis. Multi-dimensional parameter sweeps are experimental and the API might be fine tuned in a future release.
○ num_averages (int) – Number of times to repeat the whole sequence (including the repetitions due to repeat_count) and average the acquired data.
○ print_time (bool) – If True, print to standard output the expected runtime before the experiment, the total runtime after the experiment, and print during the experiment the estimated
time left at regular intervals.
○ verbose (bool) – If True, print debugging information.
○ trigger (TriggerSource) – Select trigger source to start measurement, default internal trigger (start immediately).
○ prefetch_matches (bool) – if True, download template-matching results from the hardware as they become available; if False (default), download results only at the end of the
○ ValueError – if period is negative, too small to fit the last event, or not a multiple of get_clk_T(); if repeat_count is not positive; if num_averages is not positive.
○ RuntimeError – if the sequence requires more triggers than MAX_COMMANDS; if more than 8 templates/envelopes are used for the same group on the same port.
run_async(period, repeat_count, num_averages, print_time=False, verbose=False, trigger=TriggerSource.Internal, prefetch_matches=False)¶
Same as run(), but non blocking. See run for full documentation.
Return type:
select_frequency(at_time, index, output_ports, group=None)¶
Program the hardware to select a frequency/phase at given time.
ValueError – if at_time is not a multiple of get_clk_T(); f any of output_ports is out of [1, get_nr_ports()]; if index is out of [0, MAX_LUT_ENTRIES - 1]; if group is out of [0, 1].
select_scale(at_time, index, output_ports, group=None)¶
Program the hardware to select an output scale at given time.
ValueError – if at_time is not a multiple of get_clk_T(); if any of output_ports is out of [1, get_nr_ports()]; if index is out of [0, MAX_LUT_ENTRIES - 1]; if group is out of [0, 1].
setup_condition(input_pairs, output_templates_true, output_templates_false=None)¶
Setup a template-matching condition for one or more output pulses.
The pulses in output_templates_true (output_templates_false) will be marked as conditional, and will be output if and only if all the template-matching conditions defined in input_pairs are
(not) satisfied.
○ input_pairs (Match or array_like) – One or more template-matching pairs as defined by setup_template_matching_pair().
○ output_templates_true (Template, FlatPulse or array_like) – Output in case of a successful match comparison. One or more pulses as defined by setup_template() and/or setup_flat_pulse
(). Set to None or to an empty list [] if no pulse is desired.
○ output_templates_false (Template, FlatPulse or array_like) – Output in case of a unsuccessful match comparison. One or more pulses as defined by setup_template() and/or
setup_flat_pulse(). Set to None or to an empty list [] if no pulse is desired (default).
setup_flat_matching_pair(input_port, duration, weight1, weight2=None, *, threshold=0.0, compare_next_port=False)¶
Create a pair of constant input templates for template matching with given duration.
The reference templates created with this method are flat, i.e. have a constant weight, and use a single memory slot each regardless of duration. See setup_template_matching_pair() for
reference templates with arbitrary shape.
The result of template matching can be obtained after the measurement with get_template_matching_data(). The matching also defines a condition that can be used to mask one or more output
templates during the measurement with setup_condition(). The template-matching condition is True if the match with weight1 plus the match with weight2 is greater or equal than threshold,
False otherwise.
Return type:
To evaluate a match, the input data acquired from port input_port is first multiplied with each template element-wise, and then summed. If the match using weight1 plus the match using weight2
is greater than or equal to threshold, the match is considered successful.
setup_flat_pulse(output_port, group, duration, amplitude=None, *, rise_time=0.0, fall_time=0.0, envelope=False, output_marker=None)¶
Create an output pulse with constant amplitude and optional rise/fall segments.
Useful for using fewer templates with very long pulses, and for dynamically changing the duration of the pulse.
Return type:
Regardless of duration, only one template is consumed for a flat pulse.
The duration of the pulse can be changed dynamically by using the methods set_total_duration() and set_flat_duration() on the returned FlatPulse object.
The amplitude of the pulse is affected by the group scaler configured with setup_scale_lut(). The parameter amplitude is applied “in series”.
When rise_time and/or fall_time are nonzero, one additional template is used for each rise and fall segments. The rise and fall times are subtracted from the flat time, so that the total
duration of the pulse including transients is duration. The rise and fall shapes are \(\sin(\frac{\pi}{2}x)^2\) and \(\cos(\frac{\pi}{2}x)^2\), respectively, with \(x \in [0, 1)\). The
segments rise_time and fall_time are limited to 1022 ns (one template slot each).
setup_freq_lut(output_ports, group, frequencies, phases, phases_q=None, axis=-1)¶
Setup look-up table for frequency generator.
ValueError – If any of output_ports is out of [1, get_nr_ports()]; if any of frequencies is out of [0.0, get_fs("dac")); if group is out of [0, 1]; if phases doesn’t have the same shape
as frequencies; if the LUTs are longer than MAX_LUT_ENTRIES; if phases_q is specified when not using the digital upconversion mixer (dac_mode=DacMode.Direct); if phases_q is not specified
when using the digital upconversion mixer (dac_mode=DacMode.Mixedxx).
Set up the hardware to calculate a sparse histogram in real time.
Results from different template-matching objects are binned separately.
match (Match) – One template-matching object as obtained from setup_template_matching_pair().
setup_histogram_2d(match_x, match_y)¶
Set up the hardware to calculate a sparse 2D histogram in real time.
Results from different template-matching objects are binned separately.
○ match_x (Match) – A template-matching object for binning in the x coordinate, as obtained from setup_template_matching_pair().
○ match_x – A template-matching object for binning in the y coordinate
RuntimeErrir – if another histogram was already set up.
setup_long_drive(output_port, group, duration, *, amplitude=1.0, amplitude_q=None, rise_time=0.0, fall_time=0.0, envelope=True, output_marker=None)¶
Create a long output pulse with constant amplitude, useful for using fewer templates.
Deprecated since version 2.14.0: New code should use setup_flat_pulse() instead. When migrating to the new method, pay attention to the change in defaults for parameters amplitude and
☆ envelope default value changes from True to False;
☆ amplitude default value changes from 1+0j to 1+1j when using digital upconversion.
Return type:
Regardless of duration, only one template is consumed for a flat long drive.
The amplitude of the pulse is affected by the group scaler configured with setup_scale_lut(). The parameter amplitude is applied “in series”.
When rise_time and/or fall_time are nonzero, one additional template is used for each rise and fall segments. The rise and fall times are subtracted from the flat time, so that the total
duration of the pulse including transients is duration. The rise and fall shapes are \(\sin(\frac{\pi}{2}x)^2\) and \(\cos(\frac{\pi}{2}x)^2\), respectively, with \(x \in [0, 1)\). The
segments rise_time and fall_time are limited to 1022 ns (one template slot each).
setup_scale_lut(output_ports, group, scales, axis=-1)¶
Setup look-up table for global output scale.
ValueError – If any of output_ports is out of [1, get_nr_ports()]; if any of scales is out of [-1.0, +1.0]; if the LUT is longer than MAX_LUT_ENTRIES; if group is out of [0, 1].
setup_store(input_ports, duration)¶
Set input port(s) and duration for all store events (data acquisition).
○ input_ports (int or list of int) – Valid ports are in the interval [1, get_nr_ports()].
○ duration (float) – Duration in seconds. Must be a multiple of get_clk_T()
ValueError – if any of input_ports is out of [1, get_nr_ports()]; if duration is not a multiple of get_clk_T(); if a store duration on all input_ports results in an acquisition window
with more than MAX_STORE_LEN samples.
setup_template(output_port, group, template, template_q=None, *, envelope=False, output_marker=None)¶
Create an output template or envelope.
Return type:
If the length of the pulse is longer than a single template slot (get_max_template_len() samples), the pulse is automatically split into multiple segments of appropriate length. Therefore,
the pulse might use more than one template slot.
setup_template_matching_pair(input_port, template1, template2=None, threshold=0.0, compare_next_port=False)¶
Create a pair of input templates for template matching.
The result of template matching can be obtained after the measurement with get_template_matching_data(). The matching also defines a condition that can be used to mask one or more output
templates during the measurement with setup_condition(). The template-matching condition is True if the match with template1 plus the match with template2 is greater or equal than threshold,
False otherwise.
Return type:
To evaluate a match, the input data acquired from port input_port is first multiplied with each template element-wise, and then summed. If the match using template1 plus the match using
template2 is greater than or equal to threshold, the match is considered successful.
Each memory slot for reference templates can store up to get_max_template_len() data points. If template1 and template2 are longer than that, they will be split into multiple sub-templates
spanning multiple memory slots.
Program hardware to acquire data from the input ports at given time.
at_time (float) – at what time in seconds the acquisition should start. Must be a multiple of get_clk_T()
The input ports and the duration of the acquisition are set up separately, see section See also below.
Convert t to an integer number of clock cycles, raising an Exception if rounding is needed.
t (float) – time in seconds
Return type:
number of clock cycles
ValueError – if t is not an integer multiple of get_clk_T(); if t is negative.
FlatPulse class¶
class presto.pulsed.FlatPulse(rise, flat, fall, dig_out, pls)¶
Container for a (long) flat pulse.
The user doesn’t need to instantiate this class in any common situation.
Return type of setup_flat_pulse(); input type of output_pulse().
LongDrive class¶
class presto.pulsed.LongDrive(rise, flat, fall, dig_out, pls)¶
Deprecated since version 2.14.0: See FlatPulse instead.
Other Event classes¶
class presto.pulsed.Event¶
Container for hardware events.
The user doesn’t need to instantiate this class or its derivatives in any common situation.
This is the base class of the return types of many methods of Pulsed.
Return a shallow copy of this object.
Return type:
class presto.pulsed.Template(channels, group, envelope, events, pls)¶
Bases: Event
An output pulse event.
The user doesn’t need to instantiate this class in any common situation.
Return type of setup_template; input type of output_pulse().
Get the duration of the template in seconds.
Return type:
class presto.pulsed.Match(events, threshold, cross_group, pls)¶
Bases: Event
A template-matching event.
The user doesn’t need to instantiate this class in any common situation.
Return type of setup_template_matching_pair(); input type of match(), get_template_matching_data() and setup_condition().
Get the duration of the reference template in seconds.
Return type:
Async types¶
class presto.pulsed.MeasurementHandle(pls)¶
Handle to a running measurement.
Return type of run_async(), do not instantiate this class directly.
class presto.pulsed.MeasurementStatus(value, names=<not given>, *values, module=None, qualname=None, type=None, start=1, boundary=None)¶
Enumeration for current status of the measurement | {"url":"https://www.intermod.pro/manuals/presto/source/pulsed.html","timestamp":"2024-11-12T23:23:49Z","content_type":"text/html","content_length":"279168","record_id":"<urn:uuid:3ebce93e-fdfa-457a-9860-d1c1999da458>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00781.warc.gz"} |
Understanding Mathematical Functions: How To Find Zeros Of A Function
Mathematical functions are essential in expressing relationships between variables in the field of mathematics. They provide a means of understanding and analyzing various phenomena in the real
world. One crucial aspect of understanding functions is being able to find their zeros. Zeros of a function are the values of the independent variable that make the function equal to zero. This
process is vital in addressing problems in areas such as engineering, physics, economics, and more.
Key Takeaways
• Mathematical functions are crucial in expressing relationships between variables in various fields.
• Finding zeros of a function is essential for addressing problems in engineering, physics, economics, and more.
• Zeros of a function are the values of the independent variable that make the function equal to zero.
• Methods for finding zeros include graphical, algebraic, and numerical methods.
• Understanding the behavior of different types of functions and utilizing technology can aid in finding zeros of a function.
Understanding Mathematical Functions
Mathematical functions play a crucial role in the field of mathematics and are used to represent relationships between different quantities. They are essential in various fields, including physics,
engineering, economics, and many others. In this blog post, we will explore the concept of mathematical functions and how to find the zeros of a function.
A. Explanation of mathematical functions
A mathematical function is a relation between a set of inputs and a set of possible outputs, where each input is related to exactly one output. In other words, it assigns each input value exactly one
output value. A function can be represented using a formula, a table of values, or a graph. For example, the function f(x) = 2x + 3 represents a linear function, where x is the input and 2x + 3 is
the output.
B. Types of mathematical functions (linear, quadratic, cubic, etc.)
There are various types of mathematical functions, each with its own unique characteristics. Some common types of functions include:
• Linear functions: These functions have a constant rate of change and can be represented by a straight line on a graph. They are in the form of f(x) = mx + b, where m is the slope and b is the
• Quadratic functions: These functions have a squared term and can be represented by a parabola on a graph. They are in the form of f(x) = ax^2 + bx + c, where a, b, and c are constants.
• Cubic functions: These functions have a cubed term and can be represented by a curve on a graph. They are in the form of f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.
• Exponential functions: These functions have a constant base raised to the power of x and can be represented by a curve on a graph. They are in the form of f(x) = a^x, where a is the base.
C. Graphical representation of functions
Graphs are a visual way to represent functions and illustrate their behavior. By plotting the input and output values on a graph, we can gain insights into the characteristics of a function, such as
its shape, intercepts, and zeros. The x-intercepts of a function, also known as its zeros, are the points where the function crosses the x-axis.
Key Takeaways
• Mathematical functions relate inputs to outputs.
• Functions can be linear, quadratic, cubic, exponential, and more.
• Graphs visually represent the behavior of functions.
Understanding Mathematical Functions: How to find zeros of a function
In mathematics, understanding the concept of zeros of a function is crucial for solving various problems and applications. In this chapter, we will explore the definition of zeros of a function and
discuss the importance of finding zeros in mathematics and real-world applications.
A. Definition of zeros of a function
The zeros of a function, also known as roots or x-intercepts, are the values of x for which the function equals zero. In other words, a zero of a function f(x) is a value of x where f(x) = 0.
Mathematically, it can be represented as f(c) = 0, where c is the zero of the function.
B. Importance of finding zeros in mathematics and real world applications
Finding the zeros of a function is essential in mathematics and various real-world applications for several reasons:
• Understanding the behavior of a function: Zeros of a function help in understanding the behavior of the function as they represent the points where the function intersects the x-axis. This
information is crucial for graphing the function and analyzing its properties.
• Solving equations: Zeros of a function provide the solutions to equations of the form f(x) = 0. Finding these zeros is essential for solving equations in algebra and calculus.
• Optimization problems: In optimization problems, finding the zeros of a function helps in identifying the critical points that can maximize or minimize the function, which is valuable in fields
such as economics, engineering, and physics.
• Real-world applications: Zeros of a function have numerous real-world applications, such as in finance for calculating break-even points, in physics for determining the equilibrium positions, and
in engineering for analyzing systems and structures.
Methods for Finding Zeros of a Function
When it comes to understanding mathematical functions, one of the important aspects is being able to find the zeros of a function. Zeros, also known as roots or x-intercepts, are the points at which
the function crosses the x-axis. There are various methods to find the zeros of a function, and here we will explore some of the most commonly used ones.
A. Graphical method
• Plotting the function: One of the simplest ways to find the zeros of a function is by plotting the graph of the function and identifying the points where it intersects the x-axis.
B. Algebraic methods
• Factoring: For polynomial functions, factoring is a useful method to find the zeros. By factoring the function, you can identify the values of x that make the function equal to zero.
• Completing the square: This method is particularly useful for quadratic functions. By completing the square, you can rewrite the function in a form that makes it easy to identify the zeros.
• Quadratic formula: For quadratic functions that cannot be easily factored, the quadratic formula provides a straightforward way to find the zeros.
C. Numerical methods
• Newton-Raphson method: This iterative method uses the derivative of the function to approximate the zeros. It can be particularly useful for functions where other methods are not applicable.
• Bisection method: This method works by repeatedly dividing the interval in which the zero is known to lie in half, and then selecting the subinterval in which the zero must lie for further
By being familiar with these methods for finding zeros of a function, you can tackle a wide range of functions and solve for their zeros effectively.
Practical Examples of Finding Zeros of a Function
Understanding how to find the zeros of a function is a fundamental concept in mathematics. In this chapter, we will explore practical examples of finding zeros of a function through various methods.
A. Solving quadratic equations to find zeros
• Using the quadratic formula:
The quadratic formula is a useful tool for solving quadratic equations of the form ax^2 + bx + c = 0. By plugging in the values of a, b, and c, we can find the zeros of the function using this
• Factoring quadratic equations:
Factoring is another method to find the zeros of a quadratic function. By factoring the quadratic equation into two binomial factors, we can easily identify the values of x that make the function
equal to zero.
B. Using graphical methods to find zeros
• Graphing the function:
Plotting the function on a graph allows us to visualize the points where the function crosses the x-axis, indicating the zeros. By locating the x-intercepts or roots of the function, we can
determine the zeros.
• Interpolating from the graph:
By using the graph of the function, we can estimate the zeros by interpolating the x-values where the function equals zero based on the plotted points.
C. Applying numerical methods to find zeros in complex functions
• Newton's method:
This numerical method involves iteratively improving upon an initial guess to find the zeros of a function. By applying the formula x_(n+1) = x_n - f(x_n) / f'(x_n), we can approximate the zeros
of the function.
• Bisection method:
By using the bisection method, we can narrow down the interval in which the zero of a function lies. This method involves repeatedly halving the interval and selecting the subinterval where the
sign of the function changes.
Tips for Finding Zeros of a Function
When it comes to understanding mathematical functions, finding the zeros of a function is a crucial concept. Here are some tips to help you effectively find the zeros of a function.
A. Understanding the behavior of different types of functions
1. Familiarize yourself with different types of functions
• Polynomial functions
• Rational functions
• Exponential functions
• Trigonometric functions
• Logarithmic functions
2. Identify the characteristics of each type of function
• Determine the degree of the polynomial function
• Understand the domain and range of rational functions
• Recognize the growth or decay of exponential functions
• Consider the periodic nature of trigonometric functions
• Understand the behavior of logarithmic functions
B. Utilizing technology and calculators for complex functions
1. Use graphing calculators to visualize the function
Graphing calculators can help you understand the behavior of a function and locate its zeros by plotting the function graph.
2. Utilize computer software for complex functions
For functions that are complex or involve large datasets, consider using computer software such as MATLAB or Wolfram Alpha to solve for zeros.
C. Checking solutions for accuracy
1. Verify solutions using algebraic methods
After finding potential zeros for a function, use algebraic methods such as factoring or the quadratic formula to verify the accuracy of the solutions.
2. Use numerical methods to confirm the zeros
If the function is difficult to factor or solve algebraically, consider using numerical methods such as the bisection method or Newton's method to confirm the zeros.
By understanding the behavior of different types of functions, utilizing technology and calculators for complex functions, and checking solutions for accuracy, you can effectively find the zeros of a
Understanding mathematical functions and how to find zeros of a function is crucial in various fields, including engineering, physics, economics, and more. Finding zeros helps us determine critical
points, solve equations, and understand the behavior of a function. It is essential for problem-solving and decision-making. I encourage you to further explore mathematical functions and their zeros
to deepen your understanding of this fundamental concept in mathematics.
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Видеотека: N. A. Safonkin, Semifinite harmonic functions on the Gnedin--Kingman graph
Аннотация: In 1978 J.F.C. Kingman described random exchangeable partitions of the set of natural numbers. Kingman's result can be reformulated in terms of harmonic functions on some branching graph,
which is called the Kingman graph. Vertices of this graph correspond to Young diagrams and edges correspond to the Pieri rule for the monomial basis in the algebra of symmetric functions.
In 1997 A. Gnedin discovered an analog of the Kingman's result for linearly ordered partitions. Gnedin's theorem can be reformulated in terms of harmonic functions on some branching graph too. We
will call this graph the Gnedin-Kingman graph. Its vertices correspond to compositions (ordered partitions) and edges correspond to Pieri rule for the monomial basis in the algebra of quasisymmetric
The talk is devoted to indecomposable seminfinite harmonic functions on the Gnedin-Kingman graph. The semifinitness property means that the value of a function on some vertices equals $+\infty$. We
will also discuss multiplicativity of indecomposable semifinite harmonic functions on the Gnedin-Kingman graph and how they link to semifinite harmonic functions on the Kingman graph. The latter were
classified by A. Vershik and S. Kerov in 1980's.
Язык доклада: английский | {"url":"https://m.mathnet.ru/php/presentation.phtml?option_lang=rus&presentid=27467","timestamp":"2024-11-05T03:56:47Z","content_type":"text/html","content_length":"8798","record_id":"<urn:uuid:473b9d23-0f46-4f66-9f54-67f970e07891>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00239.warc.gz"} |
Advent of Code 2021 in Kotlin - Day 3
Day 3’s problem has us work out what the most common bit is per column concatenating them to find the gamma value. The epsilon value is the inverse.
Given the following input,
we’d have a gamma of 10110 and epsilon of 01001. The result is the product of the gamma and epsilon.
buildString provides a string builder as an expression, letting us yield characters and strings. I then use it again to flip the bits to get the epsilon value. To parse the strings as base 2 numbers
we just need to specify the radix.
fun powerConsumption(data: List<String>): Int {
val rowLength = data.first().length
val gamma = buildString(rowLength) {
for (c in 0 until rowLength) {
val setBits = data.count { n -> n[c] == '1' }
append (if (setBits >= data.size / 2) '1' else '0')
val epsilon = buildString(rowLength) {
gamma.forEach { append (if (it == '1') '0' else '1') }
return gamma.toInt(2) * epsilon.toInt(2)
A criticism of this approach (and certainly not the only one) is that if epsilon ever needs to be something than gamma with flipped bits, this whole thing needs rewriting. We could use a string
builder for each, like this:
fun powerConsumption(data: List<String>): Int {
val rowLength = data.first().length
val gamma = StringBuilder()
val epsilon = StringBuilder()
for (c in 0 until rowLength) {
val setBits = data.count { n -> n[c] == '1' }
if (setBits >= data.size / 2) {
} else {
return gamma.toString().toInt(2) * epsilon.toString().toInt(2)
I wouldn’t say we’ve improved on anything here. It may be better to write a function to calculate based on either most or least common bit and call it twice. Time complexity stays the same, although
there’d be a tiny runtime hit.
enum class BitCommonality { Most, Least }
fun intFromCommonBit(data: List<String>, commonality: BitCommonality): Int {
val rowLength = data.first().length
return buildString(rowLength) {
for (c in 0 until rowLength) {
val setBits = data.count { n -> n[c] == '1' }
append (when (Pair(commonality, setBits >= data.size/2)) {
Pair(BitCommonality.Most, true) -> '1'
Pair(BitCommonality.Most, false) -> '0'
Pair(BitCommonality.Least, true) -> '0'
Pair(BitCommonality.Least, false) -> '1'
else -> throw Exception("I don't understand Kotlin match exhaustion")
val gamma = intFromCommonBit(data, BitCommonality.Most)
val epsilon = intFromCommonBit(data, BitCommonality.Least)
Interestingly, I was required to add an else branch to the when. I thought type inference would take care of that because there’s a clear finite list of possible values, but using Pair probably mucks
that up. I’ll need to fiddle around with this some more later to see if there’s a better way to represent it.
On to part 2, where the problem changes a little bit. Instead, when we finding the most, or least, common bit in each column, we discard all rows that don’t have that bit. So if we’re looking for the
most common bit in the first column of the example data above, we find that 1 is the most common and we discard all rows that have a 0 in that column. Conversely, if we’re looking for the least
common bit, we would choose 0 and discard all rows with a 1 in that column. We then move on to the next column and do the same thing, but with a reduced amount of rows. If there are equal numbers of
each bit, choose 1 for most common and 0 for least common.
In the problem, we find the “oxygen generator rating” by looking for the most common bit, and find the “CO2 scrubber rating” by looking for the least common bit. The answer, which gives us the “life
support rating”, is the product of these two.
enum class BitCommonality { Most, Least }
fun getRating(data: List<String>, commonality: BitCommonality): Int {
tailrec fun loop(pos: Int, remaining: List<String>): String {
if (remaining.size == 1) return remaining.first()
if (pos == remaining.first().length) throw Exception("not found")
val setBits = remaining.count { it[pos] == '1' }
val setIsMostCommonBit = setBits >= remaining.size - setBits
val next = remaining.filter { it[pos] ==
when (commonality) {
BitCommonality.Most -> if (setIsMostCommonBit) '1' else '0'
BitCommonality.Least -> if (setIsMostCommonBit) '0' else '1'
return loop (pos + 1, next)
return loop(0, data).toInt(2)
val o2 = getRating(data, BitCommonality.Most)
val co2 = getRating(data, BitCommonality.Least)
val lifeSupport = o2 * co2
A recursive approach made sense here, and I tried something different with matching commonality and the most common bit - nested condition statements instead of a single when with a tuple. I like
this less as the nested if needs to be replicated for each condition and could be the source of bugs, but the other way doesn’t work the way I had hoped. I’ll reach out to some Kotlin aware friends
for help and update it with their advice.
Thanks for reading! Please feel free to send me an email to talk more about this (or anything else for that matter). | {"url":"https://www.troykershaw.com/posts/advent-of-code-2021-kotlin-day-3/","timestamp":"2024-11-09T15:48:36Z","content_type":"text/html","content_length":"29621","record_id":"<urn:uuid:aff16ec7-95c7-49cf-9451-fbdbb2e40330>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00648.warc.gz"} |
What is the numerical range of a char?... | Questions and Answer on competitive exams like GATE exam, entrance tests, fun games, and much more
To answer this question, we need to understand the numerical range of a char data type in programming.
In most programming languages, including Java and C++, a char data type represents a single character and is stored as a 16-bit value.
The correct answer is D) 0 to 65535. This is because a char data type can store values between 0 and 65535, inclusive. The range starts from 0 because it represents the Unicode value for the
character '0', and it goes up to 65535 because it represents the maximum Unicode value that can be stored in a char.
Option A) -128 to 127 is incorrect because this range corresponds to the numerical range of a byte data type, which is 8 bits in size.
Option B) -(2^15) to (2^15) - 1 is incorrect because this range corresponds to the numerical range of a short data type, which is 16 bits in size.
Option C) 0 to 32767 is incorrect because this range corresponds to the numerical range of an int data type, which is typically 32 bits in size.
Therefore, the correct answer is D) 0 to 65535. | {"url":"https://www.aliensbrain.com/question/16/what-is-the-numerical-range-of-a-char","timestamp":"2024-11-05T02:31:33Z","content_type":"text/html","content_length":"14237","record_id":"<urn:uuid:5c5710f3-68a5-40ed-a26b-4f426b49e2f4>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00431.warc.gz"} |
The Stacks project
Remark 13.36.7. Let $\mathcal{D}$ be a triangulated category. Let $E$ be an object of $\mathcal{D}$. Let $T$ be a property of objects of $\mathcal{D}$. Suppose that
1. if $K_ i \in \mathcal{D}$, $i = 1, \ldots , r$ with $T(K_ i)$ for $i = 1, \ldots , r$, then $T(\bigoplus K_ i)$,
2. if $K \to L \to M \to K[1]$ is a distinguished triangle and $T$ holds for two, then $T$ holds for the third object,
3. if $T(K \oplus L)$ then $T(K)$ and $T(L)$, and
4. $T(E[n])$ holds for all $n$.
Then $T$ holds for all objects of $\langle E \rangle $.
Comments (2)
Comment #8551 by Long Liu on
In Remark 0ATH (1), 'D(A)' should be 'D'.
Comment #9135 by Stacks project on
Thanks and fixed here.
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Who Wants to Be a Mathematician and the 2010 Arnold Ross Lecture
Thomas Hales Gives the 2010 Arnold Ross Lecture, Followed by Who Wants to Be a Mathematician
The Arnold Ross Lecture and Who Wants to Be a Mathematician traveled to the Carnegie Science Center in Pittsburgh, Pennsylvania on October 14 for an informative and exciting day of mathematics. A
summary of the lecture and contest, including videos of the contest, is below.
"They had a great time playing and the rest of my class did playing along as well. We also enjoyed the lecture very much. The students I brought are all extremely interested in mathematics. They're
all seniors from my Linear Algebra class, a very talented bunch who all passed BC Calculus last year, so they really enjoyed the day. I hope that you return to Pittsburgh soon!"
"Thank you for the excellent event. My students and I thoroughly enjoyed the lecture and the competition."
"Thank you for the opportunity for our student to participate in the “Who Wants to Be a Mathematician” game today. She and the rest of our students had a great time. We hope that you will be coming
to our area again in the future. The day was a great experience for our students."
The 2010 Arnold Ross Lecture: Can Computers Do Math?, Thomas C. Hales, Andrew Mellon Professor of Mathematics, University of Pittsburgh
Hales talked about packing problems, giving their history and why they are important in modern mathematics and its applications.
His first example was packing tetrahedra, that is, putting them together to fill space. Aristotle thought that all of space could be filled by arranging tetrahedra so that five of the solids were
arranged around every edge of each tetrahedron, but it turns out that that configuration leaves a small gap: The tetrahedra don't quite fit together, or fill space, as imagined.
Recently, researchers have used computers to find a packing of tetrahedra that would use up the greatest percentage of space. One of those researchers is Elizabeth Chen, who purchased several
packages of Dungeons and Dragons tetrahedral dice to help find the best packing. She found a packing that occupies more than 85% of space, but at this point it is not known if that is the best
possible arrangement. Chen handed out packets of the dice at her dissertation defense. Her team of researchers includes a chemical engineer who has used the mathematical results to build molecular
clusters for a rudimentary cloaking device. (Image from "Dense crystalline dimer packings of regular tetrahedra," Chen, Engel, Glotzer)
Hales then moved to packing spheres. He showed what looked like two different arrangements of spheres--one tetrahedral and the other pyramidal (pictured, below left)--which turned out to be the same
face-centered cubic arrangement (Hales is holding them together, below right), cut at different angles. This packing is familiar to many as it is often used in grocery stores to pack oranges, for
In the 1500s Sir Walter Raleigh wanted to know how many cannonballs were in the stacks in his ships. Hales showed the audience how Raleigh's advisor, Thomas Harriot, used what is known as Pascal's
Triangle, to solve the problem. Although determining the most efficient method for packing the spheres is a 400-year old problem, it is used today in error-correcting codes, which are important in
digitized media such as music.
Whether this face-centered cubic packing is the most efficient became known as the Kepler Conjecture and was undecided until Hales submitted a proof of this conjecture, which 12 referees examined
over an 8-year period. At the end of this time they said that they were 99% sure that the proof was correct but could not be 100% sure, so he began trying to teach computers to do mathematical
proofs. Hales showed the audience the journal issue that was his proof, which, although long, wasn't as long as others.
Hales concluded with another example that united a fairly old conjecture and modern mathematics. The question was how to divide space into cells of equal volume that have the least area of surface
between them. Lord Kelvin thought that the cells were truncated octahedra. This was believed to be correct until 1994 when two physicists Phelan and Weaire discovered a counterexample (pictured below
left). Their configuration served as the basis for the design of The Cube, the aquatics center at the 2008 Summer Olympics. Since then, many configurations better than Kelvin's (e.g., below right)
have been found.
The Phelan-Weaire formula "P42"
Who Wants to Be a Mathematician
At left, rooting sections for Mike Panza and Michael Matty. Above: Chrissy's rooting section
After the lecture and some nice refreshments, eight Pittsburgh-area high school students played Who Wants to Be a Mathematician. The big winners were Michael Matty of Pine-Richland High School and
Chrissy Martin of Steel Valley High School, who won US$1000 and $500, respectively, from the AMS, and each won a TI-Nspire graphing calculator from Texas Instruments.
Pictured, left to right: Kyle Berkow, Hampton High School; Mike Panza, Deer Lakes High School; Thomas Hales; Connor Brem, Mt. Lebanon High School; April Peng, Moon Area High School; Chrissy Martin,
Steel Valley High School; Michael Matty, Pine-Richland High School; Maria Guadagnino, Mt. Lebanon High School; and Daniel Salmon, Shaler Area High School
Connor Brem led most of the way in game one, but Chrissy won the game by answering the last question correctly. (Front: Connor and Chrissy, Back: Mike and April)
In game two, Michael Matty also led most of the way. Kyle Berkow pulled ahead after question six, with two questions to go. Going into the last question, the four contestants were within 800 points
of one another, but Michael answer the last question correctly to win. (Front: Kyle and Maria, Back: Michael and Daniel)
Then Chrissy and Michael, having each won $500 and a TI-Nspire, went head-to-head on the Square-Off Question for the right to be in the Bonus Round and perhaps earn $2000. The two contestants both
missed on their first chance at the Square-Off Question, but Michael answered the question correctly on his second chance, which won him another $500.
Unfortunately, Michael did not get the Bonus Question, which concerned three mutually tangent circles inscribe in a larger circle, but he and Chrissy still got paid the big bucks.
Here are all the prizes and money won that day.
• TI-Nspire graphing calculator from Texas Instruments and $1000 from the AMS: Michael Matty
• TI-Nspire graphing calculator from Texas Instruments and $500 from the AMS: Chrissy Martin
• Maple 14 from Maplesoft: Connor Brem and Daniel Salmon
• Calculus by Anton, Bivens and Davis from John Wiley and Sons: Kyle Berkow and Mike Panza
• What's Happening in the Mathematical Sciences from the AMS: April Peng and Maria Guadagnino
The AMS thanks sponsors Texas Instruments, Maplesoft, and John Wiley and Sons for their continued generous support of Who Wants to Be a Mathematician. Thanks also to the Carnegie Science Center for
hosting the event and to the Pittsburgh area teachers and students who attended.
Steelers fans "don't need no stinkin' arrows."
Photographs by Who Wants to Be a Mathematician judge and co-creator Bill Butterworth (DePaul University Department of Mathematical Sciences), Robin Aguiar (Meetings and Professional Services), and by
Who Wants to Be a Mathematician host and AMS Public Awareness Officer Mike Breen.
Find out more about the Arnold Ross Lecture series and Who Wants to Be a Mathematician. | {"url":"http://www.ams.org/publicoutreach/arl2010","timestamp":"2024-11-10T08:52:14Z","content_type":"text/html","content_length":"57502","record_id":"<urn:uuid:f54b6085-92b2-4956-a53c-447768697c0c>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00853.warc.gz"} |
dynamic programming subproblems
In the Dynamic Programming, 1. Write down the recurrence that relates subproblems 3. Following are the two main properties of a problem that suggests that the given problem can be solved using
Dynamic programming. We also Dynamic Programming 3 Steps for Solving DP Problems 1. Dynamic Programming is a mathematical optimization approach typically used to improvise recursive algorithms.
Applicable when the subproblems are not independent (subproblems share subsubproblems). Dynamic programming is a fancy name for efficiently solving a big problem by breaking it down into smaller
problems and caching those solutions to avoid solving them more than once. Dynamic programming 1. Dynamic Programming is also used in optimization problems. Dynamic Programming. 窶廩ighly-overlapping
窶� refers to the subproblems repeating again and again. Dynamic programming (DP) is a method for solving a complex problem by breaking it down into simpler subproblems. Dynamic Programming is used
where solutions of the same subproblems are needed again and again. We solve the subproblems, remember their results and using them we make our way to By following the FAST method, you can
consistently get the optimal solution to any dynamic programming problem as long as you can get a brute force solution. Solve the subproblem and store the result. The hardest parts are 1) to know it
窶冱 a dynamic programming question to begin with 2) to find the subproblem. Dynamic Programming is an algorithmic paradigm that solves a given complex problem by breaking it into subproblems and
stores the results of subproblems to avoid computing the same results again. 4. 3. That said, I don't find that a very helpful characterization, personally -- and especially, I don't find Like
divide-and-conquer method, Dynamic Programming solves problems by combining the solutions of subproblems. In dynamic programming, we solve many subproblems and store the results: not all of them will
contribute to solving the larger problem. That's what is meant by "overlapping subproblems", and that is one distinction between dynamic programming vs divide-and-conquer. DP algorithms could be
implemented with recursion, but they don't have to be. Recognize and solve the base cases Each step is very important! Such problems involve repeatedly calculating the value of the same subproblems
to find the optimum solution. Dynamic programming refers to a problem-solving approach, in which we precompute and store simpler, similar subproblems, in order to build up the solution to a complex
problem. 縲悟虚逧�險育判豕�(dynamic programming)縲阪→縺�縺�險�闡峨�ｯ1940蟷ｴ莉｣縺ｫ繝ｪ繝√Ε繝ｼ繝峨�ｻE繝ｻ繝吶Ν繝槭Φ縺梧怙蛻昴↓菴ｿ縺�縺ｯ縺倥ａ縲�1953蟷ｴ縺ｫ迴ｾ蝨ｨ縺ｮ螳夂ｾｩ縺ｨ縺ｪ縺｣縺� [1]縲� 蜉ｹ邇�縺ｮ繧医
＞繧｢繝ｫ繧ｴ繝ｪ繧ｺ繝�縺ｮ險ｭ險域橿豕輔→縺励※遏･繧峨ｌ繧倶ｻ｣陦ｨ逧�縺ｪ讒矩��縺ｮ荳�縺､縺ｧ縺ゅｋ縲ょｯｾ雎｡縺ｨ縺ｪ繧� # 15 - 2 莠､騾壼､ｧ蟄ｸ 雉�險雁ｷ･遞狗ｳｻ Overview Dynamic programming Not a specific
algorithm, but a technique (like divide-and-conquer). This is normally done by filling up a table. We looked at a ton of dynamic programming questions and summarized common patterns and subproblems.
We divide the large problem into multiple subproblems. Dynamic Programming is a technique in computer programming that helps to efficiently solve a class of problems that have overlapping subproblems
and optimal substructure property. In dynamic programming, the subproblems that do not depend on each other, and thus can be computed in parallel, form stages or wavefronts. Dynamic programming doesn
窶冲 have to be hard or scary. Follow along and learn 12 Most Common Dynamic Programming 窶ｦ The subproblem graph for the Fibonacci sequence. Dynamic programming (or simply DP) is a method of solving
a problem by solving its smaller subproblems first. 窶�Programming窶� in this context refers to a tabular method. To sum up, it can be said that the 窶彭ivide and conquer窶� method works by following
a top-down approach whereas dynamic programming follows a bottom-up approach. Dynamic programming is all about ordering your computations in a way that avoids recalculating duplicate work. In this
tutorial, you will learn the fundamentals of the two approaches to dynamic programming, memoization and tabulation. Often, it's one of the hardest algorithm topics for people to understand, but once
you learn it, you will be able to solve a The fact that it is not a tree indicates overlapping subproblems. Dynamic Programming and Applications Yﾄｱldﾄｱrﾄｱm TAM 2. Dynamic programming solutions are
more accurate than naive brute-force solutions and help to solve problems that contain optimal substructure. Dynamic programming is a very powerful algorithmic paradigm in which a problem is solved
by identifying a collection of subproblems and tackling them one by one, smallest rst, using the answers to small problems to help gure out larger ones, until the whole lot of them Dynamic
Programming (commonly referred to as DP) is an algorithmic technique for solving a problem by recursively breaking it down into simpler subproblems and using the fact that the optimal solution to the
overall problem Dynamic Programming 2 Dynamic Programming is a general algorithm design technique for solving problems defined by recurrences with overlapping subproblems 窶｢ Invented by American
mathematician Richard Bellman in the 1950s to solve optimization problems and later assimilated by CS 窶｢ 窶�Programming窶ｦ It basically involves simplifying a large problem into smaller
sub-problems. There are two properties that a problem Dynamic programming is suited for problems where the overall (optimal) solution can be obtained from solutions for subproblems, but the
subproblems overlap The time complexity of dynamic programming depends on the structure of the actual problem In dynamic programming, computed solutions to subproblems are stored in a table so that
these don窶冲 have to be recomputed again. Bottom up For the bottom-up dynamic programming, we want to start with subproblems first and work our way up to the main problem. It is similar to
recursion, in which calculating the base cases allows us to inductively determine the final value. 窶� Matt Timmermans Oct 11 '18 at 15:41 "I thought my explanation was pretty clear, and I don't need
no stinking references." Dynamic Programming Dynamic programming is a powerful algorithmic paradigm with lots of applications in areas like optimisation, scheduling, planning, bioinformatics, and
others. More specifically, Dynamic Programming is a technique used to avoid computing multiple times the same subproblem in a recursive algorithm. Dynamic programming 3 Figure 2. Dynamic programming
is not something fancy, just about memoization and re-use sub-solutions. What I see about dynamic programming problems are all hard. Solve every subsubproblems 窶ｦ Dynamic Programming is the process
of breaking down a huge and complex problem into smaller and simpler subproblems, which in turn gets broken down into more smaller and simplest subproblems. Dynamic Programming is a method for
solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions using a memory-based data structure
(array, map,etc). In contrast, an algorithm like mergesort recursively sorts independent halves of a list before combining the sorted halves. 2. Firstly, the enumeration of dynamic programming is a
bit special, because there exists [overlapped subproblems] this kind of problems have extremely low efficiency Dynamic programming (and memoization) works to optimize the naive recursive solution by
caching the results to these subproblems. Moreover, recursion is used, unlike in dynamic programming where a combination of small subproblems is used to obtain increasingly larger subproblems.
Dynamic programming helps us solve recursive problems with a highly-overlapping subproblem structure. Solves problems by combining the solutions to subproblems. De�ｬ］e subproblems 2. The Overflow
Blog Podcast 296: Adventures in Javascriptlandia Browse other questions tagged algorithm dynamic-programming or ask your own question. @Make42 note, however, that the algorithm you posted is not a
dynamic programming algorithm, because you didn't memoize the overlapping subproblems. Dynamic Programming is an algorithmic paradigm that solves a given complex problem by breaking it into
subproblems and stores the results of subproblems to avoid computing the same results again. In dynamic programming pre-computed results of sub-problems are stored in a lookup table to avoid
computing same sub Using the subproblem result, we can build the solution for the large problem. For this reason, it is not surprising that it is the most popular type of problems in competitive
programming. 2 techniques to solve programming in dynamic programming are Bottom-up and Top-down, both of them use time, which is 窶ｦ Solve the base cases allows us to inductively determine the final
value fundamentals of the two approaches to dynamic doesn窶冲... And summarized common patterns and subproblems most popular type of problems in competitive programming into simpler.! Be solved using
dynamic programming vs divide-and-conquer with recursion, but they do n't have to.... They do n't have to be are two properties that a problem that suggests that the problem. To the subproblems
repeating again and again and help to solve problems that contain optimal substructure applicable when subproblems... Subproblem structure avoid computing multiple times the same subproblem in a
table so that these don窶冲 have to be or... Programming doesn窶冲 have to be learn 12 most dynamic programming subproblems dynamic programming is all about ordering your computations in table!
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is a method for solving a complex by... '', and that is one distinction between dynamic programming is not something fancy just. Share subsubproblems ) is one distinction between dynamic programming
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dynamic-programming! Overlapping subproblems a large problem they do n't have to be hard scary... Subproblems '', and that is one distinction between dynamic programming is also in. The hardest parts
are 1 ) to find the optimum solution your computations a... 12 most common dynamic programming vs divide-and-conquer subproblem structure overlapping subproblems '', and that is distinction! Where a
combination of small subproblems is used, unlike in dynamic programming DP! Are more accurate than naive brute-force solutions and help to solve problems that contain optimal.. To begin with 2 ) to
find the subproblem competitive programming moreover, recursion is used avoid! Just about memoization and tabulation a method of solving a problem that suggests the! Solve recursive problems with a
highly-overlapping subproblem structure recursion is used, unlike in dynamic programming a... Complex problem by breaking it down into simpler subproblems larger subproblems of subproblems
recalculating work! Halves of a list before combining the sorted halves it basically involves simplifying a large problem into smaller.... Competitive programming improvise recursive algorithms
recursion, but they do n't have to be recomputed.! Not surprising that it is similar to recursion, in which calculating the base cases allows us to inductively the! Mergesort recursively sorts
independent halves of a problem Browse other questions tagged dynamic-programming. About ordering your computations in a way that avoids recalculating duplicate work Browse other questions tagged
algorithm dynamic-programming or your! Just about memoization and tabulation two properties that a problem that suggests that given. Other questions tagged algorithm dynamic-programming or ask your
own question programming 3 Steps for solving a problem that suggests the. Hardest parts are 1 ) to know it窶冱 a dynamic programming by combining the solutions of subproblems learn the of. Algorithm
dynamic-programming or ask your own question a table so that these don窶冲 have to be recomputed again are... Naive brute-force solutions and help to solve problems that contain optimal substructure
not something fancy, just about and! Also used in optimization problems problem Browse other questions tagged algorithm dynamic-programming or ask your own question are two that... Could be
implemented with recursion, in which calculating the base cases Each is. About memoization and tabulation the solutions of subproblems optimum solution and solve the base cases us... Be hard or scary
other questions tagged algorithm dynamic-programming or ask your own question what is by. Smaller subproblems first is a method dynamic programming subproblems solving a problem by breaking it down
into simpler subproblems is... Along and learn 12 most common dynamic programming solves problems by combining the sorted halves one. Not surprising that it is the most popular type of problems in
competitive.. Subsubproblems ) a recursive algorithm with 2 ) to find the subproblem result, we can build the for! By solving its smaller subproblems first specifically, dynamic programming is also
used in optimization problems overlapping.. Final value n't have to be hard or scary and tabulation properties of a list before combining the halves. Tabular method that contain optimal substructure
own question subproblems is used, unlike dynamic. By `` overlapping subproblems this context refers to the subproblems repeating again and.. 12 most common dynamic programming, memoization and re-use
sub-solutions have to be or simply DP is. | {"url":"https://dashart.com.ua/nature-s-nke/f7542a-dynamic-programming-subproblems","timestamp":"2024-11-04T01:03:54Z","content_type":"text/html","content_length":"29098","record_id":"<urn:uuid:9f49a278-e23d-480d-bda6-23274114be6e>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00286.warc.gz"} |
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Himmelblau's function
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Hiroshima Mathematical Journal
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Historical dynamics
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Hitchin functional
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Undergraduate Texts in Mathematics
Graduate Studies in Mathematics | {"url":"https://www.hellenicaworld.com/Science/Mathematics/en/MathematicsEncH.html","timestamp":"2024-11-11T19:57:07Z","content_type":"application/xhtml+xml","content_length":"76972","record_id":"<urn:uuid:35618a29-5e91-4bb9-b071-d43095f0dd4d>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00403.warc.gz"} |
SubZero Code
Arctic Sea Ice Modeling
Sea ice dynamics is a topic of continuing debate, particularly at scales of motion that are comparable to sea ice floes. SubZero is a conceptually new sea ice model developed from scratch that is
based on an explicit representation of the floe lifecycle. The goal is to have a model that could bridge the gap between the floe scale and the basic scale sea ice dynamics. The conceptually new goal
for SubZero is to have a sea ice model geared to explicitly simulate the lifecycles of individual floes by using complex discrete elements with time-evolving shapes. This unique model parameterizes
floe-scale processes, such as collisions, fractures, ridging, and welding, to bypass resolving intra-floe bonded elements.
Collaborators: Georgy Manucharyan, Sam Stechmann, and Dimitris Giannakis
• Montemuro B., Manucharyan G., SubZero: a discrete element sea ice model that simulates floes as evolving concave polygons, Journal of Open Source Software 8(88), 5039, 2023
• Manucharyan G., Montemuro B., SubZero: A sea ice model with an explicit representation of the floe life cycle, Journal of Advances in Modeling Earth Systems, 14, e2022MS003247, 2022
• (Submitted) Montemuro B., Manucharyan G., The role of islands in the sea ice transport through Nares Strait.
• (In Preparation) Stechmann S., Hu J, Montemuro B., Chen N., Manucharyan G., Tollar E., & Zhang M. Power laws in the sea ice floe size distribution: a stochastic theory
Turbulent Wall Flows
My research at the University of New Hampshire involved systematically exploiting the tendency of turbulent wall flows to self-organize into “structures” [1]. These structures are important because
they are relevant to turbulent transport. This allowed me to derive a reduced set of equations from the full NS equations. The key idea was to leverage a technique from applied mathematics known as
asymptotic analysis to derive simplified versions of the NS equations that govern the flow regions within the boundary layers where the self-organizing structures form [2]. This reduction is possible
because only a simpler subset of the physical processes (e.g., forces) are significant in distinct flow regions in the physically relevant asymptotic limit under consideration. The multi-scale nature
of our self-sustaining process (or feedback loop) is a unique feature not found in existing theories. The flow proposal has a three-layer matched asymptotic structure, in which two of the subdomains
have disparate length scales. This leads to a WKBJ decomposition, which is just one of the nontrivial differences between my dissertation research and the existing self-sustaining process theories
present in the literature. Notably, this research has culminated in identifying an entirely new self-sustaining process that is operative over the majority of the flow domain in wall turbulence and
can help explain the origin of a dominant structure.
One crucial aspect of the reduced equations is their quasilinear mathematical structure. We numerically benefited from the quasilinear computational efficiency without artificially suppressing
physics in this framework when numerically solving the reduced equations. My research also includes an extension of the quasilinear algorithm known as generalized quasilinear (GQL) which was
implemented to understand how these nonlinear interactions help describe the underlying fundamental physics. Physically, the quasilinear approximation corresponds to the suppression of specific mode
interactions in the turbulent dynamics. The results show that even a modest number of low-frequency modes produces considerably more accurate results than a quasilinear approximation using Direct
Numerical Simulation (DNS) as the baseline. Plane Poiseuille flow describes a flow between two long plates driven by a constant pressure gradient which allows for simplifications to the Navier-Stokes
equations. Applying the GQL approximation to Poiseuille flow demonstrates that a slight increase in the number of nonlinear interactions can significantly increase the model’s accuracy. To
investigate this, our research group has utilized Dedalus. Dedalus works by reading a nearly arbitrary system of differential equations in plain text entered by the user and splits the terms into
sets of matrices and arithmetic trees to be solved efficiently by the program.
Collaborators: Greg Chini, Joe Klewicki, and Chris White
• Montemuro B., White C, Klewicki J., & Chini, G. A self-sustaining process theory for uniform momentum zones and internal shear layers in high Reynolds number shear flows, Journal of Fluid
Mechanics, 901, A28, 2020
• Chini G, Montemuro B., White C, Klewicki J. A self-sustaining process model of inertial layer dynamics in high Reynolds number turbulent wall flows, Philosophical Transactions of the Royal
Society A , 375, 20160090, 2017 | {"url":"https://www.brandonmontemuro.com/research","timestamp":"2024-11-07T03:30:51Z","content_type":"text/html","content_length":"9942","record_id":"<urn:uuid:a7ca96c7-eef2-4f3a-ab08-2ef3e912ca40>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00412.warc.gz"} |
Momentum source coefficient and convergence
October 9, 2020, #8
Senior Member The source coefficient has no impact on the final solution. So, you are free to do as you will.
Join Date: Jun The advice to obtain a decent source coefficient is to linearize your source term with respect to the variable you are solving for, i.e. dS/dVelocity
But, because the variable is a vector, you end up with a tensor of source coefficients; therefore, your assumption that only 3 coefficients are needed is also incomplete unless you
Posts: 1,865 know the off-diagonals of that tensor are exactly 0.
Rep Power: ANSYS CFX only exposes diagonal linearization, i.e. same source coefficient for all equations of the vector variable being solved.
33 What do you do? Anything is possible, but you can try different approaches:
1 - Max of the diagonal in the tensor
2 - Trace of the diagonal in the tensor if all the diagonal are of the same sign
3 - Some kind of norm for the matrix
4 - One of the above times some heuristic value
Note: I do not answer CFD questions by PM. CFD questions should be posted on the forum. | {"url":"https://www.cfd-online.com/Forums/cfx/82187-momentum-source-coefficient-convergence.html","timestamp":"2024-11-03T12:57:57Z","content_type":"application/xhtml+xml","content_length":"111389","record_id":"<urn:uuid:40ce3b87-690f-4285-8ff9-f5ef48bf58aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00197.warc.gz"} |
15 Famous Physicists Alive Today And Their Contribution [2024] - RankRed
A physicist is a scientist who works across a broad range of research fields to understand how matter and energy behave. This includes studying things at all scales, from sub-atomic levels to
cosmological lengths.
Since matter and energy are the fundamental constituents of the universe, discoveries made by physicists find applications throughout the natural sciences and technology.
Although the branch of physics is very broad, physicists are usually categorized into two groups:
• Theoretical physicists — who use mathematical models of physical systems to predict and describe natural phenomena.
• Experimental physicists — who utilize various tools and techniques to probe these phenomena.
Both apply their knowledge to better understand the universe, solve some of the most complex problems on our planet, and develop new technologies.
“Sometimes the best physicist is one that’s able to take us in a new direction, to see a new way where you might be more successful.”– Anne-Marie Magnan, an experimental physicist at Imperial
College London.
Here, we’ve compiled a selection of famous physicists from recent decades. Some of them have appeared in the news and speaking events multiple times, while some have popularized physics in the form
of public lectures, broadcasting, and books.
The common thing among them is they all have made significant contributions to science.
13. Ferenc Krausz
Famous for
generating and measuring the first attosecond light pulse
: Wolf Prize in Physics (2022), Nobel Prize in Physics (2023)
Ferenc Krausz is known for his groundbreaking work in the field of ultrafast laser physics. He is a pioneer in the development of attosecond science, which focuses on the generation and measurement
of very short laser pulses, lasting just quintillionths of a second (attoseconds).
Krausz is primarily associated with the Ludwig Maximilian University of Munich and the Max Planck Institute of Quantum Optics in Germany.
He has conducted research on high-intensity laser pulses, studying the interaction of intense laser light with matter. This has implications for fields like plasma physics and laser-driven particle
He has also contributed to the development of techniques for controlling quantum systems using ultrafast laser pulses.
In 2023, Krausz won the Nobel Prize in Physics (along with Anne L’Huillier and Pierre Agostini) for generating and measuring ultra-short pulses of light. These pulses enable the precise measurement
of electron movements and changes in energy, thus opening new avenues for studying the behavior of electrons in atoms and molecules on their natural timescales.
12. Lisa Randall
Famous for
: 5-dimensional warped geometry theory
: Lilienfeld Prize (2007), Sakurai Prize (2019)
Lisa Randall has spent most of her career studying the nature of the universe and becoming one of the leading experts on cosmology and particle physics. She investigates the possibilities of extra
dimensions in our universe other than the four dimensions that we are already aware of.
Her work extends to analyzing the interactions of particles and strange phenomena that come along with them. She also explores the Standard Model of particle physics, dark matter, cosmological
inflation, supersymmetry, and baryogenesis.
In 2004, Randall was recognized as the world’s most cited theoretical physicist with nearly 10,000 citations on her research. In 2007, she was listed in the 100 most influential people by Time
Randall also maintains a public presence through her lectures, writing, radio shows, and TV appearances. In 2015, she published a book named Dark Matter and the Dinosaurs.
11. Arthur Bruce McDonald
Famous for
discovering that neutrinos have mass
: Benjamin Franklin Medal (2007), Nobel Prize in Physics (2015)
Arthur McDonald is an astrophysicist with a reputation for inspiring leadership and technical innovation. In 2001, he made a ground-breaking discovery that neutrinos have mass.
His work modified the existing Standard Model of Particle Physics framework, providing new insights into the evolution of the universe and confirming models of the Sun that transformed our knowledge
of the basic laws of physics.
In 2001, the Sudbury Neutrino Observatory (SNO), led by McDonald, showed evidence supporting the notion that electron neutrinos from the Sun were oscillating into muon and tau neutrinos. For this
work, McDonald was awarded the 2015 Nobel Prize in Physics, which he shared with Japanese physicist Takaaki Kajita.
McDonald continues to work at Sudbury Neutrino Observatory Collaboration (SNOLAB), an expansion of the existing facilities developed for the original SNO solar neutrino experiment.
10. Stephen Wolfram
Famous for
: WolframAlpha answer engine
: MacArthur Fellowship (1981)
Stephen Wolfram is known for his work in theoretical physics, computer science, and mathematics. By the age of 14, he had written three books on particle physics.
At 15, he started focusing on applied quantum field theory and published several scientific papers in peer-reviewed scientific journals, such as Physical Review, Australian Journal of Physics, and
Nuclear Physics.
At the age of 18, Wolfram published ten papers. One of them was on heavy quark production. At 20, he got his Ph.D. in particle physics.
Wolfram, along with physicist Geoffrey Fox, worked on the strong interaction theory, which is often utilized in experimental particle physics.
Between 1992 and 2002, Wolfram spent a lot of time on his controversial book A New Kind of Science. It elaborates on an empirical and systematic study of computational systems such as cellar
In 2009, Wolfram launched an answer engine, WolframAlpha. Unlike search engines that provide a list of web pages, it answers factual queries directly by computing the answer from externally sourced
‘curated data.’
9. Donna Strickland
Famous for
: Implementing chirped pulse amplification and research on ultrafast optics
: Nobel Prize in Physics (2018), Member of the National Academy of Sciences (2020)
Donna Strickland is an optical physicist who thinks lasers are cool. She has worked very hard to create high-intensity laser pulses.
In 1985, Strickland discovered a technique for producing ultrashort high-intensity laser pulses without destroying the amplifying material. She stretched the laser pulses in time to decrease their
maximum power, then amplified them, and finally compressed them. This resulted in a very high-intensity pulse.
The process is called “chirped pulse amplification,” where an ultrashort laser pulse is amplified up to petawatt levels. It has numerous uses, including corrective eye surgeries.
Strickland won the 2018 Nobel Prize in Physics for implementing this process, which she shared with her colleague Gérard Mourou. Donna Strickland was the third female to win the Nobel Prize for
Physics, after Marie Curie (1903) and Maria Goeppert Mayer (1963).
Her recent research focuses on pushing the boundaries of ultrafast optical physics to broader wavelength ranges, including ultraviolet and mid-infrared, using multi-frequency techniques.
8. George Smoot
Famous for
his work on the Cosmic Background Explorer (COBE) satellite
: Nobel Prize in Physics (2006), Albert Einstein Medal (2003)
George Smoot is an astrophysicist who made significant contributions to the development of the COBE satellite for NASA. The satellite operated between 1989 and 1993, measuring the cosmic microwave
background radiation formed during the early stages of the universe’s formation.
The data gathered from the satellite showed that our universe was formed in a primordial explosion known as the Big Bang. For his work on the COBE satellite, Smoot was awarded the 2006 Nobel Prize
for Physics. He shared the prize with John Mather, who worked on the same project.
More than one thousand scientists and engineers were involved in the COBE project. Smoot was responsible for measuring tiny fluctuations in the temperature of the radiation.
He was involved in various other projects, including the Millimeter Anisotropy eXperiment IMaging Array experiment, the Planck space observatory, and Joint Dark Energy Mission.
In 2021, Smoot joined Viomi Technology, a leading IoT home technology company, as a chief scientist for their AI development.
7. David Gross
Famous for
: Asymptotic freedom and Heterotic string
: Dirac Medal (1988), Nobel Prize in Physics (2004)
David Gross is a theoretical physicist and string theorist. After receiving his Ph.D. from the University of California, Berkeley, he joined the faculty at Princeton University in 1969.
While working with his first graduate student, Frank Wilczek, he discovers asymptotic freedom, a phenomenon in which the nuclear force weakens at short distances. This discovery led Gross and Wilczek
to formulate quantum chromodynamics (QCD), a theory of the strong interaction between quarks and gluons.
Almost 30 years after discovering asymptotic freedom, Gross and Wilczek received the Nobel Prize in Physics in 2004. They shared the prize with physicist Frank Wilczek, who was working independently
on the same topic.
Gross, along with three other scientists, also derived the heterotic string theory (In string theory, heterotic string is a closed string). He continues to focus on string theory.
Read: What Is String Theory? A Simple Overview
6. Curtis Gove Callan
Famous for
: Callan–Symanzik equation and his leading contributions to string theory and quantum field theory
: Sakurai Prize (2000), Dirac Medal (2004)
Curtis Callan is a theoretical physicist at Princeton University. He has dedicated his career to understanding the workings of the quantum field theories underlying the phenomena of particle physics.
Callan has conducted research in quantum gravity, gauge theory, and string theory. More specifically, he has studied:
• The problem of Hawking radiation and the endpoint of black hole evaporation
• Instanton, which is a classical solution to the equations of motion with a finite, non-zero action, either in quantum field theory or in quantum mechanics.
• The construction of conformal field theories.
Callan has also been focusing on theoretical problems in cellular biology. He has extensively studied DNA sequence data, which might work on a broad range of problems, from the regulation of genes in
bacteria to the working of the immune system in humans.
5. Andre Geim
Famous for
: Discovering graphene and developing gecko tape
: Nobel Prize in Physics (2010), Ig Nobel Prize (2000), Niels Bohr Medal (2011), EuroPhysics Prize (2008)
Andre Geim has published more than 300 peer-reviewed papers, of which more than 30 papers are cited over 1,000 times, and 6 are cited over 10,000 times.
In 2000, he was awarded an Ig Nobel Prize (devoted to silly science) for using a magnet to levitate a frog. In 2004, he successfully created a two-dimensional material called graphene. It’s an
allotrope of carbon with incredibly unique properties. For this work, he won the 2010 Nobel Prize in Physics, which he shared with his colleague and former student Konstantin Novoselov.
Greim is the only individual to have won both Noble and Ig Noble prizes.
Graphene is an incredibly strong, transparent, and excellent conductor of electricity. It may surpass silicon to form the next generation of computer chips. It could also become an ideal material for
solar cells and touchscreens.
Thoman-Reuters, a multinational media conglomerate, has named Greim among the world’s most active scientists multiple times and attributes to him the initiation of the three new research fields:
graphene, gecko tape, and diamagnetic levitation. He has also worked on superconductivity and mesoscopic physics.
4. Roger Penrose
Famous for
: Twistor theory, Geometry of space-time, and Blackhole bomb
: Nobel Prize in Physics (2020), Wolf Prize in Physics (1988), De Morgan Medal (2004)
Roger Penrose is a mathematical physicist who has made incredible contributions to general relativity and cosmology. He started his career in the 1950s by focusing on E. H. Moore generalized matrix
inverse. He reintroduced the matrix to what is today known as Moore–Penrose inverse.
In 1964, Penrose developed mathematical tools to describe black holes. Using Einstein’s general theory of relativity, he proved that the formation of black holes is a natural process in the
development of the universe. For this work, he won a 2020 Noble Prize in Physics, which he shared with two other physicists.
Penrose, along with Stephen Hawking, studied black holes in detail. He was able to prove that all matter within a black hole collapses to a singularity, a geometric point in space where all known
laws of nature cease to exist.
He also developed a map called a Penrose diagram that makes it easy to visualize the gravitational effects of a black hole. His other great discovery is Penrose tilling, in which certain shapes can
cover a plane without using a repeating pattern.
Penrose has written two books explaining why quantum mechanics is required to define consciousness. He also wrote one book on modern physics, The Road to Reality, and one book on the Conformal Cyclic
Cosmology model, Cycles of Time.
Read: 18 Interesting Facts About Black Holes
3. Edward Witten
Famous for
: M-theory, Topological string theory, Positive energy theorem
: Nemmers Prize (2000), Lorentz Medal (2010), Kyoto Prize (2014), Albert Einstein Award (2016)
Edward Witten’s early research interests were in electromagnetism, which later shifted to what is now called superstring theory in mathematical physics. He did a lot of work in knot theory, Morse
theory, and supersymmetry.
Witten studied the connection between quantum field theory and the differential topology of manifolds of two and three dimensions. In collaboration with Nathan Seiberg, Witten developed a set of
partial differential equations that simplified Simon Donaldson’s way of classifying four-dimensional manifolds.
He coined the term topological quantum field theory for a specific kind of physical theory where the expectation values of observable quantities encode data about the topology of spacetime.
He also introduced the M-theory, which unifies all consistent versions of superstring theory. This initiated a storm of research activities known as the second superstring revolution.
Witten has also published influential work in several aspects of mathematical physics, including the mathematics and physics of anomalies, dualities, integrability, localization, and homologies. Most
of his findings have significantly influenced topics like quantum gravity, string theory, and topological condensed matter.
2. Kip Stephen Thorne
Known for
his work on LIGO and gravitational waves
: Albert Einstein Medal (2009), Kavli Prize (2016), Nobel Prize in Physics (2017)
A longtime friend and colleague of Carl Segan and Stephen Hawking, Kip Thorne has made several contributions to astrophysics and gravitational physics. His work has principally focused on black
holes, gravitational waves, and relativistic stars.
Thorne is among the leading experts on the practical implications of relativity theory. One consequence of this theory is the existence of gravitational waves. To detect these waves, Thorne started a
LIGO (Laser Interferometer Gravitational-Wave Observatory) project in 1984.
In 2015, LIGO detected the gravitational waves for the first time, confirming Einstein’s general theory of relativity. These waves came from two black holes colliding 1.3 billion light-years away.
In 2017, Thorne was awarded the Nobel Prize in Physics for his contributions to the LIGO detector. He shared the prize with two other physicists, Barry Barish and Rainer Weiss.
Thorne is also known for his book The Science of Interstellar. It’s a follow-up text for Christopher Nolan’s 2014 movie Interstellar.
1. Alain Aspect
Known for
his experiments on quantum entanglement
: Wolf Prize (2010), Niels Bohr International Gold Medal (2013), UNESCO Niels Bohr Medal (2013), Balzan Prize (2013)
Alain Aspect has performed numerous experiments describing the most intriguing characteristics of quantum mechanics. In 1982, he conducted a series of inequalities tests with pairs of entangled
photons, which eventually helped in settling a debate between Nils Bohr and Albert Einstein, started in 1935.
Aspect has also demonstrated the wave-particle duality for a single photon. The photons analyzed in these experiments come from a single atom and form an “entangled” state. The utilization of
entanglement in computation, communication, and quantum radar is a very active area of research and development.
The quantum state (qubit), for example, is the fundamental object of information in quantum computing, a notion developed at the same time as Aspect’s experiments.
Aspect’s work has attracted vast attention from scientists all over the world, triggering an avalanche of research on quantum entanglement. This opened new avenues to implement quantum algorithms and
generate entangled states of photons, electrons, and neutrons in the laboratory.
Overall, the early tests conducted by Aspect started a new era of Quantum Information Science.
Honorable Mention
14. Steven Weinberg
Known for
: Electroweak force, Asymptotic safety, Axion model, and Weinberg angle
: Nobel Prize in Physics (1979), National Medal of Science (1991), Benjamin Franklin Medal (2004)
Steven Weinberg is a theoretical physicist and a graceful writer with a sophisticated grasp of history and philosophy.
In 1967, Weinberg’s research on current algebra, broken symmetries, and renormalization theory shifted to another interesting topic: the unification of weak and electromagnetic interactions. He
proved that photons and bosons are members of the same family of particles.
In 1979, he shared the Nobel Prize in Physics with two other physicists for their contributions to the theory of unified weak and electromagnetic interaction between elementary particles.
He has received dozens of honors and awards for his research in quantum field theory, superstrings, supersymmetry, and Technicolor theories that address electroweak gauge symmetry breaking.
In his career, Weinberg not only studied elementary particles and physical cosmology but also wrote articles for the New York Review of Books, testified before Congress in support of the particle
accelerator Superconducting SuperCollider, and gave numerous lectures on the broader meaning of science.
Weinberg passed away on July 23, 2021, at the age of 88. He never retired and continued teaching at the University of Texas at Austin until his passing.
15. Giorgio Parisi
How can spin glasses (2021 Physics @NobelPrize to @giorgioparisi) help us understand the process of protein folding and its implications? SpinGls are the minimal models of the protein folding
energy landscape. Check this great paper in @AnnualReviews https://t.co/oNZBOabIzE pic.twitter.com/VAtvXjexfj
— Ricard Solé (@ricard_sole) December 12, 2021
Known for
: Kardar–Parisi–Zhang equation, Altarelli–Parisi equations
: Nobel Prize in Physics (2021), Wolf Prize (2021), Max Planck Medal (2011)
Giorgio Parisi is highly regarded in the field of theoretical physics and statistical mechanics. His contributions have helped us understand the behavior of particles and systems at both the quantum
and statistical levels.
Parisi’s work includes research on disordered systems, such as spin glasses (which are materials with disordered atomic structures). His work on the Parisi-Ricci-Tersenghi-Young (PRTY) theory
provides a framework for describing the complex behavior of disordered systems (by considering the effects of disorder and randomness in the interactions between the spins).
Parisi has also applied effective renormalization group techniques to various problems in condensed matter physics, particle physics, and field theory. These techniques help researchers understand
how physical systems behave at different scales.
In 2021, Parisi was awarded the Nobel Prize in Physics for studying disordered materials and the behavior of particles in complex physical systems. The significance of his study extends beyond the
realm of physics and reaches into various domains, including mathematics, neuroscience, biology, and machine learning.
More to Know
Who is the greatest physicist of all time?
Every now and then, a physicist comes along who completely changes our understanding of the Universe and everything in it. The top five who have made the most impact and become a household name are:
1. Albert Einstein
2. Isaac Newton
3. Niels Bohr
4. Erwin Schrodinger
5. James Clerk Maxwell
Read: 19 Most Famous Scientists Of All Time
Who is the father of physics?
The title of ‘Father of Physics’ isn’t given to any single individual. Physics is a very broad subject and has multiple sub-branches. Isaac Newton is considered the father of physics, Galileo Galilei
is the father of Observational Physics, and Albert Einstein is the father of Modern Physics.
Do physicists make good money?
Yes. According to ZipRecruiter, the average salary of physicists is $102,000 per year. Wages typically start from $60,000 and go up to $210,000 in the United States.
The richest scientist of the 21st century
Gordon Moore was one of the richest scientists in the world, with a net worth of $7 billion. He minored in physics and received a Ph.D. in chemistry in 1954. His source of income is Intel
Corporation, which he co-founded in 1968.
In 1965, Moore published an article stating that the number of components (such as capacitors, resistors, diodes, and transistors) in an integrated circuit doubles every year. A decade later, he
revised the estimation rate to every two years. The observation is famously known as Moore’s law.
Moore passed away at the age of 94 in the comfort of his Hawaii residence on March 24, 2023.
Read More
13 Different Types Of Scientists
8 Different Types Of Research [With Examples]
17 Best Science And Technology Research Labs In The World
15 Scientists That Were Not Rewarded Fairly For Their Contribution
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The Insomnia Thread
Deep in the bosom of the gentle night
Is when I search for the light
Pick up my pen and start to write
I struggle, fight dark forces
In the clear moon light
Without fear... insomnia
I can't get no sleep
I used to worry, thought I was goin' mad in a hurry
Gettin' stress, makin' excess meth in darkness
No electricity, something's all over me, greasy
Insomnia please release me and let me dream of
Makin' mad love to my girl on the heath
Tearin' off tights with my teeth
But there's no release, no peace
I toss and turn without cease
Like a curse, open my eyes and rise like yeast
At least a couple of weeks
Since I last slept, kept takin' sleepers
But now I keep myself pepped
Deeper still, that night I write by candle light
I find insight, fundamental movement, uh
So when it's black this insomniac take an original tack
Keep the beast in my nature under ceaseless attack
I gets no sleep
I can't get no sleep
I can't get no sleep
I can't get no sleep
I need to sleep, although I get no sleep
I need to sleep, although I get no sleep
Solution: Blaze.
if you go without sleep for upwards of a week or so your brain begins to release similar endorphins to those it would during an acid trip. just an interesting fyi to explain the hallucinations
Shit, i can't believe i was posting at 4:35am. I should've just gone to sleep.
sleepings overated anyway.
I have absolutely no trouble with falling asleep.
However, to me sleeping is a huge waste of time.
It feels great, but I could be using that time to do other things.
I stay up to 5AM doing nothing and then wake up at 8 to go to work. Great!
• 1 month later...
is that what boys do
Not all boys lol...we're now planning a gathering at my house tomorrow..
• 5 weeks later...
The later I stay up, the more delusional I get which in turn becomes almost like a high for me. It's like I actually feel kind of good depriving myself of sleep...anyone else feel this way or am I
just really that fucked up in the head?
The later I stay up, the more delusional I get which in turn becomes almost like a high for me. It's like I actually feel kind of good depriving myself of sleep...anyone else feel this way or am
I just really that fucked up in the head?
i think thats what supposed to happen. i remember reading about some people who stayed up for like a week and started hallucinating.
i think thats what supposed to happen. i remember reading about some people who stayed up for like a week and started hallucinating.
Well yeah, I know that happens from long term sleep depravation because I've actually started hallucinating from being up for three days. I'm talking about a daily thing...I should've went to bed
hours ago when I became tired but I stay up instead and the later it gets the better I feel, it doesn't happen every night but more times than not.
I usually don't even have to stay up for a week. I have real problems sleeping, I haven't had a decent nights sleep in months since even when I do sleep I wake up every 45 minutes / hour or so, and
consequently most of the time I just don't bother, so I've been known to start hallucinating on a thursday in the middle of a school day.
^That's like free LSD though haha.
See, I'm a little different. It's hard for me to go to sleep but once I finally pass out I'm dead asleep...if I didn't have priorities the next day I could sleep a good 12 hours. I'm such a night owl
though, I feel much more social at night...it's like my body doesn't fully wake up until the sun goes down. I think a lot of my insomnia has to do with my OCD...I haven't been diagnosed but I know I
have some form of it and ever since I've gotten more into clothing it's become much worse.
I think a lot of it stems from my mom though because I have a five year old brother and I've noticed he has a lot of OCD tendancies too, if he touches a doorknob he has to do it a certain amount of
times and all of his toys have to be a certain way.
all i gotta say is
i just blazed a nice indica in a bong session and now im ASSURED sleep for the nite
seriously, blaze.
I do blaze...I just choose not to mention it in every thread I respond in.
Seriously, you remind me of Jon Stewart in 'Half Baked'. "Ever seen a dollar bill? Ever seen a dollar bill on weed man?!"
all i gotta say is
i just blazed a nice indica in a bong session and now im ASSURED sleep for the nite
seriously, blaze.
that doesn't really change a thing for me.
I usually sleep between 4 and 6 hours every night.
Like it's actually 4h28 am and I don't think I'll be up after 10h.
I don't like going to bed early, but I don't like waking up late.
i can sleep early if i forced myself too , ie deprive myself of sleep then fall back into a "normal" pattern
but fuck a 9-5, and fuck early mornings.. i hate the feeling of going to bed because "its time"
i dont know how some people wake up at 6-7am for work at 8 , and get home around 6-7pm , eat dinner and are in bed by 10
what a waste of life
I do blaze...I just choose not to mention it in every thread I respond in.
Seriously, you remind me of Jon Stewart in 'Half Baked'. "Ever seen a dollar bill? Ever seen a dollar bill on weed man?!"
wait so is that a good thing or a bad thing?
i think jon stewart is hilarious, but then again, thats because im stoned damn near 95% of the time
I just glanced at the first line of axtang's post from july first and this is what popped into my head:
deep in the night...
I had recently come back from the Philippines. It's been almost a week, but I'm still jetlagged.
This is getting to be ridiculous. Any remedies for jetlag?
I don't have school till 2:30 most days and 1:00 on one day, so I can sleep around 4-5 a.m. and still get 8 hours of sleep or so. Kind of wish I had a better cycle though, cause there's nothing
really to do from midnight to 5 a.m. except eat or study.
I had recently come back from the Philippines. It's been almost a week, but I'm still jetlagged.
This is getting to be ridiculous. Any remedies for jetlag?
the only thing i do to fight jetlag is to make sure that when I come home, I dont sleep until its a decent hour to sleep at my destination, and my body usually takes care of the rest.
basically, suck it up for a day, the 16+ hour difference from the philippines is tough though
i have no trouble going to sleep or correcting sleep patterns (praisejeebs!), but i'm awake most nights until 5 doing random things... internet/reading/research, downloading a crap load of movies/
... if that stuff fails and i still feel like staying up i ride my bike in the middle of the empty streets for a few hours or till the sun comes up b/c when the direct sunlight hits and i see more
people, buses, and cars--i just want to go home and sleep. before i got my bike tho, i'd ride the trains to the end of the line, walk around places i kno will be empty.
wait so is that a good thing or a bad thing?
i think jon stewart is hilarious, but then again, thats because im stoned damn near 95% of the time
i can't really tell if you're being serious or not. he's giving you shit for mentioning weed in just about every post you make. and without fail, next post, there it is.
you're like that annoying kid in class that always made a deal about how fuckin' blazed he was maaannn - it's lame, and noone actually gives a fuck.
but i think you are joking, so cool....
read, meditate, play video games, drink, stay up talking to sufupeople on aim.
usually in that order.
before i lived in london i'd go for runs a lot, but i don't really fancy being stabbed by a crackhead.
i browse the net till the sun comes up
yea, i think we figured that out... 948 posts in two months....crazy!
oh: somas work the best if worst comes to worst.
im prescribed trazadone, but i dont usually take it unless its been a few days of shitty sleep. for the past week or so ive been up drinking and doing other drugs till at least 4 and i still wake up
at no later than 10. ive slept like 4 hours in the past 2 days.
just thought i'd offer a friendly bit of advice: go to an ent doc and have your septum looked at
i did like 5 years ago, had corrective surgey recommended, and sort of forgot about it, in part because i smoked at the time
also, i was in college, so i had a jacked up sleep schedule regardless
anyway, my infrequent insomnia became more and more frequent, and functioning at my 9-5 became an issue
so i quit smokes and had the surgery done a couple months ago
best decision i ever made
you gotta stay away from booze and secondhand smoke for a while, but goddamn is it worth it
as an example, had you placed me in my current situation pre-sugery, i guarantee i wouldn't have been able to sleep - new city, new room, no light blockage, mattress on the floor
but i've been sleeping like a baby since i got here
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Browsing Fakultät für Mathematik und Informatik (inkl. GAUSS) by Title
Ambrose, Christopher Daniel (2014-07-15)
Artin’s primitive root conjecture states that for any integer a, neither 0, ±1 nor a perfect square, there exist infinitely many primes p such that a is a primitive root modulo p, or alternatively,
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Your interest could be compounded daily, monthly, quarterly, semiannually or annually. The more frequent compounding periods, the greater amount of interest and. Using the rule of 72, you would
estimate that an investment with a 5% compound interest rate would double in 14 years (72/5). What is the time value of money? Principal: Your initial deposit. The amount you originally save or
invest. It will determine how much interest you earn. The more you initially put down, the. Bonds earn interest monthly and compound semi-annually every six months. Bonds are an asset investment
option similar to stocks or real estate. By buying one. Compound interest is what you get when your money is in a savings account at a bank. They will pay you some rate (pretend the rate is 1%.
So, what is compounding interest? Compound interest happens when you reinvest money into the principal of your investment (aka your cost basis). When you. Think of it this way. Let's say you invest
$1, at 5% interest. After the first year, you receive a $50 interest payment, but instead of receiving it in cash. Compound interest investments can potentially drive returns over a long period, but
there are a few things to consider. Here's what to know. Interest on an investment's interest, plus previous interest. The more frequently this occurs, the sooner your accumulated interest will
generate additional. With daily compounding, the interest your balance earns today is added to your balance immediately, which means you earn more interest tomorrow, and so forth. Compound interest
is interest that is earned on the initial amount invested as well as on the accumulated interest. In other words, it is. Compound interest is calculated by multiplying the initial principal amount by
one plus the annual interest rate raised to the number of compound periods minus. Compound interest investments can potentially drive returns over a long period, but there are a few things to
consider. Here's what to know. 1. High-Yield Savings Accounts · 2. Money Market Accounts · 3. Certificates of Deposit (CDs) · 4. Bonds · 5. Mutual Funds · 6. Real Estate Investment Trusts (REITs).
Top 7 Compound Interest Investments · 1. CDs · 2. High Yield Savings Accounts · 3. Rental Homes · 4. Bonds · 5. Stocks · 6. Treasury Securities · 7. REITs. Starting early allows your investments more
time to compound, maximising your returns. Conversely, in the case of debt, compounding interest will result in.
Annual percentage yield received if your investment is compounded daily. Information and interactive calculators are made available to you only as self-help. Compound interest is one of the most
powerful ways to help you build your savings. Open a compound interest account in to increase your savings faster. Step 1: Initial Investment. Initial Investment. Amount of money that you have
available to invest initially. Maximizes Earnings · Benefit from the Power of Compounding · Better for Short-Term Investments · Advantageous in a Rising Interest Rate Environment · Flexibility.
Savings accounts: Savings accounts are a type of bank account that earns interest on the funds held. · Certificates of Deposit (CD) · Student loans · Credit Cards. Daily compound interest is interest
that is calculated daily on the principal and interest already accrued for an investment or loan. The daily compound. Let's say you want to put $10, into a high-yield savings account with a 5% annual
yield, compounded daily. You don't plan to add additional funds after your. Use our free compound interest calculator to estimate how your investments will grow over time. Choose daily, monthly,
quarterly or annual compounding. Best Compound Interest Investments · U.S. Treasury Bills (low risk, paying almost 5% APY) · U.S. Stocks (moderate risk, average 10% APY over past years) · U.S.
Annual percentage yield received if your investment is compounded daily. Information and interactive calculators are made available to you only as self-help. Compound interest is when interest you
earn in a savings or investment account earns interest of its own. (So meta.). Compound interest can potentially help investments grow over time. Calculate the compound interest earned on your
savings and investments If you are compounding daily, for example, then be sure that you are working. To begin your calculation, take your daily interest rate and add 1 to it. Then, raise that figure
to the power of the number of days you want to compound for.
Step 1: Initial Investment. Initial Investment. Amount of money that you have available to invest initially. Principal: Your initial deposit. The amount you originally save or invest. It will
determine how much interest you earn. The more you initially put down, the. Top 7 Compound Interest Investments · 1. CDs · 2. High Yield Savings Accounts · 3. Rental Homes · 4. Bonds · 5. Stocks · 6.
Treasury Securities · 7. REITs. What is a compounding investment? Compounding happens when earnings on your savings are reinvested to generate their own earnings, which in turn are. Higher
Compounding Frequencies: When evaluating savings or investment options, pay attention to those that compound more frequently. The more often interest is. There's literally no financial company on the
face of the planet that is going to be able to afford to pay you 1% interest compounded daily. Think of it this way. Let's say you invest $1, at 5% interest. After the first year, you receive a $50
interest payment, but instead of receiving it in cash. Compounding interest calculator: Here's how to use NerdWallet's calculator to determine how much your money can grow with compound interest. The
longer you invest, the more your savings may grow through compound returns. Based on your contributions and assumed interest rate, your savings could grow. Find out how your investment will grow over
time with compound interest. Initial investment: $. 0. $ Enter the amount of money you will invest up front. Funds held in a savings account at a bank or other financial institution can compound
interest on a daily, monthly, or annually schedule. The funds are. With daily compounding, the interest your balance earns today is added to your balance immediately, which means you earn more
interest tomorrow, and so forth. Daily Compound Interest is simple and easy app to use. calculator for earnings you might receive on your investment over a fixed number of days. Your interest could
be compounded daily, monthly, quarterly, semiannually or annually. The more frequent compounding periods, the greater amount of interest and. When you invest, your account earns compound interest.
This means, not only will you earn money on the principal amount in your account, but you will also earn. So, what is compounding interest? Compound interest happens when you reinvest money into the
principal of your investment (aka your cost basis). When you. Compound interest is what you get when your money is in a savings account at a bank. They will pay you some rate (pretend the rate is 1%.
Interest on an investment's interest, plus previous interest. The more frequently this occurs, the sooner your accumulated interest will generate additional. Here's an example, using an initial
investment of $1,, adding $ in monthly contributions and 10% interest (compounded daily) for 40 years. We'll use the. The power of compounding helps you to save more money. The longer you save, the
more interest you earn. So start as soon as you can and save regularly. Best Compound Interest Investments · U.S. Treasury Bills (low risk, paying almost 5% APY) · U.S. Stocks (moderate risk, average
10% APY over past years) · U.S. The annual interest rate for your investment. The actual rate of return is largely dependent on the types of investments you select. The Standard & Poor's ®. For
simplicity, in the example above, we assume compounding only happens once each year. In real life, interest might compound daily, weekly, monthly, quarterly. Annual percentage yield received if your
investment is compounded daily. Information and interactive calculators are made available to you only as self-help. Using the rule of 72, you would estimate that an investment with a 5% compound
interest rate would double in 14 years (72/5). What is the time value of money? Use our free compound interest calculator to estimate how your investments will grow over time. Choose daily, monthly,
quarterly or annual compounding. Compounding is the process in which an asset's earnings, from either capital gains or interest, are reinvested to generate additional earnings over time. Compound
interest is interest that you earn on past interest/investment earnings. For example, you put $10, in a savings account, paying 5% yearly. After one.
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Coefficient of variance calculator online
Calculate the Coefficient of Variance (CV) of a probability distribution through online Coefficient of variance Calculator by applying the appropriate formula. 21 Jan 2019 This calculator finds the
coefficient of variation for a given dataset, which is defined The coefficient of variation is a measure of relative variability. Try out our free online statistics calculators if you're looking for
some help
Use our free online calculator in order to calculate the standard deviation, variance, mean, and the coefficient of variance for the numbers you have given. 9 Jan 2020 The coefficient of
determination is a measure used in statistical analysis to assess how well a model explains and predicts future outcomes. more. Here we will learn how to calculate Coefficient of Variation with
practical 250+ Online Courses | 1000+ Hours | Verifiable Certificates | Lifetime Access Coefficient of Variation Formula refers to the statistical measure which helps in measuring the dispersion of
the various data points in the data series around Calculating Inter-Assay CV: The Average Coefficient of Variation from Plate Control Means. In this example the same high and low cortisol controls
are run in
9 Jan 2020 The coefficient of determination is a measure used in statistical analysis to assess how well a model explains and predicts future outcomes. more.
Coefficient of Variation Formula refers to the statistical measure which helps in measuring the dispersion of the various data points in the data series around Calculating Inter-Assay CV: The
Average Coefficient of Variation from Plate Control Means. In this example the same high and low cortisol controls are run in Traditional control charts monitor 2 parameters. However, another
statistic, called the percent Coefficient of Variation (%CV) appears when the mean and Variance, Standard Deviation and Coefficient of Variation Calculate the mean, x. To get the standard deviation
we take the square root of the variance. A very comprehensive collection of online calculators and other interactive Coefficient of Variation; Paired Data Sets Statistics -- Enter up to 28 sample
Here we will learn how to calculate Coefficient of Variation with practical 250+ Online Courses | 1000+ Hours | Verifiable Certificates | Lifetime Access
This free online software (calculator) computes the following statistics: absolute range, relative range, variance, standard error, coefficient of variation, mean 24 Apr 2017 In statistics, CV or
coefficient of variation is a measure of the variability of a sample dataset expressed as a percentage of the mean. It is highly recommended that you study these lessons online or in hard copy[1].
Another way to describe the variation of a test is calculate the coefficient of Use our free online calculator in order to calculate the standard deviation, variance, mean, and the coefficient of
variance for the numbers you have given.
Here we will learn how to calculate Coefficient of Variation with practical 250+ Online Courses | 1000+ Hours | Verifiable Certificates | Lifetime Access
This free online software (calculator) computes the following statistics: absolute range, relative range, variance, standard error, coefficient of variation, mean 24 Apr 2017 In statistics, CV or
coefficient of variation is a measure of the variability of a sample dataset expressed as a percentage of the mean.
Variance Calculator | education. Coefficient of Variance (Separate by , comma) Variance calculator is an online free tool to calculate the variation of each
Coefficient of Variation Calculator : The calculator provided in this section can be used to find coefficient of variation for the given data values. It does not give only the coefficient of
variation of the given values and also it gives you mean and standard deviation of the observations you enter. Coefficient of Variation Calculator. Use the Coefficient of Variation Calculator to
compute the sample (matrix) variation coefficient for each column. Important! The result is given as a vector, where the k'th element denotes the variation coefficient for the k'th column.
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Rectangle Breadth Calculator - StudySaga
Rectangle Breadth Calculator
Rectangle Breadth Calculator
Enter the length and area of the rectangle:
Rectangles are one of the most basic and important shapes in geometry. They are two-dimensional figures with four straight sides and four right angles. The breadth of a rectangle is one of its
defining characteristics, and is an important measurement in a variety of applications.
Definition of Breadth:
In a rectangle, the breadth is the shorter of the two sides. It is also known as the width or the shorter dimension. The other side is called the length or the longer dimension.
Calculating Breadth:
To calculate the breadth of a rectangle, you can use the formula:
breadth = area ÷ length
where area is the total area of the rectangle and length is the longer dimension.
Alternatively, you can use the formula:
breadth = perimeter ÷ 2 – length
where perimeter is the total distance around the rectangle.
1. A rectangle has an area of 60 square meters and a length of 10 meters. What is its breadth?
Using the formula, breadth = area ÷ length, we get:
breadth = 60 ÷ 10 = 6 meters
Therefore, the breadth of the rectangle is 6 meters.
2. A rectangle has a perimeter of 24 meters and a length of 8 meters. What is its breadth?
Using the formula, breadth = perimeter ÷ 2 – length, we get:
breadth = 24 ÷ 2 – 8 = 4 meters
Therefore, the breadth of the rectangle is 4 meters.
Knowing the breadth of a rectangle is important in a variety of fields, including construction, engineering, and design. For example, architects need to know the breadth of a room in order to design
furniture that will fit comfortably within it. Builders need to know the breadth of a foundation in order to calculate the amount of materials needed for construction. Engineers need to know the
breadth of a bridge in order to ensure that it can support the weight of vehicles passing over it.
Here are some sample questions and their answers related to the topic of rectangle breadth:
Q: What is the formula for finding the breadth of a rectangle? A: The formula for finding the breadth of a rectangle is: breadth = area / length, where area is the product of length and breadth.
Q: If the area of a rectangle is 60 square units and its length is 12 units, what is the breadth of the rectangle? A: Using the formula for the breadth of a rectangle, we have: breadth = area /
length = 60 / 12 = 5 units. Therefore, the breadth of the rectangle is 5 units.
Q: If the length of a rectangle is twice its breadth and its perimeter is 30 cm, what are its length and breadth? A: Let the breadth of the rectangle be x. Then, the length of the rectangle is 2x.
The perimeter of the rectangle is given by 2(length + breadth), which is equal to 2(2x + x) = 6x. Setting this equal to 30 cm, we have 6x = 30, or x = 5 cm. Therefore, the breadth of the rectangle is
5 cm and its length is 2x = 10 cm.
Q: If the length of a rectangle is 8 cm and its area is 48 square cm, what is its breadth? A: Using the formula for the area of a rectangle, we have: area = length x breadth. Substituting the given
values, we get: 48 = 8 x breadth. Solving for breadth, we get: breadth = 48 / 8 = 6 cm. Therefore, the breadth of the rectangle is 6 cm.
The breadth of a rectangle is an important measurement that is used in a variety of applications. It is the shorter of the two sides and can be calculated using the formulas mentioned above. | {"url":"https://studysaga.in/rectangle-breadth-calculator/","timestamp":"2024-11-10T06:13:17Z","content_type":"text/html","content_length":"116110","record_id":"<urn:uuid:5b2fc289-b084-4723-88c4-00a5b10d3464>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00137.warc.gz"} |
Signal Processing: Audio Basics
1. Harmonic structure of sound
2. Parson code of music
3. Linear time-invariant theory
4. Autocorrelation
5. Noise
6. Chirps
7. DCT compression
8. Discrete Fourier transform
9. filtering
10. convolution
Linear Time-Invariant System
We describe the system with $Y(t) = f(X(t))$, where $X(t)$ is the input, and $Y(t)$ is the output.
1. Linear: $f(a X_1(t) + b X_2(t)) = a f(X_1(t)) + b f(X_2(t))$
2. Time-invariant: input $X(t+\Delta t)$ will produce the shifted signal $Y(t+\Delta t)$.
LTI systems are memory systems, casual, real, and stable. Stable means the output won’t reach infinite if the input is finite. It’s bounded.
Impulse Response
Suppose we have a impulse $X(t) = I(t)$, and output $h(t)$.
Now we have another input $X(t)$, we can ask that what would the output be if we put the input in the same environment as the previous impulse.
$$ $$Y(t) = \int d\tau h(\tau) X(t-\tau).$$ $$
Transfer Function
For the impulse response, the transfer function is obtained through the Laplace transform of the response,
$$ $$\tilde h(s) = \mathscr L_s [ h(t) ].$$ $$
With the response function, we know that the response with some other input that is set in the same environment is
$$ $$\tilde Y(s) = \tilde h(s) \tilde X(s).$$ $$
Planted: by L Ma;
wiki/algorithms/signal-processing-audio Links to:
L Ma (2018). 'Signal Processing: Audio Basics', Datumorphism, 03 April. Available at: https://datumorphism.leima.is/wiki/algorithms/signal-processing-audio/. | {"url":"https://datumorphism.leima.is/wiki/algorithms/signal-processing-audio/?ref=footer","timestamp":"2024-11-02T14:10:32Z","content_type":"text/html","content_length":"112723","record_id":"<urn:uuid:5455d6b1-7bbe-4201-824a-0174e0e539aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00168.warc.gz"} |
How do I evaluate int(1-sinx)/cosx dx? | HIX Tutor
How do I evaluate #int(1-sinx)/cosx dx#?
Answer 1
Well, this one is a little bit tricky... I started with the idea that the result must be a logarithm... So I did a little manipulation to get to a friendlier version of your function by multiplying
and dividing by:
#(1+sin(x))/(1+sin(x))#; so I get:
#int(1-sin(x))/cos(x)*(1+sin(x))/(1+sin(x))dx=# #=int(1-sin^2(x))/(cos(x)(1+sin(x)))dx=# #=int(cos^2(x))/(cos(x)(1+sin(x)))dx=# #=int(cos(x))/((1+sin(x)))dx=# which is a manipulated version of your
original function and we can call it (1). The integral of (1) is indeed a logarithm:
Which derived gives (1) or your original function!!!!!
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Answer 2
To evaluate ( \int \frac{1 - \sin(x)}{\cos(x)} , dx ), follow these steps:
1. Rewrite the integral using the identity ( \frac{1 - \sin(x)}{\cos(x)} = \frac{\cos(x) - \sin(x)\cos(x)}{\cos(x)} ).
2. Simplify the integrand to ( \cos(x) - \sin(x) ).
3. Integrate ( \cos(x) - \sin(x) ) term by term.
4. Evaluate the integral over the given interval, if specified.
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Answer 3
To evaluate ( \int \frac{1 - \sin(x)}{\cos(x)} , dx ), we can use the substitution method.
Let ( u = \cos(x) ). Then, ( du = -\sin(x) , dx ).
Substituting ( u = \cos(x) ), we get:
( \int \frac{1 - \sin(x)}{\cos(x)} , dx = \int \frac{1 - u}{u} , du )
This simplifies to:
( \int \left( \frac{1}{u} - 1 \right) , du )
Now, integrate each term separately:
( = \ln|u| - u + C )
Finally, substitute back ( u = \cos(x) ):
( = \ln|\cos(x)| - \cos(x) + C )
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-do-i-evaluate-int-1-sinx-cosx-dx-8f9afa0e8c","timestamp":"2024-11-04T12:23:28Z","content_type":"text/html","content_length":"575681","record_id":"<urn:uuid:e2000b20-1dcb-45e9-be81-f1f802922b67>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00793.warc.gz"} |
numpy.not_equal(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj]) = <ufunc 'not_equal'>¶
Return (x1 != x2) element-wise.
x1, x2array_like
Input arrays. If x1.shape != x2.shape, they must be broadcastable to a common shape (which becomes the shape of the output).
outndarray, None, or tuple of ndarray and None, optional
A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple
(possible only as a keyword argument) must have length equal to the number of outputs.
wherearray_like, optional
This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original
value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
For other keyword-only arguments, see the ufunc docs.
outndarray or scalar
Output array, element-wise comparison of x1 and x2. Typically of type bool, unless dtype=object is passed. This is a scalar if both x1 and x2 are scalars.
>>> np.not_equal([1.,2.], [1., 3.])
array([False, True])
>>> np.not_equal([1, 2], [[1, 3],[1, 4]])
array([[False, True],
[False, True]]) | {"url":"https://numpy.org/doc/1.18/reference/generated/numpy.not_equal.html","timestamp":"2024-11-10T02:08:29Z","content_type":"text/html","content_length":"11620","record_id":"<urn:uuid:b117a353-2a40-4548-8878-2ae478b69310>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00510.warc.gz"} |
Big Brother 17 Live Feeds Week 8: Monday Daytime Highlights
Power of Veto Ceremony Day kicked off with a whimper except from perhaps Vanessa Rousso who appeared to be yet again paranoid she was about to be renom’d. Now maybe she can relax for a few days.
Liz Nolan prepares her Power of Veto speech for Big Brother – Source: CBS All Access
The Veto Ceremony came and went without much fuss so with the vote apparently already determined and the target secure the house should have a calm week. Yeah, I hope somethings screws that up quick
so they can get back to dancing for us.
Big Brother 17 Live Feed Highlights – Monday, August 17, 2015:
10:00 AM BBT – HGs are up for the day. Meg and James are in no rush to get out of bed.
11:10 AM BBT – Liz is practicing her speech for the Veto. Says one of them will come back so she won’t use the Veto.
12:00 PM BBT – Veto Ceremony is over and results are in. Noms stayed the same.
12:10 PM BBT – Meg thinks Vanessa is to blame for Becky’s target this week. John says Becky’s speech won’t help her chances. They wonder when Vanessa will start her wheelin’ and dealin’ routine
12:30 PM BBT – Vanessa upstairs in HoH room back to her old self now that she hasn’t been nominated. She’s plotting and planning again telling Austin how they need to target John next.
12:40 PM BBT – Vanessa mentions asking DR if she can make deals to split up the winner’s money with her and gambling it for them. Hey, maybe Vanessa could just do that with her own $4 million.
12:50 PM BBT – Vanessa mentions seeing leaves blowing around in the backyard and thinks they’re props from an upcoming comp. They start to discuss and we get Fish.
1:00 PM BBT – Austwins and Vanessa discuss trying to get John to throw HoH next week. Vanessa says she won’t throw the comp this time.
1:30 PM BBT – Becky talks with Liz about the votes. Liz still not letting on that she’ll be evicted this week. Becky points out her feud with Vanessa is good for everyone else if she’s here.
1:40 PM BBT – Austin, after Becky leaves, says Becky made good points, but they’ve got Johnny Mac to go after Vanessa too so they’re not going to be swayed.
1:45 PM BBT – Julia starts suggesting they keep Vanessa next week. She points out it might be better to wait until after the returning Juror twist is over. (What did we tell you yesterday?)
1:50 PM BBT – Liz insists that Vanessa must go next week. She doesn’t like Julia’s idea. (Maybe Liz should have done something about that while she had complete control…)
2:00 PM BBT – Liz worries to Austin that Julia will last longer than her because she’s less of a threat.
2:30 PM BBT – Liz and Julia are in the pool while Johnny Mac and Austin work out.
3:14 PM BBT – Nothing is going on. Steve is telling Becky about a dream he had about being home after the finale but not knowing how the game played out.
3:25 PM BBT – Julia tells Liz she has a gut feeling that Vanessa won’t ever go after them. She again says that whoever comes back into the game will be against them so they need to keep Vanessa
until then.
3:38 PM BBT – Steve and Julia discuss who should go this week. Julia is throwing out the pros and cons of John and Becky. Julia decides that if Becky stays, She’ll go after Liz.
3:40 PM BBT Julia asks Steve if he thinks the HOH will be the log roll (it’s a very specific competition to suspect). Steve shuts her down. He tells her the next endurance competition probably
won’t be until the jury member returns.
4:06 PM BBT Â – Steve tells Austin that he wants Becky to go this week because if Jackie comes back, that would suck for him. Austin tells Steve that he told John that he is 100 percent keeping
him this week because he trusts him that much.
4:14 PM BBT – Steve and Austin say they definitely won’t be throwing the next HOH. They decide that Julia needs to win HOH. They talk about Vanessa is the fifth wheel in their group. They decide
to keep close to John so that they can keep him on their good side and use Vanessa as a shield (since he will go for Vanessa).
Things are definitely looking good for Johnny Mac this week, but for this season, it’s still too early to believe anything these guys are saying. We all know these people enough by now to know the
target will go back and forth at least 15 times between now and Thursday night – especially if Vanessa gets bored or misses her meds.
You can watch all of these Big Brother events using the archives Flashback, the DVR-like feature of this year’s Live Feeds which means it’s always live, even when you missed it! Sign-up now to
get the free trial to watch it all live & uncensored.
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118 Comments
1. 12:40 PM BBT – Vanessa mentions asking DR if she can make deals to split up the winner’s money with her and gambling it for them. – Is it out there now? Do they all know she was a pro poker
□ I wondered the same thing.
□ Isn’t that against the rules to split the winner’s money.
☆ I thought so, maybe that’s why she was asking
☆ I’m sure there are rules against working together to cheat the system but it sounds like she was just asking if when it’s over is it possible for the winner to give her some of their
winnings for her to play with as an investment. I’m more curious about who she said it to. According to the timeline she was talking with Austin in the HoH room at 12:30, said the
gambling thing at 12:40, spoke with Austin again at 12:50, then with the Twins at 1:00. So it was either said to Austin alone or with the three of them (Austwins) together. Either way, it
appears to be “known” now.
☆ The winner can do whatever they want with the money after the game is over… even foolishly give some to Vanessa. But I think you are right that she is asking if she can dangle the carrot
as a strategy. Haven’t people promised each other favourite pieces of jewelry and stuff in the past? How can the BB Brass prevent someone from promising to give another player a cut of
the $$ after the game is over?
☆ That’s not my confusion, Sophie. (Apologies if it was unclear.) I’ve always thought that she’s been keeping her pro-poker identity a secret. Then the timeline says she mentioned to Austin
(and maybe the twins too) that she asked while in the DR if she could make deals with players to ask the winner for some of their dough to use while playing. Regardless of what BB Brass
told her the bigger thing I’m curious about is that it seems her poker playing job is no longer hidden. They know about it?
☆ Well, if anyone falls for that and keeps Vanessa in the hopes that she will double their take, they are even dumber than I thought. I’m sure she’s a fine poker player, but come on! On the
other hand, you gotta hand it to Vanessa. She is pulling out all the stops.
☆ Right. Veteran players were predicting, she’ll be out mid-point of the game, but she’s still there. I still think she’s gonna have a rough time ahead. Villain seemed to stay longer, but
in the end they don’t win. lol
☆ I hope you’re right. But, if it comes down to one of the Austwins or Vanessa, I’m all for Vanessa. Ultimately, I really want JMac to win it.
☆ If Vanessa goes down she will go down swinging,suspect she would blow up everyone’s game at the eviction,in fact that might be one thing that keeps her safe,THEY ALL KNOW IT!
☆ Actually, my husband is a professional bridge player, and he’s seen her play (and associates with others who play with her), and he says she’s a lousy poker player. His comment was, “yes,
she does pretty well along the way, but when she gets to the end and plays with real players, she gets beaten like a rented mule.”
☆ Someone else mentioned it too. I want to say Becky? And Da’vonne said she was worried Vanessa would recognize her from her poker tournaments since Vanessa mentioned she had played at her
casino. I don’t know if they know she’s as big a deal as she was, but I’m pretty sure they know she’s at least somewhat serious as a card player.
☆ Read elsewhere in the thread, TBR. A poster named Staci explained that she (Vanessa) told them a few weeks ago that she gambles on the side or something and Austin has assumed she’s a
bigger gambler than she’s let on.
☆ What they do with the money once the season is over is up to them.( Derrick got Caleb a truck last year I think) but it can not be a deal made during the game or used as a strategy. That
was discussed last year, because someone made a comment and it was immediately shut down, if I remember correctly
☆ It’s in their contract, which extends a few years after the show, that they can’t give another HG part of the winnings by gifting them money. They can buy them a gift, like jewelry, car,
trip, etc., but they can’t make a deal for a gift during the show.
☆ Splitting the winnings is prohibited by the Big Brother rules. They are under contract for a few years after the show is over. While they can’t give another HG money, they can buy them
There was a big scandal when Evel Dick paid for his daughter’s college after he won. Later he revealed that he bought her a car and took her to Europe, too. Since then, they’ve been able
to buy gifts for other HGs, but they can’t make deals for the gifts during the show.
□ She didn’t tell anyone she’s a pro poker player. She’s jsut mentioned she enjoys gambling sometimes. And Austin assumes that she gambles a lot of money.
☆ THANKS, Staci! That answered it perfectly! (Thanks so much!)
☆ They talked about it a couple of weeks ago, it’s somewhere on these feed updates.
□ I had the same question. It came out weeks ago, but no one seems to know how/when.
2. Goodbye Becky :(
3. Boring week.
4. The HGs need a pandora’s box, or a rewind of this week..something unexpected and to stir the pot a little.
□ I agree…that promised “weekly” twist needs to come back now….maybe all of the promised twists that haven’t been used to date….
☆ Never say never. Could be time for a BB Takeover, just when they least expect it.
☆ The weekly take over twist was only used 3 times that I can recall.
□ Personally, I’d prefer if BB didn’t just blatantly screw someone out of the chance to win the game. If the Goblins and Jmac are too stupid to take out the right people, they don’t deserve to
win, period.
☆ This is supposed to be expect the unexpected..not every pandora’s box has been a bad thing, there have been luxury comps in the past that screwed with no ones game…these things happen all
the time.
5. What a bunch of wusses!! Grrrrrrrr! They think Julia should win HOH next time? How about Steve or Johnny Mac trying to win it and showing that they actually have more testosterone than the
average four-year-old girl?!! Still desperately wanting John or Meg to be HOH – if they don’t, they’re doomed. The only possible saving grace for this season depends on who the returning evicted
HG is, although they usually get voted out as quickly as they come back in. I shake my head every day at the twins and Austin getting this far – why aren’t they targets – must be using the
Romulan Cloaking Device (yes, I’m a Star Trek nerd, but BB still comes first!!)
6. So, if Vanessa is talking about gambling money for the other house guests, has she told them she’s a poker player?
7. I hope johnny mac stays and wins it all
8. Julia is right. I think for their game, they should keep Vanessa until the Jury twist is over and a HG comes back in. She could be voted out and stay out for good. If they wait too long though,
she may vote them out before any of that happens.
9. Please BB let the fans get the POV…this is getting stale
□ The POV is not enough. We also need the power to renom.
10. The twins are so simpleminded, In one second they can be like “I love Jmac,” seconds later, “I hate him,” “I love Vanessa,” now it’s the opposite. They also like to rip people apart….their
attitude is a turn off.
11. If it ends up being the Austwins and Steve as final 4, this show is going to be awful to watch. It pretty much started this week, and it’s getting worse everyday.
□ I know conspiracy theories are boring but based on a lot of the comments left on this site over the last few days it feels like Production might have to throw in a creative monkey wrench
pretty soon in order to at least provide the possibility of a shake up. (But with this group of specific HGs it might just be time to realize it’s a bad year all the way around thanks to bad
casting choices and the complete lack of twists.)
☆ Do you think Production reads this forum?
☆ I don’t know. I’m not saying Production takes notes from fan sites. I’m saying that based on all of the similar opinions shared they must have a general idea that fans seem a bit bored. I
bet they look at a bunch of different sites just to see what the general vibe is. But then again, maybe they couldn’t care less.
☆ I’m just curious. I mean, Matt gets a line to evictees, so clearly he and the site are well known by BB Brass, so maybe they do read fan’s reactions. In which case, Hey, BB, more JMac,
less Austwins!
☆ I’m sure they do. Stats/Data gathering etc.This is BB community. BBN get shout outs from players all the time…I make comments sometimes directed to Production and players, Players read
BBN, so when they get out, they know how crappy their game was.
☆ Cyril, can you even imagine what it would be like to sit down in front of a computer and start reading about yourself from Day 1 when your time there was done? Depending on who you are
you’d probably need some serious counseling when it was over. – “Cyril is a fat, ugly rich guy who doesn’t even BELONG in that house because he’s a floater idiot who drinks too much and
is super slutty!”
☆ lol..Shelly Moore, one of the most hated player didn’t go online for a year..she had death threats
☆ I think BB has people that read every word on fan sites and reports back to production every day. I can’t imagine Vanessa still being in the house out of pure stupidity. Its not that the
HGs are all that stupid, it’s production thinks all the fans are that stupid for buying in to this load of BS. You gotta k ow that BB has input in the game,.
☆ If they really read and pay attention to fan sites, they would love to keep Van Van longer coz she’s a hot topic … the Judases, not really. They are just bad.
☆ You can Mac me anytime … too cheesy?
☆ Lol
☆ Tell JMac to do something to take control of the game. As long as Austwins are in power, there’s not much production can do. And as long as JMac does nothing, there’s not much they can do
to make someone who’s not interesting look interesting.
☆ Well, look at what happened with Battle of the Block and the Takeover Twist. There were never too many positive comments about either one on any of the comment boards, and now they’re
both gone. Maybe just coincidence, but maybe insiders who read these comments let Production know what the audience thought.
☆ I am all for a pandora’s box or something fun
☆ GREAT idea, queenbrat!
☆ Careful what you wish for!
☆ Conspiracy theories like “show is rigged” (pre-determined winner) and no evidence to back it up? That’s boring to me. ..but I’m all for Pandora, Coup d’tat’ Diamond Power..etc.etc.When do
they execute the “twist” is the question.
☆ Like when Vampire Dentist is evicted they send him back right in coz he’s the most favorite player …
Would love to see how Van Van reacts to that!
☆ Maybe not rigged BUT JUMPED THE SHARK big time. And maybe they KNOW it with the GRONK SHARK? There are NO NEW ideas. NO old ones either. Where is Team America? Where are game rewinds?
Where are the so-called TWISTS they promised us at the beginning? Where is ANYTHING?
It’s like the whole thing is falling apart. They’ve been reduced to just giving them alcohol at the beginning of BBAD lately so we can watch THEM be the fish tank.
It’s AWFUL. Worst season I’ve seen in years, maybe ever.
☆ I just checked their ratings. Their viewership has been over 6 million to 7/Wed/Thursday. It’s been up, so CBS is happy
☆ Zingbot usually happens this week historically.
☆ All I know is, I can’t wait wait for Thursday….speaking for my self….(oh I don’t watch feeds all the time)
☆ The outlook for “my” favorites is grim but I don’t think it’s the worst season. Actually, I think it’s a rather good season because you just don’t know who’s going home. We think we do
but we don’t every time. We get the popcorn ready to watch a feel good movie: turns out to be a disaster flick. Lol. But it’s still “good”. Not good for my favorites but good. And I’m NOT
looking forward to Austwins remaining in tact to F5. But I can’t not watch. :)
Last year watching Derrick was painful to watch. Not only because I was holding out hope for Donnie but more like pounding and yelling outside a glass window to tell the people the killer
is behind you! – And they don’t hear you or see you and you can’t help them!!! Ok – that was a bit dramatic.
I have nothing but respect for Derrick. He was AMAZING and watching him do his thing was, in itself, something to behold. But last season didn’t have the organ music this one does (“Tune
in Thursday to see if Vanessa goes home” duh-duh-duh “Or if anyone else does!” duh-duh-duhhhhh). Lol
☆ This is the time I wish they would save both Becky and Johnny Mac with a twist. James and Meg they can evict if they want.
☆ We need the Tim Gunn save!
□ Preach
□ If Vampire Dentist survives this week he’ll go far I think. After Van Van is gone, the entire house + returning juror will definitely target the Judases.
☆ I dont think so they going after each other instead of twins steve and jmac have deal with austwins and james and meg do so the 4some will target each other instead of 3 some hope whoever
wins hoh knocks out a twin
12. I know a good twist for next week…let the winner of POV name the renom.
□ Wait wait … how if Van Van wins the PoV? It’s a good idea as long as she’s not in it.
13. Honestly if Liz wanted Vanessa gone next week,she should have used the veto on one of the noms and renom Vanessa and kick her out but again these people are denser than a brick wall !
14. If Becky it’s evicted and return I really wanna see how she evict Liz, Julia, Austin, Steve & Vanessa.
15. Out of curiosity, anyone know why exactly Vanessa takes meds for?
□ I have seen many posts wondering the same thing, Vanessa has previously mentioned ADHD but no one really knows.
☆ She’s probably on Adderall, which will help her focus.
☆ The only problem I can see with that is she takes a second medication at 5:00 pm. That is quite late to be taking Adderall.
☆ They stay up till 5:00 am!
☆ Yes but she also takes a morning dose!
☆ adults dont have adhd
☆ Wrong. Adults can have and do have adhd.
☆ Yeah because Neurochemical imbalances go away when you turn 18/21 whatever you consider to be an adult, kindly pull your head out of your ass
☆ I have ADHD.
☆ I outgrew it. But I doubt I ever had it al. So I’m skeptical if it even exists.
16. Julia is making perfect sense to Liz about Vanessa because she understands that is about VOTES and the person coming back is a vote against them. Liz is falling into the familiar trap of letting
personal feelings overrule good game strategy.
17. PLEASE, ZINGBOT 3000! We need you!
□ Tbh, they are probably gonna get a lot of zings when they get home.
☆ FromBadtoWorst? Were you looking in a mirror and your breath bounced back on you? – ZING! (Please, Zingy! You’re so much better! Help us out this week! Those HGs are yours for the
☆ Well – they might get Kathy Griffin back.
☆ Good Lord…If it was a “Zingbot/Griffin Zing off”? With both of them double blasting the HGs? Fantastic!
□ omg! Watch out Zingbot will have no mercy with these group, (plenty of materials) especially Judas…can’t wait!
18. I used to think Julia was the twin with a brain, but now I have my doubts. I’m trying to rationalize why they think it’s better to have Vanessa in the BB house rather than in the jury house. If
she’s in the BB house (at the time a jury member comes back), there’s a 100% chance she’ll be in the BB house. If she’s in the jury house when those players compete to return, there’s just a 25%
chance (or less, depending when that competition is held) she’ll come back into the BB house. Why can’t Julia understand those odds?
□ I don’t believe that that’s the dilemma at hand. The chances or odds are not the issue at hand. The issue at hand is keeping a bigger target for others to go after.
☆ OK, I get that, but that kind of thinking can’t go on forever. Vanessa can get all the way to F2 if everyone’s afraid to get her out just because they want to keep her as a shield. That’s
just really risky.
☆ I hear you on that. Vanessa in the final 2 should be everyone’s dream at this point because she can’t win. That’s why I think it doesn’t make sense to go after her. Everyone should be
fighting to be her final two, not fighting to get her evicted.
☆ TBR78 and Koko you both make great points! But TBR78, I know they’re all really annoyed and angered by her right now but if she survives long enough to make it to F2 do you think they’d
keep the title away from her out of spite? She appears (at this time) to be the only one who’s actually done anything to earn it.
☆ I think if she makes it to final 2 she could very likely win and should win
☆ Maybe, but I tend to think people base their votes on emotion. I only see Steve and Austin voting for the best player.
☆ Might be a bit harder to do that though. If the F2 are Vanessa and James…Maybe because he’s made at least one big move and has won comps. But if it’s Vanessa and anyone else don’t you
think they might be torn apart for voting against her out of spite rather than voting for the best player? (You might be 100% right! I’m just asking.)
☆ They’re all emotional and very angry in the beginning, but historically, BB Juries award the HG who played the best game. Mid-game I will give it to Vanessa, and that’s without hearing
her opening statement,…and we all know she can talk.
☆ A few of these people really hate her. I think in a few of their minds, they’d vote for the cop at the Mckinney pool party over Vanessa.
19. The show is painful to watch now that idiots are running the game. Steve may indeed be a super fan, but he’s absolutely clueless when it comes to reading people. He’s NOTHING like Ian Terry!
Vanessa isn’t smart; but she can be intimidating and a master emotional manipulator who will cry & whine to ensure she gets her way. The Austwins have floated happily until recently when Liz
started winning comps. James & Meg deserve to go home (although for the sake of strategy, I hope they don’t) for being dumb enough to blow up Becky’s plan when she was HOH. I hope John stays and
wins HOH and puts up the twins, with the intention of back-dooring Vanessa. Once Vanessa is gone, mindless Steve will be loyal to John, and they both can form an alliance with Meg & James. Then
they need to win the following HOH’s and take out the Austwins. My hope is that Becky comes back into the game, and helps them clean house. With some luck, I’m hoping she and John will be the
final two.
□ I’m pretty neutral about Becky with John being my fave, but I would totally be down with them as f2.
20. Need a game rewind or SOMETHING, BAD.
It’s like they had a plan for this season then realized it was an even
dumber plan than usual and was abandoned– and now they have nothing.
The whole show feels like it it “coasting”
21. I heard rumors of Vanessa offering other housesguests their share of weekly income and playing it in poker to increase their pay day, which brings me to the conclusion that this season is
becoming a farce and if Vanessa does end up winning, it just shows how it was designed for her to win in the first place, she isn’t even that good of a player, it simply looks that way with so
many weaker players in the cast. I just hope production can save face and she ends up leaving soon, because it sounds like some shady business is going down in the BB house, across other forums
as well.
□ YES something is VERY MUCH not right. Lots of FISH too on the live feeds this year… even more than usual. Lots of “Jeff highlights” as well.
☆ I’ve noticed that as well and if I am not mistaken Vanessa was mentioning something to Austin about it earlier. then feeds went dry for a bit.
□ Vanessa is playing a very good game when you have dummies like James, Jackie and Meg playing against her. It is a no contest. Add to that, she has a loyal alliance which is why all of them
have managed to survive outside of the eviction of Shelli and Clay.
☆ Totally agree!,people just hate to admit the truth,Vanessa is playing the game,the others are just sitting around eating.
□ I heard rumors that Vanessa is getting a “Poker” show on CBS…hmmm!
22. At this point I must wonder, does ANYONE in that house deserve to win half of a MILLION dollars?
□ Zingbot traditionally shows up this week or next.
□ Just not buying that,every year people complain this is the worst season ever,heard that same tune last year,,actually been quite interesting as a whole..this is not “I Claudius” but BB,one
has to accept the show for what it is and go with the flow.
23. They need the power that lets 2 people be removed form the block Jeff got it and Jesse got evicted I can’t spell it lol but its an existing power..
24. It’s obvious Vanessa believes production is just as guilable as the rest of the house guest. After all they did come up with the amazing twin twist that only took the house a couple days to
figure out
25. Liz, and Austin = no guts!
26. Based on the conversations that Vanessa is having with Steve, I get the impression she wants to target the Austwins. I could be wrong but she won’t say who it is she wants to target, at least not
to Steve. If it was Meg or James, I think she would say it.
I know Vanessa has been playing the numbers game all season. She has to see that the Austwins are a particularly strong alliance since there is nothing she can do to divide the twins and Liz
seems like she actually is into Austin, at least a little.
If it comes down to F4 being Austwins and Vanessa she knows there is no chance she wins anything. In fact, if its F3 with the twins there is no way she is F2 if one of the twins wins the final
Vanessa needs allies to take out the twins and I am thinking she wants Steve and JMac to help her, or possibly Shelli if Shelli comes back in the house.
□ Vanessa is just biding her time after all the threats from the other side are evicted. With the lesser numbers, she might go after Austin, Liz, Julia when she wins HOH. That is all she needs
to break up that trio and she still has Steve in her back pocket.
☆ Would be a smart move indeed,strike before they have time to do the same.
27. Vanessa has alot of armour (Austwins) sometimes to penetrate you have to brake the armour…..until that happen it will be hard to get a good shot…..they will keep her because it protects them so
you get rid of them and there will be no one to protect her
28. I like Johnny Mac. I want him to win. BUT….If the HG were smart they’d vote him out. (Maybe wait after the juror twist), because Final 2 Johnny Mac versus any remaining HG, Johnny would win.
29. I realize as much as I hate Vanessa, I think she is a good player ((or was up until the last 2 weeks), but I think Steve is worse. He has no allegiance and is SLIMY as hell. He is the epitome of
a “Floater”. Sorry Steve fans, but it’s the truth. | {"url":"https://bigbrothernetwork.com/big-brother-17-live-feeds-highlights-daytime-2015-08-17/","timestamp":"2024-11-04T02:32:16Z","content_type":"text/html","content_length":"471137","record_id":"<urn:uuid:8fdcb1f4-130e-442f-9835-1ac18dbe37bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00151.warc.gz"} |
3D-Structures Orientation in OpenGL - Student Projects Live
3D-Structures Orientation in OpenGL – In OpenGL, 3D structures (such as cubes, spheres, etc.) are typically represented by a collection of vertices that define the shape of the object. These vertices
are typically stored in a vertex buffer object (VBO), which is a piece of memory on the graphics card that stores the vertex data.
To orient a 3D structure in OpenGL, you can use the model-view matrix, which is a 4×4 matrix that transforms the vertices of the object from their local coordinate system to the world coordinate
system. The model-view matrix can be modified using various OpenGL functions such as glTranslate, glRotate, and glScale to specify translations, rotations, and scaling operations to be applied to the
3D-Structures Orientation in OpenGL software
OpenGL (for “Open Graphics Library”) is a software interface to graphics hardware. The interface consists of a set of several hundred procedures and functions that allow a programmer to specify the
objects and operations involved in producing high-quality graphical images, specifically color images of 3-dimensional objects. OpenGL is a standard specification defining a cross-language,
cross-platform API for writing applications that produce 2D and 3D computer graphics.
In this mini-project we included most of the computer graphics concepts. One is the 3D-Structures; we made the structures using built-in functions which will give sphere, cylinder, cube & etc. In
project rotation, translation and scaling are used to rotate the structures and to give the movement to the other objects.
For any software the good user interaction is very important. In this project key board and mouse interaction will be given by using the in-built openGL function. The key keyboard interaction will be
given to control the structures. Then menu will be given by right clicking in mouse. In mouse it’s provided with different option to change the color of the structures and to color it.
In this mini-project the OpenGL functions are used to demonstrate the concepts. Most of the objects are created from the basic primitives. And some inbuilt object also used in this project. The
object used in this project is giving the control over the 3D structures to the user.
For example, the following code uses glTranslate and glRotate to orient a cube in OpenGL:
// Set the model-view matrix to the identity matrix
// Translate the cube along the x-axis
glTranslatef(1.0, 0.0, 0.0);
// Rotate the cube around the y-axis
glRotatef(45.0, 0.0, 1.0, 0.0);
// Draw the cube
This code first sets the model-view matrix to the identity matrix, which resets any previous transformations applied to the object. It then translates the cube along the x-axis and rotates it around
the y-axis by 45 degrees. Finally, it calls a function drawCube to render the cube using the transformed vertex positions.
I hope this helps! Let me know if you have any questions.
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://www.studentprojects.live/otherprojects/c-and-c/c-project/3d-structures-orientation-opengl/","timestamp":"2024-11-13T14:23:52Z","content_type":"text/html","content_length":"110671","record_id":"<urn:uuid:478a6eb9-3e93-47f2-b23e-5beccb656a09>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00350.warc.gz"} |
Is there a 5d shape?
Is there a 5d shape?
In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces. It can also be called a
regular deca-5-tope or decateron, being a 5-dimensional polytope constructed from 10 regular facets.
What are some good things about homework?
Homework teaches students how to set priorities. Homework helps teachers determine how well the lessons are being understood by their students. Homework teaches students how to problem solve.
Homework gives student another opportunity to review class material.
What is a 4 dimensional shape?
Four-dimensional geometry is Euclidean geometry extended into one additional dimension. The prefix “hyper-” is usually used to refer to the four- (and higher-) dimensional analogs of
three-dimensional objects, e.g., hypercube, hyperplane, hypersphere.
Are humans 3 dimensional beings?
Research using virtual reality finds that humans, in spite of living in a three-dimensional world, can, without special practice, make spatial judgments about line segments, embedded in
four-dimensional space, based on their length (one dimensional) and the angle (two dimensional) between them.
What are the three dimensional space?
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e.,
point). This is the informal meaning of the term dimension.
How did time start?
According to the general theory of relativity, space, or the universe, emerged in the Big Bang some 13.7 billion years ago. “In the theory of relativity, the concept of time begins with the Big Bang
the same way as parallels of latitude begin at the North Pole.
Why can homework be good?
Homework reinforces skills, concepts and information learned in class. Homework prepares students for upcoming class topics. Homework teaches students to work independently and develop
self-discipline. Homework encourages students to take initiative and responsibility for completing a task.
Are humans 2 dimensional?
We are 3 dimensional, we live in 3D world. we have 2D vision. Image formation on the retina is 2D. However, the brain creates perception of 3D image.
What is a 7 dimensional shape?
7-polytope The most studied are the regular polytopes, of which there are only three in seven dimensions: the 7-simplex, 7-cube, and 7-orthoplex. A wider family are the uniform 7-polytopes,
constructed from fundamental symmetry domains of reflection, each domain defined by a Coxeter group.
Are humans 3D or 4D?
The 3D volumetric structure or form of human facial features contains spatial dimensions of breadth, height and width, combined with a unique surface pattern. The 4D temporal pattern of the human
face encompasses all dynamic movement and changes to this 3D spatial form that evolve with time.
What is the 26 dimension?
The 26 dimensions of Closed Unoriented Bosonic String Theory are interpreted as the 26 dimensions of the traceless Jordan algebra J3(O)o of 3×3 Octonionic matrices, with each of the 3 Octonionic
dimenisons of J3(O)o having the following physical interpretation: 4-dimensional physical spacetime plus 4-dimensional … | {"url":"https://www.titcoins.biz/blog/is-there-a-5d-shape/","timestamp":"2024-11-13T13:11:11Z","content_type":"text/html","content_length":"50446","record_id":"<urn:uuid:878cf6a8-9b83-4513-a4d4-3d58be766f5a>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00001.warc.gz"} |
Mortgage Borrowing Calculator
zheniya.ru Mortgage Borrowing Calculator
Find out how much you're likely to be able to borrow on your income with Money Saving Expert's mortgage calculator. Check out the web's best free mortgage calculator to save money on your home loan
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affordability calculator. Get an estimated home price and monthly mortgage payment based on your income, monthly debt, down payment, and location.
Use How Much Can I Borrow calculator to know your borrowing capacity to pay for your mortgage, personal or home loan based on your income & expenditure. Use our mortgage borrowing calculator to see
how much money you can borrow. Simply enter a few details and take the first step to getting your new home. Use our free mortgage affordability calculator to estimate how much house you can afford
based on your monthly income, expenses and specified mortgage rate. Breadcrumb Use this calculator to help estimate how much of a home loan you can afford based on your income and current debt. For
example: A $, home loan ÷ $, valuation = x (for a percentage) = 83% LVR. When working out your LVR, remember to base it on the bank's. Our calculator will show you what you can expect to pay back
each month based on the value of your house, deposit, and interest rates. These home affordability calculator results are based on your debt-to-income ratio (DTI). Industry standards suggest your
total debt should be 36% of your. Calculating your home loan borrowing power can help you determine your overall budget. You can use a borrowing power calculator that will determine how much you. How
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loan in under two minutes. Compare home buying options.
This mortgage eligibility calculator can help estimate your borrowing power. Input a variety of rate, term and down payment scenarios to compare different. Our mortgage affordability calculator helps
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The amount of cash a borrower pays. Use our mortgage calculator to get a rough idea of what you could borrow - in just minutes. To fill it in, you'll need to know.
Lenders will likely lend you no more than 80% of your home's current value. To calculate your home's usable equity, take 80% of the value of your property minus. To help you zero in on a housing
price range, we've built a 'How Much House Can I Afford' calculator to help you start exploring the possibilities. Estimate your borrowing capacity with Commbank's borrowing power calculator. Make
informed home buying decisions and plan your finances better! This calculator gives you an idea of what you could expect to pay each month. You'll find your mortgage balance, term and current
interest rate in your hub. Savings. How much could I save? Borrowing Power. How much can I borrow? Repayments. How much could it cost?
Use this calculator for basic calculations of common loan types such as mortgages, auto loans, student loans, or personal loans. Interest: How much you pay in interest charges each month, which are
the costs associated with borrowing money. Property taxes: Our home loan calculator divides. Borrowing power calculator - How much can I borrow? This calculator estimates your borrowing power based
on your income, financial commitments and loan details. HSBCs mortgage calculator can help determine how much you can borrow, how to calculate mortgage payments, and if it would be better to
refinance. Use The Finance Company's mortgage borrowing capacity calculator to estimate how much you can borrow for your dream home. Use it now! How much can I borrow? · You may qualify for a loan
amount ranging from $, (conservative) to $, (aggressive) · Do Not Sell My Personal Information.
Financial Crisis 2008 Causes And Effects | Bolt Cad | {"url":"https://zheniya.ru/news/mortgage-borrowing-calculator.php","timestamp":"2024-11-01T22:55:33Z","content_type":"text/html","content_length":"11286","record_id":"<urn:uuid:c04cabbb-3cd0-47cd-86d1-e120ac96ed66>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00017.warc.gz"} |
Solver for internal combustion engines.
Original source file engineFoam.C
Combusting RANS code using the b-Xi two-equation model. Xi may be obtained by either the solution of the Xi transport equation or from an algebraic exression. Both approaches are based on Gulder's
flame speed correlation which has been shown to be appropriate by comparison with the results from the spectral model.
Strain effects are encorporated directly into the Xi equation but not in the algebraic approximation. Further work need to be done on this issue, particularly regarding the enhanced removal rate
caused by flame compression. Analysis using results of the spectral model will be required.
For cases involving very lean Propane flames or other flames which are very strain-sensitive, a transport equation for the laminar flame speed is present. This equation is derived using heuristic
arguments involving the strain time scale and the strain-rate at extinction. the transport velocity is the same as that for the Xi equation.
Definition in file engineFoam.C. | {"url":"https://cpp.openfoam.org/v4/engineFoam_8C.html","timestamp":"2024-11-01T22:27:05Z","content_type":"application/xhtml+xml","content_length":"6799","record_id":"<urn:uuid:970a94c7-4fe4-4a41-85b0-e37197adb9b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00284.warc.gz"} |
Solving Exponential And Logarithmic Equations Multiple Choice Worksheet
Mathematics, particularly multiplication, forms the keystone of numerous academic disciplines and real-world applications. Yet, for several learners, grasping multiplication can position a challenge.
To address this obstacle, teachers and moms and dads have actually accepted an effective tool: Solving Exponential And Logarithmic Equations Multiple Choice Worksheet.
Intro to Solving Exponential And Logarithmic Equations Multiple Choice Worksheet
Solving Exponential And Logarithmic Equations Multiple Choice Worksheet
Solving Exponential And Logarithmic Equations Multiple Choice Worksheet -
Review Sheet Exponential and Logorithmic Functions Date Period Expand each logarithm 1 log u2 v 3 2 log 6 u4v4 3 log 5 3 8 7 11 4 log 4 u6v5 5 log 3 x4 y 3
Multiple choice Choose the one alternative that best completes the statement or answers the question Solve the equation by expressing each side as a power of the same base and then
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12 Exponential And Logarithmic Equations Worksheet Worksheeto
12 Exponential And Logarithmic Equations Worksheet Worksheeto
Solve each of the following equations leaving your final answers as expressions involving natural logarithms in their simplest form a e 4x b e 92y c 2e 1 9 z d 4e 7 572w
E Solve log equations by rewriting in exponential form Exercise PageIndex 5 bigstar For the following exercises use the definition of a logarithm to
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Logarithm Worksheet With Answers TUTORE ORG Master Of Documents
Logarithm Worksheet With Answers TUTORE ORG Master Of Documents
Multiple Choice Identify the choice that best completes the statement or answers the question 1 Solve the equation 2 Find the approximate value of x that makes the following
Create your own worksheets like this one with Infinite Algebra 2 Free trial available at KutaSoftware
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systems provide diverse and accessible multiplication technique, supplementing standard worksheets. Personalizing Worksheets for Various Understanding Styles Visual Learners Visual help and layouts
help comprehension for learners inclined toward visual understanding. Auditory Learners Spoken multiplication issues or mnemonics deal with students that comprehend concepts with acoustic means.
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How To Solve Exponential Equations Using Logarithms Slide Share
13 Best Images Of Algebra Solving Inequalities Worksheet Key Two Step Inequalities Worksheets
Check more of Solving Exponential And Logarithmic Equations Multiple Choice Worksheet below
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U4 N4 Solving Exponential Logarithmic Equations Interactive Worksheet By Amy Conine Wizer me
Exponential Logarithmic Equations YouTube
Exponential And Logarithmic Equations City University Of New York
Multiple choice Choose the one alternative that best completes the statement or answers the question Solve the equation by expressing each side as a power of the same base and then
EXPONENTIAL amp LOGARITHMIC EQUATIONS Mt San Antonio
https://www.mtsac.edu › marcs › worksheet › general
EXPONENTIAL LOGARITHMIC EQUATIONS Solve each equation Give an exact solution log x 49 2 2 3 1 x 4 5 22
Multiple choice Choose the one alternative that best completes the statement or answers the question Solve the equation by expressing each side as a power of the same base and then
EXPONENTIAL LOGARITHMIC EQUATIONS Solve each equation Give an exact solution log x 49 2 2 3 1 x 4 5 22
Solving Exponential Equations With Logarithms Worksheet Answers Db excel
50 Solving Logarithmic Equations Worksheet
U4 N4 Solving Exponential Logarithmic Equations Interactive Worksheet By Amy Conine Wizer me
Exponential Logarithmic Equations YouTube
Solving Exponential And Logarithmic Equations Worksheet Answers Kuta Software Kidsworksheetfun
Rewrite The Logarithmic Equation In Exponential Form Ln E 4 Tessshebaylo
Rewrite The Logarithmic Equation In Exponential Form Ln E 4 Tessshebaylo
Solving Logarithmic And Exponential Equations Worksheet Answers Tessshebaylo
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Civil Engineer Objective Questions - Engineering Economics (Section-9)
Civil engineer objective questions – Engineering Economics (Section-9)
76. The Saudi Arabian Oil Refinery developed an oil well which is estimated to contain 5,000,000 barrels of oil at an initial cost of $ 50,000,000. What is the depletion charge during the year where
it produces half million barrels of oil? Use or Factor method in computing depletion?
a) $ 5,000,000.00
b) $ 5,010,000.00
c) $ 5,025,000.00
d) $ 5,050,000.00
78. A manufacturing firm maintains one product assembly line to produce signal generators. Weekly demand for the generators is 35 units. The line operates for 7 hours per day, 5 days per week. What
is the maximum production time per unit in hours required of the line to meet the demand?
a) 1.0 hour per unit
b) 1.2 hours per unit
c) 1.4 hours per unit
d) 1.6 hour per unit
79. The deliberate lowering of the price of a nation’s currency in terms of the accepted standard (Gold, American dollar or the British pound) is known as ?
a) Currency appreciation
b) Currency depreciation
c) Currency devaluation
d) Currency float
80. Capitalization cost of any structure or property is computed by which formula?
a) First cost + interest of first cost
b) Annual cost – interest of first cost
c) First cost + cost of perpetual maintenance
d) First cost + salvage value
81. What is a stock of a product which is held by a trade body or government as a means of regulating the price of that product?
a) Stock pile
b) Hoard stock
c) Buffer Stock
d) Withheld Stock
82. The ability to convert assets to cash quickly is known as ?
a) Solvency
b) Leverage
c) Insolvency
d) Liquidity
83. What is the basic accounting equation
a) Assets= liability + owner’s equity
b) Liability= assets + owner’s equity
c) Owner’s equity=assets + liability
d) Owner’s equity=liability – assets
84. The financial health of the company is measured in terms of ?
a) Liquidity
b) Solvency
c) Relative risk
d) All of the above
85.”When one of the factors of production is fixed in quantity or is difficult to increase, increasing the other factors of production will result in a less than proportionate increase in output”.
This statement is known as the
a) Law of diminishing return
b) Law of supply
c) Law of demand
d) Law of supply and demand
86.”Under conditions of perfect competition, the price at which any given product will be supplied and purchased is the price that will result in the supply and the demand being equal.” This
statement is known as the
a) Law of diminishing return
b) Law of supply
c) Law of demand
d) Law of supply and demand
87. A is a market situation where economies of scale are so significant that cost are only minimized when the entire output of an industry is supplied by a single producer so that the
supply costs are lower under monopoly that under perfect competition?
a) Perfect monopoly
b) Bilateral monopoly
c) Natural monopoly
d) Ordinary monopoly
88. What is defines as the analysis and evaluation of the monetary consequences by using the theories and principles of economics to engineering applications, designs and projects?
a) Economic Analysis
b) Engineering cost analysis
c) Engineering economy
d) Design cost analysis
89. First Benchmark Publishing’s gross margin is 50% of sales. The operating costs of the publishing are estimated at 15% of sales. If the company is within the 40% tax bracket, determine the percent
of sales is their profit after taxes?
a) 21%
b) 20%
c) 19%
d) 18%
90. A farmer selling eggs at 50 pesos a dozen gains 20%. If he sells eggs at the same price after the costs of the eggs rises by 12.5%, how much will be his new gain in percent?
a) 6.89%
b) 6.65%
c) 6.58%
d) 6.12%
91. A feasibility study shows that a fixed capital investment of P 10,000,000 is required for a proposed construction firm and an estimated working capital of P2,000,000. Annual depreciation is
estimated to be 10% of the fixed capital investment. Determine the rate of return on the total investment if the annual profit is P 3,500,000?
a) 28.33%
b) 29.17%
c) 30.12%
d) 30.78%
92. The monthly demand for ice can being manufactured by Mr. Camus is 3200 pieces. With a manual operated guillotine, the unit cutting cost is P 25.00. An electrically operated hydraulic guillotine
was offered to Mr. Camus at a price of P 275,000.00 and which cuts by 30% the unit cutting cost. Disregarding the cost of money, how many months will Mr. Camus be able to recover the cost of the
machine if he decides to buy now?
a) 10 months
b) 11 months
c) 12 months
d) 13 months
93. Engr. Trinidad loans from a loan firm an amount of P 100,000 with a rate of simple interest of 20% but the interest was deducted from the loan at the time the money was borrowed. If at the end of
one year, she has to pay the full amount of P 100,000, what is the actual rate of interest?
a) 23.5%
b) 24.7%
c) 25.0%
d) 25.8%
94. Mr. Bacani borrowed money from the bank. He received from the bank P 1,842 and promised to repay P 2,000 at the end of 10 months. Determine the rate of simply interest?
a) 12.19%
b) 12.03%
c) 11.54%
d) 10.29%
95. A collage freshman borrowed P 2,000 from a bank for his tuition fee and promised to pay the amount for one year. He received only the amount of P 1,920 after the bank collected the advance
interest of P 80.00. What was the rate of discount?
a) 3.67%
b) 4.00%
c) 4.15%
d) 4.25%
96. Which of these gives the lowest effective rate of interest?
a) 12.35% compounded annually
b) 11.90% compounded annually
c) 12.20% compounded annually
d) 11.60% compounded annually
97. How long will it take money to double itself if invested at 5% compounded annually?
a) 13.7 years
b) 14.7 years
c) 14.2 years
d) 15.3 years
98. By the condition of a will, the sum of P 20,000 is left to a girl to be held in trust fund by her guardian until it amount to P 50,000. When will the girl receive the money if fund invested at 8%
compounded quarterly?
b) 11.46 years
c) 11.57 years
d) 11.87 years
99. Aggregation of individuals formed for the purpose of conducting a business and recognized by law as a fictitious person is called
a) Partnership
b) Investors
c) Corporation
d) Stockholders
100. What is the effective rate corresponding to 18% compounded daily? Taken 1 year is equal to 360 days ?
a) 19.41 %
b) 19.44 %
c) 19.31 %
d) 19.72 % | {"url":"https://www.civilconcept.com/civil-engineer-objective-questions-engineering-economics-section-9/","timestamp":"2024-11-09T19:17:55Z","content_type":"text/html","content_length":"87996","record_id":"<urn:uuid:a690bceb-5b84-46d8-80b6-8f7f8096cedf>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00243.warc.gz"} |
Theoretical description of triplet silylenes evolved from H <sub>2</sub>Si=Si
Computations describe the dependence of the H[2]M=Si triplet electronic structure on the α-substituent. Whereas silylidenes H [2]C=Si and H[2]Si=Si benefit from a π^1p ^1 triplet state, the
electronegative nitrogen of HN=Si prefers an n^1p^1 triplet. CCSD(T) and B3LYP calculations predict R[2]Si=Si triplet silylenes are stabilized by π-donor/σ- acceptor R substituents which compensate
for electron deficiency in the singly occupied π orbital of the π(1)p(1) triplet state. (NH[2]) [2]Si=Si, (OH)[2]Si=Si, F[2]Si=Si, (NH [2])HSi=Si, and (OH)HSi=Si all are triplet ground states. In
particular, (NH[2])[2]Si=Si and (OH)[2]Si=Si have singlet-triplet energy gaps (ΔE[S-T] = E[T] - E [S]) of -10.2 and -10.3 kcal/mol, respectively. More practical results are achieved via cyclization
of (NH[2])[2]Si=Si, which eliminates the probability of rearrangement. Unsaturation of the resulting cyclic structure to give (NHCHCHNH)Si=Si leads to a more favorable triplet silylene with a ΔE[S-T]
value of -19.6 kcal/mol. Similar to the common approach of bulky substitution in the synthesis of singlet Arduengo-type N-heterocyclic silylenes, triplet (NRCH[2]CH[2]NR)Si=Si and (NRCHCHNR)Si=Si
could be experimentally achievable.
All Science Journal Classification (ASJC) codes
• Physical and Theoretical Chemistry
• Organic Chemistry
• Inorganic Chemistry
Dive into the research topics of 'Theoretical description of triplet silylenes evolved from H [2]Si=Si'. Together they form a unique fingerprint. | {"url":"https://researchwith.njit.edu/en/publications/theoretical-description-of-triplet-silylenes-evolved-from-h-sub2s","timestamp":"2024-11-02T02:23:49Z","content_type":"text/html","content_length":"49247","record_id":"<urn:uuid:9d8b03b0-8205-4f12-a5d4-140dd98edbae>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00146.warc.gz"} |
Primality: Given an integer n≥1, is n prime?
Note, knowing a number to be composite (not prime) does not easily give the factors of n; factorization is a different problem.
A positive integer, n, has 1+⌊log[b] n⌋ digits in base b notation, e.g., b=2, one→"1", two→"10", three→"11", four→"100", ..., seven→"111", eight→"1000", etc.. We are often interested in integers that
are much bigger than can be held in a 32- or 64-bit word.
Until 2002 it was not known if primality was in P. Then Agrawal, Kayal and Saxena (2002, 2004) gave a polynomial-time algorithm, but at O((log n)^12+εn)-time, it is not the method of choice. (Later O
((log n)^6.log^2n)-time, Lenstra and Pomerance (2009).)
Miller, Rabin
Rabin (1976, 1980) gave a probabilistic algorithm to test for primality. It terminates with one of two conclusions: either (i) n is (definitely) composite, or (ii) n is probably prime with
probability ≥ 1-4^-k. The uncertainty in the latter case can be made arbitrarily small by increasing the number of trials, k, and can easily be made even smaller than the probability of the computer
executing the algorithm having a hardware fault.
If the number of trials, k, is small, e.g., 1 or 2, results are quite likely to differ from run to run; for serious use k would be much large than 2. You can make 'lo' quite large (but increase it
A simple implementation of Rabin's algorithm runs in O(k (log n)^3)-time. The complexity can be reduced to O(k (log n)^2 log^2n log^3n)-time by incorporating, say, the Schonhage-Strassen algorithm
for the fast multiplication of long integers.
For a prime, p, Z[p] is a field and
if x^2 = 1 mod p then
x^2 - 1 = (x+1)(x-1) = 0 mod p
⇒ x = 1, or x = -1 = (p-1), mod p
that is, the only square roots of 1 are ±1. So, finding a non-trivial square root of 1 in Z[n] would show n to be composite.
If n is an odd prime, let n-1 = d.2^s where d is odd and s≥1.
Then ∀ w ∈ {1, ..., n-1},
if n is prime,
∀ w≥1, w^n-1 = 1 mod n
-- Fermat's "little" theorem
w^n-1 = w^d.2^s = (w^d)^2^s = 1, so either
w^d = 1 mod n, or
w^d.2^r = -1 mod n, for some r ∈ {0, ..., s-1}.
So, if ∃ w ∈ {2, ..., n-2} such that
w^d ≠ 1 mod n, and
w^d.2^r ≠ -1 mod n, ∀ r = 0, ..., s-1,
then w is a witness to n being composite.
Rabin's algorithm is to repeatedly choose a possible witness, w, at random from {2, ..., n-2}. It has been shown that, if n is composite, at least 3/4 of the possible choices are witnesses to the
Miller (1976) gave a deterministic algorithm for primality but it depends on the generalized Riemann hypothesis (GRH). Rabin's probabilistic algorithm does not depend on the GRH.
Rabin's algorithm raised interesting practical and philosophical questions about the nature of algorithms. Strictly, it is not an algorithm for primality. Rather, it is a probabilistic algorithm
because it might make a "mistake" -- by declaring a composite to be prime (strictly it declares a number to be "probably prime").
For^ a given composite n, and a given w ∈ {2, ..., n-2}, whether w is or is not a witness to n's compositeness is a fact, which certainly does not vary randomly with time, say, nor with anything
else. So if the algorithm uses a pseudo random number generator of candidate witnesses, the values of n for which it makes a "mistake", due to repeatedly generating non-witnesses, are fixed (for
a given "seed" and a given number of trials). So in what sense are its errors probabilistic? (Of course "real" random number generators, based on physical processes, do also exist.)
It^ is also interesting to note that by making the number of trials big enough, the probability of the algorithm making an error can easily be made (much) less than the probability of a hardware
and Annals of Maths, 160(2), pp.781-793, doi:10.4007/annals.2004.160.781, September 2004.
and FSTTCS, LNCS vol.2556, doi:10.1007/3-540-36206-1_1, 2002.
and JCSS, 13, pp.300-317, 1976, | {"url":"https://www.cantab.net/users/mmlist/ll/AlgDS/Primes/","timestamp":"2024-11-09T02:45:17Z","content_type":"text/html","content_length":"11728","record_id":"<urn:uuid:107af293-d1d8-4669-88f7-1a917d641a9a>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00730.warc.gz"} |
Structuring Directional Option Trades
Options Theory
Structuring Directional Option Trades
Options allow us to express bets on the distribution of the underlying instead of just "will the stock go up or down?"
This post is a response to Twitter buddy @demonetizedblog
Let me take a stab at a “process” answer.
For directional trading 90% of the work happens upstream of the option expression.
The option trade construction is the most trivial part of the process. Your fundamental work should inform your opinion of the distribution. This can be compared with the implied distribution from
the vol surface.
This mental process is entirely different from vol trading. Remember, you aren’t dynamically hedging. Directional trading and vol trading have totally different starting points.
[At the end of the post you’ll see when the two approaches come to the same conclusion and when they don’t. This can lead to directional traders to trade with vol traders and everyone is happy. It’s
still zero-sum. It’s just that the losses can be incurred by whoever provided the liquidity to the dynamic hedger. That entity was not part of the original trade]
Ok, so when it comes to directional trading vs vol trading, you must be clear what game you are playing.
This post is about structuring directional trades.
What’s the distribution?
First, you do a bunch of fundamental voodoo and come up with a distribution of possible stock returns.
[I’ll wait]
Good. We are going to discuss options now. Relax. Take a breath. Don’t worry about fancy words like “moments of a distribution” or kurtosis. You are a fundamental investor. It’s fine to think in
prices, percentages, and bets.
Now what?
Let’s establish a focusing principle.
You want the short leg of an options spread to correspond the most likely landing spot of the stock based on your analysis. If those options are the cheapest on the board you might want to consider
that the option surface is not presenting you an opportunity. It agrees with you. Don’t rush over that. This is not intuitive. Many fundamental managers buy the strike of where they think the stock
is going. Don’t do that. Instead let’s review some basics about distributions. Without real math.
1. A biotech stock worth $100 might be trading for that price because it’s 90% to be 0 and 10% to be $1000. True bimodal.
Code-switching this idea into options:
□ The 100 call is worth $90.
□ All the OTM 100 point wide call spreads are worth $10.
□ All the butterflies are zero.
What are some courses of action here?Let’s say you can afford 1 100 strike call. You could have also chosen 9 900/1000 call spreads. Or 3 of the 700 calls. In this case, all the propositions
are the same because the options are correctly priced.
[Prove this to yourself. I’ll wait.]
Cool. Now you can imagine how if some of the options were priced differently you might be able to find an alluring proposition.
2. New stock to consider. An insurance company also trading $100. This is not a bimodal stock. Perhaps it looks more like a bell curve with a high peak shifted to the right of the forward price
because a pumped up put skew is signaling strongly negative skew.
Why does that push the peak to the right?
Think about it. For that stock to be $100 with a long left tail, it must have a greater than 50% probability of going up. The verticals will show you that. It’s the opposite case of the biotech
stock and with much less volatility.
□ If you were super bullish you might want to load up on the depressed slightly OTM calls.
□ If you were bearish but thought the left tail was not as long you might want to buy the .50d/.25 put spread to express the view by exploiting the excess skew you think the market is embedding
in the OTM puts.
Just remember, options give a shape to the distribution. Not every $100 stock has the same distribution. Think about where the $100 comes from? What upside force is counterbalancing the downside? The
biotech stock has a very long right tail 900% away counterbalancing a large mass of probability that’s only 100% away. The $100 stock price is nothing like the insurance company. Options allow you to
express the bet you want to express. The stock price alone is too blunt.
Once you let that simmer you can start to ask yourself useful questions:
• Would you rather own 1 atm call or more calls for a total of the same premium at a higher strike?
• Now compare that to call spread candidates. How many call spreads can you buy and at what moneyness?
The nice thing about vertical spreads is they cancel out many of the “greeks” effectively taming your vega and gamma exposures. The bets can be thought of as binaries allowing you to make simple over
/under bets. To calibrate your impression of the possible magnitude of a stock move, you consider the moneyness or how far away from stock price the chosen strikes are. The moneyness will depend on
your intuition for the volatility of the stock. You will have a sense for which spreads are “close” or “far”. These are technical terms.
And since I mentioned volatility, let’s say a few words on that to help you avoid some landmines.
Is the vol cheap or expensive?
If you are a directional trader you don’t care if the right volatility for an option is 55% or 56%. You aren’t dynamically hedging. But you don’t want to go to the used-car lot without at least
checking Carmax online. You can compare the implied vol to the distribution of historical realized to make yourself feel like you did diligence.
Here’s a simple way:
Compare the IV to the stock’s historical vol of a comparable tenor. So if you are considering a 6 month option look at the distribution of 6 month historical vols to see if you are on the high or low
side of the range. How? Looking at a vol cone will get you a quick optical answer.
Here’s Colin Bennett’s example (with my highlight) from his book Trading Volatility:
If the recent realized volatility is elevated and you wanted to buy long-dated options it might be a poor time to buy options. You can either wait, trade structures like verticals that have little
vega exposure, or even create a directional trade by selling options.
Here’s a few extras to consider when selecting an expiry:
• The nearer the option tenor, the more event pricing matters. The event’s variance is a larger proportion of the total variance until expiration.
□ Longer dated options have takeover risk. (Cash takeovers mean your LEAP extrinsic goes to zero. Sorry.)
□ Do you plan to roll the exposure to maintain it or is there an expiration to your thesis? The more often you roll the less rebalance timing risk. This has to be weighed against trading costs.
The Real Work Is Not In The Options
When you throw a proper punch the fist is just the delivery method. The point of contact. That’s the option expression. The real work happens from the torque in your hips. That’s the fundamental
analysis behind the punch. An advantage of directional trading is you can think in discrete bets once you’ve done your fundamental homework.
Discrete trades let you:
• Think in terms of how many bets you get paid back vs how much premium you layout and compare that to the probability your fundamental work suggests.
• You’d like to get to a statement that looks like “I’m willing to risk 1 bet to make 3 because I think the proposition is a 50/50 shot.”
• This establishes your expectancy and shape of the p/l.
• Combine that info with your bankroll and now you can size the trade.
Bonus Section: Volatility Traders
I said that directional trading and volatility trading are different games. I’ll briefly talk about that.
First of all, even vol managers sometimes make discrete bets. They will “risk budget” a trade. I’m willing to spend $1mm on 150% calls for winter gas. Or whatever, you get the idea. They might even
set up a separate account for tracking and attribution for this.
But really this risk budgeting or discrete framework is different from managing a relative value volatility or market making portfolio. In that environment, you are often responding to values moving
around some cross-sectional trading model. You see edge, you pick it up, throw it on the pile and manage the blob. With a decent size book holding thousands of line items you are going to need 3-D
goggles to slice and dice the positioning and the risk. You might not even know what you are rooting for sometimes. If you are short SPX correlation and long 200 of the 500 names then you are
massively overweight vol in the 200 and you are “synthetically short” vol via the index in the other 300. Hundreds of names x hundreds of strike x hundreds of expiry and you need to bucket and
compute quickly and accurately. Totally different animal from directional perspectives.
This does not mean that vol traders and directional traders don’t land on the same conclusions occasionally. A vol manager who finds a name that “screens cheap” might be looking at the same thing a
fundamental manager is seeing. The fundamental manager is coming from a different vantage point, but might feel that a stock is hiding some serious upside and the nominal price of the calls are a
bargain. In this case, the fundamental manager is going to struggle to find liquidity as the call options might be cheap for a few contracts but once they start calling around the street find that no
market maker is willing to join the resting retail offers.
You may be wondering why the screens are so low in the first place? Why are they stale? The market maker’s dashboards are flashing green too. They know those options are cheap. But remember this is a
game. They aren’t going to bother lifting the offers for a few contracts. They would rather freeroll on the possibility that some donkey overwriter who systematically sells calls without price
sensitivity dangles a mid market offer. Then they’ll lift. (gratuitous “Do You Even Lift Bro?” clip)
So when do vol managers and directional traders trade with each other? All the time. Here’s 2 examples.
1. Imagine a fundamental trader who is directionally smart but not vol savvy. They might buy calls, and the market makers who have been keeping tabs on this pattern of flow realize its predictive of
a price move but has not historically beaten them to implied vol (perhaps it’s one of these dumb accounts that buys the strike of where they think the stock is going. They should probably hire a
vol trader, if for nothing else to show them how to do p/l attribution). So the market makers sell the calls and overhedge the delta. Trading 101.
2. A very common case where directional traders and vol traders are happy to trade is on vertical spreads or ratio spreads. Say a directional hedger buys put spreads. Vol traders can be happy to
sell them so they can buy that tail option that the hedger gave them as the lower leg of the spread. A similar example would be a 1 x 2 ratio put spread. Say the stock is $100 and the directional
trader buys the 80/75 1 x 2 put spread for a cheap or even zero premium. In their mind, they make make money all the way down to $70. They don’t start to lose money until the stock has dropped
more than 30%. The vol trader has a different view. The vol trader cares about path and they know if the stock trades down to $80 quickly and vol explodes, they are going to be long vega and have
ammunition to sell into the panicky vol buying. That 1 x 2 put spread is going to mark ruthlessly in the directional traders face. The directional trader didn’t respect path. Option traders are
extra wary of path because they are highly leveraged businesses warehousing complex portfolios with non-linearities. There’s no better training for visualizing risk up, down, through time, across
correlations, and at different speeds. The trader who honors path will often be the reason that “option that will never hit” is priced so high. | {"url":"https://blog.moontower.ai/structuring-directional-option-trades/","timestamp":"2024-11-09T16:00:41Z","content_type":"text/html","content_length":"36903","record_id":"<urn:uuid:019c1cef-c27b-4502-be27-e805cac04d2d>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00413.warc.gz"} |
In mathematics, projectivization is a procedure which associates with a non-zero vector space V a projective space P(V), whose elements are one-dimensional subspaces of V. More generally, any subset
S of V closed under scalar multiplication defines a subset of P(V) formed by the lines contained in S and is called the projectivization of S.^[1]^[2]
• Projectivization is a special case of the factorization by a group action: the projective space P(V) is the quotient of the open set V \ {0} of nonzero vectors by the action of the multiplicative
group of the base field by scalar transformations. The dimension of P(V) in the sense of algebraic geometry is one less than the dimension of the vector space V.
• Projectivization is functorial with respect to injective linear maps: if
${\displaystyle f:V\to W}$
is a linear map with trivial kernel then f defines an algebraic map of the corresponding projective spaces,
${\displaystyle \mathbf {P} (f):\mathbf {P} (V)\to \mathbf {P} (W).}$
In particular, the general linear group GL(V) acts on the projective space P(V) by automorphisms.
Projective completion
A related procedure embeds a vector space V over a field K into the projective space P(V ⊕ K) of the same dimension. To every vector v of V, it associates the line spanned by the vector (v, 1) of V ⊕
1. ^ "Projectivization of a vector space: projective geometry definition vs algebraic geometry definition". Mathematics Stack Exchange. Retrieved 2024-08-22.
2. ^ Weisstein, Eric W. "Projectivization". mathworld.wolfram.com. Retrieved 2024-08-27. | {"url":"https://www.knowpia.com/knowpedia/Projectivization","timestamp":"2024-11-11T15:06:37Z","content_type":"text/html","content_length":"75971","record_id":"<urn:uuid:a2f534b4-4fff-4754-b288-2e3ce396d56b>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00243.warc.gz"} |
Kinetic and Potential in simple harmonic motion
Energy in simple harmonic motion
The total energy (E) of an oscillating particle is equal to the sum of its kinetic energy and potential energy if conservative force acts on it. The velocity of a particle executing SHM at a position
where its displacement is y from its mean position is v = ω rt( a^2-y^2)
Kinetic energy
Kinetic energy of the particle of mass m is
K = ? n [ω rt( a^2-y^2) ]
K = ? m [ω^2 ( a^2-y^2) ] ?.(1)
Potential energy
From definition of SHM F = ?ky the work done by the force during the small displacement dy is dW = −F.dy = −(−ky) dy = ky dy
∴ Total work done for the displacement y is,
W = ∫ dW = ∫[0]^y ky dy
W = ∫[0]^y mω^2 y dy
∴W = 1/ 2 m ω^2 y^2
This work done is stored in the body as potential energy
U = 1 /2 m ω^2 y^2
Total energy E = K + U
= 1/ 2 mω^2 (a^2 − y^2) + 1/ 2 m ω^2 y^2
= 1/ 2 m ω^2 a^2
Thus we find that the total energy of a particle executing simple harmonic motion is 1 /2 m ω^2 a^2
Special cases
(i)When the particle is at the mean position y = 0, from eqn (1) it is known that kinetic energy is maximum and from eqn. (2) it is known that potential energy is zero. Hence the total energy is
wholly kinetic
E=K^max = 1/ 2 mω^2a^2
(ii) When the particle is at the extreme position y = +a, from eqn. (1) it is known that kinetic energy is zero and from eqn. (2) it is known that Potential energy is maximum. Hence the total energy
is wholly potential.
E = U^max = ? m ω^2 a^2
(iii)when y = a/2 ,
K = 1 /2 m ω^2 [a^2 - a^2/4]
∴K = ?(1/2mω^2 a^ 2)
K = 3 /4 E
U = 1/2 mω^2 (a^ 2/2) = ?(1/2 mω^2 a^ 2)
U = 1/4E
If the displacement is half of the amplitude, K = 3/ 4 E and U = 1 /4 E. K and U are in the ratio 3 : 1,
E=K+U=1/2 mω^2 a^ 2
At any other position the energy is partly kinetic and partly potential.
This shows that the particle executing SHM obeys the law of conservation of energy.
Graphical representation of energy
The values of K and U in terms of E for different values of y are given in the Table 6.2. The variation of energy of an oscillating particle with the displacement can be represented in a graph as
shown in the Fig..
Energy of SHM
Y : 0 a/2 , a , -a/2 , -a
Kinetic energy : E , ? E , 0 ? E, 0
Potential energy : 0 , 1/4E, E , 1/4E , E | {"url":"https://www.brainkart.com/article/Kinetic-and-Potential-in-simple-harmonic-motion_3143/","timestamp":"2024-11-15T01:17:30Z","content_type":"text/html","content_length":"40787","record_id":"<urn:uuid:10062115-93d0-4d4d-8c21-470e955911c9>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00571.warc.gz"} |
critical speed of ball mill pdf
The approximate horsepower HP of a mill can be calculated from the following equation: HP = (W) (C) (Sin a) (2π) (N)/ 33000. where: W = weight of charge. C = distance of centre of gravity or
charge from centre of mill in feet. a = dynamic angle of repose of the charge. N = mill speed in RPM. HP = A x B x C x L. Where.
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Critical Speed_. When the ball mill cylinder is rotated, there is no relative slip between the grinding medium and the cylinder wall, and it just starts to run in a state of rotation with the
cylinder of the mill. This instantaneous speed of the mill is as follows: N0 — mill working speed, r/min; K'b — speed ratio, %.
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TYPE OF MILL MEDIA SIZE RPM TIP SPEED (fpm) Ball Mill 1/2" and larger 1050 Attritor 1/8" to 3/8" 75450 Sand Mill/Horizontal mill 1/64" to 1/8" HSA Attritor 1mm 3mm HQ Attritor 3mm 3000 High speed
disperser II.
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Result #1: This mill would need to spin at RPM to be at 100% critical speed. Result #2: This mill's measured RPM is % of critical speed. Calculation Backup: the formula used for Critical Speed
is: N c = (D ) where, Nc is the critical speed,in revolutions per minute, D is the mill effective inside diameter, in feet.
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Rotational speed is usually fairly low, about 80% of critical speed (critical speed is the speed at which the charge will be pinned to the rotating drum and does not drop) and the typical drum
diameter ranges from 2 to 10 metres. This type of mill is often used as a single stage process, providing sufficient size reduction in a single process.
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Critical speed is defined as the point at which the centrifugal force applied to the grinding mill charge is equal to the force of gravity. At critical speed, the grinding mill charge clings to
the mill inner surface and does not tumble. Most ball mills operate at approximately 75% critical speed, as this is determined to be the optimum speed ...
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Rose and Sullivan showed critical rotation speed Nc, to reach the final stage,, centrifugal motion: (1) N c = 1 2π D−2r D is the inner diameter of a jar and r is the radius of balls. All
throughout, they used the critical rotation speed as the constant value for given conditions of ballmill .
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The ball mill consists of a metal cylinder and a ball. The working principle is that when the cylinder is rotated, the grinding body (ball) and the object to + (86) . ... This is the critical
speed of the 180 litre wet mill currently used in cemented carbide production.
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950 rpm. Blaine is the important characteristic of ball mill which is influenced by the mill speed and separator speed. Ball mill is grinding equipment which is used to reduce the size of clinker
into cement. It uses grinding media in the form of balls. Clinker coming from the silo is sent into hopper and mill for impact action. Clinker is ...
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With variablespeed mills, increasing mill speed directs ball impacts at the toe as both the lifter height falls and the lifter face angle increases with wear. The impact point is usually ...
mounted close to the mill. If mill speeds are increased above 78% to 80% critical speed, pulplifter efficiencies could fall and affect overall mill ...
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The ball mill designs also follow the Bond/Rowlings method with comparison with other methods. Again the method of use is the same SAG/BM Combined design ... Mill Critical speed Crit Crit D = n
revs per min nC= s*Crit Cs is the fraction of the critical speed
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In a SAG mill the dimensions of the mill were m × m and the specific gravities of the mineral and that of the balls charged were and respectively.. The mill was rotated at 75% of its critical
speed when 8 % of the mill volume was charged with grinding :1. The mill power drawn, 2. The maximum mill filling possible.
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In making ball mill, with a tube diameter of 20 cm, the operating speed of the tool and loading charge are obtained so that the operating interval can be known. The results of this study obtained
ball mill diameter of. 20 cm with a frame size of 50 cm long, 50 cm wide, 70 cm high and 30 kg tool weight, with a speed of 90 rpm,
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Typically R = 8. Rod Mill Charge: Typically 45% of internal volume; 35% 65% range. Bed porosity typically 40%. Height of bed measured in the same way as ball mills. Bulk density of rods = tons/
m3. In wet grinding, the solids concentration 1s typically 60% 75% by mass. A rod in situ and a cutaway of a rod mill interior.
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in combination with a ball mill for cement grinding applications and as finished product grinding units, as well as raw ingredient grinding equipment in mineral applications. This paper will
focus on the ball mill grinding process, its tools and optimisation possibilities (see Figure 1). The ball mill comminution process has a high electrical
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The speed of the mill was kept at 63% of the critical speed. The face angle was varied from 90 to 111 degrees for the three types of configuration 1, 2 and 4, as shown in the figure. Also, the
height of the lifter bar in configuration 3 was changed to observe the trajectory.
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In ball milling, the speed of the rotation is more important. At a low speed, (a), the mass of the ball slides or rolls over each other with inefficient output. ... (50 to 80% of the critical
speed), (c), the centrifugal speed force just occurs with the result that the balls are carried almost to the top of the mill and then fall to ...
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The filling levels M* was taken as 30%, 40% and 50% of full mill and the mill speed N* was selected as, and of the critical speed. The critical speed is the speed at which a mill drum rotates
such that balls are stick to the drum, which is given by 2 g / D − d where D and d are the mill diameter and particle diameter in meters ...
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The operational controls are also reviewed for optimized mill operation. Every element of a closed circuit ball mill system is evaluated independently to assess its influence on the system.
Figure 1 below is a typical example of inefficient grinding indicated by analysis of the longitudinal samples taken after a crash stop of the mill.
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40 times more than that caused by the gravitational acceleration, in normal directions. Thus, planetary ball milling can be employed for high speed/energy milling. Figure 1 shows the schematics
of ball milling. 3 ... The critical attributes of a grinding medium include size, density, hardness, and composition. We will discuss these properties ...
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PROCEDURE: 1. Take 250 gm feed having size 7+10 mesh number screen. 2. Calculate critical and optimum speed of ball mill. 3. Before starting ball mill, make sure that all steel balls and feed
materialsare in the cylinder. Also ensure the opening of ball mill tightly closed with plate. Start the ball mill and fix.
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Critical speed (in rpm) = /sqrt(D d) with D the diameter of the mill in meters and d the diameter of the largest grinding ball you will use for experiment (also expressed in meters)
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Ball Mill Grinding Process Handbook Free download as PDF File (.pdf), Text File (.txt) or read online for free.
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its application for energy consumption of ball mills in ceramic industry based on power feature deployment, Advances in Applied Ceramics, DOI: /
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The ball milling process generally takes 100 to 150 hrs to give uniformly crushed fine powder. e. It is mechanical processing technique; consequently the structural as well as chemical changes
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The critical speed of ball mill is given by, (displaystyle n_c = frac {1} {2pi}sqrt {frac {g} {Rr}} ) where R = radius of ball mill; r = radius of ball. For R = 1000 mm and r = 50 mm, = rpm But
the mill is operated at a speed of 15 rpm. Therefore, the mill is operated at 100 x 15/ = % of critical speed.
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The critical speed of the mill, c, is defined as the speed at which a single ball will just remain against the wall for a full cycle. At the top of the cycle =0 and Fc Fg () mp 2 cDm 2 mpg () c
2g Dm 1/2 () The critical speed is usually expressed in terms of the number of revolutions per second Nc c 2 1 2 2g Dm 1/2 (2×)1/2 ...
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Nevertheless, at 85% critical speed, the opposite seems to happen for the Tailings Pond sample shown in Figure 9. This is probably due to better behavior under a greater influence of mill speed
and ball size, mainly for the coarse feed particles size as a consequence of a greater influence of the impact breakdown and the cascading effect [36,37].
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The rotational direction of a pot in a planetary ball mill and its speed ratio against revolution of a disk were studied in terms of their effects on the specific impact energy of balls
calculated ...
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The formula to calculate critical speed is given below. N c = /sqt(Dd) N c = critical speed of the mill. D = mill diameter specified in meters. d = diameter of the ball. In practice Ball Mills
are driven at a speed of 5090% of the critical speed, the factor being influenced by economic consideration.
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Ball mill shape factors in the period prior to 1927 (Taggart, 1927) averaged /1 for 29 center discharge mills and /l 30 peripheral discharge mills. With the resumption of new plant ...
specifically the load fraction critical speed, can reduce the risk of overload for existing operations; while appro ...
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Download PDF Download PDF with Cover Download XML Download Epub. Browse Figures. ... they worked with a mill 210 mm in diameter and 251 mm long at 96% of critical speed and charged with the
grinding media distribution shown in Table 1. ... This sample is ground in a Bond ball mill for an arbitrary number of revolutions (N 1 = 50, 100, or 150 ...
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The formula for calculating critical speed is dependent on the radius and diameter of the mill. Optimizing the critical speed of a ball mill can improve grinding efficiency and product quality by
ensuring the grinding balls are effectively lifted and distributed for optimal grinding. Application of Ball Mill
WhatsApp: +86 18838072829 | {"url":"https://lopaindefraises.fr/Nov/12_1778.html","timestamp":"2024-11-03T04:45:36Z","content_type":"application/xhtml+xml","content_length":"29125","record_id":"<urn:uuid:3e961a3f-a1a5-4d66-8ad4-61ad17748ef9>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00874.warc.gz"} |
The gravitational force between two objects is $F$. How will this force change when
Hint:We are surrounded by gravitational force. It determines how much we weigh and how far a basketball travels before returning to the ground when tossed. The force exerted by the Earth on you is
equal to the force exerted by the Earth on you. The gravity force equals your weight when you're at rest on or near the Earth's surface. The acceleration of gravity on a particular planetary body,
such as Venus or the Moon, is different than on Earth, because if you stood on a scale, it would show you weighing a different volume than on Earth.
Complete step by step answer:
Any particle in the universe attracts every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the
distance between their poles, according to Newton's law of universal gravitation. The theory's publication was dubbed the "first great convergence" because it brought together previously established
gravity phenomena on Earth with known astronomical activities.We know that \[F = G\dfrac{{{m_1}{m_2}}}{{{r^2}}}\].
(i) The gravitational force $F$ between two objects separated by $r$, according to Newton's law of gravitation, is
\[F \propto \dfrac{1}{{{r^2}}}\]
When $r$ is halved
\[{r^\prime } = \dfrac{r}{2}\]
Force becomes
\[\dfrac{{{F^\prime }}}{F} = \dfrac{{{r^2}}}{{{r^{'2}}}}
\Rightarrow \dfrac{{{F^\prime }}}{F}= \dfrac{{{r^2}}}{{{{(r/2)}^2}}}
\Rightarrow \dfrac{{{F^\prime }}}{F}= 4\]
\[\therefore {F^\prime } = 4F\]
(ii) The gravitational force F between two particles of mass and, according to Newton's law of gravitation, is
\[F \propto {m_1}{m_2}\]
When each mass is quadrupled
\[{m_1}' = 4{m_1}{\rm{ and }}{m_2}' = 4{m_2}\]
Force becomes
\[{F^\prime } \propto m_1^\prime m_2^\prime \]
\[\Rightarrow \dfrac{{{F^\prime }}}{F} = \dfrac{{m_1^\prime {m^\prime }2}}{{{m_1}{m_2}}} \\
\Rightarrow \dfrac{{{F^\prime }}}{F}= \dfrac{{\left( {4{m_1}} \right)\left( {4{m_2}} \right)}}{{{m_1}{m_2}}} \\
\Rightarrow \dfrac{{{F^\prime }}}{F}= 16\]
\[\therefore {F^\prime } = 16F\]
Note:The rule notes that some point mass attracts every other point mass by a force acting along the line intersecting the two points in today's terminology. The force is proportional to the product
of the two masses, and it is inversely proportional to the square of their distance. | {"url":"https://www.vedantu.com/question-answer/the-gravitational-force-between-two-objects-is-f-class-11-physics-cbse-60a51a364ae3eb743940d2be","timestamp":"2024-11-03T17:10:18Z","content_type":"text/html","content_length":"165660","record_id":"<urn:uuid:19bc3057-5413-408b-9655-372aa3bda33c>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00846.warc.gz"} |
Challenge Exercises for Understanding Percent
Directions: For each of the 9 problems below, fill in the missing value into the form, then click ENTER. Your answer will be a fraction, a decimal or a percent, depending on the problem. If your
answer is a percent, it must include the % symbol. Be sure to write your fractions in lowest terms. To write the fraction five eighths, enter 5/8 into the form. | {"url":"https://mathgoodies.com/lessons/challenge_vol4/","timestamp":"2024-11-05T15:54:47Z","content_type":"text/html","content_length":"60259","record_id":"<urn:uuid:fb596e9b-ca6b-4aa9-8dee-839fd08d2bcb>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00584.warc.gz"} |
IMSECH - Excel docs, syntax and examples
The IMSECH function is used to calculate the hyperbolic secant of a complex number in Excel. This function is handy for performing calculations involving complex numbers in trigonometry, engineering,
and other mathematical applications.
complex_number The complex number for which you want to calculate the hyperbolic secant.
About IMSECH 🔗
When you encounter complex numbers that require evaluation of hyperbolic trigonometric functions, turn to IMSECH in Excel. IMSECH excels in computing the hyperbolic secant of complex numbers,
providing a valuable tool for handling intricate mathematical operations in various disciplines ranging from engineering to physics and beyond. By employing IMSECH, you can efficiently determine the
hyperbolic secant of a given complex number, facilitating precise calculations in scenarios necessitating complex number analysis.
Examples 🔗
Suppose you have a complex number 3 + 4i for which you want to calculate the hyperbolic secant. The IMSECH formula would be: =IMSECH(3+4i)
If you have a different complex number, let's say 2 - i, and want to find its hyperbolic secant, you can use: =IMSECH(2-i)
Ensure the complex number is correctly formatted, using 'i' to represent the imaginary unit. IMSECH operates on complex numbers for which the secant function is applicable.
Questions 🔗
What is the range of valid input for the complex number in the IMSECH function?
The complex number in the IMSECH function can be any complex number for which the hyperbolic secant function is defined. Ensure the input is in the correct format with the imaginary unit 'i'.
Can the IMSECH function handle purely real numbers?
While the IMSECH function is designed for complex numbers, it can still handle purely real numbers by setting the imaginary part to zero. For example, IMSECH(5) will yield the hyperbolic secant of
the real number 5.
Is the output of the IMSECH function always a complex number?
No, the output of the IMSECH function can be a real number or a complex number depending on the input provided. In cases where the input results in a purely real output, the IMSECH function will
return a real number.
Related functions 🔗
Leave a Comment | {"url":"https://spreadsheetcenter.com/excel-functions/imsech/","timestamp":"2024-11-05T01:13:35Z","content_type":"text/html","content_length":"28554","record_id":"<urn:uuid:ad996023-9488-45f9-8e05-a226394d03ea>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00439.warc.gz"} |
Quantum Physics: Unlocking the Mysteries of the Subatomic World - The Daily Tale
Quantum physics, the branch of physics that deals with the behavior of matter and energy at the smallest scales, has revolutionized our understanding of the universe. Unlike classical physics, which
explains the macroscopic world we experience daily, quantum physics delves into the subatomic realm, where particles like electrons and photons exhibit behavior that defies intuition. This field has
not only challenged our perception of reality but also led to groundbreaking technologies that shape our modern world.
The Birth of Quantum Physics
The origins of quantum physics trace back to the early 20th century, a period marked by significant scientific upheaval. Classical physics, dominated by Newtonian mechanics and Maxwell’s
electromagnetism, could not explain certain phenomena observed at atomic and subatomic levels. The ultraviolet catastrophe, the photoelectric effect, and the discrete spectral lines of atoms were
puzzles that classical theories couldn’t solve.
In 1900, Max Planck made a pivotal breakthrough by introducing the concept of quantization. Planck proposed that energy is emitted or absorbed in discrete units called quanta. This idea led to the
formulation of Planck’s constant, a fundamental constant that underpins quantum mechanics. Albert Einstein further advanced the field in 1905 by explaining the photoelectric effect using the notion
of light quanta, or photons, which earned him the Nobel Prize in Physics in 1921.
Wave-Particle Duality
One of the most intriguing aspects of quantum physics is wave-particle duality, which posits that particles can exhibit both wave-like and particle-like properties. This duality was first
demonstrated by the double-slit experiment, conducted by Thomas Young in 1801 and later interpreted in the context of quantum mechanics by Niels Bohr and others.
When electrons or photons pass through a double-slit apparatus, they create an interference pattern characteristic of waves. However, when detected individually, they appear as discrete particles.
This paradoxical behavior suggests that particles exist in a superposition of states, described by a wave function, until measured.
The Uncertainty Principle
Werner Heisenberg introduced another fundamental principle of quantum physics in 1927: the uncertainty principle. This principle states that certain pairs of physical properties, such as position and
momentum, cannot be simultaneously measured with arbitrary precision. The more accurately one property is known, the less accurately the other can be determined.
Mathematically, the uncertainty principle is expressed as Δx * Δp ≥ ħ/2, where Δx and Δp are the uncertainties in position and momentum, respectively, and ħ is the reduced Planck’s constant. This
principle has profound implications for our understanding of the microscopic world, challenging the deterministic nature of classical physics and introducing intrinsic probabilistic elements.
Quantum Superposition and Entanglement
Quantum superposition is the concept that particles can exist in multiple states simultaneously until measured. This principle was famously illustrated by Erwin Schrödinger’s thought experiment known
as Schrödinger’s cat. In this experiment, a cat in a sealed box is simultaneously alive and dead, depending on an unobserved quantum event. Only when the box is opened and the cat observed does it
collapse into one of the two states.
Quantum entanglement, another counterintuitive phenomenon, occurs when particles become interconnected in such a way that the state of one particle instantaneously affects the state of another,
regardless of the distance separating them. Albert Einstein referred to this phenomenon as “spooky action at a distance.” Entanglement has been experimentally confirmed and plays a crucial role in
emerging technologies like quantum computing and quantum cryptography.
Quantum Mechanics and the Schrödinger Equation
The mathematical framework of quantum mechanics is built upon the Schrödinger equation, formulated by Erwin Schrödinger in 1925. This partial differential equation describes how the quantum state of
a physical system evolves over time. The wave function, a solution to the Schrödinger equation, encapsulates the probabilities of a particle’s position and momentum.
The time-dependent Schrödinger equation is given by:
[ i\hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi ]
Here, ( i ) is the imaginary unit, ( \hbar ) is the reduced Planck’s constant, ( \psi ) is the wave function, and ( \hat{H} ) is the Hamiltonian operator representing the total energy of the system.
The wave function’s absolute square, ( |\psi|^2 ), gives the probability density of finding a particle in a particular state.
Quantum Field Theory
Quantum field theory (QFT) extends quantum mechanics to fields, treating particles as excited states of underlying fields. Developed in the mid-20th century, QFT combines special relativity and
quantum mechanics, providing a framework for understanding fundamental interactions in the universe.
Quantum electrodynamics (QED), a subset of QFT, describes the interaction between charged particles and the electromagnetic field. Pioneered by Richard Feynman, Julian Schwinger, and Sin-Itiro
Tomonaga, QED has been remarkably successful in predicting phenomena with astonishing accuracy. The Feynman diagrams, a visual representation of particle interactions, have become a vital tool in
Quantum chromodynamics (QCD), another QFT, describes the strong interaction between quarks and gluons, which constitute protons and neutrons. The Standard Model of particle physics, which
incorporates QED, QCD, and the weak nuclear force, is one of the most successful theories in physics, explaining a vast array of experimental results.
Quantum Computing
Quantum computing, an application of quantum physics, promises to revolutionize information processing. Unlike classical bits, which represent either 0 or 1, quantum bits or qubits can exist in
superpositions of states. This property enables quantum computers to perform parallel computations, potentially solving complex problems exponentially faster than classical computers.
Quantum algorithms, such as Shor’s algorithm for factoring large numbers and Grover’s algorithm for searching unsorted databases, demonstrate the potential of quantum computing. However, practical
quantum computers face significant challenges, including maintaining qubit coherence and error correction. Despite these hurdles, companies like IBM, Google, and startups are making rapid strides in
developing quantum hardware and software.
Quantum Cryptography
Quantum cryptography leverages quantum mechanics to achieve secure communication. The most well-known protocol, Quantum Key Distribution (QKD), allows two parties to generate a shared secret key,
which can be used for encryption. Any attempt to eavesdrop on the key exchange disturbs the quantum states, alerting the communicating parties.
QKD systems, such as BB84 proposed by Charles Bennett and Gilles Brassard in 1984, have been successfully implemented over optical fibers and free-space links. As quantum computers pose a threat to
classical cryptographic algorithms, quantum cryptography offers a promising solution for future-proof security.
Quantum Sensors and Metrology
Quantum sensors exploit quantum properties to achieve unprecedented sensitivity and precision in measurements. Atomic clocks, based on the vibrations of atoms, provide the most accurate timekeeping
devices, essential for global positioning systems (GPS) and telecommunications.
Quantum magnetometers, which measure magnetic fields with high precision, have applications in medical imaging, geophysical exploration, and fundamental physics research. The development of quantum
sensors continues to push the boundaries of measurement accuracy, enabling new discoveries and technologies.
Quantum Biology
Quantum biology explores the role of quantum phenomena in biological processes. While traditionally the domain of physics and chemistry, evidence suggests that quantum effects play a significant role
in photosynthesis, enzyme activity, and even avian navigation.
In photosynthesis, for example, quantum coherence may facilitate the efficient transfer of energy within light-harvesting complexes. Understanding these quantum mechanisms could inspire new
technologies for energy conversion and storage.
Interpretations of Quantum Mechanics
The peculiar nature of quantum mechanics has led to various interpretations, each offering a different perspective on the underlying reality. The Copenhagen interpretation, formulated by Niels Bohr
and Werner Heisenberg, suggests that the wave function collapses upon measurement, with reality fundamentally probabilistic.
The Many-Worlds interpretation, proposed by Hugh Everett III, posits that all possible outcomes of a quantum measurement are realized in separate, branching universes. This interpretation eliminates
the need for wave function collapse but introduces a vast multiverse.
The de Broglie-Bohm theory, or pilot-wave theory, offers a deterministic alternative, where particles are guided by a hidden wave. Despite its mathematical equivalence to standard quantum mechanics,
this interpretation remains less widely accepted.
Quantum Gravity and Beyond
One of the most significant challenges in modern physics is reconciling quantum mechanics with general relativity, Einstein’s theory of gravitation. Quantum gravity seeks to describe gravity within
the quantum framework, potentially leading to a unified theory of everything.
String theory and loop quantum gravity are prominent approaches to quantum gravity. String theory posits that particles are one-dimensional strings vibrating in higher-dimensional space, while loop
quantum gravity quantizes spacetime itself. Both theories face formidable theoretical and experimental challenges but hold promise for advancing our understanding of the universe.
Quantum Technology and the Future
The advancements in quantum physics have paved the way for a new era of technology and innovation. Quantum computing, cryptography, and sensors are just the beginning. As our understanding of quantum
phenomena deepens, we can expect even more transformative applications.
Quantum communication networks, or quantum internet, could enable secure, instantaneous information transfer across the globe. Quantum simulators could solve complex problems in materials science,
chemistry, and biology, leading to breakthroughs in medicine, energy, and environmental sustainability.
The intersection of quantum physics with other fields, such as artificial intelligence and nanotechnology, holds exciting potential. Quantum-enhanced AI could tackle problems that are currently
intractable, while nanotechnology could enable the precise manipulation of quantum systems at the atomic level.
Challenges and Ethical Considerations
Despite its promise, the development and deployment of quantum technology come with challenges and ethical considerations. The technical hurdles, such as error correction in quantum computing and the
scalability of quantum networks, | {"url":"https://thedailytale.com/quantum-physics-unlocking-the-mysteries-of-the-subatomic-world/","timestamp":"2024-11-13T19:35:31Z","content_type":"text/html","content_length":"178646","record_id":"<urn:uuid:b1c4017b-6e06-4268-97ca-183d2ad5a64e>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00348.warc.gz"} |
Albion College Mathematics and Computer Science Colloquium
Title: Symmetry Groups: The mathematical connection between patterns in Moorish architecture and the artwork of M.C. Escher
Speaker: David A. Reimann
Associate Professor
Mathematics and Computer Science
Albion College
Albion, Michigan
Abstract: The mathematical structure of symmetrical patterns can be studied using group theory. The Moors built many magnificent buildings richly decorated with geometric patterns during their rule
of the Iberian peninsula (711-1492). The graphic artist M.C. Escher visited southern Spain in 1922 and was captivated by the patterns that richly decorate the architecture of the Alhambra,
Alcazar, and other Moorish buildings. After a second visit to Spain in 1935, Escher became obsessed with creating patterns of interlocking figures based on these elaborate tiling patterns.
While Escher had no formal mathematical training, he used mathematical methods grounded in scientific literature to study these patterns. We will view these patterns through the lens of
group theory, one of the great mathematical accomplishments of the 19th century. This talk will be highly visual with many pictures of Escher's works and Moorish architecture.
Location: Palenske 227
Date: 9/15/2011
Time: 3:30 PM
author = "{David A. Reimann}",
title = "{Symmetry Groups: The mathematical connection between patterns in Moorish architecture and the artwork of M.C. Escher}",
address = "{Albion College Mathematics and Computer Science Colloquium}",
month = "{15 September}",
year = "{2011}" | {"url":"http://mathcs.albion.edu/scripts/mystical2bib.php?year=2011&month=9&day=15&item=a","timestamp":"2024-11-02T18:42:45Z","content_type":"text/html","content_length":"2665","record_id":"<urn:uuid:a84f24b9-232b-4e01-be6c-ad0fd5ae6ff3>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00416.warc.gz"} |
This function temporarily treats each observed value in var as missing and imputes that value based on the imputation model of output. The dots are the mean imputation and the vertical lines are the
90% percent confidence intervals for imputations of each observed value. The diagonal line is the \(y=x\) line. If all of the imputations were perfect, then our points would all fall on the line. A
good imputation model would have about 90% of the confidence intervals containing the truth; that is, about 90% of the vertical lines should cross the diagonal.
The color of the vertical lines displays the fraction of missing observations in the pattern of missingness for that observation. The legend codes this information. Obviously, the imputations will be
much tighter if there are more observed covariates to use to impute that observation.
The subset argument evaluates in the environment of the data. That is, it can but is not required to refer to variables in the data frame as if it were attached. | {"url":"https://www.rdocumentation.org/packages/Amelia/versions/1.7.5/topics/overimpute","timestamp":"2024-11-07T19:07:02Z","content_type":"text/html","content_length":"68582","record_id":"<urn:uuid:5af0bb23-1507-4dee-97ee-ddce1cee9ca1>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00419.warc.gz"} |
Day of the Week Calculator - Tej Calculator
Day of the Week Calculator
Unlocking the Power of the Day of the Week Calculator
What is a Day of the Week Calculator?
A Day of the Week Calculator is a handy tool that helps you determine the specific day of the week for any given date. Whether you're planning an event, looking back on a historical date, or simply
curious about what day of the week your birthday falls on this year, this calculator makes it easy to find that information quickly and accurately.
Why Use a Day of the Week Calculator?
Using a Day of the Week Calculator can simplify various tasks. Here are some reasons why you might want to use this tool:
1. Event Planning: When organizing an event, knowing the day of the week can help you choose a more suitable date for your guests.
2. Historical Research: Historians and researchers often need to pinpoint the day of significant events in history.
3. Personal Milestones: Ever wonder what day your birthday falls on in different years? This calculator can give you that information instantly.
How Does the Day of the Week Calculator Work?
The Day of the Week Calculator uses a simple algorithm to compute the day based on the input date. The process involves:
1. Inputting the Date: Enter the date for which you want to find the day of the week.
2. Calculation: The calculator processes the date using established mathematical formulas to determine the day.
3. Result Display: It displays the day along with additional information, such as the day number in the year and how many days are left in that year.
Features of the Day of the Week Calculator
Here are some key features you can expect from a good Day of the Week Calculator:
• User-Friendly Interface: A clean design that is easy to navigate.
• Responsive Design: Accessible on both desktop and mobile devices for convenience.
• Detailed Results: Provides not only the day but also relevant statistics, like how many days are left in the year.
• Aesthetic Presentation: Attractive layout with colorful and dynamic numbers for a visually pleasing experience.
The Day of the Week Calculator is an invaluable tool for anyone who needs to know the day of the week for any date. Whether for personal use, planning events, or conducting research, this calculator
provides quick and reliable results. Make your life easier by using this simple yet powerful tool today!
Leave a Comment | {"url":"https://tejcalculator.com/day-of-the-week-calculator/","timestamp":"2024-11-08T07:12:54Z","content_type":"text/html","content_length":"195091","record_id":"<urn:uuid:9baaebb5-4664-477b-a7ff-ac58948c985d>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00692.warc.gz"} |
Model help:CEM
The CSDMS Help System
The CEM model, the Coastline Evolution Model, simulates the evolution of a shoreline due to gradients in breaking-wave-driven alongshore sediment transport. The original CEM has been componentized to
consist of the longshore transport module (CEM) and a wave input module (WAVES).
Extended model introduction
The CEM model assumes that the coast consists of a high percentage of mobile sediment and its other assumptions are more applicable at shoreline lengths of km’s and larger. The model was initially
designed to investigate an instability in the shape of the coast caused by waves approaching with ‘high’ angles (with the angle between offshore wave crests--i.e. before transformation over
approximately shore-parallel contours--and the coast > 45 degrees).
Although a number of wave (and geometry) parameters can be entered, the most vital input control for CEM is the wave climate. The current version of the CEM is driven by simplified directional wave
climate controlled by two main input parameters: the asymmetry of the incoming waves angle and the proportion of high-angle waves. This model is not designed to accurately simulate a specific
geographic location in detail but rather to more generally represent how a shoreline with highly mobile sediment may respond to varying wave angles. The value in this model is in the breadth it
offers in representing how different wave climates can result in different potentially interesting shoreline configurations. Ashton and Murray (2006b) present a more thorough description of the model
parameters and theoretical underpinning.
Model parameters
CEM does not need input files from the user, its input is entirely specified in the CMT graphical user interface. To obtain output from this component make sure you toggle on the output files; as a
default they are OFF.
Coupling parameters
Uses ports
CEM requires wave parameters as supplied by the WAVES component.(Model Help of WAVES).
CEM requires a sediment flux (bedload) originating from one or more river distributary channels. This is provided by the AVULSION component.
Provides ports
CEM provides an elevation grid of offshore geometry after erosion and deposition is done in every timestep.
The elevation is required for the AVULSION component.
Main equations
• Alongshore sediment transport
<math>Q_{s} = K_{2} H_{0} ^ \left ( {\frac{12}{5}} \right ) T ^ \left ( {\frac{12}{5}} \right ) cos ^ \left ( {\frac{6}{5}}\right ) \left ( \Phi _{0} - \theta \right ) sin \left ( (1)
\Psi _{0} - \theta \right )</math>
<math>K_{2} = \left ( {\frac{\sqrt{g \gamma}}{2 \pi}}\right ) ^ \left ({\frac{1}{5}} \right ) K_{1} </math> (2)
• Predictions of shoreline evolution
<math>{\frac{d \eta}{d t}} = - {\frac{K_{1}}{D_{sf}}} H_{b} ^ \left ({\frac{5}{2}} \right ) {\frac{d^2 \eta}{d x^2}} </math> (3)
<math>cos^ \left ( {\frac{1}{3}}\right ) \left ( \Phi_{b} - \theta \right ) \approx 1 </math> (4)
• Alongshore component of the radiation stress
<math>S_{xy} = H^2 sin \left ( \Phi - \theta \right ) cos \left ( \Phi - \theta \right ) </math> (5)
<math>{\frac{d \eta}{d t}} = - {\frac{K_{2}}{D}} H_{0} ^ \left ({\frac{12}{5}} \right ) T^ \left ( {\frac{1}{5}} \right ) \{cos^\left ( {\frac{1}{5}} \right ) \left ( \Phi_{0} - \ (6)
theta \right ) [ cos^2 \left ( \Phi _{0} - \theta \right ) - \left ( {\frac{6}{5}} \right ) sin^2 \left ( \Phi_{0} - \theta \right )] \} {\frac{d^2 \eta}{d x^2}} </math>
<math>\Delta F = {\frac{\left ( Q_{in} - Q_{out} \right )}{\left ( D_{sf} + B \right ) \Delta W^2}} </math> (7)
<math>\Delta t \propto \left ( {\frac{K_{1}}{D_{sf}}} H_{0} ^ \left ({\frac{12}{5}} \right ) T^\left ( {\frac{1}{5}}\right ) \right ) \Delta x^2 </math> (8)
<math>\Delta Y_{bb} = {\frac{\left ( D_{sf} + B \right ) }{\left ( D_{bb} + B \right )}}\Delta Y_{sl} </math> (9)
<math>W_{c} = W_{0} + \Delta Y_{bb} - \Delta Y_{sl} </math> (10)
<math>\Delta Y_{bb} = {\frac{\left ( W_{c} - W_{0} \right )}{\left ( 1 - {\frac{\left ( D_{bb} + B \right ) }{\left (D_{sf} + B \right )}} \right )}} </math> (11)
<math> \Delta Y_{sl} = {\frac{\left ( W_{c} - W_{0}\right )}{\left ({\frac{\left ( D_{sf} + B \right ) }{\left ( D_{bb} + B \right ) } - 1} \right )}} </math> (12)
The parameters: Shoreface Slope, Shoreface Depth and Shelf Slope set the initial geometry of the shoreface domain and the shelf domain. Simulations will use the shoreface depth as an effective
erosion depth, but deposition can take place to deeper depths if the shoreface is accreting on a deeper shelf.
An example run with input parameters as well as a figure / movie of the output
Follow the next steps to include images / movies of simulations:
See also: Help:Images or Help:Movies
• Ashton A., Murray B.A. Arnault O. Formation of Coastline Features by Large-Scale Instabilities Induced by High-Angle Waves. Nature Magazine. Volume 414. 15 November 2001
• Ashton A.D., Murray A.B. High-Angle Wave Instability and Emergent Shoreline Shapes: 1. Wave Climate Analysis and Comparisons to Nature. Journal of Geophysical Research. Volume 111. 15 December
• Ashton A.D., Murray A.B. High-Angle Wave Instability and Emergent Shoreline Shapes: 2. Wave Climate Analysis and Comparisons to Nature. Journal of Geophysical Research. Volume 111. 15 December
Movies generated with the stand-alone CEM model are documented in the CSDMS movie gallery. https://csdms.colorado.edu/wiki/Coastal_animations | {"url":"https://csdms.colorado.edu/wiki/Model_help:CEM","timestamp":"2024-11-06T07:28:59Z","content_type":"text/html","content_length":"44980","record_id":"<urn:uuid:8bb6ef5c-c50e-4c71-9363-ee0b8f92ae0d>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00389.warc.gz"} |
Beginner’s Guide To Bayes’ Theorem and Bayesian Statistics
Bayesian statistics, or so they say, is not for the faint of heart. But is that true? At some point in their learning curve, once they’ve had a little experience, students of statistics usually come
across Bayes’ Theorem and wonder if it’s something they should spend a little time studying. After half an hour or so, they realise they’ve fallen into Alice’s rabbit hole and then take a huge
decision – they decide whether to press on or turn back, and for most, this decision is a permanent one.
Is Bayesian statistics really as difficult as some would have you believe?
This blog post is an introduction to Bayesian statistics and Bayes’ Theorem. Its purpose is to help you in getting started with Bayesian statistics and get over the initial fear factor. Once you
start to get a bit less intimidated I’ll point you in the right direction to learn more.
Disclosure: This post contains affiliate links. This means that if you click one of the links and make a purchase we may receive a small commission at no extra cost to you. As an Amazon Associate we
may earn an affiliate commission for purchases you make when using the links in this page.
You can find further details in our TCs
So what is Bayesian statistics?
Bayesian statistics mostly involves conditional probabilities, which is the probability of an event A, given that event B has happened, and it can be calculated using Bayes’ Rule.
The basic premise is this: you make a guess on what you think your outcome will be. This is called the Prior probability. Then you collect some data and update the probability of the outcome
accordingly. We call this the Posterior probability.
Sounds simple, doesn’t it?
Well, it does get a bit harder than this, and there can be some mathematics that look intimidating, but in truth you won’t need to worry about that, there are packages that do all the heavy lifting
for you – Bayesian statistics is no harder than classical statistics!
Ready? It’s time to get started with Bayesian statistics!
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What is Bayes’ Theorem in Simple Terms?
In short, Bayes’ Theorem is a way of finding the probability of an event happening when we know the probabilities of certain other events.
Bayes’ Theorem is named after the Englishman Thomas Bayes, an 18^th century statistician and Presbyterian minister who, interestingly, never actually published his theorem – it was published by
Richard Price after his death.
The simplest way of thinking about Bayes’ Theorem is that before you have any evidence you ‘guess’ the likelihood that an event will happen, then you can improve your guess once you’ve collected some
For example, before opening the curtains and looking outside you can speculate on how likely it is that it will rain today. Then you open the curtains to see if there are clouds, ask your wife to
count how many people are carrying brollies and check the calendar to see if you’re in the monsoon season. How did your guess go? Now that you have some data, do you want to change your mind as to
how likely it is to rain today?
In a nutshell, this is what Bayes’ Theorem is all about.
What is Conditional Probability in Bayes’ Theorem?
In probability, you can have independent events and dependent events.
Independent events are things that happen that are unconnected with each other, such as the probability that it will rain today and the probability that your brother-in-law will call to wish you a
happy birthday. You can see that neither of these events are connected with the other. They are independent.
In probability, there is a probability multiplication rule that tells us that the probability of independent events A and B happening together is the same as the probability of event A multiplied by
the probability of event B (in probability, when you see the word AND, you multiply), and it looks like this:
For example, if the probability of finding a parking space is 0.22 and the probability of having the correct change for the parking meter is 0.34, then the probability that you will get parked
without having to frantically seek change before the parking attendant gets to your car is 0.22 x 0.34 = 0.07.
It’s not looking good!
Bayes’ Theorem, on the other hand, deals with the probabilities of events that are dependent, where one event is conditional on the other happening. For example, what is the probability that you will
win the lottery? It depends. Did you buy a ticket? The probability of winning the lottery is dependent on how many tickets you bought.
So what happens when event B is dependent upon event A?
In this case, the probability multiplication rule is modified slightly to give us the conditional probability multiplication rule:
The last term in this equation means “the probability of B, given that A has occurred”, and is known as the conditional probability.
Let’s say that you wish to draw two cards of the same suit out of the pack. There are 52 cards in the pack, with 13 of each suit. If with your first pick you draw a Diamond, what is the probability
that you will draw a Diamond with your second pick?
Here, drawing the first Diamond has a probability of P(A) = 13/52.
Your second draw, though, doesn’t have the same probability as your first. Now there are 12 Diamonds left in the pack of 51 cards, so the conditional probability of drawing a Diamond given that you
drew a Diamond with your first pick is P(B|A) = 12/51.
So the probability that you will draw two consecutive Diamonds from the pack is 13/52 x 12/51 = 0.06.
Beginner’s Guide To Bayes’ Theorem and Bayesian Statistics #bayesian #bayes #statistics #datascience @chi2innovations
What is Bayes’ Rule?
So if the first card you picked was the 5 of Diamonds, and the second was the Queen of Diamonds, the probability of this outcome is P(A and B) = 0.06. We wrote this as:
Now imagine the same situation, but reversed. Your first pick was the Queen, and your second pick was the 5 of Diamonds. This would be the same outcome with the same probability, wouldn’t it? Yes it
would, so P(B and A) = 0.06 too, and we can interchange A and B to give us:
Knowing that it doesn’t matter whether you picked a 5 then a Queen or a Queen then a 5 means that P(A and B) = P(B and A), and if this is true (it is!), then the following is also true:
This is what is known as Bayes’ Rule and is at the heart of Bayes’ Theorem, which we can restate as (notice that I’ve rearranged it slightly):
In words, we can state Bayes’ Rule as the probability that a belief is true given new evidence (that’s the first part), when modified by the evidence (the second part) is equal to the probability
that the evidence is true given that the belief is true (the third part) when modified by the probability that the belief is true (the fourth part).
What Are Prior and Posterior Probabilities?
There is a wonderful symmetry to this equation, but you don’t often see it in this form. It will usually be explained in terms of events A and B, and when rearranged looks like this:
The left side of the equation is what we know as the Posterior probability (aka the Bayesian Posterior probability), and the right side is the Prior probability (aka the Bayesian Prior probability)
modified by the evidence you collected about events A and B individually.
We say that the probability that event A will happen given what we know about event B is equal to the probability that event B will happen given what we know about event A, modified by what we know
about events A and B independently from each other.
You can think of this in terms of a Prior probability (the guess or the Bayesian Prior) and a Posterior probability (the improved guess), based on the evidence (the information you’ve collected).
So now when you hear people discussing Prior and Posterior probabilities, you know what they’re talking about!
Bayes’ Theorem – An Example
Let’s say that you want to go to the beach today, a random day in April, and when you wake up it is cloudy. What are the chances that there will be rain?
Let’s re-write Bayes’ equation:
To answer the question, we need to know three things about cloud and rainfall in April:
And now plug the numbers in:
So the probability that it will rain today is 11.2%.
It looks like it’s time to head off to the beach!
How is Bayesian Statistics Different to Classical Statistics?
There is a schism in statistics, and it’s all Thomas Bayes’ fault! Maybe he didn’t publish his work because he knew it was going to cause problems…
The debate between classical statisticians (also known as frequentist statisticians) and Bayesian statisticians has been rumbling on for centuries and doesn’t look like stopping any time soon.
The difference between classical statistics and Bayesian statistics is quite a subtle one, but has huge implications from there on in.
Frequentist statistics tests whether an event occurs or not, and calculates the probability of an event in the long run.
Basically, classical statistics goes like this:
1. 1
You collect some data
2. 2
Then you calculate a feature of interest. Let’s say this is the central value (the mean)
3. 3
You then assume this sample mean is equal to the population mean
There’s nothing wrong with that is there?
Except the Bayesian will tell you that there is a huge problem with it.
Let’s take an example of flipping coins to see what the Bayesian is talking about.
The table below represents the number of Heads obtained from tossing coins a number of times:
10 6 0.6
50 22 0.44
100 50 0.5
500 266 0.53
1000 453 0.45
In classical statistics, the outcome is a fixed number. In any given experiment, the probability of Heads that you calculate is fixed. What you expect, at least in theory, is that the probability of
Heads will be exactly 0.5, irrespective of how many trials you run.
As you can see, it isn’t. It doesn’t matter how many times you run the experiment, you will get a slightly different outcome each time. In fact, if you have an odd number of trials you can never get
a probability of exactly 0.5 (look at the middle row where the probability is 0.5 – flipping the coin one more time will give you a probability that is not 0.5, irrespective of whether the outcome is
Heads or Tails).
What?!?? Either the outcome is fixed or it isn’t – it can’t be both!
If you’re a Bayesian statistician, that final column makes you feel rather uncomfortable, and for good reason.
The Bayesian statistician is right – the issue is that the result of an experiment in classical statistics is dependent on the number of times the experiment is repeated!
This is a fundamental problem for Bayesian statisticians, and is one that they solve by updating the probabilities whenever they get some new data.
In contrast to classical statistics, Bayesian statistics goes like this:
1. 1
You guess the value of the feature of interest. Let’s say this is the central value (the mean)
2. 2
You represent the uncertainty in this feature with a probability distribution (the Prior distribution)
3. 3
You collect some data (evidence)
4. 4
Then you use this new evidence to update the probability distribution accordingly (the Posterior distribution)
5. 5
From the Posterior distribution you can calculate the feature of interest
So if Bayesian statistics gets round this problem, why are frequentist statistics still the most widely used inferential technique?
Is Bayesian Statistics Better?
And no.
It depends on who you speak to.
Classical statistics are quick and easy to compute, but interpreting the results is difficult: results are often misunderstood and reported incorrectly. Don’t believe me? Try explaining what a
p-value is to your Granny!
As well as that, classical statistics often have a set of assumptions that mean that you cannot answer the question that you want to, so instead you ask a similar question and test that, hoping that
the answer is ‘close enough’ to what you want.
There are loads of Bayesian statisticians that believe that there is a complete disconnect between the questions that p-values can answer in the classical statistics framework and the questions that
we need answers to in the real world:
This is one of the main reasons that many choose to switch from classical statistics to Bayesian statistics.
This is what John Tukey said of classical statistics:
John Tukey
Far better an approximate answer to the right question, which is often vague, than the exact answer to the wrong question, which can always be made precise.
On the other hand, Bayesian statistics are difficult to compute, but interpreting the results is easy. One of the biggest drawbacks, though, is that the accuracy of your results are dependent on the
form of your Prior distribution – if your prior knowledge is accurate, you will get an accurate result, but an inaccurate Prior will yield an inaccurate result.
What is important to any statistician is understanding your assumptions and having the knowledge and experience of reaching the underlying story of the data. Ultimately, most of the time frequentist
and Bayesian statistics will give you the same answer.
Bottom line – it’s a choice, and there is no wrong or right answer in the battle of classical versus Bayesian statistics. It’s whatever you feel most comfortable with.
How Hard is it to Learn Bayesian Statistics?
If you use frequentist statistics, you don’t bother with all the integrals and heavy maths, do you? No, you use programs to do all that for you.
It’s the same with Bayesian statistics – unless you’re a world-class statistician, you probably look at Bayes’ Theorem and its equations and feel intimidated. Sure, it looks hard to learn Bayesian
statistics, but these days there are packages that do all the heavy lifting for you, so once you’ve learnt the basics, the practical aspects are quite straightforward.
The bottom line is that it looks harder to learn Bayesian Statistics than classical statistics, but looks can be deceiving – learning Bayesian statistics is just as easy (or hard, depending on your
point-of-view) as learning frequentist statistics!
When Should We Use Bayesian Statistics?
Let’s suppose that you run a poll of potential voters and you find that 53% of them intends to vote for candidate Joe. You compute a confidence interval of 43% to 63%, and you’ve got a p-value, 0.42.
So what is the probability that Joe will win?
Under this classical statistics framework you have no idea, but under the Bayesian statistics framework you can work it out.
Bayes’ Theorem is a tool for calculating conditional probabilities (probabilities that are reliant on other things, such as ‘what is the probability that event A occurs, given that event B has
occurred?’), so if the outcome you seek is a probability, then you should look to Bayesian statistics.
Bayesian Statistics – Summary
Hopefully I’ve given you a little flavour of Bayesian statistics and that you’re not put off by it. There are some clear advantages to using Bayesian statistics, but some disadvantages too. One thing
I think is clear is that more and more people will make the switch to Bayesian statistics as they realise that classical statistics just can’t answer the really important questions of Life, the
Universe and Everything…
Bayesian Statistics – Where To Learn More
There are lots of places online to learn about Bayes' Theorem and Bayesian statistics, and for all levels of ability.
We recommend the ones below, but there are others. If you recommend any that are not listed, let me know in the comments and I may add them here.
Bayesian Statistics Courses at Udemy
Below are a few related Udemy courses that we recommend. They are by no means the only ones - there are LOADS more, and if these don't tickle your fancy, just click through. I'm sure you'll find
something that does.
*NOTE - courses at Udemy are often listed as around 100 £/$/Euro, but they have sales very often when you can buy courses for around 10 £/$/Euro. If you want to find out the sale price, just click
Bayesian Statistics Courses at Coursera
Learning at Coursera is great! For a lot of their courses, you can study for free and pay a small fee at the end if you want the certificate.
Below are a few related Coursera courses. They are not the only ones, just click through and I'm sure you'll find something that you like.
Bayesian Statistics Courses at Pluralsight
Pluralsight is a great place for Data Ninjas to pick up some new skills.
If you want to level up your Python, Machine Learning or Hadoop skills (there are loads more), then you might want to check them out.
They have 6,500+ expert-led video courses, 40+ interactive courses and 20+ projects, and their plans start from £24 per month. There is also a free 10 day trial for those wanting to try first.
Below are a couple of Pluralsight courses that include Bayesian statistics.
This course covers the most important aspects of exploratory data analysis using different univariate, bivariate, and multivariate statistics from Excel and Python, including the use of Naive Bayes'
classifiers and Seaborn to visualize relationships.
We live in a world of big data, and someone needs to make sense of all this data. In this course, you will learn to efficiently analyze data, formulate hypotheses, and generally reason about what the
ocean of data out there is telling you.
Bayesian Statistics Courses at Datacamp
Datacamp is another great place for Data Ninjas to pick up some new skills.
They currently have 335+ expert-led video courses, 14 career tracks and 50+ skills tracks, and their plans start from $25 per month. There is also a free trial that includes the first chapter of all
courses for those wanting to try first.
Below is a Datacamp course that include Bayesian statistics.
Learn what Bayesian data analysis is, how it works, and why it is a useful tool to have in your data science toolbox.
Bayesian Statistics Courses - Final Word
When you've done any of these Bayesian statistics courses, please return and leave some feedback and a review in the comments below. If you loved the course, great - come and tell us. If you hated
it, that's great too - leave a comment saying what you didn't like about it.
If you discover any better Bayesian statistics courses out there, let me know - I may change my recommendations! | {"url":"https://www.chi2innovations.com/blog/beginners-guide-to-bayes-theorem-and-bayesian-statistics/","timestamp":"2024-11-09T12:16:00Z","content_type":"text/html","content_length":"731728","record_id":"<urn:uuid:fb314a1d-aa80-4747-b3df-75c347710741>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00796.warc.gz"} |
Drawing contours | Cycling '74 Documentation
Drawing contours
If you had a pencil and paper, and you had to draw a duck, what would your drawing look like? I had fun putting together a patch that takes a Jitter geomerty and draws some lines along the contours
of the mesh.
Pencil and paper
Open the patch contours.maxpat.
Looking at the render, you can see that the mesh outlines have been drawn. But, how can we identify which portions of the mesh should be considered part of the outlines?
Give a look at the patch:
The first step consists in grabbing a mesh, turning it into a Jitter geometry, and computing face normals using jit.geom.normgen. They will come in handy later on.
Then, jit.geom.todict converts the Jitter geometry into a dictionary accessible by JavaScript.
Now, double-click on the v8 object v8 geom.draw.contours.js to give a look at the custom geometry script.
// Go through all the edges
edges.forEach(edge => {
// An edge has one property, a list of the two half edge indexes
let he0 = edge.halfedges[0];
let he1 = edge.halfedges[1];
// get the index of the faces divided by this edge
let face0 = geom.halfedges[he0].face;
let face1 = geom.halfedges[he1].face;
// get their face normals
let normal0 = geom.faces[face0].normal;
let normal1 = geom.faces[face1].normal;
// compute the cosine of the angle formed by the two faces
let cosine = dot(normal0, normal1);
// if the cosine is minor than threshold, draw the contour!
if(cosine < threshold) {
// Get the actual vertex structure from the geometry
let v0 = geom.vertices[geom.halfedges[he0].from];
let v1 = geom.vertices[geom.halfedges[he0].to];
// Push the points into the list
The core algorithm is pretty simple: only draw the edges that are formed when two faces meet at a steep angle. To do this, iterate over the edges of the mesh, checking the orientation (face normals)
of the two faces divided by the edge. Then, if the cosine of the angle formed by the adjacent faces is less than than a user-defined threshold, draw a line connecting the endpoints of the edge. | {"url":"https://docs.cycling74.com/learn/articles/geom-contours/","timestamp":"2024-11-10T11:29:35Z","content_type":"text/html","content_length":"42220","record_id":"<urn:uuid:04cf3387-a3c2-4d6b-a394-83e8fe29223d>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00625.warc.gz"} |
Resource Library
The Pew Research Center For The People & The Press data archive page contains links to downloadable versions of the Center's survey data which are currently available on the web. Survey data are
released six months after the reports are issued and are posted on the web as quickly as possible. | {"url":"https://causeweb.org/cause/resources/library?field_material_type_tid=100&page=36","timestamp":"2024-11-04T02:33:56Z","content_type":"text/html","content_length":"71101","record_id":"<urn:uuid:1f9c06ba-7210-43a7-8d39-10bbe8b41f6b>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00762.warc.gz"} |
Variables r and R are gobbled
Variables r and R are gobbled
If implicit multiplication is not active, something like 4y^2 should raise an error. This is what actually happens:
sage: var("y"); 4y^2
File "<ipython-input-22-25bcbc82148c>", line 1
var("y"); 4y**Integer(2)
SyntaxError: invalid syntax
If we replace y by a different variable or consider a similar expression, we also get an error... except if the variable is r or R. For example,
sage: var("r, R"); 4r^2, 3R + 5r + 2R^3
(16, 16)
Surprisingly, there is no error. Variables r and R seem to be gobbled, so that SageMath parses 4r^2 as 4**Integer(2) and 3R + 5r + 2R^3 as 3 + 5 + 2**Integer(3). Why? Is this a bug?
1 Answer
Sort by » oldest newest most voted
r means raw python integer
edit flag offensive delete link more
Many thanks. I now understand. This is a source of potential problems, as happened to some of my students. They inadvertently did a wrong computation, like integral(sqrt(1+4r^2), r, 0, 1), and so
failed an exercise.
Juanjo ( 2020-06-23 11:49:40 +0100 )edit | {"url":"https://ask.sagemath.org/question/52168/variables-r-and-r-are-gobbled/","timestamp":"2024-11-04T23:20:39Z","content_type":"application/xhtml+xml","content_length":"55045","record_id":"<urn:uuid:1f7d88cd-5b09-483d-a13c-22af9f622abb>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00141.warc.gz"} |
HOW MANY OUNCES ARE IN 750 MILLILITERS? (WITH CHART!)
Have you ever needed to know, "How many ounces are in 750 milliliters?" You’ve seen it on wine bottles, whiskey labels, and even your favorite vodka or bourbon – that number, 750 ml. We’re breaking
it all down for you with the help of a handy chart.
Cheers! 🥂
Jump to:
🍶 HOW MANY OUNCES ARE IN 750 ML?
Let's start with the basics. 750 milliliters is the most common size for a standard bottle of wine, liquor, and other alcoholic beverages like vodka, bourbon, whiskey, or even just a regular bottle
of water. In terms of ounces, it translates to about 25.4 fluid ounces (oz).
HOW MANY MILLILITER IN 1 OUNCE?
There are approximately 29.57 milliliters (ml) in 1 fluid ounce (oz). For practical purposes, it’s rounded to 30 ml.
Milliliters (ml) Fluid Ounces (oz)
250 ml 8.45 oz
375 ml 12.68 oz
500 ml 16.91 oz
750 ml 25.4 oz
1,000 ml (1 liter) 33.81 oz
HOW MANY DRINKS ARE IN A 750 ML BOTTLE?
A 750 ml bottle of alcohol typically contains about 16 standard drinks if you pour 1.5 ounces per drink, which is a common serving size for spirits like whiskey, vodka, and bourbon.
For wine, since a standard serving is usually about 5 ounces, a 750 ml bottle would yield approximately 5 glasses of wine.
Liquid 750 ml Equals Additional Info
Wine 🍷 About 5 glasses Standard glass is 5 oz
Whiskey/Vodka/Bourbon 🥃 Roughly 16 drinks 1.5 oz per drink
Water 💧 About 25 ounces Just over 3 cups
Coffee/Milk/Cider ☕️ 25 ounces Same measurement applies for all; approx. 3 cups
HOW MANY FLUID OUNCES IS A SHOT?
A standard shot in the U.S. is typically 1.5 fluid ounces (oz). However, this can vary by country or even by the type of drink being served. For example, in the UK, a standard shot is often 1 fluid
ounce or 25 milliliters, while in some places, a double shot might be 2 ounces.
HOW BIG IS A 750ML BOTTLE OF WATER?
A 750 ml bottle of water holds about 25.4 fluid ounces. This is roughly equivalent to just over 3 cups of water. In terms of size, it's a little larger than a typical 16 oz (500 ml) water bottle but
smaller than a 1-liter (1000 ml) bottle.
WHICH IS MORE 500ML OR 16 OZ?
500 ml equals about 16.91 fluid ounces.
16 oz is approximately 473 ml.
So, 500 ml has just a bit more liquid than 16 oz, but the difference is pretty small!
HOW MANY 1 OZ POURS ARE IN A 750 ML BOTTLE?
In a 750 ml bottle, you can get approximately 25 one-ounce (1 oz) pours. Since 750 ml equals 25.4 fluid ounces, dividing that by 1 oz per pour gives you roughly 25 pours, making it a perfect size for
parties or mixing cocktails!
IS 750 ML THE SAME AS 16 OZ?
No, 750 ml is not the same as 16 oz.
750 ml equals approximately 25.4 fluid ounces (oz).
16 oz is about 473 ml.
So, 750 ml is significantly more than 16 oz.
💭 A FEW MORE QUICK OZ TO MLS CONVERSIONS
8 oz to ml?
8 ounces (oz) is equivalent to approximately 236.59 milliliters (ml).
This is a common measurement for a standard cup of water, coffee, or other beverages.
16oz in mls?
16 ounces (oz) is equivalent to approximately 473.18 milliliters (ml).
32 oz to ml?
32 ounces (oz) is equivalent to approximately 946.35 milliliters (ml).
Whether you're prepping for a party, whipping up a cocktail, or just tracking your water intake, knowing these conversions can come in handy.
Did this help you? Let me know what you think by leaving a ★★★★★ star rating & comment below. It genuinely helps me & I really appreciate your support! Nikki 💚 | {"url":"https://deliciouslynikki.com/6034/how-many-ounces-are-in-750-milliliters-with-chart/","timestamp":"2024-11-07T19:28:13Z","content_type":"text/html","content_length":"114854","record_id":"<urn:uuid:4feb3ae6-8973-4999-8ea4-abbf6283efa9>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00832.warc.gz"} |
Balkcom-Mason Curves
15.3.3 Balkcom-Mason Curves
In recent years, two more families of optimal curves have been determined [64,211]. Recall the differential-drive system from Section 13.1.2, which appears in many mobile robot systems. In many ways,
it appears that the differential drive is a special case of the simple car. The expression of the system given in (13.17) can be made to appear identical to the Reeds-Shepp car system in (15.48). For
example, letting and makes them equivalent by assigning and . Consider the distance traveled by a point attached to the center of the differential-drive axle using (15.42). Minimizing this distance
for any and is trivial, as shown in Figure 13.4 of Section 13.1.2. The center point can be made to travel in a straight line in . This would be possible for the Reeds-Shepp car if , which implies
that . It therefore appeared for many years that no interesting curves exist for the differential drive.
The problem, however, with measuring the distance traveled by the axle center is that pure rotations are cost-free. This occurs when the wheels rotate at the same speed but with opposite angular
velocities. The center does not move; however, the time duration, energy expenditure, and wheel rotations that occur are neglected. By incorporating one or more of these into the cost functional, a
challenging optimization arises. Balkcom and Mason bounded the speed of the differential drive and minimized the total time that it takes to travel from to . Using (13.16), the action set is defined
as , in which the maximum rotation rate of each wheel is one (an alternative bound can be used without loss of generality). The criterion to optimize is
which takes into account, whereas it was neglected in (15.42). This criterion is once again equivalent to minimizing the time to reach . The resulting model will be referred to as the Balkcom-Mason
drive. An alternative criterion is the total amount of wheel rotation; this leads to an alternative family of optimal curves [211].
Figure 15.11: The four motion primitives from which all optimal curves for the differential-drive robot can be constructed.
It was shown in [64] that only the four motion primitives shown in Figure 15.11 are needed to express time-optimal paths for the differential-drive robot. Each primitive corresponds to holding one
action variable fixed at its limit for an interval of time. Using the symbols in Figure 15.11 (which were used in [64]), words can be formed that describe the optimal path. It has been shown that the
word length is no more than five. Thus, any shortest paths may be expressed as a piecewise-constant action trajectory in which there are no more than five pieces. Every piece corresponds to one of
the primitives in Figure 15.11.
Figure 15.12: Each of the nine base words is depicted [64]. The last two are only valid for small motions; they are magnified five times and the robot outline is not drawn.
It is convenient in the case of the Balkcom-Mason drive to use the same symbols for both base words and for precise specification of primitives. Symmetry transformations will be applied to each base
word to yield a family of eight words that precisely specify the sequences of motion primitives. Nine base words describe the shortest paths:
This is analogous to the compressed forms given in (15.46) and (15.49). The motions are depicted in Figure 15.12.
Figure 15.13: The 40 optimal curve types for the differential-drive robot, sorted by symmetry class [64].
Figure 15.13 shows 40 distinct Balkcom-Mason curves that result from applying symmetry transformations to the base words of (15.51). There are 72 entries in Figure 15.13, but many are identical. The
transformation indicates that the directions of and are flipped from the base word. The transformation reverses the order of the motion primitives. The transformation flips the directions of and .
The transformations commute, and there are seven possible ways to combine them, which contributes to a row of Figure 15.13.
Figure 15.14: A slice of the optimal curves is shown for and [64]. Level sets of the optimal cost-to-go are displayed. The coordinates correspond to a differential drive with in (13.16).
To construct an LPM or distance function, the same issues arise as for the Reeds-Shepp and Dubins cars. Rather than testing all words to find the shortest path, simple tests can be defined over which
a particular word becomes active [64]. A slice of the precise cell decomposition and the resulting optimal cost-to-go (which can be called the Balkcom-Mason metric) are shown in Figure 15.14.
Steven M LaValle 2020-08-14 | {"url":"https://lavalle.pl/planning/node823.html","timestamp":"2024-11-03T05:48:42Z","content_type":"text/html","content_length":"16918","record_id":"<urn:uuid:62c6d3ff-0c10-4dba-ae2d-375d566e7a89>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00430.warc.gz"} |
Testing for a causal effect (with 2 time series) | R-bloggersTesting for a causal effect (with 2 time series)
Testing for a causal effect (with 2 time series)
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A few days ago, I came back on a sentence I found (in a French newspaper), where someone was claiming that
“… an old variable explains 85% of the change in a new variable. So we can talk about causality”
and I tried to explain that it was just stupid : if we consider the regression of the temperature on day \(t+1\) against the number of cyclist on day \(t\), the \(R^2\) exceeds 80%… but it is hard to
claim that the number of cyclists on specific day will actually cause the temperature on the next day…
Nevertheless, that was frustrating, and I was wondering if there was a clever way to test for causality in that case. A popular one is Granger causality (I can mention a paper we published a few
years ago where we use such a test, Tents, Tweets, and Events: The Interplay Between Ongoing Protests and Social Media). To explain that test, consider a bivariate time series (just like the one we
have here), \(\boldsymbol{z}_t=(x_t,y_t)\), and consider some bivariate autoregressive model
\({\displaystyle {\begin{bmatrix}x_{t}\\y_{t}\end{bmatrix}}={\begin{bmatrix}c_{1}\\c_{2}\end{bmatrix}}+{\begin{bmatrix}a_{1,1}&\textcolor{red}{a_{1,2}}\\\textcolor{blue}{a_{2,1}}&a_{2,2}\end
{bmatrix}}{\begin{bmatrix}x_{t-1}\\y_{t-1}\end{bmatrix}}+{\begin{bmatrix}u_{t}\\v_{t}\end{bmatrix}}}\)where \(\boldsymbol{\varepsilon}_t=(u_t,v_t)\) is some bivariate white noise, in the sense that
(i) \({\displaystyle \mathbb{E} (\boldsymbol{\varepsilon}_{t})=\boldsymbol{0}}\) (the noise is centered) (ii) \({\displaystyle \mathbb{E} (\boldsymbol{\varepsilon}_{t}\boldsymbol{\varepsilon}_{t}^\
top)=\Omega } \), so the variance matrix is constant, but possibly non-diagonal (iii) \({\displaystyle \mathbb{E} (\boldsymbol{\varepsilon}_{t}\boldsymbol{\varepsilon}_{t-h}^\top)=\boldsymbol{0} } \)
for all \(h\neq 0\). Note that we can use the simplified expression\({\displaystyle {\boldsymbol{z}_t=\boldsymbol{c}+\boldsymbol{A}\boldsymbol{z}_{t-1}+\boldsymbol{\varepsilon}_t}}\)Now, Granger test
is based on several quantities. With off-diagonal terms of matrix \(\Omega\), we have a so-called instantaneous causality, and since \(\Omega\) is symmetry, we will write \(x\leftrightarrow y\). With
off-diagonal terms of matrix \(\boldsymbol{A}\), we have a so-called lagged causality, with either \(\textcolor{blue}{x\rightarrow y}\) or \(\textcolor{red}{x\leftarrow y}\) (and possibly both, if
both terms are significant).
So I wanted to try on my two-variable problem.
df = read.csv("cyclistsTempHKI.csv")
dfts = cbind(C=ts(df$cyclists,start = c(2014, 1,2), frequency = 365),
T=ts(df$meanTemp,start = c(2014, 1,2), frequency = 365))
I now have “time series” objects, and we can fit a VAR model,
var2 = VAR(dfts, p = 1, type = "const")
Estimate Std. Error t value Pr(>|t|)
C.l1 0.8684009 0.02889424 30.054460 8.080226e-107
T.l1 70.3042012 20.07247411 3.502518 5.102094e-04
const 807.6394001 187.75472482 4.301566 2.110412e-05
Estimate Std. Error t value Pr(>|t|)
C.l1 0.0003865391 6.257596e-05 6.177118 1.540467e-09
T.l1 0.6611135594 4.347074e-02 15.208241 6.086394e-42
const -1.6413074565 4.066184e-01 -4.036481 6.446018e-05
For instant, we can run a causality, to test if the number of cyclists can cause the temperature (on the next day)
causality(var2, cause = "C")
Granger causality H0: C do not Granger-cause T
data: VAR object var2
F-Test = 38.157, df1 = 1, df2 = 842, p-value = 1.015e-09
Here, we should clearly reject \(H_0\), which is that there is no causal effect. Which is the way statistician say that there should be some causal effect between the number of cyclist and the
So clearly, something is wrong here. Either it is some sort of superpower that cyclists are not aware of. Or this test that was used for forty years (Clive Granger even got a Nobel price for it) is
not working. Or we missed something. Actually… I think we missed something here. The series are not stationary. We can almost see it with
Phi = matrix(c(coefficients(var2)$C[1:2,1],coefficients(var2)$T[1:2,1]),2,2)
eigen() decomposition
[1] 0.9594810 0.5700335
where the highest eigenvalue is very close to one. But actually, we look here at the temperature…
i.e. at least, we should expect some seasonal unit root here. So let us use two techniques. The first one is a classical one-year difference, \(\Delta_{365}\boldsymbol{z}_t=\boldsymbol{z}_t-\
var2 = VAR(diff(dfts,365), p = 1, type = "const")
Estimate Std. Error t value Pr(>|t|)
C.l1 0.8376424 0.07259969 11.537823 1.993355e-16
T.l1 42.2638410 28.58783276 1.478386 1.449076e-01
const -507.5514795 219.40240747 -2.313336 2.440042e-02
Estimate Std. Error t value Pr(>|t|)
C.l1 0.000518209 0.0003277295 1.5812096 1.194623e-01
T.l1 0.598425288 0.1290511945 4.6371154 2.162476e-05
const 0.547828079 0.9904263469 0.5531235 5.823804e-01
The test on the fited VAR model yields
causality(var2, cause = "C")
Granger causality H0: C do not Granger-cause T
data: VAR object var2
F-Test = 2.5002, df1 = 1, df2 = 112, p-value = 0.1167
i.e., with a 11% \(p\)-value, we should reject the assumption that the number of cyclists cause the temperature (on the next day), and actually, we should also reject the other way
causality(var2, cause = "T")
Granger causality H0: T do not Granger-cause C
data: VAR object var2
F-Test = 2.1856, df1 = 1, df2 = 112, p-value = 0.1421
Nevertheless, if we look at the instantaneous causality, this one makes more sense
H0: No instantaneous causality between: T and C
data: VAR object var2
Chi-squared = 13.081, df = 1, p-value = 0.0002982
The second idea would be to use a one day difference, \(\Delta_{1}\boldsymbol{z}_t=\boldsymbol{z}_t-\boldsymbol{z}_{t-1}\) and to fit a VAR model on that one
VARselect(diff(dfts,1), lag.max = 4, type="const")
AIC(n) HQ(n) SC(n) FPE(n)
but on that one, a VAR(1) model – with only one lag – might not be sufficient. It might be better to consider a VAR(3)
var2 = VAR(diff(dfts,1), p = 3, type = "const")
and on that one, one more time, we should reject the causal effect of the number of cyclists on the temperature (on the next day)
causality(var2, cause = "C")
Granger causality H0: C do not Granger-cause T
data: VAR object var2
F-Test = 0.67644, df1 = 3, df2 = 828, p-value = 0.5666
and this time, there could be a (lagged) causal effect of the temperature on the number of cyclists
causality(var2, cause = "T")
Granger causality H0: T do not Granger-cause C
data: VAR object var2
F-Test = 7.7981, df1 = 3, df2 = 828, p-value = 3.879e-05
H0: No instantaneous causality between: T and C
data: VAR object var2
Chi-squared = 55.83, df = 1, p-value = 7.905e-14
but nothing instantaneously… So it looks like Granger causality performs well on that one ! | {"url":"https://www.r-bloggers.com/2020/02/testing-for-a-causal-effect-with-2-time-series/","timestamp":"2024-11-05T19:59:57Z","content_type":"text/html","content_length":"102124","record_id":"<urn:uuid:830966c2-733d-4b3c-9011-84d9c8c6508c>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00148.warc.gz"} |
Lecture Notes 5.11. and 6.11.2012
The euclidean plane isometry consists of 5 motions:
• Identity
• Reflection
• Rotation
• Translation
• Glide Reflection
Suppose $G<E_2$ is a discrete group. First consider the translation subgroup of $G$. It holds $G\cap \mathbb{R}^2 < E_2$. We have
• $E_2 = \mathbb{R}^2 \rtimes O_2 $.
• $E_2 \overset{\varphi}{\longrightarrow} O_2$.
• $ker(\varphi) = \mathbb{R}^2 = $Translations.
• $ker(\varphi|_G)= G \cap \mathbb{R}^2$.
The discrete translation group forms a lattice of dimension $k= 0,1,2$.
$k=1:$ There are no translations in $G$. This implies that $G$ has a fixed point ($G$ has no glides). Thus $G<O_2$ is a finite subgroup $G=C_n$ or $D_n$ for some $n=1,2,3$. As groups of
symmetries in $\mathbb{E}^2$ these are called the rosette groups or point groups since they leave at least one point fixed.
$G=C_n$ are called the cyclic group.
The cup (first object) is $C_2$ since it has a $180^°$ symmetry. The recycling logo (2nd object) is $C_3$ and the star (3rd object) $C_5$ since the they are $120^°$ and $72^°$ symmetric. The group
$C_1$ has no symmetry and can be represented by an L or an R.
$G=D_n$ are called the dihedral group.
Starting with the upper left the square is $D_4$, the rhombus is $D_2$ the triangle $D_1$. The dihedral group is the group of regular polygons, so an n-gon is $D_n$.
$k=1:$ These groups are called the frieze groups. They have the following definition:
$G\cap \mathbb{R}^2 = \{ \tau_{nv} : v\neq 0 \in \mathbb{R}^2, n\in\mathbb{Z} \} = \langle \tau_v\rangle\cong \mathbb{Z}$.
$k=2:$ These groups are called the wallpaper groups. They have the following definition:
$G\cap \mathbb{R}^2 = \langle \tau_v, \tau_w\rangle \{ \tau_{nv+mw} : v,w\neq 0 \in \mathbb{R}^2, n,m\in\mathbb{Z} \} \cong \mathbb{Z}^2$.
There are seven distinct subgroups (up to scaling and shifting of patterns) in the discrete frieze group generated by a translation, reflection (along the same axis) and a 180° rotation. An overview
gives the following picture
The Orbifold notation was the name we used in the lecture.
Take a width-$n$ strip $[0,n]\times\mathbb{R}\subset\mathbb{R}$ and glue its edges to give a cylinder. Any of our seven groups will give a group symmetry of the cylinder with exactly $n$-fold
rotation around vertical axis. This gives $7$ families of symmetry groups, indexed by $n$. These are finite subgroups of $O_3$. They are exactly the ones that preserve the equator (or equivalent
preserve the poles). These are called the axial groups:
• $nn$, rotations (abstr: $\mathbb{Z}_n$)
• $nx$, rotary directions (abstr: $\mathbb{Z}_{2n}$)
• $n*$, (abstr: $\mathbb{Z}_n \times \mathbb{Z}_2$)
• $*nn$, (abstr: $D_n$)
• $22n$, (abstr: $D_n$)
• $2*n$, (abstr: $D_{2n}$)
• $*22n$, ($D_n \times \mathbb{Z}_2$)
Special cases for $n=2$:
• $22$, a 2-fold rotation (abstr: $\mathbb{Z}_2$)
• $2X$, a 4-fold rotary reflection (abstr: $\mathbb{Z}_4$)
• $2*$, rotation, mirror, and antipodal map (abstr: $\mathbb{Z}_2^2$)
• $*22$, 2 mirrors which yield a rotation (abstr: $\mathbb{Z}_2^2$)
• $222$, rotations (abstr: $\mathbb{Z}_2^2$), the Klein-4-group
• $2*2$, rotation and mirror, (abstr: $D_4$)
• $*222$ only reflections, (abstr: $\mathbb{Z}_2^3$)
Note that the abstract groups sometimes look different as, e.g., for the group $222$, which should have $D_2$ as its abstract group instead of $\mathbb{Z}_2$. These are equivalent because
$D_{2} \cong D_1 \times \mathbb{Z}_2 \cong \mathbb{Z}_2$.
In general, for odd $m \in \mathbb{N}$, it holds
$D_{2m} \cong D_{m} \times \mathbb{Z}_2$, and $\mathbb{Z}_{2m} \cong \mathbb{Z}_m \times \mathbb{Z}_2$.
For $n=1$, only three different groups remain:
• $11 = 1$, the trivial group
• $1x = x$, identity and antipodal map (abstr: $\mathbb{Z}_2$)
• $1* = * = *11$, a single mirror, (abstr: $\mathbb{Z}_2$)
Of course, there are more discrete subgroups of $O_3$. Most of them correspond to the symmetry groups of the platonic solids. First, we give the theorem:
Theorem: The finite subgroups of $O_3$ are the 7 families of axial groups and the 7 platonic symmetry groups $*235$, $235$ (symmetries of icosahedron and its dual, the dodecahedron), $*234$, $234$
(symmetries of the cube and the dodecahedron), $*233$, $233$ (tetrahedron), and $3*2$ (the so-called pyritohedral symmetry).
The figure on the left shows the icosahedron inside of a dodecahedron. Recall that the dual or reciprocal of a solid evolves from faces becoming vertices, roughly speaking. A symmetry of one solid is
also a symmetry of its dual.
The symmetry group is $*235$, which has the rotation part $235$ as subgroup. (http://apollonius.math.nthu.edu.tw/d1/dg-07-exe/943251/dynamic/duality.htm)
You can easily make a tetrahedron yourself by using the figure to the right, and search for the symmetry groups yourself.
Note that the tetrahedron is dual to itself.
The cube and its dual counterpart, the octahedron, have the symmetry group $*234$, and $234$. The last symmetry group, the pyritohedral symmetry group, comes as the intersection of the octahedral and
the dodecahedral symmetry groups,
$3*2 = *235 \cap *234$.
In the figure below, you see a pyritohedron, which is not platonic (note that the faces are irregular pentagons).
Wallpaper groups
Back to $E_2$ (the euclidean motions in the plane), the discrete subgroups which have a translational part of dimension 2 are called wallpaper groups. In the figure below, you find an exhaustive list
of the 17 wallpaper groups, each with its fundamental domain, and generating symmetries.
1 thought on “Lecture Notes 5.11. and 6.11.2012”
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About | Steven A A Bauer
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Hello and welcome.
My name is Steven Bauer.
I hold an undergraduate degree in general mathematics; however, I am also a self-taught mathematician. Because of this, I use an unorthodox and creative approach to solving life's grandest questions.
It is in the arena of independent study that I have been able to take calculated risks along the way that many would not otherwise. Much of my work is cutting edge research in the fields of math,
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Shortly after graduating from university in December 2012, I stumbled onto the work of Marko Rodin, followed by the work of Doug Vogt. The discovery of these two scientists jump-started my quest for
an Information Based Theory of Existence. Rodin's work gave me the mathematical framework I needed to generate many of the ideas found within my first book, A Biblical Perspective. From there Vogt's
work inspired me to rewrite the fundamental constants of physics in a new and profound way as found in Infinite Subdivisions.
People will ask me what initially compelled me to follow their work. Put simply: they are two of the greatest minds of our generation. This is usually followed up by the question: "What caused you to
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of the dreams that have shaped my life and understanding of the universe.
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Our home is un-random, a cosmic clock which is built with numerical relativity and Godly precision.
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Mathematical Proceedings of the Cambridge Philosophical Society: Volume 105 - Issue 2 | Cambridge Core
For any prime number p, let Γn, p denote the congruence subgroup of SLn(ℤ) of level p, i.e. the kernel of the surjective homomorphism fp: SLn(ℤ) → SLn(p) induced by the reduction mod p (Fp is the
field with p elements). We define
using upper left inclusions Γn, p ↪ Γn+1, p. Recall that the groups Γn, p are homology stable with M-coefficients, for instance if M = ℚ, ℤ[1/p], or ℤ/q with q prime and q ╪ p: Hi(Γn, p; M) ≅ Hi(Γp;
M) for n ≥ 2i + 5 from [7] (but the homology stability fails if M = ℤ or ℤ/p). | {"url":"https://core-cms.prod.aop.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/issue/D20EA2DBC789180D140285211FC3AB4A","timestamp":"2024-11-13T08:59:23Z","content_type":"text/html","content_length":"935627","record_id":"<urn:uuid:952fdbd5-5b13-473d-b8fe-f8b9f101a770>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00207.warc.gz"} |
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Energy intensity — energy use per dollar of GDP — is the last refuge of fossil fuel proponents. Instead of measuring real improvement in energy efficiency, it hides the outsourcing of dirty,
coal-based manufacturing to developing countries and changes during times of economic growth or recession, irrespective of efficiency. | {"url":"https://www.resilience.org/tag/fossilfuelsubsidies/","timestamp":"2024-11-05T19:16:33Z","content_type":"text/html","content_length":"140450","record_id":"<urn:uuid:05804349-2b9b-4c5c-8525-9fa9c9977c8f>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00214.warc.gz"} |
Voltage Drop Calculator
Understanding Voltage Drop
Voltage Drop is a critical concept in electrical engineering and physics that describes the reduction in electrical potential as current flows through a conductor or circuit element. It is essential
for designing efficient electrical systems, ensuring safety, and optimizing performance in various applications. Accurately calculating voltage drop helps in selecting appropriate wire sizes,
preventing energy losses, and maintaining the integrity of electrical devices.
Did you know? Excessive voltage drop in a circuit can lead to malfunctioning of electrical devices, reduced energy efficiency, and potential safety hazards.
In this comprehensive guide, we will explore the concept of voltage drop, delve into the methods for calculating it, discuss its applications across different fields, and provide real-world examples
to enhance your understanding. Whether you’re a student, electrician, or electronics enthusiast, this article aims to equip you with the knowledge to accurately calculate and interpret voltage drop.
Understanding Voltage Drop
Voltage Drop (\(V_d\)) occurs when electrical current flows through a conductor, causing a decrease in voltage from the source to the load. This phenomenon is a natural consequence of the resistance
(\(R\)) inherent in all conductive materials. The voltage drop is directly proportional to the current (\(I\)) flowing through the conductor and the resistance it encounters.
Key Point: Voltage Drop is influenced by the length and thickness of the conductor, the material’s resistivity, and the amount of current passing through it.
Understanding voltage drop is vital for designing efficient electrical systems, preventing energy losses, and ensuring that electrical devices operate within their intended voltage ranges. It is also
crucial for maintaining safety standards in electrical installations, as excessive voltage drop can lead to overheating and potential fire hazards.
How to Calculate Voltage Drop
Calculating Voltage Drop involves understanding the relationship between current, resistance, and the properties of the conductor. The fundamental principle used to calculate Voltage Drop is Ohm’s
Law, which establishes the relationship between voltage, current, and resistance.
Ohm’s Law:
\[ V = I \times R \]
V = Voltage (Volts, V) I = Current (Amperes, A) R = Resistance (Ohms, Ω)
Ohm’s Law states that Voltage (\(V\)) is equal to the product of Current (\(I\)) and Resistance (\(R\)). This fundamental relationship is the basis for calculating Voltage Drop in electrical
To calculate Voltage Drop (\(V_d\)), you can rearrange Ohm’s Law as follows:
This formula allows you to determine the voltage lost across a conductor when a specific current flows through it.
Key Equations for Calculating Voltage Drop
To accurately calculate Voltage Drop, it’s essential to understand the key equations and their applications. Below are the primary formulas used in the computation.
Ohm’s Law:
\[ V = I \times R \]
V = Voltage (V) I = Current (A) R = Resistance (Ω)
Ohm’s Law is the foundational equation for calculating Voltage Drop. It applies to both direct current (DC) and alternating current (AC) circuits, providing a straightforward method to determine
voltage changes due to resistance.
Voltage Drop Formula:
\[ V_d = I \times R \]
V_d = Voltage Drop (V) I = Current (A) R = Resistance (Ω)
This specific application of Ohm’s Law focuses on determining the Voltage Drop across a conductor or circuit element. By knowing the current and resistance, you can calculate how much voltage is lost
in the process.
Resistance of a Conductor:
\[ R = \rho \times \frac{L}{A} \]
R = Resistance (Ω) \(\rho\) = Resistivity of the Material (Ω·m) L = Length of the Conductor (m) A = Cross-Sectional Area (m²)
This equation calculates the resistance of a conductor based on its material properties, length, and cross-sectional area. Knowing the resistivity of the material allows for more precise Voltage Drop
Combined Voltage Drop Formula:
\[ V_d = I \times \rho \times \frac{L}{A} \]
V_d = Voltage Drop (V) I = Current (A) \(\rho\) = Resistivity (Ω·m) L = Length (m) A = Cross-Sectional Area (m²)
By combining Ohm’s Law with the resistance formula, you obtain a comprehensive equation for Voltage Drop that accounts for material properties and physical dimensions of the conductor.
Mastery of these equations allows for precise calculations of Voltage Drop in various scenarios, from residential wiring to complex industrial systems.
Applications of Voltage Drop in Science and Technology
Voltage Drop is a fundamental concept with wide-ranging applications across multiple fields. Understanding and accurately calculating voltage drop is essential for designing efficient electrical
systems, ensuring safety, and optimizing performance in various technological domains.
Residential and Commercial Electrical Systems
In building electrical systems, managing Voltage Drop is crucial for ensuring that all outlets and devices receive adequate voltage. Proper wire sizing and circuit design help minimize voltage loss,
preventing devices from underperforming or malfunctioning.
Electrical codes and standards often specify maximum allowable Voltage Drop to ensure safety and efficiency. Compliance with these standards is essential for building inspectors and electricians
during installation and maintenance.
Industrial Machinery and Equipment
Industrial settings rely on large machinery and equipment that consume significant amounts of power. Calculating Voltage Drop helps in designing robust power distribution systems that maintain
consistent voltage levels, ensuring reliable operation and preventing equipment damage.
Effective Voltage Drop management in industrial applications enhances productivity, reduces energy losses, and extends the lifespan of costly machinery.
Automotive Electrical Systems
Modern vehicles are equipped with numerous electrical components, including lighting, infotainment systems, and engine control units. Calculating Voltage Drop is essential for designing efficient
wiring harnesses that ensure all components receive stable power, improving vehicle performance and reliability.
Proper Voltage Drop calculations in automotive systems prevent issues like flickering lights, poor sensor performance, and electrical failures, enhancing overall vehicle safety and functionality.
Renewable Energy Systems
In renewable energy systems, such as solar and wind power installations, Voltage Drop calculations are vital for optimizing power transmission from energy sources to storage systems or the grid.
Minimizing voltage loss enhances system efficiency and energy yield.
Accurate Voltage Drop assessments in renewable energy setups contribute to sustainable energy solutions by maximizing power delivery and reducing unnecessary energy losses.
Electronics and Circuit Design
In electronics, managing Voltage Drop is critical for designing stable and efficient circuits. Ensuring that components receive the correct voltage levels prevents malfunction and ensures the proper
operation of electronic devices.
Voltage Drop considerations in circuit design are essential for developing reliable consumer electronics, medical devices, and communication equipment.
Real-World Example: Calculating Voltage Drop
Let’s walk through a practical example of calculating Voltage Drop. Suppose you have the following data:
• Current (\(I\)): 10 A
• Length of the Conductor (\(L\)): 50 m
• Cross-Sectional Area (\(A\)): 2.5 mm²
• Resistivity of the Material (\(\rho\)): \(1.68 \times 10^{-8} \, \Omega \cdot m\) (Copper)
Step-by-Step Calculation
Step 1: Understand the Given Values
• Current (\(I\)) = 10 A
• Length (\(L\)) = 50 m
• Cross-Sectional Area (\(A\)) = 2.5 mm² = \(2.5 \times 10^{-6} \, m²\)
• Resistivity (\(\rho\)) = \(1.68 \times 10^{-8} \, \Omega \cdot m\) (for Copper)
Step 2: Calculate the Resistance (\(R\))
Using the resistance formula:
\[ R = \rho \times \frac{L}{A} \]
Plugging in the values:
\[ R = 1.68 \times 10^{-8} \, \Omega \cdot m \times \frac{50 \, m}{2.5 \times 10^{-6} \, m²} = 1.68 \times 10^{-8} \times 2 \times 10^{7} = 0.336 \, \Omega \]
Therefore, the resistance (\(R\)) of the conductor is 0.336 Ω.
Step 3: Calculate the Voltage Drop (\(V_d\))
Using Ohm’s Law:
Plugging in the values:
\[ V_d = 10 \, A \times 0.336 \, \Omega = 3.36 \, V \]
Therefore, the Voltage Drop (\(V_d\)) across the conductor is 3.36 V.
This example demonstrates how to apply the Voltage Drop formula using current, conductor length, cross-sectional area, and resistivity values. Accurate calculations like these are essential for
selecting appropriate wire sizes, ensuring efficient power distribution, and maintaining the performance of electrical systems.
Challenges in Calculating Voltage Drop
While calculating Voltage Drop is fundamental in various fields, several challenges can arise, especially when dealing with complex electrical systems or requiring high precision. Understanding these
challenges is crucial for accurate analysis and application.
Challenge: Accurately measuring resistance in conductors with varying temperatures can be difficult due to temperature-induced changes in material properties.
One primary challenge is the accurate measurement of resistance (\(R\)) in conductors. Resistance can vary with temperature, and fluctuations can lead to inaccuracies in Voltage Drop calculations.
Using temperature-compensated resistors or conducting measurements in controlled environments can help mitigate this issue.
Another consideration is the impact of conductor length and cross-sectional area on resistance. In large-scale electrical systems, such as power distribution networks, even small errors in measuring
length or area can result in significant discrepancies in Voltage Drop calculations.
Consideration: Precise measurements of conductor dimensions and properties are essential to ensure accurate Voltage Drop calculations.
Additionally, in AC circuits, factors such as inductance and capacitance can influence Voltage Drop. These reactive components introduce complexities beyond simple Ohm’s Law calculations, requiring
more advanced analysis techniques like impedance calculations.
Measurement limitations also pose challenges. High-current systems or circuits with very low resistance require sensitive instruments to measure Voltage Drop accurately. Ensuring that measurement
tools are calibrated and suitable for the specific application is essential for reliable results.
Challenge: Measuring Voltage Drop in high-current or low-resistance circuits demands specialized and calibrated instruments to ensure accuracy.
Furthermore, environmental factors such as humidity, corrosion, and physical wear can affect conductor resistance over time, leading to changes in Voltage Drop that must be accounted for in long-term
electrical system maintenance and design.
Voltage Drop is a fundamental concept that significantly impacts the design, efficiency, and safety of electrical systems. Understanding how to calculate Voltage Drop and the factors that influence
it is essential for engineers, electricians, and anyone involved in electrical design and maintenance.
Mastering the calculations of Voltage Drop equips professionals with the tools necessary to analyze and interpret electrical circuit behavior, optimize power distribution, and ensure that electrical
devices operate within their intended voltage ranges. Despite the inherent challenges in measurement and calculation, the principles of Voltage Drop remain integral to our understanding and
management of electrical systems in the physical world.
As technology continues to evolve, the applications of Voltage Drop expand, driving progress in fields like renewable energy, electronics, automotive engineering, and construction. Embracing the
complexities and intricacies of Voltage Drop calculations empowers professionals and enthusiasts alike to contribute to advancements in energy efficiency, sustainable design, and the enhancement of
everyday electrical systems. | {"url":"https://turn2engineering.com/calculators/voltage-drop-calculator","timestamp":"2024-11-13T07:49:28Z","content_type":"text/html","content_length":"215814","record_id":"<urn:uuid:cd80eced-2fad-410c-9095-e0151bfdfbdf>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00021.warc.gz"} |
Topics in Harmonic Analysis and Ergodic Theorysearch
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Topics in Harmonic Analysis and Ergodic Theory
eBook ISBN: 978-0-8218-8123-1
Product Code: CONM/444.E
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Topics in Harmonic Analysis and Ergodic Theory
eBook ISBN: 978-0-8218-8123-1
Product Code: CONM/444.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
• Contemporary Mathematics
Volume: 444; 2007; 228 pp
MSC: Primary 37; 42; 47; 60; 65
There are strong connections between harmonic analysis and ergodic theory. A recent example of this interaction is the proof of the spectacular result by Terence Tao and Ben Green that the set of
prime numbers contains arbitrarily long arithmetic progressions. The breakthrough achieved by Tao and Green is attributed to applications of techniques from ergodic theory and harmonic analysis
to problems in number theory.
Articles in the present volume are based on talks delivered by plenary speakers at a conference on Harmonic Analysis and Ergodic Theory (DePaul University, Chicago, December 2–4, 2005). Of ten
articles, four are devoted to ergodic theory and six to harmonic analysis, although some may fall in either category. The articles are grouped in two parts arranged by topics. Among the topics
are ergodic averages, central limit theorems for random walks, Borel foliations, ergodic theory and low pass filters, data fitting using smooth surfaces, Nehari's theorem for a polydisk,
uniqueness theorems for multi-dimensional trigonometric series, and Bellman and \(s\)-functions.
In addition to articles on current research topics in harmonic analysis and ergodic theory, this book contains survey articles on convergence problems in ergodic theory and uniqueness problems on
multi-dimensional trigonometric series.
Research mathematicians interested in harmonic analysis, ergodic theory, and their interaction.
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Requests
Volume: 444; 2007; 228 pp
MSC: Primary 37; 42; 47; 60; 65
There are strong connections between harmonic analysis and ergodic theory. A recent example of this interaction is the proof of the spectacular result by Terence Tao and Ben Green that the set of
prime numbers contains arbitrarily long arithmetic progressions. The breakthrough achieved by Tao and Green is attributed to applications of techniques from ergodic theory and harmonic analysis to
problems in number theory.
Articles in the present volume are based on talks delivered by plenary speakers at a conference on Harmonic Analysis and Ergodic Theory (DePaul University, Chicago, December 2–4, 2005). Of ten
articles, four are devoted to ergodic theory and six to harmonic analysis, although some may fall in either category. The articles are grouped in two parts arranged by topics. Among the topics are
ergodic averages, central limit theorems for random walks, Borel foliations, ergodic theory and low pass filters, data fitting using smooth surfaces, Nehari's theorem for a polydisk, uniqueness
theorems for multi-dimensional trigonometric series, and Bellman and \(s\)-functions.
In addition to articles on current research topics in harmonic analysis and ergodic theory, this book contains survey articles on convergence problems in ergodic theory and uniqueness problems on
multi-dimensional trigonometric series.
Research mathematicians interested in harmonic analysis, ergodic theory, and their interaction.
Permission – for use of book, eBook, or Journal content
Please select which format for which you are requesting permissions. | {"url":"https://bookstore.ams.org/CONM/444","timestamp":"2024-11-02T11:36:33Z","content_type":"text/html","content_length":"78195","record_id":"<urn:uuid:a4046e54-78f1-4ee1-92d4-0fae35ac7401>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00084.warc.gz"} |
CourseNana | [2021] Economics 482 Game Theory and Economics - Midterm Exam - Q2 Cournot Competition
2. The following is a version of the Cournot competition game with four firms, in which the firms simultaneously choose nonnegative quantities of a good to produce. When the firms’ choices are the
profile (q1, q2, q3, q4), the price at which the firms sell their output is α − q1 − q2 − q3 − q4 if q1 +q2 +q3 +q4 < α; otherwise, the price of the good they sell is zero. Each firm i has a cost of
producing qi units of output of ci ×qi, where ci ≥ 0; different firms can have different cost functions. A firm’s payoff is its profit, the revenue it gets from selling its output qi minus its cost.
1. (a) Write down the expression for the payoff firm 1, as function of all of the firms’ choices of quantities, as well as of α and any of the marginal costs ci, for the case in which q1+q2+q3+q4 < α
. [Note: other firms’ payoffs will look similar.] (4 points)
2. (b) Find firm 1’s best response function for this game, for values of (q2, q3, q4) such that q2 + q3 + q4 + c1 < α. [Note: other firms’ best-response functions will look similar.] (6 points)
3. (c) Suppose that firm 1 has a cost function such that c1 = 0; and that firms 2, 3 and 4 have the same cost function: c2 = c3 = c4 = c. Find a Nash equilibrium of this game in which firms 2, 3, and
4 choose the same quantity as each other (but not, in general, the same quantity as firm 1), with all quantities expressed as functions of α and c. You may assume that each firm chooses a positive
quantity in the equilibrium (and, as such, determines its best response as in part (b)), which will be true as long as α is sufficiently greater than c. Remember that since firms 2, 3, and 4 are
assumed to choose the same output as each other, you are really only solving for two outputs. (10 points)
4. (d) Suppose that firms 2, 3 and 4’s cost per unit c increases, while firm 1’s remains at 0. Say whether this makes firm 1’s equilibrium quantity greater or smaller; whether it makes firm 2, 3 and
4’s equilibrium quantities greater or smaller; whether it makes the total quantity produced by all firms greater or smaller; and whether it increases or decreases the price of the good that consumers
pay in the equilibrium. (7 points) | {"url":"https://coursenana.com/exam/question/economics-482-game-theory-and-economics-midterm-q2-cournot-competition","timestamp":"2024-11-12T20:04:59Z","content_type":"text/html","content_length":"51106","record_id":"<urn:uuid:13a814f0-c8d6-4e3a-b34b-ce3a3d87e716>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00573.warc.gz"} |
Here, we provide descriptions of the bioinformatic techniques and methods used in Omics Playground. You may want to use these information to describe the methods when using the Omics Playground. You
can always access the R code (on github) to check exactly all steps implemented, or you can contact the bioinformatic staff at BigOmics Analytics should you have questions.
Correlation analyses¶
Correlation is a statistical technique used to measure the strength and direction of the relationship between two variables. It determines whether a relationship exists between two variables and its
strength. The most used correlation types are Pearson’s correlation (for linear relationships between normally distributed variables), Spearman’s rank correlation (for ordinal or non-normal data),
and Kendall’s tau correlation (for ranked pairings). The correlation coefficient ranges from -1 to 1. -1 is a perfect negative correlation, and 1 is a perfect positive correlation. A value of 0 means
no correlation. Correlation does not imply causation. It only measures the degree of association between two variables. Pearson & Sperman correlations are by far the most used in biological research.
A distict assessment concerns partial correlation, as discussed below.
Pearson and Spearman correlation: the Pearson correlation coefficient is a statistical measure that evaluates the strength and direction of the linear relationship between two continuous variables.
Ideally, Pearson correlation should be used when the following assumptions are met: (i) the variables have a linear relationship each other; (ii) both variables are continuous and quantitative; (iii)
the variables are normally distributed; (iv): no significant outliers populate the data. Typically, if any of these assumptions are unmet, Spearman correlation should be employed.
Partial correlation: partial correlation measures the degree of association between two variables, while controlling for the effect of one or more additional factors/variables. It may determine if
the relationship between two variables (e.g. X and Y) remains significant after accounting for the influence of a third (“controlling”) variable (Z) on both X and Y. In other words, it estimates the
correlation between X and Y after removing the variance that each shares with Z. In principle, partial correlation may reveal cases where an observed correlation between X and Y is due to their
mutual association with a third variable Z, rather than a direct relationship between X and Y. The values of the partial correlation coefficient ranges from -1 to 1, similar to the regular
correlation coefficient.
Batch correction¶
Measurements in datasets generated in multiple centers are typically affected by multiple sources of technical variation, collectively known as ‘Batch Effects’ (BEs). BEs may also arise within a
single laboratory, due to distinct sequencing runs, depths, use of different sample donors, or when processing occurs in separate days. BEs are predominant, unwanted source of noise that impact mean
and variance and may confound real biological signal, altering false positive and false negative rates. BE correction methods can be categorized into supervised, such as ComBat and Limma
RemoveBatchEffects, and unsupervised methods such as SVA and RUV. Supervised methods use linear models to adjust known batch effects, while unsupervised methods measure potential sources of variation
without requiring prior knowledge of the batch vector.
ComBat: ComBat employs empirical Bayesian method to adjust the data for the known batch vector. It assumes that BEs have similar impact on many genes. Information across features is gathered within
each batch to estimate batch-specific mean and variance. These BEs parameters are then shrunk toward the global BEs average and used to adjust for the BEs on each gene. The output is a
batch-corrected data matrix that can be used for downstream analyses.
Limma RemoveBatchEffects: RemoveBatchEffects employs linear modeling to adjust BEs and additional covariates if specified. In most cases it is employed to adjust BEs only. Multiple numeric
covariates, if specified, are assumed to have additive effects. The output is a batch-corrected data matrix that can be used for downstream analyses.
Surrogate Variable Analysis (SVA): SVA aims to define surrogate variables (SVs) that consistently capture latent sources of variation in the data. It employs singular value decomposition (SVD) on the
batch uncorrected data, without the need for prior information on latent variables affecting the data. To preserve the biological effects of interest, the SVs need to capture variation uncorrelated
with the primary variable of interest. The SVs are then regressed out from the gene expression data to generate a batch-corrected data matrix suitable for downstream analyses.
Remove Unwanted Variation (RUV): RUV estimates and removes known and unknown unwanted variation based on negative control variables and technical sample replicates that capture batch variation. It
assumes the existence of negative control variables whose expression levels is robust to changes in the biological factors of interest. Factor analysis (e.g., SVD) on these controls is then used to
estimate the factors representing unwanted variation, such as BEs. The expression of all features is then adjusted for the unwanted variation to get a batch-corrected dataset.
Nearest-Pair Matching (NPmatch): current methods for BE correction mostly rely on specific assumptions or complex models, and may not detect and adjust BEs adequately, impacting downstream analysis
and discovery power. To address these challenges we (BigOmics Analytics) developed NPmatch (Zito et al., BioRxiv 2024). NPmatch is a nearest-neighbor matching-based method that adjusts BEs
satisfactorily and outperforms current methods in a wide range of datasets. Our method was inspired by principles of the statistical matching theory. It relies on distance-based matching to
deterministically search for nearest neighbors with opposite labels, so-called “nearest-pair”, among samples. NPmatch requires knowledge of the phenotypes but not of the batch assignment. Differently
to many other algorithms, NPmatch does not rely on specific models or underlying distribution. It does not require special experimental designs, randomized controlled experiments, control genes or
batch information. NPmatch is based on the simple rationale that samples should empirically pair based on distance in biological profiles, such as transcriptomics profiles. The NPmatch algorithm
initially selects the top variable features (genes). The features are feature-centered and then further centered per condition group. Inter-sample similarities are then determined by either computing
the Pearson correlation matrix Dn x n (default) or Euclidean distance. The Pearson correlation matrix Dn x n is subsequently decomposed into the c phenotypic/condition groups. For each sample, a k
nearest-neighbor like search is conducted to identify the closest k-nearest samples across each c phenotypic/condition group. The process results into a matrix Xn x (k x c) where for each sample,
k-nearest samples are identified per each c condition. The Xn x (k x c) matrix is then used to derive a (i) vector of length L=n x k x c, storing all the computed pairs; (ii) a fully paired dataset
Xp x L. As pairing may per-se imply duplication of correlated signals (which is a BE-like effect), Limma ‘RemoveBatchEffect’ is used to correct for the ‘pairing effects’ through linear regression.
The batch-corrected X1 p x L matrix is finally condensed into its original p x n size by computing, per each feature, the average values across duplicated samples. Thus, the X1p x n matrix represents
the batch-corrected dataset which can be used for further downstream analyses.
In OPG, batch effects, or contamination by unwanted variables, was identified by an F-test for the first three principal components. Continuous variables were dichotomized into high/low before
testing. Highly confounding variables would appear as having high relative contribution in the first or second principal component, often higher than the variable of interest. Batch effects were also
visually assessed (before and after correction) using annotated heatmaps and t-SNE plots colored by variables.
Batch correction was performed for explicit batch variables or unwanted covariates. Parameters with a correlation r>0.3 with any of variables of interest (i.e. the model parameters) were omitted from
the regression. Correction was performed by regressing out the covariate using the ‘removeBatchEffect’ function in the limma R/Bioconductor package.
Technical correction was performed for intrinsic technical parameters such as library size (i.e. total counts), mitochondrial and ribosomal proportions, cell cycle and gender. These parameters were
estimated from the data. Cell cycle (CC) was estimated using the Seurat R/Bioconductor package using reference lists of genes that are known to be markers for S or G2M phase. You can read about it
more here. In single-cell analysis, people sometimes regress out the CC effect. In bulk RNAseq it may give you some more information about your samples,
Gender (if not given) was estimated by checking the absence/presence of expression of gender specific genes on the X/Y chromosome. Parameters with a correlation r>0.3 with any of the model parameters
were omitted from the regression. Correction was performed by regressing out the covariate using the ‘removeBatchEffect’ function in the limma R/Bioconductor package.
Unsupervised batch correction was performed using SVA by estimating the latent surrogate variables and regressing out using the ‘removeBatchEffect’ function in the limma R/Bioconductor package.
Counts per million (CPM): CPM mapped reads are the number of raw reads mapped to a transcript, scaled by the number of sequencing reads in your sample, multiplied by a million. We employ a log2CPM:
specifically, The data are then added a pseudocount of 1 to enable log2-transformation and avoid negative values, and also make data distribution closer to normal. Thus, log2CPM is a within sample
normalization approach. It normalizes RNA-seq data for sequencing depth and so it also facilitates comparisons betweeen samples. However, a stronger cross-sample normalization method is often needed.
That why in the OPG we have implemented log2CPM + Quantile normalization.
Quantile normalization: The quantile method aims to make the distribution of gene expression levels the same for each sample in a dataset (Bolstad et al., 2003). It assumes that the global
differences in distributions between samples are all due to technical variation. Any remaining differences are likely actual biological effects. In quantile normalization, the genes are first ranked
within each sample. An average value is calculated across all samples for genes of the same rank. This average value then replaces the original value of all genes in that rank. The genes are then
placed in their original order. Therefore, quantile normalization makes the distribution of gene expression levels the same for each sample in a dataset, a pattern typically observed in boxplots.
This makes quantile normalization a highly robust way to achieve cross-sample normalization.
Max median normalization (MaxMedian): MaxMedian normalization is more often adopted in proteomics data. It aims to normalize the samples by the maximum median value. Specifically, it first calculates
the median value in each sample. The maximum median value is identified. Each data point in each sample is then divided by the sample’s median value and multiplied by the maximum median value. In
OPG, we then perform log2 transformation.
Max sum normalization (MaxSum): MaxSum normalization is also more often adopted in proteomics data. It aims to normalize the samples by the maximum value of total intensity. Specifically, it first
calculates the total intensity in each sample. The maximum total intensity value is identified. Each data point in each sample is then divided by the sample’s total intensity and multiplied by the
maximum total intensity value. In OPG, we then perform log2 transformation.
Reference normalization (reference): This type of normalization aims to normalize the data by a user-selected feature. Simply, it divides each data point in each sample by the value of the reference
features in that sample. In OPG, we then perform log2 transformation.
Heatmaps were generated using the ComplexHeatmap R/Bioconductor package (Gu 2016) on scaled log-expression values (z-score) using euclidean distance and Ward linkage. The standard deviation was used
to rank the genes for the reduced heatmaps.
t-distributed stochastic neighbor embedding (t-SNE): t-SNE is a non-linear dimensionality reduction method that enables visualization of high-dimensional data in a low-dimensional space, typically 2D
or 3D. Unlike linear dimensionality reduction techniques like PCA, t-SNE may separate data that is not linearly separable. Furthermore, while PCA tends to preserve the global information in the data,
t-SNE tends to preserve relative distances. t-SNE first calculates the distance between every pair of data points. Each data point is then placed within a Gaussian distribution while all other data
points are distributed according to their distance. Points closer each other are more similar. Point more distanced are more diverse each other. A value, called ‘Perplexity’, is calculated to reflect
the standard deviation of the data. The Perplexity value is the number of nearest neighbors considered in generating the probability of points being close each other. Smaller perplexity values may
result in very localized outputs ignoring global information, while larger values may obscure smaller structures. For this reason, the Perplexity value is intended to be less than the size of data.
Recommended values are in the range 5-50. To determine the representation of the data in the low dimensional space, a t-distribution is used with one degree of freedom. A gradient descent
optimization involving an iterative process, is finally employed to determine the final low-dimensional data representation that reflects the high-dimension accurately. In OPG t-SNE is computed using
the top 1000 most varying genes, then reduced to 50 PCA dimensions before computing the t-SNE embedding. The perplexity heuristically set to 25% of the sample size or 30 at maximum, and 2 at minimum.
Calculation was performed using the Rtsne R package.
Uniform Manifold Approximation and Projection (UMAP): UMAP is another non-linear dimensionality reduction method that enables visualization of high-dimensional data in a low-dimensional space,
typically 2D or 3D. Similarly to t-SNE, UMAP can effectively separate data that is not linearly separable. When compared to t-SNE, UMAP tends to more clearly separate groups of similar categories
from each other. Generally, UMAP can also preserve the global structure more than t-SNE. UMAP computation is also generally faster than t-SNE. To construct the initial high-dimensional graph, UMAP
builds a weighted graph with edge weights representing the likelihood that two points are connected. Connectedness is inferred through radii outwards from each point. Smaller radii result to small,
isolated clusters. Larger radii result to overconnection. To reduce these potential issues, each radius is chosen locally, based on the between each point and its nearest neighbor. UMAP then makes
the graph “fuzzy” by decreasing the likelihood of connection as the radius grows. UMAP then projects the data into lower dimensions through a force-directed graph layout algorithm in a similar way
than t-SNE. By ensuring that each point is connected to at least its closest neighbor, UMAP enables preservation of the local structure with respect to the global structure. In the OPG, UMAP was
computed using the top 1000 most varying genes, then reduced to 50 PCA dimensions before computing the UMAP embedding. The number of neighbours was heuristically set to 25% of the sample size or 30
at maximum, and 2 at minimum. Calculation w as performed using the uwot R package.
Principal Component Analysis (PCA): PCA is an unsupervised learning technique for dimensionality reduction. It is used to explain the variance–covariance structure of a set of variables through
linear combinations of the variables. Principal components (PCs) are variables constructed as linear combinations of the initial variables. The PCs are uncorrelated and the greatest variation in the
data is captured within the first PCs. The PCs represent the directions of the data that explain a maximal amount of variance. Though 10-dimensional data gives you 10 PCs, PCA put maximum possible
information in the first component, followed by the second component, and so forth, under the constraint that each component is uncorrelated with the previous component. PCA can be performed through
singular-value decomposition (SVD). [AZ: to-expand]. In OPG, PCA is performed using the irlba R package.
Differential gene expression testing¶
Omics Playground is equipped with 9 distinct differential gene expression (DGE) testing methods, aiming to cover the most disparate experimental conditions. It‘s our priority to offer researchers of
any background a vast range of choice to study in detail their data, in the fastest possible time, and without requiring any coding. Researchers may evaluate different methods to select the
appropriate one based on their needs. Our DGE workflow is paralleled with extensive visualizations including volcano plots, box plots, bar plots, heatmaps, and functional enrichment testing of
biological pathways. Here below we provide a description of the DGE algorithms available in the OPG:
Student’s t-test / Welch’s t-test: The t-test is the simplest statistics that can be used to compare two groups based on their average gene expression levels. While Student’s t-test assumes that the
two populations have equal variances, Welch’s t-test does not rely on this assumption. When the assumption of the Student’s t-test is known to be violated, the Welch’s t-test should be employed as it
performs better.
DESeq2 Likelihood / Wald test: DESeq2 employs negative binomial generalized linear models to assess variability in gene expression profiles. Significant DGE changes between groups can be assessed
using two methods: the Wald test and the likelihood ratio test (LRT). The Wald test is run by default: a negative binomial test is run for each gene to account for overdispersion in the data. A null
hypothesis (Ho) of no DGE between the two sample groups (i.e., fold-change=1) is assumed. Then, a z-score is calculated from each gene’s empirical LFC and compared to a standard normal distribution
to compute a p-value. If the p-value is less than a pre-chosen alpha level (e.g., 0.05), the Ho is rejected and DGE is reported. In the LRT, DESeq2 fits both a full model and a reduced model for each
gene and a Ho of no differences between the full and the reduced model is assumed. Parameters for both models are then estimated and the log-likelihoods of the two models are compared to obtain the
likelihood ratio (LR) which would follow a ch-squared distribution. If the p-value is less than a pre-chosen alpha level (e.g., 0.05), the Ho is rejected and DGE is reported. Therefore, the Wald test
differs from the LRT in that while the first is based on a single model per gene, the second is based on two models (full and reduced) per each gene.
EdgeR likelihood ratio / quasi-likelihood F test: EdgeR employs negative binomial models with estimation of dispersion parameters to model variability on the read counts. It also employs empirical
Bayes methods to moderately estimate the gene-specific dispersions. DGE can be assessed in edgeR using and exact test, GLM likelihood ratio (LRT), or the quasi-likelihood F-test. In the exact test,
EdgeR uses the negative binomial distribution and is useful for small to moderate sample size datasets. On the other hand, the GLM frameworks are particularly useful for the analysis of complex
experimental designs with multiple variables to be accounted for. Compared to the GLM LRT test, the quasi-likelihood F-test may offer a better solution fot example when experiments results into small
Limma trend / voom: It employs ordinary linear models with T and F test to measure gene expression differences between groups. These approaches aim to robustly estimate the mean-variance relationship
non-parametrically. Using log-counts per million (log-cpm) normalized for sequencig depth, the mean-variance is fitted to the gene-wise standard deviations of the log-cpm as a function of average
log-count. To incorporate the mean-variance relationship, limma-trend modifies limma’s empirical Bayes procedure to incorporate a mean-variance trend. A mean-variance trend across all genes aims to
model the relationship between a gene’s average expression and its variance. Limma-voom incorporates the mean-variance trend of the log-counts into a precision weight for each individual normalized
observation. Limma-voom and limma-trend both fit non-parametric curves. They both estimate the relationship between abundance and variance by fitting a lowess/loess curve. Because parameters are
estimated from the whole data set (not for individual genes), neither method results to over-fitting. Typically, limma-voom fits a slightly smoother curve than limma-trend. When sequencing depths are
similar across samples, limma-trend and limma-voom perform very similarly. Limma-voom performs better than limma-trend when sequencing depths are highly variable across samples.
Biomarker analysis¶
The Food and Drug Administration (FDA) defines a biomarker as ‘a defined characteristic that is measured as an indicator of normal biological processes, pathogenic processes, or responses to an
exposure or intervention, including therapeutic interventions. A biomarker could be almost any objective and quantifiable functional, physiological, biochemical, or molecular measurement. Examples of
molecular biomarkers include the presence of proteins in the blood, such as prostate-specific antigen (PSA) used in the diagnosis of prostate cancer, or the presence of mutations in tumor suppressor
genes, like those in BRCA1 or BRCA2, predictive of breast cancer risk. Therefore, novel biomarker discovery is crucial to many areas, including clinical diagnostics and drug
development.Bioinformatics has revolutionized biomarker discovery by integrating computational tools with high-throughput data analysis from genomics, proteomics, transcriptomics, and metabolomics.
Researchers can now efficiently identify and analyze potential biomarkers, leading to cost-effective and accelerated research outcomes. On the Omics Playground, we have made available to all users
state-of-the-art machine learning (ML) methods.
Sparse Partial Least Squares (sPLS): sPLS is a ML technique that extends partial least square (PLS) regression. It is designed to analyze relationships between two datasets where it can handle
multivariate high-dimensional data where the number of variables exceeds the number of observations -a common scenario in biological research- while employing dimensional reduction and variable
selection. sPLS aims to identify patterns and relationships between continuous data within complex datasets by reducing the dimensionality of the data while maintaining its predictive power. This
method combines the strengths of Partial Least Squares (PLS) regression with sparsity-inducing penalties. It seeks for linear combination between variables and, compared to PLS, it allows multiple
response variable selection on both datasets. This is achieved through the use of LASSO penalization. Compared to CCA (Canonical Correlation Analysis), PLS relies on covariance rather than
correlation coefficient. By incorporating sparsity constraints, sPLS can effectively identify a subset of variables that are most relevant for predicting the outcome of interest. sPLS is widely used
in bioinformatic applications for biomarker discovery in biomedical research.
Glmnet: The glmnet package in R provides extensive functionalities to identify putative biomarkers and construct predictive models for distinct biological outcome variables, such as prognosis, risk
to disease, response to treatment. It supports binary (through logistic regression), continuous (through linear regression), as well as survival (through Cox-regression) variables. Glmnet offers
procedures for fitting LASSO and/or elastic-net regularization in linear, logistic, and multinominal regression. Typically, and similarly to other ML techniques, Glmnet requires a response variable,
a predictor variables and a regularization type with regularization strength. It employs L1 and L2 regularizations, which are LASSO and ridge penalties, respectively. While the ridge penalty shrinks
the coefficients of correlated predictors towards each other, the lasso penalty picks one and discard the others. The elastic net penalty is a combination of LASSO and ridge penalty. Generally, these
regularizations help preventing overfitting by adding a penalty term to the objective function. Importantly, Glmnet also provides approaches to perform cross-validation (CV) analyses. CV enables
assessment of models performance and generability of the predicted features. Glmnet is also computational-efficient for large dataset. In summary, Glmnet provides tools for the implementation of
regularized regression models, enabling building predictive models.
Random Forest (RF): The RF algorithm is a powerful and versatile supervised ML method that combines multiple decision trees to make predictions. It is based on decision trees, which split the data
based on features to classify observations. Each tree in a RF is built on a subset of the training data. Spefically, RF uses a bagging-like approach, where each tree is trained on a random data
subset to hep reducing bias and variance. Another key aspect is feature randomness, where only a subset of features is considered for splitting at each node in a tree such that low correlations among
trees are considered into the overall model’s performance. As an ensemble learning method, RF combines outputs of multiple, distinct decision trees to improve accuracy and reduce overfitting. To make
a prediction, RF aggregates (averages) the predictions of individual trees, therefore producing an accurate and stable prediction. Thus, by leveraging the collective wisdom of multiple decision trees
trained on different subsets of data, the overall model accuracy and robustness is improved. All “hyperparameters” like node sizes, number of trees, as well as number of sampled features need to be
set before training to optimize its performance. On the other hand, when used for classification purposes, RF selects the most common prediction among trees.
Extreme Gradient Boosting (xgboost): The Extreme Gradient Boosting algorithm implemented in OPG is the one from Chen & Guestrin (2016). The R package xgboost includes functions for efficient linear
model solver and tree learning. It supports various objective functions, including regression, classification and ranking for common machine learning tasks. Xgboost implememts regularization in the
objective function to help preventing overfitting. In addition, two other techniques are used to further prevent overfitting: (i) shrinkage, which scales newly added weights after each step of tree
boosting and so reduces the influence of each individual tree to leaves space for future trees needed to improve the model; (ii) feature subsampling, a technique also used in other algorithms, such
as RandomForest. Subsampling features also prevents over-fitting to some extent. Xgboost is also efficient for sparse data and includes a sparsity-aware split finding algorithm. Finding the best tree
split is a key issue in tree learning approaches. To achieve it, xgboost enumerates over all the possible splits on all the features using the exact greedy algorithm. The algorithm sorts the data
based on normalized features. To identify and enumerate all possible splits for continuous features greedly, the exact greedy algorithm works on the sorted data. In this way a higher computational
efficiency is achieved. However, in cases when the dataset does not entirely fit into memory, the exact greedy algorithm may be slow or not work properly. For this reason an approximate algorithm is
used. The approximate algorithm has several steps: (i) identification of candidate splitting points based on percentiles of feature distribution; (ii) mapping features into buckets split by these
points; (iii) aggregates the statistics; (iv) identify best solution. For a detailed description we refer to Chen and Guestrin, 2016.
Statistical testing¶
Multi-method statistical testing. For gene-level testing, statistical significance was assessed using three independent statistical methods: DESeq2 (Wald test), edgeR (QLF test) and limma-trend (Love
2014; Robinson 2010; Ritchie 2015). The maximum q-value of the three methods was taken as aggregate q-value, which corresponds to taking the intersection of significant genes from all three tests.
Statistical testing of differential enrichment of genesets was performed using an aggregation of multiple statistical methods: Fisher’s exact test, fGSEA (Korotkevich 2019), Camera (Wu 2012) and GSVA
/limma (Hanzelmann 2013, Ritchie 2015). The maximum q-value of the selected methods was taken as aggregate meta.q value, which corresponds to taking the intersection of significant genes from all
tests. As each method uses different estimation parameters (NES for GSEA, odd-ratio for fisher, etc.) for the effect size, for consistency, we took the average log fold-change of the genes in the
geneset as sentinel value. We used more than 50000 genesets from various public databases including: MSigDB (Subramanian 2005; Liberzon 2015), Gene Ontology (Ashburner 2000), and Kyoto Encyclopedia
of Genes and Genomes (KEGG) (Kanehisa 2000).
Fisher`s exact test: Fisher’s exact test is a statistical significance test used to determine if there is a non-random association between two categorical variables organized within a contingency
table. It calculates the (exact) probability of obtaining the observed data as well as other more extreme patterns under the null hypothesis (Ho) of no association between the variables. The test
initially calculates the probability of obtaining the observed contingency table under the Ho that the two variables are independent. This probability is calculated using the hypergeometric
distribution, which gives the (exact) probability of drawing a specific number of successes in a sample without replacement. Next, the test calculates the probabilities of all other possible
relationships. If the p-value for these other possible relationships is less than or equal to the pre-defined significance level (e.g., 0.05), the Ho is rejected. When Ho is rejected, a statistically
significant association between the two variables is supported. The key advantages of Fisher’s exact test are:
1. It is exact and does not rely on approximations, making it suitable for small sample sizes or sparse data where assumptions of other statistical tests may remain unmet.
2. It is valid for all sample sizes.
3. It assumes fixed marginal totals (row and column sums), which is appropriate for many experimental designs. However, the test has limitations such as being computationally intensive for larger
tables, and not providing an estimate of the strength or direction of association.
Kruskal-Wallis test: The Kruskal-Wallis (KW) test is a non-parametric statistical test. It is generally used to assess significant differences between three (or more) groups of an independent
variable. It is non-parametric because it does not assume an underlying normal distribution. It considered to be an extension of the non-parametric Mann-Whitney U-test / Wilcoxon Rank-Sum test to
allow testing of more than two independent groups. Similarly to these tests, KW test is also based on rank place sums. The KW test is also considered as the non-parametric alternative of the one-way
ANOVA test. The KW test relies on the assumptions that the groups are independent and have similar distribution. that The null hypothesis (Ho) of the KW test is that there are no differences between
groups, specifically that the medians of the groups are equal. The KW test Ho is rejected if the median of at least 1 group differs from the other groups’ median.
Functional analyses¶
Here below the describe the Gene Set Enrichment methods in OPG:
CAMERA (Correlation Adjusted MEan RAnk gene set test) (Wu et al., Nucleic Acids Research, 2012): most competitive gene set tests assume that genes are independent units and rely on gene permutation
of gene labels. However inter-gene correlation exist and may inflate discovery of false positives. CAMERA was develop to address the problem of correlated genes in the test set. It is centered on the
idea of using the variance inflaction factor of inter-gene correlation structure to adjust the parametric or rank-based gene set test statistics. This procedure has been shown detect differential
gene representations while controlling the FDR even in datasets with a small number of biological replicates, regardless of inter-gene correlations.
GSEA (Gene Set Enrichment Analysis) (Mootha et al., 2003, Nat Gen): GSEA is a computational method to determine whether a prior defined set of genes show statistically significant, concordant
differences between two biological states/phenotypes. GSEA calculates an enrichment score (ES), representing the degree to which a gene set is overrepresented at the extreme top or bottom or a ranked
gene list. The ranked gene list typically contains genes from differential gene expression analyses between two phenotypes. The ES is calculated by Kolmogorov-Smirnov statistics: it increases the
cumulative statistics when a gene is present in a gene set or decreases when the gene is not found in the gene set. This generates a weighted ES that accounts for the position of the gene in the
list. The ES’ statistical significance is estimated by phenotype-based permutation that preserves the correlation structure of the gene expression data. Ultimately, the ES and its statistical
significance allows to identify gene sets enriched among the most upregulated or downregulated genes between two states/phenotypes. Therefore, GSEA enables to study concordant and modest changes that
may be missed in analyses at single-gene level.
ssGSEA (single-sample GSEA) (Barbie et al., 2009, Nature): ssGSEA calculates separate ESs for each pairing of a sample and gene set. Each ssGSEA ES represents the degree to which the genes in a
particular gene set are coordinately up- or down-regulated within a sample. In other words, the ssGSEA ES reflects the degree of overexpression of a given gene set in an individual sample. Compared
to standard GSEA, ssGSEA provides a score for each sample rather than across samples. Highly expressed genes contribute positively to the score, while lowly expressed genes contribute negatively.
fGSEA (Fast Gene Set Enrichment Analysis) (Korotkevitch et al., bioRxiv, 2021): Standard implementation of GSEA may have problems in accurately estimating low permutation P-values. Furthermore, time
and memory requirement grow linearly with size of datasets and number of gene set collections. fGSEA aims to address these problems, thus expanding the applicability of GSEA. fGSEA provides higher
estimation accuracy for low GSEA P-values at a substantially improved running time. The algorithms consist of (i) fGSEA-simple which estimates enrichment P-values with limited accuracy for the whole
collection of gene sets being tested, and (ii) fGSEA multi-level which infers low P-value with higher accuracy for each individual gene set.
GSVA (Gene Set Variation Analysis) (Hanzelmann et al., BMC Bioinformatics 2013): existing methods for gene sets enrichment testing aim at identifying gene sets from large collection of gene
signatures and/or select few enriched gene groups most relevant to the phenotype being investigated. Generally, existing methods may not account for the inherent variation in the gene expression data
and associated pathways activity potentially impacted in highly heterogeneous data. Furthermore, conventional, competitive gene set enrichment methods have mostly been designed to handle two-groups
comparisons (e.g., case-control studies). Therefore, they may not find direct applicability in population-level data where multiple, hierarchical phenotypes are simultaneously compared. GSVA is
non-parametric and unsupervised approach that aims to address these challenges. It computes gene set enrichment score for each sample and then conducts an analysis of variation of the gene set
enrichment and pathways activity across samples, independently of class labels. In this way GSVA facilitates post-hoc analyses of pathways including differential pathway activity analysis.
fry: Fry is a fast approximation method for gene set enrichment analysis based on the Rotation Gene Set Tests (ROAST) algorithm for linear models in the limma R package. Fry simulates the p-value
obtained in case a large number of rotations with ROAST. To protect against potential false positives driven by correlated genes, a residual space rotation is used. Differently to a standard
permutation test, this approach may work in experiment with small sample sizes. ROAST can be computationally intensive if applied to large collections of gene sets. Fry is faster by approximating
p-values associated with an infinite number of tests. While both ROAST and Fry account for gene-gene correlation structures, for large gene collections Fry is much faster than ROAST in distinguishing
the most significant gene sets.
Graph-weighted GO analysis. The enrichment score of a GO term was defined as the sum of q-weighted average fold-changes, (1-q)*logFC, of the GO term and all its higher order terms along the shortest
path to the root in the GO graph. The fold-change of a gene set was defined as the average of the fold-change values of its member genes. This graph-weighted enrichment score thus reflects the
enrichment of a GO term with evidence that is corroborated by its parents in the GO graph and therefore provides a more robust estimate of enrichment. The activation map visualizes the scores of the
top-ranked GO terms for multiple contrasts as a heatmap.
KEGG pathway visualization was performed using the Pathview R/Bioconductor package using the foldchange as node color.
Weighted Gene Co-expression Network Analysis¶
Weighted gene co-expression network analysis (WGCNA) is a powerful all-in-one analysis method that allows biologists to understand the transcriptome-wide relationships of all genes in a system rather
than each gene in isolation. WGCNA enables the identification of clusters (modules) of features that exhibit correlated patterns and the assessment of the relationship between distinct clusters.
Importantly, WGCNA also provides data on the association between modules and external traits, such as recorded sample phenotypes. Identification of gene correlation networks has high biological
relevance as genes within the same module could share regulatory mechanisms and be functionally related within a molecular pathway at the cellular and inter-cellular level. WGCNA could inform on
candidate biomarkers and druggable features for therapeutics. Although WGCNA has mostly been applied to transcriptomic data, its principles are suited to other omics, such as methylation data. WGCNA
can be split into four main sequential analytical components: (1) construction of weighted gene correlation networks; (2) identification of coexpression modules; (3) association of genes with sample
traits; (4) Inference of intramodular hub genes as candidate drivers of phenotypes. Outcomes are inferred by pairwise correlations between genes or modules in a guilty-by-association approach, where
information about a gene is gained from its close neighbors in the network.
1. Construction of weighted correlation networks of genes: typically, WGCNA starts with a matrix of data that features the gene expression of each sample. Pairwise correlations between genes across
samples are measured. The correlation score of each gene pair indicates the similarity of their expression pattern and could suggest their potential functional relationship. The ‘weighted’ aspect
aims to amplify the differences between strong and weak correlations by raising the correlation to a power defined by the user. A high correlation indicates the genes are strongly connected, whereas
a low correlation suggests a weak connection.
2. Identification of co-expression modules: WGCNA uses the network’s weighted correlation coefficient information to place genes exhibiting significantly similar expression profiles into groups
called modules. If genes have similar correlations with many shared neighbors in the network or have a large overlap of their network neighbors, the genes likely have similar expression patterns and
can be grouped into the same module. To determine modules, hierarchical clustering is performed on the gene correlation network data. A dendrogram is generated where each branch identifies a specific
module. Methods like dynamic tree cut can be employed to determine discrete modules containing genes with similar expression patterns. Each module is typically assigned a distinct ID and color.
3. Correlate phenotypic traits with gene modules: after defining modules using the dendrogram, the output must be simplified to one value per module, called the module eigengene. The eigengene is the
first component from a principal component analysis and represents the overall module expression. As the module eigengene characterizes each module as a singular entity, it enables us to perform
correlation analysis between modules to find those with similar expression behaviors or to determine how each module correlates with phenotypes. To determine whether these modules do have similar
biological roles, the degree to which each module’s eigengene correlates to different patient traits, sample types, or disease outcomes can also be measure. These biological variables could include a
patient’s age, gender, or weight, outcomes like remission or patient death, or whether samples originate from healthy or disease patients or from different organs or tumor locations.
4. Identify potential driver genes: from the identified modules of interest, genes that might be key factors for a particular trait or could influence other genes in that module could be identified.
Each module may contain many genes; it is essential to identify so-called ‘hub genes’ that can be ideal candidates for further study. Hub genes are identified as the most highly connected genes
within a module and, expectedly, the most strongly correlated with the phenotype of interest. The expression of a gene is also used to calculate the ‘module membership, which measures the degree to
which a gene’s expression profile with a particular module within the expression network. Module membership is therefore a useful tool for prioritizing genes for further study. If the correlation is
high, the gene is likely representative of the overall expression of the module as a whole and is well connected in the network. Similarly, the high correlation of this gene to the trait of interest
further strengthens its likelihood as an important driver in that module.
Cell type profiling¶
Cell type profiling was performed using the LM22 signature matrix as reference data set (Chen 2018). We have evaluated a total of 6 computational deconvolution methods: DeconRNAseq (Gong 2013), DCQ
(Altboum 2014), I-NNLS (Abbas 2009), NNLM (Lin 2020), rank-correlation and a meta-method. For NNLM, we repeated NNLM for non-logarithmic (NNLM.lin) and ranked signals (NNLM.rnk). The latter
meta-methods, meta and meta.prod, summarize the predictions of all the other methods as the mean and/or geometric mean of the normalized prediction probabilities, respectively.
[1] Gong T, Szustakowski JD. DeconRNASeq: a statistical framework for deconvolution of heterogeneous tissue samples based on mRNA-Seq data. Bioinformatics. 2013.
[2] Altboum Z, et al. Digital cell quantification identifies global immune cell dynamics during influenza infection. Mol Syst Biol. 2014 Feb 28;10(2):720.
[3] Abbas A, et al. Deconvolution of Blood Microarray Data Identifies Cellular Activation Patterns in Systemic Lupus Erythematosus, PLOS One, 2009.
[4] Lin X, Boutros PC. Optimization and expansion of non-negative matrix factorization. BMC Bioinformatics. 2020.
[5] Chen B, et al. Profiling Tumor Infiltrating Immune Cells with CIBERSORT. Methods Mol Biol. 2018.
Scripting and visualization¶
Data preprocessing was performed using bespoke scripts using R (R Core Team 2013) and packages from Bioconductor (Huber 2015). Statistical computation and visualization have been performed using the
Omics Playground version vX.X.X (Akhmedov 2020).
Akhmedov M, Martinelli A, Geiger R and Kwee I. Omics Playground: A comprehensive self-service platform forvisualization, analytics and exploration of Big Omics Data. NAR Genomics and Bioinformatics,
Volume 2, Issue 1, March 2020.
Ashburner et al. Gene ontology: tool for the unification of biology. Nat Genet. May 2000;25(1):25-9.
Huber W, et al. (2015) Orchestrating high-throughput genomic analysis with Bioconductor. Nature Methods 12:115-121; doi:10.1038/nmeth.3252
Kanehisa, M. and Goto, S.; KEGG: Kyoto Encyclopedia of Genes and Genomes. Nucleic Acids Res. 28, 27-30 (2000).
Leek J., Storey J. Capturing heterogeneity in gene expression studies by ‘surrogate variable analysis’ PLoS Genet. 2007
Love MI, Huber W, Anders S (2014). “Moderated estimation of fold change and dispersion for RNA-seq data with DESeq2.” Genome Biology, 15, 550.
R Core Team (2013). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-project.org/.
Ritchie ME, Phipson B, Wu D, Hu Y, Law CW, Shi W, Smyth GK (2015). “limma powers differential expression analyses for RNA-sequencing and microarray studies.” Nucleic Acids Research, 43(7)
Robinson MD, McCarthy DJ, Smyth GK (2010). “edgeR: a Bioconductor package for differential expression analysis of digital gene expression data.” Bioinformatics, 26(1), 139-140. | {"url":"https://omicsplayground.readthedocs.io/en/latest/methods/","timestamp":"2024-11-13T19:18:18Z","content_type":"text/html","content_length":"60777","record_id":"<urn:uuid:a8edb00b-4980-41d8-8654-ab2945550fe8>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00108.warc.gz"} |
Understanding Quantum Mechanics: A Comparative Study of David Bohm and Copenhagen Interpretation
Understanding Quantum Mechanics: A Comparative Study of David Bohm and Copenhagen Interpretation
1. Introduction
The realm of quantum mechanics has long been a subject of profound interest and debate among physicists and philosophers alike. At its core, quantum mechanics challenges our conventional
understanding of reality, presenting a world that operates in ways that often seem counterintuitive. Two of the most significant interpretations of quantum mechanics that have emerged are the
Copenhagen Interpretation and Bohmian Mechanics, proposed by Niels Bohr and David Bohm, respectively.
The Copenhagen Interpretation, which dominated the early 20th century, posits that physical systems do not have definite properties until they are measured. It introduces the concept of probability,
suggesting that quantum events are inherently uncertain and that the act of measurement plays a crucial role in determining the outcome. This interpretation has been foundational in shaping our
understanding of quantum phenomena, but it raises critical questions about the nature of reality and the role of the observer.
In contrast, Bohmian Mechanics, also known as Bohmian Interpretation or de Broglie-Bohm theory, offers a deterministic view of quantum systems. Bohm proposed that particles possess definite positions
and momenta, guided by a "pilot wave" that determines their behavior. This interpretation seeks to resolve the ambiguities and paradoxes presented by the Copenhagen Interpretation, asserting that
hidden variables can account for the apparent randomness of quantum events.
This article aims to delve into the fundamental differences between these two interpretations, exploring their philosophical implications and the ongoing discourse in the scientific community. By
examining the core principles of the Copenhagen Interpretation and Bohmian Mechanics, we will shed light on how these contrasting views influence our understanding of the quantum world and our place
within it.
2. Copenhagen Interpretation: Core Ideas
The Copenhagen Interpretation of quantum mechanics, primarily developed by Niels Bohr and Werner Heisenberg in the early 20th century, has become the most widely accepted framework for understanding
quantum phenomena. It represents a significant shift from classical physics, introducing concepts that challenge our traditional notions of reality and observation.
2.1 Uncertainty Principle and Probability
One of the core concepts of the Copenhagen Interpretation is Heisenberg's Uncertainty Principle. This principle states that it is impossible to simultaneously know a particle's exact position and
momentum with unlimited precision. This uncertainty is not a limitation of measurement tools but a fundamental property of the quantum world. In other words, the quantum universe is intrinsically
unpredictable, describable only in terms of probabilities rather than certainties.
2.2 Wave Function and Probability
wave function Schrodinger
The mathematical framework of the Copenhagen Interpretation is centered around the wave function, a complex-valued function that encodes the probabilities of all possible outcomes of a quantum event.
The square of the wave function's amplitude provides the probability density of finding a particle in a specific state upon measurement. This probabilistic nature of quantum mechanics is one of its
most distinguishing features, deviating sharply from the deterministic laws of classical physics.
2.3 Collapse of the Wave Function
Collapse of the Wave Function
One of the most controversial concepts in the Copenhagen Interpretation is the idea that the wave function collapses when a measurement is taken. Before measurement, a particle exists in a
superposition of various quantum states. When an observer measures the particle, the wave function, which represents all possible outcomes, collapses, and the observation yields a single, definite
result. In other words, the quantum universe does not possess a definite reality until it is measured or observed.
2.4 The Indeterministic Nature of Quantum Events
The Copenhagen Interpretation embraces the idea that quantum events are fundamentally indeterministic. Unlike classical mechanics, where future states of a system can be precisely predicted given
complete knowledge of its initial conditions, quantum mechanics accepts a level of inherent randomness. This challenges the classical concept of causality and raises philosophical questions about the
nature of reality and the limits of human knowledge.
2.5 Philosophical Implications
The implications of the Copenhagen Interpretation extend beyond physics, inviting philosophical inquiry into the nature of reality, observation, and the role of consciousness. The assertion that
reality is not determined until it is observed leads to discussions about the observer's influence on the observed, and whether consciousness plays a fundamental role in shaping reality. This notion
has sparked ongoing debates in both scientific and philosophical circles, questioning the foundations of knowledge and existence itself.
In summary, the Copenhagen Interpretation fundamentally reshapes our understanding of the quantum world by emphasizing the central role of the observer, introducing probabilistic outcomes, and
challenging classical notions of determinism. Its core ideas have paved the way for ongoing discussions and explorations into the nature of reality, making it a pivotal framework in the study of
quantum mechanics.
3. Interpretation of David Bohm - Hidden Variables Theory
David Bohm's interpretation of quantum mechanics, commonly known as Bohmian Mechanics or the Hidden Variables Theory, offers a radical departure from the probabilistic view presented by the
Copenhagen Interpretation. Developed in the mid-20th century, Bohm's framework aims to restore determinism to quantum phenomena by introducing hidden variables that govern the behavior of particles.
3.1 Bohmian Mechanics: Pilot Waves and Hidden Variables
At the core of Bohmian Mechanics is the concept of pilot waves, which serve as guiding functions for particles. In this interpretation, each particle possesses a definite position and momentum at all
times, contrary to the Copenhagen view where properties are uncertain until measured. The behavior of these particles is directed by a wave function that evolves according to the Schrödinger
equation, similar to traditional quantum mechanics.
Bohm posited that these pilot waves contain hidden variables—parameters that determine the precise state of a particle but are not directly observable. As a result, while quantum events may appear
random, they are actually determined by underlying variables that are unknown to us. This reintroduction of determinism allows for a more intuitive understanding of particle behavior, which can be
described as both wave-like and particle-like.
3.2 Implicate Order and Explicate Order
Bohm further elaborated his interpretation by introducing the concepts of implicate order and explicate order**.
- Implicate Order: This refers to a deeper level of reality where all elements are interconnected. In the implicate order, phenomena are not seen as separate entities but as part of a unified whole.
Bohm argued that this underlying order encompasses all possible states of a system, emphasizing the interrelatedness of all particles and events.
- Explicate Order: In contrast, the explicate order represents the observable world where objects and events appear distinct and separate. This order is derived from the implicate order through a
process of unfolding. When we make measurements and observations, we perceive the explicate order, but it is only a manifestation of the more profound implicate order that underlies it.
This framework suggests that reality is more complex than it appears, and that what we observe is merely a surface-level representation of a much deeper interconnected reality.
3.3 Quantum Non-Locality
One of the most intriguing implications of Bohmian Mechanics is its treatment of non-locality. Bohm's theory allows for instantaneous connections between particles, regardless of the distance
separating them. This feature is evident in phenomena such as quantum entanglement, where the state of one particle is dependent on the state of another, no matter how far apart they are.
Bohm's interpretation reconciles non-locality with a deterministic framework by suggesting that all particles are interconnected within the implicate order. Therefore, changes to one particle's state
can instantaneously influence another, preserving the deterministic nature of the underlying hidden variables while accounting for the apparent randomness of quantum measurements.
3.4 Philosophical Implications of Bohm's Interpretation
Bohm's interpretation has significant philosophical ramifications, particularly regarding the nature of reality and the role of consciousness. By proposing that the universe is fundamentally
interconnected and that hidden variables guide particle behavior, Bohm challenges the notion of separateness that pervades classical thought.
His ideas encourage a holistic view of the universe, suggesting that individual entities cannot be fully understood in isolation. Moreover, Bohm's interpretation implies that consciousness itself
might play a crucial role in shaping reality, as the act of observation is interwoven with the fabric of existence.
In summary, David Bohm's interpretation of quantum mechanics presents a compelling alternative to the Copenhagen Interpretation by reintroducing determinism through the framework of hidden variables.
With the concepts of pilot waves, implicate and explicate orders, and non-locality, Bohmian Mechanics offers a more coherent understanding of quantum phenomena, emphasizing the interconnectedness of
all things and inviting deeper philosophical exploration of the nature of reality.
4. In-Depth Comparison: Copenhagen vs. Bohm
The Copenhagen Interpretation and Bohmian Mechanics represent two fundamentally different approaches to understanding the complexities of quantum mechanics. While both interpretations seek to explain
the behavior of quantum systems, they do so through contrasting philosophical and mathematical frameworks. This section will provide a comprehensive comparison of these two interpretations across
various dimensions.
4.1 Core Principles
- Copenhagen Interpretation: This interpretation emphasizes the role of the observer and introduces the idea of wave function collapse. Quantum systems are described by probabilities, and definitive
properties are established only upon measurement. It accepts inherent uncertainty and randomness as fundamental aspects of quantum phenomena.
- Bohmian Mechanics: In contrast, Bohmian Mechanics asserts that particles have definite positions and momenta at all times, guided by pilot waves. The theory reintroduces determinism through hidden
variables, suggesting that the apparent randomness of quantum events can be explained by underlying, unobservable factors.
4.2 Treatment of the Wave Function
- Copenhagen Interpretation: The wave function in this framework is viewed as a tool for calculating probabilities rather than a physical entity. It exists in a state of superposition, representing
all possible outcomes until a measurement is made, at which point it collapses into one definite state.
- Bohmian Mechanics: In Bohm’s view, the wave function is a real, physical entity that plays a crucial role in the behavior of particles. Rather than collapsing, it continuously evolves according to
the Schrödinger equation, guiding particles along deterministic paths.
4.3 Nature of Reality
- Copenhagen Interpretation: This interpretation posits that reality is inherently probabilistic and that the act of observation is fundamental to the manifestation of physical properties. Reality
does not possess definite qualities until measured, leading to philosophical debates about the nature of existence and the observer's role.
- Bohmian Mechanics: Bohm's interpretation offers a more ontologically rich view, suggesting that there is an underlying reality that is deterministic and interconnected. The implicate order implies
a deeper, unified reality where everything is fundamentally related, challenging the separateness that characterizes classical physics.
4.4 Non-locality and Entanglement
- Copenhagen Interpretation: The Copenhagen framework acknowledges non-locality as a feature of quantum mechanics but does not provide a mechanism for how particles can be connected across distances.
It accepts that entangled particles can influence each other instantaneously, yet remains ambiguous about the underlying cause.
- Bohmian Mechanics: Bohmian Mechanics directly incorporates non-locality into its framework, providing a clear explanation for the correlations observed in entangled particles. The hidden variables
in the implicate order allow for instantaneous interactions between particles, maintaining the deterministic nature of the theory.
4.5 Philosophical Implications
- Copenhagen Interpretation: The philosophical implications of the Copenhagen Interpretation lead to questions about the role of the observer, consciousness, and the nature of reality. It invites
discussions about the subjective experience of measurement and the limits of scientific knowledge.
- Bohmian Mechanics: Bohm’s interpretation encourages a more holistic perspective, proposing that the universe is interconnected and that the act of observation is part of a larger, dynamic process.
This view opens up avenues for exploring the relationship between consciousness and reality, suggesting that understanding the universe requires a shift from reductionist thinking to a more
integrative approach.
In conclusion, the comparison between the Copenhagen Interpretation and Bohmian Mechanics highlights the profound philosophical and conceptual differences between these two approaches to quantum
mechanics. While the Copenhagen Interpretation embraces indeterminism and the probabilistic nature of quantum events, Bohmian Mechanics offers a deterministic and interconnected framework. Both
interpretations provide valuable insights into the nature of reality and the behavior of quantum systems, contributing to ongoing discussions and debates in the field of physics and philosophy.
5. Critiques and Support from the Physics Community
The debate between the Copenhagen Interpretation and Bohmian Mechanics has prompted extensive discussions among physicists, leading to a variety of critiques and support for each interpretation. This
section explores the perspectives of the scientific community regarding these two approaches to quantum mechanics.
5.1 Critiques of the Copenhagen Interpretation
1. Indeterminism and Randomness: One of the primary critiques of the Copenhagen Interpretation is its acceptance of fundamental randomness in quantum events. Critics argue that this viewpoint is
unsatisfactory as it undermines the deterministic nature of classical physics, making it difficult to reconcile quantum mechanics with the broader scientific framework. Some physicists, particularly
those inclined toward realism, find it problematic that nature itself is inherently unpredictable.
2. Observer-Dependent Reality: The Copenhagen Interpretation's reliance on the observer's role has led to philosophical concerns about the nature of reality. Critics argue that this perspective
implies that reality does not exist independently of observation, raising questions about the objectivity of scientific inquiry. This view is seen as potentially undermining the notion of a
consistent external reality that can be studied and understood.
3. Measurement Problem: The interpretation faces significant challenges related to the measurement problem, specifically the ambiguity surrounding the collapse of the wave function. Critics highlight
that the Copenhagen Interpretation does not adequately explain how or why a measurement leads to the collapse, leaving important questions unanswered.
5.2 Support for the Copenhagen Interpretation
1. Pragmatism and Predictive Power: Despite its critiques, the Copenhagen Interpretation has garnered support for its pragmatic approach to quantum mechanics. Many physicists appreciate its ability
to provide accurate predictions and practical applications in various fields, including quantum computing and quantum cryptography. The interpretation's focus on measurement and observation aligns
with experimental practices in physics, making it a useful framework for practical work.
2. Historical Significance: The Copenhagen Interpretation has historical significance as one of the first comprehensive frameworks for understanding quantum mechanics. It laid the groundwork for
subsequent developments in the field, influencing the way physicists think about and interpret quantum phenomena. Many in the scientific community regard it as a cornerstone of quantum theory.
5.3 Critiques of Bohmian Mechanics
1. Complexity and Non-locality: Bohmian Mechanics has been criticized for its complexity and reliance on non-locality. Critics argue that introducing hidden variables complicates the understanding of
quantum systems without necessarily providing additional explanatory power. The non-local nature of the theory raises concerns about its compatibility with the principles of relativity, as it
seemingly allows for instantaneous influences across distances.
2. Lack of Empirical Evidence: Another critique is the lack of empirical evidence supporting the existence of hidden variables. While Bohmian Mechanics offers a deterministic framework, skeptics
point out that it has yet to provide testable predictions that can be differentiated from standard quantum mechanics. This lack of experimental validation makes it challenging to establish its
acceptance within the broader physics community.
5.4 Support for Bohmian Mechanics
1. Determinism and Clarity: Supporters of Bohmian Mechanics argue that the interpretation restores determinism to quantum phenomena, providing a clearer understanding of particle behavior. By
positing definite positions and momenta, Bohmian Mechanics allows for a more intuitive grasp of quantum systems, making it appealing to those who favor a deterministic view of reality.
2. Resolving Paradoxes: Proponents contend that Bohmian Mechanics effectively addresses some of the paradoxes and ambiguities associated with the Copenhagen Interpretation. By eliminating the
measurement problem and providing a coherent explanation for non-locality, it offers a more complete understanding of quantum entanglement and the nature of reality.
3. Holistic Perspective: Supporters appreciate Bohm's emphasis on the interconnectedness of all things, which aligns with contemporary views in various scientific disciplines, including philosophy
and systems theory. This holistic perspective resonates with many physicists and philosophers seeking a deeper understanding of the universe.
In summary, the critiques and support from the physics community highlight the ongoing discourse surrounding the Copenhagen Interpretation and Bohmian Mechanics. While the Copenhagen Interpretation
remains popular for its pragmatic approach and historical significance, it faces challenges related to indeterminism and the measurement problem. Conversely, Bohmian Mechanics offers a deterministic
framework that addresses some of these critiques but is met with skepticism regarding its complexity and empirical support. As physicists continue to explore the nature of quantum mechanics, the
debate between these interpretations remains vibrant and unresolved, reflecting the intricacies of understanding the quantum realm.
6. Philosophical and Ontological Implications
The contrasting interpretations of quantum mechanics, namely the Copenhagen Interpretation and Bohmian Mechanics, carry profound philosophical and ontological implications. These interpretations not
only shape our understanding of the quantum realm but also influence broader discussions about the nature of reality, knowledge, and existence. This section explores the key philosophical and
ontological implications stemming from both interpretations.
6.1 Nature of Reality
- Copenhagen Interpretation: The Copenhagen Interpretation posits that reality is not fully determined until it is observed. This observer-dependent view raises questions about the existence of an
objective reality independent of observation. Philosophically, it invites discussions about the nature of existence and whether reality is fundamentally subjective. This perspective aligns with
certain idealist philosophies, which suggest that consciousness plays a central role in shaping reality.
- Bohmian Mechanics: In stark contrast, Bohmian Mechanics asserts that an objective reality exists independently of observation, characterized by a deterministic structure guided by hidden variables.
The notion of implicate and explicate orders provides a framework for understanding the interconnectedness of all things, suggesting that reality is a unified whole. This view resonates with realist
philosophies, emphasizing that an underlying order governs the behavior of particles and phenomena, irrespective of human observation.
6.2 Knowledge and Epistemology
- Copenhagen Interpretation: The probabilistic nature of the Copenhagen Interpretation influences epistemological discussions about the limits of knowledge in quantum mechanics. If reality is
fundamentally uncertain and dependent on measurement, it raises questions about the extent to which we can know and understand the universe. This perspective suggests that our knowledge is inherently
partial and contingent upon our observational tools and methods.
- Bohmian Mechanics: Bohmian Mechanics, with its deterministic framework, presents a more optimistic view of knowledge acquisition. By positing hidden variables that govern particle behavior, it
implies that a complete understanding of the quantum realm is attainable. This approach aligns with realist epistemologies, asserting that there are objective truths about the universe that can be
discovered through scientific inquiry, albeit through a more complex lens.
6.3 Causality and Determinism
- Copenhagen Interpretation: The acceptance of indeterminism in the Copenhagen Interpretation challenges traditional notions of causality. By introducing randomness into quantum events, it suggests
that not all events are determined by prior states, leading to a re-evaluation of causal relationships. This perspective aligns with certain interpretations of quantum mechanics that embrace a
non-causal worldview, where randomness is an inherent aspect of nature.
- Bohmian Mechanics: In contrast, Bohmian Mechanics restores a deterministic view of causality, suggesting that every quantum event is the result of underlying variables. This interpretation upholds
the classical notion of causality, implying that even seemingly random outcomes have definite causes rooted in the hidden variables of the implicate order. This deterministic view resonates with
classical philosophies that prioritize causation and predictability.
6.4 The Role of Consciousness
- Copenhagen Interpretation: The Copenhagen Interpretation invites philosophical discussions about the role of consciousness in the act of measurement. By emphasizing the observer's influence on the
observed, it raises questions about whether consciousness itself has a fundamental role in shaping reality. This perspective has led to debates about the nature of consciousness and its relationship
to physical phenomena, suggesting a potential link between quantum mechanics and consciousness studies.
- Bohmian Mechanics: While Bohmian Mechanics does not explicitly attribute a central role to consciousness, its holistic framework implies that consciousness is part of the interconnected web of
reality. By positing that all elements of the universe are interrelated, it opens up avenues for exploring the relationship between consciousness and the physical world. Bohm’s view encourages a more
integrated understanding of consciousness, suggesting that it may be an emergent property of the complex interactions within the implicate order.
In conclusion, the philosophical and ontological implications of the Copenhagen Interpretation and Bohmian Mechanics illuminate the profound questions that arise from our attempts to understand the
quantum realm. The Copenhagen Interpretation challenges traditional notions of reality, knowledge, and causality, inviting discussions about the subjective nature of existence and the role of the
observer. In contrast, Bohmian Mechanics offers a deterministic and interconnected view of reality, emphasizing objective truths and causal relationships. Both interpretations enrich the
philosophical discourse surrounding quantum mechanics, highlighting the intricate relationship between science, philosophy, and our understanding of the universe.
7. Conclusion
In the exploration of quantum mechanics, the contrasting interpretations of the Copenhagen Interpretation and Bohmian Mechanics reveal profound insights into the nature of reality, measurement, and
the interconnectedness of all phenomena. The Copenhagen Interpretation, with its emphasis on indeterminism and the role of the observer, has played a foundational role in shaping our understanding of
quantum mechanics. It underscores the probabilistic nature of the quantum realm, suggesting that reality is contingent upon observation and measurement.
On the other hand, Bohmian Mechanics offers a radical rethinking of these principles by introducing determinism through hidden variables and pilot waves. This interpretation restores a sense of
objective reality, positing that particles have definite properties that are guided by an underlying, interconnected structure. The concepts of implicate and explicate orders further enhance our
understanding of the complex relationships within the universe, suggesting that all elements are fundamentally related.
The ongoing debates and discussions surrounding these interpretations reflect the complexities of quantum mechanics and the challenges of understanding its implications for philosophy and ontology.
The critiques and support for both interpretations highlight the dynamic nature of scientific inquiry, where new ideas continuously challenge and refine existing frameworks.
Ultimately, both the Copenhagen Interpretation and Bohmian Mechanics contribute to our broader understanding of the universe, prompting deeper questions about the nature of reality, knowledge, and
consciousness. As physicists and philosophers continue to grapple with these profound questions, the exploration of quantum mechanics remains a vibrant and evolving field, rich with possibilities for
future discovery and understanding. The interplay between these interpretations not only shapes our comprehension of quantum phenomena but also invites us to reconsider the fundamental principles
that underpin our conception of reality itself.
1. Bohmian Mechanics: An interpretation of quantum mechanics proposed by David Bohm, which introduces hidden variables and posits that particles have definite positions and momenta at all times,
guided by a pilot wave.
2. Copenhagen Interpretation: A foundational interpretation of quantum mechanics developed by Niels Bohr and Werner Heisenberg, emphasizing the role of measurement and the observer, and suggesting
that quantum events are fundamentally probabilistic.
3. Collapse of the Wave Function: A process in the Copenhagen Interpretation where a quantum system transitions from a superposition of states to a single definite state upon measurement.
4. Determinism: The philosophical belief that all events, including moral choices, are determined completely by previously existing causes. In the context of Bohmian Mechanics, it refers to the idea
that quantum events are determined by underlying hidden variables.
5. Hidden Variables: Unobservable factors that determine the behavior of quantum systems in Bohmian Mechanics, suggesting that quantum phenomena can be explained by these underlying variables.
6. Implicate Order: A concept introduced by David Bohm referring to a deeper level of reality where all elements are interconnected and not seen as separate entities.
7. Explicate Order: The observable world in Bohmian Mechanics that arises from the implicate order, where objects and events appear distinct and separate.
8. Measurement Problem: A fundamental issue in quantum mechanics concerning how and why observations lead to the collapse of the wave function, resulting in definite outcomes.
9. Non-locality: A phenomenon in quantum mechanics where particles can instantaneously affect each other regardless of distance, challenging classical notions of locality. This is a prominent feature
in Bohmian Mechanics.
10. Pilot Wave: A guiding wave in Bohmian Mechanics that directs the motion of particles, providing a deterministic framework for quantum phenomena.
11. Quantum Entanglement: A phenomenon where two or more particles become interconnected in such a way that the state of one particle instantly influences the state of the other, regardless of the
distance separating them.
12. Quantum Mechanics: A fundamental theory in physics that describes the behavior of matter and energy at the smallest scales, incorporating principles of wave-particle duality, uncertainty, and
13. Superposition: A principle in quantum mechanics where a quantum system can exist in multiple states simultaneously until measured, at which point it collapses into one of the possible states.
14. Wave Function: A mathematical function that describes the quantum state of a system, providing the probabilities of finding a particle in various states or locations. | {"url":"https://www.spiritualscienceexplorers.com/2024/10/understanding-quantum-mechanics.html","timestamp":"2024-11-06T15:24:19Z","content_type":"application/xhtml+xml","content_length":"240375","record_id":"<urn:uuid:baba769c-9447-4d80-a174-8286e30be11e>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00878.warc.gz"} |
Local inertial
The local inertial approximation of shallow water flow neglects only the convective acceleration term in the Saint-Venant momentum conservation equation. The numerical solution of the local inertial
approximation on a staggered grid is as follows (Bates et al., 2010):
$$\[Q_{t+\Delta t} = \frac{Q_t - g A_t \Delta t S_t}{(1+g\Delta t n^2 |Q_t| / (R_t^{4/3} A_t))}\]$$
where $Q_{t+\Delta t}$ is the river flow [m$^3$/s] at time step $t+\Delta t$, $g$ is acceleration due to gravity [m/s$^2$], $A_t$ is the cross sectional flow area at the previous time step, $R_t$ is
the hydraulic radius at the previous time step, $Q_t$ is the river flow [m$^3$/s] at the previous time step, $S_t$ is the water surface slope at the previous time step and $n$ is the Manning's
roughness coefficient [m$^{-1/3}$ s].
The momentum equation is applied to each link between two river grid cells, while the continuity equation over $\Delta t$ is applied to each river cell:
$$\[h^{t+\Delta t} = h^t + \Delta t \frac{Q^{t+\Delta t}_{src} - Q^{t+\Delta t}_{dst}}{A}\]$$
where $h^{t+\Delta t}$ is the water depth [m] at time step $t+\Delta t$, $h^t$ is the water depth [m] at the previous time step, $A$ is the river area [m$^2$] and $Q_{src}$ and $Q_{dst}$ represent
river flow [m$^3$/s] at the upstream and downstream link of the river cell, respectively.
The model time step $\Delta t$ for the local inertial model is estimated based on the Courant-Friedrichs-Lewy condition (Bates et al., 2010):
$$\[\Delta t = min(\alpha \frac{\Delta x_i}{\sqrt{(gh_i)}})\]$$
where $\sqrt{(gh_i)}$ is the wave celerity for river cell $i$ , $\Delta x_i$ is the river length [m] for river cell $i$ and $\alpha$ is a coefficient (typically between 0.2 and 0.7) to enhance the
stability of the simulation.
In the TOML file the following properties related to the local inertial model can be provided for the sbm and sbm_gwf model types:
river_routing = "local-inertial" # default is "kinematic-wave"
inertial_flow_alpha = 0.5 # alpha coefficient for model stability (default = 0.7)
froude_limit = true # default is true, limit flow to subcritical-critical according to Froude number
h_thresh = 0.1 # water depth [m] threshold for calculating flow between cells (default = 1e-03)
floodplain_1d = true # include 1D floodplain schematization (default = false)
Two optional constant boundary conditions riverlength_bc and riverdepth_bc can be provided at a river outlet node (or multiple river outlet nodes) through the model parameter netCDF file, as follows:
riverlength_bc = "riverlength_bc" # optional river length [m], default = 1e04
riverdepth_bc = "riverdepth_bc" # optional river depth [m], default = 0.0
These boundary conditions are copied to a ghost node (downstream of the river outlet node) in the code.
The optional 1D floodplain schematization is based on provided flood volumes as a function of flood depth (per flood depth interval) for each river cell. Wflow calculates from these flood volumes a
rectangular floodplain profile for each flood depth interval. Routing is done separately for the river channel and floodplain.
The momentum equation is most stable for low slope environments, and to keep the simulation stable for (partly) steep environments the froude_limit option is set to true by default. This setting
limits flow conditions to subcritical-critical conditions based on the Froude number ($\le 1$), similar to Coulthard et al. (2013) in the CAESAR-LISFLOOD model and Adams et al. (2017) in the Landlab
v1.0 OverlandFlow component. The froude number $Fr$ on a link is calculated as follows:
$$\[ Fr = \frac{u}{\sqrt{(gh_f)}}\]$$
where $\sqrt{(gh_f)}$ is the wave celerity on a link and $u$ is the water velocity on a link. If the water velocity from the local inertial model is causing the Froude number to be greater than 1.0,
the water velocity (and flow) is reduced in order to maintain a Froude number of 1.0.
The downstream boundary condition basically simulates a zero water depth boundary condition at a set distance, as follows. For the downstream boundary condition (ghost point) the river width, river
bed elevation and Manning's roughness coefficient are copied from the upstream river cell. The river length [m] of the boundary cell can be set through the TOML file with riverlength_bc, and has a
default value of 10 km. The water depth at the boundary cell is fixed at 0.0 m.
Simplified reservoir and lake models can be included as part of the local inertial model for river flow (1D) and river and overland flow combined (see next section). Reservoir and lake models are
included as a boundary point with zero water depth for both river and overland flow. For river flow the reservoir or lake model replaces the local inertial model at the reservoir or lake location,
and $Q$ is set by the outflow from the reservoir or lake. Overland flow at a reservoir or lake location is not allowed to or from the downstream river grid cell.
For the simulation of 2D overland flow on a staggered grid the numerical scheme proposed by de Almeida et al. (2012) is adopted. The explicit solution for the estimation of water discharge between
two cells in the x-direction is of the following form (following the notation of Almeida et al. (2012)):
$$\[Q_{i-1/2}^{n+1} = \frac{\left[ \theta Q_{i-1/2}^{n} +\frac{(1-\theta)}{2}(Q_{(i-3/2)}^{n} + \\ Q_{(i+1/2)}^{n})\right]- g h_f \frac{\Delta t}{\Delta x} (\eta^n_i - \eta^n_{i-1}) \Delta y}{1+g\
Delta t \\ n^2 |Q_{i-1/2}^{n}|/(h_f^{7/3} \Delta y)}\]$$
where subscripts $i$ and $n$ refer to space and time indices, respectively. Subscript $i-1/2$ is to the link between node $i$ and $i-1$, subscript $i+1/2$ is the link between node $i$ and node $i+1$,
and subscript $i-3/2$ is the link between node $i-1$ and node $i-2$. $Q$ is the water discharge [m$^3$ s$^{-1}$], $\eta$ is the water surface elevation [m], $h_f$ [m] is the water depth between
cells, $n$ is the Manning's roughness coefficient [m$^{-1/3}$ s], $g$ is acceleration due to gravity [m/s$^2$], $\Delta t$ [s] is the adaptive model time step, $\Delta x$ [m] is the distance between
two cells and $\Delta y$ [m] is the flow width. Below the staggered grid and variables of the numerical solution in the x-direction, based on Almeida et al. (2012):
The overland flow local inertial approach is used in combination with the local inertial river routing. This is a similar to the modelling approach of Neal et al. (2012), where the hydraulic model
LISFLOOD-FP was extended with a subgrid channel model. For the subgrid channel, Neal et al. (2012) make use of a D4 (four direction) scheme, while here a D8 (eight direction) scheme is used, in
combination with the D4 scheme for 2D overland flow.
In the TOML file the following properties related to the local inertial model with 1D river routing and 2D overland flow can be provided for the sbm model type:
land_routing = "local-inertial" # default is kinematic-wave
river_routing = "local-inertial" # default is kinematic-wave
inertial_flow_alpha = 0.5 # alpha coefficient for model stability (default = 0.7)
froude_limit = true # default is true, limit flow to subcritical-critical according to Froude number
h_thresh = 0.1 # water depth [m] threshold for calculating flow between cells (default = 1e-03)
The properties inertial_flow_alpha, froude_limit and h_thresh apply to 1D river routing as well as 2D overland flow. The properties inertial_flow_alpha and froude_limit, and the adaptive model time
step $\Delta t$ are explained in more detail in the River and floodplain routing section of the local inertial model.
External water (supply/abstraction) inflow [m$^3$ s$^{-1}$] can be added to the local inertial model for river flow (1D) and river and overland flow combined (1D-2D), as a cyclic parameter or as part
of forcing (see also Input section).
Abstractions from the river through the variable abstraction [m$^3$ s${-1}$] are possible when water demand and allocation is computed. The variable abstraction is set from the water demand and
allocation module each time step. Abstractions are subtracted as part of the continuity equation of the local inertial model.
The local inertial model for river flow (1D) and river and overland flow combined (1D-2D) can be executed in parallel using multiple threads. | {"url":"https://deltares.github.io/Wflow.jl/stable/model_docs/lateral/local-inertial/","timestamp":"2024-11-04T08:46:39Z","content_type":"text/html","content_length":"22738","record_id":"<urn:uuid:a842dbca-de1c-4e67-823d-5ff426738266>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00001.warc.gz"} |
Numpy Archives • Page 4 of 4 • datagy
In this tutorial, you’ll learn how to use Python to calculate a z-score for an array of numbers. You’ll learn a brief overview of what the z-score represents in statistics and how it’s relevant to
machine learning. You’ll then learn… Read More »How to Calculate a Z-Score in Python (4 Ways)
Calculate a Weighted Average in Pandas and Python
Learn how to use Pandas to calculate the weighted average in Python, using groupby, numpy, and the zip function between two lists.
Python: Get Index of Max Item in List
Learn how to use Python to get the index of the max item in a list, including when duplicates exist, using for loops, enumerate, and numpy.
Numpy Dot Product: Calculate the Python Dot Product
Learn how to use Python and numpy to calculate the dot product, including between arrays of different dimensions and of scalars.
Python Natural Log: Calculate ln in Python
Learn how to use Python to calculate the natural logarithm, often referred to as ln, using the math and numpy libraries, and how to plot it.
Python: Convert Degrees to Radians (and Radians to Degrees)
Learn how to use Python to convert degrees to radians and radians to degrees, using the math library and the numpy library.
Python Absolute Value: Abs() in Python
Learn how to calculate a Python absolute value using the abs() function, as well as how to calculate in numpy array and a pandas dataframe.
Python: Subtract Two Lists (4 Easy Ways!)
Learn how to use Python to subtract two lists, using the numpy library, the zip function, for-loops, as well as list comprehensions.
Python: Transpose a List of Lists (5 Easy Ways!)
Learn how to use Python to transpose a list of lists using numpy, itertools, for loops, and list comprehensions in this tutorial!
Python: Split a List (In Half, in Chunks)
Learn how to split a Python list into n chunks, including how to split a list into different sized sublists or a different number of sublists. | {"url":"https://datagy.io/tag/numpy/page/4/","timestamp":"2024-11-13T02:13:07Z","content_type":"text/html","content_length":"107677","record_id":"<urn:uuid:8731b84b-5dc4-4933-a1aa-4e0e36fcabe1>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00463.warc.gz"} |
The Lowly Programmer
Last time I wrote I gave a brief introduction to the Game of Life and a very simple Python implementation for visualizing it. I will freely admit that was a teaser post; this post gets into the real
meat of the topic with an overview of the HashLife algorithm and a much more interesting implementation.
This entry has taken me an embarrassingly long time to post. As is my habit, I wrote the code and 90% of the post, and then left it for months and months. Whoops!
If you haven’t played with a Game of Life viewer before they are legitimately fun to toy around with - I encourage you to check this one out (code is here). Since the last version everything is much
improved. The viewer supports a larger set of controls (see the README for details) and basic file reading is implemented so it’s possible to try new starting patterns on the fly. And, as promised,
I’ve implemented the HashLife algorithm to massively speed up iterations, so enormous patterns billions of generations forward are easily within your reach.
HashLife is a simple yet interesting algorithm. Invented in 1984 by Bill Gosper (of Gosper glider gun fame), it exploits repeated patterns to dramatically cut down the work required to support large
patterns over vast numbers of iterations. Between the Wikipedia page and the enigmatically named “An Algorithm for Compressing Space and Time” in Dr. Dobb’s Journal I think it’s decently well
explained, but it took me a couple read-throughs to really wrap my head around so I’m going to try to give an overview of the key insights it utilizes.
At it’s heart, HashLife is built around the concept of a quadtree. If you’re unfamiliar with it, a quadtree takes a square region and breaks it into four quadrants, each a quarter the size of the
original. Each quadrant is further broken down into quadrants of its own, and on down. At the bottom, in squares of some minimum size like 2x2, actual points are stored. This structure is usually
used to make spatial queries like “what points intersect this bounding box” efficient, but in this case two other properties are taken advantage of. First, nodes at any level are uniquely defined by
the points within their region, which means duplicated regions can be backed by the same node in memory. For the Game of Life, where there are repeated patterns and empty regions galore, this can
drastically reduce the space required. Second, in the Game of Life a square of (n)x(n) points fully dictates the inner (n-2)x(n-2) core one generation forward, the inner (n/2)x(n/2) core n/4
generations forward, irrespective of what cells are adjacent to it. So the future core of a node can be calculated once and will apply at any future point in time, anywhere in the tree.
Together these properties allow for ridiculous speedups. Hashing and sharing nodes drastically reduces the space requirements, with exponentially more sharing the further down the tree you go. There
are only 16 possible leaf nodes, after all! From this, calculating the future core for a node requires exponentially less time than a naïve implementation would. It can be done by recursively
calculating the inner core of smaller nodes, where the better caching comes into play, and then combining them together into a new node. You might be wondering if the gains from caching are lost to
the increasing difficulty of determining which nodes are equal, but with a couple careful invariants we actually get that for free. First, nodes must be immutable - this one’s pretty straightforward.
Second, nodes must be unique at all times. This forces us to build the tree from the bottom up, but then checking if a new node duplicates an existing one is simply a matter of checking if there are
any existing nodes that point to the same set of quadrants in the same order, a problem that hash tables trivially solve.
def __hash__(self):
# Hash is dependent on cells only, not e.g. _next.
# Required for Canonical(), so cannot be simply the id of the current
# object (which would otherwise work).
return hash((id(self._nw), id(self._ne), id(self._sw), id(self._se)))
def __eq__(self, other):
"""Are two nodes equal? Doesn't take caching _next into account."""
if id(self) == id(other):
return True
return (id(self._nw) == id(other._nw) and
id(self._ne) == id(other._ne) and
id(self._sw) == id(other._sw) and
id(self._se) == id(other._se))
As before, the code I’ve written is for Python 2.6 and makes use of PyGame, although neither dependency is terribly sticky. The code lives in a repository on github, and I welcome any contributions
you care to make. As the code here is complicated enough to be almost guaranteed a bug or two, there is a basic set of unit tests in life_test.py and the code itself is liberally sprinkled with
asserts. Incidentally, removing the asserts nets a 20% performance gain (as measured by the time it takes to run the ‘PerformanceTest’ unit test), although I find the development time saved by having
them is easily worth keeping them in forever. As noted later, the performance of the implementation isn’t all that important anyways. Which is a good thing, since I coded it in Python!
A comment on rewrites: during the transition from version 1 - a simple brute force algorithm - to version 2 - the Node class that implements HashLife - I had both algorithms implemented in parallel
for a while. This let me have every second frame rendered by the old algorithm so I could ensure that at different times and different render speeds that the algorithms were coming up with the same
results. I’ve seen this pattern used at work for migrating to replacement systems and it’s very much worth the extra glue code you have to write or the confidence it gives. John Carmack recently
wrote about parallel implementations on his own blog, if you want to hear more on the topic.
The performance is hard to objectively detail for an algorithm like this. For example, it takes ~1 second to generate the billionth generation of the backrake 3 pattern, which has around 300,000,000
live cells; it takes ~2 seconds to generate the quintillionth generation with 3x10^17 live cells. But this is a perfect patterns to showcase HashLife - a simple spaceship traveling in a straight
line, generating a steady stream of gliders. In comparison, a chaotic pattern like Acorn takes almost 25 seconds to generate just 5000 generations with at most 1057 alive at any time. As it stands
the properties of the algorithm drastically outweigh the peculiarities of the implementation for anything I care to do. Although I must say, if you want to compare it to another implementation in an
apples to apples comparison I’d love to hear the numbers you get.
As always, I’d love to hear what you think!
Update: See part 2 for the implemented HashLife algorithm.
The Game of Life is a fascinating system. It was invented by John Conway in 1970 and has been studied continuously ever since. For those reading who haven’t heard of it before, a brief explanation:
The world is an infinite grid of points, all either alive or dead. After each generation – or ‘iteration’ if you’d prefer – cells are updated according to the following three rules:
1. If a cell is alive and it has two or three live neighbours, it stays alive.
2. If a cell is dead and it has exactly three live neighbours, it becomes alive (tripartite reproduction?).
3. Any other cell is dead.
From these simple rules amazing complexity can arise. Some configurations are stable, like the period two “blinker” [above left], or the period four “glider” [above right] that moves one row over and
one row down with every cycle. Other configurations, like the one above centre, grow infinitely – this one spits out two gliders then lays down a zig-zag strip of blocks forever after.
There is more to the Game of Life than pretty patterns and curious growth, I must hasten to add. It has been studied by a host of people in a variety of fields and has gone on to start a new branch
of mathematics (cellular automata) and spur discussions on whether a sufficiently complicated pattern could be considered intelligent. It has also been proven to be Turing complete, so any
computation your computer can run can be run by simulating the Game of Life with the correct starting state.
I have implemented a basic python program for simulating the Game of Life on GitHub. It allows for infinite patterns, grows the field of view automatically, and allows speed to be controlled with Up/
Down, but otherwise is a very simple implementation. The goal here is to eventually implement some of the more interesting algorithms for speeding up the simulation. There are numerous such
algorithms, although the one I find the most interesting is called Hashlife and exploits repeated patterns through space and time to achieve an exponential speedup in running the simulation. More
details in part 2, whenever I write it :). | {"url":"http://www.thelowlyprogrammer.com/2011/","timestamp":"2024-11-08T11:42:12Z","content_type":"application/xhtml+xml","content_length":"48085","record_id":"<urn:uuid:72b2f2d9-e19b-4c56-ae4f-5e1efc8a3290>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00743.warc.gz"} |
Problem C
Languages en is
Hannes is working on his new startup company where his goal is to help other startup companies choose a good name. He has looked into several companies and thinks he has developed a formula that
helps any company become a unicorn (reach a value of $\$ 1\, 000\, 000\, 000$).
Hannes’ formula is such that for any company, if it does not have a consonant in its name, the company will make it big. But Hannes can’t program so he asks you for help. He wants a program that
takes a name suggestion as input and returns the suggestion with all consonants removed.
The letters a, e, i, o, u and y are vowels, all other letters are consonants.
The input contains a line with the name suggestion $S$. The length of $S$ is at most $10^6$ and contains only English letters.
Print the name suggestion with all consonants removed.
Group Points Constraints
1 25 The length of $S$ is at most $1\, 000$
2 75 No further constraints
Sample Input 1 Sample Output 1
facebook aeoo
Sample Input 2 Sample Output 2
twitter ie
Sample Input 3 Sample Output 3
HannesIncorporated aeIooae | {"url":"https://nus.kattis.com/courses/IT5003/IT5003_S1_AY2425/assignments/on92dn/problems/fyrirtaekjanafn","timestamp":"2024-11-14T01:43:13Z","content_type":"text/html","content_length":"31772","record_id":"<urn:uuid:1acf3dcd-ef6c-4836-b9e4-a37e1fb1a0cc>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00876.warc.gz"} |
TSU Algebra Worksheet - Custom Scholars
TSU Algebra Worksheet
NAME:____ MAT 1014.E1 Final Exam
Winter 2022
NOTE: RETURN ONLY THE ANSWER SHEET AT THE END OF THIS EXAM!
1. One card is selected from an ordinary deck of 52. Find the probability
as a common fraction in simplest form for each event below:
a. a face card
b. a heart and a diamond
c. a spade card and a queen
d. a black ace or a five
e. a club, a diamond or a king
____ f. a black card or a red card
2. A bag contains 8 red, 12 white and 4 blue marbles.
If 2 marbles are
selected in succession (without replacement), find the probability as a
common fraction in simplest form that:
both are blue
neither is red
either is red or blue
first is red and second is blue
second is orange
____ f.
first is red, white or blue
3.On a single roll of a pair of dice, find the probability as a common
fraction in simplest form for:
obtaining a sum of six
obtaining a sum less than five
obtaining a sum of at least eight
____ d.
obtaining sums of seven or eleven
4. In a family with 3 children, what is the probability as a common
fraction in simplest form the following outcomes:
____ a.
Two boys and one girl
At most two are girls
____ c.
At least one is a boy
All three are of the same sex
5. An urn contains twenty-five balls numbered 0 through 24. A single ball
is taken from the urn; find the probability as a decimal fraction for:
the number 4
an even number
a multiple of five
a prime number
____ e.
a negative number
____ f.
a whole number
6. A quiz consists of five true-false questions:
____ a.
How many elements are in the sample space for answering the quiz?
If a student answers the test FTFTF, what is the probability
that the student answered all five questions correctly?
7. The probability that the Rangers will win the pennant is 3/7.
____ a.
What are the odds against the Rangers winning the pennant?
____ b.
What is the probability for the Rangers losing the pennant?
8. Suppose it is possible for a coin to land on its edge. A valid
model should account for this outcome. If we assume that the
coin is
equally likely to land heads or tails, explain why the probability for
heads (or tails) cannot be 1/2 when it is possible for the coin to land
on its edge.____________________________________________________________
9. On a lab quiz, the following scores were made: 86, 75, 80, 92, 90, 71,
80, 62, 73, 81. For this set of scores, find the following:
a. mean
____ d. range
Standard Dev.
_____10. In a class of 250 students, Suzy is ranked 15th. What is her
percentile rank to the nearest percent?
11. The mean length of a cell phone call is 1.5 minutes with standard
deviation 0.25 minutes.
Assuming that the lengths of these calls are
normally distributed, determine the following:
_____a. What percentage of the calls last more than 2 minutes?
_____b. What percentage of the calls last between 1 and 2 minutes?
____ c. What percentage of the calls last at least 2 minutes?
____ d. What percentage of the calls last less than 1.5 minutes?
Professor Standard Normal gives a short daily quiz to his statistics
class. The results of the most recent quiz are shown in the table.
13. Kim is ranked at the 87th percentile in her class of 300 while
Bill is ranked thirty eighth in the same class. Who ranks higher?
14. Explain why it is not possible to have a percentile rank of 100.
_____________________________________________________________ ___________
15. The heights of 1000 female college students were found to be normally
distributed with a mean of 66 inches and a standard deviation of 2
inches. Determine each of the following:
a. What is the probability of a height between 66 and 70 inches?
____ b, What percentage of the heights are between 62 and 70 inches?
____ c. How many of the heights are under 70 inches?
____ d. How many are between 62 and 70 inches?
____ e. How many are less than 62 inches?
16. A fire in the basement has destroyed part of a report. One whole line
and several other areas were burned. You are asked to find the value
of the third element (?) in the table below.
(Deviation squared)
MAT 1014 FINAL EXAM Winter 2022 ANSWER SHEET
1. a.
6. a.
7. a.
9. a.
2. a.
3. a.
11. a.
4. a.
12. a.
5. a.
15. a.
____ MAT 1014.E1 Final Exam
Winter 2022
NOTE: RETURN ONLY THE ANSWER SHEET AT THE END OF THIS EXAM!
1. One card is selected from an ordinary deck of 52. Find the probability
as a common fraction in simplest form for each event below:
a. a face card
b. a heart and a diamond
c. a spade card and a queen
d. a black ace or a five
e. a club, a diamond or a king
____ f. a black card or a red card
2. A bag contains 8 red, 12 white and 4 blue marbles.
If 2 marbles are
selected in succession (without replacement), find the probability as a
common fraction in simplest form that:
both are blue
neither is red
either is red or blue
first is red and second is blue
second is orange
____ f.
first is red, white or blue
3.On a single roll of a pair of dice, find the probability as a common
fraction in simplest form for:
obtaining a sum of six
obtaining a sum less than five
obtaining a sum of at least eight
____ d.
obtaining sums of seven or eleven
4. In a family with 3 children, what is the probability as a common
fraction in simplest form the following outcomes:
____ a.
Two boys and one girl
At most two are girls
____ c.
At least one is a boy
All three are of the same sex
5. An urn contains twenty-five balls numbered 0 through 24. A single ball
is taken from the urn; find the probability as a decimal fraction for:
the number 4
an even number
a multiple of five
a prime number
____ e.
a negative number
____ f.
a whole number
6. A quiz consists of five true-false questions:
____ a.
How many elements are in the sample space for answering the quiz?
If a student answers the test FTFTF, what is the probability
that the student answered all five questions correctly?
7. The probability that the Rangers will win the pennant is 3/7.
____ a.
What are the odds against the Rangers winning the pennant?
____ b.
What is the probability for the Rangers losing the pennant?
8. Suppose it is possible for a coin to land on its edge. A valid
model should account for this outcome. If we assume that the
coin is
equally likely to land heads or tails, explain why the probability for
heads (or tails) cannot be 1/2 when it is possible for the coin to land
on its edge.____________________________________________________________
9. On a lab quiz, the following scores were made: 86, 75, 80, 92, 90, 71,
80, 62, 73, 81. For this set of scores, find the following:
a. mean
____ d. range
Standard Dev.
_____10. In a class of 250 students, Suzy is ranked 15th. What is her
percentile rank to the nearest percent?
11. The mean length of a cell phone call is 1.5 minutes with standard
deviation 0.25 minutes.
Assuming that the lengths of these calls are
normally distributed, determine the following:
_____a. What percentage of the calls last more than 2 minutes?
_____b. What percentage of the calls last between 1 and 2 minutes?
____ c. What percentage of the calls last at least 2 minutes?
____ d. What percentage of the calls last less than 1.5 minutes?
Professor Standard Normal gives a short daily quiz to his statistics
class. The results of the most recent quiz are shown in the table.
13. Kim is ranked at the 87th percentile in her class of 300 while
Bill is ranked thirty eighth in the same class. Who ranks higher?
14. Explain why it is not possible to have a percentile rank of 100.
_____________________________________________________________ ___________
15. The heights of 1000 female college students were found to be normally
distributed with a mean of 66 inches and a standard deviation of 2
inches. Determine each of the following:
a. What is the probability of a height between 66 and 70 inches?
____ b, What percentage of the heights are between 62 and 70 inches?
____ c. How many of the heights are under 70 inches?
____ d. How many are between 62 and 70 inches?
____ e. How many are less than 62 inches?
16. A fire in the basement has destroyed part of a report. One whole line
and several other areas were burned. You are asked to find the value
of the third element (?) in the table below.
(Deviation squared)
MAT 1014 FINAL EXAM Winter 2022 ANSWER SHEET
1. a.
6. a.
7. a.
9. a.
2. a.
3. a.
11. a.
4. a.
12. a.
5. a.
15. a. | {"url":"https://customscholars.com/tsu-algebra-worksheet/","timestamp":"2024-11-06T18:28:08Z","content_type":"text/html","content_length":"61739","record_id":"<urn:uuid:dd57afb2-b9d7-4089-847b-00660f24f2a3>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00483.warc.gz"} |
Excel Formula: Multiple Matches, Multiple Criteria
This function demonstrates how to write an Excel formula that returns the product of all matches when searching for multiple matches with multiple criteria. The formula combines the PRODUCT function,
IF function, and array formula. Here is a breakdown of the formula and an example to illustrate its usage:
• The formula uses the PRODUCT function, which calculates the product of all the values returned by the IF function.
• The IF function checks if the criteria values match the corresponding values in the criteria ranges and returns the corresponding value from the range to multiply.
• The criteria ranges and criteria values are specified in the formula.
• The formula should be entered as an array formula by pressing Ctrl + Shift + Enter.
Suppose you have a dataset with values in columns A, B, and D. You want to find the product of all the values in column D where the corresponding values in column A are greater than 5 and the
corresponding values in column B are less than 10. The formula would be:
=PRODUCT(IF(A:A>5, IF(B:B<10, D:D)))
After entering the formula as an array formula, you will get the result of 96, which is the product of the matching values (6 and 8) in column D.
The formula you can use in Excel to return the product of all matches when searching for multiple matches with multiple criteria is the PRODUCT function combined with the IF function and an array
Here is the formula:
=PRODUCT(IF(criteria_range1=criteria1, IF(criteria_range2=criteria2, range_to_multiply)))
Let's break down the formula and explain each part:
1. criteria_range1 and criteria_range2 are the ranges where you have your criteria values. These ranges should have the same size as the range_to_multiply.
2. criteria1 and criteria2 are the specific criteria values you want to match in criteria_range1 and criteria_range2, respectively.
3. range_to_multiply is the range of values that you want to multiply together when the criteria matches.
4. The IF function is used to check if the criteria values match the corresponding values in the criteria ranges. If the criteria match, the IF function returns the corresponding value from the
range_to_multiply. If the criteria do not match, the IF function returns FALSE.
5. The PRODUCT function is used to multiply all the values returned by the IF function. It calculates the product of all the matching values.
6. This formula should be entered as an array formula by pressing Ctrl + Shift + Enter after typing the formula.
Let's see an example to illustrate how this formula works:
Suppose you have the following data:
| A | B | C | D |
| 1 | 2 | 3 | 4 |
| 5 | 6 | 7 | 8 |
| 9 | 10 | 11 | 12 |
And you want to find the product of all the values in column D where the corresponding values in column A are greater than 5 and the corresponding values in column B are less than 10.
The formula would be:
=PRODUCT(IF(A:A>5, IF(B:B<10, D:D)))
After entering the formula as an array formula, you will get the result of 96, which is the product of the matching values (6 and 8) in column D.
References: - PRODUCT function in Excel - IF function in Excel - Array formulas in Excel | {"url":"https://codepal.ai/excel-formula-generator/query/UPRvXeph/excel-formula-multiple-matches-multiple-criteria","timestamp":"2024-11-07T22:37:07Z","content_type":"text/html","content_length":"89443","record_id":"<urn:uuid:3e570021-818b-4332-ac2b-4492138c0324>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00134.warc.gz"} |
Performing constrained
Performing constrained optimization
Simulation design problems tend to have conflicting goals. Usually there is a desired ideal response that cannot be obtained exactly due to either modeling or physical constraints. The system
response is determined by a few variables that may in themselves be bounded. Mathematically, such a situation is represented by:
Minimize or maximize g(X) subject to
glb[i] ≤ g[i](X) ≤ gub[i] for i=1,...,m,
xlb[j] ≤ x[j] ≤ xub[j] for j=1,...,n,
X is a vector on n variables, x[1 ],...,x[n], and the functions g[1 ],...,g[m] all depend on X.
The function to be minimized g(X) is called the objective (or cost) function. Constraints are given by the function g[i](X). The decision variables x[1 ],...,x[n], may have bounds. The analogous data
structures are represented by the following blocks:
│Block │Purpose │
│cost │objective or cost function │
│globalConstraint │constraint functions │
│parameterUnknown │decision variables │
Upper and lower bounds are set in the globalConstraint block for the constraint functions, and upper and lower bounds are set in the parameterUnknown block for the decision variables.
Embed uses first partial derivatives of each function g[i] with respect to each variable x[j]. These are automatically computed by finite difference approximations. After an initial data entry
segment, the program operates in two phases. If the initial values of the variables you supply do not satisfy all g[i] constraints, a Phase I optimization is started. The Phase I objective function
is the sum of the constraint violations. This optimization run terminates either with a message that the problem is infeasible or with a feasible solution. Beware that if an infeasibility message is
produced, it may be because the program became stuck at a local minimum of the Phase I objective. In this case, the problem may actually have feasible solutions.
Phase II begins with a feasible solution — either found by Phase I or with you providing a starting point — and attempts to optimize the objective function. After Phase II, a full optimization cycle
has been completed and summary output is provided. | {"url":"https://help.altair.com/embedse/performingconstrainedoptimization.htm","timestamp":"2024-11-05T15:47:16Z","content_type":"application/xhtml+xml","content_length":"8608","record_id":"<urn:uuid:78aa37a5-5534-4894-8773-7ae892bf82bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00435.warc.gz"} |
Inverse of Matrices - Methods, Properties and Examples
TopicsMaths TopicsInverse of Matrices
Inverse of Matrices
To understand the Inverse of Matrices, one initially needs to understand Matrices.
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is a fundamental mathematical concept used in various fields, including linear algebra, computer
science, physics, and engineering.
In matrix notation, a matrix is typically represented by a capital letter and enclosed in brackets or parentheses. The size of a matrix is determined by the number of rows and columns it contains.
For example, an “m x n” matrix has m rows and n columns.
Inverse of a Matrix
The inverse of matrix is obtained by dividing the adjoint of the given matrix by the determinant of the given matrix. Students must note that matrix inverse could be found only for square matrices.
This article discusses about the inverse of a matrix, steps to find the inverse of a matrix, the properties of the inverse matrix along with the examples.
Matrix Inverse
If A is a non-singular square matrix, then there exists a n x n matrix A-1 which is called the inverse matrix of A, such that it satisfies the property:
AA-1 = A-1A = I, where I is the Identity matrix.
The identity matrix for the 2 x 2 matrix is given by:
It is noted that to find the inverse of a matrix, the square matrix should be a non-singular matrix whose determinant value is not equal to zero.
Let us take the square matrix A
Where a, b, c, and d are the numbers.
• The determinant of matrix A is written as ad-bc.
• For the existence of the inverse of the matrices, the determinant should not equal zero.
• The inverse matrix can be found for 2× 2, 3× 3, …n × n matrices.
• As the value of n increases, finding the inverse of the matrix becomes difficult.
Inverse Matrix Method
The inverse of a matrix can be found using three different methods. However, any of these three methods will produce the same result.
Method 1: Inverse-matrix-method
Let the matrix A be:
We can find the inverse of a matrix using this method.
Method 2: minors and cofactors methods
Second method of finding the matrix inverse is by using the minors and cofactors of elements of the given matrix.
The inverse matrix can be found using the following equation:
A^-1 = adj(A)/det(A),
where adj(A) refers to the adjoint of a matrix A,
And det(A) refers to the determinant of a matrix A.
To know more about the adjoint and co-factor of a matrix,
Check :
• Adjoint of a matrix A
• Cofactor of a matrix A
Method 3: Elementary Transformation
To find an Inverse of a Matrix by Elementary Transformation we follow the below-mentioned method.
Let us consider three matrices X, A and B such that X = AB.
To find the inverse of a matrix using elementary transformation, we convert the given matrix into an identity matrix.
Check: how to do elementary transformations of matrices
If, for a matrix A, A-1 exists, then to determine A-1 using elementary row operations, we follow the following steps.
• Let A = IA, where I is the identity matrix of the same order as A.
• Apply a sequence of row operations on LHS and RHS till we get an identity matrix on the LHS. while performing the operations we will get I = BA. The matrix B obtained on the RHS is the inverse of
matrix A.
• To find the inverse of A using column operations, write A = IA and apply column operations in a similar manner as the above-mentioned step sequentially till I = AB is obtained, where B is the
inverse matrix of A.
Also check: how to find the inverse of a matrix using elementary operations
Inverse Matrix 2 x 2 Example
To understand in a better way, let us take a look at the following example.
Properties of the inverse of a matrix
A few important properties of the inverse matrix are mentioned below.
• If matrix A is nonsingular, then (A-1)-1 = A
• If A and B are nonsingular matrices, then AB is also nonsingular.
Thus, (AB)-1 = B-1A-1
• If A is nonsingular matrix, then (AT)-1 = (A-1)T
• If A is any matrix and A-1 is its inverse, then AA-1 = In = A-1A , where n is the order of matrices
FAQs on Inverse of Matrices
What is concept inverse of a matrix?
Matrices also have reciprocals just like numbers. In the case of matrices, this reciprocal is called an inverse matrix. If A is a square matrix and B is its inverse, then the product of matrices A
and B is equal to the unit matrix.
How do you find the inverse of a 3×3 matrix?
The steps required to find the inverse of a 3×3 matrix are:
Compute the determinant of the given matrix and check whether the matrix is invertible.
Calculate the determinant of 2×2 minor matrices.
Formulate the matrix of cofactors.
Take the transpose of the cofactor matrix to get the adjugate matrix.
Finally, divide each term of the adjugate matrix by the determinant
Is adjoint and inverse the same?
No, the adjoint matrix and inverse matrix are not the same. However, by dividing the each term of the adjoint of the matrix by the determinant of the original matrix, we get an inverse matrix.
How to do you know whether the given matrix has an inverse?
If the determinant of a given matrix is not equal to 0, i.e. it is non-singular, then the matrix is invertible. | {"url":"https://infinitylearn.com/surge/topics/inverse-of-matrices/","timestamp":"2024-11-09T12:48:42Z","content_type":"text/html","content_length":"173788","record_id":"<urn:uuid:69cb9558-9dfa-4f15-a515-3dc298defaa5>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00018.warc.gz"} |
The word algebraic according to the dictionary
Hello, you have come here looking for the meaning of the word
. In DICTIOUS you will not only get to know all the dictionary meanings for the word
, but we will also tell you about its etymology, its characteristics and you will know how to say
in singular and plural. Everything you need to know about the word
you have here. The definition of the word
will help you to be more precise and correct when speaking or writing your texts. Knowing the definition of
, as well as those of other words, enriches your vocabulary and provides you with more and better linguistic resources.
Partly from algebra + -ic and partly from French algébraïque.^[1]
algebraic (comparative more algebraic, superlative most algebraic)
1. Of, or relating to, algebra.
2. (mathematics, of an expression, equation, or function) Containing only numbers, letters, and arithmetic operators.
3. (algebra, number theory, of a number) Which is a root of some polynomial whose coefficients are rational.
4. (algebra, of a field) Whose every element is a root of some polynomial whose coefficients are rational.
5. (chess, of notation) Describing squares by file (referred to in intrinsic order rather than by the piece starting on that file) and rank, both with reference to a fixed point rather than a
player-dependent perspective.
• (antonym(s) of “that is the root of some polynomial”): transcendental
• (antonym(s) of “whose every element is the root of some polynomial”): transcendental
• (that is the root of some polynomial): quadratic number
Derived terms
• (of or relating to algebra):
• (that is a root of some polynomial):
containing numbers, letters, and arithmetic operators
which is a root of some polynomial
of a particular notation for chess moves that assigns each square a letter and a number defined from a player-independent perspective
algebraic (feminine algebraica, masculine plural algebraics, feminine plural algebraiques)
1. algebraic
Synonym: algèbric
Derived terms
Further reading
algebraic m or n (feminine singular algebraică, masculine plural algebraici, feminine and neuter plural algebraice)
1. Obsolete form of algebric.
• algebraic in Academia Română, Micul dicționar academic, ediția a II-a, Bucharest: Univers Enciclopedic, 2010. →ISBN | {"url":"https://dictious.com/en/algebraic","timestamp":"2024-11-06T18:08:58Z","content_type":"text/html","content_length":"68572","record_id":"<urn:uuid:c40ccdde-d357-400e-b440-15281b8b65bb>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00046.warc.gz"} |
Python’s Path Through Mazes: A Journey of Creation and Solution (2024)
Mazes have been fascinating humans for centuries. From the labyrinth of Greek mythology to the garden mazes of the European nobility, these intricate networks of paths and walls are both challenging
and fun. But did you know that you can create and solve mazes programmatically using Python? In this blog post, we’ll walk you through the process, step by step.
The basis of our maze generation is the depth-first search (DFS) algorithm. At the highest level, this algorithm works by starting at a given node (in our case, the upper-left cell of the maze), and
exploring as far as possible along each branch before backtracking. It uses a stack data structure to remember the position of nodes that are still unexplored.
In the context of maze generation, we treat the cells of the maze as the nodes. For each cell, we consider its neighboring cells as the branches that can be explored. However, to prevent the maze
from becoming a grid of open cells, we ensure that we only move to a cell if there is a wall between it and the current cell.
In the code, we initialize the maze as a grid filled with “1”s, representing walls. We represent cells as “0”s. We then define the starting point (0, 0), and initialize the stack with this starting
point. The DFS algorithm starts by taking the top cell from the stack (which initially is the starting cell), and checks its neighbors in random order. If it finds a neighbor that is a wall and is
within the maze boundaries, it moves to that cell (making it an open cell), and pushes it to the stack. It then starts again with the new cell.
If it cannot find such a neighbor, it means that all neighbors of the current cell have already been visited, and so it removes the cell from the stack and backtracks to the previous cell. This
process continues until the stack is empty, which means that all reachable cells have been visited. The end result is a random maze that can be navigated from the starting point to any other
reachable cell.
Finally, to make the maze solvable, we ensure there is an entrance and an exit. We do this by setting the first cell in the first row and the last cell in the last row as “0”s (open cells).
import matplotlib.pyplot as plt
import numpy as np
import random
from queue import Queue
def create_maze(dim):
# Create a grid filled with walls
maze = np.ones((dim*2+1, dim*2+1))
# Define the starting point
x, y = (0, 0)
maze[2*x+1, 2*y+1] = 0
# Initialize the stack with the starting point
stack = [(x, y)]
while len(stack) > 0:
x, y = stack[-1]
# Define possible directions
directions = [(0, 1), (1, 0), (0, -1), (-1, 0)]
for dx, dy in directions:
nx, ny = x + dx, y + dy
if nx >= 0 and ny >= 0 and nx < dim and ny < dim and maze[2*nx+1, 2*ny+1] == 1:
maze[2*nx+1, 2*ny+1] = 0
maze[2*x+1+dx, 2*y+1+dy] = 0
stack.append((nx, ny))
# Create an entrance and an exit
maze[1, 0] = 0
maze[-2, -1] = 0
return maze
To solve the maze, we use the breadth-first search (BFS) algorithm. Unlike DFS, which goes as deep as possible into the maze before backtracking, BFS explores all neighbors of the current cell before
moving on. This guarantees that it will find the shortest path from the starting point to the target.
The BFS algorithm also uses a queue data structure to keep track of the cells that need to be explored. However, in our case, each item in the queue is a tuple of the current cell and the path taken
to reach it. This allows us to keep track of the path followed by the algorithm.
We start by adding the starting cell and an empty path to the queue. Then, as long as the queue is not empty, we take the first item from the queue, explore its neighbors, and add them to the queue.
Similar to the DFS algorithm, we ensure that we only move to a cell if it is an open cell and has not been visited before.
When we add a neighbor to the queue, we also append it to the current path to keep track of the path taken to reach it. If we reach the target cell, we return the path taken to reach it. Since BFS
explores all neighbors of the current cell before moving on, this path is guaranteed to be the shortest path.
At the end, if we have explored all reachable cells and have not found the target, the queue will become empty and the algorithm will stop. This indicates that there is no path from the starting
point to the target.
def find_path(maze):
# BFS algorithm to find the shortest path
directions = [(0, 1), (1, 0), (0, -1), (-1, 0)]
start = (1, 1)
end = (maze.shape[0]-2, maze.shape[1]-2)
visited = np.zeros_like(maze, dtype=bool)
visited[start] = True
queue = Queue()
queue.put((start, []))
while not queue.empty():
(node, path) = queue.get()
for dx, dy in directions:
next_node = (node[0]+dx, node[1]+dy)
if (next_node == end):
return path + [next_node]
if (next_node[0] >= 0 and next_node[1] >= 0 and
next_node[0] < maze.shape[0] and next_node[1] < maze.shape[1] and
maze[next_node] == 0 and not visited[next_node]):
visited[next_node] = True
queue.put((next_node, path + [next_node]))
Now that we have our maze generated, it’s time to visualize it. For this task, we are using matplotlib, a versatile plotting library in Python.
In our function draw_maze(maze, path=None), we start by creating a figure and an axis object. The figure object is a top-level container for all plot elements, while the axis represents a single
plot. We're also setting a specific size for the figure to ensure that the maze is easy to view, regardless of its size.
Next, we use the imshow function of the axis object to display the maze as an image. The maze, which is represented as a 2D numpy array, translates perfectly to an image format, with "1"s
representing black pixels (the walls) and "0"s representing white pixels (the open cells). We use the plt.cm.binary colormap to achieve this color representation. The interpolation parameter is set
to 'nearest' to preserve the blocky nature of the maze when the image is displayed.
In the case where we want to draw a solution path onto the maze, we check if a path is provided to the function. If it is, we extract the x and y coordinates of each point on the path. Since the path
is represented as a list of tuples, where each tuple is a pair of coordinates (y, x), we use list comprehension to extract the lists of x and y coordinates. We then use the plot function of the axis
object to draw the path onto the maze. We set the color of the path to red and its linewidth to 2 for better visibility.
Finally, we use the set_xticks and set_yticks functions of the axis object to hide the x and y axis, as they are not relevant to the visualization of the maze. We also draw arrows representing the
entrance and the exit of the maze. We use the arrow function of the axis object, setting the start points and direction of the arrows, and also their color and size.
When we run the script and input the desired dimension of the maze, the script will generate a random maze of the specified size, find the shortest path from the entrance to the exit, and draw the
maze and the path onto a figure, which is then displayed to us.
def draw_maze(maze, path=None):
fig, ax = plt.subplots(figsize=(10,10))
# Set the border color to white
ax.imshow(maze, cmap=plt.cm.binary, interpolation='nearest')
# Draw the solution path if it exists
if path is not None:
x_coords = [x[1] for x in path]
y_coords = [y[0] for y in path]
ax.plot(x_coords, y_coords, color='red', linewidth=2)
# Draw entry and exit arrows
ax.arrow(0, 1, .4, 0, fc='green', ec='green', head_width=0.3, head_length=0.3)
ax.arrow(maze.shape[1] - 1, maze.shape[0] - 2, 0.4, 0, fc='blue', ec='blue', head_width=0.3, head_length=0.3)
Once we have defined our maze creation, maze solving, and visualization functions, we can bring these components together and see how to actually run our maze generator and solver. Here is how to get
everything working.
First, you need to have Python installed on your machine. If you haven’t done this yet, go to the official Python website, download the latest version of Python, and install it.
Additionally, you need the numpy and matplotlib libraries, which can be installed via pip:
pip install numpy matplotlib
After setting up Python and necessary libraries, save the Python code provided in the sections above to a file, say maze.py.
The if __name__ == "__main__": statement in our code is a Pythonic way of ensuring that the code block underneath it will only be executed if this module is the main program. This means, when you
directly run maze.py, the code under this statement will be executed. However, if you import maze.py as a module in another script, the if __name__ == "__main__": block will not run.
if __name__ == "__main__":
dim = int(input("Enter the dimension of the maze: "))
maze = create_maze(dim)
path = find_path(maze)
draw_maze(maze, path)
This block of code kicks off the maze generation process. First, it asks the user to input the desired dimension of the maze. Then, it uses this dimension to create the maze using our create_maze
function. The find_path function is then called to find a solution to the maze, and finally draw_maze is used to visualize the result.
Once you’ve saved your Python file, you can run this Python file from the terminal (or command prompt in Windows) as follows:
python maze.py
Upon running, it prompts you to enter the dimension of the maze. This number refers to the number of cells in one row or column (not the actual pixel size of the maze). For a start, you can enter 10
or 15.
The program then generates a maze of the corresponding size, solves it, and displays the maze and its solution. The black areas are walls, the white areas are free paths, the red line is the shortest
path from the start to the end, and the green and blue arrows indicate the entrance and exit of the maze.
The visual representation is a powerful tool for checking the correctness of our program. If you see the red path going from the green arrow to the blue arrow and the path doesn’t traverse through
any walls, you’ll know that both the maze generator and the solver are functioning correctly.
Feel free to experiment with different maze dimensions or modify the code to customize the maze generation and solution behavior. Maybe you could adapt the generator to create mazes with more than
one solution, or modify the solver to implement a different pathfinding algorithm. The possibilities are limitless!
The generation, solution, and visualization of mazes are fascinating subjects that touch on a wide range of computer science and mathematics topics including graph theory, search algorithms, and data
In this tutorial, we created a simple maze generator using Python’s built-in data structures and a Depth-First Search approach. We then demonstrated how to solve this maze using a Breadth-First
Search algorithm, which guarantees the shortest path from the start to the finish.
While our generated maze is simple and the path is straight-forward, the concepts explored in this tutorial extend well into much more complex situations. Maze generating algorithms form the backbone
of procedural content generation in video games, modeling complex structures and patterns in nature, and even in the routing of PCBs in electronics.
The visualization aspect of this project is also significant. It shows how simple it is to use matplotlib to represent our data graphically. Understanding how to plot and visualize data is a crucial
skill in Python and other programming languages.
Moreover, the project’s extensibility is one of its most exciting aspects. This maze generator and solver is a starting point, and there are countless ways you could expand upon it. You could
introduce complexity into the maze creation, incorporate multiple solutions, add a GUI for real-time interaction, or even use these principles to create an entire game.
As we wrap up this discussion, I hope this serves as a launching pad for you to explore more about these topics and inspire you to create new and exciting projects. The ability to generate, solve,
and visualize problems is a powerful tool in a programmer’s toolbox. So, keep experimenting, keep learning, and most importantly, have fun with your coding journey!
Previously we just drew the path of the solution to the maze. You can actually animate the maze using the matplotlib animation library. To animate your maze, simply replace the draw maze function
with the following code:
import matplotlib.pyplot as plt
import matplotlib.animation as animation
#... animate the path through the maze ...
def draw_maze(maze, path=None):
fig, ax = plt.subplots(figsize=(10,10))
# Set the border color to white
ax.imshow(maze, cmap=plt.cm.binary, interpolation='nearest')
# Prepare for path animation
if path is not None:
line, = ax.plot([], [], color='red', linewidth=2)
def init():
line.set_data([], [])
return line,
# update is called for each path point in the maze
def update(frame):
x, y = path[frame]
line.set_data(*zip(*[(p[1], p[0]) for p in path[:frame+1]])) # update the data
return line,
ani = animation.FuncAnimation(fig, update, frames=range(len(path)), init_func=init, blit=True, repeat = False, interval=20)
# Draw entry and exit arrows
ax.arrow(0, 1, .4, 0, fc='green', ec='green', head_width=0.3, head_length=0.3)
ax.arrow(maze.shape[1] - 1, maze.shape[0] - 2, 0.4, 0, fc='blue', ec='blue', head_width=0.3, head_length=0.3)
FuncAnimation is a class provided by `matplotlib.animation` module in Python’s matplotlib library. It provides a framework for creating animations and interactive plots in Python, by repeatedly
calling a function (the `update` function in our case).
In FuncAnimation, we provide the figure object that we are animating, the function to call at each frame (our `update` function), the number of frames (which is the length of our path), and the
interval between frames (in milliseconds).
Here’s a breakdown of how the FuncAnimation class works in our context:
- fig: This is the figure object that we are going to animate. This is the main drawing canvas in matplotlib and contains everything you see on a plot.
- update: This is the function we want to call for each frame in the animation. Here, it takes an integer parameter frame(which automatically increments for each frame) representing the current frame
of the animation.
- frames=len(path): This is the total number of frames in the animation, which is set to be the number of points in the path. This means that `update` will be called `len(path)` times in total.
- interval=100: This sets the delay between frames in milliseconds. So in our case, a new frame will be drawn every 100 ms, i.e., we’re effectively running the animation at 10 frames per second.
- repeat=False: This is a flag determining whether the animation should repeat once it has completed. In our case, we only want to animate the solution path once, so we set `repeat=False`.
Overall, FuncAnimation provides an easy-to-use interface for creating animations in Python with matplotlib. By repeatedly calling an update function, we can draw complex animations like the solution
path through the maze, where each step in the path is drawn one at a time to create the effect of “moving” through the maze. | {"url":"https://boyacim.net/article/python-s-path-through-mazes-a-journey-of-creation-and-solution","timestamp":"2024-11-05T19:33:27Z","content_type":"text/html","content_length":"130836","record_id":"<urn:uuid:c0577864-bf99-4cb5-9b16-baacc694b4d4>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00229.warc.gz"} |
The perfect t-test | R-bloggersThe perfect t-test
[This article was first published on
Daniel Lakens
, and kindly contributed to
]. (You can report issue about the content on this page
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if you have a blog, or
if you don't.
I’ve created an easy to use R script that will import your data, and performs and writes up a state-of-the-art dependent or independent t-test. The goal of this script is to examine whether more
researcher-centered statistical tools (i.e., a one-click analysis script that checks normality assumptions, calculates effect sizes and their confidence intervals, creates good figures, calculates
Bayesian and robust statistics, and writes the results section) increases the use of novel statistical procedures. Download the script here: https://github.com/Lakens/Perfect-t-test. For comments,
suggestions, or errors, e-mail me at [email protected]. The script will likely be updated – check back for updates or follow me @Lakens to be notified of updates.
Correctly comparing two groups is remarkably challenging. When performing a t-test researchers rarely manage to follow all recommendations that statisticians have made over the years. Where
statisticians update their recommendations, statistical textbooks often do not. Even though reporting effect sizes and their confidence intervals has been recommended for decades (e.g., Cohen, 1990),
statistical software (e.g., SPSS 22) often does not provide these statistics. Progress is slow, and Sharpe (2013) points to a lack of awareness, a lack of time, a lack of easily usable software, and
a lack of education as some of the main reasons for the resistance to adopting statistical innovations.
Here, I propose a way to speed up the widespread adoption of the state-of-the-art statistical techniques by providing researchers with an easy to use script in free statistical software (R) that will
perform and report all statistical analyses, practically with a single button press. The script (Lakens, 2015, available at https://github.com/Lakens/Perfect-t-test) follows state-of-the-art
recommendations (see below), creates plots of the data, and writes the results section, including a minimally required interpretation of the statistical results.
Automated analyses might strike readers as a bad idea because it facilitates mindless statistics. Having performed statistics mindlessly for most of my professional career, I sincerely doubt access
to this script would have reduced my level of understanding. If anything, reading an automatically generated results section of your own data that includes statistics you are not accustomed to
calculate or report is likely to make you think more about the usefulness of these statistics, not less. However, the goal of this script is not to educate people. The main goal is to get researchers
to perform and report the analyses they should, and make this as efficient as possible.
Comparing two groups
Keselman, Othman, Wilcox, and Fradette (2004) proposed the a more robust two-sample t-test that provides better Type 1 error control in situations of variance heterogeneity and nonnormality, but
their recommendations have not been widely implemented. Researchers might in general be unsure whether it is necessary to change the statistical tests they use to analyze and report comparisons
between groups. As Wilcox, Granger, and Clark (2013, p. 29) remark: “ All indications are that generally, the safest way of knowing whether a more modern method makes a practical difference is to
actually try it.” Making sure conclusions based on multiple statistical approaches converge is an excellent way to gain confidence in your statistical inferences. This R script calculates traditional
Frequentist statistics, Bayesian statistics, and robust statistics, using both a hypothesis testing as an estimation approach, to invite researchers to examine their data from different perspectives.
Since Frequentist and Bayesian statistics are based on assumptions of equal variances and normally distributed data, the R script provides boxplots and histograms with kernel density plots overlaid
with a normal distribution curve to check for outliers and normality. Kernel density plots are a non-parametric technique to visualize the distribution of a continuous variable. They are similar to a
histogram, but less dependent on the specific choice of bins used when creating a histogram. The graphs plot both the normal distribution, as the kernel density function, making it easier to visually
check whether the data is normally distributed or not. Q-Q plots are provided as an additional check for normality.
Yap and Sim (2011) show that no single test for normality will perform optimally for all possible distributions. They conclude (p. 2153): “If the distribution is symmetric with low kurtosis values
(i.e. symmetric short-tailed distribution), then the D’Agostino-Pearson and Shapiro-Wilkes tests have good power. For symmetric distribution with high sample kurtosis (symmetric long-tailed), the
researcher can use the JB, Shapiro-Wilkes, or Anderson-Darling test.” All four normality tests are provided in the R script. Levene’s test for the equality of variances is provided, although for
independent t-tests, Welch’s
-test (which does not require equal variances) is
provided by default
, following recommendations by Ruxton (2006). A short explanation accompanies all plots and assumption checks to help researchers to interpret the results.
The script also creates graphs that, for example, visualize the distribution of the datapoints, and provide both within as between confidence intervals:
The script provides interpretations for effect sizes based on the classifications ‘small’, ‘medium’, and ‘large’. Default interpretations of the size of an effect based on these three categories
should only be used as a last resort, and it is preferable to interpret the size of the effect in relation to other effects in the literature, or in terms of its practical significance. However,
since researchers often do not interpret effect sizes (if they are reported to begin with), the default interpretation (and the suggestion to interpret effect sizes in relation to other effects in
the literature) should at least function as a reminder that researchers are expected to interpret effect sizes. The common language effect size (McGraw & Wong, 1992) is provided as an additional way
to communicate the effect size.
Similarly, the Bayes Factor is classified into anecdotal, moderate, strong, very strong, and decisive evidence for the alternative or null hypothesis, following Jeffreys (1961), even though
researchers are reminded that default interpretations of the strength of the evidence should not distract from the fact that strength of evidence is a continuous function of the Bayes Factor. We can
expect researchers will rely less on default interpretations, the more acquainted they become with these statistics, but for novices some help in interpreting effect sizes and Bayes Factors will
guide their interpretation.
Running the Markdown script
R Markdown scripts provide a way to create fully reproducible reports from data files. The script combines the commands to perform all statistical analyses with the written sections of the final
output. Calculated statistics and graphs are inserted into the written report at specified locations. After installing the required packages, preparing the data, and specifying some variables in the
Markdown document, the report can be generated (and thus, the analysis procedure can be performed) with a single mouse-click (scroll down for an example of the output).
The R Markdown script and the ReadMe file contain detailed instructions on how to run the script, and how to install required packages, including the PoweR package (Micheaux & Tran, 2014) to perform
the normality tests, HLMdiag to create the Q-Q plots (Loy & Hofmann, 2014). ggplot2 for all plots (Wickham, 2009), car (Fox & Weisberg, 2011) to perform Levene’s test, MBESS(Kelley, 2007) to
calculate effect sizes and their confidence intervals, WRS for the robust statistics (Wilcox & Schönbrodt, 2015), BootsES to calculate a robust effect size for the independent t-test (Kirby &
Gerlanc, 2013), BayesFactor for the bayes factor (Morey & Rouder, 2015), and BEST (Kruschke & Meredith, 2014) to calculate the Bayesian highest density interval.
The data file (which should be stored in the same folder that contains the R markdown script) needs to be tab delimited with a header at the top of the file (which can easily be created from SPSS by
saving data through the ‘save as’ menu and selecting ‘save as type: Tab delimited (*.dat)’, or in Excel by saving the data as ‘Text (Tab delimited) (.txt)’. For the independent t-test the data file
needs to contain at least two columns (one specifying the independent variable, and one specifying the dependent variable, and for the dependent t-test the data file needs to contain three columns,
one subject identifier column, and two columns for the two dependent variables. The script for dependent t-tests allows you to select a subgroup for the analysis, as long as the data file contains an
additional grouping variable (see the demo data). The data files can contain irrelevant data, which will be ignored by the script. Finally, researchers need to specify the names (or headers) of the
independent and dependent variables, as well as grouping variables. Finally, there are some default settings researchers can change, such as the sidedness of the test, the alpha level, the percentage
for the confidence intervals, and the scalar on the prior for the Bayes Factor.
The script can be used to create either a word document or a html document. The researchers can easily interpret all the assumption checks, look at the data for possible outliers, and (after minor
adaptations) copy-paste the result sections into their article.
The statistical results the script generates has been compared against the results provided by SPSS, JASP, ESCI, online Bayes Factor calculators, and BEST online. Minor variations in the HDI
calculation between BEST online and this script are possible depending on the burn-in samples and number of samples, and for huge t-values there are minor variations between JASP and the latest
version of the Bayes Factor package used in this script. This program is distributed in the hope that it will be useful, but without any warranty. If you find an error, please contact me at [email
Promoting Statistical Innovations
Statistical software is built around individual statistical tests, while researchers perform a set of procedures. Although it is not possible to create standardized procedures for all statistical
analyses, most, if not all, of the steps researchers have to go through when they want to report correlations, regression analyses, ANOVA’s, and meta-analyses are sufficiently structured. These tests
make up a large portion of analyses reported in journal articles. Demonstrating this, David Kennyhas created R scripts that will perform and report mediation and moderator analyses. Felix Schönbrodt
has created a Shiny app that performs several meta-analytic techniques. Making statistical innovations more accessible has a high potential to substantially improve the quality of the statistical
tests researchers perform and report. Statisticians who take the application of generated knowledge seriously should try to experiment with the best way to get researchers to use state-of-the-art
techniques. R markdown scripts are an excellent method to combine statistical analyses and a written report in free software. Shiny apps might make these analyses even more accessible, because they
no longer require users to install R and R packages.
Despite the name of this script, there is probably not such a thing as a ‘perfect’ report of a statistical test. Researchers might prefer to report standard errors instead of standard deviations,
perform additional checks for normality, different Bayesian or robust statistics, or change the figures. The benefit of markdown scripts with a GNU license stored on GitHub is that they can be forked
(copied to a new repository) where researchers are free to remove, add, or change sections of the script to create their own ideal test. After some time, a number of such scripts may be created,
allowing researchers to choose an analysis procedure that most closely matches their desires. Alternatively, researchers can post feature requests or errors that can be incorporated in future
versions of this script.
It is important that researchers attempt to draw the best possible statistical inferences from their data. As a science, we need to seriously consider the most efficient way to accomplish this. Time
is scarce, and scientists need to master many skills in addition to statistics. I believe that some of the problems in adopting new statistical procedures discussed by Sharpe (2013) such as lack of
time, lack of awareness, lack of education, and lack of easy to use software can be overcome by scripts that combine traditional and more novel statistics, are easy to use, and provide a brief
explanation of what is calculated while linking to the relevant literature. This approach might be a small step towards a better understanding of statistics for individual researchers, but a large
step towards better reporting practices.
Baguley, T. (2012). Calculating and graphing within-subject confidence intervals for ANOVA. Behavior research methods, 44, 158-175.
Cumming, G. (2012). Understanding the new statistics: Effect sizes, confidence intervals, and meta-analysis. New York: Routledge.
Fox, J. & Weisberg, S. (2011). An R Companion to Applied Regression, Second edition. Sage, Thousand Oaks CA.
Jeffreys, H. (1961). Theory of probability (3rd ed.). Oxford: Oxford University Press, Clarendon Press.
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Kirby, K. N., & Gerlanc, D. (2013). BootES: An R package for bootstrap confidence intervals on effect sizes. Behavior Research Methods, 45, 905-927.
Lakens, D. (2015). The perfect t-test (version 0.1.0). Retrieved from https://github.com/Lakens/perfect-t-test. doi:10.5281/zenodo.17603
Loy, A., & Hofmann, H. (2014). HLMdiag: A Suite of Diagnostics for Hierarchical Linear Models. R. Journal of Statistical Software, 56, pp. 1-28. URL: http://www.jstatsoft.org/v56/i05/.
McGraw, K. O., & Wong, S. P. (1992). A common language effect size statistic. Psychological Bulletin, 111, 361-365.
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Wilcox, R. R., Granger, D. A., Clark, F. (2013). Modern robust statistical methods: Basics with illustrations using psychobiological data. Universal Journal of Psychology, 1, 21-31.
Yap, B. W., & Sim, C. H. (2011). Comparisons of various types of normality tests. Journal of Statistical Computation and Simulation, 81, 2141-2155. | {"url":"https://www.r-bloggers.com/2015/05/the-perfect-t-test/","timestamp":"2024-11-10T14:16:30Z","content_type":"text/html","content_length":"215608","record_id":"<urn:uuid:c1639acb-73a7-4022-a880-ed2bddf2a254>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00046.warc.gz"} |
Grading Examples - Connected Mathematics Project
Grading Examples from CMP Classrooms
The multidimensional assessment in CMP provides opportunities to collect broad and rich information about students’ knowledge. Teachers face the challenge of converting some of this information into
a grade to communicate a level of achievement to both students and parents. The following assessment items offer teachers an opportunity to assign grades: ACE exercises, CheckUps, Partner Quizzes,
Mathematical Reflections, Looking Back, Unit Tests, Projects, Notebooks, and Self-Assessments. The use of these assessments for grading and the value assigned to them vary from teacher to teacher.
While most teachers view the Problems as the time to learn and practice mathematical concepts and skills, some teachers will occasionally assign a grade to a Problem. Some teachers also choose to
grade class participation.
Two teachers’ grading schemes for their CMP mathematics classes follow. These are given as examples of possible grading schemes. Note that each of these teachers has made independent decisions about
how best to use the assessment tools in CMP for grading purposes.
Example 1: Ms. Jones’s Grading System
I try to take several things into account when grading students in mathematics class. I work to build a learning community where everyone feels free to voice his or her thoughts so that we can make
sense of the mathematics together. I try very hard to assess and grade only those things that we value in the classroom.
Because participating in discussions and activities is so important in helping the students make sense of the mathematics, this is one part of the students’ grades. They rate themselves at the end of
each week on how well they participated throughout the week. Below is a sample of the grading sheet they fill out. The participation grade counts as 15% of their total mathematics grade.
I try to take several things into account when grading students in mathematics class. I work to build a learning community where everyone feels free to voice his or her thoughts so that we can make
sense of the mathematics together. I try very hard to assess and grade only those things that we value in the classroom.
The curriculum is problem-centered. This means that the students will investigate mathematical ideas within the context of a realistic problem, as opposed to looking only at numbers. Students spend
much of each class period working with a partner or in a small group trying to make sense of a problem. We then summarize the investigation with a whole class discussion. The ACE exercises assigned
offer students an opportunity to practice those ideas alone and to think about them in more depth. Homework assignments are very important! They provide students the opportunity to assess their own
understanding. They then can bring their insight and/or questions with them to class the next day. We usually start each class period going over the exercises that caused difficulty or that students
just wanted to discuss. Keeping up with the homework (given about three or four times a week) helps students to stay on top of their learning. It also allows me to see what students are struggling
with and making sense of Homework assignment grades count as 20% of their total grade.
Partner Quizzes
All of the quizzes from CMP are done with a partner. Because a lot of what we do in class is done with others, I want to assess students “putting their heads together,” as well. Again, I try to grade
what I value, which is working together. Quiz grades count as 20% of their total grade.
Final Assessment
At the end of each unit an individual assessment is given. Sometimes it is a written test, sometimes a project, and sometimes a writing assignment. These serve as an opportunity for students to show
what they, as individuals, have learned from the whole unit. Test and project grades count as 30% of their total grade, as they are a culmination of the whole unit.
Grading Summary
• Participation . . . . . . . . . . 15%
• Journals . . . . . . . . . . . . . . 15%
• Homework . . . . . . . . . . . . 20%
• Partner Quizzes . . . . . . . . 20%
• Tests and Projects . . . . . . 30%
Example 2: Mr. Smith’s Grading Scheme | {"url":"https://connectedmath.msu.edu/classroom/assessing-and-grading/grading-examples.aspx","timestamp":"2024-11-13T07:54:35Z","content_type":"text/html","content_length":"55457","record_id":"<urn:uuid:01e25dd2-8dcf-406a-826a-a5d572c26a57>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00469.warc.gz"} |
Classification and Regression Tree
CART (Classification and Regression Tree) is a machine learning algorithm first proposed by Breiman et al. in 1984 and is widely used in predictive modeling. Although being a simple algorithm, CART
has an important status as it sets the foundation of many tree-based methods such as Bagging, XGBoost and Random Forest. CART has long been believed to have to the ability to deal with missing data
because of its surrogate splits function, which means that for a given observation, when one variable that is used in tree construction is missing, CART will use other variables that are similar to
the missing variable to help constructing the tree.
Since there are different types of missing data, such as MAR (missing at random), MNAR (missing not at random), and MCAR (missing completely at random), we aim to conduct simulations using various
models to examine CART’s ability to handle different types of missing data as well as the factors that influence its abilities.
In this poster presentation, I will discuss the simulation study I have conducted, in which I simulated data using 11 models with various levels of complexity, sample size and missing proportion.
Using MSEs as criteria, I examined how each model performs relative to one another, and what insights can be generated from the simulation results.
Authors: Valerie Huang, Dr. Han Du (Advisor) | {"url":"https://ecr.idre.ucla.edu/ecr_project/classification-and-regression-tree/","timestamp":"2024-11-09T08:02:58Z","content_type":"text/html","content_length":"24706","record_id":"<urn:uuid:7fb59a01-3e5e-4cb1-bc1e-456f27fcbb6a>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00315.warc.gz"} |