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Math 1070Q — Mathematics for Business and Economics (Fall 2017) | Math Courses
Math 1070Q — Mathematics for Business and Economics (Fall 2017)
Linear equations and inequalities, exponents and logarithms, matrices and determinants, linear programming. Applications.
Recommended preparation: MATH 1010 or the equivalent.
There are several sections for this course.
Discussion Sections 101D-115D: Lectures in person taught by Professor Benjamin Russo.
Discussion Sections 131D-142D: Lectures in person taught by Professor Michael Biro.
Discussion Sections 161D-175D: Lectures in person taught by Professor David McArdle.
Students are also required to attend a discussion section that meets once per week.
Required Textbook:
Applied Finite Mathematics by Edmond C. Tomastik and Janice L. Epstein (1^st Edition)
You can purchase the bundled version of Applied Finite Mathematics with a Webassign code from the UConn Bookstore or you can purchase the bundle at a discount from the publisher at http://
services.cengagebrain.com/course/site.html?id=2000499. Alternatively, you may also purchase a WebAssign access code directly from WebAssign and obtain a copy of the text elsewhere.
i>clicker Registration:
Clickers will be used in the lectures. You must register your i>clicker by visiting the link through the lecture section of your instructor in HuskyCT. This only needs to be done once for any
class, but if you do not register your clicker, you will not be able to receive credit for your responses!
It is your responsibility to register your clicker properly and use the correct frequency for your classroom. Failure to do so will cost you points for any questions that you miss, and there will
be no opportunity to make up these points. 5% of your grade is not much overall and the points are easy to earn, but this can make the difference between a B and an A-, for example.
Homework and WebAssign:
WebAssign: Online homework and MATH 1070 are assigned and completed using WebAssign. WebAssign must be accessed through HuskyCT. To get to WebAssign, go to the HuskyCT site for your discussion
section (not your lecture section), and click the link on the left navigation menu which says “WebAssign Homework”. This will take you directly to the WebAssign homework assigned for your class.You
will usually have 5 attempts to answer each non-multiple choice question. For multiple choice questions the number of attempts will vary based on the content. After each attempt, you will be told
whether your answer is correct or not. If you are not able to get the correct answer after a couple attempts, come talk to your professor or TA. We’re here to help! You might also find some help at
the Q-Center.
Homework: There will be homework assignments for each section of the text in WebAssign. These will be due on Wednesday nights at 11:59 pm on the Wednesday after the material was covered. Be sure to
start the assignments early so that you can ask questions.
In-Class Quizzes: Short, weekly in-class quizzes will be given in discussion sections on material covered during the previous week.
Late Work Policy: Homework extensions will not be granted, and there will be no makeup quizzes, except for in extenuating circumstances. However, at least one homework and quiz score will be
dropped at the end of the semester to accommodate any issues that may arise. It is important that you manage your time effectively to complete the homework before it is due and attend discussion to
take quizzes when they are given. Choosing not to do either is up to you, but there will not be any opportunities to recover missed points.
Calculator Policy:
A scientific calculator is allowed and recommended for quizzes and exams. You may not use a graphing calculator (TI-84, TI-Nspire, etc) on quizzes or exams, but these devices can be helpful while
doing the homework and practicing the concepts. However, keep in mind that you will not be able to use them on quizzes or exams, so make sure to learn how to use the functions on a scientific
calculator as well. Sharing of calculators will not be permitted, so make sure to have your own if you want one.
│ Online Homework │ WebAssign │ 10% │
│ In-Class Quizzes │ Discussion │ 10% │
│ Clicker Questions │ Lecture │ 5% │
│ Exam 1: (Oct. 4th and 5th) │ Lecture │ 25% │
│ Exam 2: (Nov. 8th and 9th) │ Lecture │ 25% │
│ Final Exam: (Date TBA) │ Common exam │ 25% │
Exam 1 is scheduled for Oct. 4th and 5th , and Exam 2 is scheduled for Nov. 8th and 9th. The exams will be in lecture and multiple choice in format. You must show work or explain your thought
process on each question to receive credit.
The final exam will be scheduled by the Registrar at some point during the semester.
Note: If you think a mistake has been made in grading or in recording any grades in WebAssign, please bring this to your instructor’s attention as soon as possible. All grades must be corrected and
updated before the final exam is administered; no changes will be made after that time. | {"url":"https://courses.math.uconn.edu/fall2017/math-1070/","timestamp":"2024-11-08T07:58:27Z","content_type":"text/html","content_length":"57862","record_id":"<urn:uuid:5951c5b4-64c3-4ec2-a274-d3ccbb4ad867>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00188.warc.gz"} |
An infinite sequence of CSTRs is equivalent to what?
An infinite sequence of CSTRs is equivalent to what?
In an ideal cstr there is perfect mixing. Also as soon as the reactant enters the reactor it mixes and gets converted to product but as soon as it enters, it gets converted and the concentration
suddenly drop. as keep on adding CSTR in series, we will have perfect 100 % in each reactor
This can be looked as a PFR where in we have no axial mixing but 100% radial mixing. Each radial section of a PFR can be looked as a CSTR as each CSTR will have full mixing taking place.
Hence infinite sequence of CSTR will be nothing but a PFR | {"url":"https://justaaa.com/other/151815-an-infinite-sequence-of-cstrs-is-equivalent-to","timestamp":"2024-11-07T01:16:00Z","content_type":"text/html","content_length":"38927","record_id":"<urn:uuid:bfdd3bcf-02b1-4483-a858-b93280787179>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00166.warc.gz"} |
csyswapr.f - Linux Manuals (3)
csyswapr.f (3) - Linux Manuals
csyswapr.f -
subroutine csyswapr (UPLO, N, A, LDA, I1, I2)
Function/Subroutine Documentation
subroutine csyswapr (characterUPLO, integerN, complex, dimension( lda, n )A, integerLDA, integerI1, integerI2)
CSYSWAPR applies an elementary permutation on the rows and the columns of
a symmetric matrix.
UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**T;
= 'L': Lower triangular, form is A = L*D*L**T.
N is INTEGER
The order of the matrix A. N >= 0.
A is COMPLEX array, dimension (LDA,N)
On entry, the NB diagonal matrix D and the multipliers
used to obtain the factor U or L as computed by CSYTRF.
On exit, if INFO = 0, the (symmetric) inverse of the original
matrix. If UPLO = 'U', the upper triangular part of the
inverse is formed and the part of A below the diagonal is not
referenced; if UPLO = 'L' the lower triangular part of the
inverse is formed and the part of A above the diagonal is
not referenced.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
I1 is INTEGER
Index of the first row to swap
I2 is INTEGER
Index of the second row to swap
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
November 2011
Definition at line 103 of file csyswapr.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://www.systutorials.com/docs/linux/man/3-csyswapr.f/","timestamp":"2024-11-10T20:57:17Z","content_type":"text/html","content_length":"8416","record_id":"<urn:uuid:6a4829b2-d62b-4067-855c-93146f5c2dee>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00544.warc.gz"} |
About Us
Our goal at Hours-In is to make converting hours to other units of time as quick and easy as possible.
Our small team of developers has created a calculator that instantly converts hours to seconds, minutes, days, and many other units. We are dedicated to providing you with accurate and efficient
tools to help you manage your time more effectively. | {"url":"https://sitemap.hillhouse4design.com/about-us/","timestamp":"2024-11-01T20:22:04Z","content_type":"text/html","content_length":"43247","record_id":"<urn:uuid:69a7b297-725e-4ec7-b472-d4fc1aab2a9c>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00141.warc.gz"} |
C++ Fast Track for Games Programming Part 9: Colours
In this article we will return to the numbers that represent colours, as demonstrated in the second instalment of this series. This can only be done through the wonderful world of bit magic, which is
by the way a very nice place to be, so we will explore it thoroughly.
Previous Part: AddressesNext Part: Arrays
For this tutorial, we’ll be using the same template from the 2nd tutorial:
As an introduction, try the following Game::Tick function in a fresh template:
27 void Game::Tick( float deltaTime )
28 {
29 screen->Clear( 100 );
30 }
What you get is a somewhat dark-bluish backdrop. Change 100 to 255, and it will be brighter blue. Increase it to 256 and it will be black. Well, not really black. Actually it’s very dark green, as we
will see in a minute.
Ingredients of an Integer
Let me introduce you to some new C things. Try the following Game::Tick function:
27 void Game::Tick( float deltaTime )
28 {
29 union
30 {
31 int i;
32 unsigned char b;
33 };
34 i = 255;
35 i++;
36 }
Set a breakpoint at on line 35 (the i++ line) and start the application. Use the debugger to view the value of variable i and b. At the breakpoint position, i and b both contain 255, but when you
proceed one line (F10), i is 256 as expected, but b is now 0.
To view the contents of a variable, make sure turn on the Locals window in the Visual Studio IDE. While running the program in Debug mode, select Debug > Windows > Locals from the main menu (or use
the shortcut key Ctrl+Alt+V,L).
A word about this weird ‘twinning’ of variables: i and b are in a union. You can use a union so that two (or more) variables that share the same memory location. The result is that changing one of
them will change the other as well.
In this case, i is an integer (int), but b is a char. Both can contain numbers, but there is a difference: an int is 32-bit, and a char is 8-bit. Since a bit can contain 0 or 1, it can contain 2
unique values; 8-bits can therefore contain \(2^8 = 256\) unique values (\(0 \cdots 255\)). After that, it’s back to zero… If we look at the separate bits (binary numbers), we get this situation:
Binary Decimal
00000000 00000000 00000000 11111111 = 255
00000000 00000000 00000001 00000000 = 256
What does this have to do with colors? Hang in there, we’ll get to that in a minute!
There’s another thing about those binary numbers. Have a look at the following table:
Binary Decimal
00001 = 1
00010 = 2
00100 = 4
01000 = 8
10000 = 16
So, if we take a number, and multiply it by 2, we effectively shift the bit to the left. Or, better said: shifting the bit to the left, multiplies the number by 2; whereas shifting a bit to the right
divides it by 2 (that’s the powers of 2!).
Now let’s return to colours. To store a colour in a 32-bit integer, we in fact store four values: red, green, blue and alpha. Each of them needs 8 bits. Blue goes in the lowest (rightmost) 8 bits,
and that is why any value between 0 and 255 will get you a shade of blue. Green goes in the next 8 bits. To get there, we need to shift 8 bits to the left, which we do by multiplying by 256. So, to
get shades of green, we take a number between 0 and 255, and multiply it by 256. And to get to red, we multiply by 256 twice. So try each of the individual lines below and see the result:
36 screen->Clear( 100 ); // yields dark blue
37 screen->Clear( 100 * 256 ); // yields dark green
38 screen->Clear( 100 * 256 * 256 ); // yields dark red
And thus, the brightest red that you can get is 255 * 256 * 256. You can also mix red, green and blue, by adding them together:
36 screen->Clear( (200 * 256 * 256) + (200 * 256) ); // yellow
So, to blend some colours, you will be doing plenty of multiplications. There is a slightly easier way: using bit-shifting.
Try this:
37 int i = 1;
38 i = i << 1;
Using the debugger, you can see that << 1 multiplies a value by 2. Likewise, << 2 will multiply it by 4. For colours, we can use << 8 and << 16:
36 screen->Clear( (200 << 16) + (200 << 8) ); // yellow
Now it is a bit more clear what’s happening: We are taking a value of 200 here for red, and we shift it in the right position (which is 16 bits to the left). Likewise, we put 200 for green in the
right position by shifting it 8 bits to the left. And, consider this to be a little teaser, you can also store some stuff in the invisible alpha channel, by shifting it to the left by 24 bits…
Now that we can get red, green and blue exactly where we want them to be, we can start answering the question how to access them
Suppose you have a yellow pixel. It’s stored in a 32-bit variable, which looks like this, in binary:
So: Garbage in the leftmost eight bits (bits 31-24, alpha), then 255 for red in bits 23-16, then 255 for green in bits 15-8, and finally zeroes in bits 7..0. The garbage might not actually be there,
it could be zeroes, or ones, but that doesn’t matter. Now suppose we want to know what green is for this colour. Of course you can see it right away, but we need to do this in a program, so let’s see
how we can make C++ extract that colour.
The answer is: using bitmasking. We will be using the &-operator (pronounced as And) in this case. In code, it looks like this. Note that I used a different value for green (yielding a bright orange)
to illustrate the concept.
32 int colour = (255 << 16) + (237 << 8); // orange
33 int mask = 255 << 8; // mask for green
34 int green = colour & mask;
35 screen->Clear( green );
In binary, the following happens:
11010111 11111111 11101101 00000000 Orange (garbage in alpha).
00000000 00000000 11111111 00000000 Mask for green.
----------------------------------- &
00000000 00000000 11101101 00000000 The result of masking just the green bits.
What & does is this: Every individual bit that was 1 in the first value, and also 1 in the second value, will be 1 in the result. All other bits will be 0 in the result. The bottom line is that we
extracted the value for green. Well, almost: it’s still shifted to the left by 8 bits, so to get the correct value for green, we need to move it back:
34 int green = (colour & mask) >> 8;
Red, blue and alpha values can be extracted in the same manner.
There’s a lot more to explore when it comes to bit magic, and trust me, you will learn to love it. But now, let’s make things practical, in the assignment for this week.
Here are your tasks for today:
1. Load an image, display it, and fade it (slowly) to black.
Note: there are two ways to do this. The easy way is using
floating point operations. The slightly harder approach uses integers only. Try
both, and measure the speed difference.
2. Draw a 200x200 checkerboard pattern with black and white pixels. Next to it, draw a 200x200 pixel solid grey bar. Adjust the brightness of the grey colour so that it matches the brightness of the
checkerboard pattern. Is the result what you expected?
3. EXTRA / HARD: draw the colours of the rainbow in vertical lines:
This will require some research; what you are looking for is a conversion from wavelength to red/green/blue.
Once you have completed these, you may continue with the next part.
Previous Part: AddressesNext Part: Arrays
4 thoughts on “C++ Fast Track for Games Programming Part 9: Colours”
1. At the start it mentions that if you change the clear value to 254 that the screen will be black (or actually, very dark green). I think this shoudl be 256 since that turns black and 254 is still
bright blue.
□ You’re right! I’ve updated the article.
2. In the beginning of this section you ask the user to increase the number in screen->Clear(255) to 254, but it appears you meant to say 256 because that illustrates your point better and matches
up with the dark green described.
Love the series and tutorial. Thank you.
□ Thanks Michael, I’ve updated the article.
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://www.3dgep.com/cpp-fast-track-9-colours/","timestamp":"2024-11-04T06:03:49Z","content_type":"text/html","content_length":"107973","record_id":"<urn:uuid:291128d0-e71b-48ff-aeae-7c51a093bbf2>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00239.warc.gz"} |
Kempe equivalent list colorings revisited
A Kempe chain on colors (Formula presented.) and (Formula presented.) is a component of the subgraph induced by colors (Formula presented.) and (Formula presented.). A Kempe change is the operation
of interchanging the colors of some Kempe chains. For a list-assignment (Formula presented.) and an (Formula presented.) -coloring (Formula presented.), a Kempe change is (Formula presented.) -valid
for (Formula presented.) if performing the Kempe change yields another (Formula presented.) -coloring. Two (Formula presented.) -colorings are (Formula presented.) -equivalent if we can form one from
the other by a sequence of (Formula presented.) -valid Kempe changes. A degree-assignment is a list-assignment (Formula presented.) such that (Formula presented.) for every (Formula presented.).
Cranston and Mahmoud asked: For which graphs (Formula presented.) and degree-assignment (Formula presented.) of (Formula presented.) is it true that all the (Formula presented.) -colorings of
(Formula presented.) are (Formula presented.) -equivalent? We prove that for every 4-connected graph (Formula presented.) which is not complete and every degree-assignment (Formula presented.) of
(Formula presented.), all (Formula presented.) -colorings of (Formula presented.) are (Formula presented.) -equivalent.
• graph coloring
• kempe change
• reconfiguration
ASJC Scopus subject areas
• Geometry and Topology
• Discrete Mathematics and Combinatorics
Dive into the research topics of 'Kempe equivalent list colorings revisited'. Together they form a unique fingerprint. | {"url":"https://nyuscholars.nyu.edu/en/publications/kempe-equivalent-list-colorings-revisited","timestamp":"2024-11-07T19:32:27Z","content_type":"text/html","content_length":"50374","record_id":"<urn:uuid:6eb2ba66-154e-4b79-b678-1576f03db053>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00715.warc.gz"} |
What are examples of not acceleration?
What are examples of not acceleration?
Leaving case C. A car with it’s cruise control set, traveling in a constant direction at a constant speed, is not accelerating.
What is not considered acceleration?
Acceleration has to do with changing how fast an object is moving. If an object is not changing its velocity, then the object is not accelerating.
When can you say that there is no acceleration?
Explanation: If there are no forces acting upon the object, then there is no acceleration. If there is no acceleration, then the object will move with a constant velocity.
What cases have zero acceleration?
Theoretically, when a particle moves in constant velocity, there is no change in velocity with time. Then its acceleration is called zero acceleration. Mathematically, since velocity is constant then
the first time derivative of velocity will be zero which indicates the acceleration of a moving object.
What is the implication to a body when its acceleration is zero?
When acceleration is zero (that is, a = dv/dt = 0), rate of change of velocity is zero. That is, acceleration is zero when the velocity of the object is constant. Motion graphs represent the
variations in distance, velocity and acceleration with time.
What is constant acceleration equal to?
If the velocity of the particle changes at a constant rate, then this rate is called the constant acceleration. For example, if the velocity of a particle moving in a straight line changes uniformly
(at a constant rate of change) from 2 m/s to 5 m/s over one second, then its constant acceleration is 3 m/s2.
What are some examples of acceleration in physics?
Acceleration describes any change in velocity (which refers to an object’s speed and direction of travel). Thus, speeding up, slowing down and turning are all examples of acceleration. A simple
example would be dropping a ball: as it falls its speed increases, which is a type of acceleration.
What are three examples of velocity?
Five different examples of velocity 1.Turning South in a car. 2.Accelerating to 50mph from 45mph. 3.Walking to the back of a bus while it is moving foward. 4.Running on a treadmill. 5.A bucket
falling off a building.
What is acceleration divided by time?
Acceleration has the dimensions of velocity (L/T) divided by time, i.e. L. T−2. The SI unit of acceleration is the metre per second squared (m s −2 ); or “metre per second per second”, as the
velocity in metres per second changes by the acceleration value, every second.
What is uniform acceleration?
Uniform Acceleration. Uniform acceleration occurs when the speed of an object changes at a constant rate. The acceleration is the same over time. | {"url":"https://short-fact.com/what-are-examples-of-not-acceleration/","timestamp":"2024-11-02T07:38:40Z","content_type":"text/html","content_length":"140227","record_id":"<urn:uuid:7e21cb00-1118-4f4e-a34f-e32144c35494>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00236.warc.gz"} |
Conjugate Gradient Minimization
Xplor-NIH home Documentation
Next: Syntax Up: Energy Minimization Previous: Energy Minimization
Conjugate Gradient Minimization
X-PLOR uses the conjugate gradient method by
Powell (1977)
. The minimization is started from the atom properties X,Y,Z, and the minimized coordinates are returned in X,Y,Z. Only the coordinates of free atoms (i.e., not fixed atoms) will be modified during
the minimization (cf. Section
). SHAKE constraints are possible (cf. Section
). Upon completion of the last energy calculation, symbols are declared that contain the computed energy terms. The name of the symbols is given by $energy-term (see Section
). The overall energy (Eq.
) is stored in the symbol $ENER; the rms gradient is stored in $GRAD. The value of the second energy function (Eq.
) is returned in the symbol $PERT.
Subsections Xplor-NIH 2024-09-13 | {"url":"https://nmr.cit.nih.gov/xplor-nih/doc/current/xplor/node201.html","timestamp":"2024-11-03T13:43:46Z","content_type":"text/html","content_length":"14793","record_id":"<urn:uuid:0245916c-39ba-4766-856e-0135a47da0cf>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00822.warc.gz"} |
I. General Information
1. Course Title:
Calculus I
2. Course Prefix & Number:
MATH 1477
3. Course Credits and Contact Hours:
Credits: 5
Lecture Hours: 5
Lab Hours: 0
Internship Hours: 0
4. Course Description:
Review of the concept and properties of a function. Emphasis on the graphing and behavior of a function. Limits are introduced and developed. The derivative of a function is defined and applied to
algebraic and trigonometric functions. Anti-differentiation and elementary differential equations. Definite integral as a limit of a sum and as related to anti-differentiation via the Fundamental
Theorem of Calculus. Applications to maximum, minimum and related rates. Differentiation and integration of exponential and logarithmic functions.
5. Placement Tests Required:
Accuplacer (specify test): College Mathematics Score: 86
6. Prerequisite Courses:
MATH 1477 - Calculus I
All Credit(s) from the following...
│Course Code │Course Title │Credits│
│MATH 1472 │Precalculus │5 cr. │
9. Co-requisite Courses:
MATH 1477 - Calculus I
There are no corequisites for this course. | {"url":"https://catalognavigator.clcmn.edu/Catalog/ViewCatalog.aspx?pageid=viewcatalog&topicgroupid=3085&entitytype=CID&entityid=159&loaduseredits=True","timestamp":"2024-11-05T06:11:47Z","content_type":"text/html","content_length":"56782","record_id":"<urn:uuid:3df71018-f352-4627-98f9-cc47fcc44539>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00657.warc.gz"} |
Modelling effective antiretroviral therapy that inhibits HIVproduction in the liver.
Hasifa Nampala1,∗, Livingstone S. Luboobi1, Joseph Y.T. Mugisha1, Celestino Obua3, MatyldaJab lo´ nska2, Matti Heili¨ 1 Department of Mathematics, Makerere University, Kampala, Uganda2 Department of
Mathematics and Physics, Lappeenranta University of Technology,Lappeenranta, Finland3 Department of Pharmacology and Therapeutics, Makerere University, Kampala, Uganda∗ E-mail: Corresponding [email
protected] CD4+ and hepatocytes cells, both found in the liver, support all stages that lead to HIV production.
Among people infected with HIV, liver disease has become the second leading cause of morbidity andmortality.
Considering HIV infection and replication in hepatocytes as well as CD4+ cells, a mathematical model was developed and analysed to investigate the ability of different combinational therapy to
inhibitviral production in liver cells. Therapy efficacy in form of a dose-response function was incorporated.
Analysis of the model suggested that it is possible to have the effective reproductive number Re belowunity provided the therapy efficacy is more than 90%. Within some range of parameter values, Re
canalso be reduced below unity at an efficacy of 50%.
Simulation results showed that combinational therapy of DDI, 3TC, ATV and NFV as the most effective while AZT, d4T, ATV and NFV as the least effective, in terms of inhibiting viral production.
The findings showed that this model can possibly be used to recognize which of the current treatmentprotocols perform best in controlling HIV replication in the liver.
Infectious diseases are the second leading cause of death among humans worldwide, and the number onecause of death in developing countries [?]. Among all other previous pandemic like cholera and
influenza,the Human Immunodeficiency Virus (HIV) has for three decades socially and economically affected theworld and has claimed over 25 million lives [?].
Among people infected with HIV, liver disease has become the second most cause of morbidity and mortality [?]. Various research have revealed that liver disease can occur solely due to HIV infection
[?, ?, ?, ?].
During HIV infection, the virus uses envelope glycoprotein 120 (gp120) to access entry into the host cell by binding on the CD4 receptor or a coreceptor on the host cell. The most coreceptors for HIV
areC-X-C chemokine receptor type 4 (CXCR4) and C-X-C chemokine receptor type 5 (CXCR5) [?, ?, ?, ?].
Recent studies have revealed that HIV infection can occur in other cells other than CD4+ cells providedthose cells posses either of the coreceptors [?].
Human hepatocytes possess CXCR4 making them susceptible to HIV invasion and hence causing hepatocytes apoptosis by viral signaling through CXCR4 [?]. A study by Kong et al [?] found thatalthough
there has been a number of contradictions regarding HIV replication in hepatocytes [?, ?], thecells support the first and last stages of HIV production. Virions produced by hepatocytes can
infectother cells. However, Kong et al [?], revealed that replication in hepatocytes is low compared to viral replication in CD4+ cells. In addition to hepatocytes, HIV productively infects other
hepatic cells andmacrophages, especially, kupffer cells [?, ?].
Since the introduction of antiretroviral therapy (ART) scientists have aimed at getting the drug that can limit HIV replication and hence reduce the viral load in the body. To date, no drug with
100%efficacy and ability to eradicate the virus from within HIV infected bodies has been found.
Mathematical models have been used to study within-host dynamics of HIV. Gumel et al [?] used a heaviside function to investigate the effects of intermittent IL-2 plus ART on the dynamics of HIV,
afterusing therapy for 200 days. The findings showed that in spite of the combined effect of the theoreticalmaximum, ant-HIV cytotoxic T-lymphocytes (CTLs) action and 100% efficacies of therapy
coupled withIL-2 therapy, the virus continues to persist.
Rong et al [?] included a combination of reverse transcriptase inhibitors (RTIs) and protease inhibitors (PIs) in a mathematical model of HIV infection with two stains of HIV. They assessed the
progressionrate of exposed CD4+ cells (eclipse phase) back to uninfected stage and viral production on the evolutionof drug-resistant virus. They further investigated the evolution of drug resistant
strains in the presenceof antiretroviral treatment and the range of drug efficacies under which drug-resistant strain will be ableto invade and out-compete the wild-type strain. Results showed that
when the drug efficacy is not highenough to exert sufficient selective pressure (RTI efficacy of 0.5 and PI efficacy of 0.3), the resistant strainwill be unable to invade the established sensitive
Arnaout et al [?] used a basic within host model as by Perelson and Nelson [?] to analyse HIV dynamics in vivo. He incorporated treatment as drug effectiveness parameter between 0 and 1 to assessthe
dynamics of infection below and above the threshold efficacy. Results from model analysis showedthat if effectiveness is below a certain threshold (1%), viral load may bounce back after a
transientreduction. They further deduced that if effectiveness is below but sufficiently near the threshold, viralload may still be reduced to quite low level.
Liver disease in HIV infected people who are not co-infected with viral hepatitis, has been linked to use of ART [?] because of the toxic nature of all classes of ART. However, in recent studies it
has beenfound that HIV infection and replication in the liver cells can cause liver disease in HIV mono-infectedprior to initiation of ART [?, ?, ?]. Despite their unwanted effects, antiretroviral
drugs have improvedthe long term outlook of HIV infected patients. A number of wit-in host mathematical models of HIVdynamics have focused on viral progression in CD4+ cells [?, ?, ?]. Since there is
evidence that HIVinfect other cells, we therefore study the progression of HIV in hepatocytes when antiretroviral therapyis administered.
This study therefore intends to use a mathematical model coupled with numerical simulations to study the ability of individual drugs as well as recommended therapy combinations, to inhibit viral
replicationin liver cells. Unlike most studies, [?,?,?,?], this study considers drug efficacy as a dose-response functionas recommended by Perelson and Deeks [?].
Model development Despite HIV's high affinity for CD4+ cells as compared to hepatocytes [?], and based on the existenceof CD4+ cells in the liver [?], the study assumes that when HIV infects the
liver, the virus eitherinfects CD4+ cells or hepatocytes. Both CD4+ cells [?] and hepatocytes [?, ?] support all stages of viralproduction. Like many researchers who have modelled HIV dynamics in
vivo [?, ?, ?, ?, ?], the studyconsiders CTLs killing of infected cells.
In the model formulation, we define the eight variables as follows: uninfected CD4+ (Tc), exposed CD4+ cells (Ec), infectious CD4+ cells (Ic), uninfected hepatocytes (Th), latently infected and not
acti-vated hepatocytes (If ) [?], productively infected hepatocytes (Ia), HIV-specific cytotoxic T lymphocytes(L) and viral load (V ).
Model parameters are as follows: CD4+ cells and hepatocytes are produced from within the body at rates λ1 and λ2, and die naturally at rates b1 and b3 respectively. At infection, the virus infects
targethepatocytes with probability q at rate β2 and target CD4+ with probability 1 − q at rate β1. WhenHIV enters a resting CD4+ cell, the RNA may not be completely reverse transcribed into DNA and
theun-integrated virus cell may decay before reverse transcription [?]. This results in a proportion of exposedcells reverting to the uninfected state at a rate α. If reverse transcription takes
place, the cell becomesinfectious at a rate π. This implies that if reverse transcription takes place in a period 1/α, where 1/α <1/π then the exposed cell will revert to uninfected state, otherwise
it will proceed to the infectious state.
Infected CD4+ die at rate b2 where b2 > b1 and are cleared by HIV-specific CTLs at a rate k1.
When a hepatocyte is exposed to the virus, there is a probability p that it becomes productively infected (viral replication will take place after successful reverse transcription) and probability (1
− p)that the cell will be latently infected, such that there is no viral production until cell activation (the extentof stimulation of cellular processes initiated as a response to external stimuli
reaching the cell). Latentlyinfected hepatocytes are activated to become productively infected at rate µ. Decay rates for productivehepatocytes and latently infected hepatocytes are b4 and b3,
respectively, where b4 > b3 [?]. Productivelyinfected hepatocytes are killed by HIV-specific CTLs at rate k1 and, until activated, latently infectedhepatocytes will not trigger the action of CTLs.
This study assumes that latently infected hepatocyteswill either get activated to become infectious or die. There is no possibility of them becoming uninfectedagain [?].
With or without any pathogen in the body CTLs proliferate naturally at rate x and in the presence of HIV infection, they proliferate at rate k2 proportional to the number of infectious cells; they
are clearedat rate b5. HIV is produced by infectious CD4+ and productively infected hepatocytes at average rates s1and s2 per cell, respectively. In addition to CD4+ and hepatocytes, HIV productively
infects other cellsand macrophages, like kupffer cells in the liver, [?, ?, ?]. These cells produce virions at rate m. Virionsdie naturally at rate b6.
There are three types of antiretroviral drugs currently used as therapy for HIV. Non-nucleoside re- verse transcriptase inhibitors (NNRTI), nucleoside reverse transcriptase inhibitors (NRTI) and
proteaseinhibitors (PI). NNRTIs prevent the enzyme (reverse transcriptase) from converting RNA of HIV toDNA, thus the HIV will not multiply. NRTI latches onto the new strand of DNA that reverse
transcrip-tase is trying to build and PIs prevent final assembly and completion of new HIV viruses within the cell,resulting in the infected cells producing noninfectious virus. However, this study
considers only infectiousvirus.
Mathematical models have been used to try and address issues in HIV infection during antiretroviral therapy [?, ?, ?, ?]. Therapy efficacy has been modeled as a number between 0 and 1. There are
howevera number of underlying dynamics especially the pharmacokinetics of the medication that influence thedrug efficacy. In recent research by Perelson and Deeks [?], it was asserted that
"non-nucleoside reversetranscriptase inhibitors and protease inhibitors exhibit cooperative dose-response curves; a finding thathas implications for the treatment of HIV as well as other viral
infections. The notion that a drug's doseand effect are related is a basic tenet of pharmacology and is generally summarized by an experimentallyderived dose-response curve. Determining the dose that
gives 50% of the maximum response is one way toquantify the potency of a drug". This assertion followed an earlier research by Shen et al [?] who used aHill equation to describe the effectiveness of
HIV medication. Perelson and Deeks [?] hence recommendedthat it would be of great contribution if the efficacy of ART would be modeled as a dose-response function.
[?] validated the importance of the dose-response curves in terms of predicting which medication works perfectly in inhibiting viral replication. They found that PIs have higher gradientsimplying
better efficacies and they concluded that, this would explain why some times PIs alone couldbe effective in treating HIV.
However, in a typical dose-response relationship, the response to a dose depends on a number of factors including the dose administered, the frequency of dosing as well as the pharmacokinetics of
theparticular drug. In this research, we have assumed that dose and rate of dosing will lead to a "steady state" dose response, that is, after administering a particular dose repeatedly, the drug
concentrationwill reach a steady state. We have therefore assumed a steady effective therapeutic exposure to the drugbecause with infections such as HIV, "Steady-State" pharmacokinetics as opposed to
initial or loadingdoses is more reliable for the effects of the treatment.
The study further assumes sufficient exposure to the drug (no under dosing or poor exposure due to use of poor or substandard drugs) thus ruling out the possibility of partial suppression which leads
toselection pressure. The model also considers early stages of treatment where the infection is presumedto be sensitive to the drugs thus assuming drug resistance as being negligible.
Taking φ1 as the therapeutic response of reverse transcriptase inhibitors and φ2 as the therapeutic response of protease inhibitors, where 0 ≤ φ1, φ2 ≤ 1, the therapeutic response function is defined
as aHill equation (1) to describe the effectiveness of the drug [?] where φ = φ1orφ2, d is variable drug dose concentration, IC50 is variable drug concentration that leads to50% of the maximal viral
inhibition and m is variable gradient of the dose-response curve correspondingto individual drugs. The response in this case is the drug efficacy or ability to inhibit viral replication [?].
The gradients of the dose-response curves of HIV drugs are given by Shen [?].
This study assumes that reverse transcription in CD4+ cells does not occur immediately at infection [?]. So reverse transcriptase inhibitors (RTIs) reduce the rate of transfer of cells from exposed
to infectiousclass (π). In hepatocytes, it is the infection rate that gets reduced by RTIs, because it is assumed thatat infection, reverse transcription takes place and then the cell will become
latent or productive. It is forthis reason that a cell cannot become uninfected again like the case of CD4+ cells.
In hepatocytes however, it is assumed that in the latent class, reverse transcription has already taken place though the final stage has not yet been attained. Hence, if protease inhibitors are 100%
effective,no latently infected cell will become productive, otherwise, some will become infectious depending on theefficacy of the drug (PIs). The study therefore assumes that PIs will reduce the
rate of activation fromlatent to infectious (µ). It is further assumed that the effect of medication is translated generally intominimal viral load. Thus viral production from macrophages is also
inhibited by both RTIs and PIs. Thecombined response of PIs and RTIs in macrophages is therefore (1 − φ1)(1 − φ2) [?].
From the assumptions and description above we have the following system of ordinary differential equations (2)-(9).
λ1 − (1 − q)β1TcV − b1Tc + αEc (1 − q)β1TcV − b1E − αEc − (1 − φ1)πEc (1 − φ1)πEc − b1Ic − b2Ic − k1IcL λ2 − (1 − φ1)qβ2ThV − b3Th (1 − φ1)(1 − p)qβ2ThV − b3If − (1 − φ2)µIf (1 − φ1)pqβ2ThV − b4Ia −
k1IaL + (1 − φ2)µIf x + k2(Ic + Ia)L − b5L (1 − φ2)s1Ic + (1 − φ2)s2Ia + (1 − φ1)(1 − φ2)m − b6V The system of equations (2)-(9) settles to a a disease-free equilibrium point A0(Tc, E0, Ic, Th, Ia,
If , L, V ) =(λ1/b1, 0, 0, λ2/b3, 0, 0, x/b5, 0). The effective reproduction number for the system (2)-(9), calculated usingthe next generation method as in [?] is (1 − φ1)(1 − φ2)b5s1πβ1(1 − q) b1b6
(b1 + α + (1 − φ1)π)(k1x + b5(b1 + b2)) (1 − φ1)(1 − φ2)b5s1β1(1 − q) (b1 + α + (1 − φ1)π) b1b6(k1x + b5(b1 + b2)) b5(1 − φ1)(1 − φ2)s2qβ2λ2[p(b3 + (1 − φ2)µ) + (1 − p)(1 − φ2)µ] b3b6(b4b5 + k1x)(b3
+ (1 − φ2)µ) b5(1 − φ1)(1 − φ2)2s2(1 − p)qµβ2λ2 b5(1 − φ1)(1 − φ2)s2pqβ2λ2 b3b6(b3 + (1 − φ2)µ)(b4b5 + k1x) Rf and Ra is the number of secondary infections from latently and productively infected
hepatocytes respectively. Rc1 and Rc2 is the number of secondary infections produced by cells in the eclipse phase(exposed) and virus producing CD4+ cells respectively. Rc and Rh is the number of
secondary infectionsproduced by one virus in CD4+ and hepatocyte, respectively. R0 is the total number of secondaryinfections in the liver. The total number of secondary infections is directly
proportional to the clearancerate of CTLs and inversely proportional to the clearance rate of virions. Secondary infections in eithertype of cells largely depend on the drug efficacy. It can be seen
that if the drug is 100% effective, thenthere is no secondary infections in either cell type.
Generally, the number of secondary infections (R0) is dependent on antigen-independent CTLs prolif- eration rate (x) and independent of antigen-dependant proliferation rate (k2). This indicates that
if theCTLs are boosted prior to infection, then the body can handle infection better than when they proliferatein the presence of infection.
We then study the behaviour of the effective reproduction number for specific model parameter values as presented in Table 1. When every exposed hepatocyte becomes latently infected for some time
beforeit is activated to produce virions, we have p = 0. Then with the activation rate µ = 0.019 and 50%effectiveness of both protease inhibitors and reverse transcriptase inhibitors and all other
parameters asshown in Table 1, numerical simulations show that the effective reproductive number of hepatocytes canbe reduced below unity, as shown in Figure 1. However, if the probability p = 1,
that is, every exposedcell becomes infectious at infection, then the number of secondary infections are greater than unity. Itcan also be seen that to keep Rh below unity, p should be less than
0.4093. Thus, it can be consideredimportant to increase drug efficacy as well as reducing activation rate of latently infected hepatocytesin order to reduce the basic reproductive number below unity.
It is also shown in Figure 1 that withonly 30% of hepatocytes becoming productive at infection, the threshold activation rate below which thehepatocytes effective reproductive number is below unity
is 0.0096.
Analysing the combined dependence of the effective reproductive number on p and µ at the same time, it can be seen from the right panel of Figure 2 that, there are multiple parameter value
combinations forp and µ at which the effective reproductive number is unity. That is, with the probability of a hepatocytebecoming productive at infection less than 0.6, corresponding activation rate
µ lower than 0.019, therapy efficacy of 50% and all other parameters as stated in Table 1, HIV infection in hepatocytes can possiblybe managed.
Considering CD4+ cells, if the rate of transfer of exposed to infectious stage is π = 0.23 and the therapy is 50% effective, then the effective reproductive number is below unity when the probability
pis above 0.9266, as presented in Figure 3. In other words it could be possible to manage HIV infectionin CD4+ cells given the parameter values given in Table 1 and if almost all HIV infect
hepatocytescells. However, it has been stated that HIV has higher affinity for CD4+ cells than hepatocytes [?]. Wetherefore assume a probability of 0.8 that HIV infects a CD4+ as shown in the right
panel of Figure3. The basic reproductive number is seen to be below unity given the rate of transfer from exposed toinfectious CD4+ cells is below 0.019.
The range of values of q and π that will give the effective reproductive number below unity are as shown in Figure 4. This suggests a possibility to manage HIV in CD4+ cells given the range of
parametervalues shown in Figure 4 and Table 1 with therapy efficacy of 50%.
In all the previous simulations, the therapy efficacy has been fixed at 50% for both drug classes.
However, medically it is not the case that all classes of ART are 50% effective. We therefore investigatedrug efficacy that leads the effective reproductive number (R0) of liver cells less than
unity. Figure 5shows that, given µ = 0.0096, p = 0.4093, q = 0.9266 and π = 0.019, it is possible to have R0 < 1provided the therapy efficacies are greater than 90%.
In Figure 6 we study the dependence of the effective reproduction number R0 on the infection rates β1 and β2. Apparently, the infection might not proceed to endemic state if drug efficacies are fixed
at50% given the infection rates β1 < 0.0015 and β2 < 0.00015 for CD4+ cells are hepatocytes respectively.
This raises a big challenge given that research has revealed infection rates as high as 0.005 [?].
Numerical simulations In this section we present numerical simulations of the model equations (2)-(9) proposed in this work.
The dynamics of infection are first considered when there is no therapy. This is shown in Figure 7. Theviral load (V ) grows steeply in the first days, leading to increased number of latently
infected CD4+ (Ec),productively infected CD4+ (Ic), latently infected hepatocytes (If ) and productively infected hepatocytes(Ia). This results in a clear drop in the numbers of uninfected cells,
both CD4+ (Tc) and hepatocytes(Th). That significant decrease takes place within the first day of infection and it can be seen that mostof those previously uninfected cells start to contribute to all
the classes of infected cells. Following theprogression of the infection, there is a significant response of HIV-specific CTLs to infection at a ratek2 as shown in equation (8). That helps the liver
to reduce the viral population but cannot eliminateit completely. As we have seen in the R0 analysis in Figure 5, the number of secondary infections willalways be greater than unity when the drug
efficacies are φ1 = φ2 = 0. The graphs show that the infectionis apparently destructive to the liver without any medical intervention. Numerical simulations show adiscrepancy between equilibrium
population ratios of hepatocytes and CD4+ cells prior to therapy andtheir physiological concentrations in the liver. This could be due to high concentration of hepatocytes ascompared to CD4+ cells
yet HIV production is highest the latter.
Analysing the model with therapeutic effect of the drugs (equations (2)-(9)), the study considers medication as listed in Tables 2-4. The sampled drugs under study are representatives of all classes
ofART currently used as medication for HIV, namely, NRTIs, NNRTIs and PIs. Tables 2-4 present allthe parameters which were used in calculating of drug efficacy as shown in equation (1). All doses
areexpressed as concentrations in moles per liter.
Antiretroviral treatments are always used in combinations of three or four drugs from specific classes.
However, we first simulate the infection dynamics when each drug is administered individually with itsusual dose. The goal is to verify how every individual drug is able to inhibit viral production
and hencereduce the viral load. Figure 8 depicts the dynamics. The first immediate observation is that the infection level reduces when either drug is used. However, the severity of the influence
varies significantly fromone medicine to another. Apparently, a drug that reduces the number of infected CD4+ most effectivelydoes not perform equally well in heapatocytes. The most distinct aspects
of infection dynamics are thetime delay before the infection peaks and the maximum level reached by the infection.
In particular, the study considers the detailed Figures 9, 10 and 11 for Ic, Ia and V respectively for individual drugs. If drug efficacy is measured by reduction in the number of productively
infectedhepatocytes or the reduction in viral population, then atazanavir (ATV) is clearly the best performingdrug. However, it actually reduces the number of productively infected CD4+ the least of
all. On theother hand, stavudine (d4T) provides the least improvement in productive hepatocytes and viral load.
It is only the CD4+ that benefit most from the use of d4T.
Of all medications considered in this study, ATV is clearly the treatment which is capable of delaying and dampening the peak of infection. Considering 2007 World Health Organisation
recommendationsof using ART, that is, two NRTI and one NNRTI drug (2NRTI+1NNRTI) or two NRTI and two PIdrugs (2NRTI+2PI). The study presents simulation results when the aforementioned combinations
areconsidered. Out of the drugs used in the study, as shown in Tables 2-4, we obtain six different pairs ofNRTI drugs combined with a single NNRTI drug or with the two PI drugs. Figure 12 shows the
infectiondynamics for the 2NRTI+1NNRTI combinations.
Combinations have higher efficacy than each single drug on its own. The number of uninfected cells remain at higher levels when combinations are used as compared to single drug. Consequently, the
numberof infected cells and viral populations are reduced more with combinations than with single drugs. Inmost of the cell types it is visible that the best combination in all aspects is
DDI+3TC+EFV, whereasthe worst one is AZT+d4T+EFV.
We now consider the drug combinations of 2NRTI+2PI. As presented in Figure 13, these options are even more efficient in infection reduction. This is consistent with the previous simulations of
individualdrugs that revealed how ATV is the best among the considered drugs, in viral reduction. When combinedwith another PI drug and two more drugs from NRTI class, ATV proves strongest of all
treatmentsstudied in this paper. Simulation results show that DDI+3TC+ATV+NFV is the best combination andAZT+d4T+ATV+NFV is the worst.
We finally analyse the dynamics of the variables when the best medication in the previous simulations is used (ATV). This is considered in a situation while more (p = 0.8) and less (p = 0.3)
hepatocytes becomeproductive at infection as shown in Figure 14. When more hepatocytes become productive at infection ascompared to latency (p = 0.8), viral load will peak in the first 10 days.
Consequently, uninfected CD4+cells and hepatocytes will become fewer due to viral production from the many productive hepatocytes.
However, in the long run the infected CD4+ cells will become more when fewer hepatocytes are productiveat infection, as compared to latency (p = 0.3). This could be explained by earlier findings in
this workthat the use of ATV leads to most infectious CD4+ cells. On the other hand, even when we consider thebest combination in the previous simulations and use it, Figure 15 shows that there is
not much differencewhen less infected cells become latently infected during combination therapy. The significant differenceis only in Ia and If . This shows that during HIV infection, whether a
hepatocyte becomes productive atinfection or latent, it does not create much difference in infectious CD4+ cell. This could be so becauseit is the CD4+ cells that produce the biggest number of
virions in the liver. This could be consistentwith one dilemma perturbing researchers in the field of HIV medication, 'the virus's ability to harbourin cells for a long time'.
The aim of this study was to understand HIV infection dynamics in the liver with administration ofantiretroviral therapy. The work was based on a number of biological facts as well as feasible
assump-tions. HIV dynamics in the liver were analysed based on three main cell types; CD4+, hepatocytes and HIV-specific T-lymphocytes. Nevertheless, an aggregated effect of infection in macrophages
was also con-sidered. In CD4+ cells, reverse transcription was considered not to occur immediately at infection. Thesame consideration was taken for hepatocytes, though unlike CD4+ cells that can
return to uninfectedstate , hepatocytes could only die or proceed to infectious state after being exposed to the virus.
The model proposed in this study was formulated as a system of ordinary differential equations with eight variables. Drug efficacy was considered as a dose-response function with parameters obtained
exper-imentally in pharmaceutical studies [?, ?]. HIV therapy included the three classes of enzyme inhibitors,namely, NRTIs, NNRTIs and PIs.
Analysis of the model's basic reproduction number revealed that the key parameters to control the infection are: p – the probability that at infection, a hepatocyte becomes productively infected, q –
theprobability that HIV infects hepatocyte and not CD4+, µ – the rate at which latently infected hepatocytesbecome productive, and π – the rate at which exposed CD4+ become infectious. In particular,
consideringall the other parameters as shown in Table 1 and fixing each drug efficacy at 50% it was revealed thatin order to possibly keep the number of secondary infections below unity, the crucial
parameters need tosatisfy the conditions, p < 0.4093, q > 0.9266, µ < 0.0096 and π < 0.019. However, most of the knownvalues of those parameters are significantly outside the these limits [?, ?]. The
effective reproductionnumber was found to be below unity only when the infection rates for CD4+ cells and hepatocytes wererespectively β1 < 0.0015 and β2 < 0.00015, whereas these values are sometimes
suggested as high as0.005 [?]. Finally, with all the parameters fixed at their theoretical values it was seen that strict controlof the infection may be possible when the drug efficacy of either
therapy exceeds 90%.
The model was used to simulate infection progression scenarios in the cases of no medical treatment, single drug administration and full ART combinations. Results showed that all enzyme inhibitors in
HIVinfection significantly reduced viral load. The ability to inhibit was significantly higher when combinationtherapy was used as opposed to single drug.
Simulations results further suggested that atazanavir (ATV) was possibly the best single drug and Stavudine (4dT) the worst in terms of viral load reduction. This is consistent with the claims that
proteaseinhibitors are more effective than reverse transcriptase inhibitors in terms of viral load reduction. Amongthe considered full ART combinations, with effectiveness measured in terms of
reducing the viral loadas well as the number of infectious hepatocytes, then DDI+3TC+ATV+NFV proved to be possibly thebest option and AZT+d4T+ATV+NFV the worst. This was however not consistent with
with reducingthe number of infectious CD4+ cells. Simulation results also suggested that it is not of any advantagehaving more hepatocytes becoming latent than infectious at the point of infection
because in the longrun the implications is not significantly different.
We therefore conclude that mathematical modelling creates hints that can be used as basis for under- standing the dynamics of HIV in the liver during antiretroviral therapy. The model used in this
study,under the assumptions mentioned, suggested a possible tool for proposing the best and worst antretroviralcombinations for reduction of viral load in the liver. We therefore recommend that this
approach couldbe improved and used to optimize antiretroviral combinations.
The authors would like to acknowledge the Sida/SAREC bilateral research cooperation programme ofMakerere University for funding this research. We thank the Center for International Mobility (CIMO)
Finland for funding research visits at Lappeenranta University of Technology.
Figure 1. Basic reproduction number Rh of hepatocytes. The number is calculated withvarying parameter p and fixed µ = 0.006 (left panel) and varied µ with fixed p = 0.3 (right panel), withall the
other parameters as given in Table 1 and the drug efficacies assumed as φ1 = 0.5 and φ2 = 0.5.
Figure 2. Basic reproduction number Rh of hepatocytes (left panel) and its correspondinglevel lines (right panel). The number is calculated with varying parameters p and µ, and all theother
parameters as given in Table 1 and the drug efficacies assumed as φ1 = 0.5 and φ2 = 0.5.
Figure 3. Basic reproduction number Rc of CD4+ cells. The number is calculated with variedparameter q and fixed π = 0.23 (left panel) and varied π with fixed q = 0.2 (right panel), with all theother
parameters given in Table 1 and drug efficacies assumed as φ1 = 0.5 and φ2 = 0.5.
Figure 4. Basic reproduction number Rc of CD4+ cells (left panel) and its correspondinglevel lines (right panel). The number is calculated with varying parameters q and π, with all theother
parameters as given in Table 1 and drug efficacies assumed as φ1 = 0.5 and φ2 = 0.5.
Figure 5. Basic reproduction number R0 (left panel) and its corresponding level lines(right panel). The number is calculated with varying drug efficacies φ1 and φ2, with values of p, µ, q,π and m as
optimized with respect to Rh = 1 and Rc = 1, and with all the other parameters as given inTable 1.
Figure 6. Basic reproduction number R0 (left panel) and its corresponding level lines(right panel). The number is calculated with varying drug infection rates β1 and β2, with values of p,µ, q and π
as optimized with respect to Rh = 1 and Rc = 1, and with all the other parameters as givenin Table 1.
Figure 7. Dynamics of HIV mono-infection in the liver with no medical treatment. Verticalaxes represent the variables and horizontal axes are time in days. Parameter values are as indicated inTable
Figure 8. Dynamics of HIV mono-infection in the liver on single drug therapy. Vertical axesrepresent the variables and horizontal axes are time in days. Parameter values are as indicated in Table1.
Figure 9. Productively infected CD4+ in the liver in HIV mono-infection on single drugtherapy. Vertical axis representa the Ic variable and the horizontal axis is time in days. Parametervalues are as
indicated in Table 1.
Figure 10. Productively infected hepatocytes in the liver in HIV mono-infection on singledrug therapy. Vertical axis represents the Ia variable and the horizontal axis is time in days.
Parameter values are as indicated in Table 1.
Figure 11. Viral load in the liver in HIV mono-infection on single drug therapy. Verticalaxis represents the V variable and the horizontal axis is time in days. Parameter values are as indicatedin
Table 1.
Figure 12. HIV mono-infection dynamics in the liver on 2NRTI+1NNRTI combinationtherapy. Vertical axes represent the variables and the horizontal axes are time in days. Parametervalues are as
indicated in Table 1.
Figure 13. HIV mono-infection dynamics in the liver on 2NRTI+2PI combinationtherapy. Vertical axes represent the variables and the horizontal axes are time in days. Parametervalues are as indicated
in Table 1.
Figure 14. Infection dynamics with the best single drug therapy. The simulation is done withvarying probability that a hepatocyte gets infectious at infection. Vertical axes represent the
variablesand the horizontal axes are time in days. Parameter values are as indicated in Table 1.
Figure 15. Infection dynamics with the best combination therapy. The simulation is donewith varying probability that a hepatocyte gets infectious at infection. Vertical axes represent thevariables
and the horizontal axes are time in days. Parameter values are as indicated in Table 1.
Table 1. Parameters for the basic model of HIV in the liver.
rate of creation of CD4+ from within the body natural death rate of uninfected CD4+ probability that HIV infects hepatocytes probability that at infection, hepatocyte becomes productively infected
rate at which latently infected hepatocytes become rate of transmission of HIV in CD4+ antigen-independent CTLs proliferation rate antigen-dependent proliferation rate of CTLs rate at which exposed
CD4+ become uninfected rate at which exposed CD4+ become infectious death rate of infected CD4+ due to infection rate at which CTLs kill infected CD4+ and hepato- rate of creation of hepatocytes from
within the body natural death rate of hepatocytes rate of transmission of HIV in hepatocytes death rate of hepatocytes due to infection rate of clearance of CTLS by all means average rate of
production of virions by an infected average rate of production of virions by an infected death rate of HIV rate of production of virions from macroghages The time in this study is considered in days
and, therefore, all the rates are per day.
Table 2. Example NRTI medications used in antiretroviral therapy, and their parameters.
Table 3. Example NNRTI medication used in antiretroviral therapy, and its parameters.
Table 4. Example PI medications used in antiretroviral therapy, and their parameters.
Source: http://personal.lut.fi/wiki/lib/exe/fetch.php/en/jablonska/plos_model2_revision1.pdf
Origin and of Teleosts Honoring Gloria Arratia Joseph S. Nelson, Hans-Peter Schultze & Mark V. H. Wilson (editors) More advanced teleosts stem-based Verlag Dr. Friedrich Pfeil • München
Acknowledgments . Gloria Arratia's contribution to our understanding of lower teleostean phylogeny and classifi cation – Joseph S. Nelson .
I am sorry to hear you have been troubled by symptoms of imbalance and dizziness. Your query appears to contain two elements: 1) dizziness and imbalance related to the time of year, and 2)
complementary therapies or supplements for dizziness and imbalance. I will cover these in two sections below. I admit I do not routinely provide much advice in these areas, and don't think other
clinicians allied to a more 'medical' model commonly would either. I think this is because there is not enough evidence to back up any recommendations, although I would hope anyone would be open
minded to the possibility of stronger evidence becoming available. I have looked at the scientific literature for you, in case I have been missing something. Both of these areas appear to be poorly
understood and controversial. The scientific research proves confusing and contradictory, but I will attempt to summarise what I have discovered. Some of the difficulty in carrying out research in
these areas relates to different definitions of dizziness, imbalance and vertigo. Often people use the term dizziness to cover feeling light-headed, woozy, giddy, floaty, or unsteady. Vertigo is a
term more often used by clinicians and scientists with a stricter definition of an illusion of movement of one's self or the environment, often spinning or rotational sensations. Individuals often
mean very different things when they say they are dizzy, and research studies have often defined and categorised dizziness differently when you compare them. Patients get allocated into treatment
groups in the studies in quite varying ways. Dizziness and vertigo can be related to many different underlying conditions and it is likely that these conditions need to be considered separately in
research studies, but this is often not the case. Finding effective treatments for specific conditions can be compromised when patients are grouped in research studies into a general ‘dizzy patient'
category. Vertigo is often the subject of research studies rather than dizziness as vertigo is often considered a defining feature of vestibular disorders (conditions related specifically to the
vestibular or balance system, including the vestibular or balance organs in the inner ear), which it is sometimes possible to more clearly define. Any understanding of vertigo might not relate to
symptoms of dizziness or imbalance though. I have put my overall summary of these areas first so that you might obtain some quick guidance. More specific and detailed information is then provided if
you wish to read further, but these sections are denser to read. In summary There are relatively small numbers of studies into both the seasonality of dizziness and balance disorders, and
complementary medicine and therapy in dizziness and imbalance. The studies tend to present conflicting results. Those that have been carried out often use unsatisfactory research methods, use small
numbers of subjects, and are liable to publications bias (for example, studies published by the manufacturers of supplements). There is a distinct lack of the ‘gold standard' randomised,
double-blind, controlled studies. If you are experiencing seasonal symptoms it is difficult to advise how to manage this as it is difficult to assert any control over the seasons, the weather, or
barometric changes. If you are able to determine any triggers that are controllable or avoidable then that would be recommended. Hain (2015) advises to treat any allergy or migraine triggers
appropriately, and to try any appropriate treatment prior to the anticipated onset of symptoms. If someone has migraine then we would expect there might be triggers that cause symptoms. Being able to
identify triggers might help someone determine that they suffer from migraine. It is important to note that an individual can have migraine or vestibular migraine without any headache, which can mean
this diagnosis can be missed. VEDA (2015) advice to patients is to avoid anxiety, as we know this can potentially make any symptoms of dizziness worse, and to educate themselves. I have attached a
leaflet from the Meniere's Society that outlines some possible causes of dizziness and imbalance which might help you to decide whether any of the conditions fit with your symptoms. I have also
attached a leaflet on | {"url":"http://marysfamilymedicine.org/p/personal.lut.fi1.html","timestamp":"2024-11-04T20:14:12Z","content_type":"text/html","content_length":"46925","record_id":"<urn:uuid:e0e1574f-b247-4a5c-9560-8023ccc167f0>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00829.warc.gz"} |
Forex Trading Contest, $5000 in Prizes!
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7 thoughts to “Forex Trading Contest, $5000 in Prizes!”
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2. كل الشكر والتوفيق لكم ولجميع القائمين وكادر هذه الشركة العملاقة
3. i am interesting in this contest.
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6. i am interesting in this contest
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Elements of Geometry and Trigonometry
From inside the book
Page 17 ... vertices of two angles not adjacent . DEFINITIONS OF TERMS . 1. An axiom is a self - evident truth . 2. A demonstration is a train of logical arguments brought to a conclusion . 3. A
theorem is a truth which becomes evident by means of ...
Page 58 ... vertices of its three angles in the circumference . And generally , a polygon is said to be inscribed in a circle , when the vertices of all the angles are in the circumfer- ence . The
circumference of the circle is then said to ...
Page 103 ... vertices B and C in a line paral- lel to the base : and therefore , we have ( B. II . , P. 4 , C. ) AD : DB :: AE : EC . Cor . 1. Hence , by composition , we have ( B. II . , P. 6 ) , AD
+ DB : AD :: AE + EC : AE , or AB : AD :: AC : AE ...
Page 104 ... hence , they have the same altitude ( P. 6 , c . ) ; and consequently , their vertices D and E lie in a parallel to the base BC ( B. I. , P. 23 ) . PROPOSITION XVII . THEOREM . The line
which bisects the 104 GEOMETRY .
Page 128 ... vertices D and F , are situated in a line DF parallel to the base : these triangles are therefore equivalent ( P. 2 , C. ) Add to each of them the figure AECB , and there will result the
polygon AEDCB , equivalent to the polygon AFCB ...
BOOK VI 156
BOOK VII 174
BOOK VIII 202
BOOK IX 227
PAGE 245
PLANE TRIGONOMETRY 255
Napiers Analogies 329
Of Quadrantal Triangles 335
MENSURATION OF SURFACES 347
PAGE 358
Popular passages
If two triangles have two sides and the included angle of the one, equal to two sides and the included angle of the other, each to each, the two triangles will be equal.
That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on
which are the angles less than the two right angles.
A spherical triangle is a portion of the surface of a sphere, bounded by three arcs of great circles.
The circumference of every circle is supposed to be divided into 360 equal parts, called degrees...
Hence, the interior angles plus four right angles, is equal to twice as many right angles as the polygon has sides, and consequently, equal to the sum of the interior angles plus the sum of the
exterior angles.
The surface of a sphere is equal to the product of its diameter by the circumference of a great circle.
If two triangles have two angles of the one equal to two angles of the other, each to each, and also one side of the one equal to the corresponding side of the other, the triangles are congruent.
The area of a parallelogram is equal to the product of its base and altitude.
The angles of spherical triangles may be compared together, by means of the arcs of great circles described from their vertices as poles and included between their sides : hence it is easy to make an
angle of this kind equal to a given angle.
F, be respectively poles of the sides BC, AC, AB. For, the point A being the pole of the arc EF, the distance AE is a 'quadrant ; the point C being the pole of the arc DE, the distance CE is likewise
a quadrant : hence the point E is...
Bibliographic information | {"url":"https://books.google.com.jm/books?id=SmaklsSnwHgC&vq=vertices&dq=editions:ISBN3337155871&output=html&source=gbs_navlinks_s","timestamp":"2024-11-09T19:38:19Z","content_type":"text/html","content_length":"79473","record_id":"<urn:uuid:4f226119-39f3-45b1-9b86-884e4dfe191a>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00425.warc.gz"} |
On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems
Title data
Grüne, Lars ; Guglielmi, Roberto:
On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems.
In: Mathematical Control and Related Fields. Vol. 11 (2021) Issue 1 . - pp. 169-188.
ISSN 2156-8472
DOI: https://doi.org/10.3934/mcrf.2020032
This is the latest version of this item.
Project information
Project financing: Deutsche Forschungsgemeinschaft
Abstract in another language
The paper is devoted to analyze the connection between turnpike phenomena and strict dissipativity properties for continuous-time finite dimensional linear quadratic optimal control problems. We
characterize strict dissipativity properties of the dynamics in terms of the system matrices related to the linear quadratic problem. These characterizations then lead to new necessary conditions for
the turnpike properties under consideration, and thus eventually to necessary and sufficient conditions in terms of spectral criteria and matrix inequalities. One of the key novelty of these results
is the possibility to encompass the presence of state and input constraints.
Further data
Available Versions of this Item | {"url":"https://eref.uni-bayreuth.de/id/eprint/55642/","timestamp":"2024-11-09T04:33:14Z","content_type":"application/xhtml+xml","content_length":"24303","record_id":"<urn:uuid:00536e31-765b-40b2-86e1-019cee3883df>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00799.warc.gz"} |
Quantum Algorithms for Invariants of Triangulated Manifolds
Title Quantum Algorithms for Invariants of Triangulated Manifolds
Publication Journal Article
Year of 2012
Authors Alagic, G, Bering, EA
Journal Quantum Info. Comput. Vol.
Volume 12
Issue 9-10
Pages 843-863
One of the apparent advantages of quantum computers over their classical counterparts is their ability to efficiently contract tensor networks. In this article, we study some implications
of this fact in the case of topological tensor networks. The graph underlying these networks is given by the triangulation of a manifold, and the structure of the tensors ensures that the
overall tensor is independent of the choice of internal triangulation. This leads to quantum algorithms for additively approximating certain invariants of triangulated manifolds. We
Abstract discuss the details of this construction in two specific cases. In the first case, we consider triangulated surfaces, where the triangle tensor is defined by the multiplication operator
of a finite group; the resulting invariant has a simple closed-form expression involving the dimensions of the irreducible representations of the group and the Euler characteristic of the
surface. In the second case, we consider triangulated 3-manifolds, where the tetrahedral tensor is defined by the so-called Fibonacci anyon model; the resulting invariant is the
well-known Turaev-Viro invariant of 3-manifolds.
URL http://dl.acm.org/citation.cfm?id=2481580.2481588 | {"url":"https://quics.umd.edu/publications/quantum-algorithms-invariants-triangulated-manifolds","timestamp":"2024-11-15T04:05:22Z","content_type":"text/html","content_length":"21465","record_id":"<urn:uuid:7d76f2dc-a416-43c9-93d1-5a1322c342e6>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00617.warc.gz"} |
6.11 The Empirical Rule
Course Outline
• segmentGetting Started (Don't Skip This Part)
• segmentStatistics and Data Science: A Modeling Approach
• segmentPART I: EXPLORING VARIATION
• segmentChapter 1 - Welcome to Statistics: A Modeling Approach
• segmentChapter 2 - Understanding Data
• segmentChapter 3 - Examining Distributions
• segmentChapter 4 - Explaining Variation
• segmentPART II: MODELING VARIATION
• segmentChapter 5 - A Simple Model
• segmentChapter 6 - Quantifying Error
• segmentChapter 7 - Adding an Explanatory Variable to the Model
• segmentChapter 8 - Digging Deeper into Group Models
• segmentChapter 9 - Models with a Quantitative Explanatory Variable
• segmentPART III: EVALUATING MODELS
• segmentChapter 10 - The Logic of Inference
• segmentChapter 11 - Model Comparison with F
• segmentChapter 12 - Parameter Estimation and Confidence Intervals
• segmentChapter 13 - What You Have Learned
• segmentFinishing Up (Don't Skip This Part!)
• segmentResources
list High School / Advanced Statistics and Data Science I (ABC)
6.11 The Empirical Rule
The cool thing about normal distributions is that they all basically follow this pattern. In the smooth perfect version of the normal distribution (i.e., the theoretical probability distribution),
Zone 1 covers about .68, Zone 2 covers .95, and Zone 3 covers .997. This .68-.95-.997 pattern is called the empirical rule.
The empirical rule tells us:
• Approximately 68 percent of the scores in a normal distribution are within one standard deviation, plus or minus, of the mean.
• Approximately 95 percent of the scores are within two standard deviations.
• Approximately 99.7 percent of scores are within three standard deviations of the mean (in other words, almost all of them).
The smooth normal distribution is something that is so perfect that it doesn’t really exist. It’s a mathematical object, kind of like how there are straight lines in the world, but a mathematical
straight line is this perfect thing that has no mass, no jitter, and goes on forever. In the same way, a mathematical normal distribution is perfect with no mass, no jitter, and it goes on forever.
The tails of the normal distribution never quite hit 0, they just go on forever and ever. This is why the normal distribution is sometimes called asymptotic. This feature is important because it
allows us to predict the very tiny probabilities of very unlikely events such as a person with a thumb length of 1,000 mm.
You probably have never even heard of a thumb so long. But, if we assume the normal probability distribution, we could quantify exactly how low the probability would be of finding such a rare event.
You can try making up a standard deviation for your own game (we’ll call it Zargle) and simply run the code. It will show you the histograms and proportions for the three zones. Try some different
standard deviations to try and break the empirical rule.
require(coursekata) simulate_scores <- function(game, n, mean, sd) { scores <- rnorm(n, mean, sd) z <- (scores - mean) / sd interval <- ifelse(z > 0, trunc(1 + z), trunc(z - 1)) data.frame(game =
game, scores = scores, z = z, interval = interval, zone = abs(interval)) } compare_score_distributions <- function(sd = 3500, mean = 35000, n = 1000, ..., .seed = 5) { set.seed(.seed) kargle <-
simulate_scores("Kargle", 1000, 35000, 5000) bargle <- simulate_scores("Bargle", 1000, 35000, 1000) zargle <- simulate_scores("Zargle", n, mean, sd) games <- vctrs::vec_c(kargle, bargle, zargle) #
combine all zones > 3 into a single "outside 3" zone games$zone <- ifelse(games$zone > 3, "outside 3", games$zone) # convert the proportions to cumulative proportions for all except "outside 3" props
<- data.frame(tally(zone ~ game, data = games, format = "proportion")) props <- purrr::map_dfr(split(props, props$game), function(x) { x$Freq <- c(cumsum(x$Freq[1:3]), x$Freq[4]) x }) # re-format the
table to be wide (one column per game) zone_table <- tidyr::pivot_wider(props, names_from = game, values_from = Freq) gf_histogram(~scores, fill = ~zone, data = games, bins = 160, alpha = .8) %>%
gf_facet_grid(game ~ .) %>% print() data.frame(zone_table) } # change the standard deviation to whatever you'd like it to be # try to break the empirical rule! compare_score_distributions(sd = 3500,
mean = 35000, n = 1000) # just run the function a few times with different SDs; no solution ex() %>% check_error()
CK Code: B2_Code_Empirical_01
This is what we would get for the Zargle distribution if the standard deviation was set for 3,500.
zone Bargle Kargle Zargle
1 1 0.686 0.690 0.675
2 2 0.950 0.948 0.944
3 3 0.998 0.996 0.997
4 outside 3 0.002 0.004 0.003
The empirical rule can be very useful when trying to make a quick interpretation of a specific score. If a friend has a baby and tells you it was 54 cm long, how would you interpret that measurement?
As an experienced statistician, you should ask: what is the mean, and what is the standard deviation, of the distribution of baby length at birth?
As it turns out, the mean baby length is roughly 50 cm, and the standard deviation is 2 cm. Using the empirical rule, you would say, “Wow! Your baby is like two standard deviations above the mean!
That’s a huge baby! Only .05 of babies are longer than 54 cm (the mean plus two standard deviations). You’ve got yourself a big one!”
Actually, you’d be slightly wrong. (Sorry, I know we set you up!) According to the empirical rule, .95 scores in a normal distribution are within plus or minus two standard deviations from the mean.
It follows from this that .05 of the scores are more extreme than this, or outside plus or minus two standard deviations.
But note, in the figure, that if .05 of the scores are outside plus or minus two standard deviations, half of those would be expected to be more than two standard deviations above the mean, and half
less than two standard deviations below the mean.
So, only .025 of scores would be higher than two standard deviations above the mean. That baby is even more impressive than we thought! He or she is longer than 97.5% of all babies!
What Counts as Unlikely?
We have seen how modeling the error distribution (in the case of the empty model, the distribution of scores around the mean) can help us to calculate probabilities and make predictions. The problem
with a probability, though, is that it’s just a number. It doesn’t tell us what to do. We still have to think about it even after all our fancy R code calculations.
For example, if we wanted to use a model of finger lengths to design stretchy one-size-fits-all gloves, how big should we make the gloves? After all, even though very long thumbs are unlikely, they
are still possible. But if we make these gloves too big, then we’ll alienate short-fingered folks.
What would be the right glove size? To answer questions like this, we have to figure out what are the most likely lengths of people’s fingers, and that means we need to make a judgment call about
what “likely” and “unlikely” mean. We might be able to agree on the best way to estimate a probability, but people will differ on what counts as “unlikely.”
For example, someone who is very risky might look at a .01 probability and say, “Hey! At least it is still possible.” But someone who likes being very certain might say, “Even .40 is unlikely because
it’s less likely than a coin toss!” So in being part of a statistics community, it’s helpful to have an agreement about what counts as unlikely.
Statisticians, as a community, have decided to count .05 and lower probabilities as unlikely. So in the case of a DGP that produces a fairly normal population, we would count scores that are outside
of Zone 2 (+/- two standard deviations from the mean) as unlikely scores, and the scores within Zone 2 as likely. Note that this decision doesn’t result from a calculation. Human statisticians just
sort of agree—yeah, .05 is a pretty low likelihood. | {"url":"https://coursekata.org/preview/book/9824c414-fefa-4dad-b4a4-6f16900d1f53/lesson/9/10","timestamp":"2024-11-09T06:59:56Z","content_type":"text/html","content_length":"99604","record_id":"<urn:uuid:7e08420c-f49e-42d4-8337-52743edee84a>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00465.warc.gz"} |
ball mill motor hp calculation
T14:10:02+00:00 Motor Rating Calculation In Cement Ball Mill lerepitfr. Jan 07, 2015 Raw mills usually operate at 7274 critical speed and cement mills at 7476 32 Calculation of the Critical Mill
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Step 1: Find FLA per Table 430150. 125 hp = 156A 40 hp = 52A 30 hp = 40A. Step 2: Calculate total VA per Sec. 2202. VA = V x I x (where is the square root of 3)Example Calculation. A motor with
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how to calculate motor power for crusher machine April 18th, 2019 25 horsepower motor for a grinding mill prices Ball Mill Design Power Calculation The ball mill motor power requirement
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Mill Power required at pinionshaft = (240 x x ) ÷ = 5440 Hp Speed reducer efficiency: 98% 5440 Hp ÷ = 5550 HP (required minimum motor output power). Therefore select a mill to draw at least 5440
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Ball Mill Motor/Power Sizing Calculation Ball Mill Design/Sizing Calculator The power required to grind a material from a given feed size to a given product size can be estimated by using the
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Aforementioned basic parameters used in ball mill design (power calculations), rod mill or any tumbling mill dimension are; material to shall floor, item, Bond Work Index, bulk density, special
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Ball Mill Design Power Calculation. The ball mill motor power requirement calculated above as 1400 HP is the power that must be applied at the mill drive in order to grind the tonnage of feed
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Ball Mill Motor Hp Calculation [PDF] BALL MILL DRIVE MOTOR CHOICES Artec Machine. The mill used for this comparison is a 44meter diameter by 136 meter long ball mill with a 5000 HP drive motor It
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Thus the power to drive the whole mill. = + = = 86 kW. From the published data, the measured power to the motor terminals is 103 kW, and so the power demand of 86 kW by the mill leads to a
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CPM Homework Help
Joyce’s dad packs her lunch and always packs a yogurt. Joyce knows that there are five yogurts in the refrigerator: one raspberry, two strawberry, one blueberry, and one vanilla. Her dad usually
reaches into the refrigerator and randomly grabs a yogurt.
1. Which flavor is she most likely to have in her lunch today?
The chances of picking a particular flavor of yogurt are greater when there is more than one container of that flavor in the refrigerator. Can you tell if any one yogurt flavor has a higher
probability, or chance, of getting picked by Joyce's dad?
Strawberry is the flavor most likely to be chosen because there are two containers of this flavor in the refrigerator.
2. What are her chances of finding a vanilla yogurt in her lunch bag?
The probability of something happening can be determined by placing the number of times a ''successful outcome'' is possible over the total number of possible outcomes.
In this case, the ideal outcome would be picking vanilla yogurt. This is only possible one time and there are five possible yogurt choices, or ''outcomes.'' Can you find the probability of
picking vanilla? | {"url":"https://homework.cpm.org/category/CC/textbook/cc2/chapter/1/lesson/1.2.1/problem/1-57","timestamp":"2024-11-05T01:09:41Z","content_type":"text/html","content_length":"37222","record_id":"<urn:uuid:8d34b329-1e68-4627-8cd5-7b1512601d33>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00391.warc.gz"} |
Free Printable Times Table Worksheets
Free Printable Times Table Worksheets - Practising timetables is much more fun! Free multiplication chart pdfs can be used at home or at school. These multiplication times table worksheets are
appropriate for kindergarten, 1st grade, 2nd grade, 3rd grade, 4th grade, and 5th grade. Web here you can find the worksheets for the 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 times tables. Get the
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Times Table Worksheets 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14
Multiplication charts & times tables [free & printable!] | prodigy education Sample grade 3 multiplication worksheet Exercises also include multiplying by whole tens and whole hundreds and some
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Multiplication Worksheets 6 7 8 Printable Multiplication Flash Cards
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Printable Multiplication Tables No Answers
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Related Post: | {"url":"https://dl-uk.apowersoft.com/en/free-printable-times-table-worksheets.html","timestamp":"2024-11-14T21:09:47Z","content_type":"text/html","content_length":"31624","record_id":"<urn:uuid:cb624eba-7c4d-4f07-89f6-03c459ebc942>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00614.warc.gz"} |
Carl Friedrich Gausss Influence on Modern Mathematics
To understand better the influence that Johann Carl Friedrich Gauss, Mathematician and Physicist, had on modern mathematics, you might need to understand the man himself. He was born the only child
to a poor family in Brunswick, Germany in 1777. Gauss was often portrayed as a child prodigy when it came to math. The stories range from correcting his father on a math problem at the age of three,
to being able to figure out large sums quickly in his head in Primary School. While in school he was awarded the chance to study at the Collegium Carolinum from 1792 to 1795 by the Duke of Brunswick.
He later attended college at the University of Gottingen from 1795 to 1798. He continued to amaze those around him with his mathematical genius. His love and understanding of math set him up for the
path that his life would take.
While still in college in 1796 at the age of 19, he proved that one could use a straight-edge and compass in order to construct certain types of polygons, using the heptadecagon in his example. He
showed that you could use basic math and square roots to express trigonometric functions. This was an important discovery to mathematicians everywhere in that the actual construction of such objects
had often been wondered about.
In that same year he also invented the method of modular arithmetic, which is used most notably in the 24 hour clock. This method gets it’s name from the fact that numbers will “modulo” or “wrap
around” once they reach a certain value. He was also the first one to show proof of the number theory law of quadratic reciprocity in that same year. Quadratic reciprocity is used to determine if
certain quadratic equations can be solved. The importance of this is that it gives the mathematician the ability to know if they can even be solved without first having to solve them. Think of it
like this, would you want to spend all your time trying to fit a square peg in a round hole; or would you rather know right off the bat that they don’t go together? The fact that it can save time in
knowing which quadratic equations go together can give mathematicians the opportunity to avoid using unsolvable equations when trying to reach a solution in complex problems.
There are many more things that Johann Carl Friedrich Gauss contributed to the world of mathematics. For instance the prime number theorem and the fundamental theorem of algebra. Throughout his
lifetime he used math to solve problems in various fields from physics and astronomy to optics, geometry, and magnetism. He has been awarded several honors; in 1804 he was elected to the Fellows of
the Royal Society of London; in 1820 he was elected to the Fellows of the Royal Society of Edinburgh; in 1838 he was awarded the Copley Medal, this medal is the highest award given by the Royal
Society of London. Along with his numerous awards and honors, he also has several lunar features named after him. From the Crater Gauss on the moon to the asteroid 1001 Gaussia, which was discovered
in 1923. The first expedition ship from Germany to explore Antarctica was named the Gauss, and that same ship discovered an extinct volcano which they named Gaussberg.
His proven theories are still in use today and his influence is everywhere in the world of mathematics and beyond. If you would like to do some more research on this noted genius, please visit the
sites that I used for references. These are as follows: | {"url":"http://www.actforlibraries.org/carl-friedrich-gausss-influence-on-modern-mathematics/","timestamp":"2024-11-14T08:48:39Z","content_type":"text/html","content_length":"22421","record_id":"<urn:uuid:bf98e8d9-07a7-476b-aea8-3881498dc31b>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00003.warc.gz"} |
The Stacks project
Lemma 5.8.5. Let $f : X \to Y$ be a surjective, continuous map of topological spaces. If $X$ has a finite number, say $n$, of irreducible components, then $Y$ has $\leq n$ irreducible components.
Proof. Say $X_1, \ldots , X_ n$ are the irreducible components of $X$. By Lemmas 5.8.2 and 5.8.3 the closure $Y_ i \subset Y$ of $f(X_ i)$ is irreducible. Since $f$ is surjective, we see that $Y$ is
the union of the $Y_ i$. We may choose a minimal subset $I \subset \{ 1, \ldots , n\} $ such that $Y = \bigcup _{i \in I} Y_ i$. Then we may apply Lemma 5.8.4 to see that the $Y_ i$ for $i \in I$ are
the irreducible components of $Y$. $\square$
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Problem E: Catch Neko
Neko is running away! Eve wants to catch him. At first, Eve is standing at the point with coordinates \((x_1,y_1)\), while Neko is standing at the point with coordinates \((x_2,y_2)\). For every
minute, Eve can choose to go up, down, left, or right with 1 unit distance. For instance, if she is at \((x,y)\) now, she can go to \((x,y+1),(x,y-1),(x-1,y)\) or \((x+1,y)\).
Eve noticed that Neko also moves 1 unit distance every minute, but he only moves according to a sequence periodically. The sequence only contains 'U','D','L','R', denoting that Neko moves up, down,
left, right respectively.
Eve is now wondering how many minutes she needs at least to catch Neko. | {"url":"https://acm.sustech.edu.cn/onlinejudge/problem.php?cid=1053&pid=4","timestamp":"2024-11-15T02:42:20Z","content_type":"text/html","content_length":"9932","record_id":"<urn:uuid:0b6965e6-42cd-4b5a-9f32-cf49502dc6db>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00511.warc.gz"} |
Entropy Contractions in Markov Chains: Half-Step, Full-Step and Continuous-Time
This paper considers the speed of convergence (mixing) of a finite Markov kernel P with respect to the Kullback-Leibler divergence (entropy). Given a Markov kernel one defines either a discrete-time
Markov chain (with the n-step transition kernel given by the matrix power P^n) or a continuous-time Markov process (with the... Show more | {"url":"https://synthical.com/article/Entropy-Contractions-in-Markov-Chains%3A-Half-Step%2C-Full-Step-and-Continuous-Time-4b8f1c75-b3dc-4254-9c17-37ee48352839?","timestamp":"2024-11-12T16:57:43Z","content_type":"text/html","content_length":"71896","record_id":"<urn:uuid:cf8e47c8-eff3-4001-b003-f53ac30316db>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00077.warc.gz"} |
[2024/01 oracel sde OA] 60分钟四个题两个选择两个算法题 - csOAhelp|代码代写|面试OA助攻|面试代面|作业实验代写|考试高分代考
1. 4th Bit
A binary number is a combination of 1s and 0s. Its n[th] least significant digit is the n[th] digit starting from the right starting with 1. Given a decimal number, convert it to binary and determine
the value of the 4th least significant digit.
number = 23
• Convert the decimal number 23 to binary number: 2310=24+23+21+20=(10111)22310=24+23+21+20=(10111)2.
• The value of the 4th index from the right in the binary representation is 0.
Function Description
Complete the function fourthBit in the editor below.
fourthBit has the following parameter(s):
• int number: a decimal integer
• int: an integer 0 or 1 matching the 4th least significant digit in the binary representation of number.
2. Break a Palindrome
A palindrome reads the same from left or right, mom for example. There is a palindrome which must be modified, if possible. Change exactly one character of the string to another character in the
range ascii[a-z] so that the string meets the following three conditions:
• The new string is lower alphabetically than the initial string.
• The new string is the lowest value string alphabetically that can be created from the original palindrome after making only one change.
• The new string is not a palindrome.
Return the new string, or, if it not possible to create a string meeting the criteria, return the string IMPOSSIBLE.
palindromeStr = 'aaabbbaaa'
• Possible strings lower alphabetically than aaabbbaaa after one change are [aaaabaaaa].
• aaaabaaaa is not a palindrome and is the lowest string that can be created from palindromeStr.
Function Description
Complete the function breakPalindrome in the editor below.
breakPalindrome has the following parameter(s):
• string palindromeStr: the original string
3. Which of the following operators has the lowest precedence?
• Ternary operator (?:)
• Comma operator (,)
• Sizeof operator (sizeof)
• Member access operator (.)
4. Are you an expert on data structures?
Which of the following data structures can erase from its beginning or its end in O(1) time?
• vector
• deque
• stack
• segment tree
更多OA真题咨询，辅助VO OA，欢迎联系我 | {"url":"https://csoahelp.com/2024/01/09/2024-01-oracel-sde-oa-60%E5%88%86%E9%92%9F%E5%9B%9B%E4%B8%AA%E9%A2%98%E4%B8%A4%E4%B8%AA%E9%80%89%E6%8B%A9%E4%B8%A4%E4%B8%AA%E7%AE%97%E6%B3%95%E9%A2%98/","timestamp":"2024-11-13T12:05:04Z","content_type":"text/html","content_length":"90598","record_id":"<urn:uuid:39d4b1d0-5738-422c-8c6f-afc1f117318a>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00848.warc.gz"} |
Résumé en français · 1adf1f8f34
Résumé en français
This commit is contained in:
@ -1,9 +1,31 @@
I would like to thank
Je tiens tout d'abord à remercier mon directeur de thèse Benoît Libert, qui m'a accompagné durant toute la durée de ma thèse et m'a gratifié de sa présence et de sa disponibilité tout au long de
cette aventure. Je le remercie aussi de son support lors des phases de soumissions, des possibles \textit{rebuttals} qui s'en suivaient et lors des présentations en conférences.
Ses enseignements et ses conseils m'ont aidé à avoir une vision plus claire de la recherche et me permettront désormais de prendre mon envol.
I would also like to thank Dario Catalano and David Pointcheval, who accepted to review this thesis during the summer and granted me with their useful remarks. I also thank the rest of the
committee: Shweta Agrawal, who also invited me for a visit in the nice city of Chennai and taught me to be cautious with monkeys; Pierre-Alain Fouque, who also accepted to be part of my half-way
thesis committee; Philippe Gaborit and our discussions at \textit{La-Londe-lès-Maures}; and Carla Ràfols, who I met in several conferences and for the organization of the insightful \textsc{
Cost-Iacr} School on Randomness in Cryptography.
I also want to thank my coauthors for their ideas and our useful discussions: Benoît Libert, San Ling, Khoa Nguyen, Thomas Peters, Huaxiong Wang and Moti Yung. Without you, my research would have
be less productive!
For inviting me and their warm reception, I would like to express my gratitude to Shweta Agrawal from IIT Madras; Adeline Langlois and Pierre-Alain Fouque from Rennes; Satya Lokam from Microsoft
Bangalore; Abderrahmane Nitaj and Brigitte Vallée from Caen; and Ali El Kaafarani from Oxford.
It was the opportunity for me to discover other research environments and our interesting discussions.
Without having to move so far, I also want to thank the AriC team in Lyon and especially the crypto part with its dynamic that is very supportive to research:
Damien, Benoît, Fabien,
Alexandre, Alice, Alice, Alonso, Benjamin, Chen, Chitchanok, Elena, Gottfried, Ida, Jiangtao, Jie, Junqing, Laurent, Marie, Miruna, Radu, Sanjay, Shi, Somindu and Weiqiang.
I especially want to thank to Damien Stehlé, who introduced me to cryptography with his course in master, and helped me going further in this topic by introducing me to Fré Vercauteren from KU
Leuven who was my first contact with research in cryptography.
Where the boundaries are blurry, I would like to thank Guillaume Hanrot, who was there to guide me in solving the administrative conundrum and also for showing his interest for my research.
From the rest of the permanent members of AriC team, I want to thank
Nicolas Brisebarre, Claude-Pierre Jeannerod, Vincent Lefèvre, Nicolas Louvet, Jean-Michel Muller, Nathalie Revol, Bruno Salvy and Gilles Villard.
Je tenais aussi à remercier mes collègues, à commencer par ceux qui ont eu la chance de partager mon bureau et mes lubies: Valentina pour sa joie de vivre
@ -2,7 +2,7 @@ In the last fifty years, the use of cryptography has shifted from military and c
For instance, the Enigma machine had a design for military purposes, and another one for companies (Enigma A26).
As of today, about $60\%$ of the first million most visited websites propose encrypted and authenticated communications (via \texttt{https}), and so are most of the communications channels used
by electronic devices (like \textit{Wifi Protected Access}).
At the same time, the growth of exchanged data and the sensitivity of transferred information make the urge of procecting these data efficiently even more critical.
At the same time, the growth of exchanged data and the sensitivity of transferred information make the urge of protecting these data efficiently even more critical.
While we are reaching the Moore's law barrier, other threats exist against nowadays' cryptosystems.
For instance, the existence of a quantum computer with sufficient memory~\cite{Sho99} would break most of real-world cryptographic designs, which mostly rely on modular arithmetic assumptions.
In this context, it is crucial to design cryptographic schemes that are believed to be quantum-resistant.
@ -0,0 +1,217 @@
Les cinquante dernière années, l'utilisation de la cryptographie s'est éloignée de ses origines militaires et de son usage pour le secret commercial pour se démocratiser à un public plus large.
Par exemple, la machine Enigma initialement conçue pour un usage militaire a été déclinée pour un usage commerciale (la machine Enigma A26).
Aujourd'hui, environ $60\%$ du premier million des sites les plus visités propose une connexion chiffrée et authentifiée (à l'aide du protocole \texttt{https}), tout comme les canaux de
communication des appareils électronique portatifs (comme la norme \texttt{WPA}, en anglais \textit{Wifi Protected Access}).
Dans le même temps, la croissance des données échangée en ligne et la sensibilité de ces informations rend de plus en plus urgent la protection de ces canaux.
Pendant que la loi de Moore\footnote{La loi qui prédit la puissance de calcul des processeurs modernes.} atteint ses limites, d'autres menaces existent sur nos cryptosystèmes actuels.
Par exemple, l'existence d'un ordinateur quantique possédant suffisamment de mémoire~\cite{Sho99} casserait la majorité des constructions cryptographiques utilisées en pratique, puisqu'elles
reposent sur des hypothèses d'arithmétique modulaire dont la structure peut-être exploitée par un adversaire quantique.
Dans cette situation, il est crucial de construire des schémas cryptographiques qui résisteraient à une menace quantique.
Pour répondre à ce problème, la \textit{cryptographie post-quantique} est née au début des années 2000.
Les différents candidats reposent sur différents objets mathématiques, comme les réseaux euclidiens, les codes correcteurs d'erreurs, les systèmes polynomiaux multivariés, etc.
Récemment, le NIST (\textit{National Institude of Standards and Technology}) a organisé une compétition pour évaluer les différentes solutions post-quantiques en terme de chiffrement et de
Dans cette compétition, 82 protocoles ont été proposés, parmi lesquels 28 reposent sur les réseaux euclidiens, 24 sur les codes correcteurs, 13 sur des systèmes multi-variés, 4 sur des fonctions
de hachages et 13 sur d'autres objets.
Si la cryptographie pratique vise principalement à fournir des schémas de signatures et de chiffrement, comme l'atteste la compétition du NIST,
la recherche théorique propose des solutions à des problèmes plus spécifiques, comme la construction de systèmes de monnaie électronique\footnote{À ne pas confondre avec les cryptomonnaies\ldots}
~\cite{CFN88}, qui sont l'équivalent numérique de notre monnaies échangée. Les pièces sont délivrées par une autorité centrale (la banque), et les dépenses restent intraçables. En cas de
comportement malhonnête (comme une double-dépense), l'identité de l'utilisateur malicieux est révélée.
Les constructions cryptographiques doivent en plus vérifier des propriétés de sécurités.
Par exemple, un schéma de chiffrement doit pouvoir cacher un message en présence d'un attaquant passif voire actif (c'est-à-dire qui peut modifier certains messages).
Pour garantir ces exigences, les cryptographes fournissent des preuves de sécurité dans le cadre de modèles de sécurité précis.
Une preuve de sécurité précise principalement qu'une construction cryptographique est au moins aussi difficile qu'un problème supposé dur par la littérature.
Finalement, l'importance de la préservation de la vie privé et la protection des données a été un sujet qui a fait couler beaucoup d'encres, comme en atteste le développement de la règlementation
générale sur la protection des données (RGPD) en 2016, mise en application ce 25 mai.
Il est donc intéressant pour les cryptographes de fournir des solutions qui resisteraient, dans le meilleurs des mondes, à un adversaire quantique.
Néanmoins, la construction de ces protocoles reposent de manière décisive sur les «\,preuves à divulgation nulles de connaissances\,». Ce sont des protocoles interactifs entre un prouveur et un
vérificateur où le prouveur cherche à convaincre le vérificateur de la véracité d'une affirmation sans rien divulguer de plus que la valeur de vérité de cette affirmation.
Dans le contexte de la cryptographie post-quantique, de tels systèmes de preuves sont limités en expressivité ou en terme de coût de calcul (en temps ou en mémoire).
\section{Privacy-Preserving Cryptography}
In this context, `privacy-preserving' refers to the ability of a primitive to provide some functionalities while holding sensitive information private.
An example of such primitives are \textit{anonymous credentials}~\cite{Cha85,CL01}.
Informally, this primitive allows users to prove themselves to some verifiers without telling their identity, nor the pattern of their authentications.
To realize this, this system involves one (or more) credential issuer(s) and a set of users who have their own secret keys and pseudonyms that are bound to their secret.
Users can dynamically obtain credentials from an issuer that only knows users' pseudonyms and obliviously sign users' secret key as well as a set of attributes.
Later on, users can make themselves know to verifiers under a different pseudonym and demonstrate possession of a certification from the issuer, without revealing neither the signature nor the
secret key.
This primitive thus allows a user to authenticate to a system (e.g., in anonymous access control) while retaining its anonymity.
In addition, the system is guaranteed that users indeed possess a valid credential.
Interests in privacy-based cryptography date back to the beginning of public-key cryptography~\cite{Rab81,Cha82,GM82,Cha85}.
A reason for that could be the similarities between the motivations of cryptography and the requirements of privacy protection.
Additionally, cryptographers' work in this field may have direct consequences in term of services that could be developed in the real-world.
Indeed, having a practical anonymous credential scheme will enable its use for access control in a way that limits security flaws.
Whereas, nowadays' implementations are based on more elementary building blocks, like signatures, whose manipulations may lead to different security holes~\cite{VP17}.
Similarly, \textit{advanced primitives} often involve simpler building blocks in their design.
The difference lies in that provable security conveys security guarantees for the construction.
As explained before, these proofs make the security of a set of schemes rely on hardness assumptions.
Thus, the security relies on the validity of those assumptions, which are independently studied by cryptanalysts.
Hence, security is guaranteed by the study of those assumptions.
For example, the security analysis of multilinear maps in~\cite{CHL+15} made obsolete a large amount of candidates at this time.
This example reflects the importance of relying on well-studied and simple assumptions as we will explain in~\cref{ch:proofs}.
In the context of this thesis, the developed cryptographic schemes rely on lattices and bilinear maps over cyclic groups.
Lattice-based cryptography is used to step towards post-quantum cryptography, while the latter proves useful in the design of practical schemes.
The details of these two structures are given in~\cref{ch:structures}.
\subsection{Zero-Knowledge Proofs}
As explained before, zero-knowledge proofs are a basic building block for privacy-preserving cryptography.
They require completeness, soundness and zero-knowledge properties.
Completeness captures the correctness of the protocol if everyone is honest. In the case of a dishonest prover, soundness asks the probability that the verifier is convinced to be negligible.
On the contrary, if the verifier is cheating, the zero-knowledge property guarantees that the prover's secret remains hidden.
In the case of identification schemes, the nature of the secret remains simple and solutions exist under multiple assumptions~\cite{Sch96,Ste96,KTX08,Lyu08}.
For more complex statements, such as proving correct computation, a gap appears between post-quantum schemes and modular arithmetic-based schemes.
In the case of pairing-based cryptography, there exist non-interactive zero-knowledge proofs which can prove a large variety of statements~\cite{GOS06,GS08} without idealized assumptions.
Such proofs are still missing in the context of post-quantum cryptography so far.
In the lattice world, there are two main families of proof systems: Schnorr-like proofs~\cite{Sch96,Lyu09} and Stern-like proofs~\cite{Ste96}, named after their respective authors.
The first family works on some structured lattices. Exploiting this structure allows for more compact proofs, while the expressiveness of statements is quite restricted.
The second kind of proofs is combinatorial and works on the representation of lattice elements (as matrix and vectors).
By nature, these proofs are quite expensive in term of communication complexity.
However, they can be used to prove a wide variety of statements as we will explain in more details along this thesis and especially in~\cref{sse:stern}.
More generally, zero-knowledge proofs are detailed in~\cref{ch:zka}.
\subsection{Signatures with Efficient Protocols}
To enable privacy-preserving functionalities, a possible avenue is to couple zero-knowledge proofs with signature schemes.
One of such signatures are \textit{signatures with efficient protocols}.
This primitive extends the functionalities of ordinary digital signature schemes in two ways: (i)~It provides a protocol to allow a signer to obliviously sign a hidden message and (ii)~Users are
able to prove knowledge of a hidden message-signature pair in a zero-knowledge fashion.
These two properties turn out to be extremely useful when it comes designing efficient anonymity-related protocols such as anonymous credentials or e-cash.
The design of effective signatures with efficient protocols is thus important for privacy-preserving cryptography.
In this thesis, we provide two of these signature schemes.
One of them, described in~\cref{ch:sigmasig}, based on pairings, shifts the~\cite{LPY15} signature scheme to an idealized but practically acceptable model, aiming at efficiency.
The other, described in~\cref{ch:gs-lwe}, adapts a variant of Boyen's signature~\cite{Boy10,BHJ+15} along with the Kawachi-Tanaka-Xagawa commitment scheme~\cite{KTX08} to provide a lattice-based
signature schemes that is compatible with Stern-like proofs.
This scheme has also been relaxed in the context of adaptive oblivious transfer where, in some places, it is only required to have random-message security instead of security against
chosen-message security as described in~\cref{ch:ot-lwe}.
\section{Pairings and Lattices}
In this thesis, the proposed constructions rely on the assumed hardness of assumptions over pairing-friendly groups and lattices.
These two objects have widely been used in cryptography since the early 2000s~\cite{SOK00,Reg05}.
Even since, they attracted much attention from cryptographers, leading to multiple constructions in advanced cryptography (as in~\cite{Jou00,BBS04,BN06,GS08,LYJP14,LPQ17} for pairings, and~\cite{
GPV08,ABB10,BV11,GSW13,dPLNS17} for lattices).
\subsection{Pairing-Based Cryptography}
A pairing is a bilinear map from two cyclic source groups to a target group.
This bilinear property takes advantage of a rich structure to groups that are compatible with such a map.
It is then not surprising to see the variety of schemes that stems from pairing-based cryptography.
In the context of privacy-based cryptography, an important breakthrough was the introduction of Groth-Sahai proofs~\cite{GOS06,GS08} that allow proving in a non-interactive zero-knowledge fashion
a large class of statements in the standard model.
For instance, Groth-Sahai proofs have been used in group signatures and anonymous-credential schemes~\cite{Gro07,BCKL08,BCC+09}, or e-cash systems in the standard model~\cite{BCKL09}.
In this thesis, however, our pairing-based constructions focus on practicality.
Thus, they are instantiated in the random oracle model, where Schnorr's proof are made non-interactive through the Fiat-Shamir transform when the statement to prove is simple enough.
A recent line of work in cryptanalysis of bilinear maps~\cite{KB16,MSS17,BD18} led to a change in the panorama of practical pairing-based cryptography.
This affects us in the sense that security parameter has to be increased in order to achieve the same security level.
Nevertheless, pairing-based cryptography offers a nice tradeoff between its capabilities and efficiency.
As an example, we can cite the work of Döttling and Garg~\cite{DG17}, who closed the problem of providing an identity-based encryption scheme which only relies on the Diffie-Hellman assumption
(it is construction on cyclic groups that does not need pairings, as defined in~\cref{de:DDH}).
While their construction relies on a simpler mathematical object, it does not reach the efficiency of pairing-based ones~\cite{BB04}.
\subsection{Lattice-Based Cryptography}
From an algebraic point of view, a lattice is a discrete subgroup of $\RR^n$,
which leads to a simple additive structure.
The core difference with number-theoretic cryptography, such as discrete-logarithm-based cryptography, is the existence of the geometrical structure of the lattice.
From this geometry rises some problems that are believed to withstand quantum computers.
Despite this apparently simple structure, some advanced primitives are only known, as of today, to be possible under lattice assumptions, such as fully-homomorphic encryption~\cite{Gen09,GSW13}.
The versatility of lattice-based cryptography is enabled by the existence of lattice trapdoors~\cite{GPV08,CHKP10,MP12}, as we explain in~\cref{sse:lattice-trapdoors}.
Informally, the knowledge of a short basis for a lattice allows sampling short vectors, which is believed to be hard without such a short basis.
Furthermore, knowing a short basis for the lattice $\{\mathbf{v} \in \ZZ^m \mid \mathbf{A} \mathbf{z} = 0 \bmod q\}$ described by matrix $\mathbf{A} \in \ZZ_q^{n \times m}$ makes it possible to
generate a short basis for a related lattice described by $[ \mathbf{A} \mid \mathbf{B}] \in \ZZ_q^{n \times m'}$.
An application for this property is Boyen's signature scheme~\cite{Boy10}.
In this scheme, a signature for message $m$ is a short vector in the orthogonal lattice of the matrix $\mathbf{A}_m = [\mathbf{A} \mid \mathbf{B}_m]$, where $\mathbf{B}_m$ is publicly computable.
Hence, knowing a trapdoor for $\mathbf{A}$ makes the computation of this short vector possible, and the message is bound to the description of the lattice $\mathbf{A}_m$.
Indeed, some extra care has to be taken to avoid multiplicative attacks.
Still, the use of lattice trapdoors comes at a price, as it significantly decreases the efficiency of cryptographic designs that use them~\cite{Lyu12,LLNW16}.
Given that we provide the first lattice-based construction for the scheme we present, we focused on designing provably-secure scheme under well-studied assumptions.
\section{Our Results}
In this thesis, we present several cryptographic constructions that preserve privacy.
These constructions are the result of both improvements we made in the use of zero-knowledge proofs and the ability to prove the security of our constructions under standard assumptions.
We believe that these advances on zero-knowledge proofs are of independent interest and that the given schemes are a step towards quantum-secure privacy-preserving cryptography.
In the following, we detail four contributions that are developed in this thesis.
These results are taken from four published articles: \cite{LMPY16,LLM+16,LLM+16a,LLM+17}.
\subsection{Dynamic Group Signatures and Anonymous Credentials}
In~\cref{pa:gs-ac}, we present two primitives: dynamic group signatures and anonymous credentials.
We already described the behavior of anonymous credential in~\cref{se:privacy-preserving-crypto}.
As for dynamic group signatures, they are a primitive that allows a group of users to authenticate messages on behalf of the group while remaining anonymous inside this group.
The users still remain accountable for their actions, as another authority knows a secret information that gives it the ability to lift anonymity of misbehaving users.
By itself, this primitive can be used to provide anonymous authentications while providing accountability (which is not the case with anonymous credentials).
For instance, in the Internet of things, such as smart cars, it is important to provide authenticated communication channels as well as anonymity. For car communications, if exchanged data may
not be sensitive by themselves, the identity of the driver could be.
We can imagine a scenario where some burglars eavesdrop a specific car to know whenever a house is empty.
In this thesis, we present in~\cref{ch:sigmasig} pairing-based group signatures that aims at efficiency while relying on simple assumptions.
The resulting scheme shows competitive signature size with other schemes that rely on more ad-hoc assumptions, and its practicality is supported by an implementation.
This scheme is presented in~\cite{LMPY16}, which is joint work with Benoît Libert, Thomas Peters an Moti Yung presented at AsiaCCS'16.
\cref{ch:gs-lwe} presents the first \textit{dynamic} group signature scheme relying on lattice assumptions.
This has been made possible by adapting Stern-like proofs to properly interact with a signature scheme: a variant of Boyen's signature~\cite{Boy10,BHJ+15}.
It results in a \textit{signature with efficient protocols} that is of independent interest.
Later, it has been adapted in the design dynamic group encryption~\cite{LLM+16a} and adaptive oblivious transfer~\cite{LLM+17}.
This work is described in~\cite{LLM+16}, made with Benoît Libert, San Ling, Khoa Nguyen and Huaxiong Wang and presented at Asiacrypt'16.
\subsection{Group Encryption}
Group encryption schemes~\cite{KTY07} are the encryption analogue of group signatures.
In this setting, a user is willing to send a message to a group member, while keeping the recipient of the message hidden inside the group.
In order to keep user accountable for their actions, an opening authority is further empowered with some secret information allowing it to un-anonymize ciphertexts.
More formally, a group signature scheme is a primitive allowing the sender to generate publicly verifiable proofs that: (1) The ciphertext is well-formed and intended to some registered group
member who will be able to decrypt; (2) The opening authority will be able to identify the receiver if necessary; (3) The plaintext satisfies certain properties, such as being a witness for some
public relation, or the private key that underlies a given public key.
In the model of Kiayias, Tsiounis and Yung~\cite{KTY07}, the message secrecy and anonymity properties are required to withstand active adversaries, which are granted access to decryption oracles
in all security definitions.
A natural application is to allow a firewall to filter all incoming encrypted emails except those intended for some certified organization members and the content of which is additionally
guaranteed to satisfy certain requirements, like the absence of malware.
Furthermore, group encryption schemes are motivated by privacy applications such as anonymous trusted third parties, key recovery mechanisms or oblivious retriever storage system.
In cloud storage services, group encryption enables privacy-preserving asynchronous transfers of encrypted datasets.
Namely, it allows users to archive encrypted datasets on remote servers while convincing those servers that the data is indeed intended to some anonymous certified client who has a valid account
to the storage provider.
In case of suspicions on the archive's content, a judge should be able do identify the recipient of the archive.
To tackle the problem of designing lattice-based group encryption, we needed to handle ``quadratic relations''.
Indeed, lattice-based zero-knowledge proof systems were able to handle only relations where witnesses are multiplied by a public value.
Let us recall that, in Learning-With-Errors schemes, an encryption have the form $\mathbf{A} \cdot \mathbf{s} + \mathbf{e} + \mathbf{m} \lceil \frac{q}{2} \rceil \bmod q$, where $\mathbf{A}$ is
the recipient public-key.
As group encryption requires this public-key $\mathbf{A}$ to be private, a way to achieve this is to have a zero-knowledge proof system which handles relations where the witness is multiplied
with a private matrix.
We address this issue introducing new technique to handle this kind of relations.
These techniques, based on a \textit{divide-and-conquer} strategy, are described in~\cref{ch:ge-lwe}, as well as the construction of the group signature scheme proven fully-secure in the standard
This work have been presented at Asiacrypt'16~\cite{LLM+16a} and have been done with Benoît Libert, San Ling, Khoa Nguyen and Huaxiong Wang.
\subsection{Adaptive Oblivious Transfer}
Oblivious transfer is a primitive coined by Rabin~\cite{Rab81} and later extended by Even, Goldreich and Lempel~\cite{EGL85}.
It involves a server with a database of messages indexed from $1$ to $N$ and a receiver with a secret index $\rho \in \{1,\ldots,N\}$.
The protocol allows the receiver to retrieve the $\rho$-th message from the receiver without letting him infer anything on his choice.
Furthermore, the receiver only obtains the $\rho$-th message and learns nothing about the other messages.
In its adaptive flavor~\cite{NP99}, oblivious transfer allows the receiver to interact $k$ times with the server to obtain $k$ messages in such a way that, each request may depend on the
previously retrieved messages.
From a theoretical point of view, oblivious transfer is known to be a \textit{complete building block} for cryptography in the sense that, if it can be realized, then any secure multiparty
computation can be.
In its adaptive variant, oblivious transfer has applications in privacy-preserving access to sensitive databases (such as medical records or financial data) stored in an encrypted form on a
remote server.
In its basic form, (adaptive) oblivious transfer does not restrict in any way the population of users who can obtain specific records.
In many sensitive databases (e.g., DNA samples or patients' medical history), however, not all users should be able to access the whole database.
It is thus crucial to protect the access to certain entries conditioned on the receiver holding suitable credentials delivered by authorities.
At the same time, privacy protection requires that authorized users should be able to query database records while leaking as little as possible about their interests or activities.
This requirements is handled by endowing oblivious transfer with access control, as stated by Camenish, Dubovitskaya and Neven~\cite{CDN09}.
In this variant, each database record is protected by a different access control policy.
Based on their attributes, users can obtain credentials from pre-determined authorities, which entitle them to anonymously retrieve database records of which the access policy accepts their
certified attributes.
During the transfer phase, the user demonstrates, in a zero-knowledge manner, possession of an attribute string compatible with the policy of a record in the database, as well as a credential for
this attribute.
The only information that the database holder eventually learns is that some user retrieved some record which he was authorized to obtain.
To achieve this, an important property is the expressiveness of such access policies.
In other words, the system should be able to handle complex attribute policies while keeping time and memory consumption reasonable\footnote{Here, ``\textit{reasonable}'' means (probabilistic)
polynomial time.}.
In this thesis, we propose in~\cref{ch:ot-lwe} a zero-knowledge protocol to efficiently handle any access policy that can be described with a logarithmic-depth boolean circuit, also known as $\
mathsf{NC}1$, based on lattices.
In the context of adaptive oblivious transfer with access control, most of the schemes (based on pairing assumptions) manage to handle the case of conjunctions under reasonable assumptions~\cite{
CDN09,CDNZ11,ACDN13}. Under strong assumptions, however, the case of $\mathsf{NC}1$ can be taken care of~\cite{ZAW+10}.
This joint work with Benoît Libert, San Ling, Khoa Nguyen and Huaxiong Wang was presented at Asiacrypt'17~\cite{LLM+17}.
@ -1405,7 +1405,7 @@ We stress that the proofs can be easily adapted to the case where the opening a
Delerabl\'ee-Pointcheval & $4$ & $5$ & $2304$ bits & $q$-\textsf{SDH} + \textsf{XDH} & Dynamic & CCA \\ \hline
Bichsel {\em et al.} & $3$ & $2$ &$1280$ bits & \textsf{LRSW} + \textsf{SDL} & Dynamic & CCA- \\ \hline
Pointcheval-Sanders & $2$ & $2$ & $1024$ bits & \textsf{LRSW} & Dynamic & CCA- \\ \hline
Pointcheval-Sanders & $2$ & $2$ & $1024$ bits & \textsf{LRSW}-like & Dynamic & CCA- \\ \hline
\caption{Comparison between different group signature schemes}
@ -23,7 +23,7 @@
N\textsuperscript{o} d'ordre NNT : xxx
N\textsuperscript{o} d'ordre NNT : 2018LYSEN060
@ -36,7 +36,7 @@ opérée au sein de\\
\textbf{École Doctorale ED512\\% rectifier le numéro d'accréditation
\textbf{École Doctorale N$^{\mathrm{o}}$512\\% rectifier le numéro d'accréditation
École Doctorale en Informatique et Mathématiques de Lyon}% nom complet de l'école doctorale
@ -48,7 +48,7 @@ opérée au sein de\\
Soutenue publiquement le jj/mm/aaaa, par :\\
Soutenue publiquement le 18/10/2018, par :\\
\textbf{Fabrice Mouhartem}
@ -99,7 +99,7 @@
\input abstract
\input acknowledgements
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@ -112,6 +112,13 @@
\chapter{Résumé substantiel en Français}
\input chap-resume
\addcontentsline{tof}{chapter}{\protect\numberline{\thechapter} Introduction}
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Planck theory in Engineering Physics by | Tech Glads
Planck theory
sem1 | 0 comments
Planck theory
Max Planck suggested that the energy of light is proportional to its frequency, also showing that light exists in discrete quanta of energy
Planck’s law describes the electromagnetic radiation emitted by ablack body in thermal equilibrium at a definite temperature. The law is named after Max Planck, who originally proposed it in 1900. It
is a pioneering result of modern physics and quantum theory.
The spectral radiance of a body, $B_u$, describes the amount of energy it gives off as radiation of different frequencies. It is measured in terms of the power emitted per unit area of the body, per
unit solid angle that the radiation is measured over, per unit frequency. Planck theory showed that the spectral radiance of a body at absolute temperature T is given by
$Planck theory$
where k[B] the Boltzmann constant, h the Planck constant, and c thespeed of light in the medium, whether material or vacuum.^[1]^[2]^[3] The spectral radiance can also be measured per unit wavelength
instead of per unit frequency. In this case, it is given by
$Planck theory$.
The SI units are W·sr^−1·m^−2·Hz^−1 of B[ν] and W·sr^−1·m^−3 for B[λ]. The law may also be expressed in other terms, such as of the number of photons emitted at a certain wavelength, or of the energy
density in a volume of radiation.
In the limit of low frequencies (i.e. long wavelengths), Planck’s law tends to the Rayleigh–Jeans law, while in the limit of high frequencies (i.e. small wavelengths) it tends to the Wien
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What is the washer method formula?
Hint: In this problem, we have to find what is the washer method formula. Here we should first know about disc integration, in which the washer method is a part. We should know that the disc
integration models the resulting three-dimensional shape as a stack of an infinite number of discs. It is possible to use the same principle with rings instead of disc, which is the washer method to
obtain hollow solids of revolution. Here we can see about the definitions and the formulas.
Complete step-by-step solution:
We can write the formula for disc method which is used to find the volume of solid of revolution with the function of x
\[\pi \int\limits_{a}^{b}{R{{\left( x \right)}^{2}}dx}\]
Where \[R\left( x \right)\] is the distance between the function and the axis of rotation, which works only if the axis of rotation is horizontal (if the function is y, then the axis of rotation is
We can now write the formula for the washer method which is used to obtain hollow solids of revolution.
It is the subtraction of the volume of the inner solid of revolution from the outer solid of revolution, which can be calculated in a single integral.
\[\pi \int\limits_{a}^{b}{{{R}_{O}}{{\left( x \right)}^{2}}-{{R}_{I}}{{\left( x \right)}^{2}}dx}\]
Where \[{{R}_{O}}\left( x \right)\] is the function that is farthest from the axis of rotation and \[{{R}_{I}}\left( x \right)\] is nearest from the axis of rotation.
Note: We should know that the disc integration models the resulting three-dimensional shape as a stack of an infinite number of discs. It is possible to use the same principle with rings instead of
disc which is the washer method to obtain hollow solids of revolution. | {"url":"https://www.vedantu.com/question-answer/what-is-the-washer-method-formula-class-12-maths-cbse-60aa70be3811760ea6ae208b","timestamp":"2024-11-05T11:05:15Z","content_type":"text/html","content_length":"177669","record_id":"<urn:uuid:0310d794-c9e1-42ef-99f7-aba7ec4d1d87>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00825.warc.gz"} |
Concurrent Zero Knowledge without Complexity Assumptions
Authors: Daniele Micciancio, Shien Jin Ong, Amit Sahai, Salil Vadhan.
Theory of cryptography conference - TCC 2006. New York, NY, USA. March 2006. LNCS 3876, Springer. pp. 1-20..
[BibTex] [Postscript] [PDF]
Abstract: We provide unconditional constructions of concurrent statistical zero-knowledge proofs for a variety of non-trivial problems (not known to have probabilistic polynomial-time algorithms).
The problems include Graph Isomorphism, Graph Nonisomorphism, Quadratic Residuosity, Quadrati Nonresiduosity, a restricted version of Statistical Difference, and approximate versions of the (coNP
forms of the) Shortest Vector Problem and Closest Vector Problem in lattices.
For some of the problems, such as Graph Isomorphism and Quadratic Residuosity, the proof systems have provers that can be implemented in polynomial time (given an NP witness) and have ~O(log n)
rounds, which is known to be essentially optimal for black-box simulation.
To the best of our knowledge, these are the first constructions of concurrent zero-knowledge proofs in the plain, asynchronous model (i.e. without setup or timing assumptions) that do not require
complexity assumptions (such as the existence of one-way functions). | {"url":"https://cseweb.ucsd.edu/~daniele/papers/ConcZK.html","timestamp":"2024-11-07T16:06:04Z","content_type":"text/html","content_length":"2279","record_id":"<urn:uuid:cf3c21f4-f571-4287-bd6e-ffd65d830fe0>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00309.warc.gz"} |
4.10 Quantitative Explanatory Variables
Course Outline
• segmentGetting Started (Don't Skip This Part)
• segmentStatistics and Data Science: A Modeling Approach
• segmentPART I: EXPLORING VARIATION
• segmentChapter 1 - Welcome to Statistics: A Modeling Approach
• segmentChapter 2 - Understanding Data
• segmentChapter 3 - Examining Distributions
• segmentChapter 4 - Explaining Variation
□ 4.10 Quantitative Explanatory Variables
• segmentPART II: MODELING VARIATION
• segmentChapter 5 - A Simple Model
• segmentChapter 6 - Quantifying Error
• segmentChapter 7 - Adding an Explanatory Variable to the Model
• segmentChapter 8 - Digging Deeper into Group Models
• segmentChapter 9 - Models with a Quantitative Explanatory Variable
• segmentPART III: EVALUATING MODELS
• segmentChapter 10 - The Logic of Inference
• segmentChapter 11 - Model Comparison with F
• segmentChapter 12 - Parameter Estimation and Confidence Intervals
• segmentFinishing Up (Don't Skip This Part!)
• segmentResources
list High School / Advanced Statistics and Data Science I (ABC)
4.10 Quantitative Explanatory Variables
Up to this point we have been using Height as though it were a categorical variable. First we divided it into two categories, then three.
When we do this, we are throwing away some of the information we have in our data. We know exactly how many inches tall each person is. Why not use that information instead of just categorizing
people as either tall or short?
Let’s try another approach, a scatterplot of Thumb length by Height. Try using gf_point() with Height rather than Height2Group or Height3Group. Note: when making scatterplots, the convention is to
put the outcome variable on the y-axis, the explanatory variable on the x-axis.
require(coursekata) Fingers <- Fingers %>% mutate(Height2Group = factor(ntile(Height, 2), 1:2, c("short", "tall"))) # create a scatterplot of Thumb by Height # create a scatterplot of Thumb by Height
gf_point(Thumb ~ Height, data = Fingers) ex() %>% check_or( check_function(., "gf_point") %>% { check_arg(., "object") %>% check_equal() check_arg(., "data") %>% check_equal() }, override_solution(.,
"gf_point(Fingers, Thumb ~ Height)") %>% check_function("gf_point") %>% { check_arg(., "object") %>% check_equal() check_arg(., "gformula") %>% check_equal() }, override_solution(., "gf_point
(Fingers$Thumb ~ Fingers$Height)") %>% check_function("gf_point") %>% check_arg("object") %>% check_equal(), override_solution(., "gf_jitter(Thumb ~ Height, data = Fingers)") %>% check_function
("gf_jitter") %>% { check_arg(., "object") %>% check_equal() check_arg(., "data") %>% check_equal() }, override_solution(., "gf_jitter(Fingers, Thumb ~ Height)") %>% check_function("gf_jitter") %>% {
check_arg(., "object") %>% check_equal() check_arg(., "gformula") %>% check_equal() }, override_solution(., "gf_jitter(Fingers$Thumb ~ Fingers$Height)") %>% check_function("gf_jitter") %>% check_arg
("object") %>% check_equal() )
CK Code: ch4-16
The same relationship we spotted in the boxplots when we divided Height into three categories can be seen in the scatterplot. In the image below, we have overlaid boxes at three different intervals
along the distribution of Height.
Each box corresponds to one of the three groups of our Height3Group variable. On the x-axis you can see the range in height, measured in inches, for each of the three groups.
Remember that we used ntile() to divide our sample into three groups of equal sizes. Because most people in the sample are clustered around the average height, it makes sense that the box in the
middle is the narrowest. There aren’t that many people taller than 70 inches, so to get a tall group that is exactly one-third of the sample means we have to include a wider range of heights.
The heights of the boxes represent the middle of the Thumb distribution for that third of the sample, just like in a boxplot. So, the bottom of the box is Q1 and the top is Q3. You can see that the
thumb lengths of people who are taller tend to be longer. You can also see that height explains only some of the variation in thumb length. Within each band of Height, there is variation in thumb
length (look up and down within each box).
So, just as when we measured Height as a categorical variable, although there appears to be some variation in Thumb that is explained by Height, there is also variation left over after we have taken
out the variation due to Height.
We can try to explain variation with categorical explanatory variables (such as Sex and Height3Group) but we can also try to explain variation with quantitative explanatory variable (such as Height).
Let’s stretch our thinking further. What if you wanted to have two explanatory variables for thumb length? For example, if we wanted to think about how variation in Thumb might be explained by
variation in both Sex and Height, we could represent this idea as a word equation like this.
Thumb = Sex + Height + Other Stuff
The variation in thumb length is the same whether we try to explain it with Sex, Height, or both! The total variation in Thumb doesn’t change. But how about that unexplained variation? The better the
job done by the explanatory variables, the less left over variation.
Summary: Visualizations to Help You Explore Variation
You’ve learned many R functions that can be used to help you visualize distributions of data. In Chapter 3, you learned how to create visualizations of a single outcome variable. In Chapter 4, you
learned how to create visualizations that show the relationship between an outcome variable and an explanatory variable. Let’s review when each type of visualization is appropriate to use.
Visualizations with One Variable
Variable Visualization Type R Code
Categorical Frequency Table tally
Bar Graph gf_bar
Quantitative Histogram gf_histogram
Boxplot gf_boxplot
Visualizations with Two Variables
Outcome Variable Explanatory Variable Visualization Type R Code
Frequency Table tally
Categorical Categorical Faceted Bar Graph gf_bar %>%
Faceted Histogram gf_histogram %>%
Quantitative Categorical Boxplot gf_boxplot
Jitter Plot gf_jitter
Scatterplot gf_point
Categorical Quantitative
Quantitative Quantitative Jitter Plot gf_jitter
Scatterplot gf_point | {"url":"https://coursekata.org/preview/book/f84ca125-b1d7-4288-9263-7995615e6ead/lesson/6/9","timestamp":"2024-11-05T13:49:33Z","content_type":"text/html","content_length":"97267","record_id":"<urn:uuid:a76fa56e-07af-4ea6-a7d6-7e54a83d74ff>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00299.warc.gz"} |
2nd Grade Comparing and Ordering Numbers
2nd Grade Comparing and Ordering Numbers
Our 2nd grade comparing numbers worksheets include practice in both comparing numbers and ordering numbers.
The first section below includes six worksheets where students place three numbers in order. We’ve included practice ordering numbers from least to greatest AND from greatest to least.
The second group asks students to put four numbers in order, and the last section asks students to compare two numbers by writing a greater than (>), less than (<), or equal to sign (=). For all of
the worksheets on this page your student(s) will be working with numbers between 100 and 1,000.
Are these 2nd grade comparing numbers worksheets too advanced? Check out our 1st Grade Worksheets.
Ordering Three Numbers from 100 to 1,000 (Printable PDFs)
In these free ordering numbers worksheets, students will practice ordering three numbers ranging from 100-1000. Each worksheet focuses on one of the following: They’ll either order the numbers from
least to greatest or from greatest to least. All worksheets include answer keys.
Ordering Four Numbers from 100 to 1,000 (Printable PDFs)
In these ordering numbers worksheets, students will practice ordering four numbers ranging from 100-1000. Each worksheet focuses on one of the following: They’ll either order the numbers from least
to greatest or from greatest to least. All worksheets include answer keys.
Compare Numbers and Write <, >, or = (printable PDF)
In these comparing numbers worksheets, students will practice comparing numbers from 100-1000. Students will look at two different numbers, decide which one is bigger, and indicate their answer using
the correct symbol (>,<, or =). All worksheets include answer keys. | {"url":"https://k12mathworksheets.com/worksheets/2nd-grade-comparing-numbers/","timestamp":"2024-11-05T13:47:42Z","content_type":"text/html","content_length":"122402","record_id":"<urn:uuid:87522de4-021b-4a2f-b85a-bd85089bc144>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00374.warc.gz"} |
Monoidal t-norm based logic $\mathbf {MTL}$ is the weakest t-norm based residuated fuzzy logic, which is a $[0,1]$-valued propositional logical system having a t-norm and its residuum as truth
function for conjunction and implication. Monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ that consists of the formulas with unary predicates and just one object variable, is the monadic
fragment of fuzzy predicate logic $\mathbf {MTL\forall }$, which is indeed the predicate version of monoidal t-norm based logic $\mathbf {MTL}$. The main aim of this paper is to give an algebraic
proof of the completeness theorem for monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and some of its axiomatic extensions. Firstly, we survey the axiomatic system of monadic algebras for t
-norm based residuated fuzzy logic and amend some of them, thus showing that the relationships for these monadic algebras completely inherit those for corresponding algebras. Subsequently, using the
equivalence between monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and S5-like fuzzy modal logic $\mathbf {S5(MTL)}$, we prove that the variety of monadic MTL-algebras is actually the
equivalent algebraic semantics of the logic $\mathbf {mMTL\forall }$, giving an algebraic proof of the completeness theorem for this logic via functional monadic MTL-algebras. Finally, we further
obtain the completeness theorem of some axiomatic extensions for the logic $\mathbf {mMTL\forall }$, and thus give a major application, namely, proving the strong completeness theorem for monadic
fuzzy predicate logic based on involutive monoidal t-norm logic $\mathbf {mIMTL\forall }$ via functional representation of finitely subdirectly irreducible monadic IMTL-algebras. | {"url":"https://core-cms.prod.aop.cambridge.org/core/search?filters%5Bkeywords%5D=completeness","timestamp":"2024-11-13T00:21:35Z","content_type":"text/html","content_length":"1049980","record_id":"<urn:uuid:607349a8-f1ef-4ebd-8201-4b3693f31fc4>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00754.warc.gz"} |
Model construction · RxInfer.jl
Model creation in RxInfer largely depends on GraphPPL package. RxInfer re-exports the @model macro from GraphPPL and defines extra plugins and data structures on top of the default functionality.
The model creation and construction were largely refactored in GraphPPL v4. Read Migration Guide for more details.
Also read the Model Specification guide.
RxInfer operates with so-called graphical probabilistic models, more specifically factor graphs. Working with graphs directly is, however, tedious and error-prone, especially for large models. To
simplify the process, RxInfer exports the @model macro, which translates a textual description of a probabilistic model into a corresponding factor graph representation.
@model function model_name(model_arguments...)
# model description
@model macro generates a function that returns an equivalent graph-representation of the given probabilistic model description. See the documentation to GraphPPL.@model for more information.
Supported aliases in the model specification specifically for RxInfer.jl and ReactiveMP.jl
• a || b: alias for ReactiveMP.OR(a, b) node (operator precedence between ||, &&, -> and ! is the same as in Julia).
• a && b: alias for ReactiveMP.AND(a, b) node (operator precedence ||, &&, -> and ! is the same as in Julia).
• a -> b: alias for ReactiveMP.IMPLY(a, b) node (operator precedence ||, &&, -> and ! is the same as in Julia).
• ¬a and !a: alias for ReactiveMP.NOT(a) node (Unicode \neg, operator precedence ||, &&, -> and ! is the same as in Julia).
Note, that GraphPPL also implements @model macro, but does not export it by default. This was a deliberate choice to allow inference backends (such as RxInfer) to implement custom functionality on
top of the default GraphPPL.@model macro. This is done with a custom backend for GraphPPL.@model macro. Read more about backends in the corresponding section of GraphPPL documentation.
A backend for GraphPPL that uses ReactiveMP for inference.
After model creation RxInfer uses RxInfer.condition_on function to condition on data. As an alias it is also possible to use the | operator for the same purpose, but with a nicer syntax.
condition_on(generator::ModelGenerator; kwargs...)
A function that creates a ConditionedModelGenerator object from GraphPPL.ModelGenerator. The | operator can be used as a shorthand for this function.
julia> using RxInfer
julia> @model function beta_bernoulli(y, a, b)
θ ~ Beta(a, b)
y .~ Bernoulli(θ)
julia> conditioned_model = beta_bernoulli(a = 1.0, b = 2.0) | (y = [ 1.0, 0.0, 1.0 ], )
beta_bernoulli(a = 1.0, b = 2.0) conditioned on:
y = [1.0, 0.0, 1.0]
julia> RxInfer.create_model(conditioned_model) isa RxInfer.ProbabilisticModel
ConditionedModelGenerator(generator, conditioned_on)
Accepts a model generator and data to condition on. The generator must be GraphPPL.ModelGenerator object. The conditioned_on must be named tuple or a dictionary with keys corresponding to the names
of the input arguments in the model.
Sometimes it might be useful to condition on data, which is not available at model creation time. This might be especially useful in reactive inference setting, where data, e.g. might be available
later on from some asynchronous sensor input. For this reason, RxInfer implements a special deferred data handler, that does mark model argument as data, but does not specify any particular value for
this data nor its shape.
An object that is used to condition on unknown data. That may be necessary to create a model from a ModelGenerator object for which data is not known at the time of the model creation.
After the model has been conditioned it can be materialized with the RxInfer.create_model function. This function takes the RxInfer.ConditionedModelGenerator object and materializes it into a
Materializes the model specification conditioned on some data into a corresponding factor graph representation. Returns ProbabilisticModel.
A structure that holds the factor graph representation of a probabilistic model.
Returns the underlying factor graph model.
Returns the value from the return ... operator inside the model specification.
Returns the (nested) dictionary of random variables from the model specification.
Returns the random variables from the model specification.
Returns the data variables from the model specification.
Returns the constant variables from the model specification.
Returns the factor nodes from the model specification.
RxInfer implements several additional pipeline stages for default parsing stages in GraphPPL. A notable distinction of the RxInfer model specification language is the fact that RxInfer "folds" some
mathematical expressions and adds extra brackets to ensure the correct number of arguments for factor nodes. For example an expression x ~ x1 + x2 + x3 + x4 becomes x ~ ((x1 + x2) + x3) + x4 to
ensure that the + function has exactly two arguments.
An additional pipeline stage for the @model macro from GraphPPL. Notify the user that the datavar, constvar and randomvar syntax has been removed and is not be supported in the current version.
An additional pipeline stage for the @model macro from GraphPPL. This pipeline converts simple multi-argument operators to their corresponding bracketed expression. E.g. the expression x ~ x1 + x2 +
x3 + x4 becomes x ~ ((x1 + x2) + x3) + x4). The operators to compose are + and *.
A pipeline stage for the @model macro from GraphPPL. This pipeline applies the aliases defined in ReactiveMPNodeAliases to the expression.
Syntaxic sugar for ReactiveMP nodes. Replaces a || b with ReactiveMP.OR(a, b), a && b with ReactiveMP.AND(a, b), a -> b with ReactiveMP.IMPLY(a, b) and ¬a with ReactiveMP.NOT(a).
To get an access to an internal ReactiveMP data structure of a variable in RxInfer model, it is possible to return a so called label of the variable from the model macro, and access it later on as
the following:
using RxInfer
@model function beta_bernoulli(y)
θ ~ Beta(1, 1)
y ~ Bernoulli(θ)
return θ
result = infer(
model = beta_bernoulli(),
data = (y = 0.0, )
Inference results:
Posteriors | available for (θ)
graph = RxInfer.getmodel(result.model)
returnval = RxInfer.getreturnval(graph)
θ = returnval
variable = RxInfer.getvariable(RxInfer.getvarref(graph, θ)) | {"url":"https://reactivebayes.github.io/RxInfer.jl/stable/library/model-construction/","timestamp":"2024-11-14T08:03:40Z","content_type":"text/html","content_length":"40106","record_id":"<urn:uuid:45c77039-be1e-4c7f-852c-a8a0a90340b2>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00504.warc.gz"} |
Adding and Subtracting Fractions with Unlike Denominators
In adding and subtracting fractions with unlike denominators, you have to get a least common denominator which is exactly like the least common multiple it just so happens to be in the denominator.
But first, looking at each one of these denominators, we must make sure we have them completely factored. The x minus 2 and the x plus 2 are prime. However, this x squared minus 4 can actually be
written as a product of two primes of x minus 2 times x plus 2. So, let's find our least common denominator. So, look at the first one, and we place x minus 2 in the least common denominator. So, we
are done with the first one. Looking at the second one, we have this x plus 2 that's not in in the least common denominator, so we have to place that there as well. Notice, once we get to this last
one over here, the x minus 2 and the x plus 2 are already in the least common denominator, therefore we don't have to place extra ones in. So, the least common denominator happens to be x minus 2
times x plus 2. Now, actually doing the calculations, we are well aware that our least common denominator must contain x minus 2 times x plus 2. So, for each of these fractions, we have to write an
equivalent fraction. On this first one, we are missing the x plus 2. On the second one, we’re missing the x minus 2. And on this last one, we are not missing anything. But, if you feel like you
should put something there, you can put 1 and that will be fine. So, now what we must do is we just have to multiply all of the numerators together to get these into equivalent fractions. So, we have
7 times x plus 2. Note the negative here. Make sure you place that down in the next step. We have negative 1 times x minus 2. We also have the symbol here, the plus, and then times 4. Now all we need
to do from here is just simplify. And what I mean by simplify, is just using the distributive property. Distributing the 7 through the x plus 2, and the negative 1 through the x minus 2. So, we have,
7 times x plus 14 minus x plus 2, and then plus four, all over x minus 2 times x plus 2. We are just going to keep simplifying down. So, now all we have to do is mark our like terms and add those
together. We have 7x and negative x, which gives us 6x. Then we have 14 plus 2 plus 4, which is basically 14 plus 6, which gives us 20, all divided by x minus 2 times x plus 2. Now, if we want, we
could factor the 6x plus 20. The greatest common factor out of those two terms is actually going to be a 2. So, we have 2 times 3x plus 10 all divided by x minus 2 times x plus 2. If you feel like
you don't have full understanding of the least common denominator, including the least common multiple, or you're having trouble factoring, please refer back to the other videos. | {"url":"https://ung.edu/learning-support/video-transcripts/adding-and-subtracting-fractions-with-unlike-denominators.php","timestamp":"2024-11-08T07:38:38Z","content_type":"application/xhtml+xml","content_length":"48311","record_id":"<urn:uuid:88a6e565-ed75-4844-b305-44c75a640cc4>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00420.warc.gz"} |
Scale-stack barchart in paper.js
To empty my head after a hard day’s work on grant proposals: another quick graph test in paper.js: the scale-stack barchart (see here for a description of what it is). This type of visual encoding
solves the issue of showing data with vastly different orders of magnitude. Although other solutions have been used for a long time (e.g. using a log-scale), these have many issues. The log-scale,
for example, does make all bars fit in the same order of magnitude, but is very difficult for the end-user to interpret. From the paper referenced above, comparing cut-off bars, scale breaks, log
scales, and scale-stack bar chart.
The key in looking at such graph is to look at one order of magnitude at a time. Just looking at the top of the graph it is clear that the 5th value is much larger than all the other ones. Similarly,
at the bottom we can see that the 6th value is 2 to 3 times as high as the 11th one, and that all the others are at least one order of magnitude higher.
var verticalOffset = 700;
var horizontalOffset = 50;
var levelScaling = 5; // number of pixels per unit. If = 5: value of 3 => 15 pixels
var levelHeight = 10*levelScaling;
var data = [13,123,3617,627,2938172,3,509,8261,19,29128,1,28];
var calculateComponents = function(x) {
var maxPowerOfTen = 0; // 4->0; 15->1; 18272->4
var currentX = x;
while ( currentX >= 10 ) {
maxPowerOfTen += 1;
currentX = currentX/10;
var valueAtMaxPowerOfTen = x/Math.pow(10,maxPowerOfTen);
return {
orig: x,
val: valueAtMaxPowerOfTen,
lvl: maxPowerOfTen};
var dataInComponents = [];
for (var i = 0; i < data.length; i++ ) {
var maxLevel = Math.max.apply(Math,dataInComponents.map(function(o){return o.lvl;}))
for (var i = 0; i <= maxLevel; i++) {
var line = new Path();
line.add(new Point(20,verticalOffset-levelHeight*i));
line.add(new Point(20+data.length*25,verticalOffset-levelHeight*i));
line.strokeColor = 'lightgrey';
for (var i = 0; i < dataInComponents.length; i++ ) {
// The thin bar in the orders of magnitude smaller than the current number
var thinBar = new Path();
thinBar.add(new Point(horizontalOffset + i*20, verticalOffset));
thinBar.add(new Point(horizontalOffset + i*20, verticalOffset-levelHeight*dataInComponents[i].lvl));
thinBar.strokeColor = 'grey';
thinBar.strokeWidth = 2;
// The thick bar in the order of magnitude of the current number
var thickBar = new Path();
thickBar.add(new Point(horizontalOffset + i*20, verticalOffset-levelHeight*dataInComponents[i].lvl));
thickBar.add(new Point(horizontalOffset + i*20, verticalOffset-levelHeight*dataInComponents[i].lvl-(dataInComponents[i].val*levelScaling)));
thickBar.strokeColor = 'grey';
thickBar.strokeWidth = 10;
// The thick but flat bar in the orders of magnitude larger than the current number
for (var j = dataInComponents[i].lvl+1; j<=maxLevel; j++) {
var placeHolder = new Path();
placeHolder.add(new Point(horizontalOffset + i*20, verticalOffset-levelHeight*j));
placeHolder.add(new Point(horizontalOffset + i*20, verticalOffset-levelHeight*j-1));
placeHolder.strokeColor = 'grey';
placeHolder.strokeWidth = 10; | {"url":"http://vda-lab.github.io/2014/01/scale-stack-barchart-in-paperjs","timestamp":"2024-11-14T15:11:52Z","content_type":"text/html","content_length":"24897","record_id":"<urn:uuid:fa010b1f-2c58-414d-9eed-5e77e7b39225>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00169.warc.gz"} |
Suppression of crystallization in saline drop evaporation on pinning-free surfaces
For sessile droplets of pure liquid on a surface, evaporation depends on surface wettability, the surrounding environment, contact angle hysteresis, and surface roughness. For non-pure liquids, the
evaporation characteristics are further complicated by the constituents and impurities within the droplet. For saline solutions, this complication takes the form of a modified partial vapor pressure/
water activity caused by the increasing salt concentration as the aqueous solvent evaporates. It is generally thought that droplets on surfaces will crystallize when the saturation concentration is
reached, i.e., 26.3% for NaCl in water. This crystallization is initiated by contact with the surface and is thus due to surface roughness and heterogeneities. Recently, smooth, low contact angle
hysteresis surfaces have been created by molecular grafting of polymer chains. In this work, we hypothesize that by using these very smooth surfaces to evaporate saline droplets, we can suppress the
crystallization caused by the surface interactions and thus achieve constant volume droplets above the saturation concentration. In our experiments, we used several different surfaces to examine the
possibility of crystallization suppression. We show that on polymer grafted surfaces, i.e., Slippery Omniphobic Covalently Attached Liquid-like (SOCAL) and polyethyleneglycol(PEGylated) surfaces, we
can achieve stable droplets as low as 55% relative humidity at 25°C with high reproducibility using NaCl in water solutions. We also show that it is possible to achieve stable droplets above the
saturation concentration on other surfaces, including superhydrophobic surfaces. We present an analytical model, based on water activity, which accurately describes the final stable volume as a
function of the initial salt concentration. These findings are important for heat and mass transfer in relatively low humidity environments.
The evaporation of sessile droplets of non-pure liquids is the subject of intense study due to its wide applications in printing,^1 virology,^2 microfluidics,^3 and heat exchangers.^4 Despite this
interest, many of the models currently used do not comprehensively cover the evaporation dynamics and predict their behavior given substrate surface properties, fluid composition, and surrounding
environment.^5 For all liquids on a surface, the surrounding atmosphere plays a key role in dictating the rate of phase change from liquid to gas. For pure liquids, such as water, this evaporation
rate is often limited by the diffusion of the water vapor through the surrounding gas phase.
In non-pure liquids, such as saline solutions, the mechanisms dictating the evaporation rate are more complicated. Since the diffusion-limited evaporation is governed by the relationship between the
vapor pressure at the interface and that far away from the droplet, for non-pure liquids, the vapor pressure at the droplet interface must be adapted to capture the effect of the solute. In the work
by Soulie et al.,^6 a spatiotemporal relationship, g(c) based on the different vapor pressures of the saline solution and that of the pure fluid as well as the ambient relative humidity has been
proposed. This relationship can be used to determine a water activity that accurately describes the change in the liquid vapor pressure, which, in turn, dictates the evaporation rate. For example,
the addition of salt (and, e.g., solutions of glycerol or sulfuric acid) is used by the food industry to control the water content when drying or packaging food. Saturated salt solutions maintain a
constant humidity as long as the amount of salt present is above the saturation level. There is also a critical water activity below which no micro-organisms can grow. For most foods, this is in the
0.6–0.7 water activity range. Pathogenic bacteria cannot grow below a water activity of 0.85–0.86, whereas yeast and molds are more tolerant to a reduced water activity of 0.80, but usually, no
growth occurs below a water activity of about 0.62 (see, e.g., the work of Rahman & Labuza^7).
In a non-pure liquid, evaporation also changes the balance of the liquid composition so, for example, in a saline drop, salt concentration increases and crystallization may occur under suitable
conditions. There are two possible regimes for crystal nucleation: homogeneous and heterogeneous nucleation.^8 Homogeneous crystal nucleation is a random process in which precipitates form a perfect
lattice; this is rare and only occurs when the strain and surface energy creating the ionic interactions are small.^9 For sessile droplets, the mechanism for crystallization has been attributed to
heterogeneous surface nucleation.^10 Such nucleation involves processes occurring at the solid surface due to surface heterogeneity can be caused by roughness and variations in surface wetting, which
can be characterized by contact angle hysteresis.
Recently, smooth surfaces with extremely low contact angle hysteresis (CAH)^11 below 2° have been shown to have such little contact line pinning of droplets of water that the ideal constant contact
angle (CCA) mode of evaporation^12 identified by Picknett and Bexon^13 can be observed. The smoothness, attributed to chemical homogeneity, of polymer brush surfaces has previously been seen to have
a significant effect in scaling experiments.^17,18 In these studies, the nucleation rate is reported as being proportional to the available number of nucleation sites; therefore, we believe that the
surface hysteresis can be used to represent the suppression of salt crystallization. It has also previously been shown that polymer brush surfaces outperform fluorinated coatings for anti-scaling
experiments and therefore provide the ideal study for salt crystallization.^19 However, the implication of this for evaporation-induced crystallization with sessile saline drops has not been
In this work, we aim to reduce the heterogeneity of the surface that allows for crystals to form during the evaporation of sessile saline droplets by evaporating in the CCA mode. By suppressing these
mechanisms, we show that a sessile saline droplet’s evaporation behavior is eventually dominated by the effect of the solute on the water activity at the liquid–vapor interface. In this regime,
stable droplets that stop evaporating under controlled environmental conditions without crystallization are created. This phenomenon has previously been seen in levitating droplets^14,15 but with the
crucial difference of a lack of contact with a solid surface. The surfaces shown in the work of Armstrong et al.^12 show no contact line pinning and a contact angle (CA) of ∼105°. Here, we use the
same surface properties to attain behavior similar to evaporating levitating saline droplets but for sessile saline droplets on a surface. We also use other surfaces and see similar behavior,
specifically for other low CAH surfaces such as PEGylated surfaces but for sessile saline droplets on a solid surface.^16
A. Picknett and Bexon model for sessile droplets of water
When the size of a droplet is much less than the capillary length of the evaporating liquid,
, where
is the surface tension,
is the density of the liquid, and
is the acceleration due to gravity, the droplet adopts a spherical cap shape. For a given volume of liquid,
, there are then well-defined geometric parameters, the spherical radius
, contact radius
, and contact angle,
, which can be measured from side profile images. Geometrically, these parameters are related by
In general, the rate for diffusion-limited loss of a liquid mass by evaporation through a liquid–vapor interface is
is the diffusion coefficient of the vapor and
is the concentration of the vapor. Combining the geometrical assumptions with Eq.
and a concentration gradient model allows data on the evaporation of sessile droplets to be analyzed.
Picknett and Bexon
provided an exact solution for Eq.
for sessile droplets assuming constant temperature (isothermal model) in the form of an infinite series and provided interpolation formulas using a power series in the contact angle, one for 0° ≤
< 10° and a second for 10° ≤
≤ 180°, to provide simple evaluation. Subsequently, Stauber
et al.^20
provided an alternative solution using an integral formulation. Erbil
et al.^21
considered the Picknett and Bexon solution and introduced a function
) to take account in a common notational format of the dependence of the concentration gradient of the vapor, between the surface of the droplet and its surroundings, on the contact angle arising
from different models,
) is the difference in the vapor concentration at the liquid–vapor interface of the droplet
, which is assumed to be its saturation value, and that far removed from the droplet surface
, which is assumed to be its ambient value with
the relative humidity. An integral formulation was given by Stauber
et al.
and for ease of use, we provide an interpolation function for
) (see the
supplementary material
, Eqs. S1–S3).
B. Non-isothermal model for sessile droplets of water
Recently, Nguyen
et al.^22
reported an analytical solution of the sessile droplet evaporation coupled with interfacial cooling on a substrate heated at a constant temperature (non-isothermal model). Their results were uniquely
quantified by a dimensionless evaporative cooling number,
, defined by
= 0.032 K
is the latent heat of evaporation, and
is the thermal conductivity of water. At 295 K, the evaporative cooling number of water is estimated to be
= 0.11; for methanol,
= 0.84; and for acetone,
= 1.03. This solution incorporating evaporative cooling is different to that of an isolated spherical droplet in free space. For the isolated case, where the droplet is not supported by a substrate,
the heat flux into the droplet arises from the surrounding vapor rather than a substrate and involves a constant,
, similar to Eq.
but using the thermal conductivity of air,
(see the work of Netz and Eaton
and Netz
et al.^25
used the non-isothermal model of sessile droplet evaporation to provide a correction factor,
), to the evaporation rate from the iso-thermal model (i.e., to the right-hand side of Eq.
. This correction is expressed in an integral form and depends on the contact angle,
, and the evaporative cooling number,
. We have evaluated this integral form and provide a simple interpolation for the range 0° ≤
≤ 120° for
= 0.11,
is in degrees. [We note that eq. (4) in Shen
et al.^25
has a typographical error and should use a sinh() and not a sin() in the numerator.] For contact angles above 120°, which is particularly useful for superhydrophobic surfaces, numerical evaluation of
the integral equations is problematic and does not converge. A discussion of numerical simulations of Shen
et al.
and an interpolation of their results is given in the
supplementary material
(Sec. 2).
C. Water activity for sessile droplets of salt solutions
It has long been known that the presence of salt alters the concentration of water vapor at a water vapor surface to c[s]^surface = a[w](x)c[s], where a[w](x) is the water activity and x = m[s]/(m[s]
+ m[w]) is the mass fraction of the salt relative to the total mass of the droplet. The water activity is defined as the ratio of the vapor pressure of the solute/vapor pressure of pure water and is
related to the Equilibrium Relative Humidity (ERH) of air surrounding a system by a[w] = ERH/100, where the ERH is expressed as a %.
As discussed by Soulié et al.^6 and Seyfert et al.,^10 the evaporation rate of a sessile droplet of salt solution is reduced by a factor (a[w] − H[r])/(1 − H[r]) compared to that of a pure sessile
droplet of water. This introduces a time dependence because the activity of water depends on the salt concentration, which changes during evaporation and leads to an equilibrium droplet if a[w] = H[r
] is achieved, i.e., the ERH of the droplet becomes equal to the surrounding relative humidity. Seyfert et al.^10 noted that because there is a minimum value of water activity, a[w]^min = a[w](x[sat
]), determined by the saturated solution concentration, x[sat], some droplets containing salt may reach a stable volume. Thus, at 20 °C, the minimum water activity is a[w]^min ≈ 0.76 and droplets,
which start with a[w] > 0.76, will not evaporate completely when H[r] > 75%. Seyfert et al.^10 identified three asymptotic regimes of sessile droplet evaporation. In regime I, for a relative humidity
H[r] < 0.75, the liquid phase evaporates completely and a dry deposit is created. In regime II, for relative humidity H[r] in the upper vicinity of ∼75%, the droplet volume remains stable at a
maximum salt concentration. In regime III, for larger H[r], the droplet volume remains stable at larger volumes with a lower salt concentration due to a larger amount of liquid water.
D. Non-isothermal model for sessile droplets of salt solutions
The modification of the Picknett and Bexon
isothermal model for the evaporation of a sessile droplet of water [Eq.
] to include evaporative cooling and water activity due to the salt depressing the vapor pressure at the droplet–vapor interface is
where for the range 0° ≤
≤ 120° and
= 0.11,
, 0.11) is given by Eq.
) is given in the
supplementary material
in Sec. 1, Eqs. S1–S3. Equation
is similar to the equations previously provided by Soulié
et al.^6
and Seyfert
et al.
but with the inclusion of the non-isothermal correction provided by Shen
et al.^25
and with interpolating polynomials replacing integral formulations. The equation describes the rate of change of the mass of water, which is also the rate of change of the mass of the droplet.
can be re-written using the density,
, of the solution as
is a contact angle dependent factor in the iso-thermal model drop lifetime previously defined and discussed in Armstrong
et al.^26
This factor has a maximum value of
= 90° and decreases by not more than 11% of that value over the range 40° ≤
≤ 180°. For this reason, the average evaporation rate and drop lifetime in the isothermal droplet of the water model tends to be insensitive to the precise value of the contact angle (provided it is
sufficiently high) giving a quasi-constant contact angle mode of evaporation. In contrast, as the contact angle decreases, the correction factor,
), to the evaporation rate in Eq.
increases, and so their product increases.
Figure 1
shows the evaporation rate factor,
, and the dependence on the contact angle,
. The dashed line in
Fig. 1
shows the iso-thermal model contact angle dependent evaporation rate factor with the limits for a ±10% variation shown as the shaded orange region. In contrast, the solid blue line showing the
non-isothermal contact angle dependent evaporation rate factor,
, has a stronger variation in the higher contact angle range from 40° to 120°. Physically, this is as expected because the effect of evaporative cooling is strongest at higher contact angles, where
the distance to transfer thermal energy from the substrate through the droplet to the droplet–vapor surface is larger on average.
On a SOCAL surface with an initial contact angle of
= 104°, the changes as the contact angle reduces to 60° and 40° are 2.8% and 10.8%, respectively, for the isothermal model. However, for the non-isothermal model, the changes are 10.5% and 18.8%,
respectively. Thus, for a droplet of pure water, Eq.
predicts that the slope of a graph of
is approximately constant to within ±10% for contact angles above 40° but increases rapidly as the contact angle reduces below 40°. This corresponds to a quasi-constant contact angle mode of
evaporation with a contact area that changes linearly with time. Thus, we expect the initial evaporation rate behavior to follow a quasi-constant contact angle regime with
where the angular brackets 〈⋯〉 imply an average value over the relevant evaporation time and
are the initial density and volume of the droplet, respectively.
E. Time dependence of evaporation of droplets of salt solutions
For droplets of pure water, the water activity
and density
remain constant at their initial values, and so if the contact angle reduces as evaporation proceeds, the slope in a graph of
will become larger due to the contact angle factor,
). For sodium chloride (NaCl) salt solutions, the density term, 1/
, decreases by at most ∼6% and 1/
by at most ∼17% as the concentration in a droplet increases. This decrease in the density factor partially compensates for increases in the slope of Eq.
caused by the contact angle dependent factor,
). For solutions of NaCl, the water activity
also decreases as the salt concentration increases
Fig. 2
) and can be described by a quadratic fit valid to within 3 decimal places for temperatures between 15 and 50 C.
is the concentration (by % w/w). In regime I of Seyfert
et al^10
< 0.75), the water evaporates completely from the droplet without ever reaching a value of
giving an equilibrium with the surrounding relative humidity and so leaves a salt deposit. In this case, as a droplet approaches the end of its lifetime, we would expect the slope in Eq.
to become steeper due to the contact angle dependent factor
). In regimes II and III, the droplet eventually reaches an equilibrium volume due to the term (
) and so the slope in Eq.
should become shallower.
A. Surface preparation
The different surface coatings used in our experiments were SOCAL, PEGylated, Glaco, polytetrafluoroethylene (PTFE), and SU8. We also used a bare glass substrate as a high hysteresis low contact
angle substrate. They were all chosen to give a wide range of different static contact angles and contact angle hysteresis values (Table I).
TABLE I.
Surface coating . Contact angle (°) . CAH (°) . No. of droplets studied . Droplets producing crystals at t < 8000 s) .
SOCAL 105 <2 25 0
PEG 35–45 0.5–3.5 25 7
Glaco 170 5 5 0
PTFE 120 12 3 0
SU8 75 25 4 1
Glass 9.5 High 3 3
Surface coating . Contact angle (°) . CAH (°) . No. of droplets studied . Droplets producing crystals at t < 8000 s) .
SOCAL 105 <2 25 0
PEG 35–45 0.5–3.5 25 7
Glaco 170 5 5 0
PTFE 120 12 3 0
SU8 75 25 4 1
Glass 9.5 High 3 3
To achieve a constant contact angle evaporation, which requires surfaces that exhibit no contact line pinning, we used SOCAL surfaces. The surfaces are based on the work of McCarthy et al. using
modifications by Armstrong et al. and are smooth, hydrophobic (CA > 90°), and have low CAH (∼2°).^11,12 These surfaces were produced by plasma treating a chemically cleaned glass slide to activate
the surface. The slide was placed in a plasma cleaner (Henniker) at 30% power for 20 min using air as the reactive gas. The slide was then dip coated in a reactive solution made of a 100:10:1 mass
ratio of isopropyl alcohol, dimethyoxydimethylsilane, and sulfuric acid, respectively. The slide was dipped in the solution and held for 10 s. It was then withdrawn at 3 mm s^−1 before being left to
react in a chamber at a relative humidity of 60 ± 1% for 20 min. The excess unreacted material was washed away with deionized water, isopropyl alcohol, and toluene and then air dried. The surfaces
were characterized by an averaged value of three hysteresis measurements and were deemed to be fit for use in the evaporation experiments when they achieved the values of <2° CAH and standard
deviation of < ± 0.5°.
To achieve a CCA mode of evaporation on a smooth and hydrophilic surface (CA < 90°), we used a PEGylated coating with low CAH (∼3°);^16,30 we refer to these PEGylated surfaces as our PEG surfaces
throughout. The PEG surfaces were produced by activating the surface of a cleaned glass slide in a plasma oven for 40 min at 60 W power. The activated surfaces were then reacted in a reagent solution
of toluene, 2-[methoxy(polyethyleneoxy)6-9propyl]trimethoxysilane, and hydrochloric acid by immersing the sample in a bath of the reagent for 18 h. The samples were then rinsed with deionized water,
isopropyl alcohol, and toluene to remove any unreacted material.
The superhydrophobic Glaco sample was made by spraying a cleaned glass slide with a Glaco Mirror Coat^TM and leaving it to dry in air for an hour. The spraying process was repeated 5 times to build a
more uniform structure across the sample. The Glaco coated surfaces are referred to as Glaco surfaces throughout.
The polytetrafluoroethylene (PTFE) sample was prepared by spinning a PTFE solution (Teflon^™ AF1600) onto a cleaned glass slide at 500 rpm for 10 s and then 2000 rpm for 1 min and by baking at 155 °C
for 20 min. The samples coated with SU8 photoresist were made by spin coating SU8-3035 on a 3″ silicon wafer at 500 rpm for 10 s and then 2000 rpm for 30 s. They were then baked at 95 °C for
10 min flood exposed at 5000 mJ/cm^2 and post-baked for 2 min at 95 °C
For experiments that used a bare glass substrate, the glass slide (Sigma-Aldrich) used was taken fresh from the packet. The hysteresis of the glass slide was high, and it could not be characterized
through the same stringent method as the other samples due to the high wettability and low contact angle of the surface.
B. Salt solution preparation
NaCl solutions used 99.9% pure reactant grade sodium chloride salt (Sigma-Aldrich) and ultra-pure water (Sigma-Aldrich). 10 ml solutions were prepared in a 12 ml scintillation vial, and the solution
was mixed using a vortexer (Fisher Mini Vortexer) for 2 min continuously to ensure homogeneous dissolution of the salt.
C. Contact angle hysteresis measurements
The pinning-free properties of these surfaces were characterized through CAH measurements. These measurements were made at high relative humidities in line with the contact-line relaxation method
demonstrated by Barrio-Zhang et al.^31
D. Droplet imaging, deposition, and humidity control
To control the evaporation conditions around the droplet, a bespoke droplet shape analysis experiment was constructed. The experiment uses a 12 MP camera (Raspberry PI) with a microscope lens
(Raspberry Pi C-mount microscope lens) to capture an image of the droplet. The recording quality was 1080p to reduce analytical errors, and single images were captured at 10 s intervals (0.1 FPS).
The open source software used (PyDSA) tracks the contact radius, r, throughout the evaporation and calculates the contact angle, θ, from a fit to a third-degree polynomial between the spherical cap
fitting and the set baseline. From these data, the volume of the droplet can be calculated assuming the axial symmetry. A microfluidic syringe pump (Cellix Exigo) is used to accurately dose the
droplets of 4 ± 0.4 μl. A PID temperature-controlled stage (Thorlabs, PTC1/M) was used to control the temperature, and a bespoke humidity controller was used to control the humid environment. The
temperature of the surface was regulated to ±0.2 °C, and the glass surface of the samples was assumed to be in thermal equilibrium with the heated surface. The humidity controller was regulated to
±1% relative humidity. We used four air inlets and the hole for the needle as the outlet. This prevents any humidity gradients from forming within the chamber.
E. Single droplet water evaporation on SOCAL
To determine that CCA evaporation was achievable on SOCAL surfaces, we used water as a baseline for the evaporation dynamics. An amount of 4 μl droplets of ultra-pure water (Sigma-Aldrich) was
evaporated on SOCAL surfaces at 25 °C and 60% relative humidity (red squares). The captured images of the droplets were analyzed using a bespoke open-source tool (pyDSA) to calculate the
instantaneous contact angle and contact radius. Using the contact radius data and contact angle, assuming a spherical cap of the droplet, we can calculate the volume of the droplet numerically. From
these data, the evaporation dynamics were analyzed and the evaporation mode was determined by plotting $(V/V0)2/3$ as a function of time. According to Eq. (11), this plot should show a linear
relationship for CCA evaporation. By first ensuring that a CCA evaporation is achievable using water as the fluid, we can observe the effects of the non-pure liquid droplet as the surface behavior is
already known.
F. Single saline droplet variable humidity evaporations
To evaluate the response of saline solutions to different humidities, droplets of 0.108 weight fraction saline solution were taken at different relative humidities. A 4 μl droplet of 0.108 weight
fraction solution was evaporated at 40%, 55%, 60%, 65%, and 75% relative humidity and at a fixed temperature of 25 °C. Each different droplet was evaporated at a different location on a SOCAL sample.
The evaporation dynamics were observed in real-time and performed until the volume of the droplet remained stationary.
G. Single saline droplet variable concentration evaporations
To investigate the effects of water activity on evaporation, droplets of varied concentrations from 0.008 to 0.108 weight fraction were evaporated at 60% relative humidity and 25 °C. Each condition
was repeated three times and evaporated for at least 8000 s. The experiments were all run on the same SOCAL sample and performed non-sequentially to prevent any aging effects of the surface.
H. Single saline droplet variable surface evaporations
To study the effect of the contact angle and contact angle hysteresis on whether the droplet is crystallized or not, we used a variety of surfaces with different contact angle and contact angle
hysteresis properties (Table I) and evaporated droplets under the same conditions as for droplets on SOCAL coated glass.
I. Multiple droplet (multiplexed) crystallization study
PEG and SOCAL surfaces have very low contact angle hysteresis but very different wettability. To test the effect of the surface on the stability of the droplets, we carried out a study involving the
evaporation of arrays of multiple droplets. The conditions were kept the same as in the previous study; however, the dosing and observation of the evaporations changed, for the SOCAL, arrays of 10 or
5 droplets of 4 μl were deposited in a zig-zag pattern to relatively reduce their interference of the local humidity around the droplet (approximately one droplet diameter apart).The distance between
the droplets at this distance coupled with the long duration of the experimental observations ensures that at the end of the observations, all saline droplets are in equilibrium with the surrounding
vapor phase. These arrays were then left for 9000 s, an extra 1000 s to the singular droplet for extra confidence in the assertions. After the 9000 s, the number of crystals in the array was counted.
For the PEG surfaces, five droplets were deposited and observed under the same conditions and once 25 droplets had been observed for each surface, we compared the number of crystals deposited. The
surfaces were chosen for their comparative hysteresis values (see Table I).
A. Response of saline droplets to humidity
To investigate the ability of smooth low contact line pinning surfaces such as SOCAL to mitigate the heterogeneous nucleation of crystals, we first performed the experiments to determine how the
relative humidity of the chamber affects the evaporation of the droplets and crystal formation. We placed 0.108 weight fraction droplets of initial volume V[o] = 4 µl on a smooth SOCAL surface (CA
105 ± 1°, CAH 1.11 ± 0.28) and adjusted the relative humidity to 55%, 60%, 65%, and 75% (preliminary results on 40% and 50%; all crystallized). Figure 3(a) shows$V/Vo2/3$ as a function of time for
each different humidity. For a droplet on a SOCAL surface, which has been shown to be smooth and stable,^11 it is possible to create droplets (b) with a stable volume for at least 8000 s. As shown in
Fig. 3(a), the droplet reaches an equilibrium volume with the evaporation rate eventually reaching zero. This is consistent with Eq. (11) and the $aw−Hr$ term reaching zero as the water activity
becomes equal to the surrounding relative humidity. The lower humidity stable droplets including 55%, 60%, and 65% suggest that the water activity at the surface of the droplet can be much lower than
the limiting value of H[r] = 0.76 previously mentioned for a sessile droplet.^10
It can also be seen that as expected, the volume at which the droplet stabilizes is dependent on the relative humidity. This shows that SOCAL surfaces are able to mitigate heterogeneous nucleation
and stabilize the droplets of the constant volume at a humidity as low as 55% which is much lower than the expected deliquescence limit of ∼75%.^8
B. Effect of the initial concentration on the stable volume
The final volume of an evaporating droplet is also determined by the weight fraction of salt in the solution at which the water activity has equalized with the relative humidity. This final volume
will, thus, be dependent on the initial weight fraction of salt in the droplet. Therefore, varying the initial weight fraction of salt in the solution leads to different final volumes. Figure 4 shows
$(V/V0)2/3$ as a function of time for droplets with different initial weight fractions. The red squares show the calibration experiment for water on SOCAL, showing a CCA diffusion limited
evaporation. Figure 7(a) shows that it is possible to stabilize droplets with an initial weight fraction as low as (0.008) (brown diamonds) and that increasing the weight fraction increases the
final, stable volume. Figure 7(b) shows that the stable droplets are indeed liquid solutions even when the final volume is only 3% of the original volume. We note that perturbing the stable droplets
with the needle did not induce crystallization and that the droplets did remain liquid [see the supplementary material, Fig. S4 (Multimedia View)]. In addition, it is shown that for low weight
fraction, the droplets follow the diffusion limited model of evaporation [Eq. (11)]^13 and only deviate as the volume gets close to the stable volume. However, for much higher initial weight
fraction, the droplet deviates from diffusion limited evaporation almost immediately and follows a different dynamic path to the stable volume.
C. Evaporation on different surfaces
We assume that for sessile droplets in contact with a surface, crystal nucleation takes place due to micro or nanoscopic surface roughness. SOCAL surfaces have been shown to be very smooth and to
have very low contact angle hysteresis hindering crystal nucleation for relative humidity concentrations below that established by its water activation coefficient as shown in earlier Sec. IV A. To
test the ability of other coated surfaces to mitigate heterogeneous nucleation, we recorded evaporation sequences for 4 µl droplets of a 0.108 weight fraction salt solution, at 60% relative humidity,
on several different substrates, namely, PEG, PTFE, Glaco, and SU8 with static contact angles and contact angle hysteresis reported in (Table I). Figure 5 shows the outcome of evaporation sequences
on these surfaces. $(V/V0)2/3$ as a function of time is shown in the central graph panel, and the left and right image strips show the initial droplet and the final droplet shapes, respectively. From
the evolution of the volume in time data reported in Fig. 5, there is a rather good agreement and all of the data collapse for substrates with contact angles equal to or below 120°. In addition, we
can see from the volumetric data that we were able to stabilize saline droplets, without crystallization and for at least 8000 s, on all of the surfaces. For PEG, SOCAL, SU8, and PTFE coated
surfaces, the static average contact angle during the evaporation is between 38° and 110° [see Fig. 5(b) inset]. For the Glaco superhydrophobic surface, the average contact angle is ≈150°. According
to Fig. 1 for the non-isothermal model of evaporation, we would expect to see a change in the evaporation rate of ∼20% between 38° and 110° (Fig. 1). This is not very evident for PEG, SOCAL, PTFE,
and SU8, during the volume loss phase of the evaporation in Fig. 5. This could be because for the 20% change in the evaporation rate expected on these surfaces, the water activity is the dominant
effect controlling the evaporation rate. There is, however, a significant change in the evaporation rate for the droplet on the Glaco coated surface. Although 150° is outside the range of the plot in
Fig. 1, we can see that the evaporation rate begins to significantly deviate from the isothermal model and slows the evaporation as the contact angle gets very large. This would account for the
significant change in the evaporation rate seen for the droplet on a Glaco coated surface.
The ability to stabilize a droplet on different surfaces is more difficult to achieve as the hysteresis values increase. To test our hypothesis that low CAH surfaces with constant contact angle
evaporation may suppress the occurrence of crystallization, we performed studies using the droplet arrays on the SOCAL and PEG surfaces (see experimental methods). Table I shows the outcome of these
experiments, along with data for the experiments on other surfaces with >2° CAH.
For SOCAL surfaces, 25 of the 25 droplets in the array remained stable and showed no crystallization for at least 9000 s, and in one experiment, we were able to stabilize a single droplet for at
least 20 320 s (see supplementary Figs. S2 and S3). For PEG surfaces, the number of stable droplets dropped to 18 of the 25. The difference in the fraction of the droplets that formed crystals might
be attributed to a much larger contact footprint giving a higher probability of nucleation sites and/or to the different spatial distribution of the evaporative flux generating local concentrations
gradients near the contact line.^32 Although Glaco and PTFE surfaces showed no crystals and stable droplets for every experiment (only 3 of each) at t < 8000 s, they both proved that it is very
difficult to produce stable droplets in the long term. We were not able to stabilize a droplet on any of the glass surface experiments.
D. Weight fraction and water activity analytical model
The nature of the evaporation paths of droplets toward the final stable volume can be explained by the increasing concentration as the water evaporates, which, in turn, decreases its activity. The
percentage weight fraction of a droplet of known molarity is given by
is the molarity of the solution used in the experiment,
is the molar mass of the salt,
is the initial volume of the droplet,
is the dynamic volume of the droplet, and
is the density of the droplet solution. To analyze how the concentration and water activity vary as a function of the droplet volume, we use an upper and lower bound for the density in Eq.
Figure 6(a) shows an analytical plot of the weight fraction as a function of (V/V[o])^2/3 for droplets with different initial weight fractions matching those in the experiments. The bands represent
the space bounded by how weight fraction changes given a minimum and maximum density. In the case of saline droplets, the minimum possible density would be that of pure water (1000 kg/m^3) with no
salt in the solution (black dotted line on each band). Assuming that creating a very smooth surface mitigates heterogeneous crystallization, we assume supersaturation of the droplets, as previously
mentioned. Since the earlier experiments were performed at a relative humidity of 60%, corresponding to H[r] = 0.6, the weight fraction of NaCl at which the water activity matches the relative
humidity is 0.4. Thus, the upper limit of the density used in this plot is 1295 kg/m^3, equivalent to a saline solution of 0.40 weight fraction (this value is extrapolated from density vs weight
fraction data which is plotted between 0 and 0.26 weight fraction; see the supplementary material, Fig. S1).^29 The bands in Figs. 5 and 6 show the regions bounded by using these two densities in Eq.
By substituting the weight fraction as a function of density [Eq. (13)] into the quadratic fit to water activity [Eq. (12)], we can see how the water activity changes as a function of volume [Fig. 6
(b)]. The red dotted line in Fig. 6(b) is the water activity at the expected deliquescence limit at a[w] = 0.75, and the green line is the water activity at a[w] = 0.60. Using the analytical
solution, it is possible to predict the final volume for a stable volume droplet on the SOCAL surface and compare it with the experimental measurements. The crossing points of the different initial
concentration bands with the horizontal dotted green line should coincide with a steady state volume of the final droplet.
Figure 7 shows the data for all the experiments for different surfaces as well as the experiments for different concentrations of the saline solution. The yellow filled region shows the upper and
lower bounds which are determined using the same densities as reported in Fig. 6, this shows that the theoretical prediction is in good agreement with the experimental data. Figure 7(a) shows the
final volume of the droplet when it is stable as a function of the initial weight fraction. The blue filled circles show the concentration experiments carried out only on SOCAL surfaces, and the
orange triangles show the experiments for different surfaces for which the highest of the saline concentrations was chosen. The data in the experiments are in good agreement with the analytical
model; hence, we can predict the final volume of the droplets. From the surface data, it is also clear that this model works for different initial contact angles. Figure 7(b) shows the values of the
concentrations of the final droplets as a function of the initial weight fraction. This concentration is calculated using Eq. (13) with the density set to 1147.5 kg/m^3, which is the average density
used in the model. The horizontal red dotted line in Fig. 7(b) shows the expected saturation concentration of 26% for an NaCl solution. All the data in Fig. 7(b) show that the droplets that remain on
the surface are supersaturated when stable. According to the model, we would expect the droplet to stabilize at a concentration of 0.40 (green dotted line) weight fraction salt (equivalent to 0.60
weight fraction water) if the water activity has reached equilibrium with the surrounding environment. In all experiments, the final concentration of the stable droplet is above the expected
concentration of 26%. We note that the smallest initial concentration has a large error bar because, as shown in Fig. 6(a), a very small error in volume calculation, arising from axisymmetric
assumptions, can lead to large errors in the weight fraction. This effect becomes smaller as the initial concentration increases.
In this work, we have shown that it is possible to create sessile droplets of a supersaturated sodium chloride salt solution whose volume remains constant on different solid surfaces and at different
relative humidities without crystallization over significant periods of time. This relative humidity, at which the droplet stabilizes, is above the expected equilibrium relative humidity for a NaCl
solution and appears to occur when the water activity has equalized with the surrounding humidity. We have also shown that on a smooth surface of SOCAL, with a relatively high static contact angle
(105°) and a low contact angle hysteresis (<2°), it is possible to create stable droplets at a relative humidity as low as 55%. At a slightly higher relative humidity (60%), this is a highly
reproducible effect. On PEG surfaces, we have achieved similar results to that of SOCAL at relative a relative humidity of 60% but with slightly lower reproducibility of stable droplets on larger
timescales. We have shown that, although more difficult, other surface coatings are capable of creating stable droplets at 60% relative humidity. On superhydrophobic surfaces, we show a significant
deviation from the diffusion limited evaporation model throughout the droplet lifetime and we attribute this to a lower water activity and non-isothermal evaporation. The deviation from diffusion
limited evaporation to a stable droplet is due to the decreasing water activity within the droplet and due to an increase in the concentration of salt as the droplet evaporates. We have also
presented an analytical model that predicts the final stable volume of the droplets and have shown that it is in good agreement with the experiments.
In the supplementary material, we provide the details of a new interpolation for the function f(θ) from the data of Stauber et al., to take account in a common notational format of the dependence of
the concentration gradient of the vapor, between the surface of the droplet and its surroundings, on the contact angle arising from different models. We also provide an interpolation for the
non-isothermal evaporation factor for angles greater than 120°. The data plot of the saline solution density as a function of concentration and the extrapolation used in the analytical model in Eq.
(13) is included, and we also include an extra plot of the very long evaporation experiment mentioned in Sec. IV C as well as the droplet images from the same experiment.
GM is grateful to Dr. Yongpan Cheng for data and discussions on the evaluation of the correction factor for the evaporation rate of a non-isothermal droplet. AJ would also like to thank Dr. Hernán
Barrio-Zhang and Michele Pelizzari for providing PEGylated and Glaco surfaces, surface preparation training, and valuable insight. SA would like to acknowledge the support of the Engineering and
Physical Sciences Research Council (Grant Number EP/T025158/1).
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Alex Jenkins: Conceptualization (supporting); Data curation (lead); Formal analysis (lead); Investigation (equal); Methodology (equal); Software (equal); Writing – original draft (equal); Writing –
review & editing (equal). Gary G. Wells: Conceptualization (equal); Methodology (lead); Project administration (lead); Supervision (lead); Writing – original draft (lead); Writing – review & editing
(lead). Rodrigo Ledesma-Aguilar: Conceptualization (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). Daniel Orejon: Supervision (equal); Writing – review &
editing (equal). Steven Armstrong: Conceptualization (supporting); Data curation (supporting); Formal analysis (supporting); Methodology (supporting); Writing – review & editing (equal). Glen McHale:
Conceptualization (equal); Investigation (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Published open access through an agreement with University of Edinburgh | {"url":"https://pubs.aip.org/aip/jcp/article/158/12/124708/2881833/Suppression-of-crystallization-in-saline-drop?searchresult=1","timestamp":"2024-11-03T18:17:34Z","content_type":"text/html","content_length":"655209","record_id":"<urn:uuid:59d7bacc-9562-4eeb-81e1-8932a6fa59b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00410.warc.gz"} |
Arc Lengths
Lesson Video: Arc Lengths Mathematics • First Year of Secondary School
In this video, we will learn how to find the arc length and the perimeter of a circular sector and solve problems including real-life situations.
Video Transcript
In this video, weβ ll learn how to find the arc length and the perimeter of a circular sector and solve problems including real-life situations. But letβ s first begin by recalling how we describe
parts of a circle.
An arc of a circle is defined as a section of the circle between two radii. So here we have a radius joining the circle at a point π ΄ and another radius joining the circle at point π ΅. However,
as we look at the circle, we might notice that, in fact, there are two arcs. We have this smaller arc in the shorter distance between π ΄ and π ΅, and then we have this larger arc. Both of these
arcs would be defined in the same way in that theyβ re both sections of the circle between two radii. We can get around this problem of definition by defining that the smaller arc is called a minor
arc and the larger arc is called the major arc. If we ever have the situation where the two radii are, in fact, the diameter or the central angle is 180 degrees or π radians, then we would say
that we have two semicircular arcs.
Now we can have a think at how we would actually find the length of any of these arcs. Letβ s take this example problem. We have two radii creating a section of the circle and the angle at the
center here is given as 90 degrees. What we want to do is work out the length of this minor arc. In order to help us, we can remember that the circumference, thatβ s the distance around the outside
of the circle, is calculated by two times π times the radius. But we donβ t actually want to calculate the whole way around this circle. We only want this section. And we know that this section
must be one-quarter of the whole circle. So we multiply one-quarter by two times π times the radius.
This method works for any given central angle, if instead of having a 90-degree angle we had an angle of π degrees and so the circumference would be multiplied by this proportion π over 360. We
can define this more formally by saying that the length of an arc which subtends an angle π measured in degrees in a circle of radius π is given by arc length equals two π π π over 360.
You might notice this terminology that the angle π is measured in degrees because, of course, there are other ways in which we can measure angles. One of these ways is by using a measure of
radians. And we can also find the arc length when the angle is given in radians. If this central angle is measured as π radians, then by remembering that there are two π radians in 360 degrees,
that means that we multiply the circumference two π π by the proportion of π over two π . We can simplify this calculation by taking out a factor of two π from the numerator and
denominator, which leaves us with π π .
So now we have another similar definition. This time, the length of an arc when the central angle is measured in radians is given by arc length is equal to π π . We can apply either of these
formulas, depending on whether the angle measure is given in degrees or radians. In the first example, weβ ll see how we can find an arc length when the angle is given in radians.
Find the length of the blue arc given the radius of the circle is eight centimeters, and the angle measure shown is in radians. Give the answer to one decimal place.
In this question, we need to calculate the length of this blue arc, which is the larger of the two arcs; thatβ s also called the major arc. Weβ re given that the angle measure is in radians. We can
recall that the length of an arc subtending an angle π measured in radians in a circle of radius π is given by arc length equals π π . We can then simply plug in the information that weβ
re given. The radius is eight centimeters, and the angle measure is four π over three. And we multiply these together. This gives us 32π over three. And because itβ s a length, the units will
be in centimeters.
We could leave our answer in this form. However, this question asks us to give the answer to one decimal place. So weβ ll need to use our calculators. This gives us the values 33.510 and so on
centimeters, which we can round to one decimal place as 33.5 centimeters. And so, we can give the answer that the length of the blue arc is 33.5 centimeters.
In the next question, weβ ll see how we can find the length of an arc in a real-world context.
A pendulum of length 26 centimeters swings 58 degrees. Find the length of the circular pathway that the pendulum makes giving the answer in centimeters in terms of π .
In this question, weβ re given that thereβ s a pendulum, which is 26 centimeters long. This means that the length of string from the pivot point here at the top to the ball at the end is 26
centimeters. Weβ re told that the angle that this pendulum swings through is 58 degrees. And weβ re told that it swings in a circular pathway. We could draw a smaller diagram of the pendulum, which
allows us to say that if this pendulum was to swing the entire way round, it would in fact create a circle. The length of the string, which is 26 centimeters, would in fact be the radius of the
The length that we need to work out is marked in green and thatβ s an arc of the circle. Because weβ re given that the central angle is a measurement in degrees, then we use this formula that the
arc length of a circle of radius π , with a central angle π degrees, is given by arc length equals two π π π over 360. We can remember that this formula is a result of multiplying the
circumference, which is two π π by this proportion of π over 360 degrees.
Now all we need to do is plug in the values that the radius is 26 centimeters and π , the central angle, is 58 degrees. If we wish, we can take out this common factor of two before we simplify to
give us the answer that the arc length is 377 over 45π centimeters. In some questions, we might be asked for a decimal approximation for the length. However, this question asks us for the length
in terms of π . Therefore, we leave the answer as it is. So the circular pathway has a length of 377 over 45 π centimeters.
Weβ ll now have a look at how we can use what we know about finding the arc length to find the perimeter of a circular sector. So far, weβ ve seen that there are two alternative formulas to find
the arc length of a sector. And those two formulas are dependent on whether the central angle is given in degrees or radians. But sometimes, of course, we might need to find the perimeter of a
sector. And remember that thatβ s just the distance around the outside. Because the two extra lengths are both radii of the circle, then to find the perimeter in each case, whether itβ s in degrees
or radians, then we simply add on two π to the calculation for arc length. Letβ s have a look at an example of how we do this.
The radius of a circle is seven centimeters and the central angle of a sector is 40 degrees. Find the perimeter of the sector to the nearest centimeter.
Letβ s begin by sketching this circular sector. In order to find the perimeter, thatβ s the distance around the outside of the sector, weβ ll have these two straight lengths, which will both be
radii of the circle, along with this length, which is the arc of the circle. To find the arc length of a circle when the central angle π is given in degrees, we calculate two π π π over 360
where π is the radius. We can then plug in the values into this formula. The radius is seven centimeters and the central angle is 40 and the simplified answer will be 14π over nine. And because
this is a length, then the units will be centimeters.
Remember that this is simply just the arc length that weβ ve calculated and we still need to work out the value for the perimeter. To calculate the perimeter, we take the arc length, which weβ ve
kept in terms of π to give us the most accurate answer. And then we add on two times the length of the radius, which is two times seven. When we calculate 14π over nine plus 14, we could keep
the answer in terms of π , but this time weβ re asked for the answer to the nearest centimeter, so weβ ll need to find a decimal equivalent for the value of the perimeter, which is 18.886 and so
on centimeters. Rounding this value to the nearest centimeter gives us that the perimeter of this sector is 19 centimeters.
Weβ ll now have a look at an example where weβ re given the perimeter of a sector and we need to calculate the radius.
The perimeter of a circular sector is 67 centimeters and the central angle is 0.31 radians. Find the radius of the sector giving the answer to the nearest centimeter.
We can sketch this circular sector as shown with its central angle of 0.31 radians. Weβ re given that the perimeter of this sector is 67 centimeters. And we remember that the perimeter is the
distance around the outside. To find the perimeter, weβ d have these two straight lengths, which will be the radius of the circle which we can define as π , plus this outer edge, which will be the
arc length. We can define this arc length with the letter π . To calculate the perimeter then, we would have two times the radius two π plus π . Given the information that the perimeter is 67
centimeters, we can write the equation that 67 equals two π plus π .
We canβ t do much with this equation at the minute because we donβ t know the value of π , the radius. In fact, thatβ s what we need to calculate. So letβ s see if we can do anything with π ,
which is the arc length. We remember that to find the length of an arc subtending an angle π in radians in a circle of radius π , then we calculate arc length equals π π . Here, the arc
length π is equal to π π and we know that π is 0.31 radians. We can then substitute π is equal to 0.31π into the equation above. This gives us 67 is equal to two π plus 0.31π ,
which simplifies to 67 is equal to 2.31π . Then to find the value of π , we divide both sides by 2.31.
We could leave our answer as a simplified fraction, but because weβ re asked for the answer to the nearest centimeter, letβ s find a decimal approximation. So π is equal to 29.004 and so on, and
because this is a length and weβ re dealing with centimeters, then the radius π will also be in centimeters. Rounding this to the nearest centimeter then gives us the answer that the radius of
this sector is 29 centimeters.
Weβ ll now have a look at one final example where we use information about tangents intersecting to find the length of an arc.
If the measure of angle π ΄ equals 76 degrees and the radius of the circle equals three centimeters, find the length of the major arc π ΅π Ά.
Letβ s start by filling in the information that weβ re given. The measure of angle π ΄ is 76 degrees and the radius of the circle is three centimeters. We remember that an arc of a circle is a
section of the circumference between two radii. In fact, here we would have two arcs, which could both be called arc π ΅π Ά. This is why, when weβ re dealing with arcs, the larger arc will be
referred to as the major arc and the smaller arc is called the minor arc. In this question, weβ ll need to calculate the length of the major arc. In order to work out either the minor or the major
arc π ΅π Ά, we would need to establish the measure of the central angle subtending the arc.
Letβ s see if we can work out this angle π by using the information about the tangents. We can recall that a tangent to the circle at a point π meets the radius of the circle from π at 90
degrees. This means that weβ ll have a 90-degree angle here at π Ά and a 90-degree angle at π ΅. If we labeled the center of the circle with the letter π , then we might observe that we have in
fact got a quadrilateral π ΄π ΅π π Ά. We know that the sum of the internal angles in a quadrilateral is 360 degrees. This means that we can write that the measure of the four angles in the
quadrilateral π ΄, π ΅, π , and π Ά must add up to 360 degrees.
We can then plug in the angle measurements: π ΄ is 76 degrees, π ΅ is 90 degrees, and π Ά is also 90 degrees. Simplifying, we have that 256 degrees plus the measure of angle π is 360 degrees.
Subtracting 256 degrees from both sides then gives us that the measure of angle π is 104 degrees. Now that weβ ve found this angle π , the measure of angle π , as 104 degrees, we can
calculate an arc length. Notice, however, that if we use the angle of 104 degrees, then the arc length that we calculate will be the length of the minor arc. So there are two ways to approach this
problem and find the length of the major arc instead.
The first way to approach this problem is by considering what this reflex angle π would be and using that directly to calculate the length of the major arc. Well, because the angles about a point
add up to 360 degrees, then if we subtract 104 degrees from 360, we get 256 degrees. That means that the central angle in the major arc π ΅π Ά will be 256 degrees. To calculate the arc length of a
circle of radius π with a central angle of π measured in degrees, then we calculate arc length equals two π π π over 360. We then simply substitute in the information. The radius π is
given as three centimeters. And we know that this central angle π is 256 degrees.
Simplifying this gives us the arc length as 64π over 15 centimeters. We can keep this answer in terms of π , or we can find the decimal equivalent as 13.404 and so on centimeters, which rounded
to one decimal place would give us the length of π ΅π Ά as 13.4 centimeters. Letβ s make a note of this answer and have a look at the alternative method to calculate the major arc. Letβ s return
to the point in our working where we had calculated that this obtuse angle π is 104 degrees. Instead of immediately calculating the major arc length, letβ s calculate this smaller arc, the minor
arc length. The value of the radius that we plug into the formula will still be the same at three centimeters, but the central angle this time will be 104 degrees. When we calculate this and
simplify, we get an answer of 26π over 15 centimeters.
So how do we go about getting from the length of the minor arc to the length of the major arc? Well, the relationship between the major and the minor arc is that if we add these together, we would
get the circumference of the circle. Thatβ s the distance around the outside edge. The circumference is calculated by two times π times the radius. In this case, as the radius is three, weβ d
have two times π times three, which is simplified to six π centimeters.
So now, to calculate the major arc length π ΅π Ά, we have the circumference subtract the minor arc length π ΅π Ά. When we substitute the values six π minus 26π over 15 and simplify, we get
the value 64π over 15 centimeters. This gives us the same decimal approximation as we found before, 13.4 centimeters. Therefore, we have confirmed the answer that, to one decimal place, the length
of the major arc π ΅π Ά is 13.4 centimeters.
We can now summarize the key points of this video. We began by defining that an arc of a circle is a section of the circumference of a circle between two radii. We saw that the larger of two arcs is
the major arc and the smaller is the minor arc. We can use the size of the central angles to help us define if the arc is major or minor. Next, we saw how we can derive two formulas for the length of
an arc, depending on whether the central angle π is measured in degrees or radians. Finally, we saw that the perimeter of a sector is the sum of the length of two radii, along with the arc that
makes the sector. | {"url":"https://www.nagwa.com/en/videos/916176970575/","timestamp":"2024-11-03T00:19:59Z","content_type":"text/html","content_length":"283845","record_id":"<urn:uuid:9f856257-d9ab-4757-82bd-34b014a6449e>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00242.warc.gz"} |
Subtracting Integers
We often think of subtraction as "taking away." However, when we work with integers, subtraction can look quite the opposite.
Subtraction can be done by adding the opposite. Let's take a look at a simple example to see how the two are related.
Example: 5 - 3
We can write this as 5 + (-3). Both 5 -3 and 5 + (-3) = 2. When we subtract a positive number, we move to the left on the number line. This is the same thing that happens when we add a negative
We can use this to help us subtract negatives.
Example: 6 - (-9)
Just like in the example above, we can change this question to adding the opposite.
6 - (-9) becomes 6 + 9 = 15. You might be wondering how we can subtract and end up with a larger number than we started with. Let's look at the number line. When we subtracted a positive number we
moved to the left. So when we subtract a negative number we need to move to the right.
In the number line, you can see that when a negative number is being subtracted, we actually move towards the larger numbers on the number line. Drawing out the number line can be a little tedious
every time you go to subtract. So we can use a little saying to help us remember how it works.
Keep Change Change or KCC
This means to
the first number the same.
the subtracting to adding. Then
the sign of the second number.
Check it out:
17 - (-5)
Keep the 17. Change the minus to a plus. Change the -5 to a positive 5.
Here is another:
-18 - 5
Keep the -18. Change the minus to a plus. Change the positive 5 to a -5.
One last example:
-23 - (-11)
Keep the -23. Change the minus to a plus. Change the negative 11 to a positive 11.
From these examples, we can see that subtracting with integers is the same as adding the opposite. We can solve using a number line or by using the saying "Keep, Change, Change" to help us solve. It
is important to remember that we can subtract a negative and end up with a larger number than we started with.
Related Links: Math Fractions Factors | {"url":"https://softschools.com/math/topics/subtracting_integers/","timestamp":"2024-11-03T20:11:28Z","content_type":"application/xhtml+xml","content_length":"17815","record_id":"<urn:uuid:92d49da7-473f-4b4f-bf23-90fdbf0fb4e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00198.warc.gz"} |
Convert Binary Numbers to Decimal Numbers
What is the decimal equivalent of the binary number 1100 1001?
The decimal equivalent of the binary number 1100 1001 is -28. In binary to decimal conversion, each bit of the binary number represents a power of 2. Starting from the rightmost bit, the values of
the bits are calculated by multiplying the bit with 2 raised to the power of its position from right to left. In this case, the decimal conversion follows as: 1 * 2^7 + 1 * 2^6 + 0 * 2^5 + 0 * 2^4 +
1 * 2^3 + 0 * 2^2 + 0 * 2^1 + 1 * 2^0 = 128 + 64 + 0 + 0 + 8 + 0 + 0 + 1 = 201 However, since the leftmost bit is 1, the number is considered as negative in Two's complement representation.
Therefore, the decimal equivalent of the binary number 11001001 is -28.
Binary to Decimal Conversion:
Binary to Decimal conversion is a process of converting a binary number to its equivalent decimal form. In binary numbers, each digit represents a power of 2, starting from 2^0 on the rightmost
digit. By multiplying each bit with 2 raised to the power of its position and summing up the results, we can obtain the decimal equivalent of the binary number.
Two's Complement Representation:
Two's Complement representation is used to represent negative numbers in binary form. In this representation, the leftmost bit of a binary number indicates its sign. If the leftmost bit is 0, the
number is positive; if the leftmost bit is 1, the number is negative. To find the value of a negative binary number in Two's Complement representation, first invert all bits and then add 1 to the
Therefore, when converting the binary number 1100 1001 to a decimal number, the leftmost bit indicates that the number is negative. By applying the binary to decimal conversion and considering the
Two's Complement representation, we get the result of -28 as the decimal equivalent of the given binary number. | {"url":"https://www.businessconjunctions.com/computers-and-technology/convert-binary-numbers-to-decimal-numbers.html","timestamp":"2024-11-06T14:37:30Z","content_type":"text/html","content_length":"22667","record_id":"<urn:uuid:c55cb30b-6c04-4745-852a-4a020bb870c3>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00576.warc.gz"} |
Math Is Fun Forum
Your solution is 'still' very convoluted, with so much unnecessary calculations, There is absolutely no need to test the circle against each 4 edges of the rectangle (Not to mention that your code
snippet there will not work for vertical lines, nor is it complete, as it is only intersecting against an infinite line, and not the bounded line segment which a rectangle edge is) And even testing
against the 4 edges is not enough if you go down this road, because you then also need to handle the case of the circle being completely within the rectangle.
My solution get's around all of this, to give a very fast, very stable solution, which then also opens the door to calculate the minimum displacement needed to seperate the rectangle and the circle
so that they no longer intersect, and also the normal of the collision which would be used to resolve the collision and change the velocities of the circle etc appropriately etc.
My solution is a complete axis-aligned rectangle, circle intersection method. (It includes corners,edges,containment)
Well not sure about theorem to describe it, but is it not enough to drawn an equilaterial triangle and dividing one side into 6 using the ruler, and drawn the diagonal making a 10 and 50 degree
make a point, draw a circle, from an edge of the circle, draw another circle with same radius so that it passes through centre of the first circle
at one of the points where the two circles meet, draw lines to the two centres, draw line between centres and that gives you an equilateral triangle, using ruler divide one side into 6 equal
segments, and then as above.
group like terms together, divide throughout
group like terms together, divide throughout
or to make a nicer looking equation, you can do the following:
move terms to LHS (left hand side) so that the squared term is on it's own, take the square root (remembeing that both -x and x are roots of x^2)
again, make the cubed term be on it's own, take cubed root
(There are strictly 3 cube roots of a number, which the above will allow for, but since you'll not be introduced to complex numbers, the following would suffice)
I'll leave it at that, since I do feel i'm simply doing the work for you here
I wouldn't have thought so, because one could quite easily define a language where every name for a number, has 1 more letter than the number value, and so you could never reach a finite loop.
x = -1 can be spotted as a root of the equation quite easily as the coeffecients of the x^2 term and the 5 sum to 1.
(-1)^3 - 4(-1)^2 + 5 = -1 - 4 + 5 = 0
so you have:
(x+1)(x^2 + bx + c) (coef. of x^2 is 1 since it's multiplied with x to give x^3)
x^3 + (b+1)x^2 + (c+b)x + c = x^3 - 4x^2 + 5
which gives immediately, b+1 = -4 -> b = -5,, c+b = 0 -> c = 5 (or just c = 5 immediately from other coef.)
(x+1)(x^2 - 5x + 5)
Replies: 0
Given a 2d cubic bezier curve:
define a new curve:
for some constant x, aka the curve resulting from displacing the cubic bezier curve c a constant amount along it's normal.
I've determined that the derivative of the new curve is:
which is correct, and this is how i determined the curve 'd' isn't a cubic bezier, by the fact that attempting to render it as one given start end points and tangents results in a totally different
curve from an approximation of d directly.
Is d perhaps a rational bezier curve?
because there are two square roots, a positive AND a negative
think: (-a)*(-a) = a*a
so if a^2 = b, then a = +/-sqrt(b)
I would suggest an alternative solution to your problem, one which is both simpler to implement, and with less problems associated with it:
Assuming a circle centred at C, with radius r again, and a rectangle with centre P and half width,half radius w,h
First consider the following
if C.x + r < P.x-w then the circle cannot intersect the rectangle, the following goes for:
if C.x - r > P.x+w and equivalent on the y-axis.
This is essentially testing for the intersection of the rectangle, and the bounding square of the circle.
After this, the only test left to consider, is dependant on the voronoi region the circle centre is contained in, to test the circle against the relevant vertex of the square, drawing some simply
diagrams you should see that this is true.
So this is my suggestion for your intersection method:
bool intersectCircleAARectangle(vec2 C, float r, vec2 P, float w, float h)
//test bounding square
float dx, dy, adx, ady;
dx = adx = C.x-P.x; if(adx<0) adx = -adx;
dy = ady = C.y-P.y; if(ady<0) ady = -ady;
if(adx < (r+w) || ady < (r+h)) return false;
//vertex test
float px, py;
if(dx<0) px = P.x-w;
else px = P.x+w;
if(dy<0) py = P.y-h;
else py = P.y+h;
dx = C.x-px;
dy = C.y-py;
return (dx*dx+dy*dy)<(r*r);
Excuse me if there's any errors, i have just done this from the top of my head.
Let me guess:
you are attempting to render the circle via the rearrangement:
y = b + sqrt(r^2 - (x-a)^2))
in which case, remember that there are two roots, so you need to render both
y = b + sqrt(r^2 - (x-a)^2))
y = b - sqrt(r^2 - (x-a)^2))
There are ofcourse, simpler ways of rendering a circle which will give nicer results, using the polar equation for example with a displacement:
centre of circle C, radius r, by iterating over the full circle for θ render curves through the points
P = C + (r.cosθ i + r.sinθ j)
try going from there.
do you mean:
(x^2-36) / (x+6)
x^2 - 36/x + 6
x^2 - 36/(x+6)
or something else?
without brackets, or any form of formatting it is impossible to know.
if we're approaching from the right, then we are using the middle condition, that 1 > x <= 3, since as x approaches 1, it will always be greater than 1
and lim(x->1) x+3 = 4
if you have a sequence
then the term for n = 1 will be equal to a + b + c
the first sequence of differences is:
and the second sequence of differences is:
if we start with f_1 and have f_2 and f_3, then n = 1 and we have the following:
so for example, the following sequence of 3 numbers, starting from n = 1
the sequence of differences is
which gives a second difference of 6
so we have:
x - 3y = -8
add 3y to each side
x - 3y + 3y = -8 + 3y
x = 3y - 8
add 8 to each side
x + 8 = 3y - 8 + 8
x + 8 = 3y
divide each side by 3
(x+8)/3 = 3y/3
spin and charge are inherit properties of basic particles, which cannot be changed, only observed. quantum entanglement describes what you are talking about though, if two particles for example are
entangled, and you observe one to have a positive spin, then the other particle MUST have a negative spin.
But like i said, information cannot be transferred, because you cannot control what the spin on your particle is when it is observed, all you know is that the other particle must have the opposite.
bobbym, what you are referring to in your action at a distance is quantum entanglement, which is a perfectly real phenomena, but again, no information can be transferred.
There are many occurences of things travelling faster than the speed of light; for example the path of a beam of light emitted from a distant fast rotating pulsar.
The difference, is that in none of these occurences of things travelling past c, is any information able to be transmitted.
it starts off with hour hand at 180 degrees from 12 oclock , and minute hand at 0 degrees
at time t (in minutes), you have that the hour hand makes the angle
180+t/2 (in one hour (60min) it turns 30degrees)
and the minute hand
t*6 (in one hour (60min) ut turns 360degrees)
t*6 - (180+t/2) = 180
5.5t - 180 = 180
5.5t = 360
t = 65.454545454545... = 1hr 5min 27sec 273ms and so on, to nearest second we have that the next time will be
Replies: 7
This is one i've known for many years.
Start with any number (or infact, word i might say)
At each step, take the number that represents the number of letters in the word, and use that as your next word
you inevitably end up at four, which ofcourse is the end of the line, since the word 'four' has 4 letters.
Timbuktu -> 8
Eight -> 5
Five -> 4
Four -> 4
Equivalent version in Italian, you inevitably end up with 'tre' (3)
Timbuktu -> 8
Otto -> 4
Quatro -> 6
Sei -> 3
Tre -> 3
the two equations are exactly the same in the end; both C and K are both constants, so the product CK is also a constant.
thus e^k(t-c) is essentialy exactly the same as e^(tk-c), only that c has a different value because it is not multiplied by k
You could say that your teachers solution is simplified.
In the same way, you might end up with an integration where when you manipulate it you end with a constant like:
I(x) = f(x) + 4.5C
which ofcourse, you can equally just write as
I(x) + f(x) + C'
the C is written with the apostrophe here just to explicitly show that it is not the same constant as the first equation, but both are correct
moving at 10km/h, after 30 minutes Peter will have travelled 10*0.5 = 5km (since 30minutes = 0.5hours)
at time t (hours), the sister will have moved 12t, whilst peter moves 10t, but peter is also 5km ahead of his sister
so you have the equation
12t = 5 + 10t
2t = 5
t = 2.5hrs | {"url":"https://mathisfunforum.com/search.php?action=show_user_posts&user_id=3758","timestamp":"2024-11-13T02:00:18Z","content_type":"application/xhtml+xml","content_length":"55665","record_id":"<urn:uuid:9e29221c-482b-4bb8-a47a-4cc9072368a7>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00268.warc.gz"} |
ICSA 2018 – Q# and the Microsoft Quantum Development Kit
By Martin Roetteler (Microsoft)
Why Quantum machines? Classical IT has its limits, especially in computing power. For example, modelling and understanding chemical models works on small molecules, but not on larger. Important to
realize is that QM is not going to replace our own computers we use on a daily basis. QM is based on new technologiess, including Ion traps (some metals can be used to manipulate ions with lasers to
make transitions, and still in a very early research stage, about 10 qubits), super conductors (classical chips, but then make them super conducting, a current flows in it to decode bits: clockwise
or anti clockwise flowing, 20-50 qubits), linear optics, NV centers (for storing Qbits in a lattice-type structure, not very good for computing), Quantum dots (to couple elements, it is based on
electronic spin), and Majorana zero modes (nano wires put on a super conducting material, can be shielded very well, but very hard to build). However, for a QM to work (currently), you need to cool
down the machine to 0.01K (that is really close to the absolute zero-point!!!!). Programming still is quite primitive, as they use a graphical language, where each line represents a qubit. Some
benchmarks show that the quality of qubits are not yet optimal. They work on small scale, but need to become more accurate.
What can we do with Quantum machines? The RSA problem can be expressed on a QM, moving down the execution time for cracking RSA from 1 billion years to a single second (with about 2K qubits). This
shows that we need new cryptographic algorithms. Other speedups include computational chemistry and linear algebra.
How do we program Quantum machines? The reason why it exponentially scales has to do with the qubits: 30 qubits is something like 16 Gb, 40 qubits compares to 16 Tb, and 50 qubits corresponds to 16
Pb. An important idea is interference: useless computations are canceled out, and useful ones can be amplified. The basic idea is that a bit is replaced by a unit vector [a]*0 + [b] * 1. Computing
are unitary operations. Programming is not easy, as you have all these superpositions of a qubit. From the presentation it appears that you’re programming with the logic components, as we did in the
past with physics.
Quantum machines also have drawbacks. To start, one cannot clone: quantum information cannot be copied! Thus, redundancy is not possible, and error repair is very difficult. There are I/O
limitations: setting the input state can be very costly, and output reading is probabilistic (you get a draw if you read out the state). And, how do you verify an algorithm?
ICSA 2018 – Q# and the Microsoft Quantum Development Kit | {"url":"http://www.architecturemining.org/microblog/icsa-2018-q-and-the-microsoft-quantum-development-kit/","timestamp":"2024-11-09T01:22:40Z","content_type":"text/html","content_length":"26255","record_id":"<urn:uuid:bfe56578-4528-48ec-b1ca-e98f919e906e>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00468.warc.gz"} |
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Gotcha – sequence index evaluation
13 Jul
Gotcha – sequence index evaluation
Author: Dave Cassel | Category:
Software Development
Every now and then I get caught by this little gotcha, so I figured I’d share and hopefully by writing about it, I’ll remember to do this right. Let’s start with a little something simple, shall we?
let $seq := (1 to 100)
return $seq[10]
Simple, as promised. I create a sequence of numbers from one to one hundred and I ask for the tenth one. I run this and I get 10 as a result. So far, so good.
Now, suppose that I want to get a random element of this sequence. MarkLogic Server provides an xdmp:random() function, so this should be easy, too:
let $seq := (1 to 100)
return $seq[xdmp:random(99) + 1]
Randomly generate a number from 0 to 99, add one to get us into the 1 to 100 range, and return the value with that index. I run this one and I get… the empty sequence. I run it again and I get two
values. I run it again and get one. What’s going on?
To see what’s going on, let’s run this in CQ using the Profile button.
expression count
let $seq := 1 to 100 return $seq[xdmp:random(9) + 1] 1
xdmp:random(9) + 1 100
xdmp:random(9) 100
$seq[xdmp:random(9) + 1] 1
1 to 100 1
What we see is the xdmp:random() expression getting called 100 times. Yet if you run Profile on the first implementation ($seq[10]), you’ll see that $seq[10] is evaluated just once and that “10”
doesn’t show up as an expression.
When I put a constant in the index operator ([]), XQuery knows exactly which element(s) I want — no work is required. But when I put an expression there, it evaluates the expression once for each
element in the sequence and checks whether the current index matches the expression. That lets us do complicated things like
let $seq := (1 to 100)
return $seq[if (math:fmod(fn:position(), 3) = 0) then fn:position() else ()]
(return the elements whose indexes are divisible by three) but it comes at the cost of evaluating that expression more often than you might expect.
So what should we do instead? Happily, there is a simple solution:
let $seq := (1 to 100)
let $index := xdmp:random(99) + 1
return $seq[$index]
This approach returns one value every time it’s called, and profile shows us that each expression is evaluated only once.
Moral of the story: if you have an expression as a sequence index, make sure it’s not doing more work than you intend. Profiling, as always, is your friend.
Tags: gotcha, marklogic, xquery | {"url":"https://blog.davidcassel.net/2010/07/gotcha-sequence-index-evaluation/","timestamp":"2024-11-15T02:20:19Z","content_type":"application/xhtml+xml","content_length":"40478","record_id":"<urn:uuid:31e5e8e9-5c40-4c8d-a08f-2383ca302e0e>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00052.warc.gz"} |
In 100 questions, how did I explain dynamic programming to my 5-year-old niece? | Denver annual essay - Moment For Technology
In my impression, dynamic programming is the most difficult one. In my opinion, dynamic programming is the most difficult.
My niece is 5 years old and now she starts to learn addition and subtraction. Every time she does math, she can’t do without her little fingers. If she is more than 5, she starts to count the fingers
of the other hand
I asked her one day
“What’s 1+1+1+1+1?”
“Pick up little finger, start to count, five!”
“What about adding another one?”
“Six, of course.” – Blurt it out
“How did you get so fast this time?”
“That was just five. Add one to make six.”
“So you don’t have to recalculate because you remember the answer was 5! Dynamic programming is: you save time by remembering things from the past.”
Get into the business
$\color{red}{don’t make fun of dp! }$
Take a look at the entry for the Popular Science Encyclopedia of China
Dynamic Programming (DP) is a branch of operations research. It is the process of optimizing the decision-making process. In the early 1950s, THE American mathematician R.Bellman and others put
forward the famous optimization principle when studying the optimization problem of the multi-stage decision process, thus creating the dynamic programming. Dynamic programming is widely used,
including engineering technology, economy, industrial production, military and automation control and other fields, and has achieved significant results in the backpack problem, production and
operation problem, capital management problem, resource allocation problem, shortest path problem and complex system reliability problem.
See not over of children’s shoes to skip, the hour we simple point
Actually, this is probably the easiest dynamic programming problem you can do.
What can we see about this problem?
We know that is the last step 1 will be calculated so fast, most thanks to calculate before the answer is 5, if we put the question as a child, I want to calculate the answer to every step, you can
list an equation: f (x) = f (x – 1) + 1, everyone don’t to f (x) the mess, I put it as a simple way.
Where, f(x) is the value of which step, set an initial value, x > 0, then the first step
f(1) = 1;
f(2) = 2;
f(6) = 6
In the world of a program, what is used to store a data structure that can record the previous result values?
Obvious: arrays. We just use the subscripts to store the bashi, and that’s the dynamic in dynamic programming, which is to define some boundaries and initialize.
Here’s a training problem
$\color{red}{don’t look, this is also a simple problem}$
Leecode 322: You have three kinds of coins, 2 yuan, 5 yuan and 7 yuan. There are enough coins for each. You need 27 yuan to buy a book
Why does it feel like we’re back in grade school word problems?
— Simple analysis: Minimal coin combination -> Use as large a coin as possible
Who came up with this stupid problem? It’s so easy
7+7+7=21,21+2+2+2=27, 6 coins
Oh my god
7+5+5+5+5=27, 5 coins
We can think about it in a very violent way, in a small way, it doesn’t add up to more than 27, for example
7+7+7+7 > 27 (Excluded)
7+7+7+5 or 7+7+7+ 2+2+2 6
Exhaustive is too slow and involves a lot of double counting, so what if I want the smallest coin combination of any value up to 27? Think about it.
Since the computer can hold the previous contents in memory, and it’s fast, obviously, we can open an array to hold the previous state.
Focus on early warning
1.1. Dynamic programming Component 1: Determining status
To put it simply, when solving dynamic programming, we need to open an array. What does each element of the array f[I] or f[I][j] stand for, similar to what does x, y, and z stand for in math
Solving dynamic programming requires two consciences:
• The last step
• subproblems
The last step
As we said in the first problem, the last step is 5. So in this case, even though we don’t know what the optimal strategy is, the optimal strategy must be K coins, A1, a2,…. The ak values add up to
So there must be a last coin :ak.
So minus this coin, the face value of the first coin adds up to 27 minus ak
Key point 1:
• We don’t care how the first k-1 coin spelled 27-ak (there could be one or 100 ways to spell it), and we don’t even know ak and K yet, but we’re sure the first coin spelled 27-ak
Key point 2:
• Because it’s the optimal strategy, it has to have the least number of coins to spell 27-AK, otherwise it’s not the optimal strategy
• So we asked: what is the minimum number of coins to spell 27-AK
• The original question was what is the minimum number of coins to spell 27
• We turned the original problem into a subproblem on a smaller scale: 27-AK
• To simplify the definition, let’s say that state f of x is equal to the minimum number of coins to spell out x
Wait, we don’t know what ak is in the last quarter
1. Obviously, the last coin can only be 2,5 or 7
2. If ak is 2, f of 27 should be f of 27 minus 2 plus 1, where 1 is the last coin 2.
3. If ak is 5, f of 27 should be f of 27 minus 5 plus 1, where 1 is the last coin 5.
4. If ak is 7, f of 27 should be f of 27 minus 7 plus 1, where 1 is the last coin 7.
So using the fewest number of COINS = f (27) = min {f (27-2) + 1, f (27-5) + 1, + 1} f (27-7)
1.2. Dynamic Programming Component 2: Transfer equations
Let’s say state f of x is equal to the minimum number of coins to spell out x
For any x: f(x) = min{f(x-2)+1, f(x-5)+1, f(x-7)+1}
1.3. Dynamic Programming Component 2: Initial conditions and boundary cases
Ask questions
1. What if x minus 2, x minus 5, and x minus 7 are less than 0?
2. When does it stop?
If you can’t spell Y, define f[Y] = infinity
For example, f[-1], f[-2] = infinity
F [1]=min{f(-1)+1, f(-3)+1, f(-6)+1}
Initial condition: F [0] = 0
2.4. Dynamic programming Component 2: Computation sequence
Calculation: F [1], F [2]… f[27]
When we get to f[x], f[x-2], f[x-5], f[x-7], we’ve got the result
As shown in figure:
F [x] = the minimum number of coins to spell out x
F [x] = infinity means you can’t spell x with a coin
Reference code
public static int coinChange(int [] A, int M ) {
int[] f = new int[M+1];
int n = A.length;
f[0] = 0;
int i,j;
for (i = 1; i<=M; i++) {
f[i] = Integer.MAX_VALUE;
for (j = 0; j< n; j++) {// Boundary condition judgment
if(i >= A[j] && f[i - A[j]] ! = Integer.MAX_VALUE) { f[i] = Math.min(f[i - A[j]] +1, f[i]);
// System.out.println(i + "=" +f[i]);}}}if (f[M] == Integer.MAX_VALUE) {
f[M] = -1 ;
return f[M];
Copy the code
$\color{red}{core code is only 4 lines, is very simple ~}$
Of course, this problem can also be achieved through the method of pruning, here is not much to repeat
Let’s do another training problem
Or another question, a question is too arbitrary ~
Leecode 62: different paths
A robot is located in the upper left corner of an m x N grid (the starting point is marked “Start” in the image below).
The robot can only move one step down or to the right at a time. The robot tries to reach the lower right corner of the grid (marked “Finish” in the image below).
How many different paths are there?
Look at the above problem solving steps, step by step
2.1. Dynamic programming Component 1: Determining status
The last step
No matter how the robot reaches the lower right corner, there is always a final move: – to the right or down
If shown, let’s set the lower right coordinate to be (m-1,n-1).
So the position of the robot before the last step is m minus 2,n minus 1, or m minus 1,n minus 2.
So, if the robot has x ways to go from the top left corner to (m-2,n-1), and Y ways to go from the top left corner to (m-1,n-2), then the robot has x +Y ways to go to (m-1,n-1).
The question becomes, how many ways can the robot go from the top left corner to (m-2,n-1) or (m-1,m-2)?
If it goes to (m-2,n-1), as shown in the figure:
We can just kill the last column
Similarly, if you go to m minus 1,n minus 2 rows, you subtract one row.
Let f[I][j] be how many ways the robot can go from the top left corner to (I,j).
2.2. Dynamic programming Component 2: Transfer equations
For any lattice:
f[i][j] = f[i-1][j] + f[i][j-1]
1 represents how many ways the robot can go [I][J]
2 represents how many ways the robot can get to F [i-1][J]
3 is how many ways the robot can go to F [I][j-1]
2.3. Dynamic Programming Component 3: Initial and boundary conditions
Initial condition: F [0][0]=1, because the robot has only one way to get to the upper left corner
Boundary case: I =0 or j=0, then the previous step can only come in one direction, that is, row 0 or column 0. Each step has only one case, then f[I][j] = 1, and other regions satisfy the transition
3.4. Dynamic programming Component 4: Computation order
Row by row. Why row by row?
F [0][1] and F [1][0] have already been calculated. Similarly, these two coordinates have also been calculated by column calculation. There is no need to calculate again.
• f[0][0] = 1
• Calculate line 0: f[0][0], F [0][1],… ,f[0][n-1]
• Calculate line 1: F [1][0], F [1][1],… ,f[1][n-1]
• .
• Calculate line m-1: f[m-1][0], F [m-1][1],… ,f[m-1][n-1]
Time complexity: O(mn)
Reference code
public int uniquePaths(int m, int n) {
int[][] f = new int[m][n];
int i,j;
for (i = 0; i<m; i++) {
for (j = 0; j< n; j++) {
if (i == 0 || j == 0) {
f[i][j] = 1;
} else {
f[i][j] = f[i -1][j] + f[i][j-1]; }}}return f[m-1][n-1];
Copy the code
To summarize
What problems can WE do with dynamic programming?
Counting 1.
• How many ways to go to the bottom right
• How many ways can I pick k numbers and the sum of yes is sum
2. Find the maximum and minimum values
• The maximum number sum of paths from the top left to the bottom right
• Longest ascending subsequence length
3. Existence
• Take stone game, whether the first hand will win
• Can you pick k numbers such that the sum is sum
See here, basically be an introduction! The next few articles will be a little more difficult than today’s topic, more content, welcome to pay attention to a wave!
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Rainbow Proof Shows Graphs Have Uniform Parts
On January 8, three mathematicians posted a proof of a nearly 60-year-old problem in combinatorics called Ringel’s conjecture. Roughly speaking, it predicts that graphs — Tinkertoy-like constructions
of dots and lines — can be perfectly built out of identical smaller parts.
Mathematicians are excited that the new proof finds it’s true.
“A big reason for happiness is that this solves a very old conjecture that people couldn’t solve with other methods,” said Gil Kalai, a mathematician at the Hebrew University of Jerusalem and IDC
Herzliya who was not involved in the work.
Ringel’s conjecture predicts that certain kinds of complicated graphs — think Tinkertoy designs with trillions of pieces or more — can be “tiled,” or covered completely, by any individual copy of
certain smaller graphs. Conceptually, the statement is like looking at a kitchen and asking: Can I completely cover the floor with identical copies of any type of tile in the store? In real life,
most tiles won’t work for your particular kitchen — you’ll have to combine different shapes to cover the whole floor. But in the world of graph theory, the conjecture predicts that the tiling always
With the kitchen floor, as with graphs, where you place the first tile matters. The new work addresses this critical question about placement in a way that’s both surprising and surprisingly
The Forest and the Trees
In combinatorics, mathematicians study the way vertices (dots) and edges (lines) combine to form more complicated objects called graphs. You can ask many different questions about these graphs. One
of the most basic is this: When do smaller, simpler graphs fit perfectly inside larger, more complicated ones?
“You have puzzle pieces and you’re not sure if the puzzle can be put together from the pieces,” said Jacob Fox of Stanford University.
In 1963, a German mathematician named Gerhard Ringel posed a simple but broad question of that sort. First, he said, start with any odd number of vertices greater than 3 (the number needs to be odd
for the conjecture to be plausible, as we’ll see in a moment). Draw edges between them so that every vertex is connected to every other vertex. This creates an object called a complete graph.
Next, think about a different type of graph. It could be a simple path — edges connected in a line. Or it could be a path with other edges branching off of it. You could add branches to the branches.
You could make the graph as complicated as you want, so long as it doesn’t contain any closed loops. These types of graphs are called trees.
Ringel’s question was about the relationship between complete graphs and trees. He said: First imagine a complete graph containing 2n + 1 vertices (that is, an odd number). Then think about every
possible tree you can make using n + 1 vertices — which is potentially a lot of different trees.
Now, pick one of those trees and place it so that every edge of the tree aligns with an edge in the complete graph. Then place another copy of the same tree over a different part of the complete
Ringel predicted that if you keep going, assuming you started in the right place, you’ll be able to tile the complete graph perfectly. This means that every edge in the complete graph is covered by
an edge in a tree, and no copies of the tree overlap each other.
“I can take copies of the tree. I put one copy on top of the complete graph. It covers some edges. I keep doing this and the conjecture says you can tile everything,” said Benny Sudakov of the Swiss
Federal Institute of Technology Zurich, a co-author of the new proof with Richard Montgomery of the University of Birmingham and Alexey Pokrovskiy of Birkbeck College, University of London.
Finally, Ringel predicted that the tiling works regardless of which of the many different possible trees you use to perform it. This might seem wildly broad. Ringel’s conjecture applies equally to
complete graphs with 11 vertices and complete graphs with 11 trillion and 1 vertices. And as the complete graphs get bigger, the number of possible trees you can draw using n + 1 vertices also
skyrockets. How could each and every one of those trees perfectly tile the corresponding complete graph?
But there were reasons to think Ringel’s conjecture might be true. The most immediate one was that simple combinatoric arithmetic didn’t rule the conjecture out: The number of edges in a complete
graph with 2n + 1 vertices can always be evenly divided by the number of edges in a tree with n + 1 vertices.
“It’s an important thing that the number of edges in the tree divides the number of edges in the complete graph,” said Montgomery.
Mathematicians quickly identified another piece of evidence that suggested the conjecture was at least feasible, and it set in motion a chain of discovery that eventually led to a proof.
Place and Rotate
One of the simplest trees of all is a star: a central vertex with edges radiating out from it. But it’s different from the typical image of a star, since the edges don’t have to be arrayed uniformly
around the vertex. They just have to extend out from the same place, and they can’t intersect each other anywhere but at the central vertex. If you wanted to probe Ringel’s conjecture, the tree that
looks like a star would be a natural place to start.
And indeed, mathematicians quickly observed that the star with n + 1 vertices can always perfectly tile a complete graph with 2n + 1 vertices. That fact alone is interesting, but the way to prove it
really got mathematicians thinking.
Consider a simple example. Start with 11 vertices. Arrange these vertices in a circle and then connect each vertex with every other vertex to form a complete graph.
Now consider the corresponding star: a central point with five edges extending out from it.
Next, place the star so that the central vertex aligns with one of the vertices in the complete graph. You’ll cover some but not all of the edges. Now reposition the star one vertex to the left, as
if you were turning the face of a compass. You’ll have a new copy of your star that overlaps an entirely distinct set of edges on the complete graph.
Keep rotating the star, one unit at a time. By the time you get back to where you started, you’ll have tiled the entire complete graph without any of your stars overlapping, just as Ringel predicted.
“We know the conjecture is not completely bogus if the tree is a star,” Sudakov said. “Moreover, we can show it in this beautiful way: Put the graph on a circle, shift the star, get new copies, and
that’s how tiling will go.”
Shortly after Ringel released his conjecture, a Slovak-Canadian mathematician named Anton Kotzig used this example to make a prediction that was even bolder than Ringel’s. While Ringel said that
every complete graph with 2n + 1 vertices can be tiled by any tree with n + 1 vertices, Kotzig conjectured that the tiling can always be done exactly the way it’s done with the star: by placing the
tree on the complete graph and then simply rotating it.
It seemed like a fanciful idea. The star is symmetric, and as a result it doesn’t matter how you place it. Most trees, though, are gnarly. They have to be placed exactly right for the rotational
method to work.
“The star has this simple structure that lets you place it by hand, but if you have a wild tree with lots of different branches of different lengths, it’s hard to imagine how to find a careful
placement of that,” Pokrovskiy said.
If mathematicians were going to solve Ringel’s conjecture using Kotzig’s rotational method, they would have to figure out how to place the first copy of the tree to avoid the thicket. Luckily, they
ended up finding a colorful solution.
Rainbow Colorings
Color-coding often makes life easier. It can help you organize your calendar or quickly distinguish among lunchboxes in a big family. Turns out, it’s also an effective way to figure out how to place
that first tree inside a complete graph.
Think again about the complete graph with 11 vertices arrayed around a circle. You’re going to color-code its edges according to a simple rule, one that concerns the distance between two vertices
connected by an edge.
That distance is defined as the number of spaces around the circle you need to go to move from one vertex to another. (No shortcuts through the inside of the circle.) Of course, you can always go one
of two ways around a circle, so consider the distance to be the shorter route between two vertices. If the vertices are adjacent to each other, the distance between them is 1, not 10. If two vertices
are separated by one other vertex, the distance between them is 2.
Now color the edges of the graph according to distance. All edges connecting vertices that are one unit apart receive the same color, say, blue. All edges connecting vertices that are two units apart
receive the same color, say, yellow.
Keep going like this, so that edges connecting vertices the same distance apart all receive the same color. You’ll also want to use a different color for each distance. On a complete graph with 2n +
1 vertices, you’ll need n different colors to carry out the scheme. At the end you’ll have a pretty design — and a useful one, too.
Soon after Ringel and Kotzig proposed their conjectures, Kotzig realized that this coloring of the complete graph could serve as a guide for how to place a tree over it.
The idea was to position the tree so that it covers one edge of each color and doesn’t cover any color twice. Mathematicians call this placement a rainbow copy of the tree. Since the coloring
requires n colors, and the tree with n + 1 vertices has n edges, right away we know it’s at least possible that there could always be a rainbow copy.
By the late 1960s, mathematicians understood that the rainbow copy of the tree has a very special property: It’s the exact right starting position from which to rotate the tree in order to tile the
“If you get a rainbow copy, the tiling always works out,” Pokrovskiy said.
Now mathematicians knew they could prove Ringel’s conjecture by proving that every complete graph with 2n + 1 vertices contains a rainbow copy of every tree with n + 1 vertices. If the rainbow copy
always exists, the tiling always works.
But proving that something always exists is hard. In order to do it, mathematicians would have to establish that complete graphs can’t help but contain rainbow copies of trees. It took more than 40
years, but that’s just what Sudakov and his co-authors did in their new proof.
Perfect Packing
Imagine you’re handed a complete graph with 11 vertices, and a tree with six. The complete graph has been colored with five different colors. The tree has five edges. Your task is to find a rainbow
copy of the tree inside the complete graph.
You could simply place the edges of the tree on the graph one at a time. The first edge is easy to place: It can go on any edge of any color. The second edge is only slightly harder. It can go almost
anywhere, just not on an edge of the complete graph that’s the same color as the one you’ve already covered. But as you keep placing edges of your tree, the job of placing the next one keeps getting
harder. By the time you get to the last edge of the tree, you no longer have any choice about which color it will cover — there’s only one left. You’d better hope you did a good job planning ahead.
This idea, that the task of finding a rainbow copy of a tree gets harder as you place more edges of the tree, was central to the way the three mathematicians approached their proof. They looked for
ways to give themselves as much flexibility as possible at the end of the process.
They knew from the outset that it’s easy to find rainbow copies of very simple trees — trees in the shape of a long path, or a long path with a few short branches. The hardest trees to place
correctly were ones with many edges converging at a single vertex, like stars, but with a more unwieldy and irregular shape. Placing those is like trying to pack a stroller in the trunk of a car when
it’s already half full.
“The difficulty comes in when you’re trying to pack the complete graph with inflexible things that look like stars,” Pokrovskiy said.
As anyone who’s ever packed a trunk knows, you should always begin with the most difficult objects: the largest suitcases and big, inflexible objects like bicycles. You can always stuff jackets in at
the end. The mathematicians adopted this philosophy, too.
Imagine a tree with 11 edges. Six come together at a central vertex. Most of the rest form a single shape coming off it, like a tendril.
The hardest part of the tree to place is the vertex with six edges. So the mathematicians separated it from the rest of the tree and placed it first, with the intention of eventually reattaching the
other part, just as you might disassemble a bed to move it upstairs, and then put it back together once you had it in your room. In fact, they didn’t just place the star portion once — they found
different places to put copies of the star inside the complete graph.
Then they randomly chose one. By doing this, they ensured that the remaining space in the complete graph was also random — meaning it had a roughly equal distribution of edges of different colors.
This was the space in which they’d need to place the rest of the tree — the path portion they’d set aside.
They faced some constraints in the way they could place it. The path would have to connect with the star part they’d already placed, and it would also have to cover colors that had not already been
covered by the star it was connected to.
But the mathematicians had given themselves options. They could connect the path to almost any one of the different copies of the star. Even better, because the space around the star was random as
well, the mathematicians had options about which colors they covered with the remaining part of the tree.
“At the end of the embedding, when things are getting difficult, instead of having one color I must use, I have a little choice,” Montgomery said.
The three mathematicians finished their proof with a probabilistic argument. They showed that once they embedded the hardest parts of a tree, if the remaining space in the complete graph was
essentially random, there would always be a way they could embed the remainder of the tree to get a rainbow copy.
“You can use what you left out at the beginning to kind of absorb what was left of the tree to create a total rainbow coloring,” said Noga Alon of Princeton University.
The mathematicians did not come up with an exact way to find a rainbow copy of every tree with n + 1 vertices in every complete graph with 2n + 1 vertices. But they did prove that a rainbow copy has
to be present.
And if the rainbow copy always exists, then it’s always possible to perform exactly the kind of tiling that Ringel predicted. The conjecture is true.
The proof also provides new tools for solving similar unsolved problems in combinatorics, such as “graceful labeling,” which predicts that the tiling of a complete graph can be performed under more
stringent conditions, where the tree has to be placed with even greater precision.
“It shows the methods people have been thinking about for a while are indeed quite powerful,” Fox said. “When you properly adapt them, you can solve these questions that had seemed out of reach.”
Correction: February 19, 2020
One of the tree examples was missing a vertex where edges branched off. The missing vertex has been added to the figure. | {"url":"https://www.quantamagazine.org/mathematicians-prove-ringels-graph-theory-conjecture-20200219/","timestamp":"2024-11-02T21:40:38Z","content_type":"text/html","content_length":"215077","record_id":"<urn:uuid:6be07135-41c6-4e6d-9a93-bb24a0fbf1b5>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00334.warc.gz"} |
3D Vectors - Vectors In 3-dimensional Space | Studywell.com
3D Vectors
All of the things we have learned so far can be applied to 3D vectors. Essentially, we add the third dimension so that 3D vectors are now capable of describing points in space rather than just
planes. Click here to revise 2D vectors including unit vectors and how to find magnitude and direction. Also recall 2D vector arithmetic and vectors in context such as position vectors and vectors in
trigonometry. At this level, we often extend a lot of the earlier mechanics problems to two dimensions – it is very simple to add the third dimension. See Calculus in Kinematics, for example.
Unit 3D Vectors
Recall the unit vectors and when working in two dimensions. We now extend to three dimensions and we introduce the third unit vector:
These vectors point in the , and directions respectively. Notice the position of the and axes relative to the -axis. We call this the right-hand rule. This means that if the -axis is your index
finger on your right hand and the -axis is your middle finger, then the -axis is your thumb and its points upwards. We often see it in different orientations (see Example 1) but usually with the
-axis pointing upwards. If you find that with your thumb pointing upwards, the and axis are on the wrong fingers, you are likely using a left-handed coordinate frame.
With these vectors, we can now identify the position vector that points from the origin to any point in 3D space. The vector we can see here is a position vector as it points from the origin. We can
It follows, by a simple application of 3D Pythagoras, that the magnitude of this vector is
It follows that a unit vector in the direction of is given by . See Example 1.
3D Vector Arithmetic and Magnitude
We can apply the operations that we saw on the vector arithmetic page in a similar fashion. That is, suppose we have the vectors and . It follows that
where is a scalar constant
We can easily extend the geometric interpretations that we saw on the vector arithmetic page to 3 dimensions. For example, the vector between two points in 3D is given by . It follows that the
distance between these two points is the magnitude of this vector and given by . Note that is not a position vector as it doesn’t point from the origin. It does however lie in the same plane as and .
We say that , and are coplanar. See Example 1.
We can also extend the concept of direction to 3D vectors. A 3D vector now lies in space and not on a plane but a simple application of SOHCAHTOA (cos=adj/hyp) tells us that for the vector that
makes an angle of with the -axis. Similarly, and where makes an angle of and with the and axes respectively. See Example 2.
AS Maths Vectors
A2 Maths Vectors | {"url":"https://studywell.com/vectors/3d-vectors/","timestamp":"2024-11-06T05:16:13Z","content_type":"text/html","content_length":"340553","record_id":"<urn:uuid:252593e1-67f5-4fb3-a11f-2da921ef1794>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00815.warc.gz"} |
Time formats in Excel
In Excel time is treated as a fraction of a 24 hour day from the number 0 to 1, with 0 being 12:00am (Midnight) and 1 representing 12:00am of the next day.
All time is treated as this underlying number, but the formatting can be changed for presentation purposes in the spreadsheet, so while the underlying serial number 0.75 represents 6:00pm (three
quarters of the way through the day) you could format this as 18:00 if you wanted to use a 24 hour clock system in your spreadsheet without affecting the underlying serial number.
Download File
If you would like to follow along download the attachment below.
Decimal Values to Represent Time in Excel
Excel represents time as a decimal value, where one full day equals 1. The decimal value represents the amount of the day that has passed.
For smaller units of time smaller decimal values can be used.
These values can then be multiplied by the desired number of seconds, minutes or hours that you want to get the underlying serial number.        Â
When combined together you can create decimal values for specific times of the day.
Why does Excel user serial numbers for time
By using serial numbers for time Excel can more easily perform calculations as it will use the serial number as the basis for calculations.
This means that whatever format is applied to the time it will always have a fixed serial number to work with.
Quickly Formatting Time Values
Most of the time, but not always, if you enter a time in Excel it will show as a time value rather than the underlying decimal value.
If you do see the decimal value and want to quickly change this to a time value with a 24 hour format you can use the format drop down on the Home tab.
To do this click on the cell you want to change the formatting of, go to the Home tab, and in the Number section click on the Number Format drop down.Â
This will drop down a list of preset number formats. Select Time to change a decimal value to a 24 hour time format.
You can also use this formatting method to change a time value to a decimal value by changing the format of a cell with a time value in it to General or Number.
More Advanced Time Formatting
If you want your time value to show in a format other than 24 hours then you can choose from more options by selecting ‘More Number Formats…’ on the dropdown or clicking on the cell with the
time value and pressing Ctrl + 1.
Either option will open the Format Cells window, where you can select Time from the Category box on the left hand side which will allow you to change the time format from a list of pre set options.
Manually Format Time Values
For more flexibility in how your time values are formatted, in the Format Cells window you can go to Custom in the Category box on the left hand side to manually enter a time format.
The manual formats are made up of a combination of codes representing the time formats.
• H or HH: Represents hours. Use a single H for 1-digit hours (e.g. 6), and HH for 2-digit hours (e.g. 06).
• M or MM: Represents minutes. Use a single M for 1-digit minutes (e.g. 5), and MM for 2-digit minutes (e.g. 05).
Note:Â M or MM is interpreted as "month" if used outside a time context, so make sure to use it within a time-based format.
• S or SS: Represents seconds. Use a single S for 1-digit seconds (e.g. 7), and SS for 2-digit seconds (e.g. 07).
If only using the letter codes the format will default to 24 hours, but by entering AM or PMÂ at the end of the format it will show in a 12 hour format followed by AM or PM depending on the time of
Below are examples of how different combinations would show the time 18:05:07 (Five minutes past six pm and seven seconds).
Excel treats all times as a decimal value between 0 and 1, with the decimal number representing the times point in the day based on hours, minutes and seconds, so three quarters of a day, 6pm, would
show as 0.75.
This allows Excel to make calculations from time values based on the underlying decimal number regardless of how the time is formatted.
The format of a time value can be changed from either the preset formats on the Number Format dropdown on the Home tab, from the Time section of the Format Cells window, or manually from the
Custom section of the Format Cells window. | {"url":"https://www.excelnavigator.com/post/time-formats-in-excel","timestamp":"2024-11-08T08:01:13Z","content_type":"text/html","content_length":"1050486","record_id":"<urn:uuid:67c8d35e-26fa-4a1b-ba52-1b7c3eb0da8a>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00578.warc.gz"} |
How it works?
You need to find a combination of 3 adjoint numbers, which combination gives the highlighted number.
Swipe the numbers to give an answer.
Combination means that you can multiply two of the numbers and add or substract the third number. The order of the numbers on the board doesn't matter. | {"url":"http://3ogame.io/","timestamp":"2024-11-09T00:12:01Z","content_type":"text/html","content_length":"6510","record_id":"<urn:uuid:89e5d5c3-1b3a-4909-9da7-f62d61b081ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00534.warc.gz"} |
normal strain
What is shear strain and normal strain?
What is shear strain and normal strain?
Strain is the deformation of a material from stress. It is simply a ratio of the change in length to the original length. Deformations that are applied perpendicular to the cross section are normal
strains, while deformations applied parallel to the cross section are shear strains.
What is shear strain in machining?
Strain is relative to the deformation of a material. Shear strain is relative to the deformation of material in shearing. The shear strain in machining is calculated by tan shear plane angle minus
rake angle, plus cos shear plane angle.
What is the YZ shear strain?
YZ—shear stress or strain acting in the Z direction on the plane whose outward normal is parallel to the Y axis. ZZ—normal stress or strain along the Z axis.
What is the difference between shear stress and shear strain?
Shear stress(τ) = Tangential Force/ Resisting cross-sectional Area. Shear strain can be defined as the ratio of deformation to its original length or shape.
Why is shear strain important?
shear stress, force tending to cause deformation of a material by slippage along a plane or planes parallel to the imposed stress. The resultant shear is of great importance in nature, being
intimately related to the downslope movement of earth materials and to earthquakes.
What is the formula of shear strain?
shear strain = Δ x L 0 . shear stress=F∥A. shear stress = F ∥ A . The shear modulus is the proportionality constant in (Figure) and is defined by the ratio of stress to strain.
What is shear strain Class 11?
Shearing strain is the measure of the relative displacement of the opposite faces of the body as a result of shearing stress.
What is shear strain rate?
Strain rate is the change in strain (deformation) of a material with respect to time. It comprises both the rate at which the material is expanding or shrinking (expansion rate), and also the rate at
which it is being deformed by progressive shearing without changing its volume (shear rate).
What are the units of shear strain?
Shear strain is measured in radians and hence has no units. Shear strain has a relationship to shear modulus, which has elasticity coefficiency of a given substance that expresses the ratio between
force per unit area (shearing stress) that deforms the substance and the shear produced by this force.
What is Max shear strain?
Maximum Shear Stress The maximum shear stress at any point is easy to calculate from the principal stresses. It is simply. τmax=σmax−σmin2. This applies in both 2-D and 3-D. The maximum shear always
occurs in a coordinate system orientation that is rotated 45° from the principal coordinate system.
What is shear strain BYJU’s?
Shearing strain is the ratio of change in angle to which it is turned to its distance from the fixed layer.
What is the difference between strain and shear?
Shear stress; When stress changes the shape, it is called shear stress. Strain. The strain is a measure of the deformation of a solid when stress is applied to it. In the case of deformation in
one-dimensional strain is defined as the fractional change in length. If Δl is the change in length and l is the original length, then strain is given by:
What is the difference between normal strain and shear strain?
– γ = shear strain (unit-less) – τ = shear stress (N/m2, or Pascals in the International System of Units, or pounds per square inch (psi) in the British Imperial System) – G = shear modulus, or
modulus of rigidity (defined as the ratio of shear stress over shear strain)
How to determine shear strain?
Shear Strain measures how much a given deformation differs from a rigid deformation is calculated using shear_strain = tan (Shear angle)+ cot (Shear angle-Rake Angle).To calculate Shear Strain, you
need Shear angle (ϕ) & Rake Angle (RA).With our tool, you need to enter the respective value for Shear angle & Rake Angle and hit the calculate button.
Tensile strain. If the strain is ε is due to tensile stress σ,it is called tensile strain.
Compressive strain. If the strain is produced as a result of compressive stress,it is called compressive strain.
Volumetric strain. When the applied stress changes the volume,the change in volume per unit volume is known as volumetric strain.
Shear strain. | {"url":"https://www.wazeesupperclub.com/what-is-shear-strain-and-normal-strain/","timestamp":"2024-11-07T22:47:57Z","content_type":"text/html","content_length":"120205","record_id":"<urn:uuid:3e946e5c-e26d-4275-8577-e0976ab2fc8d>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00262.warc.gz"} |
Is Excel Hindering Your Engineering Projects?
Every engineer has access to Excel®. It's so easy to open up a worksheet and start putting in some values. A quick calculation here, add a multiplier there, change this value because you have new
information, and maybe redo the calculation with this number to see what the results would look like. The numbers look good, so you proceed with building a prototype.
Meanwhile, the spreadsheet is shared with another team, where someone adds a few more lines, changes some of the original values, and introduces a number of untraceable errors. A few months down the
line, no one knows where any of the figures came from, yet project teams have been basing months of design work on the results of these erroneous calculations.
Cryptic formulas, coupled with a lack of visibility into where data is coming from and how equations are being solved, leave room for errors which result in undesirable and even disastrous
Consider the workflow of many engineers or technical professionals. During beginning stages, there's a lot of scratchpad work to be done as the concept moves closer to reality. Without enough care
and attention, these calculations find themselves spread out over notepads, spreadsheets, and simply in the engineer's head. As the design moves forward, these calculations become bundled into all
future decisions - for better or worse. What happens if an engineer moves on, or is on vacation? Without the original author, engineers can be left scratching their head and wondering where "that"
number came from.
Then there is the JP Morgan Chase case of 2012 where a simple calculation error caused a reported loss of $6 billion. An existing spreadsheet was used as the basis to create a new one to model the
volatility of trades for a new portfolio, and the calculations involved copying and pasting data between spreadsheets.
However, instead of dividing by the average of two given numbers to calculate the volatility of the trade, the sum of the numbers was used. Dividing by this larger number effectively reduced the risk
assessment by a factor of two, which led to more risk being taken, resulting in exceptionally high losses.
The fact remains that while Excel has its uses, it was simply not designed for advanced mathematical calculations. Engineers and scientists need interactive math systems that enable them to write
equations that describe problems using standard mathematical notation, such as ax2+bx+c=0, and then solve these problems by working with the equations in a natural way.
Maple is one of the best examples of an interactive math system. Built on a foundation of symbolic math, Maple is specifically designed for describing, investigating, visualizing, and solving
mathematical problems. It offers a comprehensive range of solvers that cover all the principal areas of engineering math in a technical document environment that combines text, calculations, images,
graphs, and more into a single document. Because Maple is designed for advanced mathematical calculations and because it allows engineers to capture the thinking as well as the results, Maple
provides many benefits that Excel cannot.
The ideal tool for engineering projects is one that can handle complex calculations across a broad range of subject areas.
Excel is a business tool, which has evolved to handle some non-business calculations. Its Function Library now includes some basic math and engineering functions such as SIN, EXP, LOG, SQRT, etc.
However these are very basic operations that do not come close to covering the scope of calculations required for a typical engineering project. Excel also enables users to write macros that extend
its capabilities and automate frequently-performed tasks. However, macros work by manipulating the spreadsheet, which is not a natural way to approach problem solving.
Maple has over 5000 functions covering virtually every area of mathematics, including calculus, differential equations, statistics, linear algebra, and transforms. It supports symbolic, exact
computations where variables do not need to be given values in advance, as well as infinite-precision numeric computations. Maple has world-leading algorithms that solve problems beyond the reach of
any other software system, and efficient algorithms and tools for high performance computing and large-scale problem solving.
Maple also includes a full-featured programming language that can be used to create scripts, programs, and full applications. Designed for mathematical computations, it includes built-in mathematical
data structures, operations, and functions specifically for manipulating mathematical objects and equations, making it ideal for advanced engineering calculations.
Don't struggle trying to describe equations, variables, etc in Excel. With Maple, you can easily express your calculations in natural math notation. Learn More: Maple's built-in equation editor
allows you to express complicated mathematical problems easily using standard mathematical notation
Engineers want to describe problems in terms of equations using variables, constants, and operands, and then work through those problems in a logical manner. An effective tool must have the ability
to support users in the way they want to work.
Excel does not support standard math notation. An expression like ((B12+2*$A$1)/A12)*2.1328 does not represent math in the way engineers express their problems, neither does Excel enable engineers to
manipulate equations naturally. There is no flow to how equations are solved, and you have to jump around from cell to cell in order to see where a calculation is performed, and where the result is
being used.
Using Maple, engineers can write equations and formulae in an intuitive and readable manner, using standard mathematical notation. Maple then lets you work through your problem in a natural way, with
every step clearly visible, and well documented. You can see where input values are coming from and where results are being used. With Maple, you are 'doing real math' because Maple was developed for
that singular purpose.
Maple enables engineers to solve mathematical problems in much the same way as they would when working them out by hand - albeit much faster, without errors, and with the ability to perform
calculations that are impossible to work out by hand.
Engineering calculations involve values that have units - denoting mass, velocity, resistance, density, etc. Tools used for engineering calculations must be robust enough to recognize and correctly
handle units in order to perform correct calculations. Many calculations also involve tolerances, where some values are known as falling within a given range. The ability to correctly manage
tolerances as part of the calculation is an important part of reaching a correct result.
Excel is not designed for scientific calculations and does not handle units in an intuitive manner. You cannot perform calculations on numbers that include units. You can only convert a number in a
cell from one unit to another using a function call such as =CONVERT(C5, "ft", "m"). Similarly, Excel cannot perform calculations that involve tolerances. For example, you cannot multiply two values,
each with its own tolerance, and get a result that comes with its own tolerance information. At best, you can find out if two numbers are within a certain tolerance range of each other using the
formula: =IF(ABS(A1-A2)<0.1,"OK", "out of range"), but this is rarely what you need to do.
Maple on the other hand, enables engineers to perform intelligent calculations that include units and tolerances. It prevents the use of incompatible units, and handles their manipulation to assign
the correct unit to the result of a calculation. For example, consider the equation F=ma. In Maple, you can multiply a given mass by an acceleration, and Maple will give you the result in Newtons.
Furthermore, with Maple, engineers can also perform calculations that involve tolerances, to account for variations in inputs and operating conditions, for example. With Maple, you can perform
intelligent calculations by simply entering values with their tolerances and units, and Maple will perform the required calculations, work out the tolerance range, and assign the correct unit.
Having a verifiable development path is crucial to a company's success. It allows engineers to verify assumptions, reproduce calculations, and understand where results come from. Given that clarity
and accuracy are vital for advanced engineering calculations, an environment that gives only the results is doing just a small part of the job.
Excel is designed for business calculations, and does not provide a clean way to add notes and comments into the workflow. It's not always clear where inputs are coming from, and why certain values
are being used. You can add comments to a cell, but they are not easily identifiable. The reader has to hunt for a red triangle in the corner of a cell to know that there is a comment, and then hover
over it to read the comment. The comment itself cannot include more than very basic mathematical expressions comprised of characters available on a standard keyboard, is not always visible, and
obscures the document it is meant to explain. It is simply not a good way to convey important information.
Maple's smart document environment provides a rich environment in which to create a complete work history of a project. Your notes, comments, visualizations, and calculations are all in one document.
You can easily see where inputs are coming from, what assumptions have been made, and understand why certain actions were taken. These documents are "live", so if the assumptions change, you can make
adjustments to your parameters and formulas and re-compute your results within the original document. Serving as a record of all project activity, the smart document provides an open audit trail that
helps to reduce the risk of errors and costly delays.
The Maple environment also helps to retain organizational knowledge. By documenting the entire project work flow, the knowledge gained over the course of a project is captured in a living document
which can be referenced at a future date. So whether employees leave a company, or questions arise about why things were done a particular way, there is a record of the entire project to refer back
Excel was designed for business calculations and produces clean visualizations to view data. Excel can produce bar graphs, line graphs, pie charts, and histograms.
Maple was developed for advanced mathematical calculations, and includes over 170 plot types and options. In addition to the same types of visualizations that Excel produces, Maple produces plots
that include implicit, contour, complex, polar, vector field, conformal, density, ODE, PDE, statistical, and more. Maple also generates dedicated engineering plots, including time and frequency
domain responses, root-locus, and root-contour plots. Maple not only lets you capture 2-D and 3-D graphs and animations, but also lets you zoom and pan them as well, for better analysis of data. You
can even perform real-time rotation of 3-D plots to literally view the data from a different angle.
Annotation tools are available for all 2-D plots - text, math and graphical annotations may be added to further illustrate the solution. You can even sketch directly onto a plot to highlight
something of interest, or drag an equation onto a plot to add it to the existing visualization.
Excel on the other hand only supports text annotation for 2-D visualizations. Maple's wide selection of plot types and available visualization options enable you to visualize solutions, understand
relationships, and communicate results in a form that is visually appealing and truly meaningful.
Different tools offer unique capabilities that enhance the outcome of a project. The ability to connect with third-party tools for data input and manipulation enables engineers to leverage the right
tools for the particular task at hand.
While Excel lets you import and export data files, and interacts with other Microsoft Office suite products, it does not connect directly with other tools to take advantage of their niche
Maple offers extensive connectivity with other tools. Maple can generate code for Visual Basic, MATLAB®, Java™, C, C#, Fortran, Perl, Python, R and JavaScript. This enables you to take your work and
implement it in other tools - royalty free. You can even deploy to Excel! Maple also supports two-way connectivity with MATLAB®, providing direct access to all of the commands, variables, and
functions of each product while working in either environment. Additionally, Maple also lets you connect to CAD systems such as SolidWorks®, Autodesk® Inventor™, and NX® - enabling you to apply
Maple's computational power to analyze and optimize designs. Other forms of connectivity that Maple supports include database connectivity, the OpenMaple API, the ability to call another application
from within Maple, and internet connectivity that enables you to retrieve information from online data sources, and incorporate that data into Maple applications. This wide range of connectivity
options enables engineers to apply the right combination of tools for the optimal outcome.
When solutions, sometimes in the form of applications, are delivered to end users in a readable and readily-usable manner, they are more likely to use those solutions, and use them correctly. Also,
when solutions can be easily modified and customized, project teams can use them as a starting point for new projects, saving time and money.
As previously discussed, Excel spreadsheets are opaque documents that can be misunderstood and misapplied by end users, and they can be challenging to modify correctly because of the difficulty in
understanding what assumptions they embed and what exactly they are doing. Maple solutions are fully-documented, highly-readable documents, making it much easier for end users to read, understand,
and use them appropriately.
Both Maple and Excel offer the ability to turn these documents into applications for end users, by including interactive elements such as input fields, drop-down lists, buttons, and check boxes.
Maple also supports dials, gauges, and 3-D plots which are not available in Excel. These interactive options enable end users to insert required values and obtain customized results and plots based
on their specified input. In Excel, free-form input is done in text-style input boxes, which can be used for numeric input or formulas in calculator-style input. Maple offers both text input and a
math input field that allows the user to enter mathematical expressions using standard math notation. With standard notation, the input is visually verified much more easily and so fewer mistakes are
In both Maple and Excel, inserting the interactive elements into the document is easy, and similar. Programming those elements to complete their tasks is not. In Excel, the programming is done in VBA
(Visual Basic for Applications) and the logic of the program involves manipulating spreadsheets - cell references and spreadsheet operations - not manipulating equations. In Maple, the programming is
done in the Maple programming language, which is designed for mathematical calculations. As a result, code in Maple tends to be faster to write and easier to understand, debug, and modify. In
addition, when it comes to modifying the applications, the Maple document already contains all the reasoning and assumptions that went into the original application, presented in a readable way, so
changes can be made safely.
Once your Excel solution is ready, you can send it to other Excel users. Once your Maple solution is prepared, you have several deployment options. One is to share your documents with other Maple
users, either directly or through a private group in the MapleCloud Document Exchange, a document sharing mechanism built into Maple.
You can also share your work with people who don't have Maple using the free Maple Player. With the Maple Player, your end users can view Maple documents, and use the interactive elements, such as
buttons, entry boxes, and sliders, to perform computations and visualize results. The Maple Player can be used royalty-free by any number of users.
Maple documents can also be shared via MapleNet - a server-based deployment tool that allows you to publish Maple applications on a corporate intranet. End users interact with these applications
through a web browser, in the same way they would in Maple. When you add or change a document on the MapleNet server, it is automatically available to all your end users, making it easy to ensure
your end users are always working with the latest versions.
Whichever method you choose, Maple ensures that your teams can easily share documents, deploy applications, and communicate efficiently.
As the drive for innovation grows, companies are under pressure to deliver better products in less time and at lower cost. As a result, it is more important than ever to use the right tools to get
the job done, the first time. While Excel is good for project budgets, it simply cannot handle the scope of mathematical computation required for advanced engineering projects. Engineering teams need
robust and powerful interactive mathematical systems - tools like Maple, which have an inherent knowledge of multiple mathematical disciplines and are dedicated to solving mathematical problems. With
the right tools, you can capture your thought process, minimize errors that lead to delays, control rising costs, and avoid unexpected outcomes. Just as you wouldn't head out to mow your lawn with a
ruler and a pair of scissors, you shouldn't let the use of inappropriate software jeopardize the success of your engineering projects.
Sprechen Sie mit unseren Produktspezialisten über eine kostenlose Demoversion von Maple
*Die Maple-Evaluation ist für Schüler und Studenten bzw. die private Nutzung zurzeit nicht verfügbar. | {"url":"https://de.maplesoft.com/products/maple/professional/excel-hindering-engineering-projects.aspx?L=G","timestamp":"2024-11-05T03:07:53Z","content_type":"application/xhtml+xml","content_length":"100256","record_id":"<urn:uuid:b2e8d828-a837-4de3-af10-bc71126c9adc>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00741.warc.gz"} |
ICC Calculator
Last updated:
ICC Calculator
The ICC calculator determines where your cricket team stands in the International Cricket Council (ICC) T20 team rankings after their current win or loss. The points calculation for ICC T20 is based
on the team's current standings before the match and who they are playing. The rankings are calculated over 12 months, and as the games get older, their weighting reduces over time.
The rankings are crucial for a national team to secure a spot in the World Cup main event or the qualifiers. The top 8 teams on the cut-off date set by the ICC qualify directly for the main event,
whereas the next set of 10 or 12 teams plays against each other for a few qualifying spots. This calculator focuses on one match per time, and it will tell you how many points a team will gain for
the match outcome.
Depending on those points, the teams move up or down the rankings ladder. With the next ICC World T20 cup scheduled for this year (2022), the rankings are crucial to making the qualifiers and super
12s. You can start by entering some numbers or continue reading to understand how to calculate ICC T20 rankings.
ICC T20 rankings
The rules to calculate ICC T20 rankings depend mainly on the team's current standings. The winning margin of runs or wickets does not affect whatsoever the points gained by the team for the win.
First, the data for wins and losses is collected over a duration, mostly 12 months. Then, the points accumulated for each win or loss are calculated. The total sum of points is divided by the number
of matches played to obtain the rating value used to rank the teams. As of 2022, there are 91 teams in the ranking table. The points gained for each match depend on the following factors:
• Pre-match ratings of both sides;
• Difference between the pre-match ratings; and
• Standing of winning team.
Say two teams, A and B, are playing against each other; firstly, the rating before the match is recorded. The rating is compared to determine the difference between the ratings. Here, the weaker team
is one with a lesser rating and vice versa. The points are as follows:
Rating difference Team Result Points gained
Team 1 wins Opponent's rating + 50
<40 Team 2 loses Opponent's rating - 50
tie Opponent's rating
Stronger team wins Own rating + 10
Weaker team loses Own rating − 10
Stronger team ties Own rating − 40
Weaker team ties Own rating + 40
Stronger team loses Own rating − 90
Weaker team wins Own rating + 90
How to calculate ICC T20 rankings — points gained per match
To calculate the points per match:
1. Enter the rating for Team 1.
2. Fill in the rating for Team 2.
3. Select the result of the match from the list.
4. Based on the match result, the ICC calculator will return the points gained by each team.
Example: Using the ICC calculator for T20 points calculation
Find the number of points gained by England and Australia playing against each other in a T20i match. Take the team ratings for England and Australia as 248 and 275, respectively.
1. Enter the rating for Team 1, England, as 248.
2. Fill in the rating for Team 2, Australia, as 275.
3. Select the result of the match from the list as Team 1 wins.
4. As per the ICC T20 points calculation:
\qquad \scriptsize \begin{align*} \text{Team 1 gains} &= 275 + 50 = 325\\ \text{Team 2 gains} &= 248 - 50 = 198 \end{align*}
Team ratings
You can also calculate the team ratings before and after the match by ticking the checkbox "Help me calculate each team's rating" at the top of the calculator.
What are the factors affecting ICC T20 team rankings?
The factors that affect ICC T20 team rankings are:
• Rating of both teams involved in the match;
• Difference between the ratings of the two teams; and
• The victor side.
If the team with a higher rating loses the match, the points gained by their opponent will be more. This system rewards the weaker teams when they win against the stronger side.
How do I calculate ratings for winning a T20 match?
You can calculate the rating gained or lost per match by:
1. Find the number of points gained.
2. Add the points to the current tally.
3. Divide the points by the number of matches played.
If the team loses the match, the points gained will be less, therefore reducing the average or team rating and vice versa.
Which team is the number 1 in men's ICC T20 rankings?
As of February 2022, England leads the men's ICC T20 rankings having 9354 points from 34 matches resulting in a team rating of 275. Cricket teams from India and Pakistan hold the following positions
with 9627 and 12207 points with 267 and 265 ratings, respectively.
Which team is the number 1 in women's ICC T20 rankings?
As of February 2022, Australia leads the women's ICC T20 rankings having 5824 points from 20 matches resulting in a team rating of 291. Cricket teams from England and India hold the following
positions with 7157 and 6081 points with 286 and 264 ratings, respectively. | {"url":"https://www.omnicalculator.com/sports/icc","timestamp":"2024-11-06T23:56:56Z","content_type":"text/html","content_length":"451803","record_id":"<urn:uuid:110b2db2-21d7-45a3-a10c-1893f0afa90c>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00259.warc.gz"} |
Analog Clock: Face and Hands | sofatutor.com
Analog Clock: Face and Hands
Analog Clock: Face and Hands
Basics on the topic Analog Clock: Face and Hands
Analog Clock – Face and Hands
An analog clock is a type of clock that uses hands to show the time. It has a round face with numbers and two or three hands that move around the face. In this text, we will learn about the different
parts of an analog clock and how to read the time.
Parts of an Analog Clock
An analog clock has three main parts: the face, the hour hand, and the minute hand. Some analog clocks also have a second hand.
The Face: The face of an analog clock is a round circle with numbers from 1 to 12. The numbers represent the hours of the day. The face is divided into two halves, one for the hours and one for the
The Hour Hand: The hour hand is shorter and thicker than the minute hand. It points to the current hour on the clock face. The hour hand moves slowly and takes 12 hours to complete one full rotation.
The Minute Hand: The minute hand is longer and thinner than the hour hand. It points to the current minute on the clock face. The minute hand moves faster than the hour hand and takes 60 minutes to
complete one full rotation.
The Second Hand: Some analog clocks have a second hand, which is the thinnest and longest hand. It moves even faster than the minute hand and points to the current second on the clock face.
Telling the Time on Analog Clocks
To read the time on an analog clock, follow these steps:
Step What to do What will it tell?
1. Look at the hour hand and see which number it is pointing to. This tells you the hour
2. Look at the minute hand and see which number it is pointing to. This tells you the minute
3. If the clock has a second hand, look at it to see the seconds. This tells you the second
For example, if the hour hand is pointing to the 3 and the minute hand is pointing to the 12, the time is 3:00. If the hour hand is pointing to the 8 and the minute hand is pointing to the 30, the
time is 8:30.
Differences between Analog Clocks and Digital Clocks
Analog clocks and digital clocks are two different types of clocks.
Analog Clocks: Analog clocks have a round face with hands that move around to show the time. They are often found in homes, schools, and public places. Analog clocks can be a bit more challenging to
read, but they can also be more visually appealing.
Digital Clocks: Digital clocks display the time using numbers. They are often found on electronic devices like smartphones, computers, and microwaves. Digital clocks are easier to read because the
time is shown in digits. Feel free to learn more about digital clocks through the learning text Digital Clocks.
The Story and Importance of Analog Clocks
Analog clocks have a fascinating story that goes back many, many years. They were the first kind of clock that lots of people used. The first analog clocks were made with gears and springs to help
tell time. These clocks were super important in helping people keep track of time and find their way around.
The way analog clocks look, with their round face and moving hands, was based on how things in the sky move. The hour hand is like the sun moving across the sky, and the minute hand moves faster,
just like we do during the day. Analog clocks are not just for telling time, they can also be beautiful pieces of art. They are often decorated in fancy ways and made with a lot of skill.
Analog clocks have always been really important in helping us organize our day, plan our work, and even in how cities are built. In the 17th century, the pendulum was invented, and it made analog
clocks much more accurate. Even though we have digital clocks now, people still love analog clocks because they look nice and remind us of old times.
Analog Clock – Summary
Let’s review what we have learned about analog clocks in this text.
We have learned that:
• Analog clocks are used as a traditional way of telling time.
• They have a round face with numbers and hands that move to show the time.
• By understanding the parts of an analog clock and how to read the time, you can become a master at telling time!
Analog Clocks – Frequently Asked Questions
Transcript Analog Clock: Face and Hands
Zuri and Freddie are having a fun day searching through items at their local landfill when they hear something odd. : "Freddie, there's something down here!" : "I found it! Look what a pretty blue it
is... what... is it?" Zuri and Freddie have found an analog clock. Today we will learn about the features of an analog clock, such as its face and hands. "Analog Clock: Face and Hands" : "Uh,
Freddie? Why is it making that noise?" : "More importantly, how do we get it to STOP?" Let's take a look at the analog clock that Zuri and Freddie have found. This type of clock displays the time
with moving arrows, typically, around a circle. It's different from the rectangular digital clock, which displays time with digits instead of arrows. Today we'll focus on the features of an analog
clock. The analog clock shows numbers one through twelve around the circle. The clock starts at one and ends at twelve. An analog clock has arrows that move to show the time, also called hands. The
first arrow is the hour arrow or the short hand. You can rermember this because the hour hand is short, and so is the word "hour". The next arrow is the minute arrow, or minute hand. You can remember
this because the minute arrow is longer, and the word "minute" is longer than the word "hour". Which arrow is pointing to the number three on Zuri and Freddie's clock? (...) The shorter arrow is
pointing to the number three, so the HOUR HAND is pointing to the number three. Which arrow is pointing to the number twelve on Zuri and Freddie's clock? (...) The longer arrow is pointing to the
number twelve, so the MINUTE HAND is pointing to the number twelve. Once the minute hand makes it all the way around the clock, sixty minutes, or, one hour has passed. On a digital clock, there is
usually a label that says AM or PM. That tells us whether it's before 12:00 noon, AM, or after 12:00 noon, PM. There isn't a label on analog clocks that shows AM or PM (...) So, it's important to pay
attention to what is happening at that time to know if the time is AM or PM. For example, if Freddie says he's going to play outside at three using an analog clock we know it is 3:00 PM, or
afternoon, NOT 3:00 AM, or morning. If the analog clock says nine while Zuri eats breakfast, do you think it is 9:00 AM or PM? (...) (...) If Zuri is eating breakfast it is probably in the morning.
That means she will be eating at 9:00 AM. 9:00 AM shows something that is happening in the morning while 9:00 PM shows something that is happening at night. Let's summarize. Today we explored an
analog clock. We learned that it has numbers from one to twelve all around it, and that as time moves forward the arrows move around the clock in a direction called clockwise. We learned about the
hour and minute hands, and explored the difference between AM and PM. Let's see if Zuri and Freddie have figured out their analog clock yet. : "DID YOU MAKE IT STOP YET?"
Analog Clock: Face and Hands exercise
Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video Analog Clock: Face and Hands .
• Can you find all of the analog clocks?
Does the clock have a long hand and a short hand?
Numbers above 12 are not on analog clocks.
These are the analog clocks!
□ are usually circular
□ have a long (minute) hand, and a short (hour) hand
□ display the numbers 1 to 12
• Can you match the times?
Think about which hand is which. What is the blue hand showing? What is the red hand showing?
On these clocks, the blue hand is the longer minute hand and the red hand is the shorter hour hand.
This clock shows 2 o' clock. The long minute hand is at 12 and the short hour hand is at 2.
□ The first clock shows three o' clock. The long minute (blue) hand is at 12 and the short hour (red) hand is at 3. When the long minute hand is at 12, it is always an o' clock time.
□ The second clock shows 12 o' clock. Both hands are at 12.
□ The third clock shows 6 o' clock. The long minute hand is at 12 and the short hour hand is at 6.
□ The fourth clock shows 1 o' clock. The long minute hand is at 12 and the short hour hand is at 1.
• Do you know the features of an analog clock?
Can you use this clock to help you?
What shape is the clock? Which numbers are displayed?
Remember, the length of the hands help us to remember what each one does. On this clock, the blue hand is the minute hand and the red hand is the hour hand.
1. An analog clock is usually in the shape of a circle.
It displays numbers one through twelve around the edge.
2. It has two hands.
The hands move around in a clockwise direction to show the time.
3. The long hand is also called the minute hand.
The short hand is also called the hour hand.
4. The analog clock does not display AM or PM, so it is important to pay attention to what is happening to know what time of day it is.
• Can you order Zuri's day?
Does the picture give you a clue as to whether it is morning or evening?
Remember, 12 noon is in the middle of the day, then the analog clock goes back to 1:00pm.
Zuri got up at 7 o'clock and brushed her teeth at 7 o'clock, but one of them happened at 7:00 AM and one of them happened at 7:00 PM. You normally get out of bed in the morning, so Zuri got up at
7:00 AM. She is wearing her pajamas and it looks like the evening when she is brushing her teeth, so this event happened at 7:00 PM.
□ Zuri got up at 7:00 AM
□ Zuri ate her breakfast at 8:00 AM
□ Zuri ate her lunch at 12:00 noon, which is 12:00 PM
□ Zuri played outside with Freddie at 2:00 PM
□ Zuri brushed her teeth at 7:00 PM
□ Zuri went to sleep at 8:00 PM
• Can you label this analog clock?
Remember, the length of each hand gives us a clue about the job it does.
This clock is showing the time 9 o' clock because the long minute hand is at 12 and the short hour hand is at 9.
Here is the fully labelled analog clock.
□ We can remember that the long hand is the minute hand because minute is a longer word than hour.
□ We can remember that the short hand is the hour hand because hour is a shorter word than minute.
□ The numbers move around from 12 in a clockwise direction.
□ The time displayed on the clock is 3 o' clock.
• Can you solve Freddie and Zuri's puzzle?
Remember, when the minute hand goes all the way around, the hour hand just moves from one number to the next.
If it is going counter-clockwise, then the hour hand will move back one number.
If we are starting at 3:00 and going around the clock once, we are adding on one hour.
The correct time is 5 o' clock!
□ The clock started at 3 o' clock.
□ Freddie and Zuri wound it forwards one hour to 4 o' clock.
□ They then wound it forwards two more hours to 6 o' clock.
□ Finally, they wound it backwards one hour to 5 o' clock.
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Dual affine invariant points
An affine invariant point on the class of convex bodies K[n] in R^n, endowed with the Hausdorff metric, is a continuous map from K[n] to R^n that is invariant under one-to-one affine transformations
A on Rn, that is, p(A(K)) = A(p(K)). We define here the new notion of dual affine point q of an affine invariant point p by the formula q(K^p(K)) = p(K) for every K K[n], where K^p(K) denotes the
polar of K with respect to p(K). We investigate which affine invariant points do have a dual point, whether this dual point is unique and has itself a dual point. We also define a product on the set
of affine invariant points, in relation with duality. Finally, examples are given which exhibit the rich structure of the set of affine invariant points.
• Affine invariant point
• Dual affine invariant point
Dive into the research topics of 'Dual affine invariant points'. Together they form a unique fingerprint. | {"url":"https://experts.umn.edu/en/publications/dual-affine-invariant-points","timestamp":"2024-11-13T18:45:50Z","content_type":"text/html","content_length":"48383","record_id":"<urn:uuid:852a639b-5e3a-4703-b662-e302c20540e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00741.warc.gz"} |
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Six Possible Routes to Noninductive Tuned Circuitry, November 15, 1965 Electronics Magazine
November 15, 1965 Electronics
[Table of Contents]
Wax nostalgic about and learn from the history of early electronics. See articles from Electronics, published 1930 - 1988. All copyrights hereby acknowledged.
I remember in one of my circuits classes in college when the gyrator was introduced, and I thought it was an ingenious invention. The gyrator circuit, implemented with an operational amplifier
(opamp) and a couple resistors and capacitors, changed its measured impedance type from that of a capacitance to that of an inductance. That is, its impedance represents an R + jX Ω format. Frequency
limits are imposed by a combination of the self-resonant frequencies of the resistors and capacitors as well as the gain-bandwidth product (GBWP) of the opamp, and power handling is primarily limited
by the opamp's voltage and current capabilities. You might ask why, with all those constraints on its use you would even want to use a gyrator circuit? The answer is that within its limitations, the
gyrator often represents a less expensive and more compact version of a physical inductor. This is particularly true with integrated circuits (ICs) where, unless it is a monolithic microwave IC
(MMIC) operating in the tens of gigahertz region, there is no space available on the die for a printed metallic inductor with enough inductance to be useful. Any inductors would need to be mounted
off-chip on the PCB with I/O pins interfacing to the IC. Gyrators onboard ICs have made filtering functions available into the tens of megahertz realm nowadays with the extremely high GBWP of modern
Six Possible Routes to Noninductive Tuned Circuitry
By Vasil Uzunoglu, Applied Physics Laboratory
Johns Hopkins University, Silver Spring, Maryland
When circuit designers began shifting from tubes to transistors, they also began to seek ways to do away with bulky transformers and coils. One approach was to use resistance-capacitance (RC)
networks as substitutes for low-frequency inductance-capacitance (LC) circuits.
With the arrival of integrated circuits, the designers no longer have a choice. No practical method has been found for putting usable amounts of inductance into an integrated circuit, despite some
qualified successes. Multilayer thin films, deposited on monolithic chips, can provide a few microhenries of inductance; however, such small inductances are not adequate for operation at frequencies
lower than a few megacycles per second. More recently, an electromechanically resonant field-effect transistor has been introduced [Electronics, Sept. 20, 1965, p. 84], but its ultimate utility is
yet to be proved.
The engineer who wants to design a frequency-sensitive integrated circuit must find some way to duplicate the effect of inductance. There is no single perfect substitute for inductance, but at least
six techniques are known; the choice depends on the requirements of the system being designed.
Three of these techniques employ resistor-capacitor networks. One uses RC notch filters in the feedback path; another, RC circuits in the forward transmission path; and the third, negative impedance
converters. These methods have the same disadvantage: in some applications, particularly those in which high Q values (100 or larger) are needed, RC networks have a tendency toward instability. In
such cases, three other techniques are possible: sampling (digital filtering), using acoustic resonators, and using semiconductor delay lines.
The choice of one of these six approaches depends on the specific requirement; it should be based on a careful evaluation of the specifications, and on the comparison of these requirements with the
inherent advantages and disadvantages of each method.
Notch-Filter Feedback
A notch-filter circuit is one whose gain-versus-frequency characteristic exhibits a steep drop or rise at resonance. A typical voltage-gain curve for a notch-filter circuit is shown on page 115. Also
shown are two examples of notch-filter circuits: the parallel-T network and the bridged-T network.
Notch-filter circuits fall into two general categories: one called a minimum phase-shift type, the other called a nonminimum type. The minimum type exhibits a phase shift less than ±90°. The latter
can produce shifts in phase from 0 to 360°. This is shown by the curves on page 115. Minimum-phase-shift circuits are usually fabricated with lumped elements, nonminimum types are made with either
lumped or distributed elements.
When high Q is desired, the nonminimum type is preferable, but such circuits can be unstable. For a high degree of stability, when lower Q can be tolerated, the minimum type is usually best. For
stable oscillator circuitry,^2 however, the nonminimum circuit is preferred because of the sharper phase shift with changes in frequency.
Either type of network can be constructed using a bridged-T arrangement. If a minimum phase-shift circuit is desired, lumped resistors are used for R[1] and R[2]. For nonminimum phase shift, R[1] and
R[2] should be distributed elements. Both minimum and nonminimum phase-shift networks can be realized with lumped elements using the parallel-T circuit but the nonminimum network requires more
reactive (capacitive) components.
A nonminimum phase-shift circuit must have at least three capacitors.^1 However, the circuit can be designed so that the distributed resistors also contribute the required capacitance values.
A simple block diagram for an amplifier that incorporates a notch filter in the feedback path is shown on this page. Regardless of whether it is a minimum or nonminimum circuit, a notch filter must
satisfy two circuit requirements: the required Q must be obtained, and the insensitivity to minor variations in operating conditions must be sufficient to prevent oscillations.
The closed-loop transfer function (gain including feedback) is given by:
where A[T] is the total closed-loop gain, e[out] is the closed-loop output voltage, e[in] is the input voltage, A(s) is the open-loop gain of the amplifier (gain without feedback), β(s) is the
feedback factor, β(s) = Δe[out]/e[out], where Δe[out] is the feedback voltage.
To achieve the required Q without oscillations, the amplifier design must satisfy the conditions that |A(s)β(s)| ≈ 1 and the phase shift over the loop is approximately 360°. If the |Aβ| is actually
unity, and the phase shift 360°, oscillation would occur. Therefore, this product and phase angle should be approached but not actually reached.
The sensitivity of the gain for a closed-loop system^1 may be defined as:
where dA[T] is the variation caused by dA(s)
etc., where T is the period of the modulating function. The multiplied signal is then fed to a low-pass filter, h(t). The frequency difference between f(t) and e[in] must lie within the bandpass
limits of h(t). The outputs from the filters are multiplied again by functions Φ(t), Φ(t - T/N, Φ(t - 2T/N), in synchronism with f(t).The outputs from all branches are added up; this sum constitutes
the final output. The entire operation simulates the functioning of a series-tuned LC network that passes a desired frequency and rejects all other frequencies.
Because the modulating signals are sinusoidal, the driving-point (input) impedance of the entire network can be represented by.
where ω[o] is 2π times the center frequency and K is a constant determined by the circuit elements. Equation 7 is also the expression for the impedance of an inductor in series with a capacitor.
If the modulating time function, f(t) is an impulse and if the input signal is supplied by a current source each modulator can be replaced by a simple switch. Then the input and output modulating
signals are equivalent to a pair of rotary switches with N contacts as a common shaft that rotates at 1/T = ω[o]/2 cycles per second. Each low-pass filter, h(t), must have a bandwidth much lower than
f[o], the center frequency desired for the entire bandpass-filter network.
When only one passband is desired, the bandpass-filter circuit must have at least three h(t) sections to get rid of the harmonics (multiples of ω[o]). A mechanical sampling section for eliminating
harmonics^8 is depicted at the left. Each filter samples at a different time. As the switch's operating speed increases, the effectiveness of stopping the harmonics improves.
The extent of time during which the brushes remain on each contract is given by:
t[1] = T/R[1] and t[2] = T/R[2] (8)
where t[1], t[2] are contact times, R[1] is the source impedance, and R[2] is the load impedance.
An advantage of this technique is that the filter can be tuned at different frequencies without altering the system. The center frequency of the filter can be changed simply by changing the frequency
of the timing source that controls the switching rate. With this method, it is possible to achieve Q's of 5,000 to 10,000 at a few hundred kilocycles per second. These values are much higher than
those that can be obtained with any type of stable filter using RC networks.
Comb Filters
Besides its use as a very narrow bandpass filter, the same system can be used to build comb filters,^8 with bandpass centered at multiples of ω[o]. In the simplified mechanical analog of a comb
system, shown on page 119, t[1] is the time required by the brush to move from one contact to another. The input signal is applied through a high-value resistor to brush A (upper arrow on diagram).
While A is in contact with the corresponding capacitor, the capacitor begins to charge, so that its potential approaches that of the input signal. However, the time constant of the RC network is much
higher than the dwell time of the brush on one segment, so that it takes a certain time for the capacitor to charge; before the capacitor can build up an appreciable charge, the brush changes
If the signal frequency is a multiple of the frequency of rotation, the signal will have the same value each time the brush comes in contact with a given segment. Thus, after a certain number of
revolutions, the potential across each capacitor will attain its maximum value; this means that the locus of charge on the capacitors is an indication of the input-signal level. However, this system
prevents the buildup of random signals and signals that are periodic, or of signals that are periodic but whose frequency is not a multiplier of the rotational frequency. This suggests that such a
sampling filter may be used in detecting weak periodic signals in the presence of noise.
An electrical circuit equivalent of the mechanical data-sampling filter just discussed is shown on this page. This circuit was introduced by G. H. Danielson."
Acoustic Resonators
The acoustic-resonator technique requires mounting piezoelectric crystal onto a monolithic silicon chip. The installation is difficult, because the crystal must be positioned so that it is allowed to
vibrate freely.
When an electric wave is applied to the resonator, traveling acoustic waves are generated at the resonant frequency.^2 These traveling waves are reflected when they reach a boundary. If the resonator
is well designed, the initial transmitted and reflected acoustic energy are added together, causing an intense standing wave. This acoustic wave is converted back to an electric wave at the point of
application. At this point the electrical circuit sees the equivalent of a parallel circuit tuned to the resonant frequency. To achieve high Q, the losses must be minimized. The acoustic wave, as it
bounces back and forth, is subject to high losses. The use of an acoustic resonator in microelectronic blocks is feasible if the supporting medium of either the resonator or substrate does not absorb
the mechanical vibrations or permit leakage of the acoustic energy. Therefore, solid mounting of a conventional piezoelectric resonator is not possible. Piezoelectric materials such as cadmium
sulfide and zinc oxide have been used for acoustic resonators.
Semiconductor Delay Lines
Semiconductors delay lines^1 are relatively easy to integrate because only one diffusion is required in their manufacture. Only resistive elements used; capacitors are eliminated. In the
semiconductor delay line shown on this page, the distance between the two n regions determines the delay time and, therefore, the frequency.
Minority carriers are injected at the junction on the left and, being subject to an electric field, are diffused and drift to the right. When minority carriers are subjected to an electric field,
they cause a phase shift Φ which is given by:
where t[0] is the time delay, ω[1] is the lower bandpass limit frequency, and ω[2] is the upper bandpass limit frequency. The delay is a function of the length of the semiconductor path, as noted
above, also of the intensity of the electric field. If a delay line is inserted in the feedback path of an amplifier, it will cause a phase shift and attenuation, which will determine the closed-loop
gain and phase.
The same stability relations discussed earlier in this article for an RC network placed in an amplifier's feedback circuit also apply if a delay line is substituted for the RC network.
In general, an RC notch filter in the feedback path provides poor stability unless the Q requirements are not formidable. RC networks in the forward transmission path provide good stability but poor
Q values. With a negative impedance converter, both the stability and Q obtainable are somewhat better. All three methods have the disadvantage that they cannot provide high stability and a high Q
simultaneously. Digital filtering can, but it has the disadvantages of complexity and associated high cost.
Acoustic resonators are a recent development. One big disadvantage of these is that fabrication of circuits containing such devices is difficult. Semiconductor delay lines can be made small and are
relatively easy to integrate.
1. Vasil Uzunoglu, "Semiconductor Network Analysis and Design," Chapters 10 and 16, McGraw-Hill Book Co., 1964.
2. W.E. Newell, "Tuned Integrated Circuits," Proceedings of the IEEE, December 1964, pp. 1603-1607.
3. J.G. Linvill, "Transistor Negative-lmpedance Converters," Proceedings of the IRE, June 1953, Volume 41, pp. 725-729.
4. A.I. Larky, "Negative Impedance Converters," IRE Transactions on Circuit Theory CT-4, September 1957, pp. 124-131.
5. W.R. Lundry, "Negative Impedance Circuits-Some Basic Limitations," IRE Transactions on Circuit Theory CT·4, September 1957, pp. 132·139.
6. T. Yanagisava, "RC Active Networks Using Current Inversion Type Negative Impedance Converters," IRE Transactions on Circuit Theory CT-4, September 1957, pp. 140-144.
7. L.E. Franks and I.W. Sandberg, "An Alternative Approach to the Realization of Network Transfer Functions," The Bell System Technical Journal, September 1960, pp. 1321-1350.
8. W.R. DePage, et al., "Analysis of a Comb Filter Using Synchronously Commutated Capacitors," Electrical Engineering, March 1953, pp. 63-68.
9. G.H. Danielson, et al., "Solid State Microelectronic Systems," General Electric Report No., AD 426938, 1963.
The Author
Vasil Uzunoglu's book, "Semiconductor Network Analysis and Design," was published last year by the McGraw-Hill Book Co. He holds six patents and has applied for six more. On Nov. 1 he joined the
Arinc Research Corp. in Annapolis, Md., as a scientist in the devices research program.
Posted December 1, 2023
(updated from original post on 11/13/2018) | {"url":"https://www.rfcafe.com/references/electronics-mag/noninductive-tuned-circuitry-electronics-mag-nov-15-1965.htm","timestamp":"2024-11-12T23:33:04Z","content_type":"text/html","content_length":"47353","record_id":"<urn:uuid:51bd2416-af77-43e0-aaa2-d3424963c8b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00737.warc.gz"} |
Interest on the deposit and inflation
Money can be put into the banking account for interest. E.g., for a year. Besides, everybody knows about inflation. And everybody intuitively understands that a hundred rubles at the end of the year
are not the same as 100 rubles at the beginning of the same year - you can buy less for the same amount. Let's try to understand whether it's profitable to put money into the bank account to get your
interest at the end of the year. In 2007, the inflation rate in Russia was 11.9%. E.g., we put 10 000 rubles in the bank for a year at 9% per annum (compound interest accrued monthly - the most
profitable option), at the end of the year we will get about 10 938 rubles. However, it's equal to 9 716 rubles at the beginning of the year. In other words, the obtained 10 938 rubles at the end of
the year equal in the purchasing power to 9 716 at the beginning of the same year. But we had 10 000 rubles, and at the beginning of the year, we could have bought more by 284 rubles :(
Anyone willing to play around with the figures can do it in the calculator below:
Similar calculators
PLANETCALC, Interest on the deposit and inflation | {"url":"https://planetcalc.com/27/","timestamp":"2024-11-09T06:02:00Z","content_type":"text/html","content_length":"33445","record_id":"<urn:uuid:57e66927-6359-4596-87e9-4bbfc0f244a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00775.warc.gz"} |
TCP/IP - Networking MCQ Questions and answers |Technical Aptitude Page-2 section-1
Which of the following is the decimal and hexadecimal equivalents of the binary number 10011101?
Which statements are true regarding ICMP packets?
1. They acknowledge receipt of a TCP segment.
2. They guarantee datagram delivery.
3. They can provide hosts with information about network problems.
4. They are encapsulated within IP datagrams.
Which of the following are layers in the TCP/IP model?
1. Application
2. Session
3. Transport
4. Internet
5. Data Link
6. Physical
Which layer 4 protocol is used for a Telnet connection?
Which statements are true regarding ICMP packets?
1. ICMP guarantees datagram delivery.
2. ICMP can provide hosts with information about network problems.
3. ICMP is encapsulated within IP datagrams.
4. ICMP is encapsulated within UDP datagrams.
Which of the following are TCP/IP protocols used at the Application layer of the OSI model?
1. IP
2. TCP
3. Telnet
4. FTP
5. TFTP
What protocol is used to find the hardware address of a local device?
Which of the following protocols uses both TCP and UDP?
What is the address range of a Class B network address in binary? | {"url":"https://www.examveda.com/networking/practice-mcq-question-on-tcp-ip/?page=2","timestamp":"2024-11-12T05:59:05Z","content_type":"text/html","content_length":"64863","record_id":"<urn:uuid:92c52b47-ff3e-43a3-b0dd-22f54f1ea0d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00685.warc.gz"} |
IntroductionInverse Finite Element Formulation for Thin Shells(a) The plane stress element (b) the DKQ plate bending element (c) the iDKQ4 elementDiscrete surface strains measured on the iQS4 elementNumerical ValidationA cantilever Plate under Static Transverse Force Near Free TipCantilever plate under transverse force applied near free tipDiscrete cantilever plate model with iDKQ4Transverse displacement field derived from (a) direct FEM analysis (b) iFEM analysisA Thin-Walled Cylinder ModelSchematic diagram of cylindrical shell modelDiscretization of cylindrical shell using 960 iDKQ4 elements with strain rosettes per each elementDisplacement field in x direction calculated by (a) FEM (b) iFEM, displacement field in y direction calculated by (c) FEM (d) iFEMDiscretization of cylindrical shell using 960 iDKQ4 elements with strain rosettes located within 201 select elementsDisplacement field in (a) Ux (b) Uy direction obtained by iFEM inversionShape Sensing of Composite TankSchematic diagram of tankMechanical properties of the CFRP laminaDiscretization of tank model with strain rosettes located within (a) all elements (b) selected elementsDisplacement field in (a)
direction calculated by FEM and (b), (c)
(e), (f)
direction reconstructed by iFEMConclusionReferences
In the last decades, curved thin-shell structure such as composite tank of spacecraft has been widely used in aerospace because of its excellent bearing capacity and weight-saving [1,2]. Structural
integrity is the key factor to ensure its function and strength, but the complex construction process makes it prone to defects. Meanwhile, the tank structure bears time-varying load conditions
during service, which may damage its structural integrity and reduce its remaining life. Therefore, the establishment of a health monitoring system for real-time monitoring structural state and
predicting damage can play an important role in the whole life cycle of structure manufacturing, service and maintenance. Traditional nondestructive testing technologies such as ultrasonic testing
[3,4] and acoustic emission technology [5,6] have the problems of time-consuming and high cost which make it not suitable for real-time monitoring of structural response. So the health monitoring
technology based on embedded strain sensors (optical fiber sensors, strain gauges, etc.) will become an important direction of structural health assessment in the future [7,8].
The dynamic reconstruction of the three-dimensional displacement field of a structure known as shape sensing is a crucial component of structural health monitoring which provides data support for
subsequent calculation of stress and strain and failure prediction. Furthermore, the real-time evaluation of the deformed shape is also a vital technology for the development of smart structures such
as morphed capability and embedded conformal antennas that require real-time shape sensing to provide feedback for their actuation and control systems.
Numerous studies on shape sensing found in the open literature can be divided into the following categories: (1) the Modal Method (MM) [9–12]; (2) analytical methods [13–15]; (3) Artificial Neural
Networks (ANN) [16,17]; (4) the inverse Finite Element Method (iFEM) [18–20]. MM, firstly developed by Foss et al. [9], is a modal transformation algorithm in which the displacement field of the
structure is expressed by modal shapes and corresponding weights. The modal shapes are known and the weights need to be computed using strain–displacement relationship and measured surface strains.
There are two different ways to calculate modal shapes. The first one is to use the finite element method or analytic method, but it requires such prior information as the material properties [11].
The other is to estimate experimentally but it could be significantly onerous [12]. Based on the Bernoulli–Euler beam theory, Ko et al. [13] constructed the displacement transfer function by fitting
the axial strain distribution with piecewise polynomials, and further obtained the bending deflection of the beam. However, Ko’s theory only considers simple bending deformation of the beam, and
could do nothing about the coupling deformation of beam structures subjected to highly coupled loading cases. Xu et al. [14,15] proposed a novel method that could decouple complex beam deformations
subject to the combination of different loading cases, including tension/compression, bending and warping torsion to reconstruct deformed shape of thin-walled beam structures. The ANN needs a large
number of parameters, such as network topology, weight and initial value of threshol and a lot of training time. Moreover, ANN can be regarded as a black box and the results calculated are difficult
to explain, which will affect the credibility and acceptability of the results. Generally speaking, each of the above methods has certain limitations, so the scope of application is limited.
In order to reconstruct the three-dimensional displacement field in real-time with strain data obtained from the structure surface, Tessler et al. [20] developed an inverse Finite Element Method
(iFEM). The iFEM is formulated based on a weighted-least-square error functional between the analytical and experimental values of strain on the structure surface. Like the classic Finite Element
Method (FEM), iFEM is also a model-based technique. Therefore, the application of iFEM is not limited by complex boundary geometry and conditions. Moreover, the formulation only makes use of
strain-displacements relations which make any information about materials or load acting on the structure is unnecessary. Numerous studies had proved the applicability and robustness of iFEM and
different elements such as iMIN3 [20], iQS4 [19], and iCS8 [21] elements and so on had been developed for different application structures. All elements are developed based on Mindlin theory, and
interpolated using the anisoparametric shape functions developed by Tessler and Hughes to avoid shear locking when modeling thin shell structures. Though it has achieved good results, the
Kirchhoff–Love shell model is well suited for thin shell analysis in fact because it disregards transverse shear deformations which reveals that the deformation of thin shells is physically dominated
by membrane and bending actions and the shear locking can be avoided completely.
The main focus of this work is to redefine the weighted-least-square error functional based on classical plate theory. Subsequently, a new four-node quadrilateral inverse-shell element, iDKQ4, is
developed for numerical implementation. The new element includes hierarchical drilling rotation degrees-of-freedom (DOF) to enhance applicability in modeling complex structures. This study is
organized as follows: the iDKQ4 element is presented in brief in Section 2. Besides, an iFEM formulation developed utilizing the kinematics of Kirchhoff–Love shell theory for thin shells, is
introduced. In Section 3, a cantilever plate model is firstly employed to demonstrate the reconstruction performance of the iDKQ4 element. Then a wallpanel is analyzed to demonstrate the robustness
for modeling complex shell structures. Finally, the deformation of a typical aerospace thin wall structure (the composite tank) is computed with a few strain data with the help of the iDKQ4 element.
The conclusions of this study are provided in Section 4.
Consistent with obtaining a flat element formulation, the inverse shell element can also be regarded as a superposition of a plate-bending element and a membrane element. In this paper, the four node
plane stress (as shown in Fig. 1a) element and the DKQ plate bending element based on discrete Kirchhoff theory [22] (as shown in Fig. 1b) are selected and add them together to get the inverse shell
element iDKQ4 (as shown in Fig. 1c). There are six degrees-of-freedom (DOFs) at each node, as shown in Fig. 1c, where u and v are in plane translations; w is transverse deflection; θx and θy are
bending rotations, and θz is in-plane rotation. Due to inclusion of drilling rotations, the adaptability of iDKQ4 element to complex structures is greatly enhanced. Furthermore, the transverse shear
is neglected in Kirchhoff–Love shell theory which reveals that the shear locking can be completely avoided.
The 4-node membrane element with drilling DOFs is derived by combining the in-plane displacements using Allman-type interpolation functions [23,24] with the standard bilinear independent normal
(drilling) rotation fields.
where N is the standard bilinear isoparametric function, and L and M are the shape functions that define the interaction between the drilling rotation fields and the displacement of the element
membrane. Details of the formulation can be found in the original literature.
At the four corner nodes, two bending rotations θx , θy are the derivatives of the displacement w.
Therefore, these kinematic variables are related using the shape functions developed by Batoz for DKQ element [22]. These interpolations are given as
From the strain-displacement relationship of linear elastic theory, we can know that
It should be noted that the plane stress assumption σzz means that the transverse-normal strain εzz has no contribution to the strain energy.
The generalized strains vector consisting of membrane strain e and bending strain k can be defined as
where the node displacement vector of iDKQ4 element can be expressed as ue=[u1e,u2e,u3e,u4e] , and each node contains six degrees of freedom uie=[ui,vi,wi,θxi,θyi,θzi]T (i = 1,…,4). The matrices B
contain derivatives of the shape functions, and the explicit expressions are as follows:
In order to decouple plane strain and curvatures, the strain rosettes need to be attached to the top and bottom surfaces of the element as shown in Fig. 2. The sensor can use traditional strain
gauges or fiber-optic sensor such as distributed optical fiber, and optical fiber can collect a large amount of strain data as input for iFEM calculation, which make it more attractive.
The counterpart of membrane strains and bending curvatures calculated from Eqs. (11) and (12) can obtain by using the following formula with measured strain data:
where {εxx+εyy+γxy+}iT and {εxx−εyy−γxy−}iT are the measured strains on the upper and lower surfaces respectively with the superscripts ‘+’ and ‘−’ denoting the quantities that correspond to the top
and bottom surface locations, respectively.
For an individual inverse element, the error functional with respect to DOFs of the entire discretization can be expressed as:
where εk(ue) and εkε represent the theoretical strain and the corresponding measured values at a given point respectively. The error function consist with membrane strain, bending strain only which
is consistent with the classical plate theory.
The squared norms expressed in Eq. (22) can be written in the form of the normalized Euclidean norms:
where Ae represents the area of the middle surface of the element and n is the number of measuring points in the element. wk is the weight coefficient that represents the strength of the constraint
between the theoretical strain and its corresponding measured value. The specific form is defined by Eq. (24).
If all the values in {εε} can be obtained, the weight coefficient (k = 1,…, 6) is set to 1, otherwise, the corresponding weight coefficient is set to a small value, for example, 10−5 .
Minimizing the error function with respect to the unknown nodal displacement DOF gives rise to
where ke is only a function of the measuring position. Once the measuring point position is determined, it is determined. fe is a function of the measuring position and the measured strains, which
are updated in real-time with the measured strains. The variational statement given in Eq. (25) results in a linear system of equations that can be solved for the unknown DOF provided that
appropriate displacement boundary conditions have been imposed.
Firstly, the cantilever plate model is used to verify the accuracy of iDKQ4 element. As shown in Fig. 3, the dimension of the cantilever plate is 254 × 76.2 × 3.175 mm. The material is aluminium
alloy (with Young’s modulus of 73.084 GPa, Poisson’s ratio of 0.33 and density of 2700 kg/m^3). A concentrated force of 25.728 n is applied along the negative direction of the z-axis near the tip.
Bogert et al. [11] initially analyzed the plate and then tested it in the mechanics laboratory. Then, Tessler et al. [25] analyzed this structure using iFEM method with iMIN3 element. Kefal et al.
[19] also adopted iFEM method to reconstructed the displacement field of this structure with iQS4 element to verify its bending performance.
In order to validate the bending capability of iDKQ4 element, the cantilever plate is discretized with 28 inverse elements to ensure that the position of the strain-rosette is coincident with the
selection in the work by Tessler and Kefal. As depicted in Fig. 4, each rectangular element has a single strain rosette and the strain rosettes are placed at the centroids of each element.
High-fidelity FEM analysis is performed with ABAQUS, a commercially available finite element software, to generate the strain data as input of iFEM calculation. Moreover, The displacement field
calculated by FEM analysis can also be used as a benchmark to examine the reconstruction accuracy of iFEM. Contour plots for the transverse displacement are compared between the iFEM and high
fidelity FEM analyses as shown in Fig. 5. It can be seen that the transverse displacement field reconstructed by iFEM is basically consistent with that calculated by FEM. The percent difference
between the iFEM and FEM predictions for the maximum deflection is only 0.01%; this result is slightly batter than the predictions of Tessler and Kafel.
In Section 2, the calculation accuracy of the iDKQ4 element is verified by a simple cantilever plate model. However, structures with complex topology are very common in practical engineering
applications. Therefore, in this section, the robustness and adaptability of the iDKQ4 element in modeling complex shell structure are verified with a quarter of a thin-walled cylinder shell (as
shown in Fig. 6). The diameter of the cylinder shell is 1 m, the height is 1.5 m and the uniform thickness is 3 mm. The cylinder is made of aluminium alloy having an elastic modulus of 73.084 GPa and
the Poisson’s ratio of 0.33. The cylinder shell adopts the boundary condition that the lower edge is fixed, and 100N concentrated force is applied at two positions of the upper edge, respectively.
The finite element convergence is studied to establish an exact reference solution of the problem. In FEM calculation, 930 rectangular S4R elements are used to discretize the cylindrical shell
uniformly. In order to facilitate the transmission of strain data, iFEM calculation adopts the same discretization with FEM analysis. As shown in Fig. 7, each element has two strain rosettes, one on
the centroid of the top surface and the other one on the centroid of the bottom surface resulting 1860 strain rosettes in total.
The displacement field calculated by direct FEM analysis and the reconstructed result by iFEM are shown in Fig. 8. It can be seen from the figure that the calculation results of FEM and iFEM are
graphically indistinguishable; The reconstruction error of iFEM in Ux direction is only 0.97%; The reconstruction error of iFEM in Uy direction is only 0.98%. Although the calculation results are
satisfactory, too many strain rosettes are used. Therefore, it is worthful to explore the accuracy of reconstruction of iFEM with sparse strain data (402 strain rosettes are used as shown in Fig. 9).
In Fig. 10, the contour plots for the Ux and Uy displacement are depicted for the iFEM analyses. And the corresponding FEM analysis result is depicted in Figs. 8a and 8b. It can be seen that iFEM can
accurately reconstruct the deformation tendency of the structure even with a few of strain data. The prediction error of the maximum displacement in Ux direction is 4.52%; The prediction error of the
maximum displacement in Uy direction is 4.61%. The iFEM predictions remain sufficiently accurate even with sparse strain-rosette data.
Although having numerous advantages of high specific strength, high specific modulus, corrosion resistance and designable performance, the composite is prone to damage such as delamination and
debonding, which often is invisible and has a fatal impact on the bearing capacity of the tank. The robustness and adaptability of the improved inverse finite element method/iDKQ4 element are
verified by cantilever plate and cylindrical shell structures. In this section, the composite tank is investigated with the improved iFEM algorithm.
The geometric dimensions of the tank are shown in Fig. 11. The height of the straight barrel section is 458 mm and the diameter is 3338 mm; The section of the head section is an ellipse with a
semi-major axis of 1669 mm and a semi-minor axis of 1043.12 mm. The radius of the upper and lower manholes is 500 mm. The reinforced composite tank adopts a symmetric stacking sequence, corresponding
to [0/±45/90/±45/90/±45/0]s , with the thickness of each layer of 0.15 mm and the total thickness of 3 mm. The material properties of the single lamina are listed in Table 1.
Young’s modulus [GPa](E1/E2/E3) Shear modulus [GPa](G12/G13/G23) Poisson’s ratio(v12/v13/v23) Density [ kg/m3 ]
135/7.579/7.579 4.49/4.49/3.2 0.32/0.32/0.49 1620
Firstly, the linear static analysis of the tank is carried out in ABAQUS using a high fidelity grid composed of 10886 S4R shear deformation shell elements. The same grids is used for iFEM calculation
and FEM analysis. The strain calculated by FEM is used as the calculation input of iFEM, and the displacement field obtained by FEM analysis is used to evaluate the prediction ability of iFEM. In
order to avoid introducing errors in calculating local strain field, when the input strain field is not fully defined, elements should have a rectangular shape aligned with the input strain field
direction. It is obvious that the elements of the covers do not meet this requirement, but fortunately we do not care about the results of the metal covers. In the first case study, the strain of all
elements except covers can be obtain as presented in Fig. 12a. However, the number of strain rosettes used is too high to be applied to practical engineering applications. In the second case study
shown in Fig. 12b, a large number of strain-rosettes are removed from elements and only the elements on 14 circumferential paths and 15 radial paths are reserved.
To assess the global displacement, it is convenient to compute the axial displacement Ua and radial displacement Ur :
Figs. 13a–13f presents the displacement results of FEM analysis and iFEM reconstruction. The tank expands outward uniformly under internal pressure and the result of iFEM and FEM obtains can
accurately describe this trend. In Figs. 13a, 13b, 13d and 13e, the iFEM and FEM contour plots for Ua and Ur are presented, showing the results that are graphically indistinguishable. The percent
difference between the iFEM and FEM solutions for the maximum values of Ua and Ur are respectively 5.85% and 1.03%. The iFEM can reconstruct the deformation of the structure relatively accurately
even with sparse strain input, as shown in Figs. 13c and 13f. The percent difference between the iFEM and FEM predictions for the maximum Ur displacement is 1.80%, whereas it is only 0.38% for
maximum total rotation. | {"url":"https://cdn.techscience.cn/ueditor/files/sdhm/TSP_SDHM-16-1/TSP_SDHM_19554/TSP_SDHM_19554.xml?t=20220620","timestamp":"2024-11-08T21:12:31Z","content_type":"application/xml","content_length":"105542","record_id":"<urn:uuid:e74f2a69-6cea-484d-b3aa-a71af74c0bc2>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00264.warc.gz"} |
2017 AMC 8 Problems/Problem 16
In the figure below, choose point $D$ on $\overline{BC}$ so that $\triangle ACD$ and $\triangle ABD$ have equal perimeters. What is the area of $\triangle ABD$?
$[asy]draw((0,0)--(4,0)--(0,3)--(0,0)); label("A", (0,0), SW); label("B", (4,0), ESE); label("C", (0, 3), N); label("3", (0, 1.5), W); label("4", (2, 0), S); label("5", (2, 1.5), NE);[/asy]$
$\textbf{(A) }\frac{3}{4}\qquad\textbf{(B) }\frac{3}{2}\qquad\textbf{(C) }2\qquad\textbf{(D) }\frac{12}{5}\qquad\textbf{(E) }\frac{5}{2}$
Solution 1
Because $\overline{BD} + \overline{CD} = 5,$ we can see that when we draw a line from point $B$ to imaginary point $D$ that line applies to both triangles. Let us say that $x$ is that line. Perimeter
of $\triangle{ABD}$ would be $\overline{AD} + 4 + x$, while the perimeter of $\triangle{ACD}$ would be $\overline{AD} + 3 + (5 - x)$. Notice that we can find $x$ from these two equations by setting
them equal and then canceling $\overline{AD}$. We find that $x = 2$, and because the height of the triangles is the same, the ratio of the areas is $2:3$, so that means that the area of $\triangle
ABD = \frac{2 \cdot 6}{5} = \boxed{\textbf{(D) } \frac{12}{5}}$.
Solution 2
We know that the perimeters of the two small triangles are $3+CD+AD$ and $4+BD+AD$. Setting both equal and using $BD+CD = 5$, we have $BD = 2$ and $CD = 3$. Now, we simply have to find the area of $\
triangle ABD$. Since $\frac{BD}{CD} = \frac{2}{3}$, we must have $\frac{[ABD]}{[ACD]} = 2/3$. Combining this with the fact that $[ABC] = [ABD] + [ACD] = \frac{3\cdot4}{2} = 6$, we get $[ABD] = \frac
{2}{5}[ABC] = \frac{2}{5} \cdot 6 = \boxed{\textbf{(D) } \frac{12}{5}}$.
Solution 3
Since point $D$ is on line $BC$, it will split it into $CD$ and $DB$. Let $CD = 5 - x$ and $DB = x$. Triangle $CAD$ has side lengths $3, 5 - x, AD$ and triangle $DAB$ has side lengths $x, 4, AD$.
Since both perimeters are equal, we have the equation $3 + 5 - x + AD = 4 + x + AD$. Eliminating $AD$ and solving the resulting linear equation gives $x = 2$. Draw a perpendicular from point $D$ to
$AB$. Call the point of intersection $F$. Because angle $ABC$ is common to both triangles $DBF$ and $ABC$, and both are right triangles, both are similar. The hypotenuse of triangle $DBF$ is 2, so
the altitude must be $6/5$ Because $DBF$ and $ABD$ share the same altitude, the height of $ABD$ therefore must be $6/5$. The base of $ABD$ is 4, so $[ABD] = \frac{1}{2} \cdot 4 \cdot \frac{6}{5} = \
frac{12}{5} \implies \boxed{\textbf{(D) } \frac{12}{5}}$.
Solution 4
Using any preferred method, realize $BD = 2$. Since we are given a 3-4-5 right triangle, we know the value of $\sin(\angle ABC) = \frac{3}{5}$. Since we are given $AB = 4$, apply the Sine Area
Formula to get $\frac{1}{2} \cdot 4 \cdot 2 \cdot \frac{3}{5} = \boxed{\textbf{(D) } \frac{12}{5}}$.
Video Solution (CREATIVE THINKING + ANALYSIS!!!)
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Video Solution by OmegaLearn
~ pi_is_3.14
Video Solutions
See Also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. | {"url":"https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_8_Problems/Problem_16&oldid=191685","timestamp":"2024-11-04T08:07:15Z","content_type":"text/html","content_length":"55827","record_id":"<urn:uuid:89d86cb7-fcb2-4e01-8369-c6cac7072b11>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00151.warc.gz"} |
[QSMS Symplectic geometry seminar 2023-10-24] Linear bounds on rho-invariants and simplicial complexity of manifolds
• Date: 2023-10-24 (Tue) 11:00 ~ 12:00
• Place: 27-116 (SNU)
• Speaker: Geunho Lim (Einstein Institute of Mathematics, Hebrew University of Jerusalem)
• TiTle: Linear bounds on rho-invariants and simplicial complexity of manifolds
• Abstract: Using L^2 cohomology, Cheeger and Gromov define the L^2 rho-invariant on manifolds with arbitrary fundamental groups, as a generalization of the Atiyah-Singer rho-invariant. There are
many interesting applications in geometry and topology. In this talk, we show linear bounds on the rho-invariants in terms of simplicial complexity of manifolds. First, we obtain linear bounds on
Cheeger-Gromov invariants, using hyperbolizations. Next, we give linear bounds on Atiyah-Singer invariants, employing a combinatorial concept of G-colored polyhedra. As applications, we give new
concrete examples in the complexity theory of high-dimensional (homotopy) lens spaces. This is a joint work with Shmuel Weinberger. | {"url":"https://qsms.math.snu.ac.kr/index.php?mid=board_sjXR83&l=en&page=4&sort_index=title&order_type=asc&document_srl=2714","timestamp":"2024-11-09T12:55:00Z","content_type":"text/html","content_length":"66067","record_id":"<urn:uuid:60399770-3e40-4c35-8f47-d7a05ccbf297>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00726.warc.gz"} |
Higher Harmonic Control and Flight Control System Interaction
Title: Model for Application to Higher Harmonic Control and Flight Control System Interaction
This dissertation addresses a system, generally known as “Higher Harmonic Control” (HHC) because it consists of superimposing high frequency rotor inputs to the conventional low frequency ones used
to control and maneuver the helicopter….
1 Introduction
1.1 Motivation
1.2 Literature review
1.2.1 Higher harmonic control technology
1.2.2 Linear models
1.3 Objectives of study
1.4 Principal contributions
1.5 Organization of the dissertation
2 Mathematical Model
2.1 History of helicopter simulation model
2.2 Helicopter model
2.3 HHC implementation
2.4 Solution methods: trim
2.4.1 Algebraic trim
2.4.2 Periodic trim
2.5 Solution methods: linearization of the equation of motion
2.6 Solution methods: time integration
2.7 Vibration calculation
2.7.1 Hub loads calculation
2.7.2 Cockpit vibration calculation with the rigid fuselage
2.7.3 Cockpit vibration calculation with the flexible fuselage
2.8 Optimization formulation
3 Active Rotor Control System for Vibration Suppression
3.1 Harmonic analyzer
3.1.1 Analog bandpass filter method
3.1.2 Fourier analyzer method
3.1.3 Effect of windowing
3.1.4 Equivalent lowpass filter
3.2 Higher harmonic control algorithm
3.2.1 T-matrix method
3.2.2 T-matrix validation
3.3 Discrete HHC update
4 Extraction of the Constant-Coefficient Linearized Model
4.1 Extraction of a linearized model without higher harmonics
4.2 Extraction of a linearized model with higher harmonics
4.2.1 Definitions
4.2.2 Extraction of the control matrix B
4.2.3 Extraction of the state matrix A
4.2.4 Extraction of the feedforward matrix D
v4.2.5 Extraction of the output matrix C
4.3 Application to simple rotor equations
4.3.1 Prescribed solution form
4.3.2 Perturbation of the equations of motion
4.3.3 Extract four/rev harmonic components
4.3.4 …
5 HHC and AFCS Interaction Study
5.1 Effect of a fixed HHC input on rigid body dynamics
5.1.1 Open-loop frequency response validation
5.1.2 Effect of an optimum three/rev input on rigid body dynamics
5.2 Interaction of HHC and AFCS
5.2.1 Broken control loop response validation
5.2.2 ……
6 Summary and Conclusions
Author: Cheng, Rendy Po-Ren
Source: University of Maryland
Leave a Comment | {"url":"http://www.projectsparadise.com/flight-control-system-interaction/","timestamp":"2024-11-06T10:41:47Z","content_type":"text/html","content_length":"67090","record_id":"<urn:uuid:820eb292-9063-4f96-beae-745ea3fe7b46>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00629.warc.gz"} |
CST3607 Class Notes 2021-09-11 - ConsciousVibes.com
CST3607 Class Notes 2021-09-11
News & Tools
Assignment #2 Debriefing
Subnetting into a Large Number of Subnets
• Incrementing subnets using the Block Size works for a small number of subnets, but is not efficient when you need hundreds or thousands or millions of subnets. It doesn’t scale.
• Using the Base-256 conversion scales.
Determine the network address of a high subnet number.
1. Multiply the target subnet number by the number of addresses per subnet, to get the number of addresses to add to the network address (subnet zero) to jump to the target subnet.
2. Convert the resulting number of addresses to its Base-256 (dotted-decimal) equivalent.
3. Add the Base-256 (dotted-decimal) equivalent to the network address/subnet zero, to determine the target subnet address.
Notes about the “target subnet”
• If you’re given subnet number x, then you use x as is to multiply by the number of addresses per subnet.
• If you’re given the n^th, subnet, e.g. 59^th, 343^rd, then you subtract one, then multiply by the number of addresses per subnet. (Because we start counting from zero.)
Converting a Decimal Number to Base 256 (Dotted-decimal)
Calculations for Base-256 Conversion
Evaluate the # Is the # greater than 256?
4th Octet .
3rd Octet .
2nd Octet .
1st Octet .
Subnetting Tips/Notes
• If no mask/prefix is given, then borrow bits starting from the “Class” boundary of the IP address.
• If a mask/prefix is given, then the given mask/prefix is the result of subnetting. (Borrow bits from the “Class” boundary to the given mask/prefix.) (e.g. Q. 7, Pg. 40)
• The total number of subnets and total number of hosts must be a power of 2.
• Is the question asking for “subnets” or “hosts”
□ If you’re asked for the # of hosts, then you must determine how many bits are needed to get that # of hosts, then subtract those bits from the 32 IPv4 bits, to determine the network bits /
mask / prefix.
• Determine the number of subnets: 2^[number of bits borrowed].
• Determine the total number of addresses: 2^[the number of host bits].
• Add the Wildcard mask to the network/subnet address to determine the broadcast/last address in the network/subnet.
• Block Size:
□ The block size (256 – [The interesting octet]) is best used to determine the increment of the subnets.
□ The interesting octet is the last octet, from the left, that you borrowed bits from.
□ The “block size” is not the number of addresses per subnet. It is the increment from one subnet to the next, within the “interesting” octet.
• Determine how many addresses to add to the network address/subnet zero to get to the target subnet.
□ 1. Multiplying (Subnet “Number”) by the (number of addresses per subnet).
(For the N^th subnet, subtract 1 before multiplying by the number of addresses per subnet.)
□ 2. Convert the result to its Base-256 equivalent
□ 3. Add the Base-256 equivalent to the original network address of the block to get the network/subnet address of the target subnet.
• The “subnet address” is an alternate term for the “network address” of a subnet.
• Subnet using the methods that work for all subnets, large or small. Switching methods depending on the size of the subnet requires more effort than is necessary.
• Practice makes improvement!
Do: Assignment #2: Due before Tues. Feb. 23, 2020 6pm EST
• Important: Make sure to read and understand the instructions on how to handle the protected PDF
• If you have any issues completing all parts of every question on the assignment, e-mail me with the question # and the specifics you need assistance with.
• No late assignments will be accepted.
Read / Watch / Do
• Read Chapter : 6 OSPF (Open Shortest Path First)
• Do the Written Labs
• Answer the Review Questions
□ Do not submit your answers for this chapter. The answers are in Appendix.
Make sure to always have access to a calculator which has an Exponent function (^key) ( x^y ) for every class. | {"url":"https://consciousvibes.com/citytech/cst3607/2021-fall/classnotes-2021-09-11/","timestamp":"2024-11-05T13:43:15Z","content_type":"text/html","content_length":"118112","record_id":"<urn:uuid:8ff80882-d5e0-40c3-9fea-4a2ea099de98>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00769.warc.gz"} |
Problem of the Week
Problem C and Solution
Five Magnets
Harlow has five magnets, each with a different number from \(1\) to \(5.\) They arranged these magnets to create a five digit number \(ABCDE\) such that:
• the three-digit number \(ABC\) is divisible by \(4\),
• the three-digit number \(BCD\) is divisible by \(5\), and
• the three-digit number \(CDE\) is divisible by \(3.\)
Determine the five-digit number that Harlow created.
Since \(ABC\) is divisible by \(4\), it follows that \(C\) must be even, so \(C=2\) or \(C=4.\)
Since \(BCD\) is divisible by \(5\), it follows that \(D=0\) or \(D=5.\) However, there is no magnet with a \(0\), so it follows that \(D=5.\)
We also know that \(CDE\) is divisible by \(3.\) We can consider the following two cases.
• Case 1: \(C=2\).
If \(C=2\), then the three-digit number \(CDE\) is \(25E.\) The only possibilities for \(E\) are \(1,\) \(3,\) or \(4\). However, none of \(251,\) \(253\) and \(254\) are divisible by \(3.\) It
follows that \(C\) cannot equal \(2.\)
• Case 2: \(C=4\).
If \(C=4\) then the three-digit number \(CDE\) is \(45E.\) The only possibilities for \(E\) are \(1,\) \(2,\) or \(3.\) Since \(451\) and \(452\) are not divisible by \(3,\) but \(453\) is
divisible by \(3,\) it follows that \(C=4\) and \(E=3.\)
Thus, the three-digit number \(ABC\) is \(AB4.\) The only magnets not used yet are numbered \(1\) and \(2,\) so this number is \(124\) or \(214.\) Since \(214\) is not divisible by \(4,\) but \(124\)
is divisible by \(4,\) it follows that \(A=1\) and \(B=2.\)
Therefore, the five-digit number must be \(12453.\) | {"url":"https://cemc.uwaterloo.ca/sites/default/files/documents/2024/POTWC-24-N-07-S-070.html","timestamp":"2024-11-09T13:53:21Z","content_type":"application/xhtml+xml","content_length":"64591","record_id":"<urn:uuid:2413b1e6-3970-4a7b-965b-d883699408d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00050.warc.gz"} |
Observation of Hawking radiation by 2040?
Resolves YES if Hawking radiation is observed by 2040. This must be Hawking radiation from a black hole in GR. An astrophysical black hole counts, but so would an artificially made black hole. A
quantum system claimed to be analogous to thus-and-such theory of gravity will not count.
The observation need not be "direct". It is sufficient for the BH's mass to be observed to be decreasing, for instance.
This question is managed and resolved by Manifold.
@TomBouley that should resolve YES.
It'll be subjective. If the observation happens by 2040, I'll wait a few years to see how the consensus evolves. | {"url":"https://manifold.markets/ScottLawrence/observation-of-hawking-radiation-by","timestamp":"2024-11-09T08:16:47Z","content_type":"text/html","content_length":"125647","record_id":"<urn:uuid:ac96ad79-7b0c-4c59-b108-bfca92507e37>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00665.warc.gz"} |
An Introduction to Mathematics for Economics
Akihito Asano
in Cambridge Books from Cambridge University Press
Abstract: An Introduction to Mathematics for Economics introduces quantitative methods to students of economics and finance in a succinct and accessible style. The introductory nature of this
textbook means a background in economics is not essential, as it aims to help students appreciate that learning mathematics is relevant to their overall understanding of the subject. Economic and
financial applications are explained in detail before students learn how mathematics can be used, enabling students to learn how to put mathematics into practice. Starting with a revision of basic
mathematical principles the second half of the book introduces calculus, emphasising economic applications throughout. Appendices on matrix algebra and difference/differential equations are included
for the benefit of more advanced students. Other features, including worked examples and exercises, help to underpin the readers' knowledge and learning. Akihito Asano has drawn upon his own
extensive teaching experience to create an unintimidating yet rigorous textbook.
Date: 2012
References: Add references at CitEc
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Book: An Introduction to Mathematics for Economics (2012)
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Bibliographic data for series maintained by Ruth Austin ( this e-mail address is bad, please contact ). | {"url":"https://econpapers.repec.org/bookchap/cupcbooks/9780521189460.htm","timestamp":"2024-11-02T08:36:15Z","content_type":"text/html","content_length":"12734","record_id":"<urn:uuid:bbc6a40c-ec95-4e6f-9001-fb7e2b6dd8bb>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00545.warc.gz"} |
Build Your First Neural Network with PyTorch
TL;DR Build a model that predicts whether or not is going to rain tomorrow using real-world weather data. Learn how to train and evaluate your model.
In this tutorial, you’ll build your first Neural Network using PyTorch. You’ll use it to predict whether or not is going to rain tomorrow using real weather information.
You’ll learn how to:
• Preprocess CSV files and convert the data to Tensors
• Build your own Neural Network model with PyTorch
• Use a loss function and an optimizer to train your model
• Evaluate your model and learn about the perils of imbalanced classification
1%reload_ext watermark
2%watermark -v -p numpy,pandas,torch
1CPython 3.6.9
2IPython 5.5.0
4numpy 1.17.5
5pandas 0.25.3
6torch 1.4.0
1import torch
3import os
4import numpy as np
5import pandas as pd
6from tqdm import tqdm
7import seaborn as sns
8from pylab import rcParams
9import matplotlib.pyplot as plt
10from matplotlib import rc
11from sklearn.model_selection import train_test_split
12from sklearn.metrics import confusion_matrix, classification_report
14from torch import nn, optim
16import torch.nn.functional as F
18%matplotlib inline
19%config InlineBackend.figure_format='retina'
21sns.set(style='whitegrid', palette='muted', font_scale=1.2)
23HAPPY_COLORS_PALETTE =\
24["#01BEFE", "#FFDD00", "#FF7D00", "#FF006D", "#93D30C", "#8F00FF"]
28rcParams['figure.figsize'] = 12, 8
30RANDOM_SEED = 42
Our dataset contains daily weather information from multiple Australian weather stations. We’re about to answer a simple question. Will it rain tomorrow?
The data is hosted on Kaggle and created by Joe Young. I’ve uploaded the dataset to Google Drive. Let’s get it:
1!gdown --id 1Q1wUptbNDYdfizk5abhmoFxIQiX19Tn7
And load it into a data frame:
1df = pd.read_csv('weatherAUS.csv')
We have a large set of features/columns here. You might also notice some NaNs. Let’s have a look at the overall dataset size:
Looks like we have plenty of data. But we got to do something about those missing values.
Data Preprocessing
We’ll simplify the problem by removing most of the data (mo money mo problems - Michael Scott). We’ll use only 4 columns for predicting whether or not is going to rain tomorrow:
1cols = ['Rainfall', 'Humidity3pm', 'Pressure9am', 'RainToday', 'RainTomorrow']
3df = df[cols]
Neural Networks don’t work with much else than numbers. We’ll convert yes and no to 1 and 0, respectively:
1df['RainToday'].replace({'No': 0, 'Yes': 1}, inplace = True)
2df['RainTomorrow'].replace({'No': 0, 'Yes': 1}, inplace = True)
Let’s drop the rows with missing values. There are better ways to do this, but we’ll keep it simple:
1df = df.dropna(how='any')
Finally, we have a dataset we can work with.
One important question we should answer is - How balanced is our dataset? Or How many times did it rain or not rain tomorrow?:
1df.RainTomorrow.value_counts() / df.shape[0]
10 0.778762
21 0.221238
3Name: RainTomorrow, dtype: float64
Things are not looking good. About 78% of the data points have a non-rainy day for tomorrow. This means that a model that predicts there will be no rain tomorrow will be correct about 78% of the
You can read and apply the Practical Guide to Handling Imbalanced Datasets if you want to mitigate this issue. Here, we’ll just hope for the best.
The final step is to split the data into train and test sets:
1X = df[['Rainfall', 'Humidity3pm', 'RainToday', 'Pressure9am']]
2y = df[['RainTomorrow']]
4X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=RANDOM_SEED)
And convert all of it to Tensors (so we can use it with PyTorch):
1X_train = torch.from_numpy(X_train.to_numpy()).float()
2y_train = torch.squeeze(torch.from_numpy(y_train.to_numpy()).float())
4X_test = torch.from_numpy(X_test.to_numpy()).float()
5y_test = torch.squeeze(torch.from_numpy(y_test.to_numpy()).float())
7print(X_train.shape, y_train.shape)
8print(X_test.shape, y_test.shape)
1torch.Size([99751, 4]) torch.Size([99751])
2torch.Size([24938, 4]) torch.Size([24938])
Building a Neural Network
We’ll build a simple Neural Network (NN) that tries to predicts will it rain tomorrow.
Our input contains data from the four columns: Rainfall, Humidity3pm, RainToday, Pressure9am. We’ll create an appropriate input layer for that.
The output will be a number between 0 and 1, representing how likely (our model thinks) it is going to rain tomorrow. The prediction will be given to us by the final (output) layer of the network.
We’ll add two (hidden) layers between the input and output layers. The parameters (neurons) of those layer will decide the final output. All layers will be fully-connected.
One easy way to build the NN with PyTorch is to create a class that inherits from torch.nn.Module:
1class Net(nn.Module):
3 def __init__(self, n_features):
4 super(Net, self).__init__()
5 self.fc1 = nn.Linear(n_features, 5)
6 self.fc2 = nn.Linear(5, 3)
7 self.fc3 = nn.Linear(3, 1)
9 def forward(self, x):
10 x = F.relu(self.fc1(x))
11 x = F.relu(self.fc2(x))
12 return torch.sigmoid(self.fc3(x))
1net = Net(X_train.shape[1])
We start by creating the layers of our model in the constructor. The forward() method is where the magic happens. It accepts the input x and allows it to flow through each layer.
There is a corresponding backward pass (defined for you by PyTorch) that allows the model to learn from the errors that is currently making.
Activation Functions
You might notice the calls to F.relu and torch.sigmoid. Why do we need those?
One of the cool features of Neural Networks is that they can approximate non-linear functions. In fact, it is proven that they can approximate any function.
Good luck approximating non-linear functions by stacking linear layers, though. Activation functions allow you to break from the linear world and learn (hopefully) more. You’ll usually find them
applied to an output of some layer.
Those functions must be hard to define, right?
Not at all, let start with the ReLU definition (one of the most widely used activation function):
$\text{ReLU}(x) = \max({0, x})$
Easy peasy, the result is the maximum value of zero and the input:
The sigmoid is useful when you need to make a binary decision/classification (answering with a yes or a no).
It is defined as:
$\text{Sigmoid}(x) = \frac{1}{1+e^{-x}}$
The sigmoid squishes the input values between 0 and 1. But in a super kind of way:
With the model in place, we need to find parameters that predict will it rain tomorrow. First, we need something to tell us how good we’re currently doing:
1criterion = nn.BCELoss()
The BCELoss is a loss function that measures the difference between two binary vectors. In our case, the predictions of our model and the real values. It expects the values to be outputed by the
sigmoid function. The closer this value gets to 0, the better your model should be.
But how do we find parameters that minimize the loss function?
Imagine that each parameter of our NN is a knob. The optimizer’s job is to find the perfect positions for each knob so that the loss gets close to 0.
Real-world models can contain millions or even billions of parameters. With so many knobs to turn, it would be nice to have an efficient optimizer that quickly finds solutions.
Contrary to what you might believe, optimization in Deep Learning is just satisfying. In practice, you’re content with good enough parameter values that give you an acceptable accuracy.
While there are tons of optimizers you can choose from, Adam is a safe first choice. PyTorch has a well-debugged implementation you can use:
1optimizer = optim.Adam(net.parameters(), lr=0.001)
Naturally, the optimizer requires the parameters. The second argument lr is learning rate. It is a tradeoff between how good parameters you’re going to find and how fast you’ll get there. Finding
good values for this can be black magic and a lot of brute-force “experimentation”.
Doing it on the GPU
Doing massively parallel computations on GPUs is one of the enablers for modern Deep Learning. You’ll need nVIDIA GPU for that.
PyTorch makes it really easy to transfer all the computation to your GPU:
1device = torch.device("cuda:0" if torch.cuda.is_available() else "cpu")
1X_train = X_train.to(device)
2y_train = y_train.to(device)
4X_test = X_test.to(device)
5y_test = y_test.to(device)
1net = net.to(device)
3criterion = criterion.to(device)
We start by checking whether or not a CUDA device is available. Then, we transfer all training and test data to that device. Finally, we move our model and loss function.
Weather Forecasting
Having a loss function is great, but tracking the accuracy of our model is something easier to understand, for us mere mortals. Here’s the definition for our accuracy:
1def calculate_accuracy(y_true, y_pred):
2 predicted = y_pred.ge(.5).view(-1)
3 return (y_true == predicted).sum().float() / len(y_true)
We convert every value below 0.5 to 0. Otherwise, we set it to 1. Finally, we calculate the percentage of correct values.
With all the pieces of the puzzle in place, we can start training our model:
1def round_tensor(t, decimal_places=3):
2 return round(t.item(), decimal_places)
4for epoch in range(1000):
6 y_pred = net(X_train)
8 y_pred = torch.squeeze(y_pred)
9 train_loss = criterion(y_pred, y_train)
11 if epoch % 100 == 0:
12 train_acc = calculate_accuracy(y_train, y_pred)
14 y_test_pred = net(X_test)
15 y_test_pred = torch.squeeze(y_test_pred)
17 test_loss = criterion(y_test_pred, y_test)
19 test_acc = calculate_accuracy(y_test, y_test_pred)
20 print(
21f'''epoch {epoch}
22Train set - loss: {round_tensor(train_loss)}, accuracy: {round_tensor(train_acc)}
23Test set - loss: {round_tensor(test_loss)}, accuracy: {round_tensor(test_acc)}
26 optimizer.zero_grad()
28 train_loss.backward()
30 optimizer.step()
1epoch 0
2Train set - loss: 2.513, accuracy: 0.779
3Test set - loss: 2.517, accuracy: 0.778
5epoch 100
6Train set - loss: 0.457, accuracy: 0.792
7Test set - loss: 0.458, accuracy: 0.793
9epoch 200
10Train set - loss: 0.435, accuracy: 0.801
11Test set - loss: 0.436, accuracy: 0.8
13epoch 300
14Train set - loss: 0.421, accuracy: 0.814
15Test set - loss: 0.421, accuracy: 0.815
17epoch 400
18Train set - loss: 0.412, accuracy: 0.826
19Test set - loss: 0.413, accuracy: 0.827
21epoch 500
22Train set - loss: 0.408, accuracy: 0.831
23Test set - loss: 0.408, accuracy: 0.832
25epoch 600
26Train set - loss: 0.406, accuracy: 0.833
27Test set - loss: 0.406, accuracy: 0.835
29epoch 700
30Train set - loss: 0.405, accuracy: 0.834
31Test set - loss: 0.405, accuracy: 0.835
33epoch 800
34Train set - loss: 0.404, accuracy: 0.834
35Test set - loss: 0.404, accuracy: 0.835
37epoch 900
38Train set - loss: 0.404, accuracy: 0.834
39Test set - loss: 0.404, accuracy: 0.836
During the training, we show our model the data for 10,000 times. Each time we measure the loss, propagate the errors trough our model and asking the optimizer to find better parameters.
The zero_grad() method clears up the accumulated gradients, which the optimizer uses to find better parameters.
What about that accuracy? 83.6% accuracy on the test set sounds reasonable, right? Well, I am about to disappoint you. But first, let’s learn how to save and load our trained models.
Saving the model
Training a good model can take a lot of time. And I mean weeks, months or even years. So, let’s make sure that you know how you can save your precious work. Saving is easy:
1MODEL_PATH = 'model.pth'
3torch.save(net, MODEL_PATH)
Restoring your model is easy too:
1net = torch.load(MODEL_PATH)
Wouldn’t it be perfect to know about all the errors your model can make? Of course, that’s impossible. But you can get an estimate.
Using just accuracy wouldn’t be a good way to do it. Recall that our data contains mostly no rain examples.
One way to delve a bit deeper into your model performance is to assess the precision and recall for each class. In our case, that will be no rain and rain:
1classes = ['No rain', 'Raining']
3y_pred = net(X_test)
5y_pred = y_pred.ge(.5).view(-1).cpu()
6y_test = y_test.cpu()
8print(classification_report(y_test, y_pred, target_names=classes))
1precision recall f1-score support
3 No rain 0.85 0.96 0.90 19413
4 Raining 0.74 0.40 0.52 5525
6 accuracy 0.84 24938
7 macro avg 0.80 0.68 0.71 24938
8weighted avg 0.83 0.84 0.82 24938
A maximum precision of 1 indicates that the model is perfect at identifying only relevant examples. A maximum recall of 1 indicates that our model can find all relevant examples in the dataset for
this class.
You can see that our model is doing good when it comes to the No rain class. We have so many examples. Unfortunately, we can’t really trust predictions of the Raining class.
One of the best things about binary classification is that you can have a good look at a simple confusion matrix:
1cm = confusion_matrix(y_test, y_pred)
2df_cm = pd.DataFrame(cm, index=classes, columns=classes)
4hmap = sns.heatmap(df_cm, annot=True, fmt="d")
5hmap.yaxis.set_ticklabels(hmap.yaxis.get_ticklabels(), rotation=0, ha='right')
6hmap.xaxis.set_ticklabels(hmap.xaxis.get_ticklabels(), rotation=30, ha='right')
7plt.ylabel('True label')
8plt.xlabel('Predicted label');
You can clearly see that our model shouldn’t be trusted when it says it’s going to rain.
Making Predictions
Let’s pick our model’s brain and try it out on some hypothetical examples:
1def will_it_rain(rainfall, humidity, rain_today, pressure):
2 t = torch.as_tensor([rainfall, humidity, rain_today, pressure]) \
3 .float() \
4 .to(device)
5 output = net(t)
6 return output.ge(0.5).item()
This little helper will return a binary response based on your model predictions. Let’s try it out:
1will_it_rain(rainfall=10, humidity=10, rain_today=True, pressure=2)
1will_it_rain(rainfall=0, humidity=1, rain_today=False, pressure=100)
Okay, we got two different responses based on some parameters (yep, the power of the brute force). Your model is ready for deployment (but please don’t)!
Well done! You now have a Neural Network that can predict the weather. Well, sort of. Building well-performing models is hard, really hard. But there are tricks you’ll pick up along the way and
(hopefully) get better at your craft!
You learned how to:
• Preprocess CSV files and convert the data to Tensors
• Build your own Neural Network model with PyTorch
• Use a loss function and an optimizer to train your model
• Evaluate your model and learn about the perils of imbalanced classification | {"url":"https://curiousily.com/posts/build-your-first-neural-network-with-pytorch/","timestamp":"2024-11-06T02:34:39Z","content_type":"text/html","content_length":"358668","record_id":"<urn:uuid:d475907e-6be0-4d35-b144-71c4965c712b>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00248.warc.gz"} |
Confirmation Theory | Encyclopedia.com
Confirmation Theory
Predictions about the future and unrestricted universal generalizations are never logically implied by our observational evidence, which is limited to particular facts in the present and past.
Nevertheless propositions of these and other kinds are often said to be confirmed by observational evidence. A natural place to begin the study of confirmation theory is to consider what it means to
say that some evidence E confirms a hypothesis H.
Incremental and Absolute Confirmation
Let us say that E raises the probability of H if the probability of H given E is higher than the probability of H not given E. According to many confirmation theorists, "E confirms H " means that E
raises the probability of H. This conception of confirmation will be called incremental confirmation.
Let us say that H is probable given E if the probability of H given E is above some threshold. (This threshold remains to be specified but is assumed to be at least one half.) According to some
confirmation theorists, "E confirms H " means that H is probable given E. This conception of confirmation will be called absolute confirmation.
Confirmation theorists have sometimes failed to distinguish these two concepts. For example, Carl Hempel (1945/1965) in his classic "Studies in the Logic of Confirmation" endorsed the following
(1) A generalization of the form "All F are G " is confirmed by the evidence that there is an individual that is both F and G.
(2) A generalization of that form is also confirmed by the evidence that there is an individual that is neither F nor G.
(3) The hypotheses confirmed by a piece of evidence are consistent with one another.
(4) If E confirms H then E confirms every logical consequence of H.
Principles (1) and (2) are not true of absolute confirmation. Observation of a single thing that is F and G cannot in general make it probable that all F are G; likewise for an individual that is
neither F nor G. On the other hand there is some plausibility to the idea that an observation of something that is both F and G would raise the probability that all F are G. Hempel argued that the
same is true of an individual that is neither F nor G. Thus Hempel apparently had incremental confirmation in mind when he endorsed (1) and (2).
Principle (3) is true of absolute confirmation but not of incremental confirmation. It is true of absolute confirmation because if one hypothesis has a probability greater than ½ then any hypothesis
inconsistent with it has a probability less than ½. To see that (3) is not true of incremental confirmation, suppose that a fair coin will be tossed twice, let H[1] be that the first toss lands heads
and the second toss lands tails, and let H[2] be that both tosses land heads. Then H[1] and H[2] each have an initial probability of ¼. If E is the evidence that the first toss landed heads, the
probability of both H[1] and H[2] given E is ½, and so both hypotheses are incrementally confirmed, though they are inconsistent with each other.
Principle (4) is also true of absolute confirmation but not of incremental confirmation. It is true of absolute confirmation because any logical consequence of H is at least as probable as H itself.
One way to see that (4) is not true of incremental confirmation is to note that any tautology is a logical consequence of any H but a tautology cannot be incrementally confirmed by any evidence,
since the probability of a tautology is always one. Thus Hempel was apparently thinking of absolute confirmation, not incremental confirmation, when he endorsed (3) and (4).
Since even eminent confirmation theorists like Hempel have failed to distinguish these two concepts of confirmation, we need to make a conscious effort not to make the same mistake.
Confirmation in Ordinary Language
When we say in ordinary language that some evidence confirms a hypothesis, does the word "confirms" mean incremental or absolute confirmation?
Since the probability of a tautology is always one, a tautology is absolutely confirmed by any evidence whatever. For example, evidence that it is raining absolutely confirms that all triangles have
three sides. Since we would ordinarily say that there is no confirmation in this case, the concept of confirmation in ordinary language is not absolute confirmation.
If E reduces the probability of H then we would ordinarily say that E does not confirm H. However, in such a case it is possible for H to still be probable given E and hence for E to absolutely
confirm H. This shows again that the concept of confirmation in ordinary language is not absolute confirmation.
A hypothesis H that is incrementally confirmed by evidence E may still be probably false; for example, the hypothesis that a fair coin will land "heads" every time in 1000 tosses is incrementally
confirmed by the evidence that it landed "heads" on the first toss, but the hypothesis is still extremely improbable given this evidence. In a case like this nobody would ordinarily say that the
hypothesis was confirmed. Thus it appears that the concept of confirmation in ordinary language is not incremental confirmation either.
A few confirmation theorists have attempted to formulate concepts of confirmation that would agree better with the ordinary concept. One such theorist is Nelson Goodman. He noted that if E
incrementally confirms H, and X is an irrelevant proposition, then E incrementally confirms the conjunction of H and X. Goodman (1979) thought that in a case like this we would not say that E
confirms the conjunction. He proposed that "E confirms H " means that E increases the probability of every component of H. One difficulty with this is to say what counts as a component of a
hypothesis; if any logical consequence of H counts as a component of H then no hypothesis can ever be confirmed in Goodman's sense. In addition Goodman's proposal is open to the same objection as
incremental confirmation: It allows that a hypothesis H can be confirmed by evidence E and yet H be probably false given E, which is not what people would ordinarily say.
Peter Achinstein (2001) speaks of "evidence" rather than "confirmation" but he can be regarded as proposing an account of the ordinary concept of confirmation. His account is complex but the leading
idea is roughly that "E confirms H " means that (i) H is probable given E and (ii) it is probable that there is an explanatory connection between H and E, given that H and E are true. The explanatory
connection may be that H explains E, E explains H, or H and E have a common explanation. Achinstein's proposal is open to one of the same objections as absolute confirmation: It allows evidence E to
confirm H in cases in which E reduces the probability of H. Achinstein has argued that this implication is in agreement with the ordinary concept, but his reasoning has been criticized, for example,
by Sherrilyn Roush (2004).
It appears that none of the concepts of confirmation discussed by confirmation theorists is the same as the ordinary concept of evidence confirming a hypothesis. Nevertheless, some of these concepts
are worthy of study in their own right. In particular, the concepts of incremental and absolute confirmation are simple concepts that are of obvious importance and they are probably components in the
more complex ordinary language concept of confirmation.
All the concepts of confirmation that we have discussed involve probability. However, the word "probability" is ambiguous. For example, suppose you have been told that a coin either has heads on both
sides or else has tails on both sides and that it is about to be tossed. What is the probability that it will land heads? There are two natural answers: (i) ½; (ii) either 0 or 1 but I do not know
which. These answers correspond to different meanings of the word "probability." The sense of the word "probability" in which (i) is the natural answer will here be called inductive probability. The
sense in which (ii) is the natural answer will be called physical probability.
Physical probability depends on empirical facts in a way that inductive probability does not. We can see this from the preceding example; here the physical probability is unknown because it depends
on the nature of the coin, which is unknown; by contrast the inductive probability is known even though the nature of the coin is unknown, showing that the inductive probability does not depend on
the nature of the coin.
There are two main theories about the nature of physical probability. One is the frequency theory, according to which the physical probability of an event is the relative frequency with which the
event happens in the long run. The other is the propensity theory, according to which the physical probability of an event is the propensity of the circumstances or experimental arrangement to
produce that event.
It is widely agreed that the concept of probability involved in confirmation is not physical probability. One reason is that physical probabilities seem not to exist in many contexts in which we talk
about confirmation. For example, we often take evidence as confirming a scientific theory but it does not seem that there is a physical probability of a particular scientific theory being true. (The
theory is either true or false; there is no long run frequency with which it is true, nor does the evidence have a propensity to make the theory true.) Another reason is that physical probabilities
depend on the facts in a way that confirmation relations do not. Inductive probability does not have either of these shortcomings and so it is natural to identify the concept of probability involved
in confirmation with inductive probability. Therefore we will now discuss inductive probability in more detail.
Some contemporary writers appear to believe that the inductive probability of a proposition is some person's degree of belief in the proposition. Degree of belief is also called subjective
probability, so on this view, inductive probability is the same as subjective probability. However, this is not correct. Suppose, for example, that I claim that scientific theory H is probable in
view of the available evidence. This is a statement of inductive probability. If my claim is challenged, it would not be a relevant response for me to prove that I have a high degree of belief in H,
though this would be relevant if inductive probability were subjective probability. To give a relevant defense of my claim I need to cite features of the available evidence that support H.
In saying that inductive probabilities are not subjective probabilities, we are not denying that when people make assertions about inductive probabilities they are expressing their degrees of belief.
Every sincere and intentional assertion expresses the speaker's beliefs but not every assertion is about the speaker's beliefs.
We will now consider the concept of logical probability and, in particular, whether inductive probability is a kind of logical probability. This depends on what is meant by "logical probability."
Many writers define the "logical probability" of H given E as the degree of belief in H that would be rational for a person whose total evidence is E. However, the term "rational degree of belief" is
far from clear. On some natural ways of understanding it, the degree of belief in H that is rational for a person could be high even when H has a low inductive probability given the person's
evidence. This might happen because belief in H helps the person succeed in some task, or makes the person feel happy, or will be rewarded by someone who can read the person's mind. Even if it is
specified that we are talking about rationality with respect to epistemic goals, the rational degree of belief can differ from the inductive probability given the person's evidence, since the rewards
just mentioned may be epistemic. Alternatively, one might take "the rational degree of belief in H for a person whose total evidence is E " to be just another name for the inductive probability of H
given E, in which case these concepts are trivially equivalent. Thus if one takes "logical probability" to be rational degree of belief then, depending on what one means by "rational degree of
belief," it is either wrong or trivial to say that inductive probability is logical.
A more useful conception of logical probability can be defined as follows. Let an "elementary probability sentence" be a sentence that asserts that a specific hypothesis has a specific probability.
Let a "logically determinate sentence" be a sentence whose truth or falsity is determined by meanings alone, independently of empirical facts. Let us say that a probability concept is "logical in
Carnap's sense" if all elementary probability sentences for it are logically determinate. (This terminology is motivated by some of the characterizations of logical probability in Carnap's Logical
Foundations of Probability.) Since inductive probability is not subjective probability, the truth of an elementary statement of inductive probability does not depend on some person's psychological
state. It also does not depend on facts about the world in the way that statements of physical probability do. It thus appears the truth of an elementary statement of inductive probability does not
depend on empirical facts at all and hence that inductive probability is logical in Carnap's sense.
It has often been said that logical probabilities do not exist. If this were right then it would follow that inductive probabilities are either not logical or else do not exist. So we will now
consider arguments against the existence of logical probabilities.
John Maynard Keynes in 1921 published a theory of what we call inductive probability and he claimed that these are logical. Frank Ramsey (1926/1980) criticizing Keynes's theory, claimed that "there
really do not seem to be any such things as the probability relations he describes." The main consideration that Ramsey offered in support of this was that there is little agreement on the values of
probabilities in the simplest cases and these are just the cases where logical relations should be most clear. Ramsey's argument has been cited approvingly by several later authors.
However, Ramsey's claim that there is little agreement on the values of probabilities in the simplest cases seems not to be true. For example, almost everyone agrees with the following:
(5) The probability that a ball is white, given only that it is either white or black, is ½.
Ramsey cited examples such as the probability of one thing being red given that another thing is red; he noted that nobody can state a precise numerical value for this probability. But that is an
example of agreement about the value of an inductive probability, since nobody pretends to know a precise numerical value for the probability. What examples like this show is merely that inductive
probabilities do not always have numerically precise values.
Furthermore, if inductive probabilities are logical (i.e., non-descriptive), it does not follow that their values should be clearest in the simplest cases, as Ramsey claimed. Like other concepts of
ordinary language, the concept of inductive probability is learned largely from examples of its application in ordinary life and many of these examples will be complex. Hence, like other concepts of
ordinary language, its application may sometimes be clearer in realistic complex situations than in simple situations that never arise in ordinary life.
So much for Ramsey's argument. Another popular argument against the existence of logical probabilities is based on the "paradoxes of indifference." The argument is this: Judgments of logical
probability are said to presuppose a general principle, called the Principle of Indifference, which says that if evidence does not favor one hypothesis over another then those hypotheses are equally
probable on this evidence. This principle can lead to different values for a probability, depending on what one takes the alternative hypotheses to be. In some cases the different choices seem
equally natural. These "paradoxes of indifference," as they are called, are taken by many authors to be fatal to logical probability.
But even if we agree (as Keynes did) that quantitative inductive probabilities can only be determined via the Principle of Indifference, we can also hold (as Keynes did) that inductive probabilities
do not always have quantitative values. Thus if there are cases where contradictory applications of the principle are equally natural, we may take this to show that these are cases where inductive
probabilities lack quantitative values. It does not follow that quantitative inductive probabilities never exist, or that qualitative inductive probabilities do not exist. The paradoxes of
indifference are thus consistent with the view that inductive probabilities exist and are logical.
How can we have knowledge of inductive probabilities, if this does not come from an exceptionless general principle? The answer is that the concept of inductive probability, like most concepts of
ordinary language, is learned from examples, not by general principles. Hence we can have knowledge about particular inductive probabilities (and hence logical probabilities) without being able to
state a general principle that covers these cases.
A positive argument for the existence of inductive probabilities is the following: We have seen reason to believe that a statement of inductive probability, such as (5), is either logically true or
logically false. Which of these it is will be determined by the concepts involved, which are concepts of ordinary language. So, since competent speakers of a language normally use the language
correctly, the wide endorsement of (5) is good reason to believe that (5) is a true sentence of English. And it follows from (5) that at least one inductive probability exists. Parallel arguments
would establish the existence of many other inductive probabilities.
The concept of probability that is involved in confirmation can appropriately be taken to be inductive probability. Unlike physical probability, the concept of inductive probability applies to
scientific theories. And unlike both physical and subjective probability, the concept of inductive probability agrees with the fact that confirmation relations are not discovered empirically but by
examination of the relation between the hypothesis and the evidence.
Explication of Inductive Probability
Inductive probability is a concept of ordinary language and, like many such concepts, it is vague. This is reflected in the fact that inductive probabilities often have no precise numerical value.
A useful way to theorize about vague concepts is to define a precise concept that is similar to the vague concept. This methodology is called explication, the vague concept is called the explicandum,
and the precise concept that is meant to be similar to it is called the explicatum. Although the explicatum is intended to be similar to the explicandum, there must be differences, since the
explicatum is precise and the explicandum is vague. Other desiderata for an explicatum, besides similarity with the explicandum, are theoretical fruitfulness and simplicity.
Inductive probability can be explicated by defining, for selected pairs of sentences E and H, a number that will be the explicatum for the inductive probability of H given E ; let us denote this
number by "p(H|E)." The set of sentences for which p(H|E) is defined will depend on our purposes.
Quantitative inductive probabilities, where they exist, satisfy the mathematical laws of probability. Since a good explicatum is similar to the explicandum, theoretically fruitful, and simple, the
numbers p(H|E) will also be required to satisfy these laws.
In works written from the 1940s to his death in 1970, Carnap proposed a series of increasingly sophisticated explications of this kind, culminating in his Basic System of Inductive Logic published
posthumously in 1971 and 1980. Other authors have proposed other explicata, some of which will be mentioned below.
Since the value of p(H|E) is specified by definition, a statement of the form "p(H|E) = r " is either true by definition or false by definition, and hence is logically determinate. Since we require
these values to satisfy the laws of probability, the function p is also a probability function. So we may say that the function p is a logical probability in Carnap's sense.
Thus there are two different kinds of probability, both of which are logical in Carnap's sense: Inductive probability and functions that are proposed as explicata for inductive probability. Since the
values of the explicata are specified by definition, it is undeniable that logical probabilities of this second kind exist.
Explication of Incremental Confirmation
Since inductive probability is vague, and E incrementally confirms H if and only if E raises the inductive probability of H, the concept of incremental confirmation is also vague. We will now
consider how to explicate incremental confirmation.
First, we note that the judgment that E confirms H is often made on the assumption that some other information D is given; this information is called backgroundevidence. So we will take the form of a
fully explicit judgment of incremental confirmation to be "E incrementally confirms H given D." For example, a coin landing heads on the first toss incrementally confirms that the coin has heads on
both sides, given that both sides of the coin are the same; there would be no confirmation if the background evidence was that the coin is normal with heads on one side only.
The judgment that E incrementally confirms H given D means that the inductive probability of H given both E and D is greater than the inductive probability of H given only D. Suppose we have a
function p that is an explicatum for inductive probability and is defined for the relevant statements. Let "E.D" represent the conjunction of E and D (so the dot here functions like "and"). Then the
explicatum for "E incrementally confirms H given D " will be p(H|E.D) > p(H|D). We will use the notation "C(H, E, D) " as an abbreviation for this explicatum.
The concept of incremental confirmation, like all the concepts of confirmation discussed so far, is a qualitative concept. For each of these qualitative concepts there is a corresponding comparative
concept, which compares the amount of confirmation in different cases. We will focus here on the judgment that E[1] incrementally confirms H more than E[2] does, given D. The corresponding statement
in terms of our explicata is that the increase from p(H|D) to p(H|E[1].D) is larger than the increase from p(H|D) to p(H|E[2].D). This is true if and only if p(H|E[1].D) > p(H|E[2]D), so the
explicatum for "E[1] confirms H more than E[2] does, given D " will be p(H|E[1].D) > p(H|E[2].D). We will use the notation "M(H,E[1],E[2],D) " as an abbreviation for this explicatum.
Confirmation theorists have also discussed quantitative concepts of confirmation, which involve assigning numerical "degrees of confirmation" to hypotheses. In earlier literature the term "degree of
confirmation" usually meant degree of absolute confirmation. The degree to which E absolutely confirms H is the same as the inductive probability of H given E and hence is explicated by p(H|E).
In later literature, the term "degree of confirmation" is more likely to mean degree of incremental confirmation. An explicatum for the degree to which E incrementally confirms H given D is a measure
of how much p(H|E.D) is greater than p(H|D). Many different explicata of this kind have been proposed; they include the following. (Here "∼H " means the negation of H.)
Difference measure: p(H|E.D) − p(H|D)
Ratio measure: p(H|E.D) / p(H|D)
Likelihood ratio: p(E|H.D) / p(E |∼H.D)
Confirmation theorists continue to debate the merits of these and other measures of degree of incremental confirmation.
Verified Consequences
The remainder of this entry will consider various properties of incremental confirmation and how well these are captured by the explicata C and M that were defined above. We begin with the idea that
hypotheses are confirmed by verifying their logical consequences.
If H logically implies E given background evidence D, we usually suppose that observation of E would incrementally confirm H given D. For example, Einstein's general theory of relativity, together
with other known facts, implied that the orbit of Mercury precesses at a certain rate; hence the observation that it did precess at this rate incrementally confirmed Einstein's theory, given the
other known facts.
The corresponding explicatum statement is: If H.D implies E then C(H,E,D). Assuming that p satisfies the laws of mathematical probability, this explicatum statement can be proved true provided that 0
> p(H|D) > 1 and p(E|D) < 1.
We can see intuitively why the provisos are needed. If p(H|D) = 1 then H is certainly true given D and so no evidence can incrementally confirm it. If p(H|D) = 0 then H is certainly false given D and
the observation that one of its consequences is true need not alter this situation. If p(E|D) = 1 then E was certainly true given D and so the observation that it is true cannot provide new evidence
for H.
If H and D imply both E[1] and E[2], and if E[1] is less probable than E[2] given D, then we usually suppose that H would be better confirmed by E[1] than by E[2], given D. The corresponding
explicatum statement is: If H.D implies E[1] and E[2], and p(E[1]|D) < p(E[2]|D), then M (H, E[1], E[2], D). Assuming that p satisfies the laws of probability, this can be proved true provided that 0
< p(H|D) < 1. The proviso makes sense intuitively for the same reasons as before.
If H and D imply both E[1] and E[2] then we usually suppose that E[1] and E[2] together would confirm H more than E[1] alone, given D. The corresponding explicatum statement is that if H.D implies E
[1] and E[2] then M (H, E[1].E[2], E[1], D). It follows from the result in the previous paragraph that this is true, provided that p(E[1].E[2]|D) < p(E[1]|D) and 0 < p(H|D) < 1. The provisos are
needed for the same reasons as before.
These results show that, if we require p to satisfy the laws of probability, then C and M will be similar to their explicanda with respect to verified consequences and, to that extent at least, C and
M will be good explicata. In addition these results illustrate in a small way the value of explication. Although the provisos that we added make sense when one thinks about them, the need for them is
likely to be overlooked if one thinks only in terms of the vague explicanda and does not attempt to prove a precise corresponding result in terms of the explicata. Thus explication can give a deeper
and more accurate understanding of the explicandum. We will see more examples of this.
Reasoning by Analogy
If two individuals are known to be alike in certain respects, and one is found to have a particular property, we often infer that, since the individuals are similar, the other individual probably
also has that property. This is a simple example of reasoning by analogy, and it is a kind of reasoning that we use every day.
In order to explicate this kind of reasoning, we will use "a " and "b " to stand for individual things and "F " and "G " for logically independent properties that an individual may have (for example,
being tall and blond). We will use "Fa " to mean that the individual a has the property F ; similarly for other properties and individuals.
It is generally accepted that reasoning by analogy is stronger the more properties that the individuals are known to have in common. So for C to be a good explicatum it must satisfy the following
(6) C (Gb, Fa.Fb, Ga).
Here we are considering the situation in which the background evidence is that a has G. The probability that b also has G is increased by finding that a and b also share the property F.
In the case just considered, a and b are not known to differ in any way. When we reason by analogy in real life we normally do know some respects in which the individuals differ, but this does not
alter the fact that the reasoning is stronger the more alike a and b are known to be. So for C to be a good explicatum it must also satisfy the following condition. (Here F′ is a property that is
logically independent of both F and G.)
(7) C (Gb, Fa.Fb, Ga.F′a.∼F′b).
Here the background evidence is that a has G and that a and b differ in regard to F′. The probability that b has G is increased by finding that a and b are alike in having F.
Another condition that C should satisfy is:
(8) C (Gb, Ga, F′a. ∼F′b).
Here the background evidence is merely that a and b differ regarding F′. For all we know, whether or not something has F′ might be unrelated to whether it has G, so the fact that a has G is still
some reason to think that b has G.
In Logical Foundations of Probability Carnap proposed a particular explicatum for inductive probability that he called c*. In The Continuum of Inductive Methods he described an infinite class of
possible explicata. The function c*, and all the functions in Carnap's continuum, satisfy (6) but not (7) or (8). Hence none of these functions provides a fully satisfactory explicatum for situations
that involve more than one logically independent property.
Carnap recognized this failure early in the 1950s and worked to find explicata that would handle reasoning by analogy more adequately. He first found a class of possible explicata for the case where
there are two logically independent properties; the functions in this class satisfy (6) and (8). Subsequently, with the help of John Kemeny, Carnap generalized his proposal to the case where there
are any finite number of logically independent properties, though he never published this. A simpler and less adequate generalization was published by Mary Hesse in 1964. Both these generalizations
satisfy all of (6)-(8).
Carnap had no justification for the functions he proposed except that they seemed to agree with intuitive principles of reasoning by analogy. Later he found that they actually violate one of the
principles he had taken to be intuitive. In his last work Carnap expressed indecision about how to proceed.
For the case where there are just two properties, Maher (2000) has shown that certain foundational assumptions pick out a class of probability functions, called P [I ], that includes the functions
that Carnap proposed for this case. Maher argued that the probability functions in P [I ] handle reasoning by analogy adequately and Carnap's doubts were misplaced.
For the case where there are more than two properties, Maher (2001) has shown that the proposals of Hesse, and Carnap and Kemeny, correspond to implausible foundational assumptions and violate
intuitive principles of reasoning by analogy. Further research is needed to find an explicatum for inductive probability that is adequate for situations involving more than two properties.
Nicod's Condition
We are often interested in universal generalizations of the form "All F are G," for example, "All ravens are black," or "All metals conduct electricity." Nicod's condition, named after the French
philosopher Jean Nicod, says that generalizations of this form are confirmed by finding an individual that is both F and G. (Here and in the remainder of this entry, "confirmed" means incrementally
Nicod (1970) did not mention background evidence. It is now well known that Nicod's condition is not true when there is background evidence of certain kinds. For example, suppose the background
evidence is that, if there are any ravens, then there is a non-black raven. Relative to this background evidence, observation of a black raven would refute, not confirm, that all ravens are black.
Hempel claimed that Nicod's condition is true when there is no background evidence but I. J. Good argued that this is also wrong. Good's argument was essentially this: Given no evidence whatever, it
is improbable that there are any ravens, and if there are no ravens then, according to standard logic, "All ravens are black" is true. Hence, given no evidence, "All ravens are black" is probably
true. However, if ravens do exist, they are probably a variety of colors, so finding a black raven would increase the probability that there is a non-black raven and hence disconfirm that all ravens
are black, contrary to Nicod's condition.
Hempel was relying on intuition, and Good's counterargument is intuitive rather than rigorous. A different way to investigate the question is to use precise explicata. The situation of "no background
evidence" can be explicated by taking the background evidence to be any logically true sentence; let T be such a sentence. Letting A be "all F are G," the claim that Nicod's condition holds when
there is no background evidence may be expressed in explicatum terms as
(9) C (A, Fa.Ga, T).
Maher has shown that this can fail when the explicatum p is a function in P[I] and that the reason for the failure is the one identified in Good's argument. This confirms that Nicod's condition is
false even when there is no background evidence.
Why then has Nicod's condition seemed plausible? One reason may be that people sometimes do not clearly distinguish between Nicod's condition and the following statement: Given that an object is F,
the evidence that it is G confirms that all F are G. The latter statement may be expressed in explicatum terms as:
(10) C (A, Ga, Fa).
This is true provided only that p satisfies the laws of probability, 0 < p(A|Fa) < 1, and p(Ga|Fa) < 1. (This follows from the first of the results stated earlier for verified consequences.) If
people do not clearly distinguish between the ordinary language statements that correspond to (9) and (10), the truth of the latter could make it seem that Nicod's condition is true.
The Ravens Paradox
The following three principles about confirmation have seemed plausible to many people.
(11) Nicod's condition holds when there is no background evidence.
(12) Confirmation relations are unchanged by substitution of logically equivalent sentences.
(13) In the absence of background evidence, the evidence that some individual is a non-black non-raven does not confirm that all ravens are black.
However, these three principles are inconsistent. That is because (11) implies that a non-black non-raven confirms "all non-black things are non-ravens," and the latter is logically equivalent to
"all ravens are black," so by (12) a non-black non-raven confirms "all ravens are black," contrary to (13).
Hempel was the first to discuss this paradox. His initial statement of the paradox did not explicitly include the condition of no background evidence but he stated later in his article that this was
to be understood. The subsequent literature on this paradox is enormous but most discussions have not respected the condition of no background evidence. Here we will follow Hempel in respecting that
The contradiction shows that at least one of (11)-(13) is false. Hempel claimed that (11) and (12) are true and (13) is false but his judgments were based on informal intuitions, not on any precise
explicatum or use of probability theory.
Our preceding discussion of Nicod's condition shows that (11) is false, contrary to what Hempel thought. On the other hand, our explicata support Hempel's view that (12) is true and (13) is false, as
we will now show.
In explicatum terms, what (12) says is: If H′, E′, and D′ are logically equivalent to H, E, and D respectively, then C(H, E, D) if and only if C(H′, E′, D′). The truth of this follows from the
assumption that p satisfies the laws of probability.
Now let "F " mean "raven" and "G " mean "black." Then (13), expressed in explicatum terms, is the claim∼C (A, ∼Fa.∼Ga, T). Maher has shown that this need not be true when p is a function in P[I] ; we
can instead have C (A, ∼Fa. ∼Ga, T). This happens for two reasons:
(a) The evidence ∼Fa.∼Ga reduces the probability of Fb.∼Gb, where b is any individual other than a. Thus ∼Fa.∼Ga reduces the probability that another individual b is a counterexample to A.
(b) The evidence ∼Fa.∼Ga tells us that a is not a counterexample to A, which a priori it could have been.
Both of these reasons make sense intuitively.
We conclude that, of the three principles (11)-(13), only (12) is true.
A predicate is said to be "projectable" if the evidence that the predicate applies to some objects confirms that it also applies to other objects. The standard example of a predicate that is not
projectable is "grue," which was introduced by Goodman (1979). According to Goodman's defnition, something is grue if either (i) it is observed before time t and is green or (ii) it is not observed
before time t and is blue. The usual argument that "grue" is not projectable goes something like this: A grue emerald observed before t is green, and observation of such an emerald confirms that
emeralds not observed before t are also green. Since a green emerald not observed before t is not grue, it follows that a grue emerald observed before t confirms that emeralds not observed before t
are not grue; hence "grue" is not projectable.
The preceding account of the meaning of "projectable" was the usual one but it is imprecise because it fails to specify background evidence. Let us say that a predicate ϕ is absolutely projectable if
C (ϕb, ϕa, T) for any distinct individuals a and b and logical truth T. This concept of absolute projectability is one possible explicatum for the usual imprecise concept of projectability. Let "Fa "
mean that a is observed before t and let "Ga " mean that a is green. Let "G′a " mean that either Fa.Ga or ∼Fa.∼ Ga. Thus "G ′" has a meaning similar to "grue." (The difference is just that G uses
"not green" instead of "blue" and so avoids introducing a third property.) Maher has proved that if p is any function in P[I] then "F ", "G ", and "G′ " are all absolutely projectable. It may seem
unintuitive that "G′ " is absolutely projectable. However, this result corresponds to the following statement of ordinary language: The probability that b is grue is higher given that a is grue than
if one was not given any evidence whatever. If we keep in mind that we do not know whether a or b was observed before t, this should be intuitively acceptable. So philosophers who say that "grue" is
not projectable are wrong if, by "projectable," they mean absolute projectability.
Let us say that a predicate ϕ is projectable across another predicate ψ if C (ϕb, ϕa, ψa.∼ψb) for any distinct individuals a and b. This concept of projectability across another predicate is a second
possible explicatum for the usual imprecise concept of projectability.
It can be shown that if p is any function in P[I] then "G " is, and "G′ " is not, projectable across "F." So philosophers who say that "grue" is not projectable are right if, by "projectable," they
mean projectability across the predicate "observed before t."
Now suppose we change the definition of "Ga " to be that a is (i) observed before t and green or (ii) not observed before t and not green. Thus "G " now means what "G′ " used to mean. Keeping the
definitions of "F " and "G′ " unchanged, "G′a " now means that a is green. The results reported in the preceding paragraph will still hold but now they are the opposite of the usual views about what
is projectable. This shows that, when we are constructing explicata for inductive probability and confirmation, the meanings assigned to the basic predicates (here "F " and "G ") need to be
intuitively simple ones rather than intuitively complex concepts like "grue."
See also Carnap, Rudolf; Einstein, Albert; Goodman, Nelson; Hempel, Carl Gustav; Induction; Keynes, John Maynard; Probability and Chance; Ramsey, Frank Plumpton; Relativity Theory.
Achinstein, Peter. The Book of Evidence. New York: Oxford University Press, 2001.
Carnap, Rudolf. "A Basic System of Inductive Logic, Part I." In Studies in Inductive Logic and Probability. Vol. 1, edited by Rudolf Carnap and Richard C. Jeffrey. Berkeley: University of California
Press, 1971.
Carnap, Rudolf. "A Basic System of Inductive Logic, Part II." In Studies in Inductive Logic and Probability. Vol. 2, edited by Richard C. Jeffrey. Berkeley: University of California Press, 1980.
Carnap, Rudolf. The Continuum of Inductive Methods. Chicago: University of Chicago Press, 1952.
Carnap, Rudolf. Logical Foundations of Probability. Chicago: University of Chicago Press, 1950. Second edition 1962.
Earman, John. Bayes or Bust? A Critical Examination of Bayesian Confirmation Theory. Cambridge, MA: MIT Press, 1992.
Festa, Roberto. "Bayesian Confirmation." In Experience, Reality, and Scientific Explanation, edited by Maria Carla Galavotti and Alessandro Pagnini. Dordrecht: Kluwer, 1999.
Fitelson, Branden. "The Plurality of Bayesian Measures of Confirmation and the Problem of Measure Sensitivity." Philosophy of Science 66 (1999): S362–S378.
Gillies, Donald. Philosophical Theories of Probability. London: Routledge, 2000.
Good, I. J. "The White Shoe qua Herring Is Pink." British Journal for the Philosophy of Science 19 (1968): 156–157.
Goodman, Nelson. Fact, Fiction, and Forecast. 3rd ed. Indianapolis, IN: Hackett, 1979.
Hempel, Carl G. "Studies in the Logic of Confirmation." Mind 54 (1945): 1–26 and 97–121. Reprinted with some changes in Carl G. Hempel. Aspects of Scientific Explanation. New York: The Free Press,
Hesse, Mary. "Analogy and Confirmation Theory." Philosophy of Science 31 (1964): 319–327.
Howson, Colin, and Peter Urbach. Scientific Reasoning: The Bayesian Approach. 2nd ed. Chicago: Open Court, 1993.
Keynes, John Maynard. A Treatise on Probability. London: Macmillan, 1921. Reprinted with corrections, 1948.
Maher, Patrick. "Probabilities for Two Properties." Erkenntnis 52 (2000): 63–91.
Maher, Patrick. "Probabilities for Multiple Properties: The Models of Hesse and Carnap and Kemeny." Erkenntnis 55 (2001): 183–216.
Maher, Patrick. "Probability Captures the Logic of Scientific Confirmation." In Contemporary Debates in Philosophy of Science, edited by Christopher R. Hitchcock. Oxford: Blackwell, 2004.
Nicod, Jean. Geometry and Induction. Berkeley and Los Angeles: University of California Press, 1970. English translation of works originally published in French in 1923 and 1924.
Ramsey, Frank P. "Truth and Probability." Article written in 1926 and published in many places, including Studies in Subjective Probability, 2nd ed., edited by Henry E. Kyburg, Jr. and Howard E.
Smokler. Huntington, New York: Krieger, 1980.
Roush, Sherrilyn. "Positive Relevance Defended." Philosophy of Science 71 (2004):110–116.
Salmon, Wesley C. "Confirmation and Relevance." In Minnesota Studies in the Philosophy of Science. Vol. VI: Induction, Probability, and Confirmation, ed. Grover Maxwell and Robert M. Anderson Jr.
Minneapolis: University of Minnesota Press, 1975.
Skyrms, Brian. Choice and Chance. 4th ed. Belmont, CA: Wadsworth, 2000.
Stalker, Douglas, ed. Grue: Essays on the New Riddle of Induction. Chicago: Open Court, 1994.
Patrick Maher (2005)
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Secrets to Solving the 11 Types of Problem Sums - KooBits
Have you ever wondered why your child’s problem sums are so complicated? Problem sums, or math word problems, are designed not to test basic mathematical ability but to impart the knowledge of
various concepts. In fact, there are a total of 11 different types of problem sums, and these can be found across all the different math topics.
This article is explains the different types of problem sums, and the most effective methods to solve them.
1. Remainder Concept
Remainder concept problem sums often contain the word “remainder” or ask about the total amount of something given the remainder. Most of the time, remainder concept questions test the knowledge of
Example Question
Mrs Lim had some chocolates. She gave 178 chocolates to her neighbour and 2/5 of the remainder to her daughter. With the remaining chocolates, she gave 1/3 of the chocolates to her son and then had
256 chocolates left. How many chocolates had she at first?
2. Repeated Identity Concept
Repeated Identity concept problem sums commonly feature one unknown or variable as a point of reference for other unknowns. Hence, the units method is used to solve these problem sums.
Example Question
Alice, Betty, Clara and Denise shared $168. Denise received 1/7 of the total amount of money received by Alice, Clara and Betty. Alice received 3/4 of the total amount of money received by Clara and
Betty. Betty received 2/5 as much as Clara. How much did Betty receive?
3. Equal Concept
Equal Concept problem sums compare fractions or percentages from different unknowns, but which represent equal amounts.
4. External Transfer (unchanged quantity)
There are four different transfer type problem sums, and it can be difficult for kids to determine which transfer type is being used in the question. The simplest way to solve transfer type problem
sums is to draw models. However, children who are more advanced can try using algebra and simultaneous equations to derive the answer.
In an external transfer with unchanged quantity, one variable has amounts added to or subtracted from it while the other remains unchanged.
Example Question
Mrs Lim baked thrice as many cakes as Mrs Loh. After Mrs Lim gave away 115cakes, Mrs Loh had twice as many cakes as her. How many cakes did Mrs Lim have left?
5. Internal Transfer
Internal transfer concept problem sums refers to questions where an amount is subtracted from one unknown and added to another, so the total amount in question remains unchanged. It is also known as
the Constant Total concept.
Example Question
Leon and his sister Lily have a total of 120 crayons. If Leon gives Lily 5 crayons, Lily will have nine times as many crayons as Leon. How many more crayons does Lily have than Leon?
6. External Transfer (same difference)
In this type of external transfer, the same amount is being transferred to or from the variables/unknowns. Hence, the difference between the unknowns remains unchanged. Such questions are almost
always used to test knowledge of ratios.
Example Question
Question: Ryan is 33 years old and his son is 5 years old now. In how many years will Ryan be thrice as old as his son?
In such problem sums, remember that age difference between two people will always remain a constant. Also, when drawing models to solve the question, It is easier to draw the final model – where the
ratio is known – and then work backwards.
7. External Transfer (changed quantity)
In an external transfer with changed quantity, both variables or unknowns have amounts added to or subtracted from them. Hence, it is difficult to solve such problem sums using the model method.
Example Question
The number of ten-cent coins in a box was 1/2 the number of fifty-cent coins. Syed took out 5 fifty-cent coins and exchanged them for ten-cent coins. Then he put the money back into the box. The
number of fifty-cent coins became 5/8 the number of ten-cent coins. How much money was there in the box?
The best method to use for solving external transfer (changed quantity) questions is the units method. This question is particularly tricky because it involves the counting of money as well as the
number of coins.
8. Pattern Concept
Pattern concept problem sums are among the most challenging, as they require students to recognize varying number patterns that may involve more than one arithmetic operation.
Example Question
Study the pattern below:
Pattern 1 Pattern 2 Pattern 3
(a) Complete the table below.
Pattern Number
(b) How many matchsticks are there in Pattern 11?
(c) Which pattern will have 76 matchsticks?
Here, you will need to find the relation between the pattern number and the number of matchsticks used (Pattern Number * 3 + 1). Order of operations is also important for getting the right answer.
9. Part-whole Concept (Proportions)
The part-whole concept is one of the main concepts for problem sums, and forms the basis for other concepts mentioned here. Read more about it in our detailed post about the Singapore math model
10. Simultaneous Concept
Simultaneous concept problem sums are a precursor to simultaneous equations in algebra, where abstract variables are represented using objects. While simultaneous concept questions can be solved
using the model method, they are the best for introducing your child to algebraic methods.
Example Question
3 files and 2 books cost $60. 2 files and 3 books cost $70. Find the cost of 1 file.
11. Gap Concept / Difference Concept
Gap concept problem sums are based on the difference between one variable and another. They can be solved using the ever-useful model method, or by arithmetic.
Example Question
Joe’s father had given Joe and Kevin an equal amount of money. Joe spent $20 each day and Kevin spent $25 each day. When Joe had $157 left, Kevin had $82 left. How much did Kevin receive?
• Each day, Kevin spends $25 – $20 = $5 more than Joe.
• After some days, the difference in money left is $157 – $82 = $75.
• So, this money was spent over 75 ÷ 5 = 15 days.
• Kevin received $82 left + 25 × 15 days = $457.
You can also encourage your child to double-check his or her answer by making sure Joe also receives the same amount of money.
Remember you should try also rewarding activities after practice.
Tags: Lower Primary (7-10), mathematics, Singapore, Upper Primary (10-12)
17 Commentsclick here to leave a comment
How to solve this problem sums :
Geeta gets 50 cents more pocket money than Hani every day. Each of them spends 60 cents a day and saves the rest. If Hani saves $40, Geeta will have saved $20 more than Hani. How much is hani’s
daily pocket money?
Since the amount they spend each day is the same, the difference in the amount saved each day is still 50 cents.
If Geeta saved $20 more than Hani,
No. of days they saved for = $20 / $0.50 = 40
Amt. saved by Hani each day = $40 / 40 = $1
Hani’s daily pocket money = $1 + $0.60 = $1.60
Library A has 50 more books than library b. Library b will have twice amount of books than library A when 120 books are transferred to library b from library A. How many books are there
@Ace I think this question has a problem.
A has more than B at first.
If books were transferred from B to A, then B should have a lot lesser.
How can B have two times as much as A in the end?
FyMath hi the question is saying to B from A. This question tricked you with English.
Please help,
had some $5 notes and $2 notes.The
ratio of the number of $5 notes to the number of $2 notes was 6 : 11.When he exchanged 10 pieces of $5 notes for
some $2 notes, the ratio of the number of $5 notes to the number of $2 notes
became 2 : 7.How much money did Johan
have altogether?
@Ace 120 – 50 = 70 = half of B
B has 140
A has 140 + 50 = 190
Total 190 +140 = 330
10 $5 notes = 25 $2 notes
Now 6U – 10 : 11U +25= 2 : 7
U = 6
Was 36 X .5 = 18; 66 X .2 = 13.2, Total = 31.2
Now 36-10 = 26; 26X .5 = 13; 66+25 = 91; 91X.2 == 18.2; total 31.2
JayaSanthi https://kparents.wpengine.com/ 5ptsFeatured
3 minutes ago
10 $5 notes = 25 $2 notes
Now 6U – 10 : 11U +25= 2 : 7
U = 6
Was 36 X .5 = 18; 66 X .2 = 13.2, Total = 31.2
Now 36-10 = 26; 26X .5 = 13; 66+25 = 91; 91X.2 == 18.2; total 31.2
@Ace The earlier answer was wrong
The correct way follows:
Ace 120 – 50 = 70
Now after the transfer the half of B = 70 + 120 = 190
So B has 380
And A has 190
Before transfer A had 310
And B had 260; Total 570
Please help…..:
On Monday, there were 280 more chairs in hall B than hall A . On Tuesday, 0.25 of the chairs were moved from hall B to hall A. On Wednesday, 0.2 of the chairs were moved back to hall B. On
Thursday, half of the chairs in hall B were moved back again to hall A. In the end, there were 520 more chairs in hall A than hall B. How many chairs were there in hall B at first???
This isn’t too hard. All you have to do is draw the model for all of the days. 0.25 is 1/4 so i draw 4 units for the model of hall B in Monday, then i draw a second bar in the same model which
represents 280. Then I draw the model for hall A, which is just 4 units. On Thursday, there will be a difference of 520.Hall B will have 2 units+ 112, while hall A will have 6 units+ 170. After
that you will have to find 4 units. First, subtract 112 from 520, then add the answer with 280,and BOOM you have your answer.
In a school, 40% of the pupils and 10 pupils go to school by car. 25% of the remaining and another 8 pupils
walk to school and the rest of them go to school by bus. If there are 152 pupils walk to school, how many pupils
go to school by bus ?
Please help. Thanks.
AGNESLIEW This is really a trick question. You do not need to solve how many go by car.
75% is thus 144*3= 432
Bus is thus 432-8=424 (ans)
P.S. I am a 10 year old not kidding
@HogRiderR6 Dun boast lah. 10 yr old so wat?
Example 11. also can use algebra 2 solve.
A group of girls shared some stickers among themselves.They tried taking 17 stickers each,but found that the last girl had only 3 stickers.
When each girl took 15 stickers,there were 6 stickers left over.
a)How many girls were there?
b)How many stickers were there althogether?
PLEASE REPLYYY
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Can magnetospheric electron temperature be inferred from whistler dispersion measurements?
An approximate expression for whistler-mode group velocity is obtained, taking into account the effects of electron temperature and anisotropy, density and ion effects, effects of oblique propagation
and a non-dipolarity of the dayside magnetospheric magnetic field. This expression is applied to the propagation of whistlers between one hemisphere and the other. It is pointed out that at
frequencies close to the upper cut-off of whistler spectra, perturbations to whistler group delay times due to temperature effects can be of the same order of magnitude as, or even higher than, the
corresponding perturbations due to finite electron density and ion effects. A method of magnetospheric electron temperature diagnostics is proposed and applied to two whistlers recorded at Halley (L
= 4.3). It is pointed out that the values of temperature obtained from the analysis of whistler spectra depend on the choice of model of electron density and temperature distribution in the
magnetosphere and on the effect of ducted ray path on whistler delay times which is difficult to take into account in the computations.
Annales Geophysicae
Pub Date:
April 1990
□ Earth Magnetosphere;
□ Electromagnetic Interference;
□ Electron Energy;
□ Magnetospheric Electron Density;
□ Whistlers;
□ Ionospheric Electron Density;
□ Lightning;
□ Magnetic Field Configurations;
□ Perturbation Theory | {"url":"https://ui.adsabs.harvard.edu/abs/1990AnGeo...8..273S/abstract","timestamp":"2024-11-04T16:49:03Z","content_type":"text/html","content_length":"37274","record_id":"<urn:uuid:0490670d-9690-43a1-8f87-8bfbab420181>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00605.warc.gz"} |
On the possibility of ill-conditioned covariance matrices in the first-order two-step estimator
The first-order two-step nonlinear estimator, when applied to a problem of orbital navigation, is found to occasionally produce first step covariance matrices with very low eigenvalues at certain
trajectory points. This anomaly is the result of the linear approximation to the first step covariance propagation. The study of this anomaly begins with expressing the propagation of the first and
second step covariance matrices in terms of a single matrix. This matrix is shown to have a rank equal to the difference between the number of first step states and the number of second step states.
Furthermore, under some simplifying assumptions, it is found that the basis of the column space of this matrix remains fixed once the filter has removed the large initial state error. A test matrix
containing the basis of this column space and the partial derivative matrix relating first and second step states is derived. This square test matrix, which has dimensions equal to the number of
first step states, numerically drops rank at the same locations that the first step covariance does. It is formulated in terms of a set of constant vectors (the basis) and a matrix which can be
computed from a reference trajectory (the partial derivative matrix). A simple example problem involving dynamics which are described by two states and a range measurement illustrate the cause of
this anomaly and the application of the aforementioned numerical test in more detail.
All Science Journal Classification (ASJC) codes
• Aerospace Engineering
• Space and Planetary Science
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On a class of descrete functions for Proof-of-Space blockchain consensus protocols
The classic blockchain design implies the mining procedure, which is essentially reflected in the following: to add a new block to the chain, a user has to solve an instance of some moderately hard
computational problem (Proof-of-Work
framework, PoW). One of the most criticized points of this approach is that users have to perform a significant amount of computational work. As a result, the PoW-blockchains involve very high energy
consumption. This led to the arising of a blockchain-related research area aimed at developing other ways to prove the right of a user to add a new block. In 2015, Dziembowski et al. suggested the
Proof-of-Space concept (PoS): instead of spending a certain amount of time on computations, users should reserve a certain amount of disk space on their computers. This requirement can be
implemented, for example, by requesting the user to invert some discrete function, so that a user who has stored the table of the function values could easily do it. In 2017, Abusalah et al., trying
to overcome some shortcomings of the original (simple) PoS, suggested to choose the function as a special composition of two discrete functions. In this paper, we analyze the idea of Abusalah et al.
It is shown that this proposal does not meet the PoS requirement to reserve the specified amount of disk space. Also, we discuss the analysis of mathematical models of blockchains as
cryptographic primitives.
Varnovsky N. P. Blockchain as a cryptographic primitive // International
Journal of Open Information Technologies (INJOIT). 2020.
Vol. 8, no. 12. P. 28–32 [in Russian].
Tapscott D., Tapscott A. The blockchain revolution: How the technology
behind bitcoin is changing money, business, and the world.
London : Penguin Books, 2016.
Drescher D. Blockchain basics: A non-technical introduction in
steps. New York : Apress, 2017.
Pass R., Seeman L., Shelat A. Analysis of the blockchain protocol
in asynchronous networks // Advances in Cryptology—
EUROCRYPT ’17. Vol. 10210 of Lecture Notes in Computer Science.
Berlin, Heidelberg : Springer, 2017. P. 643–673.
Proofs of space / S. Dziembowski, S. Faust, V. Kolmogorov,
K. Pietrzak // Advances in Cryptology—CRYPTO ’92. Vol. 9216 of
Lecture Notes in Computer Science. Berlin, Heidelberg : Springer,
P. 585–605.
Beyond Hellman’s time–memory trade-offs with applications to proofs
of space / H. Abusalah, J. Alwen, B. Cohen et al. // Asiacrypt 2017,
Part II. Vol. 10625 of Lecture Notes in Computer Science. Berlin,
Heidelberg : Springer, 2017. P. 357–379.
Dwork C., Naor M. Pricing via processing or combatting junk mail //
Advances in Cryptology—CRYPTO ’92. Vol. 740 of Lecture Notes in
Computer Science. Berlin, Heidelberg : Springer, 1993. P. 139–147.
Hellman M. E. A cryptanalytic time–memory trade-off // IEEE Trans.
on Information Theory. 1980. Vol. IT-26, no. 4. P. 401–406.
Katz J., Lindell Y. Introduction to Modern Cryptography. London :
CRC Press, 2007.
Nechayev V. I. Elementy kriptografii : Osnovy teorii zashchity
informatsii. Moscow : Vysshaya shkola, 1999 [in Russian].
Vasilenko O. N. Number-theoretic algorithms in cryptography. Providence
(Rhode Island) : American Mathematical Society, 2006.
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jach@fets201 Posted: Tuesday 03rd of Jun 09:17
Ok, after hearing so much about Algebrator, I think it definitely is worth a try. How do I get hold of it? Thanks!
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daujk_vv7 Posted: Tuesday 03rd of Jun 19:10
You don’t have to worry about calling them , it can be purchased online. Here’s the link: https://softmath.com/. They even provide an unreserved money back assurance, which is just
From: I dunno,
I've lost it.
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SjberAliem Posted: Thursday 05th of Jun 09:22
A truly piece of math software is Algebrator. Even I faced similar problems while solving trigonometry, least common measure and function definition. Just by typing in the problem
from homework and clicking on Solve – and step by step solution to my algebra homework would be ready. I have used it through several math classes - Basic Math, Algebra 1 and
Remedial Algebra. I highly recommend the program.
From: Macintosh
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How do you sketch the graph of y=8/3x^2 and describe the transformation? | Socratic
How do you sketch the graph of #y=8/3x^2# and describe the transformation?
1 Answer
This graph will be a parabola since one of the terms are squared.
The ${x}^{2}$ term means that the graph will be a parabola. There are three ways to graph this quadratic parabola.
1. Use a table: Choose some $x$ values and put them into an $x$ and $y$ table. Calculate the $y$ values by subbing the $x$ values into the given equation. Then graph the coordinates from the table.
2. Since the quadratic formula is given to vertex form, plot the vertex which is (0,0) and use the step pattern which is $\frac{8}{3}$ to graph the other coordinates.
In the end, your graph should look like the one given below.
graph{y=8/3 x^2 [-10, 10, -5, 5]}
To describe the transformation use "RST". In other words, describe the reflection of the parabola if there is one, then the stretch or compression factor and then lastly the translation.
Impact of this question
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Spontaneous emission in waveguide free-electron masers near waveguide cutoff
In this work spontaneous emission is investigated in a waveguide free-electron maser, taking into account previously untreated interaction effects in the vicinity of the waveguide cutoff frequency.
Our study is based on the exact waveguide excitation equations, formulated in the frequency domain for a single electron moving in a planar magnetostatic wiggler. An analytical solution of the
amplitude of the excited waveguide mode in the frequency domain was obtained using the Green function technique and allows us to calculate the spectral density of the radiated power and the
time-dependent radiated field with good accuracy using a numerical inverse Fourier transform. The obtained solution shows that for TE-modes the spectral density of the radiated energy tends to
infinity at the cutoff frequency of a lossless waveguide. The character of this singularity is, however, such that the total radiated energy is finite. The radiated electromagnetic field in the time
domain has the form of very long (of the order of tens of characteristic times on the scale of L[w]/c, where L[w] is the wiggler length and c is the speed of light) pulse, lagging behind the
electron, at the carrier of cutoff frequency, in addition to two finite wave packets, corresponding to the two synchronism frequencies. The results of a numerical calculation of the radiated energy
spectral density and of the radiated electromagnetic field in the time domain are presented.
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Recursive Predicates And Quantifiers
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S. C. KLEENE
This paper contains a general theorem on the quantification
of recursive
predicates, with applications to the foundations of mathematics.
The theorem
(Theorem II) is a slight extension of previous results on Herbrand-Gödel
general recursive functions(2),
while the applications
include theorems of
Church (Theorem VII)(3) and Gödel (Theorem VIII)(4) and other incompleteness theorems. It is thought that in this treatment
the relationship
the results stands out more clearly than before.
The general theorem asserts that to each of an enumeration
of predicate
forms, there is a predicate not expressible in that form. The predicates considered belong to elementary
number theory.
The possibility that this theorem may apply appears whenever it is proposed to find a necessary and sufficient condition of a certain kind for some
given property of natural numbers; in other words, to find a predicate of a
given kind equivalent to a given predicate. If the specifications on the predicate which is being sought amount to its having one of the forms listed in
the theorem, then for some selection of the given property a necessary and
sufficient condition of the desired kind cannot exist.
In particular, it is recognized that to find a complete algorithmic theory
for a predicate P(a) amounts to expressing the predicate as a recursive predicate. By one of the cases of the theorem, this is impossible for a certain P(a),
which gives us Church's theorem.
Again, when we recognize that to give a complete formal deductive theory
(symbolic logic) for a predicate P{a) amounts to finding an equivalent predicate of the form (Ex)R(a, x) where R(a, x) is recursive, we have immediately
Gödel's theorem, as another case of the general theorem.
Still another application is made, when we consider the nature of a constructive existence proof. It appears that there is a proposition provable classically for which no constructive
proof is possible (Theorem X).
The endeavor has been made to include a fairly complete exposition of
definitions and results, including relevant portions of previous theory, so that
Presented to the Society, September 11, 1940; received by the editors February 13, 1942.
In the abstract of this paper, Bull. Amer. Math. Soc. abstract 46-11-464, erratum: line 4,
for "for." read "for all.".
(l) A part of the work reported in this paper was supported by the Institute
Study and the Alumni Research Foundation of the University of Wisconsin.
(?) Gödel [2, §9] (see the bibliography at the end of the paper).
(') Church [1 ].
(«) Gödel [1, Theorem VI].
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for Advanced
S. C. KLEENE
the paper should be self-contained, although some details of proof are omitted.
The general theorem is obtained quickly in Part I from the properties of
the ju-operator, or what essentially was called the p-function in the author's
Part II contains some variations on the theme of Part I, and
may be omitted by the cursory reader. The applications to foundational questions are in Part III, only a few passages of which depend on Part II.
I. The general
1. Primitive
the informal
theorem on recursive
and quantifiers
of the natural
0, ly 2,
The discussion belongs to the context of
* * f 'X, % %* * ' .
The functions which concern us are number-theoretic
functions, for which
the arguments and values are natural numbers.
We consider the following schemata as operations for the definition of a
function <j>from given functions appearing in the right members of the equations (c is any constant natural number):
<j>{x)= x',
4>{xu ■ ■ ■ , xn) = c,
<p(Xi, •••,«„)=
• • • , Xn), • • • , X».(*l. • • ■ » Xn)),
I 4>(0)= c
= Xi,
U(y') = x(y,<t>(y)),
j 0(0,
XU ■ ■ ■ , Xn) = \p(xU ■ ■ ■ , Xn)
XU • ■ ■ , Xn) = X(y,
<t>(y, XU ■ ■ ■ , Xn), Xi, ■ ■ ■ , Xn).
Schema (I) introduces the successor function, Schema (II) the constant
functions, and Schema (III) the identity functions.
Schema (IV) is the
schema of definition by substitution,
and Schema (V) the schema of primitive
recursion. Together we may call them (and more generally, schemata reducible to a series of applications of them) the primitive recursive schemata.
A function <j>which can be defined from given functions $1, ■ • • , $k by
a series of applications
of these schemata we call primitive recursive in the
given functions; and in particular, a function <j>definable ab initio by these
means, primitive recursive.
Now let us consider number-theoretic
that is, propositional
functions of natural numbers.
(*) Kleene [l, §18].
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In asserting propositions, and in designating predicates, we use a logical
symbolism, as follows. Operations
of the propositional
calculus: & (and),
V (or), — (not), —> (implies), = (equivalent).
(x) (for all x),
(Ex) (there exists an x such that). These operations may be taken either in
the sense of classical mathematics,
or in the sense of constructive
or intuitionistic mathematics,
except where one or the other of the two interpretations
is specified.
A predicate P(xi, • • • , xn) is said to be primitive
primitive recursive function tt(x\, • • ■ , x„) such that
P(XU ■ • • , Xn) = *(Xi, ■••,«■}
if there is a
= 0.
We can without loss of generality restrict it to take only 0 and 1 as values,
and call it in this case the representing function of P.
Under classical interpretations,
which give a dichotomy
of propositions
into true and false, we can assign to any predicate P a representing function ir
which has 0 or 1 as value according as the value of P is true or false; and then
say that P is primitive recursive if x is.
2. General recursive
We shall proceed to the Herbrand-Gödel
generalization of the notion of recursive function. We start with a preliminary
account, certain features of which we shall then restate carefully.
The way in which the function <f>is defined from the given functions in
an application of one of the primitive recursive schemata amounts to this:
the values q>(xi, • • • , x„) of 4>for the various sets xi, ■ ■ ■ , xn of arguments
are determined
by the equations and the values of the given
functions, using only principles of determination
which we can formalize as
a substitution
rule and a replacement rule.
The formalization
presupposes suitable conventions governing the symbolism, which are easily supplied. In particular, we must distinguish between
the variables for numbers and the numerals, that is the expressions for the
fixed numbers in terms of the symbols for 0 and the successor operation
The rules are the following.
Rl: to substitute, for the variables
xi, • • • , x„, respectively.
xi, • • • , x„ of an equation,
R2: to replace a part f(xi, • • • , xn) of the right member of an equation by x,
where f is a function symbol, where xi, ■ ■ • , x„, x are numerals, and where
f(xi, • • • , x„) =x is a given equation.
By a given equation f(xi, • • • , x„)=x for R2, we mean an equation expressing one of the values of one of the given functions for the schema
or an equation of this form already derived by Rl and R2 from
the equations of the schema application.
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S. C. KLEENE
Now let us consider any operation or schema, for the definition of a function in terms of given functions, which can be expressed by a system of equations determining
the function values in this manner. In general the equations
shall be allowed to contain, besides the principal function symbol which represents the function defined, and the given function symbols which represent
the given functions, also auxiliary function symbols. The given function symbols shall not appear in the left members of the equations. Such a schema we
shall call general recursive.
A function q>which can be defined from given functions xpi, ■ • ■ , \pk by a
series of applications
of general recursive schemata we call general recursive
in the given functions; and in particular, a function cp definable ab initio by
these means we call general recursive.
that a function
<j> is defined, either from given functions
^1, • • • , Tpk or ab initio, by a succession of general recursive operations.
Let us combine the successive systems of equations which effect the definition into one system, using different symbols as principal and auxiliary function symbols in each of the successive systems, and in the resulting system
considering as auxiliary all of the function symbols but that representing
and those representing
\J/i, • ■ • , tyk. The restriction
imposed on a general
recursive schema that the given function symbols should not appear on the
left will prevent any ambiguity
being introduced
by the interaction
Rl and R2 of equations in the combined system which were formerly in separate systems. Thus the definition can be considered as effected in a single
general recursive operation.
In particular,
any general recursive function can be defined ab initio in
one operation, so that in the defining equations there are no given function
symbols and what we have called the given equations for an application
of R2
must all be derivable from the defining equations by previous applications
of Rl and R2. For the formal development,
it is convenient
to adopt the convention that the principal function symbol shall be that one of the function
symbols occurring in the equations
of the system which comes latest in a
list of function symbols. The function is then completely
described by giving the system of defining equations.
We now restate the definition of general recursive function from this point
of view.
A function <f>(xu■ ■ ■, xH) is GENERAL RECURSIVE, if there is a system E of equations
it recursively
in the following
system E of equations defines recursively a GENERAL
tion of n variables if, for each set xx, • ■ ■ , xn of natural
of the form f(xi, • • • , xn) =x, where f is the principal
and where xi, • • • , xn are the numerals representing
of n variables
which is defined
numbers, an equation
function symbol of E,
the natural numbers
X\, • • • , xn, is derivable from E by Rl and R2 for EXACTLY
The function
sense. A
one numeral x.
by E in this case is the func-
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tion <p,of which the value <p{x\, ■ ■ • , xn) for X\, • • • , xn as arguments
is THE
NATURAL NUMBER * REPRESENTED BY THE NUMERAL x.
A predicate P(xi, ■•',»»)
is general recursive, if there is a general recursive function w(xi, ■ • ■ , x„) taking only 0 and 1 as values such that (1) holds;
in this case, w is called the representing function of P. (Or, if we introduce the
representing function it first, P is general recursive if ir is.)
3. The /i-operator.
Consider the operator: fiy (the least y such that). If
this operator is applied to a predicate R{xi, •••,««,
y) of the «4-1 variables
Xi, • • ■ , xn, y, and if this predicate satisfies the condition
(xi) ■ ■ ■ (xn)(Ey)R(x1,
we obtain
a function
■ ■ ■ , xn, y),
p,yR(x\, • ■ • , x„, y) of the remaining
n free variables
, xn.
Thence we have a new schema,
4>(xu ■ ■ ■ , xn) = ßy[p(xu
for the definition
of a function
■ ■ ■ , xn, y) = 0],
<j>from a given function
(xi) ■ ■ ■ (x„)(Ey)[p(x1,
p which satisfies
■ ■ ■ , x„, y) = 0].
We now show that this schema, subject to the condition on p, is, like
(I)-(V), general recursive. For this purpose, we rewrite it in terms of equations, using an auxiliary function symbol "<r":
er(0, xi, •■• • , x„, y) = y
Xi, • • • , Xn, y) =
■ ■ ■ , Xn, /),
• • , X„, /)
4>(Xl, ■ ■ ■ , Xn) = <r(p(Xi, ■ • • , XH, 0), Xi, ■ ■ ■ , x„, 0).
Assuming the values of p, these equations will lead us to the values of <p as
defined by (VIi), and to only those values, as follows.
Consider informally any fixed set of values oi xi, • • • , xn (formally, this
means to substitute
the corresponding
set of numerals
for the variables
uXi", • • • , "xn"). We seek to obtain the corresponding
value of 4>(xi, • • • , xn)
by replacements
on the third equation, and this is the only possibility we
have for obtaining that value under the two principles. First we can replace
p(x\, •••,*„,
0) by its value, and this is the only first replacement step possible on that equation. According as that value is 0 or is not 0 we seek the
value of er for the next replacement
step from the first or secor
- equations, and this is the only possible source for the next replacement
value. In
the first case, we obtain 0 as that value; in the second, we use the value of
p(xi, ■••,*»,
1) in the second equation, and then seek another value of a.
We continue thus, with no choice in the procedure at any stage. The first case
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S. C. KLEENE
is first encountered when we come to use the value of p(xi, ••*,*«,
y) for the
first y for which that value is 0, and hence certainly for at most the y given by
(3). When this happens, we can complete the pending replacements to obtain
that y as the value of 4>(xi, ■ ■ ■ , x„). Thus we get the intended value; and
because we had no choice at any stage of the procedure, we can get no other
The general recursiveness
of the new schema is thus established.
Hence, if
R(xi, • ■ ■ , xn, y) is a general recursive predicate and (2) holds, by taking as p
the representing
function of R, we can conclude that pyR(xi, •••,*»,
y) is
a general recursive function.
What can we conclude if (2) is not assumed to hold? In this case,
nyR(x\, • • • , x„, y) may not be completely
defined as a function of the
but for any fixed set of values of X\, • • • , x„, the sequence of steps by which we attempt to determine a value for 4>(xu • • ■ , x„)
from the equations
remains as described for the preceding case, only with
now the matter of its termination
in doubt. If (Ey)R(xi, ■ • • , xn, y) does
hold for that set of values of xi, • • ■ , x„, then it does terminate as described,
with nyR(x\, ■ ■ • , xn, y) as the value; while conversely, if it does terminate, this can only be in consequence
of a 0 being encountered
the values of p(xi, • • • , x„, y), so that (Ey)R(x\, • • ■ , xn, y) does hold,
and nyR(xi,
y) is the value.
Hence, in formal terms, if F is the system of equations obtained by adjoining, to any system E which defines p recursively, equations of the form
(VI2), with the notation so arranged that "<p" becomes the principal function
symbol f, then: an equation of the form f(xi, • • • , xn) =x, where xi, • • • , x„
are the numerals representing
the natural numbers *i, • • • , xn, and where
x is a numeral,
is derivable
from F by Rl and R2 if and only if
(Ey)R(xi, • ■ ■ ,xn, y).
4. The enumeration
theorem. We introduce a metamathematical
predicate ©„ (for each particular n) as follows.
Y): Z is a system of equations, and Y is a formal deduction from Z by Rl and R2 of an equation of the form f(xi, • • • , xn) =x, where f
is the principal function symbol of Z, where xi, • • • , x„ are the numerals representing the natural numbers «*,•••,
xn, and where x is a numeral.
With this notation,
we can state
the last result of the preceding
■ ■ ■ , xn, y) = (£Y)®„(F,
xu ■ ■ ■ , xn, Y).
From a like exploration of the possibility that the sequence of steps does not
terminate, or simply from (4) by contraposition,
we have also:
■ ■ ■ , xn, y) = (Y)©n(F,
xu • • • , xn, Y).
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Using Gödel's idea of arithmetizing
suppose that
natural numbers have been correlated to the formal objects, distinct numbers
to distinct objects. The metamathematical
predicate ©„(Z,
Y) is
carried by the correlation into a number-theoretic
predicate Sn(z, Xi, • ■ -,xn,y),
the definition of which we complete by taking it as false for values of z, y not
both correlated to formal objects.
For a suitably chosen Gödel numbering, we can show, with a little trouble
that Sn is primitive recursive.
Now (4) translates
under the arithmetization
■ ■ ■ , xn, y) = (Ey)S„(f, xu ■ ■ ■ , xn, y)
with / as the Gödel number
is obtained
of the system
of equations
■ ■ ■ , xn, y) = (y)S„(g,
likewise from (5), after changing
F. The formula
xlt ■ ■ ■ , xn, y)
the notation
so that R is inter-
changed with R.
In stating these results for reference, we shall go over from S„ to a new
predicate Tn, which entails no present disadvantage
and proves to be of convenience in some further investigations(7).
The predicate Tn is defined from
Sn as follows.
Tn(z, xu ■ ■ ■ , Xn, y):
Sn(z, xu ■ ■ , xn, y) & (t)[t
< y —> Sn(z,
xu ■ • • , xn, t)].
By a theorem of Gödel(8), the primitive recursiveness of Tn follows from that
of Sn. The formulas (6) and (7) in the theorem follow from (6a) and (7a) by
the definition of T„ in terms of Sn.
I. Given a general recursive predicate
numbers f and g such that
R(xx, • ■ ■ , x„, y), there are
, xn, y) = {Ey)Tn{f, xu • • • , x„, y),
, xn, y) =
xn, y).
Now (Ey)Tn{z,
xn, y) is a fixed predicate
of the form
(Ey)R(z, x\, • • • , xn, y) where R is general recursive (in fact, as it happens,
primitive recursive). By the theorem, if we take successively z = 0, 1, 2, • • • ,
we obtain an enumeration
(with repetitions)
of all predicates
of the form
(Ey)R(xi, ■ ■ • , xn, y) where R is general recursive(9). Likewise, the theorem
gives us a fixed predicate of the form (y)R(z, Xi, • • • , xn, y) where R is general recursive which enumerates
all predicates of the form (y)R(xi, • ■ ■ ,xn,y)
(«)Gödel [iL
(") A revision, April 13, 1942.
(8) Gödel [1, IV].
(") This result entered partly into the last theorem of Kleene [2], but the advantage of
using it at an earlier stage was overlooked. In anticipation, we may remark that XI-XVI of that
paper are essentially special cases of Theorem II below (with now a constructive proof for XVI).
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S. C. KLEENE
where R is general recursive. These enumerations
form the basis for the application of Cantor's diagonal method in the next section.
5. The general
the following
By a familiar rule of classical logic, in each of
pairs of propositions
(with a fixed R for a given pair),
{x){Ey)R(x, y)
y, z)
{x){Ey)(z)R(x, y,z)
either member is equivalent to the negation of the other. Hence we may assert
between the members of the pair. This argument is not good
in the intuitionistic
logic. However, the non-equivalence
for the case of one
(Ex)R(x) jk (x)R(x),
does hold good intuitionistically.
Consider the predicate form (x)R(a, x) where R is general recursive. This
gives a particular predicate of the variable a, whenever we specify the general
recursive predicate R(a, x) of two variables. In particular,
(x)Ti(a, a, x) is a
predicate of this form.
We shall show that this predicate is neither general recursive nor expressible in the form (Ex)R(a, x) where R is general recursive.
For this purpose, suppose we have selected any particular general recursive R(a, x), giving a particular predicate of the latter form. By (6), there is
for this R a number/such
(Ex)R(a, x) m (Ex) 7\(f, a, x).
the number
/ for the variable
(Ex)R(f, x) = (Ex)Tx{f,f, x).
By (8),
Combining (10) and (11),
(Ex)R(f, x) fi (x)T1(/, /, x).
This refutes, for a=f, the equivalence of (Ex)R(a, x) to (x)Ti(a, a, x). Since
this refutation
can be effected, whatever general recursive R we chose, for
some/ depending on the R, the predicate (x)Ti(a, a, x) is not expressible in
the form (Ex)R(a, x) where R is general recursive.
A fortiori, (x)Ti(a, a, x) is not expressible in the form R(a) where R is
general recursive. For were it so expressed, we should then have it in the form
(Ex)R(a, x) where R is general recursive, by taking as R(a, x) the predicate
R(a) &x=x.
This completes the proof of one case of the next theorem.
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For another case, consider the predicate form (Ex)R(a, x) where R is general recursive. We can show similarly, using (7) instead of (6), that the predicate (Ex)T\(a, a, x), which has this form, is neither general recursive nor expressible in the form (x)R(a, x) where R is general recursive.
To illustrate the treatment
of a case with more than one quantifier, consider the predicate form (x)(Ey)(z)R(a,
x, y, z) where R is general recursive.
The predicate (x)(Ey)(z)T3(a,
a, x, y, z) has this form. Select any particular
general recursive R(a, x, y, z). By (6), for some/depending
on this R,
(Ez)R(a, x, y, z) = (Ez)T3(f, a, x, y, z).
By corresponding
of these equivalent
x, y, z) ^ (Ex)(y)(Ez)T3(f,
a, x, y, z).
we can complete
the argument
as before, showing
a, x, y, z) is not expressible in any of the forms
(Ex)R(a, x)
(x)R(a, x)
x, y)
x, y, z)
(Ex){y)R{a, x, y)
where the R for the form is general recursive.
To obtain an alternative
phrasing of the theorem, in which it holds for all
cases intuitionistically,
we may omit in the classical proof the step which
the two kinds of quantifiers
under the operation
of negation. We thus show that the predicates
a, x), (x)Ti(a, a, x),
a, x, y, z), and so on, are neither expressible in the respective
forms (Ex)R(a, x), (x)R(a, x), (Ex)(y)(Ez)R(a,
x, y, z), and so on, where R
is general recursive, nor in any of the forms with fewer quantifiers.
II. Classically, and for the one-quantifier forms intuitionistically:
To each of the forms
(Ex)R(a, x)
x, y)
x, y, z) ■ ■ ■
x, y)
x, y, z) ■ ■ ■
where the R for each is general recursive, after the first, there is a predicate expressible in that form but not in the other form with the same number of quantifiers
nor in any of the forms with fewer quantifiers.
Classically, and intuitionistically:
To each of the forms, after the first, there
is a predicate expressible in the negation of that form but not in that form itself
nor in any of the forms with fewer quantifiers.
For simplicity,
we have given the theorem
for predicates
of one variable
but it holds:
Likewise, replacing the variable a throughout by n variables oi, • • • , an, for
any fixed positive integer n.
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S. C. KLEENE
By an elementary predicate, we shall mean one which is expressible in terms
of general recursive predicates, the operations &, V, —, —», = of the prepositional calculus, and quantifiers.
Suppose given an expression for a predicate in these terms. By the classical
predicate calculus, we can transform the expression so that all quantifiers
stand at the front. For each m, let (x)i, • • • , (x)m be a set of m primitive
recursive functions of x which as a set ranges, with or without repetitions, over
all w-tuples of natural numbers, as x ranges over all natural numbers (such
sets of functions are known). The equivalences
• • • (Exm)A(xi,
• ■ • (xm)A(xi,
■ ■ ■ , xm) = {Ex)A{{x)1, ■ ■ ■ , (x)m),
■ ■ ■ , xm) = {x)A{{x)u
■ ■ ■ , (x)m)
enable us to eliminate consecutive occurrences of like quantifiers. These transformations leave as operand of the prefixed quantifiers
a general recursive
predicate of the free and bound variables. Hence, classically, the predicate
forms listed in the theorem for a given n suffice for the expression of every
elementary predicate of n variables.
The theorem then says that no finite sublist of the forms would suffice.
Classically, we are led to a classification of the elementary predicates according to the minimum numbers of quantifiers which would suffice for their
expression in terms of general recursive predicates and quantifiers.
The analogy between the logical operations of existential
and universal
and geometrical operations of projection and intersection,
respectively, is well known(10). The possibility of a connection between present
results and theories of Borel and Baire is suggested(11)-
II. Primitive, general,
and partial
under quantification
6. Partial recursive functions. The author's definition of partial recursive
function extends the Herbrand-Gödel
definition of general recursive function
to functions <j>of n variables which need not be defined for all w-tuples of
natural numbers as arguments,
retaining the characteristic
of that definition
with respect to each w-tuple for which the function is defined(12). The partial
recursive functions include the general recursive functions as those which are
defined for all sets of arguments.
For a more complete description, take the definition of general recursive
function which is given at the end of §2, and replace the four capitalized
phrases by the following, respectively: PARTIAL RECURSIVE; PARTIAL
RECURSIVE; AT MOST; THE NATURAL NUMBER x REPRE(l0) In particular, it has been discussed by Tarski.
(u) This suggestion was made to the author by Gödel and by Ulam.
(B) Kleene [4].
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SENTED BY THE NUMERAL x IF THAT NUMERAL EXISTS, AND
In dealing with functions which may not be completely denned, we interpret the equation cj>(xi, ■ ■ ■, xn) = ^/{x\, • • • , xn) as the assertion that <pand ^
have the same value for X\, ■ ■ ■ ,xn as arguments,
taking it as undefined (nonsignificant) if either value is undefined. We write <p(xh • • • ,xn)~\p(xi,
• • -,xn)
to express the assertion that, if either of <pand \p is defined for the arguments
*i» ' • • i x„, the other is and the values are the same, and if either of <j>and ip
is undefined for those arguments,
the other is.
Similarly, in dealing with predicates which may not be completely defined,
P(xi, • • • , xn)=Q(xi,
• • • , xn) expresses equivalence
of value, and is undefined if the value of either member is undefined;
while P(x\, • • • , x„)
=Q(xi, ■ • • , x„) expresses that the definition of either implies mutual definition with equivalence,
and the indefinition
of either implies mutual in-
A predicate P(xi, • • • , xn) not necessarily defined for all w-tuples of natural numbers as arguments is partial recursive, if there is a partial recursive
function ir{xi, • • • , x„) taking only 0 and 1 as values such that
P{XU ■ ■ ■ , Xn) = 7r(Xl, • • ■ , Xn) = 0;
in this case, 7r is called the representing function of P. (Or if we first introduce
a representing function w of P, the value of which is to be 0, 1, or undefined
according as the value of P is true, false, or undefined, then P is partial recursive if 7Tis.)
In §§2, 3, we remarked
the general recursiveness
of Schemata
with (VI) subjected to the condition (3); and we also considered Schema (VI)
for the case that p is general recursive but (3) is not required to hold. The
method of those sections applies equally well without the restrictions; in explanation of the schemata when the given functions may not be completely
defined or (3) not hold for (VI), it will suffice here to remark that the conditions of definition for the functions introduced
by the schemata may be inferred a posteriori from the metamathematical
III. The class of general recursive functions is closed under applications of Schemata (I)-(VI) with (3) holding for applications of (VI).
The class of partial recursive functions is closed under applications
Schemata (I)-(VI).
Every function obtainable by applications of Schemata (I)-(VI)
with (3) holding for applications of (VI) is general recursive.
Every function obtainable by applications of Schemata (I)-(VI) is partial recursive.
7. Normal form for recursive
We shall pursue a little further
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S. C. KLEENE
the method of §4 to obtain the converse of this result. Besides the metamathematical predicate ©„, we now require a metamathematical
function as follows.
tl(Y): the natural number x which the numeral x represents, in case Y is a
formal deduction of an equation of the form t = x, where x is a numeral and t is
any term; and 0, otherwise.
According to the definition of general recursive
recursive function of n variables, there is a system
(*,) • ■• (x„)(£Y)@„(E,
function, if <pis a general
E of equations such that
xn, Y),
(19) (*0 ■• • (*„)(Y)[@»(E,*i, • • ■, *„, Y) -*U(Y) =
and the function
<p(xh ■ ■ • , xn) can be expressed
in terms of E thus
<p(xlt ■ ■ ■ , xn) = U(MY©n(E, Xi, ■ ■ ■ , xn, Y)),
if we understand
the formal objects to be enumerated
in some order, so that
the operator jx can be applied with respect to the metamathematical
variable Y; we may take the order to be that of the corresponding
Gödel numbers.
If d>is a partial recursive function of n variables, instead of asserting (18),
we can write
xu ■■ ■ , x,t, Y)
as the condition on xi, • • • , xn that the function be defined for Xi, ■ ■ ■ , xn
as arguments;
we have (19), taking the implication
to be true whenever the
first member is false, irrespective
of the status of the second member; and
our convention calls for rewriting (20) thus,
4>(xu ■■■ , xn) ~U(mY@„(E,
in order
it be true
xu ■■ ■ , x„, Y)),
(and not sometimes
for all values
* , Xn>
By the Gödel numbering already considered, the metamathematical
function U(Y) is carried into a number-theoretic
function U(y), the definition of
which we complete by taking the value to be 0 for any y not correlated to a
formal object. If the Gödel numbering was suitably chosen, U as well as Sn
is primitive recursive.
Now (20), (18) and (19) in terms of ©„ and U are carried into formulas
of like form in terms of 5„ and U. On passing over from Sn to Tn, we then
have the (22), (23) and (24) of the theorem(13). The part of the theorem which
refers to a partial
is obtained
IV. Given a general recursive function
4>(xi, • ■ • , x„), there is a
(») Kleene [2, IV], with some changes in the formulation. The present 5» corresponds to
the former Tn, using the Gödel numbering of proofs instead of the enumeration
of provable
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number e such that
4>(xu •••,*»)=
U{nyTn{e, *!,«••,
(xi) • • ■ (xn)(Ey)Tn(e,
(xi) ■ ■ ■ (xn)(y) [Tn(e, xlt ■ ■ ■ , xn, y) —* U(y)
Given a partial
recursive function
x„, y)),
xu ■ ■ ■ , xn, y),
= 4>(xlt ■ ■ ■ , xn)\.
4>{xi, • • ■ , xn), there is a number
e such
<p(xu •••,**)
2k U(nyTn(e, x%,■ ■ ■ , x», y)),
is the condition of definition
xu ■ ■ ■ , xm y)
of the function,
and (24) holds.
Thus any general recursive function
(any partial recursive
expressible in the form \p(ßyR(xi, • ■ • , xn, y)) with (2) holding
yp(pyR(x\, • ■ • , x„, y))) where ^ and R are primitive recursive.
Every general recursive function
is obtainable
(in the form
by applications
of Schemata (I)-(VI) with (3) holding for applications of (VI).
Every partial
recursive function
is obtainable
by applications
of Schemata
Formula (25) contains the substance
of the theorem. For it implies the
condition of definition of the function; and, in the case that <p(xi, • • • , xn)
is defined for all sets of arguments,
it'gives (22) and (23). Moreover by the
definition of Tn in terms of S„, it implies (24).
We say that e defines <j>recursively, or e is a Gödel number of <p, if (25)
holds(14), in which case e has all the properties in relation to <f>which are
specified in the theorem.
It is here that the advantage of using T„ instead of Sn appears. A number e
which satisfies q>(xi, ■ ■ ■ , xn)~U(pySn(e,
Xi, • ■ • ,x„,y)) (which is equivalent
to (25)) does not necessarily
satisfy (xi) ■ • • (xn)(y) [Sn(e, xi, • • • , xn, y)
—>U(y) =(p(xi, ■ ■ ■ , xn)]. While we could get around the difficulty by imposing the latter as an additional condition on the Gödel numbers, it is more convenient simply to use Tn instead of 5n. (On the basis of Theorem III and the
results which we had in terms of 5„ before passing over to Tn, one can set
up a primitive recursive function Fsuch that, if e satisfies (25), then V(e) has
all the properties in terms of Sn.)
The numbers/and
g for Theorem I can be described now as any numbers
which define recursively
the partial recursive functions /xyR(xi, ■ ■ ■ , xn, y)
and nyR(xi, • • • , xn, y), respectively.
(u) Kleene [2, Definition 2c, p. 738] and [4, top p. 153]. We have now also the changes in
the formulation of Theorem IV.
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S. C. KLEENE
8. Consistency. Let us review the arguments used in proof of Theorems I
and III. For rigor, these have to be put in metamathematical
form. Let E
be the system of equations associated with a series of applications of Schemata
(I)-(VI). We shall review only the case that no given function symbols occur
in E.
In general, we easily establish that, for each of certain sets Xi, •••.,*«
of natural numbers, an equation of the form f(xi, • • • , x„) =x, as described
in the definitions of general and partial recursive function, is derivable from
E by Rl and R2. In particular,
if we are proving that E defines a general
recursive function, we must show this for all x\, ■ • • , xn; if we have a prior
of the schemata applications as definition of a (partial or complete) function <p(xi, •••,*„),
or require that E define a <f>(xi,• • • , x„) already known to us in some other manner, we must show this for all Xi, • •, x„
belonging to the range of definition of <p, and also show that the x in the
equation is the numeral representing
the value of <f>for Xt, * * • , #„ as arguments. This property of the equations
E and rules Rl and R2, the precise
formulation of which depends on the circumstances,
we call the "completeness property."
(When we wish merely to show that E defines a partial recursive function, the function to be determined a posteriori from E, no completeness property is required.)
The second part of the discussion consists in showing that an equation
of the described form f(xi, ■ • • , x„) =x is derivable from E for at most one
numeral x; or if we have already established completeness in one of the above
senses, that the equations f(xx, • • • , x„)=x referred to in the discussion of
for various xi, • • • , xn, are the only equations of that form
which are derivable from E by Rl and R2. This we call the "consistency
As we indicated in §2, it suffices to handle each of the schemata in turn,
assuming equations for use with R2 which give the values of the given functions. The argument for consistency which we sketched in §3 for Schema (VI)
applies as well to the other schemata.
For Schema (IV) there is indeed a
choice in the order in which the values of the several x's are introduced, but
it is without effect on the final result.
This very easy consistency
proof was gained by restricting
the replacement rule so that replacement
is only performable on the right member of an
equation, a part f(xi, • • • , x„) where f is a function symbol and xi, • • • , xn
are numerals being replaced by a numeral x. This eliminates the possibility
of deriving an equation of the form g(yi, ■ ■ ■ , ym) =y, where g is a fixed function symbol, yi, • • • ,ym are fixed numerals, andy is any numeral, along essentially different paths within the system, and therewith the possibility that
such an equation should be derivable for differenty's.
In some previous versions of the theories of general and partial recursive
functions, the replacement rule was not thus restricted. The consistency proof
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which we gave in the version with the unrestricted replacement rule was based
on the notion of verifiability of an equation(15). This notion makes presupposition of the values of the functions, and for the theory of partial recursive
functions also of the determinateness
whether or not the values are defined.
In the latter case, it is not finitary. To give a constructive
consistency proof
for the theory of partial recursive functions with the stronger replacement
rule seems to require the type of argument
used in the Church-Rosser
consistency proof for X-conversion(16),
and in the Ackermann-von
consistency proof for a certain part of number theory in terms of the Hilbert
€-symbol(17)It is easily shown, by using the method of proof of Theorem IV to obtain
the same normal form with the stronger replacement rule, that every function
partial recursive under the stronger replacement
rule is such under the weaker.
Thus we find the curious fact that the main difficulty in showing the equivalence of the two notions of recursiveness comes in showing that the stronger
rule suffices to define as many functions as the weaker. This is because the
consistency of a stronger formalism is involved. The consistency of that formalism is of interest on its own account, but is extraneous for the theory of
recursive definition, including the applications
to those of
Church in terms of the X-notation which presuppose the complicated ChurchRosser consistency
proof. All that is required for the theory of recursive
definition is some consistent formalism sufficient for the derivation of the
equations giving the values of the functions.
To this discussion we may add several supplementary
remarks. We might
in practice have a system E of equations and a method for deriving from E
by Rl and the strong replacement
rule, for all and only the w-tuples of a
certain set, an equation of the form f(xi, • • • , x„) =x with a determinate
but lack the knowledge that unlimited use of the two rules could not lead to
other such equations.
In this situation, a function is defined intuitively
the w-tuples of the set, and undefined off the set. If we can characterize metamathematically
our method of applying the two rules, we shall obtain a
limited formalism known to be consistent, and the method used in establishing Theorem IV can then be applied to obtain equations defining the function
recursively with the weak replacement
For some types of equations which define a function recursively with the
strong replacement
rule (consistency being known), a more direct method
may be available for obtaining a system defining the function recursively
with the weak replacement
rule. For example, consider (in informal language) the equation <f>(\[/(x))= x(x)- To use this in deriving equations giving
values of <f>,we need to introduce values of \p by replacement on the left. After
(16)Kleene [2, p. 731 ] and [4, §2, the bracketed portion of the fifth paragraph].
(18)Church and Rosser [l].
(") Hilbert and Bernays [l, §2, part 4, pp. 93-130, and Supplement II, pp. 396-400].
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S. C. KLEENE
expressing the equation in the form <f>(y)= (fxw[\p((w)i) =y& x((w)i) —(w)2])2,
and separating the latter into a series of equations without the ^-symbol by
the method which the theory of the schemata affords, replacement will be required only on the right. This device is applicable to any equation of which
the left member has the form f(gi(xi, • • • ,x„), ■ • ■ , gBl(xl, • • • , xj).
The precise form of the restriction which is used to weaken the replacement rule is somewhat arbitrary,
so long as it accomplishes
its purpose of
the deductions
of equations
giving the values of the functions.
The restriction as it was stated in the early Gödel version is now simplified,
since we need to consider only equations having the forms appearing in the
six schemata. Gödel provided for equations the left members of which could
have the form f(gi(xi, • • • , xn), • ■ ■ , gm(xi, • • • , x„)) where f is the principal function symbol and gi, • • • , gm are given function symbols, and therefore allowed replacement on the left in the case of the g's.
9. Predicates
for any general
in both one-quantifier
forms. By Theorem
P(xu ■ ■ ■ , xn) m (y) [Tn(e, xu ■ ■ ■ , *„, y)
xu ■ ■ ■ , xn, y) & U(y) = 0],
U(y) = 0],
where e is any Gödel number of the representing
function of P.
for a predicate
P both P(xi,
= (Ey)R(xu
■ ■ • , xn, y) and
• • ■ , xn),
■ ■ ■ , xn)=(y)S(xu
• ■ ■ , xn)
y) where
and 5 are general recursive. From the second of these equivalences,
classical interpretations,
P(xu ■ ■ • , xn) = (Ey)S(xi,
■ • ■ , xn, y). By the clas-
sical law of the excluded middle, (Ey) [R(xh ■••,*„,
(28) P(xi,
■ ■ ■ , x„) = R{xu
where the second member
■ ■ ■ ,xn, py[R(xi,
is general recursive
• • • , xn, y)].
■ ■ ■ , xn, y) \ZS(xu
■ ■ ■ , xn, y)]),
by Theorem
Theorem V. Every general recursive predicate P(xi, • • ■ , xn) is expressible
in both of the forms (Ey)R(xi,
• • ■ , xn, y) and (y)R(xh ■ ■ • , xn, y) where the R
for each is primitive recursive. Under classical interpretations, conversely, every
predicate expressible in both of these forms where the R for each is general recursive is general recursive.
Now consider any predicate expressible in one of the forms of Theorem II
after the first. According as the innermost quantifier in this form is existential
or universal, we can apply (26) or (27), and then absorb the extra quantifier
by (15) or (16), respectively, to obtain the original form but with a primitive
recursive R. For example,
x, y) m (x)(Eyi)(Ey2)[T3(e,
a, x, yu y2) & U(y2) - 0]
= (x)(Ey)[Tt(e, a, x, (y)u (yh) & U((y)2) = 0].
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The class of predicates expressible in a given one of the forms of
Theorem II after the first {for a given n variables) is the same whether a primitive
recursive or a general recursive R be allowed.
This generalizes the observation
of Rosser that a class enumerable
general recursive function is also enumerable
by a primitive
by a
tion (18).
The formulas
for the one-quantifier
cases are
■ ■ ■ , xn, y)
m (Ey)[Tn+1(e,
(y)2) & U((y%)
= 0],
(y)2) —►t/((y)2)
= 0],
xn, (y)i,
where e is any Gödel number of the representing
function of R. These afford
a new proof of the enumeration
theorem of §4, with new enumerating
predicates, and thence a new proof of Theorem II.
10. Partial recursive predicates. Let P(xi, • ■ ■ , xn) be a predicate which
may not be defined for all w-tuples of natural lumbers
as arguments.
By a
completion of P we understand
a predicate
Q such that, if P(x\, ■ ■ ■ , xn)
is defined, then Q(xi, ■ ■ ■ , xn) is defined and has the same value, and if
P(xi, •••,*«)
is undefined, then Q(xi, ■ ■ ■ , xn) is defined. In particular,
completion P+(xi, • • • , xn) which is false when P{x\, ■ ■ • , xn) is undefined,
and the completion P~(xi, • • • , xn) which is true when P{x\, •••,*»)
is undefined, we call the positive completion and negative completion of P(xi, ■ ■ - ,x„),
(In P and P+, the "positive parts" coincide; in P and P~, the
"negative parts" coincide.)
If P(xi, • • • , xn) is a partial recursive predicate, then by Theorem IV,
P+(xlt •.,«.)■
(Ey) [Tn(e, xu • ■ ■ ,
P~{xi, ■ ■ ■ , xn) = (y)[Tn(e,
y) & U(y) = 0],
xu ■ ■ ■ , xn, y) -> U(y) = 0],
where e is any Gödel number of the representing
function of P.
if R(xu • ■ ■ , xn, y) is any general recursive predicate,
by Theorem III,
■ ■ ■ , xn, y) = nyR(xu
■ ■ ■ , xn, y) = + nyR(xu
■ ■ ■ , xn, y),
■ ■ ■ , xn, y) = nyR{xi,
■ ■ ■ , xn, y) ^ ~ ßyR(xu
■ ■ ■ , xn, y).
Theorem VI. The positive completion P+(xi, ■ ■ ■ , xn) of a partial recursive
predicate P(xu • • • , xn) is expressible in the form (Ey)R{xi, ■ ■ ■ , xn, y) where
R is primitive recursive; and conversely, any predicate expressible in the form
(Ey)R(xi, ■ ■ • , xn, y) where R is general recursive is the positive completion
P+(xi, • • • , xn) of a partial recursive predicate P(xi, ■ ■ ■ , xn).
(1S) Rosser [l, Lemma I, Corollary
I, p. 88].
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S. C. KLEENE
for negative completions
• • • , xn) and the predicate form
■ ■ • ,xn,y).
It follows that, for the predicate forms of Theorem II which have an existential quantifier (universal quantifier) innermost, we may, without altering
the class of predicates expressible in that form, take R to be the positive completion (negative completion) of a partial recursive predicate.
Let us abbreviate
U{p.yTn{z, xx, • • • , x„, y)) as <P„(z,
Then 3>„is a fixed partial recursive function of «+1 variables, from which any
partial recursive function <p of n variables can be obtained thus (rewriting
<p(xu ■ ■ ■ , xn) ~
• , xn)
where e is any Gödel number of q>. Since for a constant z, $n(z, x\, ■ ■ • , xn)
is always a partial
of the remaining
n variables,
$„(z, Xx, • • • i xn) therefore gives for z = 0, 1, 2, • • • an enumeration
(with ,
of the partial recursive functions of n variables. It follows that
3V(z, xi, • • • , xn) = 0 is a partial recursive predicate of w-f-1 variables which
enumerates (with repetitions) the partial recursive predicates of n variables.
This, seen in the light of Theorem VI, has as consequence the enumeration
theorem of §2 (with other enumerating
and thence by Cantor's
diagonal method Theorem II.
Elsewhere, the enumeration
theorem for partial
by Cantor's diagonal method what may be called
for proofs of recursive definability(20).
This fundamental
theorem, and the existence
tions and predicates, no completions of which are
what occasioned the introduction
of the notion of a
recursive functions gave
the fundamental
of partial recursive funcgeneral recursive(21), are
partial recursive function.
III. Incompleteness
theorems in the foundations
of number theory
11. Introductory remarks. We entertain various propositions about natural numbers. These propositions
have meaning, independently
of or prior to
the consideration
of formal postulates and rules of proof. We pose the problem
of systematizing
our knowledge about these propositions
into a theory of
some kind. For certain definitions of our objectives in constructing
the theory,
and certain classes of propositions,
we shall be able to reach definite answers
concerning the possibility of constructing
the theory.
The naive informal approach which we are adopting may be contrasted
(19) Using the notation
of Kleene
[4, bottom
p. 152], but with the changes in the formula-
tion of Theorem IV.
(»»)Kleene [4, the last result in §2].
(21) Kleene [4, Footnote
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with that form of the postulational
approach which consists in first listing
formal postulates, which are then said to define the content of the theory
based on them. In the case of number theory, the formal approach cannot
render entirely dispensable an intuitive understanding
of propositions
of the
kind which we commonly interpret the theory to be about. For the explicit
statement of the postulates and characterization
of the manner in which they
are to determine the theory belong to a metatheory
on another level of discourse; and the ultimate metatheory
must be an intuitive mathematics
unregulated by explicit postulates, and having the essential character of num-
ber theory.
Of course the informality of our investigation
does not preclude the enumeration, from another level, of postulates which would suffice to describe it.
Indeed, such regulation may perhaps be considered necessary from an intuitive standpoint
for that part of it which belongs to the context of classical
The propositions about natural numbers which we shall consider will contain parameters.
We shall thus have infinitely many propositions
of a given
form, according to the natural numbers taken as values by the parameters.
In other words, we have predicates, for which these parameters are the independent variables. Generally, in a theory, a number of predicates are dealt
with simultaneously;
but for our investigations
it will suffice to consider a
theory with respect to some one predicate without reference to other predicates which might be present. Usually, we shall write a one-variable predicate
P(a), though the discussion applies equally well to a predicate P(ai, • • • , an)
of n variables.
12. Algorithmic theories. As one choice of the objective, we can ask that
the theory should give us an effective means for deciding, for any given one
of the propositions which are taken as values of the predicate, whether that
proposition is true or false. Examples of predicates for which a theoretical
conquest of this kind has been obtained are: a is divisible by b (that is,
in symbols, (Ex) [a = bx}), ax+by^c
is solvable for x and y (that is,
(Ex)(Ey) [ax+by = c]). We shall call this kind of theory for a predicate
a complete algorithmic theory for the predicate.
Let us examine the notion of this kind of theory more closely. In setting
up a complete algorithmic theory, what we do is to describe a procedure,
performable for each set of values of the independent
variables, which procedure necessarily terminates
and in such manner that from the outcome
we can read a definite answer, "Yes" or "No," to the question, "Is the predicate value true?"
We can express this by saying that we set up a second predicate: the procedure terminates in such a way as to give the affirmative answer. The second
predicate has the same independent
variables as the first, is equivalent
to the
first, and the determinability
of the truth or falsity of its values is guaranteed.
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S. C. KLEENE
This last property
of the second predicate
we designate
as the property
being effectivelydecidable.
Of course the original predicate becomes effectively decidable, in a derivative sense, as soon as we have its equivalence to the second; extensionally,
the two are the same. But while our terminology
is ordinarily extensional,
at this point the essential matter can be emphasized by using the intensional
language. The reader may if he wishes write in more explicit statements
referring to the (generally) differing objects or processes with which the two
predicates are concerned.
Now, the recognition that we are dealing with a well defined process which
for each set of values of the independent
variables surely terminates
so as to
afford a definite answer, "Yes" or "No," to a certain question about the manner of termination,
in other words, the recognition
of effective decidability
a predicate, is a subjective affair. Likewise, the recognition of what may be
called effective calculability
in a function.
We may assume, to begin with,
an intuitive ability to recognize various individual instances of these notions.
In particular,
we do recognize the general recursive functions as being effectively calculable, and hence recognize the general recursive predicates as be-
ing effectively decidable.
Conversely, as a heuristic principle, such functions (predicates)
as have
been recognized as being effectively calculable
(effectively decidable),
for which the question has been investigated,
have turned out always to be
general recursive, or, in the intensional language, equivalent to general recursive functions (general recursive predicates).
This heuristic fact, as well as
certain reflections on the nature of symbolic algorithmic
processes, led Church
to state the following thesis(22). The same thesis is implicit in Turing's description of computing
Thesis I. Every effectively calculable function
is general recursive.
(effectively decidable predicate)
Since a precise mathematical
definition of the term effectively calculable
(effectively decidable) has been wanting, we can take this thesis, together
with the principle already accepted to which it is converse, as a definition of
it for the purpose of developing a mathematical
theory about the term. To
the extent that we have already an intuitive notion of effective calculability
(effective decidability),
the thesis has the character of an hypothesis—a
emphasized by Post and by Church(24). If we consider the thesis and its converse as definition, then the hypothesis is an hypothesis about the application
of the mathematical
theory developed from the definition. For the acceptance
of the hypothesis, there are, as we have suggested, quite compelling grounds.
(22) Church [1 ].
(») Turing [1].
(«) Post [1, p. 105], and Church [2].
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A full account of these is outside the scope of the present paper (25). We are
here concerned rather to present the consequences.
In the intensional language, to give a complete algorithmic theory for a
predicate P(a) now means to find an equivalent
effectively decidable predicate Q(a). It would suffice that Q(a) be given as a general recursive predicate;
and by Thesis I, if Q(a) is not so given, then at least there is a general recursive predicate R(a) equivalent
to Q(a) and hence to P(a). Thus to give a
complete algorithmic
theory for P{a) means to find an equivalent
recursive predicate R(a), or more briefly, to express P(a) in the form R(a)
where R is general recursive. This predicate
form is the one listed first in
Theorem II; and Theorem II gives to each of the other forms a predicate not
expressible in that form. Thus, while under our interpretations
there is a complete algorithmic
theory for each predicate of the form R(a) where R is general recursive, to each of the other forms there is a predicate for which no
such theory is possible. We state this in the following theorem, using the particular examples for the one-quantifier
forms which were exhibited in the
proof of Theorem II.
VII. There exists no complete algorithmic
predicates (Ex)T\(a, a, x) and (x)Ti(a, a, x).
theory for either of the
Of course, once the definition of effective decidability
is granted, which
affords an enumeration
of the effectively decidable predicates, Cantor's methods immediately
give other predicates. This theorem, as additional content,
shows the elementary
forms which suffice to express such predicates.
from the particular
used here, the theorem
Church's theorem on the existence of an unsolvable
problem of elementary
number theory, and the corresponding
theorem of Turing in terms of his
machine concept(26). The unsolvability
is in the sense that the construction
called for by the problem formulation,
which amounts to that of a recursive
R with a certain property, is impossible. The theorem itself constitutes
solution in a negative sense.
13. Formal deductive theories. A second possibility for giving theoretic
cohesion to the totality of true propositions
taken as values of a predicate
P(a) is that offered by the postulational
or deductive method. We should like
all and only those of the predicate values which are true to be deducible from
given axioms by given rules of inference. To make the axioms and principles
of inference quite explicit, according to modern standards
of rigor, we shall
suppose them constituted
into a formal system (symbolic logic), in which
the propositions taken as values of the predicate are expressible. Those and
only those of the formulas expressing the true instances of the predicate
(as) por a resumei see Kleene
[4, Footnote
2 ], where further
(») Turing [1, §8].
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are given.
S. C. KLEENE
should be provable. We call this kind of theory for a predicate P(a) a complete formal deductive theory for the predicate.
This type of theory should of course not be confused with incompletely
formalized axiomatic theories, such as the theory of natural numbers itself
as based on Peano's axioms.
It is convenient in discussing a formal system to name collectively as the
the rules describing the formal axioms and the rules of inference.
Let us now examine more closely the concept of provability
in a stated
formal system. If the formalization
does accomplish its purpose of making
matters explicit, we should be able effectively to recognize each step of a
formal proof as an application
of a postulate of the system. Furthermore,
the system is to constitute a theory for the predicate P(a), we should be able
effectively to recognize, to each natural number a, a certain formula of the
system which is taken as expressing the proposition
P(a). Together, these
imply that we should be able, given any sequence of formulas
which might be submitted as a proof of P(a) for a given a, to check it, thus
determining effectively whether it is actually such or not.
Let us introduce a designation for the metamathematical
predicate with
which we deal in making this check, for a given formal system and predicate
9t(a, X): X is a proof in the formal
proposition P{a).
system of the formula
Then the concept of provability
in the system of the formula expressing
P(a), or briefly, the provability
of P(o), is expressible as (£X)9?(a, X).
As we have just argued, the predicate 9t(a, X) should be an effectively
decidable metamathematical
predicate. Here the formal objects over which X
ranges, if the notation of the system is explicit, should be given in some manner which affords an effective enumeration
of them. Using the indices in this
or generally any effective Gödel numbering of the formal objects, the metamathematical
predicate 9?(a, X) will be carried into a numbertheoretic predicate R(a, x), taken as false for any x not correlated to a formal
object, which should then also be effectively decidable. By Thesis I, the effective decidability
of the latter implies its general recursiveness.
We are thus
led to state a second thesis.
Thesis II. For any given formal system and given predicate P(a), the predicate that P(a) is provable is expressible in the form (Ex)R(a, x) where R is general recursive.
This thesis corresponds
tive system for a predicate
constitutes a proof of P{a)
P(a) does not do this, we
to the standpoint
that the role of a formal deducP{a) is that of making explicit the notion of what
for a given a. If a proposed "formal system" for
should say that it is not a formal system in the
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strict sense, or at least not one for P(a). Taken this way, the thesis has a
definitional character.
Presupposing, on the other hand, a prior conception of what constitutes a
formal system for a given predicate in the strict sense, the thesis has the character of an hypothesis, to which we are led both heuristically
and from Thesis
I by general considerations.
Conversely, if a predicate of the form (Ex)R(a, x) where R is general recursive is given, it is easily seen that we can always set up a formal system
of the usual sort, with an explicit criterion of proof, in which all true instances
of this predicate and only those are provable.
Using the thesis, and this converse, we can now say that to give a complete formal deductive theory for a predicate Pia) means to find an equivalent predicate of the form (Ex)R(a, x) where R is general recursive, or more
briefly, to express the predicate in this form. By Theorem II, there are predicates of the other one-quantifier
form, and of the forms with more quantifiers,
not expressible in this form. Hence while there are complete formal deductive
theories to each predicate of either of the forms R(a) and (Ex)R(a, x) where
R is general recursive, to each of the other forms there is a predicate for which
no such theory is possible. Specifically, using the one-quantifier
example given
in the proof of Theorem II:
Theorem VIII.
cate (x)Ti(a, a, x).
There is no complete formal deductive theory for the predi-
This is the famous theorem of Gödel on formally undecidable
propositions, in a generalized form. A proposition is formally undecidable in a given
formal system if neither the formula expressing the proposition
nor the formula expressing its negation is provable in the system. Gödel gave such a
proposition for a certain formal system (by a method evidently applying to
similar systems), subject to the assumptions of the consistency and co-consistency of the system. Later Rosser gave another proposition,
for which the
latter assumption is dispensed with(27).
In the present form of the theorem, we have a preassigned
(x)Ti(a, a, x) and a method which, to any formal system whatsoever for this
predicate, gives a number/
for which the following is the situation.
Suppose that the system meets the condition that the formula expressing
the proposition (x)Ti(J,f, x) is provable only if that proposition is true. Then
the proposition is true but the formula expressing it unprovable.
This statement of results uses the interpretation
of the formula, but if the system has
certain ordinary deductive properties for the universal quantifier and recursive predicates, our condition on the system is guaranteed by the metamathematical one of consistency.
If the system contains
also a formula
the negation
(«) Resser
[l ].
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S. C. KLEENE
(x)T\(f,f, x), and if the system meets the further condition that this formula
is provable only if true, then this formula cannot be provable, and we have
a formally undecidable proposition. The further condition, if the system has
ordinary deductive properties, is guaranteed
by the metamathematical
of co-consistency.
Moreover, we can incorporate
Rosser's elimination
of the hypothesis
co-consistency into the present treatment.
To do so, we replace the predicate
(Ex)R(a, x) for the application
of Theorem II by (Ex) [R(a, x) & (y)[y<x
—*S(a, y)]] where (Ex)S(a, y) is the predicate expressing the provability
the negation of (x)Ti(a, a, x). This changes the/for
the system.
Thus we come out with the usual metamathematical
results for a given
formal system.
For the case that a formal system is sought which should not only prove
the true instances of P(a) but also refute the false ones, if the classical law
of the excluded middle is applied to the propositions
P(a), then the Gödel
theorem (Theorem VIII) comes under the Church theorem (Theorem VII).
For had we completeness
with respect both to P(a) and to P(a), we could
obtain a general recursive R(a) equivalent
to the given predicate
by the
method used in proving the second part of Theorem V. Informally,
amounts merely to the remark that we should have the algorithm for P(a)
which consists in searching through some list of the provable formulas until
we encounter either the formula expressing P(a) or the formula expressing
The connection
between Gödel's theorem and the paradoxes
has been
much noted. The author gave a proof of Gödel's theorem along much the
present lines but as a refinement
of the Richard paradox rather than of the
Epimenides(28). That gave the undecidable
as values of a predicate of the more complicated
form (x)(Ey)R(a,
x, y) where R is general recursive. The Epimenides
paradox now appears as the more basic. Currently,
Curry has noted the same phenomenon in connection with the Kleene-Rosser
14. Discussion,
In the present form of Gödel's theo-
rem, several aspects are brought into the foreground which perhaps were not
as clearly apparent in the original version.
Not merely, to any given formal system of the type considered, can a
proposition be formulated with respect to which that system is incomplete,
but all these propositions can be taken as values of a preassignable elementary
predicate, with respect to which predicate therefore no system can be complete. This depends on the thesis giving a preassignable
form to the concept
of provability in a formal system.
(28) Kleene [2, XIII].
(29) Kleene and Rosser [l], Curry [2].
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For the interpretation
of the propositions we have required, as minimum,
only the notions of effectively calculable predicates and of the quantifiers used
It seems that lesser presuppositions,
if one is to allow any
infinite, are hardly conceivable.
Beyond that the system should fulfil the structural
expressed in Thesis II, and should yield results correct under this modicum
of interpretation,
we have need of no reference whatsoever
to its detailed
In particular, the nature of the intuitive evidence for the deductive processes which are formalized in the system plays no role.
Let us imagine an omniscient number theorist, whom we should expect,
through his ability to see infinitely many facts at once, to be able to frame
much stronger systems than any we could devise. Any correct system which
he could reveal to us, telling us how it works without telling us why, would
be equally subject to the Godel incompleteness.
It is impossible to confine the intuitive mathematics
of elementary
sitions about integers to the extent that all the true theorems will follow from
explicitly stated axioms by explicitly stated rules of inference, simply because
the complexity of the predicates soon exceeds the limited form representing
the concept of provability
in a stated formal system.
We selected as the objective in constructing
a formal deductive system
that what constitutes
proof should be made explicit in the sense that a proposed proof could be effectively checked, and either declared formally correct
or declared formally incorrect.
Let us for the moment entertain
a weaker conception
of a formal system,
under which, if we should happen to discover a correct proof of a proposition
or be presented with one, then we could check it and recognize its formal correctness, but if we should have before us an alleged proof which is not correct,
then we might not be able definitely to locate the formal fallacy. In other
words, under this conception a system possesses a process for checking, which
in the affirmative
case, but need not in the negative. Then the
concept of provability
would have the form (Ex)P+(a,
x) where P+ is the
positive completion of a partial recursive predicate P(a, x)..By Theorem VI,
P+(a, x) is expressible in the form (Ey)R{a, x, y) where R is general recursive.
Then the provability concept has the form (Ex)(Ey)R(a,
x, y), or by contraction of quantifiers (Ex)R(a, (x)i, (x)n). This is of the form (Ex)R(a, x) where
R is general recursive. Thus the concept of provability
has the usual form,
and Gödel's theorem applies as before. If we take a new concept of proof
based on R(a, x), that is, if we redesignate the steps in the checking process as
the formal proof steps, the concept of proof assumes the usual form.
We gave no attention, when we formulated the objectives both of an algorithmic and of a formal deductive theory, to the nature of the evidence for
the correctness of the theory, or to various other practical considerations,
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S. C. KLEENE
simply because the crude structural objectives suffice to entail the corresponding incompleteness
theorems. In this connection, it may be of some interest
to give the corresponding definitions, although these may not take into account all the desiderata, for the case of incomplete theories of the two sorts.
We shall state these for predicates of n variables oi, • • • , an, as we could also
have done for the case of the complete theories.
To give an algorithmic theory (not necessarily complete) for a predicate
P(<zi, • • • , an) is to give a general recursive function ir(ai, • • • , ö„), taking
only 0, 1, and 2 as values, such that
The algorithm always terminates,
but if 7r(<ii, • • • , a„) has the value 2 we
can draw no conclusion about P(ai, • • • , an).
To give a formal deductive theory (not necessarily complete) for a predicate
P(ai, • • • , o„) is to give a general recursive predicate R(ai, ■ ■ • , an, x) such
■ • • , an, x)—> P(au
• • • , a„).
In words, to give a formal deductive theory for a predicate P(ai, • • • , a„) is
to find a sufficient condition for it of the form (Ex)R(ai, • • • , an, x) where R
is general recursive. Here, according to circumstances,
the sufficiency may be
established from a wider context, or it may be a matter of postulation
(hypothesis), or of conviction (belief).
From the present standpoint,
the setting up of this sufficient condition
is the essential accomplishment
in the establishment
of a so-called metatheory
(in the constructive
sense) for the body of propositions taken as the values of
a predicate. We note that this may be accomplished without necessarily going
through the process of setting up a formal object language, from which R is
obtainable by subsequent
although as remarked above, we
can always set up the object language, if we have the R by some other means.
In the view of the present writer, the interesting variations of formal technique recently considered by Curry have the above as their common feature
with formalization
of the more usual sort(30). This is stated in our terminology, Curry's use of the terms "meta" and "recursive"
being different. He
gives examples of "formal systems," in connection with which he introduces
some predicates by what he calls "recursive definitions," but what we should
prefer to call "inductive definitions." This important type of definition, under
suitable precise delimitation
so that the individual clauses are constructive, can be shown to lead always to predicates expressible in the form
(Ex)R(ai, • ■ • , c„, x) where R is recursive in our sense. Indeed, this fact
('») Curry [l].
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can be recognized by substantially
the method indicated above for the case
of the inductive definition establishing
the notion of provability
for a formal
system of the usual sort.
Conversely, given any predicate expressible in the form (Ex)R(ai, • • • ,
a„, x) where R is recursive, we can set up an inductive definition for it.
15. Ordinal logics. In ordinal logics, studied by Turing(31), the requirement of effectiveness for the steps of deduction is relaxed to allow dependence
on a number (or X-formula) which represents
an ordinal in the ChurchKleene theory of constructive
ordinals(32). A presumptive
proof in an ordinal
logic cannot in general be checked objectively,
since the proof character depends on the number which occupies the role of a Church-Kleene
representative of an ordinal actually being such, for which there is no effective criterion.
it was hoped that ordinal logics could be used to give complete
orderings (with repetitions)
of the true propositions
of certain forms into
transfinite series, by means of the ordinals represented
in the proofs, in such
a way that the proving of a proposition in the ordinal logic (and therewith
the determination
of a position for it in the series) would somehow make it
easier to recognize the truth of the proposition.
Turing obtained a number of interesting results, largely outside the scope
of this article, but among them the following. There are ordinal logics which
are complete for the theory of a predicate of the form (x)(Ey)R(a, x, y) where
R is general recursive; however, for the example of such a logic which is given,
its use would afford no theoretic gain, since the recognition that the number
which plays the role of ordinal representative
in a proof of the logic is actually
such comes to the same as the direct recognition of the truth of the proposition proved.
Now let us approach the topic by inquiring whether, and if so where, the
property of being provable in a given ordinal logic is located in the scale of
predicate forms of Theorem II. First, it turns out that the property of a
number a of being the representative
of an ordinal is expressible in the form
(x)(Ey)R(a, x, y) where R is recursive(33). Now we may use the definition of
ordinal logic in terms of X-conversion, or we may take the notion in general
terms as described above, and state the thesis that for a given predicate P(a)
and given ordinal logic the provability
of Pia) is expressible in the form
a, x) where a ranges over the ordinal representatives
and R
is general recursive. In either case, it then follows that the provability of P{a)
is expressible in the form (Ex)(y)(Ez)R(a,
x, y, z) where R is general recursive.
Conversely, to any predicate of the latter form, we can find an ordinal logic
(31)Turing [2]. Turing gave a somewhat restricted definition of "ordinal logic" in terms of
the theory of \-conversion for predicates expressible in the form (x)(Ey)R{a, x, y) where R is
(») Church and Kleene [l], Church [2], Kleene [4].
(33) Kleene [5].
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S. C. KLEENE
in the more general sense such that provability in the logic expresses the predicate. Hence there is a complete ordinal logic to each predicate of each of the
(Ex)R(a, x)
x, y)
(x)R(a, x)
x, y)
x, y, z)
where R is general recursive, but by Theorem II, classically there are predicates of the form (x)(Ey)(z)R(a,
x, y, z) and of each of the forms with more
quantifiers, or classically and intuitionistically
of the form (Ex)(y)(Ez)R(a,
y, z) and of the negation of each of the forms with more quantifiers, for which
no complete ordinal logic is possible. Specifically:
no complete
logic for
the predicate
(Ex){y){Ez)Tz{a, a, x, y, z).
Ordinal logics form a class of examples of the systems of propositions
which have recently come under discussion, in which more or less is retained
of the ordering of propositions
in deductive reasoning, but with an extension
into the transfinite, or a sacrifice of constructiveness
in individual steps. These
may be called "non-constructive
logics," in contrast to the formal deductive
systems in the sense of §§13-14 which are "constructive
logics." In general,
the usefulness of a non-constructive
logic may be considered to depend on the
degree to which the statement
of the non-constructive
proof criterion is removed from the direct statement
of the propositions.
Theorem IX is a "Gödel theorem" for the ordinal logics. The ordinal logics
were at least conceived with somewhat of a constructive
bias. Rosser has
shown how Gödel theorems arise on going very far in the direction of nonconstructiveness(34),
and Tarski has stated the Gödel argument
for systems
of sentences in general(35). Incidental of Rosser's results for finite numbers of
of the Hilbert "rule of infinite induction,"
also called "Carnap's
rule," can easily be inferred from Theorem II, through the obvious correspondence of an application of this rule to a universal quantifier in the proof
concept. However, the proof concepts for non-constructive
logics soon outrun
the scale of predicate forms of Theorem II. This appears to be the case even
for the extension to protosyntactical
given by Quine(36). If one
is going very far in the direction of non-constructiveness,
and is not interested
in considerations
of the sort emphasized in §§12-14, there is no advantage in
starting from the theory of recursive functions. But the more general results
do not detract from the special significance which attaches to the Gödel theorems associated with provability
criteria of the forms R(a) and (Ex)R(a, x)
(») Rosser [2].
(36) Tarski
(») Quine [1].
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where R is general recursive, that is, Church's theorem and Gödel's
for which forms only it is true that a given proof is a finite object.
16. Constructive
proofs. A proof of an existential
(Ey)A{y) is acceptable
to an intuitionist,
only if in the course of the proof
there is given a y such that A (y) holds, or at least a method by which such a y
could be constructed.
Consider the case that A (y) depends on other variables.
Say that there is one of these, x, and rewrite the proposition as (x) (Ey)A (x, y).
The proposition asserts the existence of a y to each of the infinitely many
values of x. In this case, the only way in which the constructivist
could in general be met would be by giving the y as an effectively calculable
function of x, that is, by giving the function. According to Thesis I, this function would have to be general recursive. Hence we propose the following thesis
(and likewise for n variables Xi, • • .'• , x„):
Thesis III. A proposition of the form (x)(Ey)A(x, y) containing no free
variables is provable constructively, only if there is a general recursive function
4>(x)such that (x)A(x, <p(x)).
When such a <pexists, we shall say that
y) is recursively ful-
This thesis expresses what seems to be demanded from the standpoint
the intuitionists.
Whether such explicit rules of proof as they have stated do
conform to the thesis is a further question which will be considered elsewhere^8). However, in its aspect as restriction on all intuitionistic
proofs, the possibilities for which, as we know by Theorem VIII, transcend
the limitations of any preassignable
formal system, the thesis is more general
than a metamathematical
result concerning a given system.
We now examine the notion of recursive fulfillability as it applies to the
values of a given predicate of the form (x) (Ey) (z)R(a, x, y, z) where R is general recursive. Select any fixed value of a. Given a recursive c6which fulfils the
by Theorem IV there is a number e such that
{x)(Ey)Ti(e, x, y) and (x)(y) [Ji(e, x, y)—>(z)R(a, x, U(y), z)]. Conversely, if
such an e exists, the proposition is fulfilled by the general recursive function
U(nyTi(e, x, y)). Thus
(Ee) {(x)(Ey)T1(e,
x, y) & (x)(y) [T^e, x, y) -> (z)R(a, x, U(y), z)]\
is a necessary and sufficient condition for recursive fulfillability.
When the
quantifiers are suitably brought to the front and contracted,
this assumes the
form (Ex)(y)(Ez)R(a,
x, y, z) with another general recursive R depending on
the original R.
By Theorem II, classically, there is a predicate of the original form
(") A further analysis of the implications
(38) Nelson [l ].
of constructive
is given in Kleene
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S. C. KLEENE
(x) (Ey) (z)R(a, x, y, z) which is not expressible in this form (Ex) (y) (Ez)R(a, x, y, z),
in which the condition of its recursive fulfillability is expressible.
Using the example of such a predicate given in the proof of Theorem II,
we have then
for a certain
a, x, y, z) rec. fulf.} = (Ex)(y)(Ez)R(a,
R. Substituting
the number/
x, y, z)
of (14) for a in
(14) and (38),
(ExXy)(Ez)R(f, x, y, z) m (Ex)(y)(Ez)T3(f, f, x, y, z),
{(x)(Ey)(z)T3(f, f, x, y, z) rec. fulf.} m (Ex)(y)(Ez)R(f, x, y, z).
By the definition of recursive fulfillability,
{(x)(Ey)(z)T3(f, f, x, y, z) rec. fulf.} -»(x)(£y)(z)T3(/,
/, x, y, z) were recursively
could then conclude by (40) and (39), (Ex)(y)(Ez)T3(J,f,
f, x, y, z).
x, y, z), and by (41),
x, y, z). These results are incompatible.
Therefore by reductio ad absurdum, (x)(Ey)(z)T3(f, /, x, y, z) is not recursively fulfillable, and
hence by Thesis III not constructively
Now by (40) and (39), we have (Ex)(y)(Ez)T3(f, f, x, y, z); and thence
we can proceed to (x)(Ey)(z)T3(J,
/, x, y, z).
Theorem X. For a certain number/, the proposition
is true classically, but not constructively provable.
x, y, z)
Notice that we have here a fixed unprovable proposition for all constructive methods of reasoning, whereas in the preceding incompleteness
we had only an infinite class of propositions, some of which must be unprovable in a given theory.
number theory has been presented as a subsystem
of the
classical, so that the intuitionistic results hold classically, though many classical results are not asserted intuitionistically.
The possibility now appears of
extending intuitionistic
number theory by incorporating
Thesis III in the
y) —>{for some general recursive <£,(x)A(x, <t>(x))},
so that the two number theories should diverge, with the proposition
Theorem X true classically, and its negation true intuitionistically(M).
For the classical proof, an application of
(x)A(x) -* (Ex)A~(x)
suffices as the sole non-intuitionistic
step; therewith
(*•) This is perhaps hinted in Church [l, first half of p. 363].
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that law of logic would
be refuted intuitionistically,
for a certain A. Hitherto the intuitionistic
refutations of laws of the classical predicate calculus have depended on the
of the quantifiers in intuitionistic
set theory(40).
The result of Theorem X, with another proposition as example, can be
reached as follows. Consider the proposition,
This holds classically,
*, z) & y = 0] V [(*)T|(*, *, z) & y = l]}.
by application
of the law of the excluded
middle in the
(x){(Ez)A(x,z) V(z)2(x,z)\,
or the form
(x)(A(x) V K*)),
from which the other follows by substituting
(Ez)A(x, z) for A(x). But it is
not recursively fulfillable, since it can be fulfilled only by the representing
function of the predicate (Ez)Ti(x, x, z), which, as we saw in the proof of
Theorem II, is non-recursive.
17. Non-elementary
predicates. The elementary predicates are enumerable. By Cantor's methods, there are therefore non-elementary
number-theoretic predicates. However let us ask what form of definition would suffice to
give such a predicate. Under classical interpretations,
the enumeration
predicate forms given in Theorem II for n variables suffices for the expression
of every elementary predicate of n variables. By defining relations of the form
shown in the next theorem, we can introduce a predicate M(a, k) so that it
depends for different values of k on different numbers of alternating quantifiers. On the basis of Theorem II, it is possible to do this in such a way that
the predicate will be expressible in none of the forms of Theorem II.
Theorem XI. Classically, there is a non-elementary
finable by relations of the form
predicate M(a, k) de-
M(a, 0) ■ R(a)
■ M(a, 2k + l)m
M(a, 2k + 2) m (x)M(<j>(a,x), 2k + 1)
where R and <j>are primitive recursive.
We are dealing here with essentially the same fact which Hilbert-Bernays
discover by setting up a truth definition for their formal system (Z)(41).
The system (Z) has as primitive terms only ', 4-, •, = and the logical
operations. The predicates expressible in these terms are elementary.
Con(") Heyting [l, p. 65].
(") Hilbert and Bernays [l, pp. 328-340].
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S. C. KLEENE
versely, using Theorem IV and Gödel's reduction of primitive recursive functions to these terms(42), every elementary
predicate is expressible in (Z).
The Hilbert-Bernays
result is an application
to (Z) of Tarski's theorem
on the truth concept(*'), with the determination
of a particular form of relations which give the truth definition for (Z). If (Z) is consistent,
a formal
proof that the relations do define a predicate is beyond the resources of (Z).
Alonzo Church
1. An unsolvable problem of elementary
number theory, Amer. J. Math.
vol. 58 (1936) pp.
2. The constructive second number class, Bull. Amer. Math. Soc. vol. 44 (1938) pp. 224-232.
Alonzo Church and S. C. Kleene
1. Formal definitions in the theory of ordinal numbers, Fund. Math. vol. 28 (1936) pp. 11-21.
Alonzo Church and Barkley Rosser
1. Some properties
of conversion, Trans.
Amer. Math.
Soc. vol. 39 (1936) pp. 472-482.
H. B. Curry
1. Some aspects of the problem of mathematical
rigor, Bull. Amer. Math.
Soc. vol. 47 (1941)
pp. 221-241.
2. The inconsistency of certain formal logics, J. Symbolic Logic vol. 7 (1942) pp. 115-117.
1. Über formal unentscheidbare
Sätze der Principia
und verwandter Systeme I,
Monatshefte für Mathematik und Physik vol. 38 (1931) pp. 173-198.
2. On undecidable propositions
of formal mathematical
systems, notes of lectures at the In-
stitute for Advanced Study, 1934.
David Hilbert and Paul Bernays
1. Grundlagen der Mathematik, vol. 2, Berlin, Springer, 1939.
Arend Heyting
1. Die formalen
Regeln der intuitionistischen
Akad. Wiss. Sitzungsber,
Phys.-math. Kl. 1930, pp. 57-71, 158-169.
S. C. Kleene
1. A theory of positive integers in formal logic, Amer. J. Math. vol. 57 (1935) pp. 153-173,
2. General recursive functions
of natural
numbers, Math.
Ann. vol. 112 (1936) pp. 727-742.
3. A note on recursive functions. Bull. Amer. Math. Soc. vol. 42 (1936) pp. 544-546.
4. On notation for ordinal numbers, J. Symbolic Logic vol. 3 (1938) pp. 150-155.
5. On the forms of the predicates in the theory of constructive ordinals,
to appear
in Amer. J.
Math. (Bull. Amer. Math. Soc. abstract 48-5-215).
6. On the interpretation
of intuitionislic
number theory, Bull. Amer.
Soc. abstract
S. C. Kleene and Barkley Rosser
1. The inconsistency of certain formal logics, Ann. of Math. (2) vol. 36 (1935) pp. 630-636.
David Nelson
1. Recursive functions
and intuitionislic
number theory, under preparation.
E. L. Post
1. Finite combinatory processes—formulation
I, J. Symbolic Logic vol. 1 (1936) pp. 103-105.
(«) Gödel [l, Theorem VII]. See Kleene [3 (erratum: p. 544, line 11, "of" should be at the
end of the line) ].
(«) Tarski [l ].
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
W. V. Quine
1. Mathematical logic, New York, Norton, 1940.
Barkley Rosser
1. Extensions of some theorems of Gbdel and Church, J. Symbolic Logic vol. 1 (1936) pp. 87-
2. Gbdel theorems for non-constructive logics, ibid. vol. 2 (1937) pp. 129-137.
Alfred Tarski
1. Der Wahrheitsbegriff
in den formalisierten
vol. 1 (1936)
pp. 261^05. (Original in Polish, 1933.)
2. On undecidable statements in enlarged systems of logic and the concept of truth, J. Symbolic
Logic vol. 4 (1939) pp. 105-112.
A. M. Turing
1. On computable numbers, with an application
to the Entscheidungsproblem,
Math. Soc. (2) vol. 42 (1937) pp. 230-265.
2. Systems of logic based on ordinals, ibid. vol. 45 (1939) pp. 161-228.
Amherst College,
Amherst, Mass.
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Proc. London | {"url":"https://studyres.com/doc/13499496/recursive-predicates-and-quantifiers","timestamp":"2024-11-13T12:47:31Z","content_type":"text/html","content_length":"166519","record_id":"<urn:uuid:fd63cce2-37fe-4ded-8c69-3026a684fdb9>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00213.warc.gz"} |
Quantum Computing: The Future of Computer Technology - UnitedStateNews.com
Quantum Computing: The Future of Computer Technology
Quantum Computing is viewed as the technology of the future, however, its practical applications are already seen at present.
The term Quantum Computing seems very complex and vague matter to many people. However, its real-world applications are waiting to happen. And, in some cases, it would appear that in some capacity
the wait is over.
Quantum techniques are already becoming noteworthy resources in the computing field. Humans are in pursuit of valuable quantum machines that can extend our reach into the unknown.
The development of Quantum Computing was also a central point of Matthias Troyer’s keynote speech at ISC (International Science Conference) 2021.
Like Troyer, a good number of scientists are also working and thinking about the development of Quantum Computing.
What is Quantum Computing?
Before understanding the details of Quantum computing, we’ve to understand the molecules of elements. If we look deep down to the level of atoms and electrons, nature behaves quantum mechanically.
This thing is described by the laws of quantum mechanics, however, these are different from what we are used to in the conventional world.
A conventional object like a phone or a ball is a particle that sits at one location. If you go deeper into the structure of the products, for instance, if you have an electron, it can be a particle
when it hits an object.
But, it acts like a wave when it moves. The duality of objects and waves is explained by the laws of quantum physics.
Let’s imagine what if bits in a computer could be waves and have certain wave amplitude for being zero and one and exists in both states.
Then we can think of computations having registers of no fixed value but can exist in a wavelike superposition or exponential values.
We can use that condition to do the task of computing exponentially faster and solve problems that are conventionally intractable.
Any traditional problems, in fact, also be solved using quantum hardware. To solve this, the conventional register can be used.
But, this task will be practically non-feasible as quantum bits are much more expensive than conventional bits and the operation process would be much slower, because of the complexity of quantum
However, there’s a solution to this point. If you put the quantum registers into this wave-like superposition of many values, the number of operations would be fewer.
And in this way, quantum computers can work successfully in near future.
So, what would be the advantage of Quantum Computing? Truly, you will be able to solve the number of problems on future quantum hardware in a few days what a conventional computer, even the size of
the earth, would take a billion years to solve.
How Can We Implement Quantum Approaches on Traditional HPC Architecture and What Challenges May Arise?
How challenging its to implement Quantum Computing on conventional HPC architecture?
The algorithm developed for quantum computers gives us ideas for running them on a conventional computer. This is not the same for all algorithms.
If a quantum algorithm is naively stimulated on classic hardware, then it’s necessary to apply the algorithm to all the values in superposition. And for that process, need exponential time
But, if the quantum algorithm is not stimulated at the microscopic level of quantum bits or qubits and viewed abstractly, then then we can have the same algorithmic idea which can be implemented as a
probabilistic conventional algorithm.
In this way, new conventional or classical algorithms have been invented recently, which capture the same idea of the quantum algorithm almost as well on conventional hardware.
Primarily, quantum researchers may have found a quantum algorithm that’s exponentially better than the most well-known conventional algorithm. They celebrate it as a success as well.
But, looking deeper into the matter, they may think, it can also be done as well on conventional hardware. But, the fact is, exponential progress on conventional hardware is done.
And since the goal of the scientists is solving the problem actually, they think we don’t need to wait for quantum computers to realize the value of this algorithm. Rather, we can run it today. With
quantum ideas, we can solve the problems of conventional hardware today.
Famous quantum scientist Matthias Troyer
According to famous scientist Matthias Troyer, when people assume a certain intractable problem, they don’t want to go for making a new solution. Rather they want to know whether Quantum Computing
helps them.
If you want to think of radically new ways of solving problems, there’s the possibility of coming disruption, he added.
Will the application of quantum computers assist in better understanding quantum mechanics?
Definitely, it will, because there is always an open question in quantum physics. For instance, you can think about the famous Schrodinger’s cat paradox.
Imagine a radioactive atom that may decay and you place that atom in a room with a cat. And in the room, there is a vial with poison gas and a hammer.
If the atom decays, the hammer will fall, hit the vial and gas will come out. As a result, the cat will die.
But at present, with the progression of time, the equations of quantum mechanics will tell us that the atom is in a wave-quantum superposition of two conditions: having decayed and not decayed.
The same thing will happen to the hammer and vial. The hammer will be fallen or not fall. The vial is in a similar quantum state of being broken and not broken.
Hence, you will find the cat in a quantum state of being both dead and alive at the same time. This is the paradox of Schrodinger’s Cat.
Schrodinger’s Cat Paradox
Have you been Schrodinger’s cats alive and running? No. The reason behind that as we scale up a quantum system, something will interact with the system all the time.
If that happens, it will go from a wave-like state to a particle-like state. Just looking at Schrodinger’s Cat will make it into a cat that is either dead or alive.
To build the quantum computer, we now want to avoid this process that we know as ‘decoherence.’ We have to find a way to protect the quantum state in a macroscopic computer with a view to keeping the
wave-like state alive for the lifetime of computers.
And for that, we need to understand all of the mechanics that transform a wave-like quantum state into a conventional one. Thus, building a quantum computer will give us deep insight into the process
of crossing over from the quantum world to the classic or conventional one.
Shall We Go for Quantum Applications?
There’s a big potential as quantum algorithms can solve problems much better and faster than conventional systems.
So, we must go for it. From climate prediction to protein folding, there are a multi-dimensional and big amount of complex problems, where Quantum Computing can assist. However, there’re two
challenges ahead.
Firstly, quantum operations are slower than a conventional system. A conventional computer is able to process 10 billion times more operations per second than a quantum computer.
Hence, we need to look at problems where quantum computers need to perform significantly fewer operations than their conventional counterparts.
But, the interesting matter is, when the problem size gets large, quantum computers will always win as it has a scaling advantage called quantum speedup.
But, if the significant, exponential quantum speedup is not realized, the success possibility of quantum computers is just a story of the far future.
The real impactful application of Quantum Computing will be seen in solving the problems of quantum science. With the conventional way, the problems of quantum science can be exponentially hard.
However, this will map perfectly to quantum hardware. We can apply quantum computers to design new catalysts, for carbon fixation, efficient fertilizer production, clearing combustion, and many other
complex tasks.
Quantum Computing is totally new technology of the millennia. The way our laptop, and HPC machine work, is based on digital logic. The same logic was used in the abacus, the thousand-year-old
computing instrument.
Nevertheless, Quantum Computing is totally a new way of computing, the way we’ve never applied before, which beyond doubt is going to be very exciting. | {"url":"https://unitedstatenews.com/quantum-computing-the-future-of-computer-technology/","timestamp":"2024-11-13T21:51:52Z","content_type":"text/html","content_length":"144679","record_id":"<urn:uuid:26d8199c-df44-4bfa-99fd-8eec9c444a37>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00459.warc.gz"} |
What is Gordon's Model of Dividend Policy? (Formula, Example, Calculation, and More) - CFAJournal
What is Gordon’s Model of Dividend Policy? (Formula, Example, Calculation, and More)
Gordon Growth model is based on the concept of dividend growth in the future. If the company does not make some significant breakthrough/unexpected growth or faces some extreme misfortune, then the
growth depends on doing more for the same work. For instance, if the company has four retail outlets, it needs to expand from 4 outlets to 5 by investing in growth and development.
This growth and development of the company are dependent on the retention of the profit if we ignore external financing. So, If the company distributes all of its earnings as a dividend, it does not
have the capital to invest in projects for growth and development. It means the company’s growth depends on the rate of profit earned and subsequent retention, which can be shown as follows as the
formula below:
g = bR
Where g is the company’s rate of growth, b is the percentage of earnings retained by the company, and R is the profitability earned by the company on new investment. As per the given equation, if
there is an increase in the b or profit retention, the growth rate is expected to increase. It’s due to the logic that more retained earnings allow the company to invest more and earn more and pay
more like a dividend.
Let’s understand in detail that how Gordon’s growth model calculates the value of the firm. So, this model uses an expected series of dividends by the company in the future and growth rate. The
expected dividend is discounted in the present time by using the discounting technique.
Understanding Gordon Growth Model
Gordon’s growth model helps to calculate the value of the security by using future dividends. The formula for GGM is as follows,
1. D1 = Value of next year’s dividend.
2. r = Rate of return / Cost of equity.
3. g = Constant rate of growth expected for dividends in perpetuity.
This model does not consider external factors that significantly impact the company’s capitalization but is centered around market expected returns and payout retention. Suppose the value calculated
by GGM is higher than the current share price in the market. In that case, the security is considered to be undervalued compared to the dividend generating capacity, and a security purchase decision
is recommended. On the other hand, if the value calculated by GGM is less than the current trading price in the market, the security is said to be over-valued, and purchase decision is not
The logic of the whole model is that if you retain stock of the company in perpetuity. Its value will be realized by receipt of the dividend in perpetuity. So, we do a discounting of the constantly
growing dividend per share in perpetuity with a required rate of return. This gives us the potential of the security to generate dividends for the rest of its life. Hence, we can compare it with the
current share price.
However, there are certain limitations of this model. It assumes dividend grows at a constant rate for the company’s whole life, which may not be true in reality as reality is much dynamic and there
are ups and downs in the business. So, it’s a limitation of the model which limits its use just for stable companies.
Another problem with this model is the relationship between growth rate and discount factor. If some company has a required rate of return lower than the growth rate of dividend per share, the result
is a negative value calculated by GGM.
In addition to this, if the growth rate of dividend and required rate of return is the same, the value calculated gives a mathematical error and can be considered in infinity. However, the main
benefit of the GGM is to assess the under/overvaluation of stock by comparing the current trading price and internal growth potential of the business in the future.
Example of Calculation for the Gordon Growth Model
Suppose the current share price of the company’s share is $110 per share. The required rate of return by the shareholders of the company is 8%. The company intends to pay a $3 dividend per share, and
the expected growth rate is 5%. The current trading price of the share is $110.
The value of the share in the Gordon Growth Model can be calculated as follows.
1. D1 = Dividend in the next year = $3
2. R = required rate of return = 8%
3. g = Growth rate of a dividend = 5%
The value calculated with the GGM amounts to $100, and the current trading price of the shares is $110. It means there is less potential for the shares to generate dividends than the current share
price. So, security is overvalued, and purchase decision is not recommended.
Assumptions of the Model
1. Growth of the dividend is constant for whole life of the business.
2. No external factors impact the share price and earnings of the company.
3. The rate of growth is lower than the expected rate of return.
4. The company pays all of its free cash flow in the form of a dividend.
5. The business model of the company is stable alongwith constant rate of growth. Further, No significant changes are expected in the operating structure of the business.
6. The financial leverage of the company is stable.
7. The business does not avail of external financing for the rest of its life. Instead, the earnings of the business are reinvested after the payment of the dividend.
8. The rate of return remains the same for the whole life of the business.
9. The retention ratio of the dividend is constant.
When Should Gordon Growth Model be Used?
The Gordon Growth model seems to produce be useful in the following circumstances.
1. The business has established internal operations.
2. The company believes in regular payment of the dividend.
3. The company’s dividend is expected to grow at a constant rate.
4. The growth rate of the company is expected to be lower than the required rate of return.
5. The company does not retain free cash flow while pays all of its free cash flow in the form of a dividend.
Frequently asked questions
Why is the Gordon Growth model used?
This model calculates the fair value of the stock. It does not consider external environmental conditions that impact the business but dividend payout factors, expected rate of return, and the growth
of the dividend. If the value calculated by GGM is more than the company’s current trading price, the security is said to be under-valued, and purchase is recommended and vice versa.
What are the inputs required for the Gordon Growth Model?
There are three inputs in the Gordon Growth model. These models include Dividend per share (DPS), rate of growth for the dividend in the perpetuity, and the required rate of return. Dividend per
share is the dividend announced by the company against each share. The rate of growth is the expected growth rate for dividends in the future, and the required rate of return is the return desired by
investors on their investments.
What are the limitations of the Gordon Growth Model?
First of all, It does not consider the external environmental conditions of the business and depends on limited input factors. Secondly, it assumes growth of the dividend is constant in perpetuity
which may not always be true. Thirdly, if the rate of growth and required rate of return is the same, the calculations will lead to an infinite value.
Finally, this model can only be applied to the companies that pay a dividend. There may be companies that do not pay dividends instead reinvest the whole amount. In such cases, the Gordon Growth
model seems to be of no use.
Why Gordon Growth model uses dividend instead of cash flow?
In the Gordon Growth Model, we actually use cash flow while it seems to be a dividend. It’s because these models assume the company pays all of its equity-free cash in the form of a dividend. | {"url":"https://www.cfajournal.org/gordons-model-dividend-policy/","timestamp":"2024-11-07T06:01:27Z","content_type":"text/html","content_length":"156038","record_id":"<urn:uuid:9f37b682-75a9-4110-bee6-75a87815a115>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00479.warc.gz"} |
After creating a workbook and importing a data source, you are now ready to start analyzing the data present in your workbook.
The Formula Builder lets you create formulas by first clicking on a category and then selecting the appropriate function. When you click a function, the Formula Builder provides a description of the
selected function along with the required arguments and the types of data supported (such as integer, date, string, Boolean). After choosing which function to use, you then need to enter the
arguments into the function. You can either select enter the column from a worksheet in the workbookyour workbook by adding it into a value field in the inspector, or you can type the column
reference into the Formula field at the top of the workbook beside the fx symbol.
If you are familiar with Excel, you are probably interested in the differences between a Datameer workbook and Excel - Tips for Excel Users highlights these differences.
Creating a Formula
With the Formula Builder in 7.
As of Datameer v6.4
1. Click the Fx button in the
fomula bar.
formula bar and Datameer's Formula Builder pops up.
2. Select a category from the column on the left and select a function from the list on the right.
3. Read the description for the selected function at the bottom of the Formula Builder, or you can click Help to see the online help with examples.
4. Enter the arguments as shown in the Formula Builder. To do so, click the column that contains the desired data. The types of arguments required are displayed next to the arguments' names. You can
also enter a null argument by typing in null, all lower case.
5. If the function supports multiple elements for a single argument, a + (plus) button is available. Click to add additional elements.
6. Click OK to finish entering the formula. The results are shown in the column selected in from step one.
The Formula Builder can store up to 5000 characters. If you build a larger query outside of Datameer then paste the query in, the builder accept accepts it. However, if you edit the query, the
builder only displays the first 5000 characters.
Creating a Formula With the Formula Builder as of 7.2
As of Datameer 7.2
1. Access the formula builder by opening a workbook, click within the column you would like build a formula, and open the Fx tab in the workbook inspector.
2. Select a category from the column on the left and select a function from the list on the right.
3. Read the description for the selected function at the bottom of the Formula Builder. Click Online Documentation and Samples for additional information.
4. Enter the arguments as shown in the Formula Builder. To do so, type the column that contains the desired data. The types of arguments required are displayed next to the arguments' names. You can
also enter a null argument by typing in null, all lower case.
5. If the function supports multiple elements for a single argument, a + (plus) button is available. Click to add additional elements.
6. Click Add to Formula to finish entering the formula. The results are shown in the column selected from step one.
Editing a Formula
1. Click a column in your workbook that already contains a formula.
2. The formula is displayed in the Formula Bar - beside the fx symbol at the top of the workbook sheet.
3. Edit the formula as appropriate and press enter.
4. The results are now shown in the column selected in step one. | {"url":"https://datameer.atlassian.net/wiki/pages/diffpagesbyversion.action?pageId=31963974149&selectedPageVersions=4&selectedPageVersions=5","timestamp":"2024-11-10T03:06:27Z","content_type":"text/html","content_length":"66715","record_id":"<urn:uuid:2c630631-54d8-44c5-a27c-a548f60fd91d>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00504.warc.gz"} |
Power System Dynamics and Control
Power System Dynamics and Control. Instructor: Prof. A. M. Kulkarni, Department of Electrical Engineering, IIT Bombay. This course introduces a student to power stability problems and the basic
concepts of modeling and analysis of dynamical systems. Modeling of power system components - generators, transmission lines, excitation and prime mover controllers - is covered in detail. Stability
of single machine and multi-machine systems is analyzed using digital simulation and small-signal analysis techniques. The impact of stability problems on power system planning, and operation is also
brought out. (from nptel.ac.in)
Lecture 01 - Introduction
Lecture 02 - Introduction: Voltage and Frequency Stability, Control Hierarchy
Lecture 03 - Analysis of Dynamical Systems
Lecture 04 - Analysis of Dynamical Systems (cont.)
Lecture 05 - Analysis of Linear Time Invariant Dynamical Systems
Lecture 06 - Analysis of Linear Time Invariant Dynamical Systems (cont.)
Lecture 07 - Stiff Systems, Multi Time Scaling Modeling
Lecture 08 - Numerical Integration
Lecture 09 - Numerical Integration (cont.)
Lecture 10 - Numerical Integration (cont.)
Lecture 11 - Modeling of Synchronous Machines
Lecture 12 - Modeling of Synchronous Machines (cont.)
Lecture 13 - Modeling of Synchronous Machines (cont.)
Lecture 14 - Modeling of Synchronous Machines: dq0 Transformation
Lecture 15 - Modeling of Synchronous Machines: Standard Parameters
Lecture 16 - Modeling of Synchronous Machines: Standard Parameters (cont.)
Lecture 17 - Synchronous Generator Models using Standard Parameters
Lecture 18 - Synchronous Generator Models using Standard Parameters: Per Unit Representation
Lecture 19 - Open Circuit Response of a Synchronous Generator
Lecture 20 - Synchronous Machine Modelling: Short Circuit Analysis
Lecture 21 - Synchronous Machine Modelling: Short Circuit Analysis; Synchronization of a Synchronous Machine
Lecture 22 - Synchronization of a Synchronous Machine (cont.)
Lecture 23 - Simplified Synchronous Machine Models
Lecture 24 - Excitation Systems
Lecture 25 - Excitation System Modeling
Lecture 26 - Excitation System Modeling: Automatic Voltage Regulator
Lecture 27 - Excitation System Modeling: Automatic Voltage Regulator (cont.)
Lecture 28 - Excitation System Modeling: Automatic Voltage Regulator: Simulation
Lecture 29 - Excitation System Modeling: Automatic Voltage Regulator: Simulation (cont.)
Lecture 30 - Excitation System Modeling: Automatic Voltage Regulator: Linearized Analysis
Lecture 31 - Load Modeling
Lecture 32 - Induction Machines, Transmission Lines
Lecture 33 - Transmission Lines, Prime Mover System
Lecture 34 - Transmission Lines (cont.), Prime Mover Systems
Lecture 35 - Prime Mover Systems, Stability in Integrated Power System
Lecture 36 - Stability in Integrated Power System: Two Machine Example
Lecture 37 - Stability in Integrated Power System: Two Machine System (cont.)
Lecture 38 - Stability in Integrated Power System: Large Systems
Lecture 39 - Frequency/Angular Stability Programs, Stability Phenomena: Voltage Stability Example
Lecture 40 - Voltage Stability Example (cont.), Fast Transients: Tools and Phenomena
Lecture 41 - Torsional Transients: Phenomena of Sub-Synchronous Resonance
Lecture 42 - Sub-Synchronous Resonance, Stability Improvement
Lecture 43 - Stability Improvement
Lecture 44 - Stability Improvement, Power System Stabilizers
Lecture 45 - Stability Improvement (Large Disturbance Stability)
Power System Dynamics and Control
Instructor: Prof. A M Kulkarni, Department of Electrical Engineering, IIT Bombay. This course introduces a student to power stability problems and the basic concepts of modeling and analysis of
dynamical systems. | {"url":"http://www.infocobuild.com/education/audio-video-courses/electronics/power-system-dynamics-and-control-iit-bombay.html","timestamp":"2024-11-08T14:48:38Z","content_type":"text/html","content_length":"15253","record_id":"<urn:uuid:4d0da805-b85f-4256-bc94-993c13f551fa>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00818.warc.gz"} |
Stability of the uniform ferroelectric nematic phase
The recent discovery of the ferroelectric nematic phase N[F] resurrects a question about the stability of the uniform N[F] state with respect to the formation of either a standard for the solid
ferroelectric domain structure or for the often occurring liquid crystal space modulation of the polarization vector P (and naturally coupled to P nematic director n ). In this work, within Landau
mean-field theory, we investigate the linear stability of the minimal model admitting the conventional paraelectric nematic N and N[F] phases. Our minimal model (in addition to the standard terms of
the expansion over the P and director gradients) includes the standard for liquid crystals, the director flexoelectric coupling term (f ), and, often overlooked in the literature (although similar by
its symmetry to the director flexoelectric coupling), the flexodipolar coupling (β ). We find that in the easy-plane anisotropy case (when the configuration with P orthogonal to n is energetically
favorable), the uniform N[F] state loses its stability with respect to one-dimensional (1D) or two-dimensional (2D) modulation. If f ≠0 , the 2D modulation threshold (β[c 2] value) is always higher
than its 1D counterpart value β[c 1]. There is no instability at all if one neglects the flexodipolar coupling (β =0 ). In the easy-axis case (when n prefers to align along P ), both instability (1D
and 2D) thresholds are the same, and the instability can occur even at β =0 . We speculate that the phases with 1D or 2D modulations can be identified as discussed in the literature [see M. P.
Rosseto and J. V. Selinger, Phys. Rev. E 101, 052707 (2020), 10.1103/PhysRevE.101.052707] for single-splay or double-splay nematics.
Physical Review E
Pub Date:
January 2021
□ Condensed Matter - Soft Condensed Matter;
□ Condensed Matter - Statistical Mechanics
10 pages | {"url":"https://ui.adsabs.harvard.edu/abs/2021PhRvE.103a2704K","timestamp":"2024-11-12T20:42:10Z","content_type":"text/html","content_length":"41576","record_id":"<urn:uuid:e6c97d50-29f5-4286-b891-b54f7097076c>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00151.warc.gz"} |
GR9768 Q 64 Option 3? slope bounded?
Here is a old post regarding this questions below
http://www.mathematicsgre.com/viewtopic ... 8+64#p1799
Ques: Suppose that f is a continuous real-valued function defined on the closed interval [0,1]. Which of the following must be true ?
III. There is a constant E > 0 such that |f(x) -f(y)| <= E |x-y| for all x and y in [0,1]
The answer rejected III. The discussion said that not all the function are uniformly continuous.
But if we rewrite the condition as:
|f(x) -f(y)| / |x-y| = |K| <= E
So the left hand side become the abs of the slope. I am wondering for a continuous function defined on a closed interval, is it possible that the slope in not bounded?
Re: GR9768 Q 64 Option 3? slope bounded?
Any non-Lipschitz function does the job
consider the square root of x...
Re: GR9768 Q 64 Option 3? slope bounded?
blitzer6266 wrote:Any non-Lipschitz function does the job
consider the square root of x...
That's a brilliant example. Thank you | {"url":"https://mathematicsgre.com/viewtopic.php?f=1&t=650","timestamp":"2024-11-13T18:33:09Z","content_type":"text/html","content_length":"21095","record_id":"<urn:uuid:e1aeaa61-8130-4e80-86a4-37cab21803c6>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00701.warc.gz"} |
What Is the Sacrifice Ratio in Economics?
The sacrifice ratio is the percentage reduction in real disposable income required to reduce inflation by one percentage point. The sacrifice ratio is often used as a measure of the cost of reducing
inflation. What are the 5 types of ratio? 1. Liquidity Ratios
2. Activity Ratios
3. Debt Ratios
4. Margin Ratios
5. profitability Ratios What are the two dots called in a ratio? The two dots are called a colon.
What means golden ratio?
The golden ratio is a mathematical concept that describes a relationship between two numbers. The golden ratio is often represented by the symbol φ. The golden ratio has been used in art,
architecture, and design for centuries. The golden ratio is said to create a sense of harmony and balance. What is the ratio formula? The ratio formula is a mathematical formula used to calculate the
ratio between two numbers. The formula is:
ratio = (number1 / number2) * 100
For example, if you want to calculate the ratio between two numbers, such as 10 and 20, you would use the formula as follows:
ratio = (10 / 20) * 100
The resulting ratio would be 50%.
What do you understand by sacrificing ratio explain with example?
The "sacrificing ratio" is a financial ratio that compares a company's net income to its total debt. The sacrificing ratio is also sometimes referred to as the "net income to total debt ratio".
For example, let's say that Company A has a net income of $1,000 and total debt of $10,000. This gives Company A a sacrificing ratio of 10%.
Now let's say that Company B also has a net income of $1,000, but its total debt is $20,000. This gives Company B a sacrificing ratio of 5%.
In this example, Company A has a higher sacrificing ratio than Company B. This means that, all else being equal, Company A is in a better financial position than Company B because it is able to
service its debt with a larger portion of its income. | {"url":"https://www.infocomm.ky/what-is-the-sacrifice-ratio-in-economics/","timestamp":"2024-11-12T10:50:47Z","content_type":"text/html","content_length":"39817","record_id":"<urn:uuid:f8221db6-bc14-4d07-bcb9-4744a9d6d67b>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00177.warc.gz"} |
How to plot in Mathcad 15
Jul 20, 2022 04:00 PM
Jul 20, 2022 02:57 PM
The first line in your sheet specifies an equation.
The second line gives an equation, to solve, but to solve for what?
The third line factors an expression.
The fourth line gives an equation to be rewritten.
The plot: You did not define the function b(a), so there's nothing to plot.
You can define b(a), related to the equation with: | {"url":"https://community.ptc.com/t5/Mathcad/How-to-plot-in-Mathcad-15/m-p/808086","timestamp":"2024-11-09T16:14:29Z","content_type":"text/html","content_length":"188160","record_id":"<urn:uuid:872f6e34-883d-46c0-9b20-6ba9c6da8879>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00289.warc.gz"} |
Unsupervised Anomaly Detection
This topic introduces the unsupervised anomaly detection features for multivariate sample data available in Statistics and Machine Learning Toolbox™, and describes the workflows of the features for
outlier detection (detecting anomalies in training data) and novelty detection (detecting anomalies in new data with uncontaminated training data).
For unlabeled multivariate sample data, you can detect anomalies by using isolation forest, robust random cut forest, local outlier factor, one-class support vector machine (SVM), and Mahalanobis
distance. These methods detect outliers either by training a model or by learning parameters. For novelty detection, you train a model or learn parameters with uncontaminated training data (data with
no outliers) and detect anomalies in new data by using the trained model or learned parameters.
• Isolation forest — The Isolation Forest algorithm detects anomalies by isolating them from normal points using an ensemble of isolation trees. Detect outliers by using the iforest function, and
detect novelties by using the object function isanomaly.
• Random robust cut forest — The Robust Random Cut Forest algorithm classifies a point as a normal point or an anomaly based on the changes in model complexity introduced by the point. Similar to
the isolation forest algorithm, the robust random cut forest algorithm builds an ensemble of trees. The two algorithms differ in how they choose a split variable in trees and how they define
anomaly scores. Detect outliers by using the rrcforest function, and detect novelties by using the object function isanomaly.
• Local outlier factor — The Local Outlier Factor (LOF) algorithm detects anomalies based on the relative density of an observation with respect to the surrounding neighborhood. Detect outliers by
using the lof function, and detect novelties by using the object function isanomaly.
• One-class support vector machine (SVM) — One-class SVM, or unsupervised SVM, tries to separate data from the origin in the transformed high-dimensional predictor space. Detect outliers by using
the ocsvm function, and detect novelties by using the object function isanomaly.
• Mahalanobis Distance — If sample data follows a multivariate normal distribution, then the squared Mahalanobis distances from samples to the distribution follow a chi-square distribution.
Therefore, you can use the distances to detect anomalies based on the critical values of the chi-square distribution. For outlier detection, use the robustcov function to compute robust
Mahalanobis distances. For novelty detection, you can compute Mahalanobis distances by using the robustcov and pdist2 functions.
To detect anomalies when performing incremental learning, see incrementalRobustRandomCutForest, incrementalOneClassSVM, and Incremental Anomaly Detection Overview.
Outlier Detection
This example illustrates the workflows of the five unsupervised anomaly detection methods (isolation forest, robust random cut forest, local outlier factor, one-class SVM, and Mahalanobis distance)
for outlier detection.
Load Data
Load the humanactivity data set, which contains the variables feat and actid. The variable feat contains the predictor data matrix of 60 features for 24,075 observations, and the response variable
actid contains the activity IDs for the observations as integers. This example uses the feat variable for anomaly detection.
Find the size of the variable feat.
Assume that the fraction of outliers in the data is 0.05.
contaminationFraction = 0.05;
Isolation Forest
Detect outliers by using the iforest function.
Train an isolation forest model by using the iforest function. Specify the fraction of outliers (ContaminationFraction) as 0.05.
rng("default") % For reproducibility
[forest,tf_forest,s_forest] = iforest(feat, ...
forest is an IsolationForest object. iforest also returns the anomaly indicators (tf_forest) and anomaly scores (s_forest) for the data (feat). iforest determines the score threshold value
(forest.ScoreThreshold) so that the function detects the specified fraction of observations as outliers.
Plot a histogram of the score values. Create a vertical line at the score threshold corresponding to the specified fraction.
xline(forest.ScoreThreshold,"k-", ...
join(["Threshold =" forest.ScoreThreshold]))
title("Histogram of Anomaly Scores for Isolation Forest")
Check the fraction of detected anomalies in the data.
OF_forest = sum(tf_forest)/N
The outlier fraction can be smaller than the specified fraction (0.05) when the scores have tied values at the threshold.
Robust Random Cut Forest
Detect outliers by using the rrcforest function.
Train a robust random cut forest model by using the rrcforest function. Specify the fraction of outliers (ContaminationFraction) as 0.05, and specify StandardizeData as true to standardize the input
rng("default") % For reproducibility
[rforest,tf_rforest,s_rforest] = rrcforest(feat, ...
rforest is a RobustRandomCutForest object. rrcforest also returns the anomaly indicators (tf_rforest) and anomaly scores (s_rforest) for the data (feat). rrcforest determines the score threshold
value (rforest.ScoreThreshold) so that the function detects the specified fraction of observations as outliers.
Plot a histogram of the score values. Create a vertical line at the score threshold corresponding to the specified fraction.
xline(rforest.ScoreThreshold,"k-", ...
join(["Threshold =" rforest.ScoreThreshold]))
title("Histogram of Anomaly Scores for Robust Random Cut Forest")
Check the fraction of detected anomalies in the data.
OF_rforest = sum(tf_rforest)/N
Local Outlier Factor
Detect outliers by using the lof function.
Train a local outlier factor model by using the lof function. Specify the fraction of outliers (ContaminationFraction) as 0.05, 500 nearest neighbors, and the Mahalanobis distance.
[LOFObj,tf_lof,s_lof] = lof(feat, ...
ContaminationFraction=contaminationFraction, ...
LOFObj is a LocalOutlierFactor object. lof also returns the anomaly indicators (tf_lof) and anomaly scores (s_lof) for the data (feat). lof determines the score threshold value
(LOFObj.ScoreThreshold) so that the function detects the specified fraction of observations as outliers.
Plot a histogram of the score values. Create a vertical line at the score threshold corresponding to the specified fraction.
xline(LOFObj.ScoreThreshold,"k-", ...
join(["Threshold =" LOFObj.ScoreThreshold]))
title("Histogram of Anomaly Scores for Local Outlier Factor")
Check the fraction of detected anomalies in the data.
One-Class SVM
Detect outliers by using the ocsvm function.
Train a one-class SVM model by using the ocsvm function. Specify the fraction of outliers (ContaminationFraction) as 0.05. In addition, set KernelScale to "auto" to let the function select an
appropriate kernel scale parameter using a heuristic procedure, and specify StandardizeData as true to standardize the input data.
[Mdl,tf_OCSVM,s_OCSVM] = ocsvm(feat, ...
ContaminationFraction=contaminationFraction, ...
Mdl is a OneClassSVM object. ocsvm also returns the anomaly indicators (tf_OCSVM) and anomaly scores (s_OCSVM) for the data (feat). ocsvm determines the score threshold value (Mdl.ScoreThreshold) so
that the function detects the specified fraction of observations as outliers.
Plot a histogram of the score values. Create a vertical line at the score threshold corresponding to the specified fraction.
xline(Mdl.ScoreThreshold,"k-", ...
join(["Threshold =" Mdl.ScoreThreshold]))
title("Histogram of Anomaly Scores for One-Class SVM")
Check the fraction of detected anomalies in the data.
OF_OCSVM = sum(tf_OCSVM)/N
Mahalanobis Distance
Use the robustcov function to compute robust Mahalanobis distances and robust estimates for the mean and covariance of the data.
Compute the Mahalanobis distance from feat to the distribution of feat by using the robustcov function. Specify the fraction of outliers (OutlierFraction) as 0.05. robustcov minimizes the covariance
determinant over 95% of the observations.
[sigma,mu,s_robustcov,tf_robustcov_default] = robustcov(feat, ...
robustcov finds the robust covariance matrix estimate (sigma) and robust mean estimate (mu), which are less sensitive to outliers than the estimates from the cov and mean functions. The robustcov
function also computes the Mahalanobis distances (s_robustcov) and the outlier indicators (tf_robustcov_default). By default, the function assumes that the data set follows a multivariate normal
distribution, and identifies 2.5% of input observations as outliers based on the critical values of the chi-square distribution.
If the data set satisfies the normality assumption, then the squared Mahalanobis distance follows a chi-square distribution with D degrees of freedom, where D is the dimension of the data. In that
case, you can find a new threshold by using the chi2inv function to detect the specified fraction of observations as outliers.
s_robustcov_threshold = sqrt(chi2inv(1-contaminationFraction,D));
tf_robustcov = s_robustcov > s_robustcov_threshold;
Create a distance-distance plot (DD plot) to check the multivariate normality of the data.
d_classical = pdist2(feat,mean(feat),"mahalanobis");
yline(s_robustcov_threshold,"k-", ...
join(["Threshold = " s_robustcov_threshold]));
l = refline([1 0]);
l.Color = "k";
xlabel("Mahalanobis Distance")
ylabel("Robust Distance")
legend("Normal Points","Outliers",Location="northwest")
title("Distance-Distance Plot")
Zoom in the axes to see the normal points.
xlim([0 10])
ylim([0 10])
If a data set follows a multivariate normal distribution, then data points cluster tightly around the 45 degree reference line. The DD plot indicates that the data set does not follow a multivariate
normal distribution.
Because the data set does not satisfy the normality assumption, use the quantile of the distance values for the cumulative probability (1 — contaminationFraction) to find a threshold.
s_robustcov_threshold = quantile(s_robustcov,1-contaminationFraction);
Obtain the anomaly indicators for feat using the new threshold s_robustcov_threshold.
tf_robustcov = s_robustcov > s_robustcov_threshold;
Check the fraction of detected anomalies in the data.
OF_robustcov = sum(tf_robustcov)/N
Compare Detected Outliers
To visualize the detected outliers, reduce the data dimension by using the tsne function.
rng("default") % For reproducibility
T = tsne(feat,Standardize=true,Perplexity=100,Exaggeration=20);
Plot the normal points and outliers in the reduced dimension. Compare the results of the five methods: the isolation forest algorithm, robust random cut forest algorithm, local outlier factor
algorithm, one-class SVM model, and robust Mahalanobis distance from robustcov.
title("Isolation Forest")
title("Robust Random Cut Forest")
title("Local Outlier Factor")
title("One-Class SVM")
title("Robust Mahalanobis Distance")
l = legend("Normal Points","Outliers");
l.Layout.Tile = 3;
The novelties identified by the five methods are located near each other in the reduced dimension.
You can also visualize observation values using the two most important features selected by the fsulaplacian function.
idx = fsulaplacian(feat);
t = tiledlayout(2,3);
title("Isolation Forest")
title("Robust Random Cut Forest")
title("Local Outlier Factor")
title("One-Class SVM")
title("Mahalanobis Distance")
l = legend("Normal Points","Outliers");
l.Layout.Tile = 3;
xlabel(t,join(["Column" idx(1)]))
ylabel(t,join(["Column" idx(2)]))
Novelty Detection
This example illustrates the workflows of the five unsupervised anomaly detection methods (isolation forest, robust random cut forest, local outlier factor, one-class SVM, and Mahalanobis distance)
for novelty detection.
Load Data
Load the humanactivity data set, which contains the variables feat and actid. The variable feat contains the predictor data matrix of 60 features for 24,075 observations, and the response variable
actid contains the activity IDs for the observations as integers. This example uses the feat variable for anomaly detection.
Partition the data into training and test sets by using the cvpartition function. Use 50% of the observations as training data and 50% of the observations as test data for novelty detection.
rng("default") % For reproducibility
c = cvpartition(actid,Holdout=0.50);
trainingIndices = training(c); % Indices for the training set
testIndices = test(c); % Indices for the test set
XTrain = feat(trainingIndices,:);
XTest = feat(testIndices,:);
Assume that the training data is not contaminated (no outliers).
Find the size of the training and test sets.
Isolation Forest
Detect novelties using the object function isanomaly after training an isolation forest model by using the iforest function.
Train an isolation forest model.
[forest,tf_forest,s_forest] = iforest(XTrain);
forest is an IsolationForest object. iforest also returns the anomaly indicators (tf_forest) and anomaly scores (s_forest) for the training data (XTrain). By default, iforest treats all training
observations as normal observations, and sets the score threshold (forest.ScoreThreshold) to the maximum score value.
Use the trained isolation forest model and the object function isanomaly to find novelties in XTest. The isanomaly function identifies observations with scores above the threshold
(forest.ScoreThreshold) as novelties.
[tfTest_forest,sTest_forest] = isanomaly(forest,XTest);
The isanomaly function returns the anomaly indicators (tfTest_forest) and anomaly scores (sTest_forest) for the test data.
Plot histograms of the score values. Create a vertical line at the score threshold.
hold on
xline(forest.ScoreThreshold,"k-", ...
join(["Threshold =" forest.ScoreThreshold]))
legend("Training data","Test data",Location="southeast")
title("Histograms of Anomaly Scores for Isolation Forest")
hold off
The anomaly score distribution of the test data is similar to that of the training data, so isanomaly detects a small number of anomalies in the test data.
Check the fraction of detected anomalies in the test data.
NF_forest = sum(tfTest_forest)/NTest
Display the observation index of the anomalies in the test data.
idx_forest = find(tfTest_forest)
Robust Random Cut Forest
Detect novelties using the object function isanomaly after training a robust random cut forest model by using the rrcforest function.
Train a robust random cut forest model. Specify StandardizeData as true to standardize the input data.
[rforest,tf_rforest,s_rforest] = rrcforest(XTrain,StandardizeData=true);
rforest is a RobustRandomCutForest object. rrcforest also returns the anomaly indicators (tf_rforest) and anomaly scores (s_rforest) for the training data (XTrain). By default, rrcforest treats all
training observations as normal observations, and sets the score threshold (rforest.ScoreThreshold) to the maximum score value.
Use the trained robust random cut forest model and the object function isanomaly to find novelties in XTest. The isanomaly function identifies observations with scores above the threshold
(rforest.ScoreThreshold) as novelties.
[tfTest_rforest,sTest_rforest] = isanomaly(rforest,XTest);
The isanomaly function returns the anomaly indicators (tfTest_rforest) and anomaly scores (sTest_rforest) for the test data.
Plot histograms of the score values. Create a vertical line at the score threshold.
hold on
xline(rforest.ScoreThreshold,"k-", ...
join(["Threshold =" rforest.ScoreThreshold]))
legend("Training data","Test data",Location="southeast")
title("Histograms of Anomaly Scores for Robust Random Cut Forest")
hold off
The anomaly score distribution of the test data is similar to that of the training data, so isanomaly detects a small number of anomalies in the test data.
Check the fraction of detected anomalies in the test data.
NF_rforest = sum(tfTest_rforest)/NTest
The anomaly score distribution of the test data is similar to that of the training data, so isanomaly does not detect any anomalies in the test data.
Local Outlier Factor
Detect novelties using the object function isanomaly after training a local outlier factor model by using the lof function.
Train a local outlier factor model.
[LOFObj,tf_lof,s_lof] = lof(XTrain);
LOFObj is a LocalOutlierFactor object. lof returns the anomaly indicators (tf_lof) and anomaly scores (s_lof) for the training data (XTrain). By default, lof treats all training observations as
normal observations, and sets the score threshold (LOFObj.ScoreThreshold) to the maximum score value.
Use the trained local outlier factor model and the object function isanomaly to find novelties in XTest. The isanomaly function identifies observations with scores above the threshold
(LOFObj.ScoreThreshold) as novelties.
[tfTest_lof,sTest_lof] = isanomaly(LOFObj,XTest);
The isanomaly function returns the anomaly indicators (tfTest_lof) and anomaly scores (sTest_lof) for the test data.
Plot histograms of the score values. Create a vertical line at the score threshold.
hold on
xline(LOFObj.ScoreThreshold,"k-", ...
join(["Threshold =" LOFObj.ScoreThreshold]))
legend("Training data","Test data",Location="southeast")
title("Histograms of Anomaly Scores for Local Outlier Factor")
hold off
The anomaly score distribution of the test data is similar to that of the training data, so isanomaly detects a small number of anomalies in the test data.
Check the fraction of detected anomalies in the test data.
NF_lof = sum(tfTest_lof)/NTest
Display the observation index of the anomalies in the test data.
idx_lof = find(tfTest_lof)
One-Class SVM
Detect novelties using the object function isanomaly after training a one-class SVM model by using the ocsvm function.
Train a one-class SVM model. Set KernelScale to "auto" to let the function select an appropriate kernel scale parameter using a heuristic procedure, and specify StandardizeData as true to standardize
the input data.
[Mdl,tf_OCSVM,s_OCSVM] = ocsvm(XTrain, ...
Mdl is a OneClassSVM object. ocsvm returns the anomaly indicators (tf_OCSVM) and anomaly scores (s_OCSVM) for the training data (XTrain). By default, ocsvm treats all training observations as normal
observations, and sets the score threshold (Mdl.ScoreThreshold) to the maximum score value.
Use the trained one-class SVM model and the object function isanomaly to find novelties in the test data (XTest). The isanomaly function identifies observations with scores above the threshold
(Mdl.ScoreThreshold) as novelties.
[tfTest_OCSVM,sTest_OCSVM] = isanomaly(Mdl,XTest);
The isanomaly function returns the anomaly indicators (tfTest_OCSVM) and anomaly scores (sTest_OCSVM) for the test data.
Plot histograms of the score values. Create a vertical line at the score threshold.
hold on
xline(Mdl.ScoreThreshold,"k-", ...
join(["Threshold =" Mdl.ScoreThreshold]))
legend("Training data","Test data",Location="southeast")
title("Histograms of Anomaly Scores for One-Class SVM")
hold off
Check the fraction of detected anomalies in the test data.
NF_OCSVM = sum(tfTest_OCSVM)/NTest
Display the observation index of the anomalies in the test data.
idx_OCSVM = find(tfTest_OCSVM)
idx_OCSVM = 2×1
Mahalanobis Distance
Use the robustcov function to compute Mahalanobis distances of training data, and use the pdist2 function to compute Mahalanobis distances of test data.
Compute the Mahalanobis distance from XTrain to the distribution of XTrain by using the robustcov function. Specify the fraction of outliers (OutlierFraction) as 0.
[sigma,mu,s_mahal] = robustcov(XTrain,OutlierFraction=0);
robustcov also returns the estimates of covariance matrix (sigma) and mean (mu), which you can use to compute distances of test data.
Use the maximum value of s_mahal as the score threshold for novelty detection.
s_mahal_threshold = max(s_mahal);
Compute the Mahalanobis distance from XTest to the distribution of XTrain by using the pdist2 function.
sTest_mahal = pdist2(XTest,mu,"mahalanobis",sigma);
Obtain the anomaly indicators for XTest.
tfTest_mahal = sTest_mahal > s_mahal_threshold;
Plot histograms of the score values.
hold on
xline(s_mahal_threshold,"k-", ...
join(["Threshold =" s_mahal_threshold]))
legend("Training data","Test Data",Location="southeast")
title("Histograms of Mahalanobis Distances")
hold off
Check the fraction of detected anomalies in the test data.
NF_mahal = sum(tfTest_mahal)/NTest
Display the observation index of the anomalies in the test data.
idx_mahal = find(tfTest_mahal)
See Also
iforest | isanomaly (IsolationForest) | rrcforest | isanomaly (RobustRandomCutForest) | lof | isanomaly (LocalOutlierFactor) | ocsvm | isanomaly (OneClassSVM) | robustcov | pdist2
Related Topics | {"url":"https://uk.mathworks.com/help/stats/unsupervised-anomaly-detection.html","timestamp":"2024-11-05T06:01:17Z","content_type":"text/html","content_length":"129685","record_id":"<urn:uuid:4f4dbc02-9a25-42be-8d1d-f87f8e3eb27b>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00658.warc.gz"} |
Probability Theory by Curtis T. McMullen
Probability Theory
by Curtis T. McMullen
Publisher: Harvard University 2011
Number of pages: 98
Contents: The Sample Space; Elements of Combinatorial Analysis; Random Walks; Combinations of Events; Conditional Probability; The Binomial and Poisson Distributions; Normal Approximation; Unlimited
Sequences of Bernoulli Trials; Random Variables and Expectation; Law of Large Numbers; Integral-Valued Variables. Generating Functions; Random Walk and Ruin Problems; The Exponential and the Uniform
Density; Special Densities.
Download or read it online for free here:
Download link
(630KB, PDF)
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Number Gossip
(Enter a number and I'll tell you everything you wanted to know about it but were afraid to ask.)
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• If you write n consecutive digits starting with 1 and following 9 by 0, the smallest n that will give you a prime number is 171
Common Properties of 171 | {"url":"http://www.numbergossip.com/171","timestamp":"2024-11-15T01:24:35Z","content_type":"application/xhtml+xml","content_length":"9943","record_id":"<urn:uuid:67e4f7fa-b4d6-4b68-aa8c-6c58a87203c2>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00823.warc.gz"} |
Velocity Structure and Temperature Dependence of an Extreme-Ultraviolet Jet Observed by Hinode
The acceleration mechanism of EUV and X-ray jets is still unclear. In general, there are two candidates for the mechanism. One is magnetic reconnection, and the other is chromospheric evaporation. We
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23 pages, 13 figures, submitted to Solar Physics | {"url":"https://ui.adsabs.harvard.edu/abs/2019SoPh..294...74K","timestamp":"2024-11-12T20:47:07Z","content_type":"text/html","content_length":"42837","record_id":"<urn:uuid:b800ac94-a548-4da5-81db-ec556f8117f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00347.warc.gz"} |
Download and unpack qisog-20181030.tar.gz:
wget https://quantum.isogeny.org/qisog-20181030.tar.gz
gunzip qisog-20181030.tar
tar -xf qisog-20181030.tar
cd qisog-20181030
The software has been tested under Linux. Times mentioned below are on one core of a 3.5GHz Haswell CPU.
Bit-operation simulator
To run:
cd bits
This takes 8 minutes. If you don't have clang++, try changing clang++ to g++ in bits/Makefile.
Copies of the expected outputs are in bits/*.exp. In particular, bits/*cost.exp shows various bit-operation counts. For example:
• The 512 mul quad line in naturalcost.exp says 241908, meaning 241908 nonlinear bit operations to multiply 512-bit integers.
• The 511 ... (x*y)%p quad line in csidhcost.exp says 447902, meaning 447902 nonlinear bit operations to multiply integers modulo the CSIDH-512 prime.
• The 511 ... (x^-1)%p quad line says 220691666, meaning 220691666 nonlinear bit operations for inversion.
• The 511 ... iteration1 quad line says 3805535430, meaning 3805535430 nonlinear bit operations for one iteration of the main loop handling one isogeny computation.
• The 511 ... iteration2 quad line says 4969644344, meaning 4969644344 nonlinear bit operations for one iteration of the main loop handling two isogeny computations.
Some outputs also have matching Sage scripts:
cd bits
sage pointtest.sage | cmp - pointtest.exp
sage pointtest3.sage | cmp - pointtest3.exp
sage elligatortest2.sage | cmp - elligatortest2.exp
sage csidhtest.sage | cmp - csidhtest.exp
python crandom.py | sage csidhtest2.sage | cmp - csidhtest2.exp
Internally, the core of the simulator is bits/bit.h, counting the number of NOTs, XORs, ANDs, and ORs. The value method decapsulates the value of a bit.
Mathematical calculations of failure probabilities
Probabilities are computed for the 74 CSIDH-512 primes, with C = 5. These parameters are set at the top of each script. Each output line indicates the number of iterations and the failure probability
for that number of iterations. Each failure probability is printed in the form a * 2^(-b), where a (rounded to three digits after the decimal point) is between 0.500 inclusive and 1.000 exclusive,
and b is a nonnegative integer.
To compute failure probabilities for iteration1 (each iteration tries to reduce the top nonzero exponent), in a realistic model where each l-isogeny step has failure chance 1/l:
cd chances
sage top1exact.sage 210 > top1exact.out.210
Here 210 asks for results for 0, 1, ..., 209 iterations. This takes 7 seconds. Changing 210 to 500 increases the time to 40 seconds.
To compute failure probabilities for iteration2 (each iteration tries to reduce the top nonzero exponent and the next exponent having the same sign), in a realistic model where each l-isogeny step
has failure chance 1/l:
cd chances
sage top2exact.sage 110 50 upper > top2exact.out.110.50.upper
Here 110 asks for results for 0, 1, ..., 109 iterations; 50 indicates the number of bits of precision in the calculation; upper computes upper bounds on failure probabilities (while lower would have
computed lower bounds). This takes 164 seconds. Changing 110 50 to 350 512 increases the time to 6143 seconds, and then changing lower to upper increases the time to 9908 seconds.
To compute failure probabilities for iteration1 (each iteration tries to reduce the top nonzero exponent), in a pessimistic model where each isogeny step has failure chance 1/3:
cd chances
sage top1crude.sage 500 > top1crude.out.500
Here 500 asks for results for 0, 1, ..., 499 iterations. This takes 92 seconds. Changing 500 to 1000 increases the time to 790 seconds.
The following programs need libsodium installed.
To run 10000000 iteration1 experiments in the realistic model:
cd chances
./top1trials > top1trials.out
This takes 83 seconds.
To run 10000000 iteration2 experiments in the realistic model:
cd chances
./top2trials > top2trials.out
This takes 94 seconds.
To run 10000000 iteration1 experiments in the pessimistic model:
cd chances
./top1ctrials > top1ctrials.out
This takes 72 seconds.
This is version 2018.10.31 of the "Software" web page. | {"url":"https://quantum.isogeny.org/software.html","timestamp":"2024-11-11T04:42:25Z","content_type":"text/html","content_length":"7458","record_id":"<urn:uuid:f3a487e4-f5ad-4c31-a32b-b69457481f17>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00800.warc.gz"} |
A local restaurant wants to determine what proportion of their
customers prefer red wine with dinner...
A local restaurant wants to determine what proportion of their customers prefer red wine with dinner...
A local restaurant wants to determine what proportion of their customers prefer red wine with dinner instead of white wine. If they use a 99% level of confidence, how many customers do they need to
survey to be within 2.5% of the true proportion?
The following information is provided,
Significance Level, α = 0.01, Margin of Error, E = 0.025
The provided estimate of proportion p is, p = 0.5
The critical value for significance level, α = 0.01 is 2.58.
The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:
n >= p*(1-p)*(zc/E)^2
n = 0.5*(1 - 0.5)*(2.58/0.025)^2
n = 2662.56
Therefore, the sample size needed to satisfy the condition n >= 2662.56 and it must be an integer number, we conclude that the minimum required sample size is n = 2663
Ans : Sample size, n = 2663 | {"url":"https://justaaa.com/statistics-and-probability/374481-a-local-restaurant-wants-to-determine-what","timestamp":"2024-11-13T06:37:28Z","content_type":"text/html","content_length":"41570","record_id":"<urn:uuid:127a766c-eb5a-42c8-96f4-4724cf396e9e>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00581.warc.gz"} |
Genetic Algorithm-Based Optimal Design of a Rolling-Flying Vehicle
This work describes a design optimization framework for a rolling-flying vehicle consisting of a conventional quadrotor configuration with passive wheels. For a baseline comparison, the optimization
approach is also applied for a conventional (flight-only) quadrotor. Pareto-optimal vehicles with maximum range and minimum size are created using a hybrid multi-objective genetic algorithm in
conjunction with multi-physics system models. A low Reynolds number blade element momentum theory aerodynamic model is used with a brushless DC motor model, a terramechanics model, and a vehicle
dynamics model to simulate the vehicle range under any operating angle-of-attack and forward velocity. To understand the tradeoff between vehicle size and operating range, variations in
Pareto-optimal designs are presented as functions of vehicle size. A sensitivity analysis is used to better understand the impact of deviating from the optimal vehicle design variables. This work
builds on current approaches in quadrotor optimization by leveraging a variety of models and formulations from the literature and demonstrating the implementation of various design constraints. It
also improves upon current ad hoc rolling-flying vehicle designs created in previous studies. Results show the importance of accounting for oft-neglected component constraints in the design of
high-range quadrotor vehicles. The optimal vehicle mechanical configuration is shown to be independent of operating point, stressing the importance of a well-matched, optimized propulsion system. By
emphasizing key constraints that affect the maximum and nominal vehicle operating points, an optimization framework is constructed that can be used for rolling-flying vehicles and conventional
Mobile robots promise advancements in many fields, with recent research focusing on exploration and search and rescue capabilities [1]. Commercially, mobile robots offer avenues for revolutionizing
delivery [2], inspection [3], surveillance, and emergency response [4]. Despite rapid advancements in capabilities and applications, improvements in power management are needed to expand the
operating time and range of mobile robots. Achieving these improvements requires addressing the fundamental energetic costs and tradeoffs inherent to robot locomotion modalities.
Traditionally, mobile robots have relied upon a single locomotion modality, such as rolling, flying, walking, or swimming. Each modality can be further subdivided; flight can be achieved using fixed
wings, flapping wings, lighter-than-air structures, rotary wings, or combinations of these configurations. Advantages and limitations are inherent to each mode and configuration. For example, fixed
wing flight is less maneuverable than rotary wing flight but is better suited for covering long distances at high speeds. To address the fundamental limitations of unimodal locomotion, robots capable
of multi-modal transportation are being developed. These vehicles are designed with complementary modes to better operate in multiple complex environments. To this end, vehicles capable of
flying-crawling [5–7], flying-swimming [8,9], and rolling-flying-swimming [10] have been developed. Some of these vehicles rely on transforming or reconfigurable mechanisms [11]. Others have a fixed
configuration and rely on sharing actuators to achieve bimodal locomotion [12,13].
Mobility and energy efficiency are of particular importance for exploratory robots in unstructured environments. A mobile robot must be able to traverse rough or evolving terrain for long distances
and durations. The vehicle must also be maneuverable and operate at a variety of speeds: lower speeds for high-resolution data collection and higher speeds for enhanced deployment. Currently,
multi-rotor flying vehicles are very mobile, as indicated by their six-dimensional configuration space. However, as evidenced by their high cost-of-transport [14], multi-rotor flying vehicles are not
particularly energy efficient due to the constant high-power consumption required to stay aloft. Commercially available quadrotors, such as the DJI Mavic Pro, have a maximum 30 min operating time at
steady operating conditions. Researchers have proposed methods of improving unimodal operating endurance; for example, a 15% reduction in power consumption was obtained by using a single large rotor
to provide lift with smaller rotors providing maneuverability [15]. Alternatively, others have focused on optimizing specific components [16] to improve subsystem efficiency and reduce vehicle mass.
In contrast to flying vehicles, rolling vehicles tend to have very low cost-of-transport [14], but the tradeoff is decreased mobility. To leverage the low cost-of-transport of a rolling vehicle,
while maintaining the mobility of a flying vehicle, researchers have developed several rolling-flying vehicles (RFVs) [10,11,17–19], including the initial micro-aerial-vehicle (MAV) scale,
rotor-propelled hybrid terrestrial/aerial quadrotor (HyTAQ) invented by Kalantari and Spenko [20]. The RFV configuration under consideration here consists of a quadrotor suspended between two passive
wheels, which is one of the HyTAQ variants patented by Kalantari and Spenko [21]. This vehicle is capable of flight in the same fashion as a conventional quadrotor, but can also roll on its two
wheels, with the propulsive force supplied by its propellers. Appropriately combining these modalities can produce a bimodal vehicle capable of energy-efficient rolling under normal operation, but
with the ability to fly when necessitated by the environment or task at hand [22]. If successfully executed, such a configuration offers high mobility and maneuverability along with an
energy-efficient locomotion capability. The energetic analysis and power minimization of such a vehicle have been considered previously [1,22], where it was shown that in contrast to conventional
quadrotors in flight, the RFV's angle-of-attack and forward velocity are independent of one another during rolling operation; angle-of-attack can, therefore, be used to minimize power consumption
during rolling transit. This power minimization capability is a key difference in comparison with other rolling-flying bimodal approaches. Simulations reveal that the power consumption is dependent
upon complex interactions between aerodynamics, electromechanics, terramechanics, and rigid body dynamics [1].
The RFV prototype design detailed in Ref. [1] was somewhat ad hoc, with components and parameters selected based on availability and approximate sizing rules. This paper formalizes a detailed design
process involving modeling, parameterization, and simulation, using a framework that is general enough to apply to both conventional quadrotors and RFVs. The design process for an RFV differs from
that of a conventional quadrotor because the RFV must operate over a large angle-of-attack range and at operating conditions that vary from a conventional quadrotor's nominal near-hover state.
Optimal quadrotor design has a variety of approaches, depending on the design goal and subsequent formulation. Many approaches iterate through a component database until a design meets some
constraint, or use heuristic optimization [23–27], often taking spatial constraints into account. Alternative approaches parameterize components using statistical regression [28–31] and are able to
predict off-the-shelf vehicle mass to within ±5% [31] and flight time to within ±5.4% [29]. Others, instead of parameterizing and optimizing the vehicle, focus on modeling the propulsion system [32–
34]. Finally, Internet-based tools for performance estimation also exist.^^2 The approach taken here is most similar to Refs. [35,36], where the component mass and parameter correlations are used as
in Refs. [28,29,31] and are combined with first-principles models. However, key differences exist. Because of the desire for energetically efficient performance at nominal operating conditions that
vary from those of conventional multi-rotors, the first-principles models in this paper must be valid for a large angle-of-attack range and formulated in a manner conducive to RFV parameterization.
matlab simulations utilize the parameterized models to evaluate a design's range and size, allowing a multi-objective genetic algorithm (MOGA) to create Pareto frontiers of maximum-range and
minimum-size RFVs. The Pareto-optimal designs are used to better understand design variable relationships, demonstrate the importance of constraints, and explore subsystem interactions.
This paper first details the multi-physics models and system parameterization. Next, the MOGA implementation is described, and the resulting design trends are investigated. A comparison of
rolling-optimized and flying-optimized RFVs is presented alongside conventional quadrotors, with key differences noted. A case study for an optimized design lends further insight into a well-matched
propulsion system. Finally, the sensitivity of the vehicle range to changes in design variables is investigated.
Rolling-Flying Vehicle Modeling and Parameterization
There are numerous approaches to quadrotor modeling and design, generally differing based on the intended application. For example, designing for maximum thrust-to-weight ratio, maximum flight time,
or minimum size is only a small subset of potentially useful design objectives. Depending on the specific application, the design process, parameterization, and optimization are formulated
differently. The modeling approach taken here has similarities and differences to other approaches in the literature, so that the vehicle can be parameterized in a way that makes the simulation and
comparison of optimal quadrotors and RFVs tractable. For example, many quadrotor design algorithms implement different hover-based momentum theory or blade element momentum theory (BEMT) solutions
from the helicopter literature, which inherently assume propellers undergoing small deviations from a nominal horizontal rotor plane. However, because the RFV must operate with a comparatively large
angle-of-attack range and at many different operating points, the BEMT implementation offers a general formulation that allows for oblique flow, as in Refs. [1,37].
This section formulates the modeling and parameterization of the RFV such that a heuristic design tool, in this case a multi-objective genetic algorithm, can be utilized to create optimal designs. To
this end, the RFV is conceptualized as a multi-rotor with attached wheels, where the quadrotor pitch can be controlled independently of the wheel motion. To perform the detailed design of the
vehicle, first the free body diagram and mechanics model for the RFV and a conventional quadrotor are described. Next, the independent physical dimensions and component measures are parameterized for
use in simulation. The key vehicle subsystems for parameterization are the vehicle geometric model, brushless DC motor model, propeller model, battery model, and vehicle mass model.
Free Body Diagram and Terramechanics.
Consider a conventional quadrotor with angle-of-attack,
, flying up an incline with constant climb angle,
, as shown in Fig.
. Forces acting on the quadrotor include the gravitational force,
, parasitic drag,
, net propeller thrust,
, and net propeller in-plane force,
(i.e., the force acting normal to the thrust vector in the propeller plane, caused by the differing airspeeds acting on the advancing and retreating blades of the propeller). At a given steady-state
, and incline angle, the quadrotor's angle-of-attack and net thrust force are entirely prescribed because the thrust force must maintain equilibrium with the vehicle weight and drag. However, when
considering the RFV shown in Fig.
, the vehicle weight is at least partially offset by the ground normal force,
, allowing the vehicle angle-of-attack and thrust to be controlled independently. This allows for the computation of an optimal angle-of-attack, where the vehicle thrust serves to propel the vehicle
and to partially unload the wheels, therefore reducing rolling resistance losses [
]. The RFV mechanics model used here is nearly identical to that developed in Ref. [
], where the rolling resistance is conceptualized as the vehicle continuously rolling up a small step [
], and where vehicle velocity, incline angle, and the step height are assumed to be operating condition parameters. The rolling resistance,
, is
is a nondimensional terrain parameter dictated by the wheel diameter,
, and the effective terrain step size,
. For the rolling case when
> 0, the propeller thrust serves to decrease the normal force and thus reduce the rolling resistance of the vehicle. The parasitic drag force scales with airframe planform area,
, angle-of-attack, and velocity, and is computed as
is the air density and
F[q](A[0], α[P])
is the dynamic-pressure-normalized drag force obtained using experimental data from Ref. [
]. Power consumption is related to the required thrust using an oblique flow BEMT aerodynamic model and an electromechanics model. The in-plane propeller force is also computed using the BEMT
aerodynamic model. The RFV energetic modeling and power minimization are detailed in Ref. [
The propeller-driven vehicle configuration is chosen over a direct-drive vehicle for several reasons. Although a direct-drive approach may offer energetic benefits because the drive motor and gearing
can match the load to an efficient drive motor operation, the RFV can also operate efficiently by using a portion of its thrust to unload the wheels, thus reducing rolling resistance and reducing
required power [1]. The direct-drive configuration also inherently forces a discrete roll/fly decision to be made. In contrast, the propulsion-driven vehicle removes the requirement of a discrete
roll/fly decision by changing its angle-of-attack such that power consumption is minimized, as described in Ref. [1]. Finally, whereas the RFV is towed by the rotors, a direct-drive approach is
traction driven and, therefore, subject to terrain limitations due to wheel slip. A two-wheeled direct-drive configuration is analogous to a pendulum-driven robot, such as the GroundBot [40]. Due to
the mechanics of using a displaced pendulous mass in traction-driven operation, unimodal pendulum-driven robots are limited to traversing a maximum slope of approximately 30 deg with limited
acceleration. The two-wheeled direct-drive configuration requires two additional motors and a distally located center of mass for optimal operation. A four-wheeled direct-drive vehicle might be
simpler to control than the RFV but will generally require a mobility-reducing tank-drive steering, or a steering linkage with differential. Furthermore, the four-wheeled direct drive will either
require four additional motors, a drivetrain with a single motor, or a transforming mechanism allowing the drive and flight-propulsion motors to be shared. Because the RFV uses the same actuators for
flying and rolling, it is expected to have an inherently lower mass than an RFV with a four-wheeled direct-drive or a two-wheeled direct-drive configuration, allowing for improved flight
Vehicle Geometry.
The RFV geometry, which consists of four propellers in a square configuration mounted inside of two passive wheels, is shown in Fig.
. The wheels have a diameter of
and are spaced one diameter apart. The propeller locations are constrained such that the propeller discs (i.e., the two-dimensional swept area formed by a revolution of the propeller) are contained
within the outer dimensions of the wheels. The distance from the center of one propeller to its neighbor,
, can take on a range of values, with a maximum value of
constrained by the propeller disc impinging upon the wheel and a minimum value of
determined by the propellers impinging upon one another. To avoid adverse aerodynamic performance, the center of each propeller is constrained to be a minimum distance of
units away from the edge of the neighboring propeller blade or wheel [
], where
is the radius of the propeller.
are given by
The height,
, of the propeller disc plane above the wheel axle can similarly take on a range of values, with a minimum value of
equal to the motor height, and a maximum value of
is computed as
To parameterize the vehicle geometry for simulation, the design variables include the wheel diameter, the propeller radius, and two scale factors:
. The propeller center-to-center distance scale factor,
, and the propeller plane height scale factor,
, are used to define the propeller location in terms of their constraints, where
Defining the dimensions in this manner allows the design optimization algorithm to generate spatially feasible designs with the freedom to place the rotor in any practical configuration. To avoid
impeding the downstream flow of the propellers, the battery tray is sized to fit in between the propellers when considering a top view (Fig.
), such that battery side length,
, is
Motor Model.
To quantify the vehicle's electrical power consumption, a three-phase brushless DC motor is modeled. Conceptually, the motor transduces electrical power into mechanical power to drive the propeller.
As the propeller spins, the aerodynamic forces acting on the propeller create thrust and a drag moment, the latter of which counteracts the torque produced by the motor. Assuming a trapezoidal
commutation scheme standard in quadrotor electronic speed controllers and steady-state operation [
], the angular velocity of the motor,
, is related to the phase voltage,
, the winding resistance,
, the winding current,
, and the back-EMF constant,
, by
The torque,
, is proportional to the current and the back-EMF constant, such that
is the no-load current associated with overcoming internal friction and losses. The brushless DC motor performance is parameterized using the back-EMF constant, the winding resistance, and the
no-load current. Generally, motors of the size and scale typically used for quadrotors are marketed according to their
rating, such that an unloaded motor will produce a no-load speed approximately equal to the
rating multiplied by the applied voltage. Assuming nominal units, the
rating is related to the back-EMF constant by
Note that
is generally specified in rpm/V, whereas
is specified in V/(rad/s). The electrical power consumed by the motor is
To parameterize the motors for simulation,
are used as the independent design variables. Ampatis and Papadopoulos [
] aggregated a data set of maximum torques for a line of MAV scale motors and correlated this data to characteristic motor lengths. The characteristic motor length
is a nonphysical dimension equal to the cube root of motor volume and is correlated to
rating and resistance by
The no-load current is then computed via a correlated relationship [
] using the characteristic motor length, where
Some approaches in the literature also attempt to relate k[V] and R such that they are dependent on each other; however, this approach over-constrains the motor design. Two motors with identical k[V]
ratings can have different winding resistances (and therefore, different dimensions and masses). This is evident when comparing commercially available motors; many sizes of motors exist with
identical k[V] ratings but different resistances.
The motor operating point is mechanically limited by the maximum no-load motor speed and thermally limited by the current flowing through the windings. For motors common in MAV applications with a
propeller creating the load, the no-load speed is assumed to be greater than the obtainable operating speed [
], and the motor is, therefore, assumed to be thermally limited by the maximum sustainable current. Because current is proportional to torque, the maximum current,
, can be computed using the maximum sustainable motor torque,
, as parameterized in Ref. [
], where
The voltage required to produce this current is a function of the torque–speed operating point. For a motor-propeller system, the load is computed by a steady-state propeller model, which is
described later in this section.
Battery Model.
The battery subsystem provides the necessary power for rotor operation. The battery model relates the number of battery cells in series,
, battery voltage,
, battery mass,
, battery capacity,
, and stored battery energy,
. The battery is assumed to be a Lithium-Polymer (LiPo) type common in MAV designs due to its high specific energy density and high discharge rates. A LiPo cell has a nominal voltage of 3.7 V, so the
battery voltage is related to the number of cells by
The maximum battery mass is constrained such that the vehicle center of gravity lies on the wheel axis of rotation, as this greatly simplifies dynamic modeling and control of the vehicle [
]. The computation of the battery mass is shown in the “Mass Model” section. The capacity of each cell is generally given in units of milli-Amp-hours, and using the relationship from Ampatis and
Papadopoulos [
The energy stored within the battery is therefore
The relationships between battery energy, battery capacity, battery voltage, and mass are shown in Fig. 3. Note that stored battery energy is proportional to battery mass. This steady-state analysis
assumes nominal 80% maximum battery depth of discharge (indicating that the usable battery energy is 80% of the value computed in Eq. (20)) and a 3.7 V cell voltage.
Battery-Motor Subsystem Interaction.
The motor and battery may restrict the vehicle's maximum thrust due to the available, tolerable, or required current and voltage. The subsystems are well-matched when the maximum battery current and
voltage balance the motor's required voltage and tolerable current at the maximum operating point. For a given motor mass, many feasible k[V] and R combinations exist; those with lower relative k[V]
ratings will require more voltage (but less current) to produce a given torque–speed (τ–ω) operating point than a pair with higher relative k[V] rating. The battery voltage and discharge current are
dictated by the battery mass, S-rating (i.e., the number of cells in series), and C-rating (i.e., the discharge rating, equal to the battery capacity divided by the maximum discharge current). If the
battery cannot supply enough voltage or current for a motor's k[V] -R value, then the desired τ–ω operating point cannot be reached and thrust will be limited. To more concretely understand how this
information is used in designing the subsystem parameters, consider an example vehicle with 60 g motors and a 100 g battery operating at τ = 0.075 Nm and ω = 1400 rad/s. Nominal values for LiPo
batteries are assumed such that the voltage per cell is 3.7 V and the battery internal resistance is 0.005 Ω per cell. Each motor is assumed to operate at the same point, such that the motor current
is one-quarter of the battery current. To define the current-limited case, a C-rating of 40C is assumed, implying a maximum discharge current. The power system design process is illustrated in Fig. 4
. The dashed curve indicates possible k[V]-R values for the given motor mass. The shaded area denotes infeasible k[V]-R values. Using the τ–ω operating point, the voltage and current required to
drive the motor can be computed for all k[V]-R pairs, and therefore the required battery voltage and minimum C-rating are known for each k[V]-R pair. The k[V]-R pairs that utilize full battery
voltage to create τ–ω are indicated by the black markers, where each marker corresponds to a different S-rating. The lighter markers define the battery-current-limited case for each S-rating. The
solid lines connecting the markers represent k[V]-R pairs that can produce τ–ω while operating somewhere between maximum battery voltage and maximum battery current. As shown in Fig. 4, there may be
multiple sets of k[V]-R pairs and S-ratings that are capable of driving the τ–ω operating point.
Propeller Model.
The propeller model provides a method of relating the motor brake power to the propeller thrust. Many propeller models are formulated for either the axial flow case (e.g., an airplane in level
flight) or the near-hover case (e.g., a rotorcraft with zero freestream velocity). Because the RFV angle-of-attack can be anywhere between these two configurations, propulsion in oblique flow (i.e.,
when the incoming air velocity is neither normal nor parallel to the propeller disc plane) must be accounted for. Oblique flow considerably impacts propulsive forces as compared with conventional
axial or near-hover operation. At low freestream velocities, as the propeller transitions from an axial configuration (α[P] = 0 deg) to a transverse configuration (α[P] = 90 deg) at constant angular
velocity, the increasing flow angle serves to slightly increase thrust and decrease the required torque [37]. Qualitatively, this is because as the propeller approaches the horizontal configuration,
it more closely resembles a static thrust (zero freestream inflow) condition. At increased airspeeds, the literature shows that the propeller-induced power decreases due to the increased propeller
inflow [46]. Additional in-plane forces, present due to differences in relative airspeed on the retreating and advancing propeller blades, also become significant. The in-plane forces are the cause
of p-factor during traditional fixed wing aircraft climb-out, and partially necessitate the cyclic pitch control required in helicopter flight. Although not always observable at low velocities, the
model described here is of sufficient fidelity to demonstrate these effects [1].
Propeller modeling is generally based on momentum theory, which uses the conservation of momentum of the airstream passing through the propeller disk area to relate the thrust of a propeller to the
mechanical power required to turn the propeller. More complex models, such as the BEMT used in this paper, more accurately quantify propulsion by using propeller geometric and aerodynamic data. BEMT
considers the lift and drag generated by differential spanwise cross sections of the propeller blade. Each spanwise cross section is modeled as an airfoil using the geometric and aerodynamic data,
and a local air velocity vector is used to compute the differential lift and drag contributions to the thrust produced and the torque required to drive the propeller. By integrating the differential
elements along the blade radius and along the annulus formed by a blade rotation, the thrust and torque of a propeller can be related to the angular and forward velocity. BEMT methods that account
for oblique flow generally vary in how the induced velocity is computed. Different techniques are reviewed in Ref. [47]; for implementing BEMT in the RFV model, induced velocity is formulated as in
Ref. [48]. This model assumes that the induced velocity does not vary with azimuthal location, simplifying computations considerably with less than 2% differences in thrust and torque values [1] when
compared with more complex models [49]. Detailed formulation of the BEMT model used for the RFV, including the derivatives used for computationally efficient convergence via the Newton-Raphson
method, is described in their entirety in Ref. [1]. Other propeller modeling approaches include computational tools such as QPROP [50], vortex lattice methods [37], and computational fluid dynamics
(CFD). BEMT is chosen in lieu of these options because QPROP is not formulated for oblique flow, vortex lattice methods are prone to wake instabilities [37], and CFD has high setup and computational
costs, making implementation in the vehicle simulations impractical.
By making some simplifying assumptions about the propeller shape, the propeller geometry can be modeled using diameter and pitch length as design variables. To this end, the propeller pitch length is
related to the blade angle by assuming a constant pitch length
is the pitch length,
is the radial blade section station, and
is the aerodynamic pitch angle. The propeller blade is assumed to have an optimal taper ratio as described in Ref. [
], where the local chord length,
), is related to the propeller radius, the chord tip length,
, and the radial station,
, by
The chord length is limited to a maximum value to ensure a pragmatic shape for manufacturing, as in Ref. [33]. Winslow et al. tested a variety of propellers and showed experimentally that blades with
thin, cambered airfoil section shapes provide the best performance for small-scale MAV propellers. The recommended NACA 6504 airfoil shape is, therefore, chosen to represent the propeller blade
section airfoil. The propeller blade geometry is described in Fig. 5.
Due to its small size, a MAV propeller operates in a low Reynolds number (Re < ∼100,000) regime, which has been shown to reduce airfoil performance due to laminar separation bubble formation [52].
This low Reynolds number effect is often neglected in the MAV literature but limits the thrust output and propulsive efficiency. At a moderate Reynolds number, airfoil lift and drag coefficients are
nearly constant; however, at a low Reynolds number, the lift and drag coefficients are functions of the Reynolds number. To account for performance degradation, a process as in Ref. [53] is
implemented: NACA 6504 lift and drag polars are computed using XFOIL [54] for a range of Reynolds numbers from Re = 10,000 to Re = 200,000, C[l] and C[d] values are extracted from the lift and drag
polars, and the C[l] and C[d] data are interpolated as functions of Reynolds number and blade section angle-of-attack. The BEMT computation can then locally compute the Reynolds number and
angle-of-attack for the current blade section and refer the appropriate C[l] and C[d] values [55].
Mass Model.
The total vehicle mass is the sum of the component masses
Using relationships from Refs. [
], the motor and propeller masses are correlated to their design variables by
The electronic speed controller mass is correlated to the maximum sustainable current
and battery mass is computed using the battery volume and density
where the battery volume is approximated as a rectangular prism with a side length of
(as determined earlier) and length of
. The maximum battery length is determined by balancing the center of mass of the four rotors and the battery such that the resulting center of mass lies on the vehicle axis of rotation
To give the design algorithm the freedom to size the battery with a range of heights, a battery height scale factor,
, is introduced as a design variable, where
The effective battery density,
= 2113 kg/m
, is computed using the specific energy determined for LiPo batteries in Ref. [
]. Quadrotor airframes, defined here as the vehicle structure not including the wheels, are correlated to the propeller radius and battery mass [
] using the function
The avionics are assumed to have a constant mass of
= 0.6 kg for all vehicle sizes. This value was determined by weighing components and sensors, such as control electronics and a small LIDAR, which could represent one RFV implementation [
]. The RFV wheel mass is parameterized by multiplying the wheel solidity
by the wheel area and the wheel material area density
, and adding a constant value to represent the wheel hub hardware
Values of σ = 0.4, ρ[wheel] = 1.4 kg/m^2, and m[hub] = 0.05 kg are used in this study. The wheel solidity and hub mass are computed using CAD models and physical prototyping. The wheel material used
is DragonPlate (ALLRed & Associates Inc, Elbridge, NY), a stiff, lightweight composite material comprised of a balsa wood inner core laminated with thin carbon fiber sheets on either side.
Range and Endurance Evaluation.
To recap the modeling section, the design variables used to parameterize the RFV are wheel diameter, motor
rating, motor resistance, propeller diameter, propeller pitch length, and three scale factors (the propeller plane height scale factor, propeller spacing scale factor, and battery length scale
factor). Using the design variables and subsystem models, a vehicle's rolling range,
, and hover endurance,
, are computed as
is the power consumed by a single motor at the rolling operating point and
is the power consumed by a single motor at the hover operating point.
To assess the validity of the subsystem models, the parameterized models are implemented in matlab. Using manufacturer specifications^^3 [57], and published values [29] to supply propeller pitch
length, propeller diameter, motor k[V] rating, motor resistance, battery weight, and gross takeoff weight (GTOW), the predicted hover endurances of four quadrotors are computed. The quadrotors
represent a spectrum of available sizes, with GTOWs ranging from 80 g to 1375 g. The system model successfully predicts all four vehicles’ hover endurances to within 9.6%, with a mean absolute error
of 6.3%, as shown in Table 1. The GTQ-Mini calculated endurance includes published specifications for the power consumed by onboard electronics (48 W), leading to an endurance computation within 20 s
of the reported value.
Table 1
Vehicle name GTOW (g) Reported endurance (s) Calculated endurance (s) Difference (s) Difference (%)
DJI Phantom 4 Pro v2.0^^4 1375 1800 1972 172 9.6
DJI Mavic Pro^^5 743 1620 1551 −69 −4.3
GTQ-Mini [29] 496 330 314 −16 −4.9
Ryze Tello [57] 80 780 830 50 6.4
Mean abs. error, % 6.3
Vehicle name GTOW (g) Reported endurance (s) Calculated endurance (s) Difference (s) Difference (%)
DJI Phantom 4 Pro v2.0^^4 1375 1800 1972 172 9.6
DJI Mavic Pro^^5 743 1620 1551 −69 −4.3
GTQ-Mini [29] 496 330 314 −16 −4.9
Ryze Tello [57] 80 780 830 50 6.4
Mean abs. error, % 6.3
Methods—MOGA Problem Formulation and Vehicle Optimization
A MOGA is implemented to heuristically determine optimal vehicle configurations using the system model. Because larger vehicles can use larger rotors, and larger rotors improve rotor efficiency [41],
a tradeoff is expected between vehicle range and vehicle size. Pareto-optimal frontiers are, therefore, used to characterize the tradeoffs between vehicle range and vehicle size. To illuminate the
fundamental differences between a conventional, flight-only quadrotor and the RFV, three classes of optimal vehicle designs are generated. First, a Pareto frontier of rolling-range-optimized (yet
flight-capable) vehicles is created. For this vehicle class, the Pareto frontier axes are vehicle size and rolling range. Next, a Pareto frontier of hover endurance -optimized (yet rolling capable)
RFVs is created. For this vehicle class, Pareto frontier axes are vehicle size and hover time. The hover-optimized RFVs are also evaluated for their rolling range, and the rolling-optimized RFVs are
likewise evaluated for hover endurance. Finally, a conventional quadrotor class is implemented using the described parameterization framework, but with zero-wheel mass. In this case, the Pareto
frontier also uses axes of vehicle size and hover time. For the flying-optimized and rolling-optimized RFVs, the vehicle size is the wheel diameter. For the conventional quadrotor, the vehicle size
is equal to the minimum diameter wheel that could be used to turn the conventional quadrotor into an RFV.
The vehicle optimization is formulated as in Eq.
(where the design variable string
= [
k[V], R, λ, r[P], x[B], x[H], x[S], D
]), subject to the bounded design variables in Eq.
and the constraints in Eq.
The fitness function f[1](x) is equal to the vehicle size (i.e., minimize vehicle size). To maximize range, the fitness function f[2](x) = −s is used (i.e., minimize the negative of range), or to
maximize hover endurance, f[2](x) = −t[hover] is used (i.e., minimize the negative of hover endurance). In Eq. (35), the maximum and minimum design variable values are based on a survey of
commercially available components [28,29,31]. The scale factor x[B] ensures that the battery is sized so that the vehicle center of mass is located on the wheel axis of rotation. The scale factors x
[h] and x[s] ensure that the vehicle is parameterized such that the geometry is feasible. Finally, nonlinear constraints are shown in Eq. (36), where ω[max] is the maximum rotor speed, TW[min] is the
minimum thrust-to-weight ratio, ω[TW] is the rotor speed required to produce TW[min], τ(ω[TW]) is the torque required to operate the propeller at ω[TW], and V(ω[TW]) and I(ω[TW]) are the voltage and
current required to drive the rotor at ω[TW]. To ensure sufficient maneuverability during flight, a minimum thrust-to-weight ratio of 2.0 is imposed in the first constraint [58]. The final three
constraints represent the physical limitations that can prevent a vehicle from satisfying the minimum thrust-to-weight ratio: either (1) the maximum motor torque is insufficient to generate enough
rotor speed, and therefore the propeller cannot create enough thrust; (2) the battery cannot supply enough voltage or current as described previously, and thus is unable to drive the motor at the
required torque–speed operating point; or (3) the maximum propeller tip speed is aerodynamically limited by an imposed Mach limit of 0.4, as in Ref. [33]. Hypothetically, a propeller could
mechanically fail due to loading or vibration before reaching the imposed maximum speed; however, researchers have demonstrated carbon fiber propellers at this scale capable of this angular velocity
[33], so the aerodynamic limitation is imposed in lieu of a propeller structural limitation. All designs are checked for feasibility as part of the MOGA algorithm.
Using these design variables, parameterized models, and constraints, the MOGA is implemented using the matlabgamultiobj function [59] via the matlab Global Optimization Toolbox [60]. The MOGA uses a
controlled elitist approach to maintain population diversity while favoring the most fit designs, helping to ensure that a global minimum is found. The default options with adaptive feasible
mutation, intermediate crossover, and tournament selection were found to provide satisfactory performance. Once the MOGA converges as dictated by a prescribed tolerance, a local single-objective
constrained optimization solver is used to refine the final solution (matlabfmincon). Because the local optimization solver is single objective (as compared with the multi-objective GA), it is
sequentially run for discrete vehicle size values between 35 and 75 cm at 5 cm intervals. For each vehicle size, the local optimization solver is seeded with the MOGA Pareto frontier design at that
diameter. The local optimization solver slightly improves the Pareto-optimal designs, generally improving the solution less than 1%. The hybrid approach uses the MOGA to ensure a global minimum is
found and uses the single-objective solver to home in on the optimum design more quickly than is possible with only a heuristic MOGA. Although the exact amount was not measured, the most noticeable
computation time reduction came from vectorizing the innermost-nested BEMT loops. Simulations were executed on the North Carolina State University High-Performance Computing cluster [61].
Optimal designs are presented in this section as Pareto frontier curves. Each point on the Pareto frontier represents a different Pareto-optimal design. Figure 6(a) depicts the Pareto-optimal
frontier for a rolling-optimized RFV of maximum rolling range versus vehicle size, optimized for operating conditions of v = 5 m/s, x[R] = 25 mm, and θ = 0 rad (referred to as the rolling-optimized
RFV). Supplementary curves in Fig. 6(a) show the rolling range associated with an RFV optimized for the hover operating condition (hover-optimized RFV), and the flying range of a wheel-less,
flying-optimized conventional quadrotor operating at a velocity of 5 m/s (conventional QR). At small vehicle sizes, the rolling-optimized RFV and hover-optimized RFV have the same maximum range. As
size increases, the rolling-optimized RFV range begins to outperform the hover-optimized RFV range. Both RFV cases have significant range advantages over the conventional quadrotor. All three curves
show that maximum range increases as vehicle size increases, verifying the expected tradeoff between size and range. As the vehicles become larger, propellers and motors increase in size (becoming
more efficient and allowing for greater thrust production), which allows for larger battery sizes. Figure 6(b) depicts the maximum hover endurance of the same designs in Fig. 6(a). Although the
rolling-optimized RFV shows a rolling range advantage over the hover-optimized RFV and the conventional quadrotor, this advantage comes at a sacrifice of flying endurance. As will be shown in the
“Discussion” section, this is due to differences in the propulsion system and, in the case of the conventional quadrotor, not having to hover with additional weight from the wheels. Pareto-optimal
RFV configurations show ranges that are two to three times greater than the Pareto-optimal conventional quadrotor configurations, at the cost of an 18–25% reduction in hover endurance.
The designs represented by distinct points in Fig. 6 are next broken down into their respective design variables. The associated propeller diameters and pitch lengths are shown in Fig. 7. The black
dashed lines represent constraints of the design space, either imposed as bounds on the design variables or implicitly based on spatial constraints. The Pareto-optimal solutions show nearly identical
propeller diameters for the rolling-optimized, hover-optimized, and conventional cases. The diameter upper bound shown in Fig. 7(a) is for a propeller plane height collocated with the vehicle axis of
rotation; however, if the propeller plane height is non-zero, the maximum diameter is less than that shown in Fig. 7(a), as determined by the spatial constraints. The propeller diameters show little
variation between configurations, with a maximum difference of 0.8 cm. In contrast to similarities in propeller diameters, the optimal pitch lengths for the rolling-optimized case show higher values
than the other cases, with a 4.3 cm difference in the large vehicles’ pitch lengths. Rationale for this observation is provided in the “Discussion” section.
Figure 8 shows the Pareto-optimal motor masses, k[V] ratings, and resistances. The RFV motors are heavier than the conventional quadrotor motors, indicating greater maximum torques. All
configurations favor low k[V] motors, with mass differences resulting primarily from changes in resistance. This is contrary to many published models that assume mass is inversely proportional to k
[V] without considering resistance. Whether considering the RFV or a conventional quadrotor, both k[V] and resistance must be considered to design vehicles with maximum range.
Figure 9 shows the Pareto-optimal propeller plane height and the Pareto-optimal center-to-center propeller distance. All cases show nearly identical spatial configurations, an important result that
supports using a vehicle with a fixed mechanical design, a modular battery, a modular propulsion system, and removable wheels to adapt to user needs. The battery size scale factor x[B] has a unity
value for all cases, implying that the MOGA maximizes battery size for the optimal vehicle configurations.
Constraint Boundaries.
The MOGA maximizes performance by finding designs that simultaneously reach the constraint boundaries imposed by the vehicle geometry, the prescribed thrust-to-weight ratio, and the dependent
electromechanical and aerodynamic constraints, as shown in Fig. 10. Suboptimal designs that do not satisfy these relationships can improve performance by increasing battery size, using larger rotors
to increase thrust, or using larger motors to increase the maximum torque. The 45 cm and 50 cm conventional quadrotor designs do not reach the angular velocity constraint boundary; this potentially
indicates that these designs are not fully optimized, and a slight endurance increase can be realized by computing additional GA generations.
Nominal Design Case Study.
To understand the relationships between design variables and vehicle systems, the rolling-optimized and hover-optimized RFVs with 60 cm vehicle size are considered in a nominal case study. The
propulsion system design variables are investigated to demonstrate how a well-matched propeller and motor improve system performance. Table 2 shows the k[V] rating, resistance, propeller diameter,
and pitch length for these vehicles.
Table 2
Rolling-optimized RFV Hover-optimized RFV
DV 1 k[V] (rpm/V) 500 500
DV 2 R (Ω) 0.192 0.236
DV 3 r[P] (cm) 9.6 9.72
DV 4 λ (cm) 14.0 11.91
Flying time (min) 12:52 13:33
Rolling range (km) 12.0 11.5
Rolling-optimized RFV Hover-optimized RFV
DV 1 k[V] (rpm/V) 500 500
DV 2 R (Ω) 0.192 0.236
DV 3 r[P] (cm) 9.6 9.72
DV 4 λ (cm) 14.0 11.91
Flying time (min) 12:52 13:33
Rolling range (km) 12.0 11.5
To understand trends in the rotor design, electrical power loading curves are created, shown in Fig.
. Electrical power loading
is a measure of how effectively a propeller produces thrust with respect to the required motor power, and is computed as
This efficiency measure is useful in rotorcraft analysis because it allows the computation of a static (i.e., zero velocity) operating point [62], as compared with traditional efficiency which is
trivially equal to zero in the static case. The electrical power loading for the hover and rolling cases as functions of thrust produced by a single rotor are shown in Fig. 11. The curves are
generated using the Pareto-optimal motor parameters. The solid line indicates the power loading at the optimal rotor pitch. The dashed lines indicate off-design pitch values. The vertical dashed
lines indicate the thrust required at the rolling or hover design points. The endpoint of each curve represents the maximum thrust of the configuration, as dictated by a constraint (either voltage,
current, or aerodynamic). Figure 11(a) shows that the MOGA selects a propeller pitch that maximizes the electrical power loading at the thrust required for rolling operation. This propeller pitch is
suboptimal for the hover operating point. Similarly, in Fig. 11(b), the propeller pitch that maximizes electrical power loading at the hover operating point is selected by the MOGA. This propeller
pitch value is suboptimal when the hover-optimized vehicle is in a rolling configuration. Although not considered in detail here, using a variable pitch propeller with collective pitch control could
allow the propulsion system to operate at a maximum electrical power loading, regardless of the required thrust.
Both designs use the minimum allowed k[V] value. Although multiple combinations of k[V] and R are possible for a given motor mass, the optimal vehicles use the lowest feasible k[V] value for the
optimal motor mass, as it requires the least current and, therefore, reduces the electronic speed controller mass. A future implementation of the optimization model could use motor mass as a design
variable (with a minimum k[V] assumed) in lieu of k[V] and R, reducing the dimension of the optimization.
Parameter Sensitivity.
The local sensitivity of the objective function to changes in the propulsion system design variables is computed using numerical partial derivatives. The partial derivatives are computed using a
finite-difference symmetric-midpoint-quotient method. A negative sensitivity value indicates that the objective function (range) will increase by decreasing the design variable, whereas a positive
sensitivity value indicates that the objective function will increase by increasing the design variable. This method is applied for every design variable at each vehicle size, in 5 cm increments.
Sensitivities to motor parameters are shown in Fig. 12 as functions of vehicle size. Vehicle range can be improved by decreasing the motor k[V] rating or by decreasing the motor resistance; however,
decreasing either of these values will increase motor mass, potentially driving the design infeasible due to an insufficient thrust-to-weight ratio. This further demonstrates the necessity of
including component constraints in the model formulation. Vehicle performance is relatively insensitive to changes in motor resistance at smaller vehicle sizes; however, as the vehicle size
increases, the magnitude of the sensitivity to changes in k[V] and resistance increases.
Sensitivities to propeller parameters are shown in Fig. 13 as functions of vehicle size. Vehicle range can be improved by increasing the blade radius; however, this could lead to spatially infeasible
designs. Future work should experimentally determine if the merits of increasing the blade radius outweigh any adverse aerodynamic effects associated with close propeller spacing. Figure 13(b) shows
that for 35 cm and 40 cm sized vehicles, further reducing the propeller pitch will increase range—note that optimum design variable values rest on the lower constraint bound. This suggests that
custom low-pitch propellers can offer performance benefits to the smaller vehicles. For design sizes greater than 40 cm, increasing the propeller pitch will increase the vehicle range; however, this
will also require more torque, which will overload the motors at the maximum thrust condition. The complexities of propulsion system interactions and constraints must be considered when designing
optimum range vehicles.
This work describes the systematic design optimization of a rolling-flying vehicle using a multi-objective genetic algorithm to optimize a parameterized multi-physics model. By emphasizing key
constraints that affect the maximum and nominal vehicle operating points, an optimization framework is constructed that can be used for RFVs and conventional multi-rotors. A low Reynolds number BEMT
aerodynamic model is used to parameterize the propeller in conjunction with a three-phase brushless DC motor model. The optimization yields a better understanding of the interaction between design
variables and system performance by demonstrating the link between key geometric, aerodynamic, and electromechanical constraints on the system. Although discussed in the context of RFVs, the
methodology and conclusions also apply to conventional quadrotors. The resulting optimized vehicles show that the ranges and flight endurances of rolling-optimized and hover-optimized RFVs are
similar, with more significant differences for larger vehicles. Both RFV configurations result in ranges that are two to three times greater than a conventional quadrotor, at the cost of an 18–25%
reduction in hover endurance. Using electrical power loading, the relationships between propeller parameters and system performance are investigated, demonstrating that the MOGA selects parameters
that maximize electrical power loading at the required operating thrust. Finally, sensitivities to changes in the optimum design variables are examined, allowing the designer to understand where to
concentrate design effort. This work provides a baseline understanding of the desired components in a high-range multi-rotor vehicle. Future work will leverage this work to study effects due to being
constrained to commercial off-the-shelf products and investigate vehicle performance over a range of operating conditions.
Conflict of Interest
There are no conflicts of interest.
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Actual Volatility of an Option
Are their any studies of the relationship (if any) of the historical volatility of an option (considered as an asset itself; I am not talking about the volatility of the underlying) and its
implied volatility?
• 0
It would be wonderful to find such a tool, however with the short life span of options, and all of the various expirations and strikes I think it would be hard to find such a tool. Keep in mind
though that historic volatility is simply a standard deviation calculation, so if you wanted to you could track the prices of a particular option and calculate the standard deviation on Excel.
You could then compare your finding against the current implied volatility, which traditionaly is how volatile the maket makers feel the options would be until expiration.
I wasn't talking about a tool. I think all the data you need to do it is available from sources like profit.net or optionsxpress.com. I am looking for an intuitive meaning for IV. I think the
"traditional" meaning for it, you mention, is a stretch. Characterizing it as what you get by back solving Black-Scholes, after plugging in the current option price, doesn't help either. The data
showing that you get good trading results by using relative IV, for a particular option's chain, as a proxy for relative valuation, also is pretty meager.
the actual volatility of any asset is just it's trading range normalized to price. not exactly sure why you want to compare that to the IV of something else, but it's easy enough to do. others
have pointed out where to get historical IV data, so all you need is to suck that into excel along with price data, calculate the running trading volatility, and make a chart of both squiggles.
have to admit i'm at a loss why you want to do this - IV relates to the underlying, not to the option - but hey, i'm always interested in hearing something new.
I have heard something really new.
• 1
rrisch -
A couple thoughts:
1. When you say "historical volatility of the option" are you really interested in range of implied volatility the option has had? Since the option is a derivative of the underlying security, HV
of the option prices seems somewhat meaningless since their price is based on the underlyings. On the other hand, looking at the IV range the option has taken relative the underlying is useful -
you might be able to get that from Ivolatility.com
2. Keep in mind that when you're looking at historical price data for options, it's not a simple and clean task. For it to be meaningful, you have to know the price of the underlying at the time
the price of the option that you're looking at was snapshot.
You also can't depend on closing prices because the closing price quoted for an option is just the last trade price. If the last trade occurred 1/2 hour before market close and the underlying's
price moved considerably in the last 1/2 hour - then the option's "close" price has no meaningful relationship to the underlying's "close" price. And if you're looking at a fairly thinly traded
contract, the "close" you're looking at for the option might actually be from a trade yesterday and completely meaningless.
For a rational comparison, you'd need the historical last bid/ask for the contract for the day to use as a reference against the underlying's closing price. But even then you could still get
skews because some contracts may go to bid:0 ask:999 at the close as placeholders.
not sure what's troubling you. IV is part of the option price, but the volatility it (theoretically) represents isn't that of the option itself, it's of the underlying. an option on the option
would have IV representing the volatility of the, um, first option.
too many options, lol.
murray t turtle
• 4,654
Robert R;
Pondered your question;
might want to study, record in notebook , ponder paragraph # 2 in Archangel reply .
I solve the non liquid contract problem by not looking/trading them.
#8 May 1, 2004
• 0
actually what you said is true that if the stock has a big move at the end of day and the option hasnt traded, its closing price is not in line.
while that is correct, now all options are marked either last trade or if its out of range its marked on the current bid or offer.
#9 May 1, 2004
• 0
I believe each one of these posts neglected to find out the terms of the volatility sought.
Because there are so many options traded for any given asset, the investor should first determine his criteria, including range and time, around an asset. Then the investor should classify this
criteria as an asset itself.
I might be wrong, but I belive "misch" desires to have an understanding of the probability options possess for increasing or decreasing in value regardless of the underlying asset.
If we define a set of terms regarding options, as an asset itself, we can study those terms and determine the percentage of ones that increased or decreased in value, and the percentage range
those prices moved.
For example:
Say we set the terms of an option class as:
10% on either side of the underlying
3 months to expiration
Now we can apply that set of terms against any historical set of prices, and determine the volatility of the option class, as an asset itself.
#10 May 1, 2004 | {"url":"https://www.elitetrader.com/et/threads/actual-volatility-of-an-option.31702/","timestamp":"2024-11-12T08:44:00Z","content_type":"text/html","content_length":"58864","record_id":"<urn:uuid:fec56d62-df1d-4128-b124-b7dc306149b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00694.warc.gz"} |
Network Security: Principles of Public Key Cryptosystems - codingstreets
Public Key Cryptosystems
Public Key Cryptosystems hold two keys for the encryption and decryption process. The key used in encryption is the Public key whereas Decryption uses the Private key. That is the reason it is also
called topsy-turvy key cryptography.
A pair of keys (Public and Private keys) is generated by the sender and receiver. If the sender sends a message to the receiver, then the receiver can decrypt the message only with the corresponding
private key of the pair.
In other words,
The public key cryptosystem is a powerful cryptographic method that relies on key pairs. The use of public and private key pairs allows for secure encryption and decryption of messages. The public
key is shared openly, while the private key remains secret, ensuring that only the intended recipient can decrypt the encrypted message. This asymmetry in key usage provides confidentiality and
privacy in communication.
Terms in Public Key Cryptosystems
1. Plaintext – Plaintext is a simple unscrambled message.
2. Encryption and Decryption process – Public key cryptography enables secure communication by allowing encryption and decryption using different keys. The public key is used for encryption, and any
message encrypted with it can only be decrypted using the corresponding private key. This ensures the confidentiality and privacy of the communication.
3. Encryption and Decryption algorithm – This is an essential algorithm used in the Encryption and Decryption process with a key.
4. Public and Private keys – Each user has a pair of related keys. The public key is shared with others, while the private key is kept secret. The keys are generated in such a way that encrypting
with one key can only be decrypted with the other key.
5. Ciphertext – This is the scrambled message, which means a converted message of plaintext into some unreadable text.
Steps in public key cryptography
1. Each user has to generate two keys one of which will be used for encryption and the other for decryption of messages.
2. Each user has a pair of keys, among which one has to be made public by each user. And the other has to be kept secret.
3. If a user has to send a message to a particular receiver then the sender must encrypt the message using the intended receiver’s public key and then send the encrypted message to the receiver.
4. On receiving the message, the receiver has to decrypt the message using his private key.
In public key cryptography, there is no need for key distribution as we have seen in symmetric key cryptography. As long as this private key is kept secret no one can interpret the message. In the
future, the user can change its private key and publish its related public key in order to replace the old public key.
Public Key Cryptography Requirements
To accomplish public key cryptography there are the following requirements as discussed below.
• The computation of the pair of keys i.e. private key and the public key must be easy.
• Knowing the encryption algorithm and public key of the intended receiver, the computation of cipher text must be easy.
• For a receiver of the message, it should be computationally easy to decrypt the obtained cipher text using his private key.
• It is also required that any opponent in the network knowing the public key should be unable to determine its corresponding private key.
• Having the cipher text and public key an opponent should be unable to determine the original message.
• The two keys i.e. public and private keys can be implemented in both orders D[PU, E(PR, M)] = D[PR, E(PU, M)]
Public Key Cryptosystem Applications
In public key cryptography, every user has to generate a pair of keys among which one is kept secret known as a private key and the other is made public hence called a public key. Now, the decision
of whether the sender’s private key or the receiver’s public key will be used to encrypt the original message depends totally on the application.
We can classify the applications of the public key cryptosystem as below:
a. Encryption/Decryption
If the purpose of an application is to encrypt and decrypt the message then the sender has to encrypt the message using the intended receiver’s public key and the receiver can decrypt the message
using his own private key.
b. Digital Signature
If the purpose of the application is to authenticate the user then the message is signed or encrypted using the sender’s private key. As only the sender can have its private key, it assures all
parties that the message is sent by the particular person.
c. Key Exchange
The two communicating parties exchange a secret key (maybe a private key) for symmetric encryption to secure a particular transaction. This secret key is valid for a short period.
Well, some algorithms implement all three applications and some implement one or two among these applications. Below is the image showing you the details of the algorithm possessing these
Public Key Cryptanalysis
To prevent the brute force attack the key size must be kept large enough so that it would be difficult for the attacker to calculate the encryption and decryption. But the key size should not be so
large that it would become much more difficult to compute practical encryption and decryption.
Another type of attack in public key cryptography is that the attacker would try to compute the private key knowing the public key.
One more type of attack is a probable message attack. If an adversary knows that the encrypted message from a particular sender is a 56-bit key. Then he would simply encrypt all possible 56-bit keys
using the sender’s public key as the public key is known to all. And then match all the encrypted messages with the cipher text. This type of attack can be prevented by appending some random bits to
the original message.
Key Takeaways
• A public key cryptosystem is one which involves two separate keys for encryption and decryption.
• Each user participating in the communication has to generate two keys, one is to be kept secret (private key) and one is to make public (public key).
• Public key cryptosystems can achieve both confidentiality and authenticity.
• The public key cryptosystem is based on invertible mathematics so it has too much computation.
• Large key size reduces the probability of brute force attack in a public key cryptosystem
• Examples of public key cryptosystems are RSA, Diffie-Hellman, DSS, and the Elliptic curve.
Overall, the public key cryptosystem has revolutionized the field of cryptography by providing a practical solution to secure communication and data protection. It has found widespread use in various
applications, including secure online transactions, email encryption, secure file transfer, and virtual private networks (VPNs). The principles of the public key cryptosystem form the foundation of
modern secure communication systems and play a crucial role in ensuring confidentiality, integrity, and authenticity in the digital world. | {"url":"https://codingstreets.com/network-security-principles-of-public-key-cryptosystems/","timestamp":"2024-11-11T00:56:37Z","content_type":"text/html","content_length":"240315","record_id":"<urn:uuid:5fc187d4-725f-4689-ad04-535002be1454>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00842.warc.gz"} |
Cultural Diversity in the Mathematics Classroom
This page is being created for Dr. Larry Hatfields History of Mathematics (EMAT 4/6650) class at the University of Georgia in Athens, Georgia. Our group set out to discover and present material on
Cultural Diversity in Mathematics. We discovered a wealth of resources, including books, articles, websites, lesson plans, and activities, and we have compiled a list of those resources. The primary
purpose of this page is to present research into and explanation of culturally relevant pedagogy, as well as two lesson plans that reflect culturally diverse teaching practices.
Culturally Diverse Pedagogy: Research and Background
Jump_to Cultural Diversity in Mathematics Education: Current Tendencies
Jump__to Classroom Activities
Jump_to_ Compilation of Resources
Culturally Diverse Pedagogy: Research and Background
History Definition_of_CR_Teaching Elements_of_CR_Teaching Origins_of_CR_Pedagogy References
History of Culturally Diverse Education
By the year 2050, Caucasian people will lose their majority status to people of color in the United States population (Burns, Keyes, and Kusimo, 2005).
With a large percentage of teachers falling under one ethnic category, schools should be looking for ways to bridge the gap between teachers and students.
Culturally relevant teaching practices include the specific methodology that the teacher brings to the classroom to effectively teach students based on their culture.
The effects of and need to empower school culture is an essential building block for enacting relevant teaching practices for with students.
Culturally Relevant Teaching Practices Defined
In 1968, Beauchamp argued that curriculum theory and practice are driven by socio-cultural pressure and political structures instead of thoughtful analysis. This led way to the idea that curriculum
should reflect students lifeworlds.
The term culturally relevant began to appear in the 1970s (Ladson-Billings, 1995).
Culturally relevant can be defined in several different ways.
o Some researchers believe that culturally relevant teaching practices can only occur when teachers and students are from the same ethnic background (Grant, 1978). (This is not a widely held belief
because this is not practical or feasible in the educational arena and the world.)
o Websters dictionary (2003) defines culturally relevant in terms of two ideas. According to Webster (2003), culture is defined as relating to a specific group or culture.
o Relevant is defined as having some bearing on or importance for real-world issues, present-day events, or the current state of society.
There are several different terms that historically have been used interchangeably to define culturally relevant teaching practices: culturally relevant pedagogy, culturally congruent, and culturally
responsive teaching.
o Although the primary terminology is culturally relevant teaching practices, other terms may be used as well.
o Specific definitions of culturally relevant and pedagogy vary in respect to the content, methodology, and referent group orientations (Gay, 2003).
Suzuki (1984) looks at culturally relevant teaching practices as a multicultural education that includes interdisciplinary instructional programs that provide multiple learning environments to meet
the individual needs of the student.
In 1986, Parekh referred to culturally relevant teaching as multicultural education.
Parekh (1986) stated that multicultural education was a refined version of a liberal education which celebrated the plurality of the world.
Parekh (1986) defined culturally relevant pedagogy as ensuring equitable access and treatment for all groups in schools.
Hulsebosch and Koerner (1993) claim that culturally relevant teaching means that teachers have actively engaged in assimilating themselves into the mainstream culture of their students while
searching for tools, strategies and other means to enact culturally relevant pedagogy.
Banks (1990) believes that multicultural education is a framework for a way of thinking to set the criteria for making decisions that will better serve diverse population.
Banks (1995) also believes that culturally relevant teaching is a concept that some scholars have come to include as an integral part of multicultural education.
Nieto (1992) defines multicultural education as a process of comprehensive school reform.
o Neito (1992) further states that it challenges and rejects racism and other forms of discrimination in schools and society and accepts and affirms the pluralism that students, their communities,
and teachers represent (208).
o Neito is reinforcing the idea that culturally relevant teaching practices encourage and support the cultural differences that students bring to the classroom and work to include those in the daily
teaching practices.
In 2000, Gay defined culturally relevant teaching as the practice of using prior experiences, cultural knowledge, and performance styles of diverse learners to make the curriculum more appropriate
and effective for them.
o Gay (2003) states that equality reflects culturally sensitive instructional strategies that will lead to maximal academic outcomes for culturally diverse students.
o Gay also defined culturally relevant pedagogy as a set of beliefs and explanations that recognizes and values the importance of cultural diversity in shaping individuals identities (2003).
Grant (1977, 1978), Garcia (1982), Frazier (1987), and Banks (1990) define multicultural education as an education reform movement that is attempting to change the structure of all educational
o This change would involve training teachers to use methods that are effective for individual cultural groups and not follow traditional educational practices.
o Major goal of multicultural education is to reform educational institutions so that students from diverse backgrounds will experience educational equality (Banks, 1993; Matthews, 2003; Sleeter,
1991; Sadker and Sadker, 1982; Klein, 1985; Grant and Sleeter, 1986).
Common Elements of Culturally Relevant Teaching Practice
Three common elements critical to the success of culturally relevant pedagogy in middle school mathematics for African American students that emerge when examining the literature.
o The first element is the idea that beliefs of the individual school play a key factor in the implementation of relevant teaching practices (Matthew, 2003).
These beliefs can include, but are not limited to the schools attitude towards the culture of the student body, the belief that these practices are needed, as well as the belief that culture plays a
factor in instruction of students mathematics education.
o A second element is the level and quality of teacher training with respect to culturally relevant pedagogy (Matthew, 2003).
Most of the schools that have implemented these types of teaching practices have extensive teacher training programs that are on-going.
These programs have many different aspects that are essential to the success of teachers and students. Included in this is the money required to appropriately and adequately train teachers to be the
most effective when teaching mathematics to culturally diverse students.
o The third and final element includes on-going assessment of strengths and weaknesses of culturally relevant teaching practices (Matthew, 2003). As the demographics of schools are changing,
relevance must be qualified continually by asking who the curriculum is relevant to and for what purpose (Keane and Malcolm, 2003).
Origins of the Construct of Culturally Relevant Pedagogy
Culturally relevant pedagogy has a recent history.
Ladson-Billings (1990, 1992, 1993, 1994, 1995, and 2000) claimed that culturally responsive teaching methods develop intellectual, social, emotional, and political learning by using cultural
referents to impart knowledge, skills, and attitudes.
A major goal of culturally relevant teaching practices is to reform public schools and other educational institutions so that students from all diverse backgrounds will experience educational
equality (Banks, 1993).
Mathematics has always been the subject that students struggle with them most (Sleeter, 1997).
o There is a disconnect as students go through school that causes them to believe that learning math is not an experience that makes sense (Gutstein, Lipman, Hernandez, and de los Reyes, 1996).
o This is further compounded by teachers belief that integration of culture and content is not something that applies to mathematics teachers (Banks, 1993).
o To make the curriculum relevant, it must be defined in terms of the dimensions of relevance and assigning priorities (Keane and Malcolm, 2003).
o Material is only considered relevant to the audience, thus it is important to recognize the audience for who they are.
o Culturally relevant teaching practices attempt to connect the meaningfulness between home and school experiences as well as academic concepts and social realities (Gay, 2000).
Teachers need to know how children think in mathematics in order to make appropriate instructional decisions based on what each child knows and can do (Carey, Fennema, and Carpenter, 1995).
Unfortunately, as Keane and Malcolm (2003) point out, it is difficult to determine what the students know about relevant mathematics, because they are constrained by their perceptions of mathematics
and school.
Students need to experience connected, applied mathematics (Ladson-Billings, 2000 b).
Equitable schools demonstrate this by helping teachers obtain the knowledge and experience needed to connect mathematics in relevant ways to the lives of their students (Murphy, 1996).
According to Kahle (1987), many diverse students attribute their success in mathematics to a nurturing relationship with an adult who provides high expectations, mentoring, and support.
Gay (2000) points out that mathematics instruction incorporates every day life concepts, economics, employment, and consumer habits which can be used to help African American students can make
connections to their own lives.
However, Banks (1993) contends that content integration can only occur to the extent to which teachers use examples, data, and information from cultures to illustrate key mathematical concepts.
Sheppo, Hartsfield, Ruff, Jones, and Holinga (1994) emphasize that technology is an excellent resource to help connect mathematics to middle school culturally diverse students.
o Computers allow students to connect mathematics to real issues in their communities.
o Interactive software for geometry, algebra, and calculus can empower diverse students to use fundamental ideas, multiple representations, and technology assisted methods to reason mathematics
problems and ideas (Heid and Zbiek, 1995).
Cultural Diversity in Mathematics Education: Current Tendencies.
By: Victor Brunaud-Vega, Graduate Student in Mathematics Education, UGA, Athens, GA
Introduction Situated_Perspective Culturally_Relevant_Perspective Critical_Pedagogies References_
Culture generally refers to patterns of human activity and the symbolic structures that give such activity significance. There are many different definitions of "culture" and each one of them
reflects a different theoretical base for understanding, or criteria for evaluating, human activity. In general, the term culture denotes the whole product of an individual, group, or society of
intelligent beings. The term includes technology, art, science, as well as moral systems and the characteristic behaviors and habits of the selected intelligent entities. In particular, it has
specifically more detailed meanings in different domains of human activities.
Anthropologists most commonly use the term "culture" to refer to the universal human capacity to classify, codify and communicate their experiences symbolically. This capacity has long been taken as
a defining feature of humans. It can be also said that culture is the way people live in accordance to beliefs, language, history, or the way they dress.
For the purposes of this paper we will understand culture as "the ways in which a group of people make meaning of their experiences through language, beliefs, social practices, and the use and
creation of material objects" (Gutstein et.al., 1997). Nevertheless, because culture is continually being socially constructed, and because individual identities are constructed through the
intersection of racial, ethnic, class, gender, and other experiences, it cannot be reduced to static characteristics or essences (McCarthy, 1995).
The vision of current reform aiming at academic achievement for all students requires integrating disciplinary knowledge with knowledge of student diversity (McLaughlin, Shepard, & ODay, 1995).
Unfortunately, the existing knowledge base for promoting academic achievement with a culturally and linguistically diverse student population is limited and fragmented, in part because disciplinary
knowledge and student diversity have traditionally constituted separate research agendas (O. Lee, 1999). In mathematics education, although reform documents highlight mathematics for all (NCTM, 1989,
2000) as the principle of equity and excellence, they do not provide a coherent conception of equity or strategies for achieving it (Eisenhart, Finkel, & Marion, 1996; O. Lee, 1999; Rodrguez, 1997).
The multicultural education literature, on the other hand, emphasizes issues of cultural and linguistic diversity and equity, but with little consideration of the specific demands of the different
academic disciplines (Banks, 1993; Ladson-Billings, 1994).
Since mathematics usually tends to be presented as a set of objective and universal facts and rules, these subjects are often viewed as "culture free" and not considered socially and culturally
constructed disciplines (Banks, 1993; O. Lee, 1999; Peterson & Barnes, 1996; Rodrguez, 1997; Secada, 1989). Teachers need to understand what counts as knowledge in math/science as well as how
knowledge may be related to norms and values of diverse languages and cultures.
Instructional practices have traditionally relied on examples, analogies, and artifacts that are often unfamiliar to non-mainstream students (Barba, 1993; Ninnes, 2000). Teachers who provide
culturally relevant instruction capitalize on student strengths—what they do know instead of what they do not know. For example, the curriculum of the Algebra Project (Silva & Moses, 1990) uses
student knowledge of the subway system as a basis for understanding operations with integers. The focus on student strengths contrasts to a remediation model of teaching urban students, where
curriculum and instruction are predicated on what students do not know and often emphasize rote skills (Haberman, 1991; Oakes, 1990).
Dealing with integrating diverse cultures in the classroom needs a conceptual framework in order to make coherent decisions as a teacher. Here we present a summary of three well-known and respected
approaches in teaching mathematics while integrating diverse cultures.
2. Teaching mathematics in multicultural classrooms: three approaches.
In the situated perspective, learning becomes a process of changing participation in changing communities of practice in which an individuals resulting knowledge becomes a function of the environment
in which she or he operates (Stinson, 2004).
Consequently, the situated perspective (in contrast to constructivist perspectives) emphasizes interactive systems that are larger in scope than the behavioral and cognitive processes of the
individual student.
Mathematics knowledge in the situated perspective is understood as being co-constituted in a community within a context. It is the community and context in which the student learns the mathematics
that significantly impacts how the student uses and understands the mathematics (Boaler, 2000b).
Boaler (1993) suggests that learning mathematics in context assists in providing student motivation and interest and enhances transference of skills by linking classroom mathematics with real-world
mathematics. She argues, however, that learning mathematics in contexts does not mean learning mathematics ideas and procedures by inserting them into real-world textbook problems or by extending
mathematics to larger real-world class projects. Rather, she suggests that the classroom itself becomes the context in which mathematics is learned and understood: If the students social and cultural
values are encouraged and supported in the mathematics classroom, through the use of contexts or through an acknowledgement of personal routes and direction, then their learning will have more
meaning (p. 17).
The situated perspective offers different notions of what it means to have mathematics ability, changing the concept from either one has mathematics ability or not to an analysis of how the
environment co-constitutes the mathematics knowledge that is learned (Boaler, 2000a). Boaler argues that this change in how mathematics ability is assessed in the situated perspective could move
mathematics education away from the discriminatory practices that produce more failures than successes toward something considerably more equitable and supportive of social justice (p. 118).
b) The Culturally Relevant Perspective
Gloria Ladson-Billings (1994) developed the culturally relevant (p. 17) perspective as she studied teachers who were successful with African-American children. This perspective is derived from the
work of cultural anthropologists who studied the cultural disconnects between (White) teachers and students of color and made suggestions about how teachers could match their teaching styles to the
culture and home backgrounds of their students (Ladson-Billings, 2001, p. 75). Ladson-Billings defines the culturally relevant perspective as promoting student achievement and success through
cultural competence (teachers assist students in developing a positive identification with their home culture) and through sociopolitical consciousness (teachers help students develop a civic and
social awareness in order to work toward equity and social justice).
Teachers working from a culturally relevant perspective (a) demonstrate a belief that children can be competent regardless of race or social class, (b) provide students with scaffolding between what
they know and what they do not know, (c) focus on instruction during class rather than busy-work or behavior management, (d) extend students thinking beyond what they already know, and (e) exhibit
in-depth knowledge of students as well as subject matter (Ladson-Billings, 1995). Ladson-Billings argued that all children can be successful in mathematics when their understanding of it is linked to
meaningful cultural referents, and when the instruction assumes that all students are capable of mastering the subject matter (p. 141).
Mathematics knowledge in the culturally relevant perspective is viewed as a version of ethnomathematics—ethno defined as all culturally identifiable groups with their jargons, codes, symbols, myths,
and even specific ways of reasoning and inferring; mathema defined as categories of analysis; and -tics defined as methods or techniques (DAmbrosio, 1985/1997, 1997). In the culturally relevant
mathematics classroom, the teacher builds from the students ethno or informal mathematics and orients the lesson toward their culture and experiences, while developing the students critical thinking
skills (Gutstein, Lipman, Hernandez, & de los Reyes, 1997).
c) Critical Pedagogies.
Rooted in the social and political critique of the Frankfurt School, critical pedagogies perceive mathematics as a tool for sociopolitical critique.
The critical perspective in pedagogy is characterized as (a) providing an investigation into the sources of knowledge, (b) identifying social problems and plausible solutions, and (c) reacting to
social injustices. In providing these most general and unifying characteristics of a critical education, Skovsmose (1994) notes, A critical education cannot be a simple prolongation of existing
social relationships. It cannot be an apparatus for prevailing inequalities in society. To be critical, education must react to social contradictions (p. 38). Skovsmose (1994), drawing from Freires
(1970/2000) popularization of the concept conscientizao and his work in literacy empowerment, derived the term mathemacy (p.48).
Skovsmose claims that since modern society is highly technological and the core of all modern-day technology is mathematics, that mathemacy is a means of empowerment. He stated, If mathemacy has a
role to play in education, similar to but not identical to the role of literacy, then mathemacy must be seen as composed of different competences: a mathematical, a technological, and a reflective
(p. 48).
In the critical perspective, mathematics knowledge is seen as demonstrating these three competences (Skovsmose, 1994). Mathematical competence is demonstrating proficiency in the normally understood
skills of school mathematics, reproducing and mastering various theorems, proofs, and algorithms. Technological competence demonstrates proficiency in applying mathematics in model building, using
mathematics in pursuit of different technological aims, and reflective competence achieves mathematics critical dimension, reflecting upon and evaluating the just and unjust uses of mathematics.
Skovsmose contends that mathemacy is a necessary condition for a politically informed citizenry and efficient labor force, claiming that mathemacy provides a means for empowerment in organizing and
reorganizing social and political institutions and their accompanying traditions.
Barba, R. H. (1993). A study of culturally syntonic variables in the bilingual/bicultural science classroom. Journal of Research in Science Teaching, 30, 1053-1071.
Boaler, J. (1993). The role of context in the mathematics classroom: Do they make mathematics more real? For the Learning of Mathematics, 13(2), 12–17.
Boaler, J. (2000a). Exploring situated insights into research and learning. Journal for Research in Mathematics Education, 31(1), 113–119.
Boaler, J. (2000b). Mathematics from another world: Traditional communities and the alienation of learners. Journal of Mathematical Behavior, 18(4), 379–397.
DAmbrosio, U. (1997). Ethnomathematics and its place in the history and pedagogy of mathematics. In A. B. Powell & M. Frankenstein (Eds.), Ethnomathematics: Challenging Eurocentrism in mathematics
education (pp. 13–24). Albany: State University of New York Press. (Original work published in 1985)
DAmbrosio, U. (1997). Foreword. In A. B. Powell & M. Frankenstein (Eds.), Ethnomathematics: Challenging Eurocentrism in mathematics education (pp. xv–xxi). Albany: State University of New York Press.
Eisenhart, M., Finkel, E., & Marion, S. F. (1996). Creating the conditions for scientific literacy: A re-examination. American Educational Research Association, 33, 261-295.
Freire, P. (2000). Pedagogy of the oppressed (30th anniversary ed.). New York: Continuum. (Original work published 1970)
Gutstein, E., Lipman, P., Hernandez, P., & de los Reyes, R. (1997). Culturally relevant mathematics teaching in a Mexican American context. Journal for Research in Mathematics Education, 28(6),
Haberman, M. (1991). The pedagogy of poverty versus good teaching. Phi Delta Kappan, 73, 290-294.
Ladson-Billings, G. (1994). The Dreamkeepers: Successful teachers of African American children. San Francisco: Jossey-Bass.
Ladson-Billings, G. (1995). Making mathematics meaningful in a multicultural context. In W. G. Secada, E. Fennema, & L. Byrd (Eds.), New directions for equity in mathematics education (pp. 126–145).
Cambridge: Cambridge University Press.
Ladson-Billings, G. (2001). The power of pedagogy: Does teaching matter? In W. H. Watkins, J. H. Lewis, & V. Chou (Eds.), Race and education: The roles of history and society in educating African
American students (pp. 73–88). Boston: Allyn & Bacon.
Lee, O. (1999). Equity implications based on the conceptions of science achievement in major reform documents. Review of Educational Research, 69(1), 83-115.
McCarthy, C. (1995). The problems with origins: Race and the contrapuntal nature of the educational experience. In C. E. Sleeter & P. L. McLaren (Eds.), Multicultural education, critical pedagogy,
and the politics of difference (pp. 245-268). Albany, NY: SUNY Press.
McLaughlin, M. W., Shepard, L. A., & ODay, J. A. (1995). Improving education through standards-based reform: A report by the National Academy of Education Panel on Standards-based Education Reform.
Stanford, CA: Stanford University, National Academy of Education.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
Ninnes, P. (2000). Representations of indigenous knowledge in secondary school science textbooks in Australia and Canada. International Journal of Science Education, 22(6), 603-617.
Oakes, J. (1990). Opportunities, achievement, and choice: Women and minority students in science and mathematics. In C. B. Cazden (Ed.), Review of Research in Education, Vol. 16 (pp. 153-222).
Washington, DC: American Educational Research Association.
Rodrguez, A. (1997). The dangerous discourse of invisibility: A critique of the NRCs National Science Education Standards. Journal of Research in Science Teaching, 34, 19-37.
Secada, W. G. (1989). Educational equity and equality of education: An alternative conception. In W. G. Secada (Ed.), Equity in education (pp. 68-88). Philadelphia, PA: The Falmer Press.
Silva, C. M., & Moses, R. P. (1990). The Algebra Project: Making middle school mathematics count. Journal of Negro Education, 59(3), 375-391.
Skovsmose, O. (1994). Towards a critical mathematics education. Educational Studies in Mathematics, 27, 35–57.
Stinson, D. W. (2004). Mathematics as Gate-Keeper (?): Three Theoretical Perspectives that Aim Toward Empowering All Children With a Key to the Gate. The Mathematics Educator, Vol. 14, No. 1, 8-18.
Classroom Activities: Culturally Diverse Lesson Plans
Heirloom Geometry : This activity allows students to bring in an artifact from home with both cultural and geometric significance, discuss the significance of their artifacts with the class, and
potentially work with coordinatizing the patterns on the artifacts.
Sweat Shop Math: This activity examines the wages in countries including the United States. The goal is to help students look at the cost of living in various countries to compare how people live.
Students will work to understand how these wages effect living conditions and life in those countries. Part of the goal is to promote cultural understanding and tolerance to create a deeper
understanding in the classroom. Students make mathematical connections to data analysis, and statistical inference.
Sweat Shop Math Teachers Notes taken from:
Gutstein, E. & Peterson, B. (2005). Rethinking Mathematics: Teaching Social Justice
by the Numbers. Milwaukee, WI: Rethinking Schools. (pages 53-61, 160-161).
NCTM Principles and Standards References
to Equity and Cultural Diversity
The Equity Principle
Excellence in mathematics education requires equity—high expectations and strong support for all students.
Equity requires high expectations and worthwhile opportunities for all.
Equity requires accommodating differences to help everyone learn mathematics.
Equity requires resources and support for all classrooms and all students.
Flores, Alfinio. "S Se Puede, 'It Can Be Done': Quality Mathematics in More than One Language." In Multicultural and Gender Equity in the Mathematics Classroom: The Gift of Diversity, 1997 Yearbook
of the National Council of Teachers of Mathematics, edited by Janet Trentacosta, pp. 81–91. Reston, Va.: National Council of Teachers of Mathematics, 1997.
Chapter 1: A Vision for School Mathematics
URL: chapter1/index.htm.
".. and voting knowledgeably all call for quantitative sophistication. Mathematics as a part of cultural heritage. Mathematics is one of the greatest cultural and intellectual .."
URL: chapter2/equity.htm
".. help to understand the strengths and needs of students who come from diverse linguistic and cultural backgrounds, who have specific disabilities, or who possess a special talent and .."
4. Standards for School Mathematics: Representation
URL: chapter3/rep.htm
".. expressions and equations, graphs, and spreadsheet displays—are the result of a process of cultural refinement that took place over many years. When students gain access to .."
5. Grades Pre-K - 2: Communication
URL: chapter4/comm.htm
".. questions in class (Bransford, Brown, and Cocking 1999). Teachers need to be aware of the cultural patterns in their students' communities in order to provide equitable .."
6. Grades 9 - 12: Representation
URL: chapter7/rep.htm
".. point against the time from takeoff to landing Mathematics is one of humankind's greatest cultural achievements. It is the "language of science," providing a means by .."
Compilation of Cultural Diversity
Resources for Teaching Mathematics
│ Books │ NCTM │Articles│
│Lesson Plans/ │Ga Standards │Websites│
│ │ │ │
│ Activities │ │ │
Thank you for visiting our Website!!
Created for Dr. Larry Hatfields EMAT 4650/6650 Class, Summer 2007,
By: Kelli Parker, for Special Focus Group: Cultural Diversity Project. | {"url":"https://jwilson.coe.uga.edu/EMAT6680Fa06/Parker/4650%20Special%20Focus/Cultural%20Diversity%20Website.htm","timestamp":"2024-11-13T09:17:25Z","content_type":"text/html","content_length":"97787","record_id":"<urn:uuid:033f2141-db87-4201-bcf5-db233edee446>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00236.warc.gz"} |
Can a Cohen extension have an $\omega_1$-sequence of successively more generic Cohen reals?
A forcing extension $V[g]$ by the Cohen forcing ${\rm Add}(\omega,1)$ has continuum many Cohen generic reals. Indeed, as Joel Hamkins explains in this Math Overflow answer, a Cohen extension has a
perfect set of Cohen generic reals such that any two finite subsets of them are mutually generic. Since adding a Cohen real is isomorphic to adding $\omega$-many Cohen reals, a Cohen forcing
extension also has countable sequences $\langle r_n\mid n\lt\omega\rangle$ of Cohen reals such that for every $n\lt\omega$, $r_n$ is Cohen generic over $V[\langle r_i\mid i\lt n\rangle]$. Can we push
this to obtain a sequence $\langle r_\xi\mid\xi\lt\omega_1\rangle$ such that for every $\alpha\lt\omega_1$, $r_\alpha$ is Cohen generic over $V[\langle r_\xi\mid\xi\lt\alpha\rangle]$? This question
arose out of my work with Richard Matthews on ${\rm ZFC}^-$-models and we were stumped on it for some time. My intuition was that the answer should be yes; after all, we are not asking that the
sequence is generic for an $\omega_1$-product of Cohen reals. It turns out that Andreas Blass already solved the question years ago and my intuition was all wrong!
Theorem: (Blass [1]) A Cohen forcing extension cannot have a sequence of Cohen reals $\langle r_\xi\mid\xi\lt\omega_1\rangle$ such that for every $\alpha\lt\omega_1$, $r_\alpha$ is Cohen generic over
$V[\langle r_\xi\mid\xi\lt\alpha\rangle]$.
The proof sketch I give here is my modification of Blass's proof which will allow us to generalize the argument to the forcing ${\rm Add}(\kappa,1)$, adding a Cohen subset to $\kappa$, for an
inaccessible $\kappa$.
Let $\mathbb B$ be the Boolean completion of ${\rm Add}(\omega,1)$. In particular, $\mathbb B$ has a dense subset of size $\omega$. Now suppose that a forcing extension by $\mathbb B$ (equivalentely
${\rm Add}(\omega,1))$ has a sequence $\langle r_\xi\mid \xi\lt\omega_1\rangle$ of Cohen reals such that for every $\alpha\lt\omega_1$, $r_\alpha$ is Cohen generic over $V[\langle r_\xi\mid\xi\lt\
alpha\rangle]$. The model $V[\langle r_\xi\mid\xi\lt\omega_1\rangle]$ is a generic extension of $V$ by a complete subalgebra $\mathbb D$ of $\mathbb B$ by the Intermediate Model Theorem of Solovay
[2]. Let's argue that $\mathbb D$ also has a dense subset of size $\omega$. Given a condition $p\in {\rm Add}(\omega,1)$, let $q_p$ be the infima of $b$ in $\mathbb D$ such that $p\leq b$. Each $q_p$
is in $\mathbb D$ by completeness and the conditions $q_p$ are dense in $\mathbb D$. Let $\dot R$ be a $\mathbb D$-name such that it is forced by $1$ that (1) $\dot R$ is an $\omega_1$-sequence of
successively more generic Cohen reals and (2) the extension by $\mathbb D$ is equal to the extension $V[\dot R]$. Then it is easy to see that the Boolean values $[[n\in \dot R(\xi)]]$ for $n\lt\
omega$ and $\xi\lt\omega_1$ must generate $\mathbb D$. Next, observe that since $\mathbb D$ has a countable dense subset, there must be some $\alpha\lt\omega_1$ such that $\mathbb D$ is generated by
the Boolean values $[[n\in \dot R(\xi)]]$ for $n\lt\omega$ and $\xi\lt\alpha$. But this means that if $V[G]$ is $\mathbb D$-generic, then $G$ can be recovered from the sequence $\langle R(\xi)\mid\xi
\lt\alpha\rangle\, (R=\dot R_G)$, which contradicts that $R(\alpha)$ is $V[\langle R(\xi)\mid\xi\lt\alpha\rangle]$-generic. $\square$
The proof easily generalizes to give the following result for an inaccessible cardinal $\kappa$.
Theorem: A forcing extension by ${\rm Add}(\kappa,1)$ cannot have a sequence, $\langle A_\xi\mid\xi\lt\kappa^+\rangle$ of subsets of $\kappa$ such that for every $\alpha\lt\kappa^+$, $A_\alpha$ is $
{\rm Add}(\kappa,1)$-generic over $V[\langle A_\xi\mid\xi\lt\alpha\rangle]$.
I don't know whether the result holds for every regular cardinal $\kappa$ because the current argument relies on $\kappa^{\lt\kappa}=\kappa$.
Question: Does the above result hold true for any regular cardinal $\kappa$?
Question: Can Blass's theorem be proved using partial orders without reference to Boolean algebras?
1. A. Blass, “The model of set theory generated by countably many generic reals,” J. Symbolic Logic, vol. 46, no. 4, pp. 732–752, 1981.
2. S. Grigorieff, “Intermediate submodels and generic extensions in set theory,” Ann. Math. (2), vol. 101, pp. 447–490, 1975. | {"url":"https://victoriagitman.github.io/research/2022/08/15/can-a-cohen-extension-have-an-omega-1-sequence-of-successively-more-generic-cohen-reals.html","timestamp":"2024-11-08T09:21:48Z","content_type":"text/html","content_length":"21795","record_id":"<urn:uuid:468dba81-5bf1-4100-accd-182d9c31d853>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00429.warc.gz"} |
Mathematics of Deep Learning: An Introduction
English | 2023 | ISBN: 978-3111024318 | 126 Pages | PDF, EPUB | 22 MB
The goal of this book is to provide a mathematical perspective on some key elements of the so-called deep neural networks (DNNs). Much of the interest in deep learning has focused on the
implementation of DNN-based algorithms. Our hope is that this compact textbook will offer a complementary point of view that emphasizes the underlying mathematical ideas. We believe that a more
foundational perspective will help to answer important questions that have only received empirical answers so far. The material is based on a one-semester course Introduction to Mathematics of Deep
Learning” for senior undergraduate mathematics majors and first year graduate students in mathematics. Our goal is to introduce basic concepts from deep learning in a rigorous mathematical fashion,
e.g introduce mathematical definitions of deep neural networks (DNNs), loss functions, the backpropagation algorithm, etc. We attempt to identify for each concept the simplest setting that minimizes
technicalities but still contains the key mathematics.
Download from free file storage
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10 Nicolaus Copernicus Accomplishments and Achievements - Have Fun With History
10 Nicolaus Copernicus Accomplishments and Achievements
Nicolaus Copernicus (1473-1543) was a Renaissance astronomer and mathematician whose revolutionary ideas and contributions reshaped our understanding of the universe.
His most significant accomplishment was proposing the heliocentric model, which placed the Sun at the center of the solar system with the planets, including Earth, orbiting around it.
This departure from the prevailing geocentric view challenged centuries-old beliefs and paved the way for the Copernican Revolution.
Copernicus’s work, presented in his book “De revolutionibus orbium coelestium,” influenced future astronomers, sparked a paradigm shift in scientific thinking, and contributed to the development of
modern scientific methodology.
His observations, calculations, and theories laid the foundation for a more accurate and comprehensive understanding of the cosmos.
Accomplishments of Nicolaus Copernicus
1. Proposed the heliocentric model of the solar system
Copernicus’s most significant achievement was his proposal of a heliocentric model, which challenged the prevailing geocentric view that positioned the Earth at the center of the universe.
Also Read: Facts About Nicolaus Copernicus
In his model, Copernicus argued that the Sun is at the center, with the Earth and other planets orbiting around it. This revolutionary idea placed the Sun, rather than the Earth, as the central body
of the solar system.
2. Published the book “De revolutionibus orbium coelestium” on his heliocentric theory
Copernicus’s groundbreaking ideas were presented in his book “De revolutionibus orbium coelestium” (On the Revolutions of the Celestial Spheres), which was published in 1543, just before his death.
This work provided a comprehensive account of his heliocentric theory and mathematical calculations for planetary motion.
Also Read: Timeline of Nicolaus Copernicus
It included detailed explanations of the orbits and movements of the planets, as well as the Earth’s rotation. Despite facing criticism and controversy, the book laid the foundation for a new
understanding of the universe.
3. Argued for the Earth’s rotation on its axis
Copernicus also proposed that the Earth rotates on its axis, thereby explaining the daily motion of the celestial sphere. This idea countered the prevailing belief that the Earth was stationary at
the center of the universe.
By suggesting Earth’s rotation, Copernicus provided a more coherent explanation for the observed diurnal (daily) cycle of the Sun and stars. This notion of Earth’s rotation added further support to
his heliocentric model and helped shape our understanding of planetary motion.
4. Developed a system for calculating and predicting planetary motions
Copernicus developed a systematic approach to calculate and predict the motions of the planets based on his heliocentric model. He proposed that the planets move in perfect circles around the Sun,
with each planet having a unique distance and period of revolution.
Copernicus’s system incorporated intricate mathematical calculations, such as epicycles and deferents, to explain the observed retrograde motion of planets. His model provided a more accurate
representation of planetary motions compared to the geocentric models that preceded it.
5. Eliminated the need for equants in explaining planetary motion
One of Copernicus’s important contributions was eliminating the need for equants in explaining the irregular motion of planets. Equants were hypothetical points introduced in the geocentric model to
account for variations in planetary speed.
However, Copernicus’s heliocentric model, which used uniform circular motion, offered a simpler and more elegant explanation. By employing the concept of concentric circles for planetary orbits,
Copernicus removed the need for equants and provided a more coherent understanding of planetary motion.
6. Recognized the vast distances to the stars
Copernicus recognized that the stars were enormously distant from the Earth, which was a significant departure from the prevailing belief that they were fixed points relatively close to our planet.
Copernicus understood that the apparent motion of stars was due to the Earth’s rotation.
While he did not have precise measurements of stellar distances, his realization of their immense remoteness was a critical step toward later astronomers’ endeavors to gauge the vastness of the
universe. Copernicus’s insight laid the groundwork for future studies on the distances and nature of stars.
7. Made various astronomical observations and measurements
Copernicus conducted a range of astronomical observations and measurements throughout his lifetime. While his observations were not as precise as those made by later astronomers, they formed the
basis of his mathematical calculations and theories.
Copernicus observed the positions and movements of celestial bodies, such as the Sun, Moon, planets, and stars. These observations helped him refine his heliocentric model and develop a better
understanding of the celestial motions.
8. Influenced future astronomers like Galileo Galilei and Johannes Kepler
Copernicus’s work had a profound influence on subsequent generations of astronomers. His ideas inspired scientists such as Galileo Galilei, who used the telescope to provide further evidence
supporting the heliocentric model.
Johannes Kepler, another prominent astronomer, built upon Copernicus’s work and refined the heliocentric model by introducing the concept of elliptical planetary orbits. Copernicus’s ideas and
discoveries laid the groundwork for further advancements in astronomy and paved the way for a new era of scientific thinking.
9. Marked a paradigm shift in scientific thinking
Copernicus’s heliocentric model marked a significant shift in scientific thinking and challenged long-held beliefs about the nature of the universe. The prevailing geocentric view, supported by the
Catholic Church and prominent philosophers like Aristotle and Ptolemy, placed the Earth at the center of the cosmos.
Also Read: Ptolemy Facts
Copernicus’s revolutionary idea that the Earth orbited the Sun went against this established worldview. His heliocentric model introduced a new framework for understanding the solar system, sparked
debates, and prompted a reevaluation of fundamental assumptions about the cosmos.
10. Contributed to the development of scientific methodology
Copernicus’s approach to scientific inquiry set a precedent for modern scientific methodology. His work emphasized the importance of observation, measurement, and mathematical modeling in explaining
natural phenomena.
Copernicus combined careful observations of celestial bodies with mathematical calculations to develop his heliocentric model.
His approach demonstrated the power of using empirical evidence and mathematics to understand and explain the workings of the universe. Copernicus’s work laid the foundation for the scientific
revolution and shaped the way scientists approach research and discovery. | {"url":"https://www.havefunwithhistory.com/nicolaus-copernicus-accomplishments/","timestamp":"2024-11-12T10:23:47Z","content_type":"text/html","content_length":"49838","record_id":"<urn:uuid:4cf18ef5-7fdc-4545-ad2f-58382402fb3c>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00104.warc.gz"} |
NCERT Solutions for Class 5 Maths Chapter 11
NCERT Solutions for Class 5 Maths Chapter 11 Area and its Boundary in English and Hindi Medium free to download for CBSE 2024-25. Maths class 5 chapter 11 is about area and extent where students have
to find out the area of rectangles and squares, the difference between length and area, and how they can measure them. Students will use basic formulas to find problems and their solutions. The
Chapter 11 also looks at some other techniques such as multiplying and counting tiles. Other examples discussed in this chapter include finding the perimeter and area of rectangles.
NCERT Solutions for Class 5 Maths Chapter 11
NCERT Solutions for Class 5 Maths Chapter 11 Area and its Boundary
5th Standard NCERT Maths Chapter 11 Aim
You have found many such problems while playing or comparing things of yours with your friends for which you didn’t get the proper answer and your comparison end up in quarrels. However, you still
didn’t get the answers. Sound familiar? Well, we can say this chapter is going to help you to find the answer for such a solution so that your question and answer sessions can go without such worries
and you can compare anything with anyone. Well, you already guessed the way. Measurements are the right answer to compare.
Class 5 NCERT Maths Chapter 11 Comparison of Data
You can measure pretty much everything and in the previous chapter, you already created the data to prove the point to do that. You picked several things and you measured them. But what if you had to
measure something that you cannot pick then how would you measure and compare it?
CBSE Class 5 NCERT Maths Book Chapter 11 Activity
Have you tried to play Kabaddi-like games or tag in box-like games in the playground and found your box in which you were playing is smaller and due to that you are losing the game and then you
started to find a way to prove that your smaller box giving you a disadvantage. Well than you try to use your other skills to measure because clearly, you cannot have a ruler in your playground to
measure as it is too small to cover the entire area. Find out what are the ways you can use measure your base. I can suggest one.
You can measure the base by counting the steps how long is the boundary and then using the addition of all the sides you can get the measure of the boundary of your base and then similarly you can
repeat. When you used this method you can notice you have not used the traditional way of measuring things and then you have used the addition method to save the time to calculate the total number of
steps you walk around it.
5th Maths Chapter 11 Suggestions
Similarly, such methods are given in the chapter and then it will actually help you to practice some of the problems which required your creative sense of solving.
What is the main motive of Chapter 11 in Class 5 Maths?
Students of class 6th will be able to learn many important things about the areas of certain shapes and how one will be able to get measure the areas of such places. Different and important
terminologies are being used here and these terminologies will be important for upcoming chapters.
What are the major topics to consider important in unit 11 of class 5th Maths?
There are several important things to cover in chapter 11 of class 5th Maths such are calculating the area and understanding of terminologies are some of things that are considered important. Though,
It is just basic things to understand in this book but in this chapter, I found it important to have a grip on it so that upcoming chapters would be easy to understand.
Do you think students in class 5th students require more practice for unit 11 Maths?
After you study the chapter you will find out various new and important things in unit 11 of class 5th Maths to understand that properly studying isn’t going to be enough. Students are suggested to
complete various tests or practices to test their understanding level and keep that in use.
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Multiplication Is Repeated Addition Worksheets
Math, particularly multiplication, develops the keystone of countless academic disciplines and real-world applications. Yet, for several learners, mastering multiplication can present a difficulty.
To address this hurdle, educators and moms and dads have actually accepted a powerful device: Multiplication Is Repeated Addition Worksheets.
Introduction to Multiplication Is Repeated Addition Worksheets
Multiplication Is Repeated Addition Worksheets
Multiplication Is Repeated Addition Worksheets -
Worksheet 1 These multiplication and repeated addition worksheets offer elementary aged children a deep dive into this foundational math principle Repeated addition is another way students can
approach multiplication and these worksheets provide lots of practice with the concept Fun themes and instructional support help children as they
Write a multiplication fact and repeated addition number sentence for each illustration 3rd and 4th Grades View PDF Arrays as Repeated Addition Using Arrays Bar Model Worksheet 2 Repeated Addition
and Multiplication This is a slightly easier version of the bar model worksheet above 3rd and 4th Grades View PDF
Significance of Multiplication Practice Comprehending multiplication is critical, laying a solid structure for sophisticated mathematical ideas. Multiplication Is Repeated Addition Worksheets offer
structured and targeted method, fostering a deeper understanding of this basic math operation.
Development of Multiplication Is Repeated Addition Worksheets
Repeated Addition Game Worksheets Worksheet Hero
Repeated Addition Game Worksheets Worksheet Hero
Multiplication as Repeated Addition 5 Pack A prompted worksheet set for students to work with Each worksheet is set over two pages There are five version so it is ten pages in all Using
Multiplication Arrays Lesson Give us words than sums and finally products Count the number of dogs in the two groups
16 x 16 20 x 20 8 8 8 8 4 x 8 9 9 9 9 9 5
From conventional pen-and-paper exercises to digitized interactive styles, Multiplication Is Repeated Addition Worksheets have actually developed, accommodating diverse understanding styles and
Sorts Of Multiplication Is Repeated Addition Worksheets
Fundamental Multiplication Sheets Simple workouts focusing on multiplication tables, assisting learners build a strong math base.
Word Problem Worksheets
Real-life circumstances incorporated right into troubles, enhancing critical thinking and application skills.
Timed Multiplication Drills Examinations designed to boost speed and accuracy, assisting in quick psychological math.
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Multiplication And Addition Worksheets 99Worksheets
Multiplication And Addition Worksheets 99Worksheets
ID 437019 21 10 2020 Country code AE Country United Arab Emirates School subject Math 1061955 Main content Multiplication 2013181 write a multiplication sentence using addition
Repeated addition is a problem solving strategy that helps solve multiplication problems This set of assorted easy to print worksheets helps users revise and review the concept at their own pace Walk
through our printable multiplication as repeated addition worksheets whose prolific exercises help boost the kid s multiplication and
Improved Mathematical Abilities
Consistent technique develops multiplication proficiency, enhancing overall mathematics capacities.
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Word issues in worksheets develop logical reasoning and strategy application.
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Worksheets suit private understanding rates, cultivating a comfortable and versatile understanding atmosphere.
Just How to Produce Engaging Multiplication Is Repeated Addition Worksheets
Incorporating Visuals and Colors Dynamic visuals and shades catch interest, making worksheets visually appealing and involving.
Consisting Of Real-Life Circumstances
Relating multiplication to everyday scenarios adds importance and practicality to exercises.
Customizing Worksheets to Different Ability Degrees Customizing worksheets based on varying effectiveness levels guarantees inclusive discovering. Interactive and Online Multiplication Resources
Digital Multiplication Tools and Games Technology-based sources offer interactive understanding experiences, making multiplication appealing and enjoyable. Interactive Internet Sites and Applications
On the internet platforms provide diverse and easily accessible multiplication technique, supplementing standard worksheets. Tailoring Worksheets for Different Learning Styles Visual Students Visual
aids and diagrams aid comprehension for learners inclined toward aesthetic knowing. Auditory Learners Spoken multiplication issues or mnemonics cater to learners that grasp ideas with acoustic ways.
Kinesthetic Students Hands-on activities and manipulatives sustain kinesthetic students in understanding multiplication. Tips for Effective Implementation in Knowing Consistency in Practice Routine
practice enhances multiplication abilities, promoting retention and fluency. Balancing Repetition and Selection A mix of repeated workouts and varied issue layouts keeps rate of interest and
understanding. Giving Constructive Feedback Responses help in determining locations of enhancement, urging continued development. Obstacles in Multiplication Practice and Solutions Inspiration and
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creating a favorable learning environment is essential. Effect of Multiplication Is Repeated Addition Worksheets on Academic Performance Researches and Research Findings Research shows a favorable
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Repeated Addition Worksheets 3rd Grade In 2020 Repeated addition worksheets Repeated addition
Math Multiplication Worksheets 4th Grade Math Worksheets Math Workbook Array Worksheets
Check more of Multiplication Is Repeated Addition Worksheets below
15 Best Times Tables Resources Repeated Addition Images On Pinterest Times Tables worksheets
Multiplication As Repeated Addition Worksheet Pdf Times Tables Worksheets
Multiplication As Repeated Addition Worksheet Pdf Free Printable
Multiplication Arrays And Repeated Addition Worksheets Free Printable
Multiplication Repeated Addition Grade 1 Worksheets
Multiplication As Repeated Addition Worksheets Free Printable
Repeated Addition Super Teacher Worksheets
Write a multiplication fact and repeated addition number sentence for each illustration 3rd and 4th Grades View PDF Arrays as Repeated Addition Using Arrays Bar Model Worksheet 2 Repeated Addition
and Multiplication This is a slightly easier version of the bar model worksheet above 3rd and 4th Grades View PDF
Browse Printable Math Worksheets Education
These worksheets enable kids to simplify multiplication problems by breaking numbers down into smaller groups Once they are able to simplify numbers into groups students can use repeated addition to
solve multiplication equations This strategy of visualizing numbers as groups is especially useful for kids who find math more challenging
Write a multiplication fact and repeated addition number sentence for each illustration 3rd and 4th Grades View PDF Arrays as Repeated Addition Using Arrays Bar Model Worksheet 2 Repeated Addition
and Multiplication This is a slightly easier version of the bar model worksheet above 3rd and 4th Grades View PDF
These worksheets enable kids to simplify multiplication problems by breaking numbers down into smaller groups Once they are able to simplify numbers into groups students can use repeated addition to
solve multiplication equations This strategy of visualizing numbers as groups is especially useful for kids who find math more challenging
Multiplication Arrays And Repeated Addition Worksheets Free Printable
Multiplication As Repeated Addition Worksheet Pdf Times Tables Worksheets
Multiplication Repeated Addition Grade 1 Worksheets
Multiplication As Repeated Addition Worksheets Free Printable
Multiplication Repeated Addition x4 TMK Education
Worksheets On Repeated Addition For Grade 2 Basic multiplication Worksheets1000 Images About
Worksheets On Repeated Addition For Grade 2 Basic multiplication Worksheets1000 Images About
Multiplication As Repeated Addition Worksheets For Grade 2 Multiplication Worksheets
FAQs (Frequently Asked Questions).
Are Multiplication Is Repeated Addition Worksheets ideal for every age groups?
Yes, worksheets can be customized to various age and skill degrees, making them versatile for different students.
How frequently should trainees practice utilizing Multiplication Is Repeated Addition Worksheets?
Consistent technique is key. Normal sessions, preferably a couple of times a week, can produce considerable enhancement.
Can worksheets alone boost mathematics skills?
Worksheets are an useful tool but ought to be supplemented with diverse learning approaches for detailed skill advancement.
Are there on-line systems offering totally free Multiplication Is Repeated Addition Worksheets?
Yes, numerous educational internet sites use free access to a wide range of Multiplication Is Repeated Addition Worksheets.
Just how can parents sustain their youngsters's multiplication method in the house?
Motivating regular technique, giving support, and creating a favorable understanding environment are useful steps. | {"url":"https://crown-darts.com/en/multiplication-is-repeated-addition-worksheets.html","timestamp":"2024-11-06T11:37:53Z","content_type":"text/html","content_length":"29410","record_id":"<urn:uuid:7a800f40-0418-4b77-abd4-322777bd2e3d>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00806.warc.gz"} |
Megawatt-Hours to Watt-Seconds Conversion (MWh to Ws)
Megawatt-Hours to Watt-Seconds Converter
Enter the energy in megawatt-hours below to convert it to watt-seconds.
Do you want to convert watt-seconds to megawatt-hours?
How to Convert Megawatt-Hours to Watt-Seconds
To convert a measurement in megawatt-hours to a measurement in watt-seconds, multiply the energy by the following conversion ratio: 3,600,000,000 watt-seconds/megawatt-hour.
Since one megawatt-hour is equal to 3,600,000,000 watt-seconds, you can use this simple formula to convert:
watt-seconds = megawatt-hours × 3,600,000,000
The energy in watt-seconds is equal to the energy in megawatt-hours multiplied by 3,600,000,000.
For example,
here's how to convert 5 megawatt-hours to watt-seconds using the formula above.
watt-seconds = (5 MWh × 3,600,000,000) = 18,000,000,000 Ws
How Many Watt-Seconds Are in a Megawatt-Hour?
There are 3,600,000,000 watt-seconds in a megawatt-hour, which is why we use this value in the formula above.
1 MWh = 3,600,000,000 Ws
Megawatt-hours and watt-seconds are both units used to measure energy. Keep reading to learn more about each unit of measure.
What Is a Megawatt-Hour?
A megawatt-hour is a measure of electrical energy equal to one megawatt, or 1,000,000 watts, of power over a one hour period. Megawatt-hours are a measure of electrical work performed over a period
of time, and are often used as a way of measuring energy usage by electric companies.
Megawatt-hours are usually abbreviated as MWh, although the formally adopted expression is MW·h. The abbreviation MW h is also sometimes used. For example, 1 megawatt-hour can be written as 1 MWh, 1
MW·h, or 1 MW h.
In formal expressions, the centered dot (·) or space is used to separate units used to indicate multiplication in an expression and to avoid conflicting prefixes being misinterpreted as a unit
Learn more about megawatt-hours.
What Is a Watt-Second?
The watt-second is a measure of electrical energy equal to one watt of power over a one second period. One watt-second is equal to 1/3,600 of a watt-hour or one joule.
Watt-seconds are usually abbreviated as Ws, although the formally adopted expression is W·s. The abbreviation W s is also sometimes used. For example, 1 watt-second can be written as 1 Ws, 1 W·s, or
1 W s.
Learn more about watt-seconds.
Megawatt-Hour to Watt-Second Conversion Table
Table showing various megawatt-hour
measurements converted to
Megawatt-hours Watt-seconds
0.000000001 MWh 3.6 Ws
0.000000002 MWh 7.2 Ws
0.000000003 MWh 10.8 Ws
0.000000004 MWh 14.4 Ws
0.000000005 MWh 18 Ws
0.000000006 MWh 21.6 Ws
0.000000007 MWh 25.2 Ws
0.000000008 MWh 28.8 Ws
0.000000009 MWh 32.4 Ws
0.0000000001 MWh 0.36 Ws
0.000000001 MWh 3.6 Ws
0.00000001 MWh 36 Ws
0.0000001 MWh 360 Ws
0.000001 MWh 3,600 Ws
0.00001 MWh 36,000 Ws
0.0001 MWh 360,000 Ws
0.001 MWh 3,600,000 Ws
0.01 MWh 36,000,000 Ws
0.1 MWh 360,000,000 Ws
1 MWh 3,600,000,000 Ws
1. Bureau International des Poids et Mesures, The International System of Units (SI), 9th edition, 2019, https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf
More Megawatt-Hour & Watt-Second Conversions | {"url":"https://www.inchcalculator.com/convert/megawatt-hour-to-watt-second/","timestamp":"2024-11-07T18:22:05Z","content_type":"text/html","content_length":"72736","record_id":"<urn:uuid:46a96e34-626d-4363-99fb-a963c9745bf1>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00537.warc.gz"} |
Math, Grade 7, Putting Math to Work, Interpreting Graphs & Diagrams
Volume of the Great Lakes
Work Time
Volume of the Great Lakes
Use the table and Exploring the Great Lakes interactive to answer the following questions.
• How can you use the information from the table and Exploring the Great Lakes interactive to find the volume of each Great Lake? What information could you use from the interactive?
• What is the approximate volume of water in each Great Lake?
• What is the volume of water in all of the Great Lakes combined?
• Could all the water in Lakes Erie, Huron, Michigan, and Ontario fit in Lake Superior if Lake Superior was drained? If so, how much space would not be used? If not, how much water would spill out
of Lake Superior?
Think about the diagram. Which measure of depth would you use to find the volume—the surface elevation, the average depth, or the maximum depth? | {"url":"https://oercommons.org/courseware/lesson/4500/student/?section=5","timestamp":"2024-11-03T00:33:44Z","content_type":"text/html","content_length":"35709","record_id":"<urn:uuid:cc343f02-cd53-4de8-93b0-b7d5d1415cf5>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00559.warc.gz"} |
Univariate, Bivariate, and Multivariate Analysis
Businesses collect vast amounts of data daily, but extracting valuable patterns and insights for informed decision making requires knowledge of exploratory data analysis techniques. In this session,
we will discuss basic techniques based on the nature of the data and the specific requirements.
Exploratory data analysis can be classified as Univariate, Bivariate, and Multivariate. Let’s explore each of these classifications in greater detail.
Key takeaways from the blog
• What is the univariate analysis?
• What are the types of univariate analysis in machine learning?
• What is bivariate analysis?
• What are the types of bivariate analysis?
• What is multivariate analysis?
• What are the methods used for multivariate analysis?
What is Univariate Analysis?
‘Uni’ refers to one, and ‘variate’ means variable, the word univariate refers to the analysis involving a single variable. This type of analysis includes summarization, measurements of dispersion,
and measurements of central tendency. Visualizations, such as histograms, distributions, frequency tables, bar charts, pie charts, and boxplots, are also commonly used in univariate analysis. It is
important to note that the data in univariate analysis must contain only a single variable, which can be either categorical or numeric.
Types of univariate analysis
Let’s dive deeper into the different types of analysis involved in univariate analysis.
Frequency distribution analysis
This analysis is used to analyze continuous numerical data, where we try to extract the statistical summary of the feature.
• Maximum, minimum, and mean (average) analysis: Information like maximum, minimum, and mean values of any numerical data gives us a great impression of how that feature is distributed. Suppose we
are analyzing the age of our customers. We saw that the minimum age of our customers is 18, the maximum age is 26, and the average age is 22. We can extract information that our customers are
• Standard deviation and variance analysis: We have the mean value from the earlier step. To analyze each sample present in the data, we can take the reference of the mean and calculate the
deviation of that sample from it. This is known as standard deviation and is used to estimate the dispersion present in the data. High dispersion means samples are widespread, and low dispersion
means samples are very close to the mean value.
A histogram plots the distribution of a numeric variable as a sequence of bars. Each bar in a histogram covers a range of values called bins. The “total range” of the dataset is divided into a number
of equal parts, known as bins or class intervals. There’s no defined way to find the bins, but generally, we avoid using too many and too few bins. Also, changing the bin size changes the histogram.
The height of the histogram represents the frequency of values falling within the corresponding bin. Let’s implement a histogram to visualize the univariate data:
import seaborn as sns
penguins = sns.load_dataset('penguins')
sns.histplot(data=penguins['flipper_length_mm'], kde=True);
The above histogram displays the distribution of the Penguin’s flipper_length in millimeters. Here, the bin values can be confirmed using the below line.
Most of the Penguin’s flipper lengths are between 183 and 195mm.
Histograms are perfect for exhibiting the general distribution of features. Using the histogram, we can tell whether the distribution is symmetric or skewed (unsymmetric). Additionally, we can
comment on the presence of outliers. Please refer to this blog if you are still getting familiar with symmetric and skewed distributions.
Pie Charts
A Pie Chart is a visualization of univariate data that depicts the data in a circular diagram. Each pie chart slice corresponds to a relative proportion of the category versus the entire group. In
other words, the parts/slice of the graph is proportionate to the fraction of the whole in each category. The pie chart comprises 100% of all categories, while the piece represents the categories
within the data.
import seaborn as sns
import matplotlib.pyplot as plt
fig, ax = plt.subplots()
labels = ['Ocean', 'Land']
color_palette_list = ['#009ACD', '#ADD8E6']
percentages = [70.8, 29.2]
ax.pie(percentages, explode=explode, labels=labels,
colors=color_palette_list[0:2], autopct='%1.0f%%',
shadow=False, startangle=0,
ax.set_title("Land to Ocean Ratio")
ax.legend(bbox_to_anchor=(1, 1));
The above pie chart shows the percentage of earth captured by land and water. Per the pie chart, 29% of the earth is captured by land while 71% is covered with water. Informative and straightforward.
A boxplot or whisker plot is a diagram often used for visualizing the distribution of numeric values. A boxplot divides the data into equal parts using the three quartiles, which serves as an
excellent distribution visualization. A boxplot consists of the lowest value, the first quartile (Lower Quartile), the Second quartile (Median), the Third quartile (Upper Quartile), and finally, the
highest value. A quartile is a statistical term used to describe the division of observations. The mentioned three quartiles divide the data into four equal parts. This can be confirmed using the
illustration given below:
Let's implement a boxplot:
x = np.random.normal(0, 1, 10000)
mean = x.mean()
std = x.std()
q1, median, q3 = np.percentile(x, [25, 50, 75])
iqr = q3 - q1
fig, (ax1, ax2) = plt.subplots(nrows=2, sharex=True, figsize=(13,8))
medianprops = dict(linestyle='-', linewidth=2, color='yellow')
sns.boxplot(x=x, color='#009ACD', saturation=1, medianprops=medianprops,
flierprops={'markerfacecolor': 'mediumseagreen'}, whis=1.5, ax=ax1)
The above box plot is generated from a normal distribution, which is approximately symmetric with respect to the middle yellow line.
The Inter Quartile Range (IQR) represents the middle 50% values. Each quartile-to-end or quartile covers 25% of the data. Hence, IQR is the difference between the third and the first quartile.
IQR = (Third Quartile (Q3)- First Quartile (Q1))
IQR can be used to find the outliers in the data. A detailed approach has been discussed in this blog.
Boxplots can help in visualizing the distribution of data. The image below can distinguish the skewed distributions vs. the normal distribution pattern.
Bar Chart
A bar chart plots the count of categories within a feature as bars. It is only applicable to categorical data. The category level is mentioned over the x-axis, while the frequency of the categories
is mentioned over the y-axis. Each category in the feature will have a corresponding bar value stating the frequency of the class appearing in the feature. Also, the bars are plotted on a baseline
for easy comparison. Let’s implement a bar chart:
import matplotlib.pyplot as plt
fig = plt.figure(figsize=(8, 5))
ax = fig.add_axes([0,0,1,1])
langs = ['Math', 'Science', 'Economics', 'Health Education', 'English']
students = [16, 13, 15, 9, 6]
ax.bar(langs,students, color = '#ADD8E6')
ax.set_title("Subjects taken by Number of Students", fontsize = 15)
plt.xlabel("Subjects", fontsize = 14)
plt.ylabel("Number of Students", fontsize = 14)
What is Bivariate Analysis?
‘Bi’ means two, and ‘variate’ means variable. Collectively, Bivariate analysis refers to the exploratory data analysis between two variables. Now again, the variables can be either numeric or
categorical. Bivariate analysis helps study the relationship between two variables, and if the two are related, we can comment on the strength of the association. Let’s discuss and implement some
basic bivariate EDA techniques:
Types of bivariate analysis
We know the types of data can be either numerical or categorical. So there can be three types of scenarios:
• Numerical feature vs. Numerical feature
• Categorical feature vs. Categorical feature
• Numerical feature vs. Categorical feature
Let’s look at some methods to do the bivariate analysis.
Scatter Plot (Numeric vs. Numeric)
A scatter plot or scatter graph plots data points corresponding to two features. This helps explain the change in one variable with respect to the change in the other. A dot in the scatterplot
represents each row of the dataset. This also helps explain the correlation between two variables, but primarily, scatter plots are used to establish the relationship between two variables.
iris = sns.load_dataset('iris')
sns.scatterplot(data=iris, x='sepal_length', y='petal_length', hue='species')
plt.xlabel('Sepal Length')
plt.ylabel('Petal Length')
The above scatterplot clearly shows the presence of three distinct clusters of different flower species. On the X-axis, we have the Sepal length of the flower, while on the Y-axis, we have the Petal
length. The scatterplot indicates a strong positive correlation between Sepal Length and Petal Length.
How can we comment on the correlation just by looking at the scatterplot? The image below will illustrate how we can comment on the correlation between two variables by looking at the scatterplot.
Correlation varies between -1 to 1. A correlation of positive one indicates a perfect positive linear relationship, while a negative one indicates a perfectly inverse relationship between two
variables. Further, a correlation of zero indicates no connection between the two variables.
Chi-Squared Test(Categorical vs. Categorical)
Chi-Squared Test is used to describe the relationship between categorical variables. It is a hypothesis test developed to test the statistical significance of the relationship between two categorical
variables. It tells us whether the two variables are related or not. It works by calculating the Chi Statistics, which is calculated using the below formula:
X² = Σ ----------
i Ei
Here, O represents the Observed Values, and E represents the Expected Values. This Chi Statistics is calculated and compared with the critical Chi value corresponding to the degrees of freedom © and
decided significance level. In statistics, the degrees of freedom © indicate the number of independent values that can alter an analysis without breaking any restrictions. Finally, a Null Hypothesis
is tested against an alternate hypothesis which is either rejected or accepted based on the difference between chi statistics and critical chi value. Please follow this blog if you’re not aware of
null hypothesis testing.
Analysis of Variance: ANOVA (Continuous vs. Categorical)
ANOVA is a statistical test used to describe the potential differences in a continuous dependent variable by a categorical (Nominal) variable having two or more classes. It splits the observed
variability in the data into two parts:
• Systematic Factors
• Random Factors
Systematic factors statistically influence the data, while random factors don’t add any information. ANOVA can explain the impact of an independent variable over the dependent variable. When there’s
only one dependent variable and one independent variable, it is known as one-way ANOVA.
For instance, we want to find the influence of weekdays over the parameter hotel price. Naturally, the hotel’s price might be lower on weekdays to attract the crowd. Alternatively, on weekends, hotel
prices rise because demand rises. Let’s confirm if the day of the week influences the hotel prices.
import pandas as pd
import numpy as np
import statsmodels.api as sm
from statsmodels.formula.api import ols
df = pd.DataFrame({'weekday': np.repeat(['Weekday', 'Weekend'], 10),
'hotel_price': [96, 94, 89, 105, 110, 100, 102, 98, 91, 104, 122, 114, 119, 115, 122, 109, 111, 106, 107, 113]})
model = ols('hotel_price ~ C(weekday)', data=df).fit()
sm.stats.anova_lm(model, typ=1)
dí sum_sq mean_sq F PR(>F)
C(weekday) 1.0 1110.05 1110.050000 28.853285 0.000042
Residual 18.0 692.50 38.472222 NaN NaN
Now, the P-value for weekdays is 0.000042, which is less than 0.05, which means weekday is highly significant in determining Hotel Price. ANOVA’s result shows that hotel prices are highly influenced
by the day of the week, which is intuitively true.
What is Multivariate Analysis?
‘Multi’ means many, and ‘variate’ means variable. Multivariate analysis is the statistical procedure for analyzing data involving more than two variables. Alternatively, this can be used to analyze
the relationship between dependent and independent variables. Multivariate analysis has various applications in clustering, feature selection, root-cause analysis, hypothesis testing, dimensionality
reduction, etc.
Methods used for multivariate analysis
We can easily correlate the multivariate with the unsupervised learning techniques in machine learning. Unsupervised learning techniques are used to analyze patterns present in the data. The popular
methods associated with it are clustering and dimensionality reduction. Let’s have a look at these techniques.
Clustering Analysis
Clustering analysis segregates the data points into groups known as clusters. The data is grouped into clusters based on the similarity between the multivariate features. This data mining technique
allows us to understand the data distribution based on the available features. Let’s implement the K-means clustering algorithm over the Iris dataset:
We will remove the species column for the demonstration and find the optimum number of clusters using the elbow plot. Here’s a link if you are not familiar with the k-means algorithm. Remember, our
goal is to group similar data points in a cluster, but we need to find the optimum clusters before that. Let’s apply the elbow technique:
iris = sns.load_dataset("iris")
iris.drop(['species'], axis=1, inplace=True)
normalizer = MinMaxScaler().fit(iris)
iris = normalizer.transform(iris)
distortions = []
inertias = []
K = range(1, 10)
for k in K:
kmeans = KMeans(n_clusters=k).fit(iris)
distortions.append(sum(np.min(cdist(iris, kmeans.cluster_centers_,'euclidean'), axis=1)) / iris.shape[0])
plt.plot(K, distortions, 'bx-')
plt.xlabel('Number of Clusters', fontsize = 13)
plt.ylabel('Distortion or SSE', fontsize = 13)
plt.title('SSE vs Number of Clusters - Elbow Plot', fontsize = 13)
The Elbow appears at k = 3; hence, it will be the optimum number of clusters for the K-means algorithm.
kmeans = KMeans(n_clusters=3)
iris['clusters'] = kmeans.fit_predict(iris)
iris.columns = ['sepal_length', 'sepal_width', 'petal_length', 'petal_width', 'clusters']
plt.xlabel("Sepal Length", fontsize=14)
plt.ylabel("Petal Length", fontsize=14)
From the above plot, we can visualize the three clusters. We have successfully grouped similar data points.
Principal Component Analysis (PCA)
PCA is a dimensionality reduction technique frequently used to reduce the dimensions of large datasets that exhibit multicollinearity. In PCA, the original data is transformed into a new set of
features such that fewer transformed features explain the variance of the original dataset. This comes at a minimal loss of information. For a deep understanding of PCA, visit this blog.
Let’s implement PCA on the credit card dataset:
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn.decomposition import PCA
transaction_data = pd.read_csv('creditcard.csv')
transaction_data.drop("Time", axis=1, inplace=True)
transaction_feature = transaction_data.iloc[:,:-2]
This dataset contains 28 features, and we aim to reduce the number of features.
pca = PCA()
transaction_feature = pca.fit_transform(transaction_feature)
explained_variance = pca.explained_variance_ratio_
[12.48375707 8.87294517 7.48093391 6.52314765 6.19904486 5.77559233
4.97985207 4.64169566 3.92749719 3.85786696 3.39014785 3.24875815
3.22327116 2.99007578 2.72617319 2.49844761 2.34731555 2.2860303
2.15627103 1.93390711 1.7555909 1.71367096 1.26888126 1.19357733
0.88419944 0.75668372 0.53013145 0.354534361]
The initial 17 principal components contribute to 85% of the original data variance. Let’s also visualize this using the Scree plot:
PC_values = np.arange(pca.n_components_) + 1
plt.plot(PC_values, pca.explained_variance_ratio_, 'o-', linewidth=2, color='blue')
plt.axhline(y=0.023, color='r', linestyle=' - ')
plt.title('Scree Plot', fontsize=15)
plt.xlabel('Principal Component', fontsize=14)
plt.ylabel('Variance Explained', fontsize=14)
pca = PCA(n_components=17)
reduced_features = pca.fit_transform(transaction_feature)
reduced_features = pd.DataFrame(reduced_features)
## (284807, 17)
Finally, we have only 17 features in the final dataset at the cost of a 15% variance loss.
Multiple Correspondance Analysis (MCA)
Correspondence Analysis is a powerful data visualization technique frequently utilized for visualizing the relationship between categories. This is applicable when data is multinomial categorical and
highly used in surveys and questionnaires for association mining.
MCA works by separating the respondents based on their categories. For instance, respondents or individuals falling into the same categories are plotted next to each other, while respondents in
different categories are plotted as far as possible. This will form a cluster of similar respondents or individuals, which can be visualized in a plot. Also, this is a distance-based approach.
Advantages of using Multiple Correspondance Analysis (MCA)
• Explains how categorical features are associated with each other.
• Explains whether individuals or respondents shares similarity with the categorical variables.
• Provides visualization explaining the association between categories.
When do we use MCA?
• When there are no missing values or negative values in the dataset.
• All the data has the same scale.
• Data must contain at least two columns.
• When the dataset contains categorical features.
Let’s implement Multiple Correspondance Analysis:
import pandas as pd
import prince
import numpy as np
X = pd.read_csv("HarperCPC.csv")
Unnamed: 0 name membership abbr
0 INAN.1 Indigenous and Northern Affairs C Warkentin INAN
1 INAN.2 Indigenous and Northern Affairs J Crowder INAN
2 INAN.3 Indigenous and Northern Affairs C Bennett INAN
3 INAN.4 Indigenous and Northern Affairs § Ambler INAN
4 INAN.5 Indigenous and Northern Affairs D Bevington INAN
mca = prince.MCA()
mca_data = mca.fit(X)
mca_X = mca_data.transform(X)
ax = mca.plot_coordinates(
figsize=(6, 6),
Possible Interview Questions
These are some popular questions asked on this topic:
• What is the difference between univariate, bivariate, and multivariate analysis?
• What are the types of univariate, bivariate, and multivariate analysis?
• Explain the ANOVA technique and the category for which it is used.
• Explain Multiple Correspondance Analysis (MCA).
• How does the correlation between features represent the relationship between two features?
In this session, we briefly discussed the different methods used for data analysis, namely the Univariate, Bivariate, and Multivariate analysis techniques. These are classified based on the number of
variables involved in the analysis. Under each analysis, we discussed some methods used to analyze the data and implemented them in python under each analysis. Choosing the correct way for the
analysis depends on the data we are handling and the number of variables involved in the analysis. We haven’t covered more strategies in this session, but knowing the above techniques is essential
for any data analyst.
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