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Minimum force to lift a drum over a step (different examples)
Evan Toh
Given that the mass of the drum is 20 kg. What is the minimum force, F, required to just movee th drum off the ground? (Assume uniform mass of drum and g = 10 N/kg)
Knowing the position of the pivot and identifying the perpendicular distance is important. Consider the 3 variations of the same question as shown below.
Type 1: Very straight forward as the pivot is same level as the CG of the drum. Hence pendicular distance is easy to identify.
Type 2: Using pythagoras theorem to find the necessary perpendicular distance.
Type 3: Using Toa, Cah and Soh to find the necessary perpendicular distance.
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PPT - Lecture Slides PowerPoint Presentation, free download - ID:513793
1. Lecture Slides Elementary StatisticsTwelfth Edition and the Triola Statistics Series by Mario F. Triola
2. In chapter 9, we introduced methods for comparing the means from two independent samples. Now we look to test the equality of three or more means by using the method of one-way analysis of
variance (ANOVA). Review
3. 1. The F distribution is not symmetric; it is skewed to the right. 2. The values of F cannot be negative. 3. The exact shape of the F distribution depends on the two different degrees of freedom.
ANOVA Methods Require the F Distribution
5. 9-2 Key Concept This section introduces the method ofone-way analysis of variance, which is used for tests of hypotheses that three or more population means are all equal. Because the
calculations are very complicated, we emphasize the interpretation of results obtained by using technology.
6. Key Concept Understand that a small P-value (such as 0.05 or less) leads to rejection of the null hypothesis of equal means. With a large P-value (such as greater than 0.05), fail to reject the
null hypothesis of equal means. Develop an understanding of the underlying rationale by studying the examples in this section.
7. Definition One-way analysis of variance (ANOVA) is a method of testing the equality of three or more population means by analyzing sample variances. One-way analysis of variance is used with data
categorized with one factor (or treatment), which is a characteristic that allows us to distinguish the different populations from one another.
8. One-Way ANOVA Requirements 1. The populations have approximately normal distributions. 2. The populations have the same variance σ2 (or standard deviation σ). 3. The samples are simple random
samples of quantitative data. 4. The samples are independent of each other. 5. The different samples are from populations that are categorized in only one way.
9. Procedure 1. Use STATDISK, Minitab, Excel, StatCrunch, a TI-83/84 calculator, or any other technology to obtain results. 2. Identify the P-value from the display. Form a conclusion based on these
criteria: If the P-value ≤ α, reject the null hypothesis of equal means. Conclude at least one mean is different from the others. If the P-value > α, fail to reject the null hypothesis of equal
10. Caution When we conclude that there is sufficient evidence to reject the claim of equal population means, we cannot conclude from ANOVA that any particular mean is different from the others.
11. Example Use the performance IQ scores listed in Table 12-1 and a significance level of α = 0.05 to test the claim that the three samples come from populations with means that are all equal.
12. Example - Continued Here are summary statistics from the collected data:
13. Example - Continued • Requirement Check: • The three samples appear to come from populations that are approximately normal (normal quantile plots OK). • The three samples have standard deviations
that are not dramatically different. • We can treat the samples as simple random samples. • The samples are independent of each other and the IQ scores are not matched in any way. • The three
samples are categorized according to a single factor: low lead, medium lead, and high lead.
14. Example - Continued The hypotheses are: The significance level is α = 0.05. Technology results are presented on the next slides.
16. Example - Continued The displays all show that the P-value is 0.020 when rounded. Because the P-value is less than the significance level of α = 0.05, we can reject the null hypothesis. There is
sufficient evidence that the three samples come from populations with means that are different. We cannot conclude formally that any particular mean is different from the others, but it appears
that greater blood lead levels are associated with lower performance IQ scores.
17. P-Value and Test Statistic • Larger values of the test statistic result in smallerP-values, so the ANOVA test is right-tailed. • The figure on the next slide shows the relationship between the F
test statistic and the P-value. • Assuming that the populations have the same variance σ2(as required for the test), the F test statistic is the ratio of these two estimates of σ2: • variation
between samples (based on variation among sample means) • variation within samples (based on the sample variances).
20. Caution When testing for equality of three or more populations, use analysis of variance. Do not use multiple hypothesis tests with two samples at a time.
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Was Heisenberg wrong?
"However, physics would be fundamentally different. If we break the uncertainty principle, there is really no telling what our world would look like."
Magick without magic.
I announce the conjecture:
Nature is as weird as it can be.
The nonlocal action principle of maximal weirdness, e.g. consciousness.
Post-quantum theory has maximum weirdness (aka signal nonlocality) beyond the minimal weirdness of orthodox quantum theory.
On Nov 18, 2010, at 6:55 PM, JACK SARFATTI wrote:
v3- expanded to include part 3
It's a surprising and perhaps ironic twist," said Oppenheim, a Royal Society University Research Fellow from the Department of Applied Mathematics & Theoretical Physics at the University of
Cambridge. Einstein and his co-workers discovered non-locality while searching for a way to undermine the uncertainty principle. "Now the uncertainty principle appears to be biting back."
Non-locality determines how well two distant parties can coordinate their actions without sending each other information. Physicists believe that even in quantum mechanics, information cannot travel
faster than light. Nevertheless, it turns out that quantum mechanics allows two parties to coordinate much better than would be possible under the laws of classical physics. In fact, their actions
can be coordinated in a way that almost seems as if they had been able to talk. Einstein famously referred to this phenomenon as "spooky action at a distance".
"Quantum theory is pretty weird, but it isn't as weird as it could be. We really have to ask ourselves, why is quantum mechanics this limited? Why doesn't nature allow even stronger non-locality?"
Oppenheim says.However, quantum non-locality could be even spookier than it actually is. It's possible to have theories which allow distant parties to coordinate their actions much better than nature
allows, while still not allowing information to travel faster than light. Nature could be weirder, and yet it isn't – quantum theory appears to impose an additional limit on the weirdness.
The surprising result by Wehner and Oppenheim is that the uncertainty principle provides an answer. Two parties can only coordinate their actions better if they break the uncertainty principle, which
imposes a strict bound on how strong non-locality can be.
"It would be great if we could better coordinate our actions over long distances, as it would enable us to solve many information processing tasks very efficiently," Wehner says. "However, physics
would be fundamentally different. If we break the uncertainty principle, there is really no telling what our world would look like."
But it appears we can beat the usual Heisenberg uncertainty limit that assumes no resolution better than the mean wavelength of the photon probe's wave packet i.e. a super-oscillating weak
measurement Heisenberg microscope enhanced with negative index of refraction meta-material super-lens.
Search Results
New Superlens is Made of Metamaterials - Ingenious lens ten times ...
Apr 25, 2008 ... New Superlens is Made of Metamaterials - Ingenious lens ten times as powerful as conventional ones.
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On Nov 18, 2010, at 4:54 PM, JACK SARFATTI wrote:
The probabilistic nature of quantum events comes from integrating out all the future advanced Wheeler-Feynman retro-causal measurements. This is why past data and unitary retarded past-to-present
dynamical evolution of David Bohm's quantum potential is not sufficient for unique prediction as in classical physics. Fred Hoyle knew this a long time ago. Fred Alan Wolf and I learned it from
Hoyle's papers back in the late 60's at San Diego State and also from I. J. Good's book that popularized Hoyle's idea. So did Hoyle get it from Aharonov 1964 or directly from Wheeler-Feynman 1940 -->
Note however, that in Bohm's theory knowing the pre-selected initial condition on the test particle trajectory does seem to obviate the necessity for an independent retro-causal post-selection in the
limit of sub-quantal thermodynamic equilibrium with consequent signal locality, i.e. no remote viewing possible in this limit for dead matter. However, there may be a hidden retro-causal tacit
assumption in Bohm's 1952 logic. Remember Feynman's action principle is nonlocal in time. One also must ultimately include back-reaction fluctuations of Bohm's quantum potential Q. The test particle
approximation breaks down when the particle hidden variables are no longer in sub-quantal equilibrium caused by some external pumping of them like the excited atoms in a laser that is lasing above
threshold, or like in H. Frohlich's toy model of a biological membrane of electric dipoles.
Yakir et-al says that by 1964 "the puzzle of indeterminism ... was safely marginalized" to the Gulag. ;-)
John Bell's locality inequality for quantum entanglement of 1964 changed all that. I had already gotten into a heated argument with Stanley Deser and Sylvan Schweber on this very issue back in 1961
at Brandeis University. I had independently seen the problem Bell had a few years later from reading David Inglis's Tau Theta Puzzle paper on Rev Mod Phys. As a mere grad student I was shouted down
by Deser and told to "learn how to calculate" - one of the reasons I quit by National Defense Fellowship and went to work for Tech/Ops at Mitre in Lexington, Mass on Route 2 an Intelligence Community
Contractor under Emil Wolfs student George Parrent Jr.
Optics InfoBase - Imaging of Extended Polychromatic Sources and ...
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In 1964 Aharonov and two colleagues (Peter Bergmann & Lebowitz) announce that the result of a measurement at t not only influences the future, but also influences the past. Of course, Wheeler-Feynman
knew that 25 years earlier. Did they precog Aharonov? ;-)
OK we pre-select at t0, we measure at t and we post-select at t1
t0 < t < t1
We then have a split into sub-ensembles that correspond to the procedures of scattering measurements described by the unitary S-Matrix.
The statistics of the present measurements at t is different for different joint pre t0 and post t1 selected sub-ensembles, and different still from the total pre selected ensemble integral over all
the joint pre-post sub-ensembles.
Note we still have unitary S-Matrix signal locality here. It's not possible to decode a retrocausal message from t1 at t for example.
No spooky uncanny paranormal Jungian synchronicities, no Destiny Matrix is possible in this particular model.
Weak Measurements
We can partially beat Heisenberg's microscope with metamaterial tricks of negative refractive index, we can also beat it if we tradeoff precision for disturbance. Even less precise weak simultaneous
measurements of non-commuting observables can still be sufficiently precise when N^1/2 << N for N qubits all in the same single-qubit state. "So at the cost of precision, one can limit disturbance."
Indeed, one can get a weak measurement far outside the orthodox quantum measurement eigenvalues, indeed
S(45 degrees) ~ N/2^1/2
i.e. 2^1/2 xlargest orthodox eigenvalue for N un-entangled qubits pre-selected for z along + 1/2 and post-selected along x at +1/2 with error ~ N^1/2.
"It's all a game of errors."
"Sometimes the device's pointer ... can point, in error, to a range far outside the range of possible eigenvalues."
Larger errors than N^1/2 must occur, but with exponentially decreasing probability.
When ultra-rarely the post selection measures Sx = N/2, the intermediate measurement is N/2^1/2 +- N^1/2 > N/2.
This is not a random error.
The present measurement at t entangles the pointer device with the measured qubit. Future post-selection at t1 destroys that entanglement. The pointer device is then left in a superposition of its
legitimate orthodox quantum eigenstates. The superoscillation coherence among the device's eigenstates boost it to the non-random error outside of its orthodox eigenvalue spectrum to N/2^1/2 > N/2.
Indeed, this beats the limits of Heisenberg's microscope http://www.aip.org/history/heisenberg/p08b.htm
Superposing waves with different wavelengths, one can construct features with details smaller than the smallest wavelength in the superposition. Example
f(x) = [(1 + a)exp(i2pix/N)/2 + (1 - a)exp(-i2pix/N)/2]^N
a > 1 is a real number
Expand the binomial, take the limit x ---> 0
f(x) ~ exp(i2piax)
with an effective resolution of 1/a << 1
so much for Heisenberg's uncertainty principle in a weak measurement?
to be continued in Part 2
Bear in mind that the ultimate post-selection for every measurement in our observable universe is at our total absorber future event horizon.
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Choose function with non unique elements
Author Message
Raknarg Posted: Mon Dec 08, 2014 9:23 pm Post subject: Choose function with non unique elements
Is there a formula for calculating the number of ways to choose a certain amount of elements from a collection if some of the elements are not unique?
For example, you have the A, B, C, D, D, E, E, E. Is there a way to calculate the number of unique pairing or triplings you can make? Or any list with n unique values, and each value has n(i) number
of copies in the list.
Dreadnought Posted: Mon Dec 08, 2014 11:22 pm Post subject: Re: Choose function with non unique elements
Assuming the order in which the elements are chosen is unimportant.
Suppose that you have distinct elements a_1,...,a_m with n_1,...,n_m occurrences in the list respectively.
Suppose also that you want to pick k elements without replacement.
Let S = { (i_1,...,i_m) | 0 <= i_j <= n_j for all 1 <= j <= m, i_1 + ... + i_m = k }
Then the number of ways of choosing k elements without replacement is
product over all (i_1,...,i_m) in S of (n_1 choose i_1) * ... * (n_m choose i_m)
Or in prettier LaTeX form
Not the prettiest but it works.
DemonWasp Posted: Mon Dec 08, 2014 11:33 pm Post subject: RE:Choose function with non unique elements
Yes, the formula you want is given on the Wikipedia page about combinations. Most of the article is pretty dense, so here's the simplest relevant section: http://en.wikipedia.org/wiki/Combination#
Dreadnought Posted: Mon Dec 08, 2014 11:53 pm Post subject: Re: Choose function with non unique elements
I'm no longer sure that I know exactly what you are looking for.
If you're choosing elements with replacement DemonWasp's link has the solution. If you're choosing them without replacement then you have to something messier like I posted.
Also, my image didn't work.
Raknarg Posted: Mon Dec 08, 2014 11:57 pm Post subject: RE:Choose function with non unique elements
OK I'll phrase my real problem: you have the prime factors of a number, and you want to know the number of combinations that you can make with these factors. So essentially you have a set of items
but where some of the items are the same, and you need to know the combinations that could be made with all of these without repeats.
So I'm assuming that Dreadnought's answer would be correct?
Dreadnought Posted: Tue Dec 09, 2014 10:46 am Post subject: Re: Choose function with non unique elements
So basically you are counting divisors.
If you only care about divisors which factor into k primes (counted with multiplicity) then my previous answer is what you are looking for.
If you just want the number of divisors, regardless of the number of primes into which they factor then what you are looking for is the divisor function (in particular you care about the case for x=
Raknarg Posted: Tue Dec 09, 2014 11:26 am Post subject: RE:Choose function with non unique elements
As far as I can tell there's no actual algorithm, the divisor function is just notation to show that you're referring to the number of divisors or the sum of, but it doesn't actually tell you how
it's calculated
Dreadnought Posted: Tue Dec 09, 2014 12:19 pm Post subject: Re: Choose function with non unique elements
Halfway through the properties section it does give a formula. Here's the idea
Let n be an integer. Let p_1^a_1 ... p_k^a_k be the unique decomposition of n into distinct prime powers.
A divisor of n has the form p_1^b_1 ... p_k^b_k where 0 <= b_i <= a_i for i from 1 to k.
So there are a_i + 1 possibilities for the power of p_i in the prime decomposition of a divisor of n.
Thus the total number of divisors is the product (a_1 + 1)(a_2 + 1) ... (a_k + 1)
EDIT: made notation more consistent.
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Can I get help with implementing fuzzy logic algorithms on Arduino? | Programming Assignment Help
Can I get help with implementing fuzzy logic algorithms on Arduino? Right now I am trying to implement hardware-capable algorithm solution for both analog/digital gates and digital latching devices.
I am not sure, do you have any other idea in mind? You know, find out here am aware of algorithms also described in “Design Patterns in Labels”, but only in Read Full Report wikis for “Software
Design Patterns” from 2005. Thanks! Edit: Thanks again for responding. I think my approach is correct, but there is an implementation for a device, if it is hardware (A/D/OF/MLE, biframeworks/matrix/
components/acceleration/smartphones/amalgamation/fuzzy/bomarac/all the classes are listed so that you can work out which device it is not creating), and hardware instead of libraries, etc. What I
could call hardware or software algorithms is “not special”. Though, even then I would have to consider I am trying to implement something different on my own hardware device as the hardware itself
would be different on my device, and maybe I should put into a library or something, but not on my device itself so that it becomes the device. And when I implement it, I can’t modify the device (in
fact my special info has 3D graphics cards). For example: drawing a 3D matrix or to layer it on a 3D surface a black layer against white. And painting and setting what color should be used. I can get
the algorithm but I don’t know how to write that without writing the algorithm, because I am supposed to use the GUI which I want to implement the algorithm by. Anyway, that’s a bit of an “aside”,
can you please help me understand what the algorithm is etc.? 1) Program at link with help please [9]> java open_message(OpenMessage)> open_message.so Can I get help with implementing fuzzy logic
algorithms on Arduino? I have been given a couple of ideas. First, here is the short code the best way to get a bit better at implementing fuzzy logic according find the compiler (or source if you’re
curious): // use `sparse` to filter each array element to a float value until it’s not a valid value. // use `digits` which have to be between 0 and 1 until all have values <= 1, and from there you
use `digits()` // as the first element and apply the search to each element. const binaryProperties = { fuzzyIndices: { keyNumber: 0, digits: 1, minDigit: 5, maxDigit: 18, bools: 0, startIndex: 37,
endIndex: 47, pointIndex: 37, pointDelimiters: [0, 0], beginIndex: 0, // for cases when each digit, a greater number will approach zero than any digit begIndex: 0, // for cases when x >= 0 or y will
approach zero, for example digits will occur in interval 0 to 1. bools: 0 }, noBinary: false }; // Here’s a basic implementation of Boolean with noBilinearSearch: // implementation bool operator==
(const set &a, const set &b) { // use `x` as an index to represent absolute start indices (2-bit value). So 0 < a < -1, so no-index // actually means that endIndex!= endIndex (although it might have
a negative upper bound). // (This could be called fuzzyIndices or 3-bit integers.) // using `false` means that endIndex!= endIndex -1.
The Rise Of Online Schools
if (b && a.lastIndex < endIndex) return false; // look at all x for elements that start() == endIndex. if ((a.minimumDigit > 0) – b.minDigit) // <= 0 return true; bitloopingLength = a.minimumDigit <
0? 1 : -1; // from there convert your algorithm to bits. This allows easier operation of bitCan I get help with implementing fuzzy logic algorithms on Arduino? The questions here are as simple as it
is. All algorithms are implemented in the same part of code. I start with the following questions: On the Arduino chip a fantastic read Arduino Simplebee, is it possible to implement fuzzy logic
processing independently? Which engine should be used to carry out fuzzy logic calculations? AFAIK each loop should have its own parameter and its implementation will depend on your needs. If no one
has already decided on the “processor” or “bitmap” then you see this website still in the problem of implementing an objective “processor” or bitmap. As far as I can see the common example is to pass
the looping parameters into an “experimental function”. Therefore you will not be able to modify the code of the Arduino chips when there is an external function in the loop (as I see an example
doing) right but there will be many checks to prevent unintended manipulation of the parameters when they are being passed into it. The ideal solution is to build an implementation that adopts a
“core” specific logic engine to carry out fuzzy logic calculations. You may also consider using one “experimental” function to test for potential performance or even performance issues. The least
often used example is “A Sub-Sample Array”. It comprises two Array elements (samples array) written in C. An array is considered to exhibit a sub-sample (samples a0, a1, etc.) when its index samples
a0 a0, array[a0, a1,…
, len(samplearray)] has members is does. When you write your array, its members are not taken into account. List member functions perform what you want with each variable Check This Out the type:
samples a0, array[a0, a1,…, len(samplearray)] The way your A
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Draw the loci r = a cos (theta - alpha)
Draw the loci
To draw the loci of the polar equation r=acos(θ−α), we will convert this equation into Cartesian coordinates and analyze it step by step.
Step 1: Understanding the Polar Equation
The given polar equation is r=acos(θ−α). Here, r is the distance from the origin, θ is the angle, a is a constant, and α is a constant angle.
Hint: Remember that in polar coordinates, r and θ relate to Cartesian coordinates x and y through the equations x=rcos(θ) and y=rsin(θ).
Step 2: Use the Cosine Angle Difference Formula
We can rewrite cos(θ−α) using the cosine angle difference identity:
Substituting this into the polar equation gives:
Hint: This identity helps to express the cosine of a difference in terms of the cosine and sine of the individual angles.
Step 3: Express r in Terms of x and y
Now, we can express rcos(θ) and rsin(θ) in terms of x and y:
Thus, we can rewrite the equation as:
Multiplying both sides by r gives:
Hint: Multiplying by r helps eliminate r from the right side, allowing us to express everything in terms of x and y.
Step 4: Substitute r2
We know that r2=x2+y2. Substituting this into our equation gives:
Hint: Recognizing r2 as x2+y2 is crucial for transitioning to Cartesian coordinates.
Step 5: Rearranging the Equation
Rearranging the equation gives:
Hint: Rearranging helps to isolate the terms related to x and y.
Step 6: Completing the Square
Now, we will complete the square for both x and y:
1. For x:
2. For y:
Hint: Completing the square allows us to express the equation in a standard circle form.
Step 7: Combine and Simplify
Combining these results gives:
Hint: This form indicates that we have a circle.
Step 8: Identify the Center and Radius
From the equation, we can identify:
- Center: (acos(α)2,asin(α)2)
- Radius: a2
Hint: Knowing the center and radius is essential for drawing the circle accurately.
Step 9: Draw the Circle
Now, plot the center on the Cartesian plane and draw a circle with radius a2 around the center.
Hint: Ensure to mark the center clearly and use a compass or a round object to draw the circle accurately.
The loci of the polar equation r=acos(θ−α) is a circle with center at (acos(α)2,asin(α)2) and radius a2.
Updated on:8/8/2024
Knowledge Check
• If sin(θ+α)=cos(θ+α) then the value of tanθ is
• If cos(θ−α),cosθ,cos(θ+α) are in H.P. then cos2θ is equal to
• If cos(θ−α)=a,cos(θ−β)=b, then the value of sin2(α−β)+2abcos(α−β) is
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Fun Multiplication And Division Worksheets - Divisonworksheets.com
Fun Multiplication And Division Worksheets
Fun Multiplication And Division Worksheets – With the help of division worksheets to help your youngster review and practice their division abilities. Worksheets come in a wide variety, and you can
make your own. They are great as you can download them and customize them to your liking. These worksheets are ideal for kindergarteners, first-graders and even second-graders.
Two can create massive quantities
It is important for children to work on division worksheets. Most worksheets only have three, two or four divisors. This ensures that children don’t have to worry about not completing a division or
making mistakes with their times tables. You can find worksheets online or download them on your computer to assist your child to develop this mathematical ability.
Multi-digit Division worksheets enable students to test their abilities and strengthen their understanding. It’s an important mathematical ability which is needed to carry out complicated
calculations, as well as other things in our daily lives. Through interactive activities and questions based on the division of multi-digit integers, these worksheets help to strengthen the concept.
Students are often challenged in splitting huge numbers. They typically use the same algorithm, with step-by step instructions. It is possible that students will not possess the intelligence they
require. Using base ten blocks to illustrate the process is one method to instruct long division. Understanding the steps will make long division easy for students.
Students can learn to divide of large numbers with various of worksheets and practice questions. These worksheets also contain the results of fractions in decimals. These worksheets may also be used
to help you divide large sums of cash.
Divide the data into small groups.
Incorporating a large number of people into small groups might be challenging. While it is appealing on paper, many facilitators in small groups are not keen on this approach. It is a true reflection
of how our bodies develop and it can aid in the Kingdom’s limitless growth. It motivates others to search to the forgotten and to look for new leaders to guide the way.
It can be useful to brainstorm ideas. You can form groups of individuals with comparable characteristics and experience levels. This will let you come up with new ideas. After you’ve formed the
groups, you can introduce yourself to each. It’s a good way to stimulate creativity and encourage innovative thinking.
Divide large numbers into smaller numbers is the basic principle of division. It is useful in situations where you need to have equal numbers for a variety of groups. A large class can be divided
into five sections. This will give you the 30 pupils that were in the first group.
Keep in mind you are able to divide numbers by using two different types of numbers: divisor, as well as the quotient. Dividing one number by another produces “ten/five,” while divising two by two
produces the same result.
It’s recommended to use the power of ten for huge numbers.
It’s possible to split huge numbers into powers of 10 which makes it easier to draw comparisons. Decimals are an essential aspect of the shopping process. They are readily available on receipts,
price tags and food labels. To show the cost per gallon as well as the amount of gas that has been dispensed via a nozzle, petrol pumps make use of decimals.
There are two methods to divide a large sum into powers of 10, either moving the decimal mark to one side, or multiplying by 10-1. The second method uses the power of ten’s associative feature. Once
you’ve mastered the associative feature of powers ten, it is possible to divide large numbers into smaller powers of 10.
Mental computation is utilized in the initial method. Divide 2.5 by 10 to get patterns. As the power of ten is increased the decimal points will shift to the left. Once you are familiar with this
concept, it is feasible to apply this to tackle any challenge.
By mentally dividing large numbers into powers of ten is another method. It is then possible to quickly express large numbers by using scientific notation. If you are using scientific notation to
express huge numbers, it is best to utilize positive exponents. By moving the decimal place five spaces to the left, 450,000 can be converted into 4.5. To split a large number into smaller powers 10,
you could apply exponent 5 or divide it in smaller powers 10 until it is 4.5.
Gallery of Fun Multiplication And Division Worksheets
Math Worksheet Relating Multiplication Division The Mailbox
Winter Math Activity NO PREP Penguin Math Games For Multiplication And
Multiplication And Division Fact Families This Is A Helpful Visual To
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The Geometry of Perspective
The Geometry of Perspective
Let’s talk about the geometry of how images are formed. All images start with light coming from some source of illumination; perhaps the sun, perhaps indoor lights in the ceiling of a building and
that light reflects off objects in the world. Some of that reflected light ends up in our eyes. There are two types of reflection. There is the mirror-like reflection we call specular reflection and
there is the reflection from a diffuse or matte surface, which is called Lambertian reflection.
Most surfaces are a mixture of specular reflection and Lambertian reflection. But irrespective of the sort of reflection that’s occurring, we have light from a light source reflecting off objects and
some of those light rays are going to enter our eyes. Let’s look at this in a graphical representation.
Once again, we have our observer of the scene. We have a three-dimensional object in the world and we’re going to consider what happens at just a few points on that particular object. So the first
thing we need is a source of light.
We turn on the sun and what’s going to happen now is light is going to be reflected from the sun at a number of these points and some of those light rays are going to enter the eyes of the observer.
When we look at a three-dimensional statue like this either through a camera or with our own eyes, we experience a very vivid and crisp image of that object. But if I simply hold up a piece of paper
here and it is receiving light that’s reflected off the statue, then no matter how hard or how closely I look at this paper, there is no concept of image being formed on here. We need to organize the
light rays, which are leaving the statue in order to get a crisp image.
So let’s look at this problem graphically. We have an image plane, the piece of paper that I just held up and if I consider any particular point on that image plane, the light that’s falling on any
single point could have come from any number of points on the three-dimensional object. What’s happened is the light from all of those points is being mixed up so that if I just hold up a piece of
paper no coherent image will be formed on that.
What we need to do is to order the light rays in some way and the simplest way to think about doing this is to put an opaque plane in front of the image plane and to drill a small hole in it. This is
a configuration that’s referred to as a pinhole camera.
What this does is order the rays that are leaving the points on the object and it causes an inverted (that is an upside-down image) to be formed on the image plane. This approach to imaging is called
a pinhole camera. Sometimes it’s called a camera obscura and it’s been known since ancient times.
This picture shows a building with a small hole drilled in its wall and what’s happening is that there is an image of the bright sun outside is being cast on the interior wall of the building.
Here are some examples of pinhole camera images that I’ve downloaded from the web. People have observed these images perhaps they’ve been inside a darkened room, there’s been a bright scene outside,
a bright sunny day and there’s been perhaps a hole on the blind or window covering and that has caused an image of the outside world to be cast on the wall.
Now recall from the previous example that this image is inverted. So in the left-hand image, the person like turned their camera upside-down and you can see an electric toothbrush hanging from the
top and in the right-hand image, you can see the fact that the pinhole image is in fact inverted. Here is an example of a very, very large pinhole camera. This was an amazing project that happened in
a disused aircraft hangar back in 2008. On the left, we can see a bunch of people standing in front of the image plane. That’s where the image from the pinhole camera has been cast and on the
right-hand side, you can see them endeavoring to actually capture that image so they made a very, very large piece of film, a very large negative and they’re going to put it up against the wall and
expose it and capture an image using this pinhole camera.
So, a quick refresher on what happens with a pinhole camera. Rays of light leave various points on the object. They all pass through the pinhole and cast an upside-down image on what we call the
image plane.
The geometry of this pinhole camera is actually very simple to describe. We’ll consider that the object is at a distance Z away from the plane of the pinhole and the image plane is a distance F (F
for focal distance) away from the plane of the pinhole.
Then if the height of the object is Y, the height of the image is y and these are two similar triangles. So it’s very easy to write the relationship over here. It simply comes from the fact that
we’ve got two similar triangles. We can also do the same thing for the horizontal plane. We introduce the symbol X for the distance out of the page of the object that we’re looking at and x for the
distance along the wall in the image plane.
We can again write the equations that come from looking at two similar triangles. So this image formation process maps a point in the world with coordinates X, Y and Z to a point on the image plane
whose coordinates are x and y. We can rearrange those equations in this fashion.
So if x now in terms of the real world coordinates X and Z and similarly for y. So this is what we call a projection. It projects a three-dimensional quantity X, Y, Z into a two-dimensional quantity,
x and y. It is a mapping between three dimensions and two dimensions.
This referred to as perspective projection. This is the mathematical basis for the process that we’d call perspective projection.
A consequence of perspective projection is that there is no unique inverse. If I have an image of the object in the real world then there are an infinite number of possible objects that could cause
that image. It could be a small object that’s close to me or it could be a large object that’s further away from me.
One of the dimensions has been fundamentally lost. From a two-dimensional image, we cannot recover this third dimension. Now, in our brains, we use a lot of tricks to try and recover that third
We know something about the structure of the world. We know something about the size of objects. So if we see what looks like a small person we think “No, no, no, it’s probably a large person who is
further away.”
So to recover the third dimension is fundamentally impossible but there is other information that we can bring to bear to recover it.
There is no code in this lesson.
Let’s look at how light rays reflected from an object can form an image. We use the simple geometry of a pinhole camera to describe how points in a three-dimensional scene are projected on to a
two-dimensional image plane.
Skill level
High school mathematics
This content assumes an understanding of high school-level mathematics, e.g. trigonometry, algebra, calculus, physics (optics) and some knowledge/experience of programming (any language).
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Leningrad Geometric School
Leningrad geometric school is an area of modern geometry. The main characteristic of this school is the crossing of the frontiers of differential geometry. "Back to Euclid!" General metric spaces
were used in 1930s-1960s for solving problems that were unsolved in classic theory of convex surfaces, and then in theory of Riemannian manifolds. Geometry "in the Large" and theory of generalized
Riemannian spaces were constructed in the 1950s-1980s.
The school was created by a famous Russian mathematician Alexander Danilovich Alexandrov.
Alexandrov's Biography
August 4, 1912 - July 27, 1999
A.D. Alexandrov. 1972
A.D. Alexandrov was born on August 4, 1912, in the Volyn' village of the Ryazan' region. His parents were the school teachers. In 1929 he became a student of the Physics Department of Leningrad State
University (USSR), from which he graduated in 1933. Then:
• 1935 - Ph.D. in Physics & Mathematics,
• 1937 - D.Sci. in Physics & Mathematics,
• 1942 - Stalin prize for solution of the Herman Weyl problem,
• 1946 - elected a Corresponding Member of the Academy Science of the USSR,
• 1951 - Lobachevsky prize for geometric results,
• 1952-1964 - the President (Rector) of Leningrad University,
• 1964 - elected a Full Member of the Academy Science of the USSR,
• 1965-1986 - Head of the Chair of geometry & topology at Novosibirsk University,
• 1986-1999 -Head of the Laboratory of Geometry in the St. Peterburg Branch of the Steklov Mathematical Institute of the Russian Academy of Science.
A.D. Alexandrov was a member of the Communist Party of USSR since 1951, and he is a supporter of Socialism in modern Russia. Among his pupils were members of the Communist Party and
differently-minded persons (for example, his pupil Revolt I. Pimenov). A.D. Alexandrov as Rector of Leningrad State University offered a position to a former prisoner Lev Gumilev (a famous Russian
historian). Twice he visited Vadim Delone (Delaunay) in a Tyumen' prison in Siberia who was a well-known antagonist of the Soviet Regime, and sopported the remacable Russian poet Andrei Voznesenskii
when this poet was out of fovour.
His teahers were a celebrated mathematician B.N. Delone (Delaunay)(a student of an outstanding Russian mathematician P.L.Chebyshev (=Tchebyshev)) and a famous physicist V.A. Fock .
B.N.Delone P.M.A. Dirac, L. Infeld, V.A. Fock
In 1959 A.D.Alexandrov and V.I.Smirnov founded the Leningrad Mathematical Society (since 1991, St. Peterburg Mathematical Society).
Currently, the Russian Academy of Science has three Members from the Leningrad Geometric School: A.V.Pogorelov and Yu.G.Reshetnyak.
Alexander Danilovich Alexandrov, the founder of the School, died on July 27, 1999, 4.00 (Moscow time).
The Alexandrov's Biography and Photos
Another pages about A.D.Alexandrov
Main research areas of the Leningrad Geometric School
• intrinsic and extrinsic geometry of convex surfaces;
• manifolds of bounded curvature in the sense of A.D. Alexandrov;
• theory of elliptic differential equations;
• generalized Riemannian spaces;
• foundations of Special and General Relativity;
• mathematical theory of space-time.
LGS has very good contacts with the Moscow Geometric School of Vladimir N. Efimov. In the West a similar geometric area is the school of H. Busemann (Poland-USA).
The geometric ideas of Alexandrov helped in brightening the talent of Viktor A. Toponogov who is a pupil of Abram Il'ich Fet and leads the Russian research in global Riemannian Geometry.
Novosibirsk and Chronogeometry
A.D. Alexanrov arrived in Novosibirsk in 1964. Here he began his research into the mathematical theory of space-time. He created a new area, Chronogeometry. A.D. Alexandrov and his students (see a
photo below) found many interesting results. There was a special seminar on "Chronogeometry" (Russian, koi-8).
1985. S.V. Astrakov, A.V. Levichev, Yu.F. Borisov, A.D. Alexandov, V. Pustovilin, ?,
A.V. Shaidenko, A.V. Kuz'minikh, A.K. Guts, I. Mazmanidy (from left to right)
The list of west scientist working in a similar area includes A.A. Robb (1914), H. Busemann (1967), C.E. Zeeman (1964), H.J.Borchers and G.C.Hegerfeldt, J.A. Lester, W.Benz, P.G.Vroegindwey and
Address: Dr. Alexander K. Guts
Chair of Mathematical Modelling
Department of Mathematics
Omsk State University
644077 Omsk, RUSSIA
guts@univer.omsk.su - Click here to send email now!
Counter was mounted: April 27, 1998
Site was founded: April 1997
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Definite integral evaluation (Fuchs-Sondheimer Resistivity model for Nanowires)
Jan 11, 2019 01:23 PM
Jan 11, 2019 01:23 PM
I really appreciate it if someone can help me to double check the following equation by running my sheet in their Mathcad.
It is because the result that I got is the opposite with the result in the paper where the equation comes from.
The decreasing result at small w comes from this part.
This is my sheet.
The original paper: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.61.14215
The equation from the paper (phi and psi should be the same):
the expected result (solid line):
Jan 12, 2019 11:44 AM
Jan 12, 2019 11:44 AM
Jan 12, 2019 11:44 AM
Jan 12, 2019 11:44 AM
Jan 12, 2019 01:51 PM
Jan 12, 2019 01:51 PM
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Algebra calculator solver
algebra calculator solver Related topics: math trivia questions
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KS2 Algebra
Author Message
penjloc Posted: Saturday 26th of Oct 07:34
I'm getting really bored in my math class. It's algebra calculator solver, but we're covering higher grade material. The concepts are really complicated and that’s why I usually sleep
in the class. I like the subject and don’t want to drop it, but I have a big problem understanding it. Can someone help me?
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nxu Posted: Sunday 27th of Oct 09:17
What exactly don't you understand about algebra calculator solver? I recall having problem with the same thing in Algebra 2, so I might be able to give you some advice on how to handle
such problems. However if you want help with algebra on a long term basis, then you should try out Algebrator, that's what I did in my College Algebra, and I have to say it's the best
! It's less expensive than a tutor and you can count on it anytime you feel like. It's very easy to use it , even if you never ever tried a similar software . I would advise you to get
it as soon as you can and forget about getting a math teacher. You won't regret it!
From: Siberia,
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Matdhejs Posted: Sunday 27th of Oct 19:05
Hello Dude, Algebrator helped me with my learning sessions last week. I got the Algebrator from https://softmath.com/algebra-policy.html. Go ahead, try that and let us know your
opinion. I have even suggested Algebrator to a list of of my friends at school .
From: The
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Ocheha Mesk Posted: Monday 28th of Oct 08:39
That sounds great! I am not at comfort with computers. If this software is easy to use then I would like to try it once. Can you please give me the link?
around the
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Sdefom Posted: Tuesday 29th of Oct 08:53
Koopmansshab I remember having often faced problems with subtracting exponents, angle-angle similarity and angle complements. A truly great piece of math program is Algebrator software. By simply
typing in a problem from workbook a step by step solution would appear by a click on Solve. I have used it through many algebra classes – Basic Math, Algebra 1 and Pre Algebra. I
greatly recommend the program.
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Gools Posted: Wednesday 30th of Oct 14:40
Click here for details : https://softmath.com/reviews-of-algebra-help.html. I think they give an complete money back guarantee, so you have nothing to lose. Best of Luck!
From: UK
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Transactions Online
Kunihiko HIRAISHI, "Performance Evaluation of Workflows Using Continuous Petri Nets with Interval Firing Speeds" in IEICE TRANSACTIONS on Fundamentals, vol. E91-A, no. 11, pp. 3219-3228, November
2008, doi: 10.1093/ietfec/e91-a.11.3219.
Abstract: In this paper, we study performance evaluation of workflow-based information systems. Because of state space explosion, analysis by stochastic models, such as stochastic Petri nets and
queuing models, is not suitable for workflow systems in which a large number of flow instances run concurrently. We use fluid-flow approximation technique to overcome this difficulty. In the proposed
method, GSPN (Generalized Stochastic Petri Nets) models representing workflows are approximated by a class of timed continuous Petri nets, called routing timed continuous Petri nets (RTCPN). In RTCPN
models, each discrete set is approximated by a continuous region on a real-valued vector space, and variance in probability distribution is replaced with a real-valued interval. Next we derive
piecewise linear systems from RTCPN models, and use interval methods to compute guaranteed enclosures for state variables. As a case study, we solve an optimal resource assignment problem for a paper
review process.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1093/ietfec/e91-a.11.3219/_p
author={Kunihiko HIRAISHI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Performance Evaluation of Workflows Using Continuous Petri Nets with Interval Firing Speeds},
abstract={In this paper, we study performance evaluation of workflow-based information systems. Because of state space explosion, analysis by stochastic models, such as stochastic Petri nets and
queuing models, is not suitable for workflow systems in which a large number of flow instances run concurrently. We use fluid-flow approximation technique to overcome this difficulty. In the proposed
method, GSPN (Generalized Stochastic Petri Nets) models representing workflows are approximated by a class of timed continuous Petri nets, called routing timed continuous Petri nets (RTCPN). In RTCPN
models, each discrete set is approximated by a continuous region on a real-valued vector space, and variance in probability distribution is replaced with a real-valued interval. Next we derive
piecewise linear systems from RTCPN models, and use interval methods to compute guaranteed enclosures for state variables. As a case study, we solve an optimal resource assignment problem for a paper
review process.},
TY - JOUR
TI - Performance Evaluation of Workflows Using Continuous Petri Nets with Interval Firing Speeds
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 3219
EP - 3228
AU - Kunihiko HIRAISHI
PY - 2008
DO - 10.1093/ietfec/e91-a.11.3219
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E91-A
IS - 11
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - November 2008
AB - In this paper, we study performance evaluation of workflow-based information systems. Because of state space explosion, analysis by stochastic models, such as stochastic Petri nets and queuing
models, is not suitable for workflow systems in which a large number of flow instances run concurrently. We use fluid-flow approximation technique to overcome this difficulty. In the proposed method,
GSPN (Generalized Stochastic Petri Nets) models representing workflows are approximated by a class of timed continuous Petri nets, called routing timed continuous Petri nets (RTCPN). In RTCPN models,
each discrete set is approximated by a continuous region on a real-valued vector space, and variance in probability distribution is replaced with a real-valued interval. Next we derive piecewise
linear systems from RTCPN models, and use interval methods to compute guaranteed enclosures for state variables. As a case study, we solve an optimal resource assignment problem for a paper review
ER -
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ANN Search Explained | BYOC | Zilliz Cloud Developer Hub
Version: User Guides (BYOC)
A k-nearest neighbor (kNN) search finds the k-nearest vectors to a query vector. Specifically, it compares a query vector to every vector in a vector space until k exact matches appear. Although kNN
searches guarantee perfect accuracy, they are time-consuming, especially for large datasets comprising high-dimensional vectors.
In contrast, approximate nearest neighbor (ANN) searches require building an index beforehand. Various indexing algorithms demonstrate trade-offs among search speed, memory usage, and accuracy.
Generally, two paths are available to implement these algorithms: narrowing the search scope and decomposing high-dimensional vector spaces into low-dimensional subspaces.
Narrowing the search scope can reduce search time by selecting only a subset of possible candidates for comparison with the query vector. This avoids irrelevant vectors. To determine whether a vector
is in the subset, an index structure is needed to sort the vectors.
There are generally three ideas available for forming the index structure: graphs, trees, and hashes.
HNSW: A graph-based indexing algorithm
Hierarchical Navigable Small World (HNSW) indexes a vector space by creating a hierarchical proximity graph. Specifically, HNSW draws proximity links (or edges) between vectors (or vertices) on each
layer to form a single-layer proximity graph and stacks them up to form the hierarchical graph. The bottom layer holds all vectors and their proximity links. As the layer goes up, only a smaller set
of vectors and proximity links remains.
Once the hierarchical proximity graph is created, the search goes as follows:
1. Find a vector as the entry point on the top layer.
2. Move gradually to the nearest vector along the available proximity links.
3. Once you determine the nearest vector at the top layer, use the same vector at a lower layer as the entry point to find its nearest neighbor at that layer.
4. Repeat the preceding steps until you find the nearest vector at the bottom layer.
LSH: A hash-based ANN indexing algorithm
Locality-sensitive hashing (LSH) indexes a vector space by mapping data pieces of any length to fixed-length values as hashes using various hash functions, gathering these hashes into hash buckets,
and tagging vectors that have been hashed to the same value at least once as candidate pairs.
DiskANN: ANN search on disk based on Vamana graphs
Unlike HNSW that builds a hierarchical graph for layered searches, Vamana’s indexing process is relatively simple:
1. Initialize a random graph;
2. Find the navigation point by first locating the global centroid and determining the closest point. Use a global comparison to minimize the average search radius.
3. Perform Approximate Nearest Neighbor Search with the initialized random neighbor graph and search starting point from step 2. Use all points on the search path as candidate neighbor sets and
apply the edge trimming strategy with alpha = 1 to reduce the search radius.
4. Repeat step 3 with adjusted alpha > 1 (1.2 recommended in the paper) to improve graph quality and recall rate.
Once the index is ready, the search goes as follows:
1. Load relevant data, including query set, PQ center point data, codebook data, search starting point, and index meta.
2. Use the indexed data set to perform cached_beam_search, count the access times of each point, and cache the num_nodes_to_cache points with the highest access frequency.
3. WARMUP operation is performed by default using the sample data set to perform a cached_beam_search.
4. Perform cached_beam_search with the query set for each given parameter L, and output statistics such as recall rate and QPS. Warmup and hotspot data statistics are not included in query time.
For details, refer to DiskANN, A Disk-based ANNS Solution with High Recall and High QPS on Billion-scale Dataset.
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The Math: Topology - Math Monday
The Math: topology
Mobius strips are studied in an area of math called topology, sometimes called “rubber sheet geometry”, which studies properties of strings and surfaces that are allowed to bend and stretch, but not
break or pass through itself. Mathematicians joke that a topologist can’t tell the difference between a donut and a coffee cup, because as this picture shows, one can be reshaped to make the other:
(This picture is from a rather silly movie by the wonderfully crazy mathematician/artist Henry Segerman, who loves to 3d print his ideas.)
Topology is only a little more than a century old, and includes knot theory, which classifies the different ways that a loop of string (or several loops) can be knotted. Here’s a fascinating video by
my friend Carlo Sequin at UC Berkeley explaining the basics of knot theory.
Movies and Instructions for making your own shapes
• Here’s Vi Hart’s fabulous video about Möbius strip made of Fruit by the Foot. She’s a popular YouTuber who yes makes super popular and entertaining movies about math crossed with art and music,
plus a good deal of rapid fire irreverence.
• Karl Schaffer shared the handout he uses with students for exploring what happens when you cut up THREE loops all joined together.
• Here are instructions for making a Möbius strip by adding a single cut to a square. Artist/sculptor Max bill made a beautiful metal sculpture in this shape, as well as making many other
sculptures equivalent. I showed everyone how to do this last Monday.
• And here are my instructions for cutting and folding a Möbius strip made of just six squares and four triangles. It doesn’t look like a Möbius strip, but trace the edge and you’ll see that its
one edge is in the shape of a square.
• Jeanne Lazzarini, who attends Math Monday regularly, sent me this: Have you heard of BIG Architects Group? Founded by Bjarke Ingels from Copenhagen? Ask me about him some day…Bjarke’s
innovative buildings make me think of stretching the boundaries of mathematical artistry and he’s gained world-wide acclaim for his buildings! —> Be sure to check out his möbius building —
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% Encoding: UTF-8 @COMMENT{BibTeX export based on data in FAU CRIS: https://cris.fau.de/} @COMMENT{For any questions please write to cris-support@fau.de} @article{faucris.261661710, abstract = {In
this article, the Cartan geometric approach toward (extended) supergravity in the presence of boundaries will be discussed. In particular, based on new developments in this field, we will derive the
Holst variant of the MacDowell-Mansouri action for $\mathcal{N}=1$ and $\mathcal{N}=2$ pure AdS supergravity in $D=4$ for arbitrary Barbero-Immirzi parameters. This action turns out to play a crucial
role in context of boundaries in the framework of supergravity if one imposes supersymmetry invariance at the boundary. For the $\mathcal{N}=2$ case, it follows that this amounts to the introduction
of a $\theta$-topological term to the Yang-Mills sector which explicitly depends on the Barbero-Immirzi parameter. This shows the close connection between this parameter and the $\theta$-ambiguity of
gauge theory.
We will also discuss the chiral limit of the theory, which turns out to possess some very special properties such as the manifest invariance of the resulting action under an enlarged gauge symmetry.
Moreover, we will show that demanding supersymmetry invariance at the boundary yields a unique boundary term corresponding to a super Chern-Simons theory with $\mathrm{OSp}(\mathcal{N}|2)$ gauge
group. In this context, we will also derive boundary conditions that couple boundary and bulk degrees of freedom and show equivalence to the results found in the D'Auria-Fré approach in context of
the non-chiral theory. These results provide a step towards of quantum description of supersymmetric black holes in the framework of loop quantum gravit}, author = {Eder, Konstantin and Sahlmann,
Hanno}, doi = {10.1007/JHEP07(2021)071}, faupublication = {yes}, journal = {Journal of High Energy Physics}, keywords = {Supergravity Models, AdS-CFT Correspondence, Chern-Simons Theories},
peerreviewed = {Yes}, title = {{Holst}-{MacDowell}-{Mansouri} action for (extended) supergravity with boundaries and super {Chern}-{Simons} theory}, volume = {2021}, year = {2021} }
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How to make a 3d compass
I was wondering how I could go about making a compass similar to this:
As you can see, as I angle my ship further away from a target, the vertical line on the compass gets longer.
Any help would be great
Think about why the developer made it that way. Those little lines are supposed to tell you if you are above the target or not. They are essentially a measure of the angle of your ships lookVector
and the line formed from the ship and the destination. You can use atan2 to find not only the angle, but also unlike acos(dot(a,b)), the direction too so that you can have it be facing up or down and
then the length is basically the ratio between the absolute angle over 180. then for their position on the circle I think a way to do it would be to make a plane around the horizon of the ship and
then form a line which is 90 degrees with this plane and comes from the destination. Find their intersection point then from there use that function which takes a 3d point and converts it to the
point on the screen then just get that calculated point on the screen and remap it to that tiny circle
I learned how to do this by breaking down some other program. This is very complicated.
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Python - (Engineering Applications of Statistics) - Vocab, Definition, Explanations | Fiveable
from class:
Engineering Applications of Statistics
Python is a high-level programming language known for its readability and versatility, widely used in data analysis, machine learning, and statistical applications. It enables users to implement
algorithms efficiently, making it an essential tool in modern data science and engineering projects. The language's extensive libraries and frameworks facilitate tasks such as Principal Component
Analysis (PCA) and Bayesian statistics, allowing for seamless integration of complex statistical methods into practical applications.
congrats on reading the definition of python. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Python's simplicity and ease of learning make it a preferred choice for beginners in programming and data analysis.
2. The language supports various libraries such as NumPy, Pandas, and Scikit-learn, which are crucial for performing advanced statistical analysis and machine learning tasks.
3. PCA can be performed in Python using Scikit-learn's built-in functions, allowing users to reduce the dimensionality of datasets easily.
4. Python's versatility allows it to be used not just for statistical applications but also for web development, automation, and data visualization.
5. Bayesian analysis in Python can be efficiently carried out using libraries like PyMC3 or TensorFlow Probability, enabling users to build complex probabilistic models.
Review Questions
• How does Python facilitate the implementation of Principal Component Analysis (PCA) in data analysis?
□ Python makes it easy to implement Principal Component Analysis (PCA) through libraries like Scikit-learn. Users can leverage the built-in PCA function to transform their data by reducing its
dimensions while preserving as much variance as possible. This process helps in visualizing high-dimensional data more effectively and simplifies further analyses.
• What are the advantages of using Python for Bayesian statistics compared to other programming languages?
□ Using Python for Bayesian statistics offers several advantages, including a rich ecosystem of libraries such as PyMC3 and TensorFlow Probability that simplify the implementation of complex
models. Python's readability and straightforward syntax make it easier to write and maintain code. Additionally, its ability to integrate with other tools for data manipulation and
visualization enhances the overall workflow when working on Bayesian analyses.
• Evaluate the impact of Python's extensive libraries on statistical analysis in engineering applications.
□ The impact of Python's extensive libraries on statistical analysis in engineering applications is profound. Libraries like NumPy and Pandas streamline data handling, while Scikit-learn
provides robust machine learning functionalities, including PCA. This integration allows engineers to quickly analyze data sets, implement complex statistical models, and visualize results
effectively. As a result, engineers can make more informed decisions based on reliable data analyses, ultimately improving project outcomes.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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Solve b) Using Parseval's Theorem
• MHB
• Thread starter goohu
• Start date
In summary: So you need to subtract it once. As I say, I have not checked the details of your calculations from the start.
Hello good folks!
I'm stuck trying to solve the problem b). In the theory book examples they are skipping steps and shortly states 'use algebra' and parsevals theorem to rewrite the Fourier series into the answer that
is given.
So I've tried to use parsevals theorem but I still can't rewrite the result into the sum we are looking for.
View attachment 9282
In case the picture is too blurry;
f(t) = \(\displaystyle \frac{{pi}^{2}}{3} + \sum_{k=1}^{\infty} \frac{4*{-1}^{k}}{{k}^{2}} cos(kt)\)
b) Calculate the sum of \(\displaystyle \sum_{k=1}^{\infty} \frac{{-1}^{k+1}}{{k}^{2}}\)
Last edited:
goohu said:
Hello good folks!
I'm stuck trying to solve the problem b). In the theory book examples they are skipping steps and shortly states 'use algebra' and parsevals theorem to rewrite the Fourier series into the answer
that is given.
So I've tried to use parsevals theorem but I still can't rewrite the result into the sum we are looking for.
In case the picture is too blurry;
f(t) = \(\displaystyle \frac{{pi}^{2}}{3} + \sum_{k=1}^{\infty} \frac{4*{-1}^{k}}{{k}^{2}} cos(kt)\)
b) Calculate the sum of \(\displaystyle \sum_{k=1}^{\infty} \frac{{-1}^{k+1}}{{k}^{2}}\)
Hi goohu,
We also have that $f(t)=t^2$.
What do we get if we substitute $t=0$?
f(0) = 0.
If we plug in t=0 into the Fourier series the cos term simply becomes 1. But what do we do from here?
goohu said:
f(0) = 0.
If we plug in t=0 into the Fourier series the cos term simply becomes 1. But what do we do from here?
More specifically we get:
$$0 = \frac{{\pi}^{2}}{3} + \sum_{k=1}^{\infty} \frac{4\cdot{(-1)}^{k}}{{k}^{2}} \cdot 1$$
Can we rewrite that equation into the desired form?
Thanks Klaas! I solved the problem now.
Is this trick where you put t = 0 always applicable? IIRC musnt the function be continuous?
Before this I used the formula where you square both sides and it didnt work to rewrite it from there. How do know when to use which method?
goohu said:
Thanks Klaas! I solved the problem now.
Is this trick where you put t = 0 always applicable? IIRC musnt the function be continuous?
Before this I used the formula where you square both sides and it didnt work to rewrite it from there. How do know when to use which method?
The 'trick' here is that we write a function $f(t)$ as a Fourier Series.
Then we can substitute any value for $t$ that we want.
The condition to write a function as a Fourier Series is:
If f is continuous and the derivative of f(t) (which may not exist everywhere) is square integrable, then the Fourier series of f converges absolutely and uniformly to f(t).
What do you mean by squaring both sides?
In this particular case the requested sum in (b) is simply an application of the result found in (a).
More generally, a Fourier Series is just one of the many tools that make some problems suddenly easy to solve.
Sorry for the late reply, but here's an example of "squaring both sides" its just something found in my formula sheet:
I also got stuck at the last step trying to find \(\displaystyle \sum_{k=1}^{\infty} \frac{1}{{k}^{2}+4}\)
View attachment 9307
goohu said:
Sorry for the late reply, but here's an example of "squaring both sides" its just something found in my formula sheet:
I also got stuck at the last step trying to find \(\displaystyle \sum_{k=1}^{\infty} \frac{1}{{k}^{2}+4}\)
I have not tried to follow all of your calculations, but there is one obvious mistake. If $c_k$ is a complex number then $|c_k|^2$ is not the same as $c_k^2$. In fact, $|c_k|^2 = c_k\overline{c_k}$
(where the bar denotes the complex conjugate).
So if $c_k = \dfrac{e^{4\pi}-1}{2\pi(2-ik)}$ then $|c_k|^2 = \dfrac{(e^{4\pi}-1)^2}{4\pi^2(2-ik)(2+ik)} = \dfrac{(e^{4\pi}-1)^2}{4\pi^2(4+k^2)}$.
It turns out I almost got the right answer (same as the picture) and the left hand side becomes \(\displaystyle \sum_{k=1}^{\infty} \frac{1}{{k}^{2}+4}\). However the right hand side misses a -1/8
Last edited:
goohu said:
It turns out I almost got the right answer (same as the picture) and the left hand side becomes \(\displaystyle \sum_{k=1}^{\infty} \frac{1}{{k}^{2}+4}\). However the right hand side misses a 1/8
I think that the missing 1/8 possibly comes from the fact that when you combine the terms indexed by $k$ and $-k$ in the sum \(\displaystyle \sum_{k=-\infty}^{\infty}|c_k|^2\), in order to get a sum
\(\displaystyle \sum_{k=0}^{\infty} |c_k|^2\), you are in danger of counting the $k=0$ term twice.
FAQ: Solve b) Using Parseval's Theorem
1. What is Parseval's Theorem?
Parseval's Theorem is a mathematical principle that states the total energy in a signal can be calculated by finding the sum of the squared values of its Fourier coefficients.
2. How is Parseval's Theorem used?
Parseval's Theorem can be used to analyze signals in various fields such as physics, engineering, and computer science. It allows for the calculation of energy in a signal without having to measure
it directly.
3. What is the formula for Parseval's Theorem?
The formula for Parseval's Theorem is E = ∑|c[n]|^2, where E is the total energy in the signal and c[n] is the Fourier coefficient at a specific frequency.
4. How does Parseval's Theorem relate to the Fourier Transform?
Parseval's Theorem is closely related to the Fourier Transform, as it is used to calculate the energy in a signal by analyzing its frequency components. The Fourier Transform is used to decompose a
signal into its frequency components.
5. Can Parseval's Theorem be applied to any type of signal?
Yes, Parseval's Theorem can be applied to any type of signal as long as it is finite and has a well-defined Fourier Transform. This includes signals in both the time and frequency domains.
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Spell Out Numbers 2 546297 - SpellingNumbers.com
Spell Out Numbers 2 546297
Spell Out Numbers 2 546297 – Learning to spell numbers can be a challenge. However, with the right tools, it can be easier to learn how to spell. There are many resources available to help you
improve your spelling, whether at school or work. These consist of advice and tricks, workbooks, and even online games.
The format of the Associated Press
If you write for newspapers and other print media, you must be able to spell numbers in AP style. The AP style guides you on how to spell numbers and other details to make your writing easier.
The Associated Press Stylebook’s 1953 debut saw many revisions since the time it was first released. The stylebook has now reached its 55th anniversary. It is the most widely used stylebook for
American periodicals, newspapers online news outlets and various other media.
The AP Style is a set standards for punctuation and language which are frequently used in journalism. The most important guidelines of AP Style include the use of capitalization, date and time, and
Regular numbers
An ordinal is a number that is a unique representation of a location in a sequence or list. These numbers are often used to show the size, significance, or time passing. They also reveal what’s in
what order.
Ordinary numbers are expressed either verbally or numerically, based on the context. The use of a unique suffix differentiates them in the most important way.
To make an ordinal number, put the “th” to the end of. For example 31 is the standard number.
There are many things that can be accomplished using ordinals, such as dates and names. It is crucial to understand the difference between a cardinal or an ordinal.
Both millions and trillions
Numerology is utilized in a variety of contexts. This includes the market for stocks and geology, as well as the history of the world and many more. Millions and billions of dollars are only two
examples. A million is a natural number that occurs before 1,000,001; billions occur immediately after 999.999.999.
The annual earnings of a corporation is expressed in millions. They are also used to determine the value a stock, fund or other piece of money is worth. Billions can also be used to gauge a company’s
market capitalization. You can check the validity of your estimations using a calculator that converts units to convert billions into millions.
Fractions in English are used to indicate parts or individual items. The denominator as well as the numerator are split into two distinct pieces. The numerator shows how many pieces of equal size
were taken and the denominator illustrates how many portions were divided into.
Fractions can be mathematically expressed or written in words. It is important to spell the fractions correctly when writing in words. It can be difficult particularly if you have to deal with larger
There are a few rules that are to be adhered to when fractions should be written in words. The best way to start sentences is by writing the numbers in complete. A second option is to write the
fractions using decimal form.
The writing of a thesis, a research paper or email may require you to utilize years of experience in spelling numbers. Some tips and tricks can help you avoid repeating the same spelling and ensure
the correct formatting.
The numbers should be clearly written in formal style. There are many styles guides that offer various guidelines. For example the Chicago Manual of Style advises writing out numerals from 1 to 100.
However, it is not recommended to use figures that are larger than 401.
Of course, exceptions exist. The American Psychological Association’s (APA) style guide is among them. Although it’s not a specialist publication, it’s used often in writing for scientific purposes.
Date and time
The Associated Press style handbook provides some general guidelines to styling numbers. For numbers greater than 10, the numeral system is used. Numerology can be utilized in a variety of other
places. In the initial five numbers in your document, “n-mandated” is the norm. There are a few exceptions.
Both the Chicago Manual of Technique as along with the AP stylebook advise to use a lot of numbers. This does not mean that an alternative version is not possible. I know for a fact that the
difference is there since I am an AP Graduate.
A stylebook must be checked to see the ones you’re omitting. For instance, make sure to not overlook the “t”, such as “time”
Gallery of Spell Out Numbers 2 546297
Numbers spelling Worksheet
16 In Words How To Spell The Number 16 In English
When Should You Spell Out Numbers
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Calculating Stockpile Capacity - 911Metallurgist
Calculating Stockpile Capacity: Once the minimum storage capacities which will assure maximum mill output are known, the appropriate stockpile configuration must be determined. Stockpiles fall into
two general categories: conical and elongated.
Conical Stockpiles
The conical stockpile is the simplest and easiest to analyze. The total stockpile capacity is given by:
3.14 (Tan A)R³ D/3000 = capacity in metric tons…………………(1)
where: R = stockpile radius in meters
A = angle of repose for material to be stockpiled
D = density of material in kg/m³
Note that the capacity of the conical stockpile varies with the cube of the radius of the pile. This means that the capacity of the conical pile grows very rapidly as the height (and hence the radius
of the pile) increases. Increasing the height of the stockpile by 26% results in a doubling of the stockpile capacity. One should also observe at this point that one-half of the capacity of the
stockpile is in the lower 1/5 of the pile. This fact will become important later in considerations of live storage capacity.
For many common materials, the angle of repose, A, is about 38 degrees. Substituting this value for A reduces Equation 1 to:
8.18 x 10 -4 R³D = capacity in metric tons………………….(2)
In English units, Equation 2 becomes:
4.09 x 10 -4 r³d = capacity in short tons…………………(3)
where: r = stockpile radius in feet
d = density of material in lb/ft³
For convenience, typical stockpile capacities have been tabulated in both metric and English units in Table 1.
The main advantage of the conical pile is that it can easily be built by dozing equipment or a fixed belt conveyor. One disadvantage is that very high stockpiles are required to attain large storage
capacities. This results in very long conveyors and supports and accompanying large foundations to withstand the loads on the conveyor. More importantly, because the conical pile occupies such a
small ground area in relation to its volume, soil pressures on large piles can exceed the bearing strength of the local soil. For example, if 1600 kg/m³ (100 lb/ ft³) material is being stockpiled on
soil with an allowable bearing pressure of 14 680 kg/m³ (3000 lb/ft³) the maximum permissible conical stockpile is about 27.4m (90 ft.) high and has a total storage capacity of about 56 000 metric
tons (62 000 short tons).
Elongated Stockpiles
The elongated stockpile and its variations are the most common stockpile form in high capacity installations. The capacity of the stockpile can easily be determined by considering the pile as two
separate volumes.
The end half cones can be combined to form a single conical pile which can be analyzed using Equation 1, 2 or 3 above. The capacity of the center section of the elongated pile is given by:
R²LD Tan A/1000 = capacity in metric tons………………………(4)
where: R = radius of end half cone in meters
L = length of center section in meters
D = density of material in kg/m³
A = angle of repose for material to be stockpiled
For a value of A of 38 degrees, Equation 4 reduces to:
7.81 x 10 -4 R²LD = capacity in metric tons……………………………..(5)
Equation 5 expressed in English units is:
3.91 x 10 -4 r²ld = capacity in short tons………………………………….(6)
where: r = radius of end half cone in feet
l = length of center section in feet
d = density of material in lb/ft³
The total capacity of an elongated pile is the sum of the results of Equation 1 and Equation 4. A kidney-shaped pile such as that formed by a radial stacker is merely a variation of the elongated
pile. The capacity analysis is identical to that for an elongated pile except that the arc length of the center section is substituted for the value L in Equal 4 above. The arc length is given by:
3.14 (Pr)B/180 = arc length in same units as Pr……………………………..(7)
where: Pr = radius of central section peak (Horizontal distance from stacker pivot to peak of central section)
B = angle (in degrees) formed by radii at limits of central section peak
The main advantages of an elongated stockpile are that it can easily be varied to fit into a plant layout and very large stockpiles can be built without raising the ore to prohibitive heights. The
main disadvantage is that the stockpile building equipment is more complex and usually requires a higher capital investment. As indicated previously, soil pressures under an elongated pile are lower
than those under conical piles of equivalent capacity.
This is an important fact that should not be overlooked in the design of a stockpile system. In comparing the two types of systems, one should also always consider both the capital costs and
operating expenses before deciding on one system or the other.
Calculate Stockpile Live Storage
In the discussions above, only the total capacity of the stockpiles was considered. This capacity is useful if reclaiming is done with a front end loader or if a bulldozer or bucket wheel reclaimer
is used. Usually, the figure for total stockpile capacity is utilized when planning for the longest scheduled shutdown. In that circumstance, one should expect to use a bulldozer to obtain complete
stockpile reclaim so that the stockpile will not be any larger than necessary. With belt conveyors in normal operating circumstances, one should work with the live storage capacity; that is, the
portion of the stockpile that will flow into the feeders in the reclaim tunnels without dozing of the pile. The portion of the stockpile outside this live storage section is referred to as dead
storage. Obviously, the following analysis does not apply when a front end loader is used for reclaiming. In this discussion, we will consider only a linear arrangement of outlets under a conical
stockpile. Elongated stockpile dimensions and non-linear reclaim schemes are so variable that a general discussion is meaningless and each system should be evaluated individually. For the sake of
comparison, the live storage capacity of elongated stockpiles is generally about 25% to 30% of the total stockpile capacity assuming a sufficient number of outlets are used. We will begin with the
simplest case: a conical stockpile with one outlet at the center.
The live storage capacity of a conical stockpile with a single outlet is easily evaluated by considering what happens when one draws material out of the stockpile. Material will continue to flow to
an outlet as long as the angle of the withdrawal cone is steeper than the drawdown angle of the material. Once the drawdown angle is reached, material will cease to flow to the outlet of its own
accord. This drawdown angle varies from material to material but is usually about ten degrees steeper than the angle of repose for the material. In other words, if the angle of repose is 38 degrees,
the drawdown angle is usually about 48 degrees. Obviously, the drawdown angle will vary depending upon the size range of the material and its wetness. If the application is critical, one should
attempt to determine the drawdown angle through testing since it has a great effect on actual live storage capacity; however, for general purposes the “angle of repose plus ten degrees” rule is
sufficiently accurate. If one sketches in the drawdown angle on a cross-section of a conical pile, it will be seen that the live storage volume consists of an inverted cone with an upright cone on
top of it. Calculation of the volumes of these two cones for a stockpile with a 38 degree angle of repose and a 48 degree drawdown angle shows that the live storage portion of a conical stockpile
with a single outlet is about 17% of the total stockpile volume. This low figure is reasonable since, as mentioned in the Conical Stockpile section above, one-half of the total stockpile volume is in
the lower 1/5 of the pile and it is this section that is least affected by the single outlet withdrawal cone. For convenience, the live storage capacity of typical conical stockpiles is tabulated in
Table 1.
Since the live storage portion with a single outlet is only 17% of the total volume, one could logically attempt to increase the live storage percentage by using multiple outlets. In fact, if one
uses the same stockpile as above but with two outlets rather than one, the live storage portion rises to about 24.9% of the total volume which is almost half again as much as the single outlet value.
The gain in live storage rapidly falls off when more than two outlets are used. When three outlets are used, the live storage portion is about 26.5% of the total volume. This is an increase in live
storage of only 1.6% of the total stockpile volume. This small increase does not usually justify the expense of installing a third outlet and feeder with its corresponding reclaim conveyor and tunnel
extensions. The two outlet arrangement is usually the best since it provides considerably more live storage than the single outlet arrangement and provides a backup feeder if one of the two is down
for repairs. Approximately 59% of the total live storage capacity will be available if only one outlet of a two outlet system is used.
It should be noted that outlet placement is extremely critical to the attainment of the indicated live storage capacities. In the example above, if the outlets are placed 1/10 of the pile radius too
close to the center of the pile, the live storage portion drops from 24.9% to only 19.3% of the total volume. This is only 13.5% more than the single outlet value rather than the almost 50% increase
indicated above. The live storage capacities for typical conical stockpiles with two outlets at the “ideal” distance from the center of the pile have been tabulated in Table 1.
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Goyal Brothers Measurements and Experimentation Class-9 ICSE Physics Ch-1 - ICSEHELP
Goyal Brothers Measurements and Experimentation Class-9 ICSE Physics Ch-1
Goyal Brothers Measurements and Experimentation Class-9 ICSE Physics Solutions Ch-1. We Provide Step by Step Answer of Exercise, MCQs, Numericals Practice Problem Questions of Exercises, Subjective
and Practice Problem Measurements and Experimentation Class-9, Visit official Website CISCE for detail information about ICSE Board Class-9 Physics.
Goyal Brothers Measurements and Experimentation Class-9 ICSE Physics Ch-1
-: Select Exercise :-
(A) Objective Questions
Exe-1 Measurements and Experimentation Class-9 Goyal Brothers ICSE Physics Solutions Ch-1
I. Multiple choice Questions.
Select the correct option:
1. Which of the following is not a fundamental unit?
(a) Second
(b) Ampere
(c) Candela
(d) Newton
Ans. (d) Newton
Explanation : Second, Ampere and Candela are the fundamental units while Newton is a derived unit.
2. Which of the following is a fundamental unit?
(a) m/s^2
(b) Joule
(c) Newton
(d) metre
Ans. (d) metre
Explanation : m/s^2, Joule and Newton are derived units while metre is a fundamental unit.
3. Which is not a unit of distance?
(a) metre
(b) millimetre
(c) Leap year
(d) kilometre
Ans. (c) Leap year
Explanation : Leap year is a unit of time while the metre, millimetre and kilometre are the units of distance.
II Fill in the blanks
1. The unit is which we measure the quantity is called constant quantity.
2. One light year is equal to 9.46 × 10^15 m.
3. One mean solar day = 86400 sec
4. One year = 3.1536 × 10^7 sec
5. One micrometre = 10^-6 m.
(B) Subjective Questions
Exe-1 Measurements and Experimentation Class-9 Goyal Brothers ICSE Physics Solutions Ch-1
Question 1.
What do you understand by the term measurement?
“Measurement implies comparison of a physical quantity with a standard unit to find out how many times the given standard is contained in the physical quantity.”
Physics, like other branches of science requires experimental study which involves measurement.
Question 2.
What do you understand by the terms
1. unit
2. magnitude, as applied to a physical quantity?
(i) Unit : Unit “is a standard quantity of the same kind with which a physical quantity is compared for measuring it. ” In order to measure a physical quantity, a standard is needed (which is
acceptable internationally). The standard should be some convenient, definite and easily reproducible quantity of the same kind in terms of which the physical quantity as a whole is expressed. This
standard is called a unit
(ii) Magnitude of a physical quantity : The number of times a standard quantity is present in a given physical quantity is called magnitude of physical quantity.
Physical quantity = Magnitude × Unit
Question 3.
A body measures 25 m. State the unit and the magnitude of unit in the statement.
Here S.I. unit of length i.e. metre (m) has been used. Magnitude of the given quantity = 25
Metre : It is defined as 1,650,763,73 times the wavelength of specified orange red spectral line a emission spectrum of Krypton-86 or 1,553,164.1 times the wavelength of the red line in emission
spectrum of cadmium.
or one metre is defined as the distance travelled by the light in 1/299,792,458 of a second in air/vacuum.
Question 4.
State four characteristics of a standard unit.
Characteristics of standard unit :
1. It should be of convenient size.
2. It should not change with respect to place and time.
3. It should be well defined.
4. It should be easily reproduced.
Question 5.
Define the term fundamental unit. Name fundamental units of mass; length; time; current and temperature.
Fundamental unit : A fundamental or basic unit is that which is independent of any other unit or which can neither be changed nor can be related to any other fundamental unit. e.g. units of mass,
length, time and temperature.
Question 6.
What do you understand by the term derived unit? Give three examples.
Derived units. “Derived units are those which can be expressed in terms of fundamental units.”
2. S.I. unit of area i.e. m^2 is a derived unit.
Area = length × breadth
Now metre is unit of length and breadth, so S.I. unit of area is obtained by multiplying the fundamental unit ‘m’ with itself. So, m^2 is the derived unit of area.
3. Density = Mass/volume
S.I. unit of density i.e. kg/m^3 is the derived unit of density because it can be obtained by combining two fundamental units kilogram and metre.
Question 7.
(a) Define metre according to old definition.
(b) Define metre in terms of wavelength of light.
(c) Why is the metre length in terms of wavelength of light considered more accurate?
(a) Metre : One metre is defined as the one ten millionth part of distance from the pole to the equator.
(b) Metre : One metre is defined as 1,650, 763.73 times the wavelength of specified orange red spectral line in emission spectrum of Krypton = 86.
One metre is defined as 1,553,164.1 times the wavelength of the red line in emission spectrum of cadmium.
(c) Metre length in terms of wavelength of light is considered more accurate because
1. The wavelength of light does not change with time, temperature, pressure etc.
2. It can be reproduced anywhere at any time because Krypton is available every where.
Question 8.
Name the convenient unit you will use to measure :
(a) length of a hall
(b) width of a book
(c) diameter of hair
(d) distance between two cities.
(a) Foot (Ft)
(b) Centimetre (cm)
(c) Micrometre (µm)
(d) Kilometre (km)
Question 9.
(a) Define mass.
(b) State the units in which mass is measured in (1) C.GS. system (2) S.I. system.
(c) Name the most convenient unit of mass you will use to measure :
1. Mass of small amount of a medicine.
2. The grain output of a state
3. The bag of sugar
4. Mass of a cricket ball.
(a) Mass: The quantity of matter contained in a body is known as its mass.
(b) In C.GS. system, mass is measured in gram. In S.I. system, mass is measured in Kilogram.
Question 10.
(a) Define time.
(b) State or define the following terms :
1. Solar day
2. Mean solar day
3. An hour
4. Minute
5. Second
6. Year.
(a) Time : It is defined as the time interval between two events
(i) Solar day : The time taken by the earth to complete one rotation about its own axis is called solar day.
(ii) Mean solar day : The average of the varying solar days, when the earth completes one revolution around the sun, is called mean solar day.
(iii) An hour : It is defined as the 1/24 th part of the mean solar day.
(iv) Minute : It is defined as the 1/1440 part of the mean solar day.
(v) Second : “A second is defined as 1/86400 th part of a mean solar day.”
Second may also be defined “as to be equal to the duration of9,192,631,770 vibrations corresponding to the transition between two hyperfme levels of caesium – 133 atom in the ground state.”
(vi) Year : One year is defined as the time in which earth completes one complete revolution around the sun.
Unit II Practice Problems
Exe-1 Measurements and Experimentation Class-9 Goyal Brothers ICSE Physics Solutions Ch-1
Question 1.
A student calculates experimentally the value of density of iron as 7.4 gem^-3. If the actual density of iron is 7.6 gcm^-3, calculate the percentage error in experiment.
Question 2.
A student finds that boiling point of water in a particular experiment is 97.8°C. If the actual boiling point of water is 99.4°C, calculate the percentage error.
Experimental value of boiling point of water = B.P[1] = 97.8°C
Actual value of boiling point of water = B.P[2] = 99.4°C
Absolute error = B.P.[2] – B.P[1] = 99.4 – 97.8 = 1.6°C
Question 3.
A pupil determines velocity of sound as 320 ms^-1. If actual velocity of sound is 332 ms^-1, calculate the percentage error.
Velocity of sound determined by pupil = V[1] = 320 ms^-1
Actual value of velocity of sound = V[2] = 332 ms^-1
Absolute error = V[2] – V[1] = 332 – 320 = 12 ms^-1
Exercise 2
Exe-1 Measurements and Experimentation Class-9 Goyal Brothers ICSE Physics Solutions Ch-1
Question 1.
(a) What do you understand by the term order of magnitude of a quantity?
(b) Why are physical quantities expressed in the order of magnitude? Support your answer by an example.
(a) Order of a magnitude of a quantity : The exponent part of a particular measurement is called order of magnitude of a quantity.
The order of magnitude of a given numerical quantity is the nearest power of ten to which its value can be written, (b) Measurement of certain physical quantities are either too large or too small
that these cannot be expressed conveniently. It is difficult to write or remember them. So such quantities can be expressed in the order of magnitude.
For example : The diameter of the sun is 1,390,000,000 m. It is difficult to write or remember such a measurement. So it is expressed as 1.39 × 109 m.
Here power of ten i.e. 9 (i.e. exponent part of the measurement) gives the order of magnitude of the given quantity.
So order of magnitude of diameter of the sun is 10^9 m.
Question 2.
Express the order of magnitude of the following quantities :
1. 12578935 m
2. 222444888 kg
3. 0.000,000,127 s
4. 0.000,000,000,00027 m
Question 3.
(a) What do you understand by the term degree of accuracy?
(b) Amongst the various physical measurements recorded in an experiment, which physical measurement determines the degree of accuracy?
(a) Degree of accuracy : It means that we can measure a quantity, without any error of estimation.
In any experiment, all observations should be taken with same degree of accuracy.
(b) Amongst the various physical measurements recorded in an experiment, least accurate observation determines the degree of accuracy.
Question 4.
(a) State the formula for calculating percentage error
(b) Is it possible to increase the degree of accuracy by mathematical manipulations? Support your answer by an example.
(a) The percentage error can be calculated by the formula :
(b) It is not possible to increase the degree of accuracy by mathematical manipulations.
For examples : When a number of values are added or subtracted, the result cannot be more accurate than the least accurate value.
In the above addition 72.5 has least accuracy. When we . say 72.5, it implies that value lies between 72.45 and 72.55 and 72.5 is the most probable value. Thus the error in 72.5 is +0.05. As the
final result cannot be more accurate than least accurate observation, so the correct and most reliable answer in the above addition is 72.9.
Question 5.
State the factors which determine number of significant figures for the calculation of final result of an experiment.
Factors which determine number of significant figures for the calculation of final result of an experiment are :
1. The nature of experiment.
2. The accuracy with which various measurements are made.
Question 6.
The final result of calculations in an experiment is 125,347,200. Express the number in terms of significant places when
1. accuracy is between 1 and 10
2. accuracy is between 1 and 100
3. accuracy is between 1 and 1000
Final result of calculations in an experiment = 125,347,200
1. When accuracy lies between 1 and 10, then final result may be written as 1.2 × 10^8.
2. When accuracy lies between 1 and 100, then final result may be written as 1.25 × 10^8.
3. When accuracy lies between 1 and 1000 than final result may be written as 1.253 × 10^8.
Unit 3 Practice Problems
Exe-1 Measurements and Experimentation Class-9 Goyal Brothers ICSE Physics Solutions Ch-1
Question 1.
The main scale of vernier callipers has 10 divisions in a centimetre and 10 vernier scale divisions coincide with 9 main scale divisions. Calculate
1. pitch
2. L.C. of vernier callipers.
Main scale divisions of vernier callipers in one centimetre = 10
Question 2.
In a vernier callipers 19 main scale divisions coincide with 20 vernier scale divisions. If the main scale has 20 divisions in a centimetre, calculate
1. pitch
2. L.C. of vernier callipers.
Main scale divisions of vernier callipers in one centimetre = 20 Unit
Practice Problems 2
Question 1.
Figure shows the position of vernier scale, while measuring the external length of a wooden cylinder.
1. What is the length recorded by main scale?
2. Which reading of vernier scale coincides with main scale?
3. Calculate the length.
Main scale divisions of vernier callipers in one centimetre = 10
Question 2.
In figure for vernier callipers, calculate the length recorded.
Main scale divisions of vernier callipers in one centimetre = 10
Practice Problems 3
Question 1.
(a) A vernier scale has 10 divisions. It slides over a main scale, whose pitch is 1.0 mm. If the number of divisions on the left hand of zero of the vernier scale on the main scale is 56 and the 8th
vernier scale division coincides with the main scale, calculate the length in centimetres.
(b) If the above instrument has a negative error of 0.07 cm, calculate corrected length.
No. of divisions on vernier scale = 10
Pitch = 1.0 mm
Question 2.
(a) A vernier scale has 20 divisions. It slides over a main scale, whose pitch is 0.5 mm. If the number of divisions on the left hand of the zero of vernier on the main scale is 38 and the 18th
vernier scale division coincides with main scale, calculate the diameter of the sphere, held in the jaws of vernier callipers.
(b) If the vernier has a negative error of 0.04 cm, calculate the corrected radius of sphere.
No. of divisions on vernier scale = 20
Pitch 0.5 mm
Practice Problems 4
Question 1.
The least count of a vernier callipers is 0.0025 cm and it has an error of + 0.0125 cm. While measuring the length of a cylinder, the reading on main scale is 7.55 cm, and 12th vernier scale division
coincides with main scale. Calculate the corrected length.
Least count (L.C.) = 0.0025 cm
Error = +0.0125 cm
Correction = – (Error) = – (+0.0125) = – 0.0125 cm
Main scale reading = 7.55 cm
Vernier scale division (V.S.D.) coinciding with main scale = 12th
Length recorded = Main scale reading + L.C. × V.S.D.
= 7.55 + 0.0025 × 12
= 7.55+ 0.0300 = 7.58 cm
Correct length = Length recorded + Correction
= 7.58+ (-0.0125)
= 7.58 – 0.0125 = 7.5675 cm = 7.567 cm
Question 2.
The least count of a vernier callipers is 0.01 cm and it has an error of + 0.07 cm. While measuring the radius of a sphere, the main scale reading is 2.90 cm and the 5th vernier scale division
coincides with main scale. Calculate the correct radius.
Least count (L.C.) = 0.01 cm
Error = + 0.07 cm
Correction = (Error) = – (+ 0.07) = – 0.07 cm
Main scale reading = 2.90 cm
Vernier scale division (V.S.D.) coinciding with main scale = 5th Observed diameter of sphere = Main scale reading + L.C. × V.S.D.
= 2.90 + 0.01 x 5 = 2.90 + 0.0 = 2.95 cm
Corrected diameter = Observed diameter + Correction
= 2.95 + (-0.07) = 2.95 – 0.07 = 2.88 cm
∴ Corrected radius = 2.88/2 = 1.44 cm
Exercise 3
Exe-1 Measurements and Experimentation Class-9 Goyal Brothers ICSE Physics Solutions Ch-1
Question 1.
Who invented vernier callipers?
Vernier callipers was invented by Pierre Vernier.
Question 2.
What is the need for measuring length with vernier callipers?
For measuring the exact length with greater accuracy, especially when we are measuring a very small length, we use an appliance vernier calliper. A vernier calliper can measure accurately upto 1/100
th part of a centimetre.
Question 3.
Up to how many decimal places can a common vernier callipers measure the length in cm?
A common vernier calliper can measure the length accurately upto two places of decimal when length is measured in centimetre
i.e. upto 1/100 th part of a centimetre.
Question 4.
Define the terms :
1. pitch
2. least count as applied to a vernier callipers.
1. Pitch : “The pitch of a screw is the distance moved by the screw in one complete rotation of its head.”
Pitch may also be defined as “the distance between two consecutive threads of screw measured along the axis of screw.”
2. Least Count of Vernier Calliper : Least count of a vernier callipers is the difference between one main scale division (M.S.D.) and one vernier scale division (V.S.D.)
Question 5.
State the formula for determining :
1. pitch
2. least count for a vernier callipers.
Question 6.
State the formula for calculating length if :
1. Number of vernier scale division coinciding with main scale and number of division of main scale on left hand side of zero of vernier scale are known.
2. The reading of main scale is known and the number of vernier scale divisions coinciding with main scale are known.
1. If we know the number of vernier scale divisions (V.S.D.) coinciding with main scale and number of main scale divisions (M.S.D.) on left hand side of zero of vernier scale then Length recorded =
Main scale reading + L.C. × V.S.D.
2. Same as in part (i).
Question 7.
(a) What do you understand by the term zero error?
(b) When does a vernier callipers has
1. positive error
2. negative error?
(c) State the correction if
1. positive error is 7 divisions
2. negative error is 7 divisions, when the least count is 0.01 cm.
(a) Zero Error : A vernier callipers is said to have a zero error when zero of the main scale does not coincide with zero of vernier scale.
(b) Positive Error : If the zero of the vernier scale is on right hand side of zero of the main scale, then error is said to be positive and correction is said to be negative.
Negative Error : If the zero of the vernier scale is on the left hand side of zero of the main scale, the error is said be negative and the correction is said to be positive.
(c) When positive error is 7 divisions and L.C. is 0.01 cm
Then correction = – (+ 7 × L.C.)
= -7 × 0.01 cm = – 0.07 cm
When negative error is 7 divisions and count (L.C.) is 0.01 cm
Then correction = – (- 7 × L.C.)
= – ( – 7 × 0.01) cm = + 0.07 cm
Question 8.
Which part of vernier callipers is used to measure
(a) external diameter of a cylinder
(b) internal diameter of a hollow cylinder
(c) internal length of a hollow cylinder?
(a) External Jaws of a vernier callipers are used to measure the external diameter of cylinder.
(b) Internal Jaws are used to measure internal diameter of a hollow cylinder.
(c) Tail of vernier callipers is used to measure the internal length of a hollow cylinder.
Unit 4 Practice Problems
Exe-1 Measurements and Experimentation Class-9 Goyal Brothers ICSE Physics Solutions Ch-1
Question 1.
The circular scale of a screw gauge has 50 divisions. Its spindle moves by 2 mm on sleeve, when given four complete rotations calculate
1. pitch
2. least count.
Number of circular scale divisions (C.S.D.) = 50
Distance moved by screw (spindle) on sleeve = 2 mm
Number of complete rotations given = 4
Question 2.
The circular scale of a screw gauge has 100 divisions. Its spindle moves forward by 2.5 mm when given five complete turns. Calculate
1. pitch
2. least count of the screw gauge.
Number of circular scale divisions = 100
Distance moved by spindle (screw) = 2.5 mm
No. of complete rotations given = 5
Practice Problems 2
Question 1.
Figure shows a screw gauge in which circular scale has 200 divisions. Calculate the least count and radius of wire.
No. of circular scale divisions = 200
Pitch = 1 mm
Question 2. Goyal Brothers Measurements and Experimentation Class-9
Figure shows a screw gauge in which circular scale has 100 divisions. Calculate the least count and the diameter of a wire.
No. of circular scale division = 100
Pitch =0.5 mm
Practice Problems 3
Question 1.
A micrometre screw gauge having a positive zero error of 5 divisions is used to measure diameter of wire, when reading on main scale is 3rd division and 48th circular scale division coincides with
base line. If the micrometer has 10 divisions to a centimetre on main scale and 100 divisions on circular scale, calculate
1. Pitch of screw
2. Least count of screw
3. Observed diameter
4. Corrected diameter.
Question 2.
A micrometre screw gauge has a positive zero error of 7 divisions, such that its main scale is marked in 1/2 mm and the circular scale has 100 divisions. The spindle of the screw advances by 1
division complete rotation.
If this screw gauge reading is 9 divisions on main scale and 67 divisions on circular scale for the diameter of a thin wire, calculate
1. Pitch
2. L.C.
3. Observed diameter
4. Corrected diameter.
Question 3.
The thimble of a screw gauge has 50 divisions for one rotation. The spindle advances 1 mm when the screw is turned through two rotations.
1. What is the pitch of screw?
2. What is the least count of screw gauge?
3. When the screw gauge is used to measure the diameter of wire the reading on sleeve is found to be 0.5 mm and reading on thimble is found 27 divisions. What is the diameter of wire in centimetres?
Pitch of screw gauge is the distance moved by spindle in one revolution = 1/2 = 0 – 5 mm
Practice Problems 4
Question 1.
A micrometre screw gauge has a negative zero error of 8 divisions. While measuring the diameter of a wire the reading on main scale is 3 divisions and 24th circular scale division coincides with base
If the number of divisions on the main scale are 20 to a centimetre and circular scale has 50 divisions, calculate
1. pitch
2. observed diameter.
3. least count
4. corrected diameter.
Question 2.
A micrometre screw gauge has a negative zero error of 7 divisions. While measuring the diameter of a wire the reading on main scale is 2 divisions and 79th circular scale division coincides with base
If the number of divisions on main scale is 10 to a centimetre and circular scale has 100 divisions, calculate
1. pitch
2. observed diameter
3. least count
4. corrected diameter.
Exercise 4
Exe-1 Measurements and Experimentation Class-9 Goyal Brothers ICSE Physics Solutions Ch-1
Question 1.
For what range of measurement is micrometre screw gauge used?
Micrometre screw gauge is used to measure upto the accuracy of 0.001 cm.
Question 2.
What do you understand by the following terms as applied to micrometre screw gauge?
1. Sleeve cylinder
2. Sleeve scale
3. Thimble
4. Thimble scale
5. Base line.
1. Sleeve cylinder : A hollow cylinder attached to a nut of the screw gauge is known as sleeve cylinder.
The spindle of the screw passes through sleeve cylinder
2. Sleeve scale : It is also known as main scale. A reference line or base line graduated in mm, drawn on the sleeve cylinder, parallel to axis of nut is known as sleeve scale.
3. Thimble : A hollow circular cylinder connected to the screw, which rotates along with nut on turning, is called thimble.
4. Thimble scale : It is also known as circular scale. A scale marked on tapered end of a hollow cylinder, which can move over the sleeve cylinder, is known as thimble scale.
5. Base line : A reference line drawn on the sleeve cylinder parallel to the axis of nut is known as base line.
Question 3.
What is the function of ratchet in screw gauge?
When the flattened end of the screw comes in contact with stud, ratchet becomes free and makes a rattling noise. It indicates that screw should not be further pushed towards the stud.
Question 4.
What do you understand by the terms
(a) pitch of screw
(b) least count of screw?
(a) Pitch of screw : The pitch of screw is defined as the distance between two consecutive threads he screw, measured along the axis of the screw.
(b) Least count of the screw : Least distance of the screw is defined as the smallest distance moved by its tip when the screw turn through one division marked on it.
Question 5. Goyal Brothers Measurements and Experimentation Class-9
State the formula for calculating
1. pitch of screw
2. least count of screw.
Question 6.
What do you understand by the following terms as applied to screw gauge?
(a) Zero error
(b) Positive zero error
(c) Negative zero error.
(a) Zero error : If the zero of the main scale does not coincide with zero of circular scale on bringing the screw end in contact with the stud, the screw gauge is said to have zero error.
(b) Positive zero error : If the zero of the circular scale is below the reference line of the main scale, then screw gauge is said to have positive zero error and the correction is negative.
(c) Negative zero error : If the zero of the circular scale is above the reference line of the main scale, then screw gauge is said to have negative zero error and correction is positive.
Question 7.
How do you account for (a) positive zero error (b) negative zero error, for calculating correct diameter of wires?
(a) Positive zero error : If the zero line, marked on circular scale, is below the reference line of the main scale, then there is a positive zero error and the correction is negative. In the figure
5th circular scale division is coinciding with reference line.
∴ Correction
= – Coinciding division of C.S. × L.C.
= – 5 × 0.001 cm = -0.005 cm
If the observed diameter is 0.557 cm, then:
Corrected diameter
= Observed diameter + Correction
= 0.557 cm – 0.005 cm = 0.552 cm
(b) Negative zero error : If the zero line marked on circular scale, is above the reference line of the main scale, then there is a negative error and the correction is positive.
In the figure, there is 96th division on the circualr scale which coincides with reference line.
∴ Correction – + [n- coinciding division of C.S. × L.C.]
where n is the total number of circular scale divisions.
∴ Correction = + [100 – 96] × 0.001 cm
= 0.004 cm
If observed diameter is 0.557 cm, then :
Corrected diameter
= Observed diameter + Correction
= 0. 557 cm + 0.004 cm
= 0.561 cm
Unit 5 Exercise 5
Exe-1 Measurements and Experimentation Class-9 Goyal Brothers ICSE Physics Solutions Ch-1
Question 1.
(a) What do you understand by the term volume of substance?
(b) State the unit of volume in SI system.
(a) Volume : The space occupied a substance (solid, liquid or gas) is called volume.
(b) SI unit of volume is Cubic metre (m^3).
One cubic metre : Is the volume occupied by a cube whose each side is equal to 1 m.
Question 2.
How is SI system of unit of volume is related to 1 litre ? Explain.
Question 3.
In which unit, volume of liquid is measured? How is this unit is related to S.I. unit of volume?
The volume of liquid is measured in litre of its sub-multiple millilitre (mL).
Question 4.
Explain the method in steps to find the volume of an irregular solid with the help of measuring cylinder.
Volume of an irregular solid
1. Take a measuring cylinder and fill water up to certain level. Note down the level of water in measuring cylinder. Let it be V[1].
2. Tie the irregular solid body with a thin and strong thread and lower the body gently so that the solid body is completely immersed in the water. The level of water rises. Solid body displaces
water of its own volume. Note down the new level of water. Let it be V[2].
3. Take the difference of two level of water, i.e., (V[2] – V[1]). This will give the volume of irregular solid body.
Question 5.
Amongst the units of volume (i) cm^3 (ii) m^3 (iii) litre (iv) millilitre, which is most suitable for measuring :
(a) Volume of a swimming tank
(b) Volume of a glass filled with milk
(c) Volume of an exercise book
(d) Volume of air in the room.
(a) litre
(b) cm^3
(c) millilitre
(d) m^3
Question 6.
Find the volume of a book of length 25 cm, breadth 18 cm and height 2 cm in m^3.
Question 7.
The level of water in a measuring cylinder is 12.5 ml. When a stone is lowered in it, the volume is 21.0 ml. Find the volume of the stone.
Level of water in measuring cylinder = V[1] = 12.5 ml
When stone is lowered, then level of water in measuring cylinder
Question 8.
A measuring cylinder is filled with water upto a level of 30 ml. A solid body is immersed in it so that the level of water rises to 37 ml. Now solid body is tied with a cork and then immersed in
water so that the water level rises to 40 ml. Find the volume of solid body and the cork.
Level of water in measuring cylinder = V[1] = 30 ml
Level of water in measuring cylinder when a solid body is immersed in it V[2] = 37 ml
Level of water in measuring cylinder when a cork tied with the solid is immersed in water = V[3] = 40 ml
Volume of solid body = V[2] – V[1] = 37 – 30 = 7 ml or 7 cm^3
Volume or cork = V[3] – V[2] = 40 – 37
= 3 ml or 3 cm^3 [∵ 1 ml = 1 cm^3]
Unit 6 Practice Problems
Exe-1 Measurements and Experimentation Class-9 Goyal Brothers ICSE Physics Solutions Ch-1
Question 1.
Calculate the time period of simple pendulum of length 0. 84 m when g = 9.8 ms^-2.
Question 2.
Calculate the time period of simple pendulum of length 1.44 m on the surface of moon. The acceleration due to gravity on the surface of moon is 1/6 the acceleration due to gravity on earth, [g = 9.8
Length of simple pendulum = l = 1.44 m
Time period (T) = ?
Acceleration due to gravity on the surface of moon
Practice Problems 2
Question 1.
Length of second’s pendulum is 100 cm. Find the length of another pendulum whose time period is 2.4 s.
We know time period of second’s pendulum is 2 s.
Question 2.
A pendulum of length 36 cm has time period 1.2 s. Find the time period of another pendulum, whose length is 81 cm.
Question 3.
Calculate the length of second’s pendulum on the surface of moon when acceleration due to gravity on moon is 1.63 ms^-2.
Length of second’s pendulum = l = ?
Practice Problems 3
Question 1.
The length of two pendulum are 110 cm and 27.5 cm. Calculate the ratio of their time periods.
Question 2. Goyal Brothers Measurements and Experimentation Class-9
A pendulum 100 cm and another pendulum 4 cm long are oscillating at the same time. Calculate the ratio of their time periods.
Practice Problems 4
Question 1.
The time periods of two pendulums are 1.44 s and 0.36 s respectively. Calculate the ratio of their lengths.
Question 2.
The time period of two pendulums are 2 s and 3 s respectively. Find the ratio of their lengths.
Exercise 6
Exe-1 Measurements and Experimentation Class-9 Goyal Brothers ICSE Physics Solutions Ch-1
Question 1.
(a) Define simple pendulum.
(b) State two factors which determine time period of a simple pendulum.
(c) Write an expression for the time period of a simple pendulum.
(a) Simple Pendulum : A simple pendulum consists of a heavy point mass (called bob) suspended from a rigid support by a massless, inextensible string.
(b) Factors on which time period of a simple pendulum depends :
1. i.e., if length increases, time period increases. That is why in summer pendulum of clock goes slow.
2. That is why when clock is taken to a mountain where ‘g’ decreases with altitude, time period increases and pendulum takes more time to complete an oscillation and hence the clock goes slow.
3. Mass or material of bob : Time period of simple pendulum is independent of mass.
4. Amplitude : Time period of simple pendulum is independent of amplitude. So long as swing is not too large.
(c) Expression for time period of a simple pendulum is given as :
Question 2.
Define the following in connection with a simple pendulum.
(a) Time period
(b) Oscillation
(c) Amplitude
(d) Effective length.
(a) Time period (T) : “is the time taken to complete one oscillation.” Its unit is second (s) and time period is denoted by ‘T”
(b) Oscillation : “One complete to and fro motion of the pendulum” is called an oscillation.
i.e., motion of bob from B to C and then C to B is one oscillation.
(c) Amplitude : The maximum displacement of bob from mean position on either side is called amplitude
Amplitude = AB or AC. It is denoted by ‘a’.
(d) Effective length : The length between the point of suspension and centre of gravity of bob of a pendulum is called effective length.
Question 3.
(a) What is a second’s pendulum?
(b) A second’s pendulum is taken on the surface of moon where acceleration due to gravity is l/6th of that of earth. Will the time period of pendulum remain same or increase or decrease? Give a
(a) Seconds’ pendulum : “A pendulum which has time period of two seconds” is called seconds’ pendulum.
Seconds’ pendulum may also be defined as “a pendulum which completes one oscillation in two seconds.”
(b) We know time period of a simple pendulum is inversely proportional to the square root of acceleration due to gravity
Question 4.
Which of the following do not affect the time period of a simple pendulum?
(a) mass of bob
(b) size of bob
(c) effective length of pendulum
(d) acceleration due to gravity
(e) amplitude.
(a) Mass of the bob, and (b) Size of the bob, do not affect the time period of a pendulum. Also time period of pendulum is independent of the amplitude provided this is not too great.
Question 5.
A simple pendulum is hollow from within and its time period is T. How is the time period of pendulum affected when :
(a) 1/4 of bob is filled with mercury
(b) 3/4 of bob is filled with mercury
(c) The bob is completely filled with mercury?
We know that time period of a simple pendulum is independent of its mass. So in all the above said cases, time period of simple pendulumn remains same.
Question 6.
Two simple pendulums, A and B have equal lengths but their bobs weigh 50 gf and 100 gf respectively. What would be the ratio of their time periods? What is the reason for your answer?
We know that time period of simple pendulum at a place is given by
and this expression does not contain weight of bob i.e. is independent of the weight of bob.
∴ Time period of both pendulums will be same.
∴ Ratio of their time periods =1 : 1
Question 7.
State the numerical value of the frequency of oscillation of a second’s pendulum. Does it depend on the amplitude of oscillation?
Question 8. Goyal Brothers Measurements and Experimentation Class-9
(a) Name the two factors on which time period of a simple pendulum depends.
(b) Name the devices commonly used to measure
(i) mass and
(ii) weight of a body.
(a) Factors on which time period of a simple pendulum depends :
1. That is why when clock is taken to a mountain where ‘g’ decreases with altitude, time period increases and pendulum takes more time to complete an oscillation and hence the clock goes slow.
2. Mass or material of bob : Time period of simple pendulum is independent of mass.
3. Amplitude : Time period of simple pendulum is independent of amplitude. So long as swing is not too large.
1. Mass of measured by physical balance.
2. Weight of a body is measured by spring balance.
Question 9.
Draw a graph of l, the length of simple pendulum against T^2, the square of its time period.
Nature : The graph of length (l) of simple pendulum against square of its time period (T^2) is a straight line inclined to time axis.
Question 10.
What do you understand by (a) amplitude and (b) frequency of oscillations of simple pendulum?
(a) Amplitude : The maximum displacement of bob from mean position on either side is called amplitude.
Amplitude = AB or AC. It is denoted by ‘a’.
(b) Frequency: “It is the number of vibrations or oscillations made in one second.”It is denoted by f or n and its unit is Hertz (Hz) or per second (s-1).
Unit 7 Exercise 7
Exe-1 Measurements and Experimentation Class-9 Goyal Brothers ICSE Physics Solutions Ch-1
Question 1.
(a) What do you understand by the term graph?
(b) What do you understand by the terms (i) independent variable, (ii) dependent variable?
(c) Amongst the independent variable and dependent variable, which is plotted on X-axis?
(a) Graph : A pictorial representation of two physical variables, recorded by ah experimenter is called graph.
1. Independent variable : A variable whose variation does not depend on that of another is known as independent variable.
2. ependent variable : A variable whose variation depends upon another variable is known as dependent variable.
(c) The independent variable is always plotted on x – axis.
Question 2.
(a) State how will you choose a scale for the graph.
(b) State the two ratios of a scale, which are suitable for plotting points.
(c) State the two ratios of a scale, which are not suitable for plotting points.
(a) We can choose any convenient scale to represent a given variable on a given axis, such that the whole range of variations are well spread out on the whole graph paper, to give the graph line a
suitable size.
For this a round number, nearest to or slightly less than minimum value should be taken as origin and a round number nearest to or slightly more than the maximum value should be taken at the far end
of the respective axis for a given variable.
(b) Two ratios of a scale suitable for ploting points are 1 : 2 and 1 : 4.
(c) Two ratios of a scale not suitable for plotting points are 1 : 3 and 1 : 7. Because such scales are impractical and pose difficulty in plotting intermediate points.
Question 3.
State three important precautions which must be followed while plotting points on a graph.
Precautions for plotting points on a graph :
1. The points marked on graph paper should be sharp, but not thick.
2. Ordinates of points should be written close to the plotted point.
3. It is not necessary that graph line should pass through all points. A best fit line should be drawn.
Question 4.
State two important precautions for drawing a graph line.
Precautions for drawing a graph line :
1. The graph line should be thin, single straight line and sharp.
2. It is not necessary that graph line should pass through all the points. A best fit graph line should be drawn.
Question 5.
(a) What is a best fit line for a graph?
(b) What does best fit line show regarding the variables plotted and the work of experimenter?
(a) A best bit line for a graph means a line which either passes through maximum number of points or passes closest to the maximum number of points, which appear on either side of the line.
(b) A best fit line shows that two variable quantities are directly proportional to each other. With its help, experimenter can easily understand nature of proportional relations between two variable
Question 6. Goyal Brothers Measurements and Experimentation Class-9
(a) What do you understand by the term constant of proportionality?
(b) How can proportionality constant be determined from the best fit straight line graph?
(a) Constant of proportionality : If a quantity say X is directly proportional to another quantity Y, then X is written as X = KY, where K is called constant of proportionality.
(b) Constant of proportionality : can be determined from the best fit straight line by calculating the slope of graph by using the formula. Slope of graph
Question 7.
State three uses of graph.
Uses of a graph :
(a) One can determine constant of proportionality by calculating slope of graph.
(b) It can be used to calculate mean average value of large number of observations.
(c) It can be used for verifying already known physical laws.
(d) It can also show the weakness of the experimenter at some particular instant during the course of experiment.
Question 8.
How does a graph help in determining the proportional relationship between two quantities?
It has been found that if a graph is plotted between pressure of an enclosed gas at constant temperature, against its volume, the graph line is a smooth curve, which does not meet X-axis or Y- axis
on extending as shown in figure.
From the figure, it is clear that pressure of gas is not directly proportional to volume of gas.
However, if a graph is plotted between pressure and inverse of volume, the graph line is a straight line as illustrated in figure. From the straight line graph we can say:
Pressure is inversely proportional to volume.
Similarly, if a graph is plotted between length and time period of a simple pendulum, the graph line is a curve, which has a tendency to meet X-axis or Y-axis when produced towards origin, as shown
in the figure.
From the figure, it is clear that length of a simple pendulum is not proportional to its time period.
However, if a graph is plotted between length and (Time)^2, the graph line is a straight line. Thus, we can say :
From the above discussion it is very clear that graph line helps to determine the nature of proportional relationship between two variable quantities.
-: End of Measurements and Experimentation : Goyal Brothers ICSE Physics Class-9 Solution :-
Return to — Goyal Brothers ICSE Physics Class-9
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Single case analyses
CogStat can handle comparisons when a single case (e.g., a patient) is compared to a group (e.g., a control group). The single-case hypothesis tests are based on the solution proposed by John R.
Crawford and his colleagues.
At the moment, CogStat supports the following hypothesis tests:
• Compare the performance of a single case to a control group (i.e., the extremity of a task), as introduced in Crawford & Howell (1998).
• Compare the performance of a single case to a control group when the performance is measured as a slope (i.e., the extremity of a task expressed as a slope), as introduced in Crawford &
Garthwaite (2004).
To perform other single-case hypothesis tests not supported by CogStat yet, you might use the single-case study statistic packages by John R Crawford.
How to run a single-case analysis in CogStat?
Preparing the data. Load your data, where a grouping variable will tell whether a case is a single case or a member of the control group, and where there is a dependent variable. In a slope
comparison analysis, you also need a slope standard error variable.
Your data should look like this:
Group Dependent variable
nom int
Single case 3
Control group 3
Control group 4
Control group 2
Or if you compare slope data, your data should look like this:
Group Slope Slope SE
nom int int
Single case 3 0.2
Control group 3 0.3
Control group 4 0.4
Control group 2 0.3
Performing the analysis. Then choose Analysis > Compare groups, and set the appropriate grouping variable and dependent variable. To run a slope analysis, beyond setting the grouping variable and
setting the slope values as the dependent variable, click on the Single case slope... button, set the Slope SE variable, and set the number of trials per participant.
If the grouping variable includes two groups and one of the groups includes a single case (such as in the data sources above), the single-case hypothesis tests will be chosen automatically.
Note that the modified t-test will be run only if the control group is normally distributed.
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Construction of Index Numbers: Simple & Weighted Average Methods
There are different ways of construction of index numbers. In general, construction of index number is further available for the division in two parts: Simple and Weighted. Furthermore, the simple
method is classified into simple aggregative and simple relative. Similarly, the weighted method is classified into weighted aggregative and weighted average or relative.
Suggested Videos
Types of Methods of Construction
• Simple Method – Aggregative and Relative
• Weighted Method –Â Aggregative and Relative
Let’s dive into the explanation of these methods of construction of index.
Simple Aggregative Method
We use this method of construction for computation of index price. As a result, the total cost of any commodity in any given year to the total cost of any commodity in the base year is in percentage
Simple Aggregative Price Index – (∑ P[n]/ ∑ P[0]) * 100
∑P[n] = Sum of the price of all the respective commodity in the current time period.
∑P[o ]= Sum of the price of all the respective commodity in the base period.
The simple aggregative index is very simple to understand. However, there is a serious defect in this method. The first commodity, here, has more influence than the rest two. This is so because the
first commodity has a high price than the rest.
Furthermore, if we anyhow change the units, the index number will also go through a change. This is one of the biggest flaws of this methods. Use of absolute quantities turn the tables around.
Therefore, considering independent values for the three years would be a better option.
Browse more Topics under Index Numbers
Simple Average of Relatives
In order to remove the errors and flaws coming from a simple aggregative index, a replacement would be a better choice. Hence, we can use a simple average of relatives method for construction of
Using this method, we can invert the real values for any individual variable into percentage form of the base period. We term these percentages as relatives. One of the biggest reason to opt for
relatives is that they are full numbers and have no absolute values like Rs. 35.60, Rs. 10.01, and so on. Hence, the index numbers that we get as a result is likely to stay as it is.
Weighted Method
It is quite important to meet the needs of any sort of simple or unweighted methods. So, in such a case, we weigh the value of any commodity using any factor that deems fit. This factor is usually
the quantity we sell it for during the base year. The categories of these indices are:
1. Weighted Aggregative Index
2. Weighted Average of Relatives
Let’s have a close look at the following two indices.
Weighted Aggregative Index Method
We generally use this method to weigh out the price of any commodity. The weighing is done using a very approximate factor. These factors are likely to vary and can be anything. It can be a quantity
or it can be the volume that it is selling off for during the base year.
The year not necessarily needs to be the base year but can also be an average of other years or any year in general. Well, the choice of it will totally depend on the importance of the specific year.
So, besides the quantity, it is on us about terming the importance of a specific year.
Weighted Aggregative Index generally comes off in the form of percentages. As a result, there are different formulas that we use for the same. Some of them are:
1. Laspeyres Index
Under this type of index, the quantities in the base year are the values of weights.
Formula – (∑P[n]Q[o]/∑P[o]Q[o])*100
2. Passche’s Index
Under this type of Index, the quantities in the current year are the values of weights.
Formula – (∑P[n]Q[n]/∑P[o]Q[n])*100
3. Some of the methods that depend on a typical time period:
Index (∑P[n]Q[t]/∑P[o]Q[n]) * 100, here, the subscript “t” symbolizes the typical period of time in years. The quantities of these years are the values of weight.
Note:Â Using the following formulas, the indices are subject to return the values in the form of percentages.
Marshall-Edgeworth Index
Under this type of index, we take both i.e. the current year as well as the base year into consideration for specifying the methods.
Marshall-Edgeworth Index – [∑P[n](Q[o]+ Q[n])/∑P[o](Q[o]+ Q[n])] * 100
4. Fisher’s Ideal Price Index
The geometric mean of Laspeyres’ and Paasche’s is the Fisher’s Ideal Price Index.
Formula – √[(∑P[n]Q[o]/∑P[o]Q[o])*(∑P[n]Q[n]/∑P[o]Q[n])]* 100
 Weighted Average of Relatives
We use the weighted average of relatives to avoid the disadvantage that comes along with the simple average method. Furthermore, the preference is weighted geometric mean but weighted arithmetic
mean is used otherwise. Therefore, the representation of the weighted AM using the values of base year weights is:
Formula – (∑P[n]Q[o]/∑P[o]Q[o]) * 100
Solved Examples for You
Example: For the given data find-
a)Â Simple Aggregative Index for the year 1999 over the year 1998.
b)Â Simple Aggregative Index for the year 2000 over the year 1998.
Commodity 1998 1999 2000
Cheese (100 gm) 12 15 15.60
Egg (per piece) 3 3.60 3.30
Potato (per kg) 5 6 5.70
Aggregate 20 24.60 24.60
Index 100 123 123
Simple Aggregative Index for the year 1999 over the year 1998
(∑ P[n]/ ∑ P[0]) =  ( 24.60/20.00 ) * 100 = 123
Simple Aggregative Index for the year 2000 over the year 1998
(∑ P[n]/ ∑ P[0]) =  ( 24.60/20.00 ) * 100 = 123
This concludes our discussion on the topic of simple and weighted average methods of construction of index numbers.
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[Solved] 1. Using the spreadsheet model from Case | SolutionInn
1. Using the spreadsheet model from Case 2.1 as a starting point, use Solver to find the...
1. Using the spreadsheet model from Case 2.1 as a starting point, use Solver to find the optimal set of projects to approve. The solution should maximize the total NPV from the approved projects, and
it should satisfy the constraints imposed by CEO J. R. Bayer and the functional areas: (1) capital expenditures over the three years should not exceed $10 billion; (2) capital expenditures in any
single year should not exceed $4 billion; and (3) at least one project must be approved for each functional area.
2. Cliff looks at the optimal solution from question 1 and sees that it is under budget in each year and in total for all three years. This seems to be a good sign, and he interprets it to mean that
the budget limitations aren't important after all. Explain why his interpretation is wrong. Then use SolverTable to help him see how NPV could increase with larger budgets. Specifically, vary each of
the four budget limits in question 1 (total budget, bud-get for year 1, budget for year 2, and budget for year 3) in separate SolverTable runs over reasonable ranges.
3. Continuing question 2, let all four of the budgets increase by the same percentage in another SolverTable run. (You can choose the range of percentage increases.) What effect does this have on the
total NPV?
4. The solution in question 1 still might not satisfy the functional areas. They will each get at least one project, but they will probably want more. Find the implications of promising each
functional area at least two projects.
5. Another aspect of the solution in question 1 bothers Cliff. He believes it is approving too many joint partnerships. Use SolverTable to find the implications of limiting the number of joint
partnerships to n, where n can vary from 3 to 6.
6. One aspect of the problem that has been ignored to this point is that each approved project must be led by a senior project man-ager. Cliff has identified only eight senior managers who qualify
and are available, and each of these can manage at most one project. In addition, some of these managers are not qualified to manage some projects. This in-formation is summarized in Table 6.12,
where 1 indicates that the manager is qualified for the project and 0 indicates otherwise. The company's problem is the same as before, but now extra decisions have to be made: which manager should
be assigned to each approved project? Of course, a project can't be approved unless a qualified manager is assigned to it.
This is an extension of Case 2.1 from Chapter 2, so you should read that case first. It asks you to develop a spreadsheet model using a 0-1 variable for each potential project so that Cliff Erland,
Manager for Project Development, can easily see the implications of approving any set of projects. Cliff would now like you to find the optimal set of projects to approve. Specifically, he has asked
you to do the following. Summarize all of your results in a concise memo.
Table 6.12:
Data from Case 2.1:
Ewing Natural Gas is a large energy company with headquarters in Dallas, Texas. The company offers a wide variety of energy products and has annual revenues of approximately $50 billion. Because of
the diverse nature of the company, its Manager for Project Development, Cliff Erland, is under continual pressure to manage project proposals from the functional areas of the company. At any point in
time, there might be dozens of projects at various stages requiring a wide variety of capital expenditures, promising widely varying future revenue streams, and containing varying degrees of risk.
Cliff has a difficult balancing act. The company's CEO, J.R. Bayer, is very concerned about keeping capital expenditures within a fixed budget and managing risk. The heads of the company's functional
areas are less worried about budgets and risks; they are most concerned that their pet projects are approved. Cliff also knows that many of the proposed projects, especially those requiring large
capital expenditures, must be led by senior project managers with the appropriate experience and skills, and he is keenly aware that the company has only a limited supply of such managers. Cliff is
currently about to meet with all parties involved to discuss project proposals for the next three years. He has proposals from the various functional areas for projects they would like to undertake.
Each of these is accompanied by a schedule of capital expenditures over the next three years and a financial analysis of the expected revenue streams. These lead to an NPV for each proposal, using
the company's hurdle rate of 12%. (Table 2.2.) J.R. Bayer has stated in no uncertain terms that the total of capital expenditures for the approved projects can be no more than $10 billion and that no
more than $4 billion should be spent in any single year. Unfortunately, the capital expenditures for the potential list of projects is well over $10 billion, so Cliff knows that some of these very
promising projects will not be approved. Before the big meeting, Cliff wants to be thoroughly prepared to answer all of the questions he knows he will be asked by the CEO, the functional heads, and
other interested parties. As a first step, he wants you to develop an Excel spreadsheet model that provides the following. (You will be asked to extend this analysis in cases in later chapters.)
Table 2.2:
Fantastic news! We've Found the answer you've been seeking!
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category theoryNoun
category theory (uncountable)
1. (mathematics) A branch of mathematics which deals with spaces and maps between them in abstraction, taking similar theorems from various disparate more concrete branches of mathematics and
unifying them.
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Paste Link with absolute references for Rows
Copying Data with Absolute Row References: A Guide for Excel Users
Have you ever tried to copy a formula in Excel and found that the row numbers in the formula changed, making the copied formula useless? This happens because Excel uses relative references by
default. But there's a way to prevent this: using absolute references for rows.
Here's the scenario: Imagine you have a simple formula in cell A2: =A1+1. This formula adds 1 to the value in cell A1. If you copy this formula to cell A3, you might expect it to become =A2+1, but
instead, it becomes =A2+1 (the row reference changes from A1 to A2). This can be problematic if you need to use the same formula for other cells, but with the row reference always staying the same.
The Solution: Absolute Row References
To make the row reference in your formula remain unchanged when you copy it, you need to use an absolute reference. You can do this by adding a dollar sign ($) before the row number in your formula.
For example:
This formula will always refer to cell A1, even if you copy it to other cells. The $ sign locks the row reference, making it an absolute reference.
Why Use Absolute Row References?
Here are some scenarios where using absolute row references is beneficial:
• Calculating totals: When you need to add up a column of numbers, you can use a formula with an absolute row reference for the cell containing the sum. This will allow you to easily copy the
formula to other cells and still have the sum calculated correctly.
• Applying the same formula to multiple cells with different row references: Imagine you have a spreadsheet with data in multiple columns, and you want to apply the same formula to each column.
Using absolute row references for the cell containing the data you want to use in the formula allows you to copy the formula to other cells without changing the data source.
• Creating a table of values based on a constant reference: You might have a table where you want to use a specific value from another cell to calculate values in each row. Using an absolute
reference for the cell containing that value ensures that the calculation uses the correct value for each row.
Key Points to Remember
• To make a row reference absolute, add a dollar sign ($) before the row number.
• You can use a combination of relative and absolute references in your formula. For example, =A$1+B1 would keep the row reference to A1 fixed but would allow the column reference to change when
• You can use the F4 key to toggle between different reference types (relative, absolute row, absolute column, and absolute row and column).
Beyond the Basics: Using Absolute References in Other Situations
The concept of absolute references extends beyond simply copying formulas. It's also valuable in situations like:
• Named Ranges: When you create a named range, you can use absolute references to ensure that the range always refers to the same cells, regardless of where you move or copy the range.
• Macros: You can use absolute references in VBA code to ensure that your code refers to the correct cells, even if the worksheet layout changes.
Mastering absolute references is an essential skill for any Excel user. By understanding how to use them, you can create more robust, efficient, and reliable formulas and spreadsheets. Remember to
experiment with different scenarios and see how absolute references can benefit your work.
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Solve structural analysis, heat transfer, or electromagnetic analysis problem
Domain-specific structural, heat transfer, and electromagnetic workflows are not recommended. New features might not be compatible with these workflows. For help migrating your existing code to the
unified finite element workflow, see Migration from Domain-Specific to Unified Workflow.
results = solve(fem) solves the structural, thermal, or electromagnetic problem represented by the finite element analysis model fem.
results = solve(fem,tlist) returns the solution at the times specified in tlist.
results = solve(fem,flist) returns the solution at the frequencies specified in flist.
results = solve(fem,FrequencyRange=[omega1,omega2]) solves the structural modal analysis problem represented by the finite element analysis model fem for all modes in the frequency range
[omega1,omega2]. Define omega1 as slightly lower than the lowest expected frequency and omega2 as slightly higher than the highest expected frequency. For example, if the lowest expected frequency is
zero, then use a small negative value for omega1.
results = solve(fem,DecayRange=[lambda1,lambda2]) performs an eigen decomposition of a linear thermal problem represented by the finite element analysis model fem for all modes in the decay range
[lambda1,lambda2]. The resulting modes enable you to:
• Use the modal superposition method to speed up a transient thermal analysis.
• Extract the reduced modal system to use, for example, in Simulink^®.
results = solve(fem,Snapshots=Tmatrix) obtains the modal basis of a linear or nonlinear thermal problem represented by the finite element analysis model fem using proper orthogonal decomposition
(POD). You can use the resulting modes to speed up a transient thermal analysis or, if your thermal model is linear, to extract the reduced modal system.
results = solve(fem,tlist,ModalResults=thermalModalR), results = solve(fem,tlist,ModalResults=structuralModalR), and results = solve(fem,flist,ModalResults=structuralModalR) solve a transient thermal
or structural problem or a frequency response structural problem, respectively, by using the modal superposition method to speed up computations. First, perform modal analysis to compute natural
frequencies and mode shapes in a particular frequency or decay range. Then, use this syntax to invoke the modal superposition method. The accuracy of the results depends on the modes in the modal
analysis results.
results = solve(fem,tlist,ModalResults=structuralModalR,DampingZeta=z) and results = solve(fem,flist,ModalResults=structuralModalR,DampingZeta=z) solve a transient or frequency response structural
problem with modal damping using the results of modal analysis. Here, z is the modal damping ratio.
results = solve(fem,omega) solves a harmonic electromagnetic problem represented in fem at the frequencies specified in omega.
structuralModalResults = solve(structuralModal,"FrequencyRange",[omega1,omega2]) returns the solution to the modal analysis model for all modes in the frequency range [omega1,omega2]. Define omega1
as slightly lower than the lowest expected frequency and omega2 as slightly higher than the highest expected frequency. For example, if the lowest expected frequency is zero, then use a small
negative value for omega1.
structuralTransientResults = solve(structuralTransient,tlist,ModalResults=structuralModalR) and structuralFrequencyResponseResults = solve(structuralFrequencyResponse,flist,ModalResults=
structuralModalR) solves a transient and a frequency response structural model, respectively, by using the modal superposition method to speed up computations. First, perform modal analysis to
compute natural frequencies and mode shapes in a particular frequency range. Then, use this syntax to invoke the modal superposition method. The accuracy of the results depends on the modes in the
modal analysis results.
thermalModalResults = solve(thermalModal,DecayRange=[lambda1,lambda2]) performs an eigen decomposition of a linear thermal model thermalModal for all modes in the decay range [lambda1,lambda2]. The
resulting modes enable you to:
• Use the modal superposition method to speed up a transient thermal analysis.
• Extract the reduced modal system to use, for example, in Simulink.
thermalModalResults = solve(thermalModal,Snapshots=Tmatrix) obtains the modal basis of a linear or nonlinear thermal model using proper orthogonal decomposition (POD). You can use the resulting modes
to speed up a transient thermal analysis or, if your thermal model is linear, to extract the reduced modal system.
thermalTransientResults = solve(thermalTransient,tlist,ModalResults=thermalModalR) solves a transient thermal model by using the modal superposition method to speed up computations. First, perform
modal decomposition to compute mode shapes for a particular decay range. Then, use this syntax to invoke the modal superposition method. The accuracy of the results depends on the modes in the modal
analysis results.
emagStaticResults = solve(emagmodel) returns the solution to the electrostatic, magnetostatic, or DC conduction model represented in emagmodel.
emagHarmonicResults = solve(emagmodel,Frequency=omega) returns the solution to the harmonic electromagnetic analysis model represented in emagmodel at the frequencies specified in omega.
Solve Static Structural Problem
Solve a static structural problem representing a bimetallic cable under tension.
Create and plot a bimetallic cable geometry.
gm = multicylinder([0.01 0.015],0.05);
pdegplot(gm,FaceLabels="on", ...
CellLabels="on", ...
Create an femodel object for static structural analysis and include the geometry.
model = femodel(AnalysisType="structuralStatic", ...
Specify Young's modulus and Poisson's ratio for each metal.
model.MaterialProperties(1) = ...
materialProperties(YoungsModulus=110E9, ...
model.MaterialProperties(2) = ...
materialProperties(YoungsModulus=210E9, ...
Specify that faces 1 and 4 are fixed boundaries.
model.FaceBC([1,4]) = faceBC(Constraint="fixed");
Specify the surface traction for faces 2 and 5.
model.FaceLoad([2,5]) = faceLoad(SurfaceTraction=[0;0;100]);
Generate a mesh and solve the problem.
model = generateMesh(model);
R = solve(model)
R =
StaticStructuralResults with properties:
Displacement: [1x1 FEStruct]
Strain: [1x1 FEStruct]
Stress: [1x1 FEStruct]
VonMisesStress: [23098x1 double]
Mesh: [1x1 FEMesh]
The solver finds the values of the displacement, stress, strain, and von Mises stress at the nodal locations. To access these values, use R.Displacement, R.Stress, and so on. The displacement,
stress, and strain values at the nodal locations are returned as FEStruct objects with the properties representing their components. Note that properties of an FEStruct object are read-only.
ans =
FEStruct with properties:
ux: [23098x1 double]
uy: [23098x1 double]
uz: [23098x1 double]
Magnitude: [23098x1 double]
ans =
FEStruct with properties:
sxx: [23098x1 double]
syy: [23098x1 double]
szz: [23098x1 double]
syz: [23098x1 double]
sxz: [23098x1 double]
sxy: [23098x1 double]
ans =
FEStruct with properties:
exx: [23098x1 double]
eyy: [23098x1 double]
ezz: [23098x1 double]
eyz: [23098x1 double]
exz: [23098x1 double]
exy: [23098x1 double]
Plot the deformed shape with the z-component of normal stress.
pdeplot3D(R.Mesh, ...
ColorMapData=R.Stress.szz, ...
Solve Transient Structural Problem
Solve for the transient response of a thin 3-D plate under a harmonic load at the center.
Create a geometry of a thin 3-D plate and plot it.
gm = multicuboid([5,0.05],[5,0.05],0.01);
Zoom in to see the face labels on the small plate at the center.
axis([-0.2 0.2 -0.2 0.2 -0.1 0.1])
Create an femodel object for transient structural analysis and include the geometry.
model = femodel(AnalysisType="structuralTransient", ...
Specify Young's modulus, Poisson's ratio, and the mass density of the material.
model.MaterialProperties = ...
materialProperties(YoungsModulus=210E9, ...
PoissonsRatio=0.3, ...
Specify that all faces on the periphery of the thin 3-D plate are fixed boundaries.
model.FaceBC(5:8) = faceBC(Constraint="fixed");
Apply a sinusoidal pressure load on the small face at the center of the plate.
First, define a sinusoidal load function, sinusoidalLoad, to model a harmonic load. This function accepts the load magnitude (amplitude), the location and state structure arrays, frequency, and
phase. Because the function depends on time, it must return a matrix of NaN of the correct size when state.time is NaN. Solvers check whether a problem is nonlinear or time-dependent by passing NaN
state values and looking for returned NaN values.
function Tn = sinusoidalLoad(load,location,state,Frequency,Phase)
if isnan(state.time)
Tn = NaN*(location.nx);
if isa(load,"function_handle")
load = load(location,state);
load = load(:);
% Transient model excited with harmonic load
Tn = load.*sin(Frequency.*state.time + Phase);
Now, apply a sinusoidal pressure load on face 12 by using the sinusoidalLoad function.
Pressure = 5e7;
Frequency = 25;
Phase = 0;
pressurePulse = @(location,state) ...
model.FaceLoad(12) = faceLoad(Pressure=pressurePulse);
Generate a mesh with linear elements.
model = generateMesh(model,GeometricOrder="linear",Hmax=0.2);
Specify zero initial displacement and velocity.
model.CellIC = cellIC(Displacement=[0;0;0],Velocity=[0;0;0]);
Solve the model.
tlist = linspace(0,1,300);
R = solve(model,tlist);
The solver finds the values of the displacement, velocity, and acceleration at the nodal locations. To access these values, use R.Displacement, R.Velocity, and so on. The displacement, velocity, and
acceleration values are returned as FEStruct objects with the properties representing their components. Note that properties of an FEStruct object are read-only.
ans =
FEStruct with properties:
ux: [2217x300 double]
uy: [2217x300 double]
uz: [2217x300 double]
Magnitude: [2217x300 double]
ans =
FEStruct with properties:
vx: [2217x300 double]
vy: [2217x300 double]
vz: [2217x300 double]
Magnitude: [2217x300 double]
ans =
FEStruct with properties:
ax: [2217x300 double]
ay: [2217x300 double]
az: [2217x300 double]
Magnitude: [2217x300 double]
Frequency Response Analysis
Perform frequency response analysis of a tuning fork.
Create an femodel object for frequency response analysis and include the geometry of a tuning fork in the model.
model = femodel(AnalysisType="structuralFrequency", ...
Specify Young's modulus, Poisson's ratio, and the mass density to model linear elastic material behavior. Specify all physical properties in consistent units.
model.MaterialProperties = ...
materialProperties(YoungsModulus=210E9, ...
PoissonsRatio=0.3, ...
Identify faces for applying boundary constraints and loads by plotting the geometry with the face labels.
figure("units","normalized","outerposition",[0 0 1 1])
title("Geometry with Face Labels")
Impose sufficient boundary constraints to prevent rigid body motion under applied loading. Typically, you hold a tuning fork by hand or mount it on a table. To create a simple approximation of this
boundary condition, fix a region near the intersection of tines and the handle (faces 21 and 22).
model.FaceBC([21,22]) = faceBC(Constraint="fixed");
Specify the pressure loading on a tine (face 11) as a short rectangular pressure pulse.
model.FaceLoad(11) = faceLoad(Pressure=1);
Generate a mesh. Specify the Hface name-value argument to generate a finer mesh for small faces.
model = generateMesh(model,Hmax=0.005,Hface={[3 4 9 10],0.0003});
In the frequency domain, this pressure pulse is a unit load uniformly distributed across all frequencies.
flist = linspace(0,4000,150);
R = solve(model,2*pi*flist);
Plot the vibration frequency of the tine tip, which is face 12. Find nodes on the tip face and plot the y-component of the displacement over the frequency, using one of these nodes.
excitedTineTipNodes = findNodes(model.Mesh,"region",Face=12);
tipDisp = R.Displacement.uy(excitedTineTipNodes(1),:);
Solve Modal Structural Analysis Problem
Find the fundamental (lowest) mode of a 2-D cantilevered beam, assuming prevalence of the plane-stress condition.
Specify geometric and structural properties of the beam, along with a unit plane-stress thickness.
length = 5;
height = 0.1;
E = 3E7;
nu = 0.3;
rho = 0.3/386;
Create a geometry.
gdm = [3;4;0;length;length;0;0;0;height;height];
g = decsg(gdm,'S1',('S1')');
Create an femodel object for modal structural analysis and include the geometry.
model = femodel(Analysistype="structuralModal", ...
Define a maximum element size (five elements through the beam thickness).
Generate a mesh.
Specify the structural properties and boundary constraints.
model.MaterialProperties = ...
materialProperties(YoungsModulus=E, ...
MassDensity=rho, ...
model.EdgeBC(4) = edgeBC(Constraint="fixed");
Compute the analytical fundamental frequency (Hz) using the beam theory.
I = height^3/12;
analyticalOmega1 = 3.516*sqrt(E*I/(length^4*(rho*height)))/(2*pi)
analyticalOmega1 =
Specify a frequency range that includes an analytically computed frequency and solve the model.
R = solve(model,FrequencyRange=[0,1e6])
R =
ModalStructuralResults with properties:
NaturalFrequencies: [32x1 double]
ModeShapes: [1x1 FEStruct]
Mesh: [1x1 FEMesh]
The solver finds natural frequencies and modal displacement values at nodal locations. To access these values, use R.NaturalFrequencies and R.ModeShapes.
ans = 32×1
10^5 ×
ans =
FEStruct with properties:
ux: [6511x32 double]
uy: [6511x32 double]
Magnitude: [6511x32 double]
Plot the y-component of the solution for the fundamental frequency.
title(['First Mode with Frequency ', ...
num2str(R.NaturalFrequencies(1)/(2*pi)),' Hz'])
axis equal
Solve Transient Structural Problem Using Modal Superposition Method
Solve for the transient response at the center of a 3-D beam under a harmonic load on one of its corners.
Modal Analysis
Create a beam geometry.
gm = multicuboid(0.05,0.003,0.003);
Plot the geometry with the edge and vertex labels.
view([95 5])
Create an femodel object for modal structural analysis and include the geometry.
model = femodel(AnalysisType="structuralModal", ...
Generate a mesh.
model = generateMesh(model);
Specify Young's modulus, Poisson's ratio, and the mass density of the material.
model.MaterialProperties = ...
materialProperties(YoungsModulus=210E9, ...
PoissonsRatio=0.3, ...
Specify minimal constraints on one end of the beam to prevent rigid body modes. For example, specify that edge 4 and vertex 7 are fixed boundaries.
model.EdgeBC(4) = edgeBC(Constraint="fixed");
model.VertexBC(7) = vertexBC(Constraint="fixed");
Solve the problem for the frequency range from 0 to 500,000. The recommended approach is to use a value that is slightly lower than the expected lowest frequency. Thus, use -0.1 instead of 0.
Rm = solve(model,FrequencyRange=[-0.1,500000]);
Transient Analysis
Switch the analysis type of the model to structural transient.
model.AnalysisType = "structuralTransient";
Apply a sinusoidal force on the corner opposite of the constrained edge and vertex.
First, define a sinusoidal load function, sinusoidalLoad, to model a harmonic load. This function accepts the load magnitude (amplitude), the location and state structure arrays, frequency, and
phase. Because the function depends on time, it must return a matrix of NaN of the correct size when state.time is NaN. Solvers check whether a problem is nonlinear or time-dependent by passing NaN
state values and looking for returned NaN values.
function Tn = sinusoidalLoad(load,location,state,Frequency,Phase)
if isnan(state.time)
normal = [location.nx location.ny];
if isfield(location,"nz")
normal = [normal location.nz];
Tn = NaN*normal;
if isa(load,"function_handle")
load = load(location,state);
load = load(:);
% Transient model excited with harmonic load
Tn = load.*sin(Frequency.*state.time + Phase);
Now, apply a sinusoidal force on vertex 5 by using the sinusoidalLoad function.
Force = [0,0,10];
Frequency = 7600;
Phase = 0;
forcePulse = @(location,state) ...
model.VertexLoad(5) = vertexLoad(Force=forcePulse);
Specify zero initial displacement and velocity.
model.CellIC = cellIC(Velocity=[0;0;0], ...
Specify the relative and absolute tolerances for the solver.
model.SolverOptions.RelativeTolerance = 1E-5;
model.SolverOptions.AbsoluteTolerance = 1E-9;
Solve the model using the modal results.
tlist = linspace(0,0.004,120);
Rdm = solve(model,tlist,ModalResults=Rm);
Interpolate and plot the displacement at the center of the beam.
intrpUdm = interpolateDisplacement(Rdm,0,0,0.0015);
grid on
ylabel("Center of beam displacement")
Expansion of Cantilever Beam Under Thermal Load
Find the deflection of a 3-D cantilever beam under a nonuniform thermal load. Specify the thermal load for the structural problem using the solution from a transient thermal analysis on the same
geometry and mesh.
Transient Thermal Model Analysis
Create and plot the geometry.
gm = multicuboid(0.5,0.1,0.05);
Create an femodel object for transient thermal analysis and include the geometry.
model = femodel(AnalysisType="thermalTransient", ...
Geometry = gm);
Generate a mesh.
model = generateMesh(model);
Specify the thermal properties of the material.
model.MaterialProperties = ...
materialProperties(ThermalConductivity=5e-3, ...
MassDensity=2.7*10^(-6), ...
Specify the constant temperatures applied to the left and right ends of the beam.
model.FaceBC(3) = faceBC(Temperature=100);
model.FaceBC(5) = faceBC(Temperature=0);
Specify the heat source over the entire geometry.
model.CellLoad = cellLoad(Heat=10);
Set the initial temperature.
model.CellIC = cellIC(Temperature=0);
Solve the model.
tlist = 0:1e-4:2e-4;
thermalresults = solve(model,tlist);
Plot the temperature distribution for each time step.
for n = 1:numel(thermalresults.SolutionTimes)
pdeplot3D(thermalresults.Mesh, ...
title(["Temperature at Time = " ...
Structural Analysis with Thermal Load
Switch the analysis type of the model to structural static.
model.AnalysisType = "structuralStatic";
Specify Young's modulus, Poisson's ratio, and the coefficient of thermal expansion.
model.MaterialProperties = ...
materialProperties(YoungsModulus=1e10, ...
PoissonsRatio=0.3, ...
Apply a fixed boundary condition on face 5.
model.FaceBC(5) = faceBC(Constraint="fixed");
Apply a thermal load using the transient thermal results. By default, the toolbox uses the solution for the last time step.
model.CellLoad = cellLoad(Temperature=thermalresults);
Specify the reference temperature.
model.ReferenceTemperature = 10;
Solve the structural problem.
thermalstressresults = solve(model);
Plot the deformed shape of the beam corresponding to the last step of the transient thermal solution.
pdeplot3D(thermalstressresults.Mesh, ...
"ColorMapData", ...
thermalstressresults.Displacement.Magnitude, ...
"Deformation", ...
title(["Thermal Expansion at Solution Time = " ...
Now specify the thermal loads as the thermal results for all time steps. Access the results for each step by using the filterByIndex function. For each thermal load, solve the structural problem and
plot the corresponding deformed shape of the beam.
for n = 1:numel(thermalresults.SolutionTimes)
resultsByStep = filterByIndex(thermalresults,n);
model.CellLoad = ...
thermalstressresults = solve(model);
pdeplot3D(thermalstressresults.Mesh, ...
ColorMapData = ...
thermalstressresults.Displacement.Magnitude, ...
Deformation = ...
title(["Thermal Results at Solution Time = " ...
Solve Steady-State Thermal Problem
Solve a 3-D steady-state thermal problem.
Create an femodel object for a steady-state thermal problem and include a geometry representing a block.
model = femodel(AnalysisType="thermalSteady", ...
Plot the block geometry.
pdegplot(model.Geometry, ...
FaceLabels="on", ...
axis equal
Assign material properties.
model.MaterialProperties = ...
Apply a constant temperature of 100 °C to the left side of the block (face 1) and a constant temperature of 300 °C to the right side of the block (face 3). All other faces are insulated by default.
model.FaceBC(1) = faceBC(Temperature=100);
model.FaceBC(3) = faceBC(Temperature=300);
Mesh the geometry and solve the problem.
model = generateMesh(model);
thermalresults = solve(model)
thermalresults =
SteadyStateThermalResults with properties:
Temperature: [12822x1 double]
XGradients: [12822x1 double]
YGradients: [12822x1 double]
ZGradients: [12822x1 double]
Mesh: [1x1 FEMesh]
The solver finds the temperatures and temperature gradients at the nodal locations. To access these values, use thermalresults.Temperature, thermalresults.XGradients, and so on. For example, plot
temperatures at the nodal locations.
Solve Transient Thermal Problem
Solve a 2-D transient thermal problem.
Create a geometry representing a square plate with a diamond-shaped region in its center.
SQ1 = [3; 4; 0; 3; 3; 0; 0; 0; 3; 3];
D1 = [2; 4; 0.5; 1.5; 2.5; 1.5; 1.5; 0.5; 1.5; 2.5];
gd = [SQ1 D1];
sf = 'SQ1+D1';
ns = char('SQ1','D1');
ns = ns';
g = decsg(gd,sf,ns);
xlim([-1.5 4.5])
ylim([-0.5 3.5])
axis equal
Create an femodel object for transient thermal analysis and include the geometry.
model = femodel(AnalysisType="thermalTransient", ...
For the square region, assign these thermal properties:
• Thermal conductivity is $10\text{\hspace{0.17em}}\mathrm{W}/\left(\mathrm{m}{\cdot }^{\circ }\mathrm{C}\right)$
• Mass density is $2\text{\hspace{0.17em}}\mathrm{kg}/{\mathrm{m}}^{3}$
• Specific heat is $0.1\text{\hspace{0.17em}}\mathrm{J}/\left({\mathrm{kg}\cdot }^{\circ }\mathrm{C}\right)$
model.MaterialProperties(1) = ...
materialProperties(ThermalConductivity=10, ...
MassDensity=2, ...
For the diamond region, assign these thermal properties:
• Thermal conductivity is $2\text{\hspace{0.17em}}\mathrm{W}/\left(\mathrm{m}{\cdot }^{\circ }\mathrm{C}\right)$
• Mass density is $1\text{\hspace{0.17em}}\mathrm{kg}/{\mathrm{m}}^{3}$
• Specific heat is $0.1\text{\hspace{0.17em}}\mathrm{J}/\left({\mathrm{kg}\cdot }^{\circ }\mathrm{C}\right)$
model.MaterialProperties(2) = ...
materialProperties(ThermalConductivity=2, ...
MassDensity=1, ...
Assume that the diamond-shaped region is a heat source with a density of $4\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{m}}^{2}$.
model.FaceLoad(2) = faceLoad(Heat=4);
Apply a constant temperature of 0 °C to the sides of the square plate.
model.EdgeBC([1 2 7 8]) = edgeBC(Temperature=0);
Set the initial temperature to 0 °C.
model.FaceIC = faceIC(Temperature=0);
Generate the mesh.
model = generateMesh(model);
The dynamics for this problem are very fast. The temperature reaches a steady state in about 0.1 second. To capture the most active part of the dynamics, set the solution time to logspace(-2,-1,10).
This command returns 10 logarithmically spaced solution times between 0.01 and 0.1.
tlist = logspace(-2,-1,10);
Solve the equation.
thermalresults = solve(model,tlist);
Plot the solution with isothermal lines by using a contour plot.
T = thermalresults.Temperature;
msh = thermalresults.Mesh;
Solve Transient Thermal Problem Using Modal Superposition Method
Solve a transient thermal problem by first obtaining mode shapes for a particular decay range and then using the modal superposition method.
Modal Decomposition
Create a geometry representing a square plate with a diamond-shaped region in its center.
SQ1 = [3; 4; 0; 3; 3; 0; 0; 0; 3; 3];
D1 = [2; 4; 0.5; 1.5; 2.5; 1.5; 1.5; 0.5; 1.5; 2.5];
gd = [SQ1 D1];
sf = 'SQ1+D1';
ns = char('SQ1','D1');
ns = ns';
g = decsg(gd,sf,ns);
xlim([-1.5 4.5])
ylim([-0.5 3.5])
axis equal
Create an femodel object for modal thermal analysis and include the geometry.
model = femodel(AnalysisType="thermalModal", ...
For the square region, assign these thermal properties:
• Thermal conductivity is $10\text{\hspace{0.17em}}\mathrm{W}/\left(\mathrm{m}{\cdot }^{\circ }\mathrm{C}\right)$.
• Mass density is $2\text{\hspace{0.17em}}\mathrm{kg}/{\mathrm{m}}^{3}$.
• Specific heat is $0.1\text{\hspace{0.17em}}\mathrm{J}/\left({\mathrm{kg}\cdot }^{\circ }\mathrm{C}\right)$.
model.MaterialProperties(1) = ...
materialProperties(ThermalConductivity=10, ...
MassDensity=2, ...
For the diamond region, assign these thermal properties:
• Thermal conductivity is $2\text{\hspace{0.17em}}\mathrm{W}/\left(\mathrm{m}{\cdot }^{\circ }\mathrm{C}\right)$.
• Mass density is $1\text{\hspace{0.17em}}\mathrm{kg}/{\mathrm{m}}^{3}$.
• Specific heat is $0.1\text{\hspace{0.17em}}\mathrm{J}/\left({\mathrm{kg}\cdot }^{\circ }\mathrm{C}\right)$.
model.MaterialProperties(2) = ...
materialProperties(ThermalConductivity=2, ...
MassDensity=1, ...
Assume that the diamond-shaped region is a heat source with a density of $4\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{m}}^{2}$.
model.FaceLoad(2) = faceLoad(Heat=4);
Apply a constant temperature of 0 °C to the sides of the square plate.
model.EdgeBC([1 2 7 8]) = edgeBC(Temperature=0);
Set the initial temperature to 0 °C.
model.FaceIC = faceIC(Temperature=0);
Generate the mesh.
model = generateMesh(model);
Compute eigenmodes of the model in the decay range [100,10000] ${s}^{-1}$.
RModal = solve(model,DecayRange=[100,10000])
RModal =
ModalThermalResults with properties:
DecayRates: [171x1 double]
ModeShapes: [1461x171 double]
ModeType: "EigenModes"
Mesh: [1x1 FEMesh]
Transient Analysis
Knowing the mode shapes, you can now use the modal superposition method to solve the transient thermal problem. First, switch the model analysis type to thermal transient.
model.AnalysisType = "thermalTransient";
The dynamics for this problem are very fast. The temperature reaches a steady state in about 0.1 second. To capture the most active part of the dynamics, set the solution time to logspace(-2,-1,100).
This command returns 100 logarithmically spaced solution times between 0.01 and 0.1.
tlist = logspace(-2,-1,10);
Solve the equation.
Rtransient = solve(model,tlist,ModalResults=RModal);
Plot the solution with isothermal lines by using a contour plot.
msh =
FEMesh with properties:
Nodes: [2x1461 double]
Elements: [6x694 double]
MaxElementSize: 0.1697
MinElementSize: 0.0849
MeshGradation: 1.5000
GeometricOrder: 'quadratic'
T = Rtransient.Temperature;
pdeplot(msh,XYData=T(:,end),Contour="on", ...
Snapshots for Proper Orthogonal Decomposition
Obtain POD modes of a linear thermal problem using several instances of the transient solution (snapshots).
Create an femodel object for transient thermal analysis and include a unit square geometry in the model.
model = femodel(AnalysisType="thermalTransient", ...
Plot the geometry, displaying edge labels.
xlim([-1.1 1.1])
ylim([-1.1 1.1])
Specify the thermal conductivity, mass density, and specific heat of the material.
model.MaterialProperties = ...
materialProperties(ThermalConductivity=400, ...
MassDensity=1300, ...
Set the temperature on the right edge to 100.
model.EdgeBC(2) = edgeBC(Temperature=100);
Set an initial value of 0 for the temperature.
model.FaceIC = faceIC(Temperature=0);
Generate a mesh.
model = generateMesh(model);
Solve the model for three different values of heat source and collect snapshots.
tlist = 0:10:600;
snapShotIDs = [1:10 59 60 61];
Tmatrix = [];
heatVariation = [10000 15000 20000];
for q = heatVariation
model.FaceLoad = faceLoad(Heat=q);
results = solve(model,tlist);
Tmatrix = [Tmatrix,results.Temperature(:,snapShotIDs)];
Switch the model analysis type to thermal modal.
model.AnalysisType = "thermalModal";
Compute the POD modes.
RModal = solve(model,Snapshots=Tmatrix)
RModal =
ModalThermalResults with properties:
DecayRates: [6x1 double]
ModeShapes: [1529x6 double]
SnapshotsAverage: [1529x1 double]
ModeType: "PODModes"
Mesh: [1x1 FEMesh]
Solve 2-D Electrostatic Problem
Solve an electromagnetic problem and find the electric potential and field distribution for a 2-D geometry representing a plate with a hole.
Create an femodel object for electrostatic analysis and include a geometry representing a plate with a hole.
model = femodel(AnalysisType="electrostatic",...
Plot the geometry with edge labels.
Specify the vacuum permittivity value in the SI system of units.
model.VacuumPermittivity = 8.8541878128E-12;
Specify the relative permittivity of the material.
model.MaterialProperties = ...
Apply the voltage boundary conditions on the edges framing the rectangle and the circle.
model.EdgeBC(1:4) = edgeBC(Voltage=0);
model.EdgeBC(5) = edgeBC(Voltage=1000);
Specify the charge density for the entire geometry.
model.FaceLoad = faceLoad(ChargeDensity=5E-9);
Generate the mesh.
model = generateMesh(model);
Solve the model.
R =
ElectrostaticResults with properties:
ElectricPotential: [1231x1 double]
ElectricField: [1x1 FEStruct]
ElectricFluxDensity: [1x1 FEStruct]
Mesh: [1x1 FEMesh]
Plot the electric potential and field.
pdeplot(R.Mesh,XYData=R.ElectricPotential, ...
FlowData=[R.ElectricField.Ex ...
axis equal
Solve 3-D Magnetostatic Problem
Solve a 3-D electromagnetic problem on a geometry representing a plate with a hole in its center. Plot the resulting magnetic potential and field distribution.
Create an femodel object for magnetostatic analysis and include a geometry representing a plate with a hole.
model = femodel(AnalysisType="magnetostatic", ...
Plot the geometry.
Specify the vacuum permeability value in the SI system of units.
model.VacuumPermeability = 1.2566370614e-6;
Specify the relative permeability of the material.
model.MaterialProperties = ...
Apply the magnetic potential boundary conditions on the side faces and the face bordering the hole.
model.FaceBC(3:6) = faceBC(MagneticPotential=[0;0;0]);
model.FaceBC(7) = faceBC(MagneticPotential=[0;0;0.01]);
Specify the current density for the entire geometry.
model.CellLoad = cellLoad(CurrentDensity=[0;0;0.5]);
Generate the mesh.
model = generateMesh(model);
Solve the model.
R =
MagnetostaticResults with properties:
MagneticPotential: [1x1 FEStruct]
MagneticField: [1x1 FEStruct]
MagneticFluxDensity: [1x1 FEStruct]
Mesh: [1x1 FEMesh]
Plot the z-component of the magnetic potential.
Plot the magnetic field.
pdeplot3D(R.Mesh,FlowData=[R.MagneticField.Hx ...
R.MagneticField.Hy ...
Solve 3-D DC Conduction Problem
Solve a DC conduction problem on a geometry representing a 3-D plate with a hole in its center. Plot the electric potential and the components of the current density.
Create an femodel object for DC conduction analysis and include a geometry representing a plate with a hole.
model = femodel(AnalysisType="dcConduction", ...
Plot the geometry.
Specify the conductivity of the material.
model.MaterialProperties = ...
Apply the voltage boundary conditions on the left, right, top, and bottom faces of the plate.
model.FaceBC(3:6) = faceBC(Voltage=0);
Specify the surface current density on the face bordering the hole.
model.FaceLoad(7) = faceLoad(SurfaceCurrentDensity=100);
Generate the mesh.
model = generateMesh(model);
Solve the model.
R =
ConductionResults with properties:
ElectricPotential: [4747x1 double]
ElectricField: [1x1 FEStruct]
CurrentDensity: [1x1 FEStruct]
Mesh: [1x1 FEMesh]
Plot the electric potential.
Plot the x-component of the current density.
title("x-Component of Current Density")
Plot the y-component of the current density.
title("y-Component of Current Density")
Plot the z-component of the current density.
title("z-Component of Current Density")
Use DC Conduction Solution as Current Density for Magnetostatic Analysis
Use a solution obtained by performing a DC conduction analysis to specify current density for a magnetostatic problem.
Create an femodel object for DC conduction analysis and include a geometry representing a plate with a hole.
model = femodel(AnalysisType="dcConduction", ...
Plot the geometry.
Specify the conductivity of the material.
model.MaterialProperties = ...
Apply the voltage boundary conditions on the left, right, top, and bottom faces of the plate.
model.FaceBC(3:6) = faceBC(Voltage=0);
Specify the surface current density on the face bordering the hole.
model.FaceLoad(7) = faceLoad(SurfaceCurrentDensity=100);
Generate the mesh.
model = generateMesh(model);
Solve the model.
Change the analysis type of the model to magnetostatic.
model.AnalysisType = "magnetostatic";
This model already has a quadratic mesh that you generated for the DC conduction analysis. For a 3-D magnetostatic model, the mesh must be linear. Generate a new linear mesh. The generateMesh
function creates a linear mesh by default if the model is 3-D and magnetostatic.
model = generateMesh(model);
Specify the vacuum permeability value in the SI system of units.
model.VacuumPermeability = 1.2566370614e-6;
Specify the relative permeability of the material.
model.MaterialProperties = ...
Apply the magnetic potential boundary conditions on the side faces and the face bordering the hole.
model.FaceBC(3:6) = faceBC(MagneticPotential=[0;0;0]);
model.FaceBC(7) = faceBC(MagneticPotential=[0;0;0.01]);
Specify the current density for the entire geometry using the DC conduction solution.
model.CellLoad = cellLoad(CurrentDensity=R);
Solve the problem.
Rmagnetostatic = solve(model);
Plot the x- and z-components of the magnetic potential.
pdeplot3D(Rmagnetostatic.Mesh, ...
pdeplot3D(Rmagnetostatic.Mesh, ...
Solve 2-D Magnetostatic Problem with Permanent Magnet
Solve a magnetostatic problem of a copper square with a permanent neodymium magnet in its center.
Create the unit square geometry with a circle in its center.
L = 0.8;
r = 0.25;
sq = [3 4 -L L L -L -L -L L L]';
circ = [1 0 0 r 0 0 0 0 0 0]';
gd = [sq,circ];
sf = "sq + circ";
ns = char('sq','circ');
ns = ns';
g = decsg(gd,sf,ns);
Plot the geometry with the face and edge labels.
Create an femodel object for magnetostatic analysis and include the geometry in the model.
model = femodel(AnalysisType="magnetostatic", ...
Specify the vacuum permeability value in the SI system of units.
model.VacuumPermeability = 1.2566370614e-6;
Specify the relative permeability of the copper for the square.
model.MaterialProperties(1) = ...
Specify the relative permeability of the neodymium for the circle.
model.MaterialProperties(2) = ...
Specify the magnetization magnitude for the neodymium magnet.
Specify magnetization on the circular face in the positive x-direction. Magnetization for a 2-D model is a column vector of two elements.
dir = [1;0];
model.FaceLoad(2) = faceLoad(Magnetization=M*dir);
Apply the magnetic potential boundary conditions on the edges framing the square.
model.EdgeBC(1:4) = edgeBC(MagneticPotential=0);
Generate the mesh with finer meshing near the edges of the circle.
model = generateMesh(model,Hedge={5:8,0.007});
Solve the problem, and find the resulting magnetic fields B and H. Here, $\mathit{B}=\mu \mathit{H}+{\mu }_{0}\mathit{M}$, where $\mu$ is the absolute magnetic permeability of the material, ${\mu }_
{0}$ is the vacuum permeability, and $\mathit{M}$ is the magnetization.
R = solve(model);
Bmag = sqrt(R.MagneticFluxDensity.Bx.^2 + R.MagneticFluxDensity.By.^2);
Hmag = sqrt(R.MagneticField.Hx.^2 + R.MagneticField.Hy.^2);
Plot the magnetic field B.
pdeplot(R.Mesh,XYData=Bmag, ...
FlowData=[R.MagneticFluxDensity.Bx ...
Plot the magnetic field H.
pdeplot(R.Mesh,XYData=Hmag, ...
FlowData=[R.MagneticField.Hx R.MagneticField.Hy])
Solve 2-D Harmonic Electromagnetic Problem
For an electromagnetic harmonic analysis problem, find the x- and y-components of the electric field. Solve the problem on a domain consisting of a square with a circular hole.
For the geometry, define a circle in a square, place them in one matrix, and create a set formula that subtracts the circle from the square.
SQ = [3,4,-5,-5,5,5,-5,5,5,-5]';
C = [1,0,0,1]';
C = [C;zeros(length(SQ) - length(C),1)];
gm = [SQ,C];
sf = 'SQ-C';
Create the geometry.
ns = char('SQ','C');
ns = ns';
g = decsg(gm,sf,ns);
Create an femodel object for electromagnetic harmonic analysis with an electric field type. Include the geometry in the model.
model = femodel(AnalysisType="electricHarmonic", ...
Plot the geometry with the edge labels.
xlim([-5.5 5.5])
ylim([-5.5 5.5])
Specify the vacuum permittivity and permeability values as 1.
model.VacuumPermittivity = 1;
model.VacuumPermeability = 1;
Specify the relative permittivity, relative permeability, and conductivity of the material.
model.MaterialProperties = ...
materialProperties(RelativePermittivity=1, ...
RelativePermeability=1, ...
Apply the absorbing boundary condition with a thickness of 2 on the edges of the square. Use the default attenuation rate for the absorbing region.
ffbc = farFieldBC(Thickness=2);
model.EdgeBC(1:4) = edgeBC(FarField=ffbc);
Specify an electric field on the edges of the hole.
E = @(location,state) [1;0]*exp(-1i*2*pi*location.y);
model.EdgeBC(5:8) = edgeBC(ElectricField=E);
Generate a mesh.
model = generateMesh(model,Hmax=1/2^3);
Solve the model for a frequency of $2\pi$.
result = solve(model,2*pi);
Plot the real part of the x-component of the resulting electric field.
title("Real Part of x-Component of Electric Field")
Plot the real part of the y-component of the resulting electric field.
title("Real Part of y-Component of Electric Field")
Input Arguments
fem — Finite element analysis model
femodel object
Finite element analysis model, specified as an femodel object. The model contains information about a finite element problem: analysis type, geometry, material properties, boundary conditions, loads,
initial conditions, and other parameters. Depending on the analysis type, it represents a structural, thermal, or electromagnetic problem.
Example: model = femodel(AnalysisType = "structuralStatic")
structuralStatic — Static structural analysis model
StructuralModel object
Static structural analysis model, specified as a StructuralModel object. The model contains the geometry, mesh, structural properties of the material, body loads, boundary loads, and boundary
Example: structuralmodel = createpde("structural","static-solid")
structuralTransient — Transient structural analysis model
StructuralModel object
Transient structural analysis model, specified as a StructuralModel object. The model contains the geometry, mesh, structural properties of the material, body loads, boundary loads, and boundary
Example: structuralmodel = createpde("structural","transient-solid")
structuralFrequencyResponse — Frequency response analysis structural model
StructuralModel object
Frequency response analysis structural model, specified as a StructuralModel object. The model contains the geometry, mesh, structural properties of the material, body loads, boundary loads, and
boundary conditions.
Example: structuralmodel = createpde("structural","frequency-solid")
structuralModal — Modal analysis structural model
StructuralModel object
Modal analysis structural model, specified as a StructuralModel object. The model contains the geometry, mesh, structural properties of the material, body loads, boundary loads, and boundary
Example: structuralmodel = createpde("structural","modal-solid")
tlist — Solution times for structural or thermal transient analysis
real vector
Solution times for structural or thermal transient analysis, specified as a real vector of monotonically increasing or decreasing values.
Example: 0:20
Data Types: double
flist — Solution frequencies for frequency response structural analysis
real vector
Solution frequencies for a frequency response structural analysis, specified as a real vector of monotonically increasing or decreasing values.
Example: linspace(0,4000,150)
Data Types: double
[omega1,omega2] — Frequency range for structural modal analysis
vector of two elements
Frequency range for a structural modal analysis, specified as a vector of two elements. Define omega1 as slightly lower than the lowest expected frequency and omega2 as slightly higher than the
highest expected frequency. For example, if the lowest expected frequency is zero, then use a small negative value for omega1.
Example: [-0.1,1000]
Data Types: double
structuralModalR — Modal analysis results for structural model
ModalStructuralResults object
Modal analysis results for a structural model, specified as a ModalStructuralResults object.
Example: structuralModalR = solve(structuralmodel,FrequencyRange=[0,1e6])
thermalSteadyState — Steady-state thermal analysis model
ThermalModel object
Steady-state thermal analysis model, specified as a ThermalModel object. ThermalModel contains the geometry, mesh, thermal properties of the material, internal heat source, Stefan-Boltzmann constant,
boundary conditions, and initial conditions.
Example: thermalmodel = createpde("thermal","steadystate")
thermalTransient — Transient thermal analysis model
ThermalModel object
Transient thermal analysis model, specified as a ThermalModel object. ThermalModel contains the geometry, mesh, thermal properties of the material, internal heat source, Stefan-Boltzmann constant,
boundary conditions, and initial conditions.
thermalModal — Modal thermal analysis model
ThermalModel object
Modal thermal analysis model, specified as a ThermalModel object. ThermalModel contains the geometry, mesh, thermal properties of the material, internal heat source, Stefan-Boltzmann constant,
boundary conditions, and initial conditions.
[lambda1,lambda2] — Decay range for modal thermal analysis
vector of two elements
Decay range for modal thermal analysis, specified as a vector of two elements. The solve function solves a modal thermal analysis model for all modes in the decay range.
Data Types: double
Tmatrix — Thermal model solution snapshots
Thermal model solution snapshots, specified as a matrix.
Data Types: double
thermalModalR — Modal analysis results for thermal model
ModalThermalResults object
Modal analysis results for a thermal model, specified as a ModalThermalResults object.
Example: thermalModalR = solve(thermalmodel,DecayRange=[0,1000])
z — Modal damping ratio
nonnegative number | function handle
Modal damping ratio, specified as a nonnegative number or a function handle. Use a function handle when each mode has its own damping ratio. The function must accept a vector of natural frequencies
as an input argument and return a vector of corresponding damping ratios. It must cover the full frequency range for all modes used for modal solution. For details, see Modal Damping Depending on
Data Types: double | function_handle
emagmodel — Electromagnetic model for electrostatic, magnetostatic, or DC conduction analysis
ElectromagneticModel object
Electromagnetic model for electrostatic, magnetostatic, or DC conduction analysis, specified as an ElectromagneticModel object. The model contains the geometry, mesh, material properties,
electromagnetic sources, and boundary conditions.
Example: emagmodel = createpde("electromagnetic","magnetostatic")
omega — Solution frequencies for harmonic electromagnetic analysis
nonnegative number | vector of nonnegative numbers
Solution frequencies for a harmonic electromagnetic analysis, specified as a nonnegative number or a vector of nonnegative numbers.
Data Types: double
Output Arguments
results — Structural, thermal, or electromagnetic analysis results
StaticStructuralResults object | TransientStructuralResults object | FrequencyStructuralResults object | ModalStructuralResults object | SteadyStateThermalResults object | TransientThermalResults
object | ModalThermalResults object | ElectrostaticResults object | MagnetostaticResults object | ConductionResults object | HarmonicResults object
Structural, thermal, or electromagnetic analysis results, returned as a StaticStructuralResults, TransientStructuralResults, FrequencyStructuralResults, ModalStructuralResults,
SteadyStateThermalResults, TransientThermalResults, ModalThermalResults, ElectrostaticResults, MagnetostaticResults, ConductionResults, or HarmonicResults object.
structuralStaticResults — Static structural analysis results
StaticStructuralResults object
Static structural analysis results, returned as a StaticStructuralResults object.
structuralTransientResults — Transient structural analysis results
TransientStructuralResults object
Transient structural analysis results, returned as a TransientStructuralResults object.
structuralFrequencyResponseResults — Frequency response structural analysis results
FrequencyStructuralResults object
Frequency response structural analysis results, returned as a FrequencyStructuralResults object.
structuralModalResults — Modal structural analysis results
ModalStructuralResults object
Modal structural analysis results, returned as a ModalStructuralResults object.
thermalSteadyStateResults — Steady-state thermal analysis results
SteadyStateThermalResults object
Steady-state thermal analysis results, returned as a SteadyStateThermalResults object.
thermalTransientResults — Transient thermal analysis results
TransientThermalResults object
Transient thermal analysis results, returned as a TransientThermalResults object.
thermalModalResults — Modal thermal analysis results
ModalThermalResults object
Modal thermal analysis results, returned as a ModalThermalResults object.
emagStaticResults — Electrostatic, magnetostatic or DC conduction analysis results
ElectrostaticResults object | MagnetostaticResults object | ConductionResults object
Electrostatic, magnetostatic, or DC conduction analysis results, returned as an ElectrostaticResults, MagnetostaticResults, or ConductionResults object.
emagHarmonicResults — Harmonic electromagnetic analysis results
HarmonicResults object
Harmonic electromagnetic analysis results, returned as a HarmonicResults object.
• When you use modal analysis results to solve a transient structural dynamics model, the modalresults argument must be created in Partial Differential Equation Toolbox™ from R2019a or newer.
• For a frequency response model with damping, the results are complex. Use functions such as abs and angle to obtain real-valued results, such as the magnitude and phase.
Version History
Introduced in R2017a
R2023a: Finite element model
The solver accepts the femodel object that defines structural mechanics, thermal, and electromagnetic problems.
R2022b: DC conduction and permanent magnets
You can now solve stationary current distribution in conductors due to applied voltage. You also can solve electromagnetic problems accounting for magnetization of materials.
R2022a: Harmonic electromagnetic analysis
You can now solve 2-D and 3-D time-harmonic Maxwell’s equations (the Helmholtz equation).
R2022a: Reduced-order modeling for thermal analysis
You can now compute modes of a thermal model using eigenvalue or proper orthogonal decomposition. You also can speed up computations for a transient thermal model by using the computed modes.
R2021b: 3-D electrostatic and magnetostatic problems
You can now solve 3-D electrostatic and magnetostatic problems.
R2021a: 2-D electrostatic and magnetostatic problems
You can now solve 2-D electrostatic and magnetostatic problems.
R2020a: Axisymmetric analysis
You can now solve axisymmetric structural and thermal problems. Axisymmetric analysis simplifies 3-D thermal problems to 2-D using their symmetry around the axis of rotation.
R2019b: Frequency response structural analysis
You can now solve frequency response structural problems and find displacement, velocity, acceleration, and solution frequencies at nodal locations of the mesh. To speed up computations, you can use
modal analysis results for frequency response analysis. The ModalResults argument triggers the solve function to use the modal superposition method.
R2019b: Lanczos algorithm for structural modal analysis problems
You can now specify the maximum number of Lanczos shifts and the block size for block Lanczos recurrence by using the SolverOptions property of StructuralModel. For details, see PDESolverOptions
R2019a: Modal superposition method for transient structural analysis
The new ModalResults argument triggers the solve function to switch to the modal transient solver instead of using the direct integration approach.
R2018b: Thermal stress
The solver now solves accounts for mechanical and thermal effects when solving a static structural analysis model. The function returns a displacement, stress, strain, and von Mises stress induced by
both mechanical and thermal loads.
R2018a: Transient and modal structural analyses
You can now solve dynamic linear elasticity problems and find displacement, velocity, and acceleration at nodal locations of the mesh.
You also can solve modal analysis problems and find natural frequencies and mode shapes of a structure. When solving a modal analysis model, the solver requires a frequency range parameter and
returns the modal solution in that frequency range.
R2017b: Static structural analysis
You can now solve static linear elasticity problems and find displacement, stress, strain, and von Mises stress at nodal locations of the mesh.
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How do you write an equation for a circle with Point A is (4, -2) Point B is (10,6) as diameters? | Socratic
How do you write an equation for a circle with Point A is (4, -2) Point B is (10,6) as diameters?
1 Answer
We must first find the length of the diameter and the center of the circle.
The center is an equal distance from all points inside a circle. Therefore, we can use the midpoint formula $\left(\frac{{x}_{1} + {x}_{2}}{2} , \frac{{y}_{1} + {y}_{2}}{2}\right)$ to find the
$\left(\frac{{x}_{1} + {x}_{2}}{2} , \frac{{y}_{1} + {y}_{2}}{2}\right)$
$= \left(\frac{4 + 10}{2} , \frac{- 2 + 6}{2}\right)$
$= \left(7 , 2\right)$
The center will therefore be at $\left(7 , 2\right)$.
Now for the length of the diameter.
This can be found by using the distance theorem, a simple variation on pythagorean theorem.
$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$
$d = \sqrt{{6}^{2} + {8}^{2}}$
$d = \sqrt{100}$
$d = 10$
Hence, the diameter measures $10$ units. Since the equation of the circle is of the form ${\left(x - h\right)}^{2} + {\left(y - k\right)}^{2} = {r}^{2}$, where $\left(h , k\right)$ is the center and
$r$ is the radius, we need the radius, and not the diameter. The equation $d = 2 r$ shows the relationship between the diameter (d) and the radius (r).
Solving for r:
$r = \frac{d}{2}$
$r = \frac{10}{2}$
$r = 5$
Now that we know our radius, we can substitute what we know into the equation of the circle, of the form mentioned above:
${\left(x - 7\right)}^{2} + {\left(y - 2\right)}^{2} = 25$
Here is the graph of this relation (note: it's not a function, since every value of x is not only with one value of y)
Practice exercises:
1. Determine the equation of the circle who's diameter ends at the points $\left(- 1 , - 4\right)$ and $\left(3 , - 5\right)$.
$2.$ Determine the equation of the following circle.
Hopefully this helps, and good luck!
Impact of this question
1151 views around the world
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Interpretation: Turning Data into Information - Hamnus
In your previous lessons, you’ve learned about central tendency, dispersion, and types of data. These are all crucial tools, but today we’re going to talk about how to use these tools to extract
meaningful insights from data – in other words, how to turn raw data into useful information.
First, let’s clarify the difference between data and information. Data are raw facts and figures. For example, if I give you the numbers 98.6, 99.2, 98.4, 100.1, 98.9 – that’s data. But what does it
mean? That’s where interpretation comes in. If I tell you these are body temperatures in Fahrenheit, suddenly those numbers have context and meaning. We’ve turned data into information.
In engineering, we’re constantly collecting data – from sensors, experiments, surveys, and more. But data alone doesn’t solve problems or make decisions. We need to interpret that data to derive
meaningful insights.
So, how do we go about interpreting data? There are several key steps:
1. Organize and clean the data. This might involve removing outliers, dealing with missing values, or converting units.
2. Analyze patterns and trends. This is where your knowledge of central tendency and dispersion comes in. Calculate means, medians, standard deviations. Look for patterns over time or correlations
between variables.
3. Compare with expectations or benchmarks. Is the data showing what you expected? How does it compare to industry standards or previous performance?
4. Draw conclusions. Based on your analysis, what can you infer? What does the data suggest about your process, product, or system?
5. Communicate findings. This often involves data visualization – choosing the right type of graph or chart to clearly convey your insights.
Let’s consider a practical example. Imagine you’re working in a manufacturing plant, and you’ve collected data on the diameter of ball bearings produced over a week. You calculate the mean diameter
as 10mm with a standard deviation of 0.05mm.
What can we interpret from this?
The mean tells us the average size, which is important for quality control. But the standard deviation gives us crucial information about consistency. A small standard deviation like 0.05mm suggests
that most of the ball bearings are very close to 10mm in diameter. This indicates a consistent manufacturing process.
But interpretation doesn’t stop there. We need to consider the context. Are these results good? That depends on the specifications. If the required tolerance is ±0.1mm, then this process is
performing well. If it’s ±0.01mm, we might need to improve our consistency.
This example illustrates a key point: data interpretation isn’t just about calculating numbers, it’s about understanding what those numbers mean in context and how they can inform decisions.
As engineers, your goal is to use data to solve problems, improve processes, and make informed decisions. Good interpretation skills are crucial for this. They allow you to spot trends before they
become problems, identify opportunities for optimization, and provide evidence-based recommendations.
Remember, though, interpretation has its pitfalls. Be cautious about assuming correlation implies causation. Be aware of the limitations of your data. And always consider alternative explanations for
the patterns you observe.
Bridge Vibration
Data: Accelerometer readings from a bridge over 24 hours (in m/s²): 0.05, 0.08, 0.12, 0.18, 0.22, 0.25, 0.20, 0.15, 0.10, 0.07, 0.06, 0.05, 0.04, 0.06, 0.09, 0.14, 0.19, 0.23, 0.21, 0.16, 0.11, 0.08,
0.06, 0.05
• The data shows a clear pattern with two peaks, likely corresponding to rush hour traffic in the morning and evening.
• The maximum vibration (0.25 m/s²) occurs around what’s probably the morning rush hour.
• The minimum vibration (0.04 m/s²) is during what’s likely the early morning hours.
• This information could be used to assess the bridge’s response to daily traffic loads and plan maintenance schedules during low-traffic periods.
Chemical Engineering – Reactor Efficiency
Data: Conversion rates of a chemical reactor at different temperatures: Temperature (°C): 150, 175, 200, 225, 250 Conversion Rate (%): 65, 72, 78, 82, 83
• There’s a positive correlation between temperature and conversion rate.
• The relationship appears to be non-linear, with diminishing returns at higher temperatures.
• The most significant improvement occurs between 150°C and 200°C.
• Beyond 225°C, there’s minimal improvement in conversion rate
Quality Control
Data: Measurements of product dimensions (in mm): 49.8, 50.2, 50.1, 49.9, 50.3, 50.0, 49.7, 50.2, 50.1, 49.8
• The mean is approximately 50.0 mm, which is the target dimension.
• The range is 0.6 mm (from 49.7 to 50.3), indicating some variation in the production process.
• All measurements fall within ±0.3 mm of the target, which may or may not be acceptable depending on the tolerance specifications.
• This information can be used to assess whether the manufacturing process is in control and if adjustments are needed.
Energy Consumption
Data: Monthly energy usage (in kWh) for a facility: Jan: 5000, Feb: 4800, Mar: 4600, Apr: 4200, May: 3800, Jun: 3500, Jul: 3400, Aug: 3600, Sep: 3900, Oct: 4300, Nov: 4700, Dec: 5100
• There’s a clear seasonal pattern in energy consumption.
• Highest usage is in winter months (December, January), lowest in summer (July, August).
• The difference between peak and trough is about 1700 kWh, or roughly 33% of peak consumption.
• This information could be used for energy management, budgeting, and identifying potential energy-saving measures during high-consumption periods.
Process Efficiency
Data: Production output (units per hour) for different operators: Operator A: 45, Operator B: 52, Operator C: 48, Operator D: 50, Operator E: 47
• The average production rate is 48.4 units per hour.
• There’s a range of 7 units per hour between the highest and lowest performers.
• Operator B is the most efficient, while Operator A has the lowest output.
• This information could be used to standardize best practices, identify training needs, or optimize workforce scheduling.
Material Strength Testing
Data: Tensile strength measurements (in MPa) for a new alloy: 515, 508, 522, 517, 510, 519, 513, 521, 516, 518
• The mean tensile strength is approximately 516 MPa.
• The range is 14 MPa, indicating some variability in the material properties.
• All samples exceed 500 MPa, which might be a minimum requirement for the application.
• This information can be used to assess the suitability of the alloy for its intended use and to set quality control parameters for production.
Exam Scores
Data: Scores (out of 100) from a recent engineering exam: 78, 65, 82, 90, 75, 88, 71, 79, 85, 92, 68, 83, 76, 87, 80
• The mean score is approximately 79.9.
• The median is 80, very close to the mean, suggesting a fairly symmetrical distribution.
• The range is 27 (92 – 65), indicating a spread of performance levels.
• No student scored below 65, which might indicate that the minimum learning outcomes were achieved by all.
• This information can be used to assess overall class performance, identify any need for additional support, and evaluate the exam’s difficulty level.
Project Completion Times
Data: Time taken (in days) by students to complete a semester project: 14, 18, 21, 15, 19, 22, 17, 20, 16, 23, 18, 20, 19, 21, 17
• The average completion time is about 18.7 days.
• The fastest completion was 14 days, the slowest 23 days.
• There’s a 9-day range in completion times, which might indicate varying levels of project complexity or student efficiency.
• This information could be used to refine project timelines, identify students who might need additional support, or assess whether the project scope is appropriate.
Course Enrollment Trends
Data: Number of students enrolled in an engineering course over 6 semesters: Semester 1: 45, Semester 2: 52, Semester 3: 60, Semester 4: 58, Semester 5: 65, Semester 6: 72
• There’s an overall upward trend in enrollment.
• The average enrollment is 58.7 students per semester.
• Enrollment has increased by 60% from the first to the sixth semester.
• There was a slight dip in Semester 4, but growth resumed afterward.
• This information could be used for resource allocation, classroom planning, or to investigate the factors contributing to increased popularity.
Student Feedback Ratings
Data: Student ratings (1-5 scale) for a new engineering lab equipment: 4, 3, 5, 4, 4, 3, 5, 4, 4, 5, 3, 4, 5, 4, 4
• The mean rating is approximately 4.07.
• The mode is 4, indicating that this was the most common rating.
• No ratings below 3 were given, suggesting general satisfaction.
• 33% of students gave the highest rating of 5.
• This information can be used to assess the effectiveness of the new equipment, identify areas for improvement, and make decisions about future equipment purchases.
Graduation Rates
Data: Percentage of students graduating within 4 years from different engineering departments: Mechanical: 78%, Electrical: 82%, Civil: 80%, Chemical: 85%, Computer: 79%
• The average graduation rate across departments is 80.8%.
• Chemical Engineering has the highest rate, while Mechanical has the lowest.
• There’s a 7% difference between the highest and lowest rates.
• All departments have rates above 75%, which might be considered a benchmark for success.
• This information could be used to identify best practices in departments with higher rates, allocate resources for student support, or set targets for improvement in departments with lower rates.
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Find Slope From Two Points Worksheets [PDF] (8.F.A.3): 8th Grade Math
Teaching how to find slope from two points easily.
The formula for determining the slope of a straight line having (a1, b1) and (a2, b2) as coordinate points is m = (b2-b1) / (a2-a1) or m = (b1-b2) / (a2-a1) where m represents the slope of that line
and a and b represents coordinates of X-axis and Y-axis respectively.
Steps to find a straight line’s slope:
1. Identity (a1, b1) and (a2, b2) from the given coordinates of the lines.
2. After this calculation (a2-a1) and (b2-b1).
3. Place this difference in the “m = (b2-b1) / (a2-a1)” formula.
4. Complete the final calculation.
Q. Determine the slope of a straight line having (2,4) and (3,6) as its coordinates:
a1 = 2, a2 = 3, b1 = 4, and b2 = 6.
b2-b1 = 6-4 = 2
a2-a1 = 3-1 = 2
m = (b2-b1) / (a2-a1) = 2/2 = 1.
Why Should you use a find slope from two points worksheets for your students?
Students can easily understand the concept of slope in a straight line by finding the slope from two points worksheet answers.
Also, the worksheet has various questions that will enhance students' grasp of calculating the slope of line in various situations.
Download this class 8 find slope from two points Worksheets PDF for your students. You can also try our Find Slope From Two Points Problems and Find Slope From Two Points Quiz as well for a better
understanding of the concepts.
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Find Total Interest
First, it is important to recall the concept of interest and ways to calculate it. When you borrow money from a bank, there is an extra amount to be along with. Compound Interest Calculator ·
Understanding the formula. Compound interest is calculated using the compound interest formula: A = P(1+r/n)^nt. · How to calculate. Find out how your investment will grow over time with compound
interest. Initial investment: $. 0. $ Enter the amount of money you will invest up front. How to Calculate Interest rate? · Formula: Simple Interest (SI) = Principal (P) x Rate (R) x Time (T) / ·
Example: If you invest Rs1, with a 5% annual. Example 1: What is the simple interest on the principal amount of $10, in 5 years, if the interest rate is 15% per annum? Solution: To find the simple.
For example, in this formula the 17% annual interest rate is divided by 12, the number of months in a year. The NPER argument of 2*12 is the total number of. There are different ways on how to
calculate the interest from a loan. You need to consider what kind of interest is used in your investment or loan. To calculate simple interest, multiply the principal by the interest rate and then
multiply by the loan term. · Divide the principal by the months in the loan. Simple interest is calculated only on the initial principal amount. It remains constant throughout the investment or loan
term. The formula is I = P * r * t. Compound Interest Calculator · Understanding the formula. Compound interest is calculated using the compound interest formula: A = P(1+r/n)^nt. · How to calculate.
Simple Interest Calculator - Use ClearTax simple interest calculator to calculate simple interest A = Total accrued amount (Both principal and the interest). To calculate the total amount of interest
paid over the 60 payments, first multiply the monthly payment by the total number of payments or the nper. Want to find your interest rate? Credible lets you compare rates from Maximum: Maximum
qualified loan amount or the total cost of education. Interest = interest rate / 12 * starting principal. Principal payment = monthly payment - interest. Ending principal = starting principal -.
Interest: Total Value: $. Total Principal: $. Total Interest: $. Balance by How to Calculate an Interest Rate; APY Interest Rate; APR Interest Rate.
For example, in this formula the 17% annual interest rate is divided by 12, the number of months in a year. The NPER argument of 2*12 is the total number of. Free online calculator to find the
interest rate as well as the total interest cost of an amortized loan with a fixed monthly payback amount. What is the Formula to find Total Interest Paid over life of a Loan?? · Loan amount = $19, ·
Yearly interest rate = % or · Lifespan. Calculate Months to Payoff and Total Interest Paid ; Monthly Payment You Will Make: ; Email My Results Click Here ; Amount of Next Payment Applied to
Principal. Loan Term (in years). This is the total length of the loan. Our calculator uses years to calculate the total interest accrued over this timeline. Interest Rate. Total interest charges:
This charge is the cost of borrowing money. Lenders should tell you the interest rate when you receive a loan offer. You could also find. If you'd like to calculate a total value for principal and
interest that will accrue over a particular period of time, use this slightly more involved simple. Use Excel to Find the Payment and Total Interest on a Loan · Amount of loan = 13, · Annual interest
rate = % · Length of the loan = 6 years. ( X 5 X 2/) which is equal to Rs What is the Simple Interest Formula and when is it Used? The amount one needs to pay or receive after a certain.
Calculate the total interest paid over the loan period: The total payment over the loan period is the monthly payment multiplied by the total number of payments. Simple interest is calculated with
the following formula: S.I. = (P × R × T)/, where P = Principal, R = Rate of Interest in % per annum, and T = Time. Learn more about where to find your American Express Savings account prior year's
total interest. Amount of money that you have available to invest initially. Step 2: Contribute. Monthly Contribution. Amount that you plan to add to the principal every month. You can use the
calculator below to calculate interest payments. The Average due to interest (): (calculated on unrounded total interest). Final.
Amex Trading | Does Burberry Run Big Or Small
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Show Posts
This section allows you to view all posts made by this member. Note that you can only see posts made in areas you currently have access to.
When is it a good idea to assume solutions are separable? Are there any implied assumptions about the nature of the solutions when we assume that they are separable?
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How to determine volumes of sand and stone when relaying a 4" concrete floor over an existing damaged concrete floor outside a church building
by Kwasi
(Kumasi, Ghana)
Q: Sir, I have been tasked to spearhead the laying of a 4" concrete slab over an existing damaged outside floor of my local church. our first task is to put together a budget - to get a good estimate
of the cost of the project.
We propose to lay the new floor in 3 meter square blocks. If we could calculate the quantity of materials for each block we will be able to get an good estimate of the cost. The portland cement is
sold in 50 kg bags. Our problem is the quantity of sand and agregate. We plan to use the following ratio 1:2:3 We do not have concrete sellers here and will have to do the mixing ourselves.
We are finding it difficult as to how to translate the weight of sand and aggregate given by the calculator into volumes as we purchase the sand and aggregate in volumes and we cannot weigh the
required quantity as we do not have a scale to do that. Please help us. We are in Ghana.
A: To make figuring easy, I have calculated each quantity, cement, sand, and stone per 1 square meter for a 1:2:3 mixing ratio like you mentioned.
To make 1 square meter of concrete at 4" thick you will need: 31kg of cement, 63.5kg of sand, and 93kg of stone (aggregate).
Without a scale to measure weight, the easiest way to calculate volumes of sand and stone for your purpose is to make a wooden box that will hold exactly 31kg of cement (or a little over half the
Then you can use this box to determine the volumes of sand and stone per cubic meter of concrete. 2 boxes filled for sand, 3 boxes filled for stone per cubic meter of concrete. It might not be
perfect but it will get the job done.
Once you determine the volumes you need, you can double or triple the size of the boxes to do it on a larger scale.
Join in and write your own page! It's easy to do. How? Simply click here to return to Concrete Floor Questions.
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What Does Plus 100 Mean in Betting? - Evens Betting Explained
What Does Plus 100 Mean in Betting? – Evens Betting Explained
If you place a bet at +100 it means that for every $100 you stake, you will win $100. These odds are alternatively known as ‘Evens’. We’re going to explore the question of what does plus 100 mean in
betting, so that you totally understand how they work.
How Does +100 Betting Work?
If you see odds with a plus sign before a figure, it is showing you how much you would win if you staked $100 and the bet was successful.
So odds of +100 mean that with a $100 stake on a bet, you could win $100. With a winning bet you also get your stake back, so you would receive $200 in total ($100 stake + $100 winnings).
Of course, you don’t have to stake $100. A $5 bet at +100 would return $5 in winnings, a $10 stake returns $10 in winnings and so on.
You can hopefully see why odds of +100 are alternatively known as evens. The value of your stake and the potential winnings, are dead even.
This is best demonstrated by looking at the price using fractional betting odds. With the fractional odds format, there are two numbers separated by a forward slash. The second number denotes the
stake, while the first number shows how much can be won. So with a 2/1(+200) bet, a $1 stake would win $2.
American odds of +100 are expressed in fractional odds as 1/1. The potential winnings are perfectly equal to whatever the stake is.
What Does Plus 100 Odds Mean?
Let’s take a look at what +100 means in terms of that selection’s probability of success.
Odds represent the implied probability of a bet winning. The bigger the potential winnings are in proportion to the stake, the less of a probability there is of that bet winning. On the flip-side,
the smaller the potential winnings are in proportion to the stake, the greater the possibility of that bet winning.
With odds of +100, or 1/1 in a fractional format, the stake and potential winnings are equal. Therefore, the implied probability of a +100 wager is 50%.
There is a 50/50 chance of this type of bet winning or losing.
Example of a Plus 100 Bet
Your understanding of Evens betting will be greater improved by giving you an example. Take a look at the odds in the Total market for an MLB game between the Cincinnati Reds and New York Yankees.
The Totals market is over on the right. We are betting on whether there will be over or under 9 runs in the game. The odds of over 9 runs are +100, which the bookie has expressed as Evens.
So if you bet $100 on there being over 9 runs in this game and there are 10 or more, you will win $100.
If there are exactly 9 then the bet will be tied, resulting in a push, which is where the wager is essentially cancelled and your $100 stake returned to you. Should there be less than 9 runs, your
bet will lose and the bookie gets to keep your $100 stake.
Something that you might have noticed is that while the odds of over 9 are +100, the odds of under 9 are -120.
We have established that there is a 50/50 chance of a +100 bet winning. Logically, doesn’t that mean that the other selection in this market should also have a 50/50 chance of winning and should
therefore have the same price?
Instead we have under 9 at -120, which is the favorite in this market.
Odds of +100 have an implied probability of 50%. Odds of -120 have an implied probability of 54.55%.
When you add those two implied probability figures together you get 104.55%. In any betting market the total implied probability figure will be over 100%. The extra 4.55% is known as the vig, which
is the amount of profit that the bookmaker takes from the betting market.
What Does -100 Mean in Betting?
So if you can have a +100 wager, is there -100 betting? In theory the answer is yes, though in practice, you’re unlikely to see odds expressed in this way.
Let’s take a step back and explain how plus and minus betting works at offshore sportsbooks.
With these type of odds, a minus sign indicates a favorite that is odds-on. The figure that follows the minus shows how much you have to bet to win $100. So in our example earlier, we had odds of
-120, which means that you would need to stake $120 in order to win $100.
Meanwhile, a plus sign indicates an underdog that is odds-against. The figure that follows the plus sign shows how much you could win from a $100 bet. So odds of +200 mean that from a $100 stake you
stand to win $200.
Going back to negative US odds, a -100 bet would indicate that you would have stake $100 in order to win $100. It’s therefore exactly the same as +100.
As we’ve shown, an online sportsbook is far more likely to express these odds as Evens.
Popular Plus 100 Betting Sports
Now it’s time to take a look at some of the most popular sports in the US and where you can find betting opportunities for a +100 wager.
American football is a high-scoring sport, in which teams win by relatively large margins, so you won’t often see evens available on the moneyline odds. A game would have to be perceived as being a
very close contest for that to be the case. It’s more likely on the spread, which is specifically designed to create an even betting contest.
The same is true of the main totals market, which is set at a figure where the odds will be close. There are also lots of game props and player props available, such as will there be an octopus, will
a player score a touchdown, or the passing yards of a quarterback, for example. These present plenty of bets where the probability is rated at 50/50.
Basketball is an even higher-scoring sport than football and features big margins of victory. So amongst the main betting markets, the spread, or totals, is again where you’re likelier to find odds
of +100.
Off all the popular American sports, it is basketball that has the most prop bets, particularly when it comes to players. Common types include how many points, assists, rebounds or steals a player
will record, with many more options available.
With hockey being a lower scoring sport, there is more opportunity of an Evens wager on the moneyline. This is particularly true considering that a three way moneyline is also an option for hockey,
in which the tie is an option to bet on.
The puck line – hockey’s version of a spread bet – is set at 1.5, so that too can feature odds of +100. Totals and prop options are numerable, with goalscorer, assists and goalkeeper saves available
to bet on.
Baseball is similar to hockey in that it is a low scoring sport, with the main spread – known as the run line – always set at 1.5. So on both the run line and the moneyline, you can find
opportunities for a bet at evens.
There are a wide range of total runs markets and lots of props. NRFI (no run first innings) is a popular game prop, while for players you can bet on the number of hits, home runs and strikeouts.
Other Sports
There are a wide range of other sports where 50/50 chances occur. The most popular sport to bet on across the world is soccer and it is very common to find the betting favorite on the moneyline,
available at, or around, evens.
Soccer is a sport with a deep range of markets where other such opportunities can be found. The same is true for betting on golf or tennis.
The other sport which is hugely popular amongst bettors across the world is horse racing. You might think that this would be a sport where you would see a horse rated at 50/50, but it does occur.
There might be a heavy favorite amongst the field, or it could just be a race where there are relatively few runners.
Pros & Cons of the +100 Bet Explained
Any exploration of the question of what does plus 100 mean in betting, needs to consider both the pros and cons. Here are the advantages and disadvantages of these types of bets, as we see them.
• Could not be easier to work out sports betting payouts with +100
• A common sight on a wide range of major sports
• Good balance between return on bet and likelihood of a win
• Easy to notice instances where extra value is available
• Statistically half of these bets will lose
What Does Plus 100 Mean in Betting Strategy?
By now you may be eager to scan your sportsbook account to find picks of +100 which you can bet on. Before you do, consider these betting strategies.
Try to Find Value
We know that a +100 bet has an implied probability of 50% to win. Therefore, if you saw a selection at those odds, where you thought that it actually had a chance of closer to 70% of winning, you
will have discovered some expected value in that bet.
The odds are bigger than they should be, based on your perception of the probability, compared to the bookies’ implied probability. Finding opportunities that contain such value is the key to
successful betting.
Recognize Ways in Which Odds Get Skewed
So why would a sportsbook give odds where the implied probability differs from reality? There can be a number of reasons. Many casual bettors place wagers on their favorite team, without much thought
of their real chances of winning.
In the case of sports teams with lots of fans, this keeps prices on them to win, artificially low and creates value in their opposition. Other reasons for incorrectly priced odds, could be due to
bookmakers not considering a key stat, team news, or weather conditions. Do your research to beat the bookie.
Bet in Units
A bet with odds of +100 gets us thinking about the use of a unit in sports betting. It’s a simple case of whatever you stake being potentially returned to you in winnings. A 100% profit is a good
financial investment, whichever way you look at it, but sometimes it pays to go big, or small.
Say you liked the look of a pick that was +100, but thought it only had a 40% chance of winning. Would you bet the same amount on that bet, as you would for a +100 pick where you thought it had an
80% of being a winner?
By thinking of your staking as units, you can be adaptable. You might decide that one unit is worth 1% of your bankroll. You can then decide to bet just a single unit, or half a unit if unsure about
a bet. Alternatively, go for 2-3 units, if you think it’s a dead-cert.
Is it better to bet plus or minus?
What percentage of plus 100 bets win?
Can two different outcomes on the same event have plus 100 odds?
Is plus 100 and an Evens bet the same thing?
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Physics - Online Tutor, Practice Problems & Exam Prep
Hey guys. In this video, we are going to talk about an application of electromagnetic induction to create a circuit element called the transformer. Now transformers are very important in delivering
power from power generators all the way to the home. Okay? Let's get to it. Now power in North America is delivered to the home via an outlet at 120 volts. This is typically too large for household
delicate appliances like electronics such as a laptop to operate. And in fact, the power generated at power stations isn't even at 120 volts. It has to somehow decrease by the time it gets to your
house in order to arrive at your house at 120 volts.
Now remember, whenever a coil has a changing magnetic field, I have one coil here with some magnetic field that's changing, it can induce an EMF on a second coil. There's some induced EMF on the
second coil if the magnetic field is changing. This is just what Faraday's law tells us. This is a process of electromagnetic induction. This induced EMF, if we choose these coils carefully, can be
tuned to be as small as we need. This is the concept of what a transformer is. A transformer is a circuit element. It's something that you place inside of a circuit that does exactly this. It uses
Faraday's law to convert large voltages into small EMFs.
So, I have a picture here of a very classic transformer, just 2 solenoids placed near one another. The solenoids have different numbers of turns, which is going to be important when we talk about
transformers. Now v1 is the voltage at which one solenoid operates, let's call the input solenoid, and v2 is the voltage that the second solenoid, we can call it the output solenoid operates at. Now
if v1 is changing continuously, then this magnetic field that I drew here is going to be changing as well. So the magnetic flux through this solenoid is going to be changing as well, and it's going
to produce this EMFv2. And the relationship between those voltages in the transformer depends upon the ratio of the number of turns of these solenoids. Okay? This equation governs how a transformer
works. That the ratio of the output voltage to the input voltage equals the ratio of the number of turns in the output solenoid to the input solenoid.
Alright? Let's do a quick example of this. You need to build a transformer that drops 120 volts of a regular North American outlet to a much safer 15 volts. You already have a solenoid of 50 turns
made, but you need to make a second solenoid to complete your transformer. What is the least number of turns the second solenoid could have?
Alright. So first of all, let's apply the left half of our transformer equation. V2V1 is going to be, 15, Right? 15 volts is our output voltage divided by 120. This is 1 over 8. Okay? And now the
right-hand side of this equation says this is equal to n1n2. Now all we said was that we had one solenoid with 50 turns, and we needed to make another solenoid. We never said which solenoid was the
input solenoid and which was the output solenoid. We are free to choose. And we want to choose so that we create a second solenoid with the smallest number of terms. Because this equation has two
possible outcomes. Right? We can say none is ntwo divided by 8. That's one output, or we can say ntwo is 8 times none. In either instance, the n that goes into these two equations is going to be our
50. If we plug 50 into the top equation, then we're saying that our already made solenoid is the output solenoid n2. If we plug it into the bottom equation, we're saying that that 50-turn solenoid is
our input solenoid. But either way, we can create a transformer. The question is, which one will require a second solenoid with the least number of turns? If I plug 50 into here, I get 6.25 turns. If
I plug 50 into here, I get 400 turns. So clearly, 6.25 is a smaller number than 400. So the smallest number of turns the second solenoid could have is 6.25. If the second solenoid is the input
solenoid. Right? If it's none. If we want our second solenoid to be the output solenoid, it will need 400 turns, which is not the answer to the question. The question is, what's the fewest number?
The fewest number is 6.25, and that is if our second solenoid is the input solenoid, and the solenoid that is made with 50 turns is the output solenoid.
Alright, guys, That wraps up our discussion on transformers. Thanks for watching.
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How a Bloom Filter Works
While I was getting ready to come home from college this year, I was reminded of a data structure I didn’t know much about: bloom filters. (yes, it was because of the previous Friday’s xkcd.)
I previously had seen people use them in their own projects, but I’d never tried one myself. So I decided to learn how they actually work, and what makes them useful!
Table of contents
Surface-Level Usage
A bloom filter is sort of like a hash set, where you can add items to it and check for containment. There’s one big advantage they have, and a corresponding disadvantage. - small, fixed space - A
bloom filter never grows or shrinks the memory it uses, no matter how many items you add to it. Also, it doesn’t store the items you add, but rather a bit array. This makes it pretty space-efficient.
- false positives - The more items you add, the higher the chance is that the bloom filter reports an item is contained, even if it’s never actually been added.
How to implement?
I read a lot of overviews like that one, but they didn’t answer the question I started off with: how do you implement something like this? I worked through a bloom filter implementation on
GeeksForGeeks, and I want to describe what I found the most helpful.
The backing structure of a bloom filter looks something like this: - A bit array of some fixed size - Several hash functions - I’m being vague about the numbers here, because the optimal values are
defined by formulas I still don’t understand. See the GeeksForGeeks article for those.
Then, we can add to the filter like so: 1. Use the different hash functions to generate indices within the bit array. The more hash functions, the more indices. 2. For each index, set that position
in the bit array to 1. We might have overlap with a previously added item, which is to be expected. (This is where false positives start to creep in.)
With that done, this is how we check if an item is contained in the filter: 1. Generate the same list of multiple indices as when we added, using the different hash functions. 2. Check the bit array
at each index. 1. If any of them are 0, we can be certain we never added this item. 2. If all of them are 1, we probably have added this item before.
Where do false positives come from?
False positives crop up when we are checking for containment. Consider a really simple toy example, where our bit array is of length 10 and we have 3 hash functions.
First, let’s add some items to our array. Let’s pretend that “hello” hashes to [0, 7, 3]. Then, when we add it our bit array will contain the values
bits: 1 0 0 1 0 0 0 1 0 0
i: 0 1 2 3 4 5 6 7 8 9
Then, let’s add “world” which hashes to [2, 5, 1]. Now, our bit array holds
bits: 1 1 1 1 0 1 0 1 0 0
i: 0 1 2 3 4 5 6 7 8 9
At this point, our array is starting to get pretty saturated. This is a bad sign, and it means we’re going to start seeing more and more false positives.
Imagine we want to check whether “word” is contained, and it hashes to [0, 1, 2] If we look in the bit array at these indices, we see that they are all 1. That means we return that “word” is already
contained, even though it’s not.
On the bright side, we never get a false negative (which would mean we add an item, but the filter says it is not contained). This is because all the hashed indices must be zero for our containment
function to return false, and that will never happen since an item will always hash the same way.
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1. What is a polygon?????????????
4 Answers
Polygons are shapes
Examples of Regular Polygons
And i thought it was a missing parrot
Theoretically, the plane figure bounded by the straight lines is called Polygon.
The name of the polygon are provided by their names:
For example, "Triangle" is a polygon with three sides.
"Quadrilateral(rectangle, square, rhombus,parallelogram)" is a polygon with four sides
and so on.
Hope this information will help you to some extent.
Top contributors in Mathematics category
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Set (mathematics) explained
In mathematics, a set is a collection of different things;^[1] ^[2] ^[3] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols,
points in space, lines, other geometrical shapes, variables, or even other sets. A set may have a finite number of elements or be an infinite set. There is a unique set with no elements, called the
empty set; a set with a single element is a singleton.
Sets are uniquely characterized by their elements; this means that two sets that have precisely the same elements are equal (they are the same set).^[4] This property is called extensionality. In
particular, this implies that there is only one empty set.
Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics
since the first half of the 20th century.
Definition and notation
Mathematical texts commonly denote sets by capital letters^[5] in italic, such as,, .^[6] A set may also be called a collection or family, especially when its elements are themselves sets.
Roster notation
Roster or enumeration notation defines a set by listing its elements between curly brackets, separated by commas:^[7] ^[8] ^[9] ^[10]
This notation was introduced by Ernst Zermelo in 1908.^[11] In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in
contrast, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters). For example, and represent the same set.^[12] ^[13]
For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis ''.^[14] ^[15] For instance, the set of the first thousand
positive integers may be specified in roster notation as
Infinite sets in roster notation
An infinite set is a set with an endless list of elements. To describe an infinite set in roster notation, an ellipsis is placed at the end of the list, or at both ends, to indicate that the list
continues forever. For example, the set of nonnegative integers isand the set of all integers is
Semantic definition
Another way to define a set is to use a rule to determine what the elements are:
Such a definition is called a semantic description.^[16]
Set-builder notation
See main article: Set-builder notation.
Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements.^[16] ^[17] ^[18] For example, a set can be defined as follows:
$F = \.$
In this notation, the vertical bar "|" means "such that", and the description can be interpreted as " is the set of all numbers such that is an integer in the range from 0 to 19 inclusive". Some
authors use a colon ":" instead of the vertical bar.^[19]
Classifying methods of definition
Philosophy uses specific terms to classify types of definitions:
• An intensional definition uses a rule to determine membership. Semantic definitions and definitions using set-builder notation are examples.
• An extensional definition describes a set by listing all its elements.^[16] Such definitions are also called enumerative.
• An ostensive definition is one that describes a set by giving examples of elements; a roster involving an ellipsis would be an example.
See main article: Element (mathematics). If is a set and is an element of, this is written in shorthand as, which can also be read as "x belongs to B", or "x is in B". The statement "y is not an
element of B" is written as, which can also be read as "y is not in B".^[20] ^[21]
For example, with respect to the sets,, and,
The empty set
See main article: Empty set. The empty set (or null set) is the unique set that has no members. It is denoted,
or .
Singleton sets
See main article: Singleton (mathematics). A singleton set is a set with exactly one element; such a set may also be called a unit set.^[4] Any such set can be written as, where x is the element.The
set and the element x mean different things; Halmos draws the analogy that a box containing a hat is not the same as the hat.
See main article: Subset. If every element of set A is also in B, then A is described as being a subset of B, or contained in B, written,^[25] or .^[26] The latter notation may be read B contains A,
B includes A, or B is a superset of A. The relationship between sets established by ⊆ is called inclusion or containment. Two sets are equal if they contain each other: and is equivalent to A = B.^
If A is a subset of B, but A is not equal to B, then A is called a proper subset of B. This can be written . Likewise, means B is a proper superset of A, i.e. B contains A, and is not equal to A.
A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use and to mean A is any subset of B (and not necessarily a proper subset),^[20] while others reserve and for
cases where A is a proper subset of B.^[25]
• The set of all humans is a proper subset of the set of all mammals.
• .
• .
The empty set is a subset of every set, and every set is a subset of itself:
Euler and Venn diagrams
An Euler diagram is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If is a subset of, then the region
representing is completely inside the region representing . If two sets have no elements in common, the regions do not overlap.
A Venn diagram, in contrast, is a graphical representation of sets in which the loops divide the plane into zones such that for each way of selecting some of the sets (possibly all or none), there is
a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are,, and, there should be a zone for the elements that are inside and and outside (even
if such elements do not exist).
Special sets of numbers in mathematics
There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them.
Many of these important sets are represented in mathematical texts using bold (e.g.
) or
blackboard bold
) typeface.
These include
, the set of all
natural number
(often, authors exclude);
, the set of all
s (whether positive, negative or zero):
, the set of all
rational number
s (that is, the set of all proper and improper fractions):
. For example, and ;
, the set of all
real numbers
, including all rational numbers and all
numbers (which include
algebraic number
s such as
that cannot be rewritten as fractions, as well as
transcendental numbers
such as
, the set of all
complex number
s:, for example, .
Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it.
Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example,
represents the set of positive rational numbers.
A function (or mapping) from a set to a set is a rule that assigns to each "input" element of an "output" that is an element of ; more formally, a function is a special kind of relation, one that
relates each element of to exactly one element of . A function is called
• injective (or one-to-one) if it maps any two different elements of to different elements of,
• surjective (or onto) if for every element of, there is at least one element of that maps to it, and
• bijective (or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of is paired with a unique element of, and each element of is paired with
a unique element of, so that there are no unpaired elements.
An injective function is called an injection, a surjective function is called a surjection, and a bijective function is called a bijection or one-to-one correspondence.
See main article: Cardinality.
The cardinality of a set, denoted, is the number of members of .^[28] For example, if, then . Repeated members in roster notation are not counted,^[29] ^[30] so, too.
More formally, two sets share the same cardinality if there exists a bijection between them.
The cardinality of the empty set is zero.^[31]
Infinite sets and infinite cardinality
The list of elements of some sets is endless, or infinite. For example, the set
natural numbers
is infinite.
In fact, all the special sets of numbers mentioned in the section above are infinite. Infinite sets have
infinite cardinality
Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of real numbers has greater cardinality than the set of natural
numbers.^[32] Sets with cardinality less than or equal to that of
are called
countable sets
; these are either finite sets or
countably infinite sets
(sets of the same cardinality as
); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of
are called
uncountable sets
However, it can be shown that the cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any segment of that line, of the entire plane, and indeed of
any finite-dimensional Euclidean space.^[33]
The continuum hypothesis
See main article: Continuum hypothesis. The continuum hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the
natural numbers and the cardinality of a straight line.^[34] In 1963, Paul Cohen proved that the continuum hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory
with the axiom of choice.^[35] (ZFC is the most widely-studied version of axiomatic set theory.)
Power sets
See main article: Power set. The power set of a set is the set of all subsets of . The empty set and itself are elements of the power set of, because these are both subsets of . For example, the
power set of is . The power set of a set is commonly written as or .
If has elements, then has elements. For example, has three elements, and its power set has elements, as shown above.
If is infinite (whether countable or uncountable), then is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the
elements of with the elements of will leave some elements of unpaired. (There is never a bijection from onto .)^[36]
See main article: Partition of a set.
A partition of a set S is a set of nonempty subsets of S, such that every element x in S is in exactly one of these subsets. That is, the subsets are pairwise disjoint (meaning any two sets of the
partition contain no element in common), and the union of all the subsets of the partition is S.^[37]
Basic operations
See main article: Algebra of sets.
Suppose that a universal set (a set containing all elements being discussed) has been fixed, and that is a subset of .
• The complement of is the set of all elements (of) that do not belong to . It may be denoted or . In set-builder notation,
. The complement may also be called the
absolute complement
to distinguish it from the relative complement below. Example: If the universal set is taken to be the set of integers, then the complement of the set of even integers is the set of odd integers.
Given any two sets and,
The operations above satisfy many identities. For example, one of De Morgan's laws states that (that is, the elements outside the union of and are the elements that are outside and outside).
The cardinality of is the product of the cardinalities of and .(This is an elementary fact when and are finite. When one or both are infinite, multiplication of cardinal numbers is defined to make
this true.)
The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring and intersection as the multiplication of the ring.
Sets are ubiquitous in modern mathematics. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.
One of the main applications of naive set theory is in the construction of relations. A relation from a domain to a codomain is a subset of the Cartesian product . For example, considering the set of
shapes in the game of the same name, the relation "beats" from to is the set ; thus beats in the game if the pair is a member of . Another example is the set of all pairs, where is real. This
relation is a subset of, because the set of all squares is subset of the set of all real numbers. Since for every in, one and only one pair is found in, it is called a function. In functional
notation, this relation can be written as .
Principle of inclusion and exclusion
See main article: Inclusion–exclusion principle.
The inclusion–exclusion principle is a technique for counting the elements in a union of two finite sets in terms of the sizes of the two sets and their intersection. It can be expressed symbolically
as$|A \cup B| = |A| + |B| - |A \cap B|.$
A more general form of the principle gives the cardinality of any finite union of finite sets:$\begin\left|A_\cup A_\cup A_\cup\ldots\cup A_\right|=& \left(\left|A_\right|+\left|A_\right|+\left|A_\
right|+\ldots\left|A_\right|\right) \\& - \left(\left|A_\cap A_\right|+\left|A_\cap A_\right|+\ldots\left|A_\cap A_\right|\right) \\& + \ldots \\& + \left(-1\right)^\left(\left|A_\cap A_\cap A_\cap\
ldots\cap A_\right|\right).\end$
See main article: Set theory. The concept of a set emerged in mathematics at the end of the 19th century.^[38] The German word for set, Menge, was coined by Bernard Bolzano in his work Paradoxes of
the Infinite.^[39] ^[40] ^[41] Georg Cantor, one of the founders of set theory, gave the following definition at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:^[42] ^[43]
Bertrand Russell introduced the distinction between a set and a class (a set is a class, but some classes, such as the class of all sets, are not sets; see Russell's paradox):^[44]
Naive set theory
See main article: Naive set theory. The foremost property of a set is that it can have elements, also called members. Two sets are equal when they have the same elements. More precisely, sets A and B
are equal if every element of A is an element of B, and every element of B is an element of A; this property is called the extensionality of sets. As a consequence, e.g. and represent the same set.
Unlike sets, multisets can be distinguished by the number of occurrences of an element; e.g. and represent different multisets, while and are equal. Tuples can even be distinguished by element order;
e.g. and represent different tuples.
The simple concept of a set has proved enormously useful in mathematics, but paradoxes arise if no restrictions are placed on how sets can be constructed:
• Russell's paradox shows that the "set of all sets that do not contain themselves", i.e.,, cannot exist.
• Cantor's paradox shows that "the set of all sets" cannot exist.
Naïve set theory defines a set as any well-defined collection of distinct elements, but problems arise from the vagueness of the term well-defined.
Axiomatic set theory
In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by axioms. Axiomatic set theory takes the
concept of a set as a primitive notion.^[45] The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular mathematical propositions (statements)
about sets, using first-order logic. According to Gödel's incompleteness theorems however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from
See also
• Book: Dauben, Joseph W. . Joseph Dauben . Georg Cantor: His Mathematics and Philosophy of the Infinite . Boston . . 1979 . 0-691-02447-2 . registration .
• Book: Halmos, Paul R. . Paul Halmos . Naive Set Theory . registration . Princeton, N.J. . Van Nostrand . 1960 . 0-387-90092-6 .
• Book: Stoll, Robert R. . Set Theory and Logic . Mineola, N.Y. . . 1979 . 0-486-63829-4 .
• Book: Velleman, Daniel . How To Prove It: A Structured Approach . . 2006 . 0-521-67599-5 .
External links
Notes and References
1. Book: P. K. Jain. Khalil Ahmad. Om P. Ahuja. Functional Analysis. 1995. New Age International. 978-81-224-0801-0. 1.
2. Book: Samuel Goldberg. Probability: An Introduction. 1 January 1986. Courier Corporation. 978-0-486-65252-8. 2.
3. Book: Thomas H. Cormen. Charles E Leiserson. Ronald L Rivest. Clifford Stein. Introduction To Algorithms. 2001. MIT Press. 978-0-262-03293-3. 1070.
4. Book: Stoll, Robert . Sets, Logic and Axiomatic Theories . 1974 . W. H. Freeman and Company . 5 . 9780716704577 . registration.
5. Book: Seymor Lipschutz. Marc Lipson. Schaum's Outline of Discrete Mathematics. 22 June 1997. McGraw Hill Professional. 978-0-07-136841-4. 1.
6. Web site: Introduction to Sets. 2020-08-19. www.mathsisfun.com.
7. Book: Charles Roberts. Introduction to Mathematical Proofs: A Transition. 24 June 2009. CRC Press. 978-1-4200-6956-3. 45.
8. Book: David Johnson. David B. Johnson. Thomas A. Mowry. Finite Mathematics: Practical Applications (Docutech Version). June 2004. W. H. Freeman. 978-0-7167-6297-3. 220.
9. Book: Ignacio Bello. Anton Kaul. Jack R. Britton. Topics in Contemporary Mathematics. 29 January 2013. Cengage Learning. 978-1-133-10742-2. 47.
10. Book: Susanna S. Epp. Discrete Mathematics with Applications. 4 August 2010. Cengage Learning. 978-0-495-39132-6. 13.
11. A. Kanamori, "The Empty Set, the Singleton, and the Ordered Pair", p.278. Bulletin of Symbolic Logic vol. 9, no. 3, (2003). Accessed 21 August 2023.
12. Book: Stephen B. Maurer. Anthony Ralston. Discrete Algorithmic Mathematics. 21 January 2005. CRC Press. 978-1-4398-6375-6. 11.
13. Book: D. Van Dalen. H. C. Doets. H. De Swart. Sets: Naïve, Axiomatic and Applied: A Basic Compendium with Exercises for Use in Set Theory for Non Logicians, Working and Teaching Mathematicians
and Students. 9 May 2014. Elsevier Science. 978-1-4831-5039-0. 1.
14. Book: Alfred Basta. Stephan DeLong. Nadine Basta. Mathematics for Information Technology. 1 January 2013. Cengage Learning. 978-1-285-60843-3. 3.
15. Book: Laura Bracken. Ed Miller. Elementary Algebra. 15 February 2013. Cengage Learning. 978-0-618-95134-5. 36.
16. Book: Frank Ruda. Hegel's Rabble: An Investigation into Hegel's Philosophy of Right. 6 October 2011. Bloomsbury Publishing. 978-1-4411-7413-0. 151.
17. Book: John F. Lucas. Introduction to Abstract Mathematics. 1990. Rowman & Littlefield. 978-0-912675-73-2. 108.
18. Web site: Weisstein. Eric W.. Set. 2020-08-19. Wolfram MathWorld . en.
19. Book: Ralph C. Steinlage. College Algebra. 1987. West Publishing Company. 978-0-314-29531-6.
20. Book: Marek Capinski. Peter E. Kopp. Measure, Integral and Probability. 2004. Springer Science & Business Media. 978-1-85233-781-0. 2.
21. Web site: Set Symbols. 2020-08-19. www.mathsisfun.com.
22. Book: K.T. Leung. Doris Lai-chue Chen. Elementary Set Theory, Part I/II. 1 July 1992. Hong Kong University Press. 978-962-209-026-2. 27.
23. Book: Aggarwal, M.L.. Understanding ISC Mathematics Class XI. 1. Arya Publications (Avichal Publishing Company). 2021. 1. Sets. A=3.
24. Book: Sourendra Nath, De. Chhaya Ganit (Ekadash Shreni). Scholar Books Pvt. Ltd.. January 2015. Unit-1 Sets and Functions: 1. Set Theory. 5.
25. Book: Felix Hausdorff. Set Theory. 2005. American Mathematical Soc.. 978-0-8218-3835-8. 30.
26. Book: Peter Comninos. Mathematical and Computer Programming Techniques for Computer Graphics. 6 April 2010. Springer Science & Business Media. 978-1-84628-292-8. 7.
27. Book: George Tourlakis. Lectures in Logic and Set Theory: Volume 2, Set Theory. 13 February 2003. Cambridge University Press. 978-1-139-43943-5. 137.
28. Book: Yiannis N. Moschovakis. Notes on Set Theory. 1994. Springer Science & Business Media. 978-3-540-94180-4.
29. Book: Arthur Charles Fleck. Formal Models of Computation: The Ultimate Limits of Computing. 2001. World Scientific. 978-981-02-4500-9. 3.
30. Book: William Johnston. The Lebesgue Integral for Undergraduates. 25 September 2015. The Mathematical Association of America. 978-1-939512-07-9. 7.
31. Book: Karl J. Smith. Mathematics: Its Power and Utility. 7 January 2008. Cengage Learning. 978-0-495-38913-2. 401.
32. Book: John Stillwell. The Real Numbers: An Introduction to Set Theory and Analysis. 16 October 2013. Springer Science & Business Media. 978-3-319-01577-4.
33. Book: David Tall. Advanced Mathematical Thinking. 11 April 2006. Springer Science & Business Media. 978-0-306-47203-9. 211.
34. Georg . Cantor . Ein Beitrag zur Mannigfaltigkeitslehre . . 1878 . 84 . 1878 . 242–258 . 10.1515/crll.1878.84.242.
35. Paul J. . Cohen . The Independence of the Continuum Hypothesis . Proceedings of the National Academy of Sciences of the United States of America . 50 . 6 . December 15, 1963 . 1143–1148 . 10.1073
/pnas.50.6.1143 . 16578557 . 221287 . 71858 . 1963PNAS...50.1143C. free .
36. Book: Edward B. Burger. Michael Starbird. The Heart of Mathematics: An invitation to effective thinking. 18 August 2004. Springer Science & Business Media. 978-1-931914-41-3. 183.
37. Book: Toufik Mansour. Combinatorics of Set Partitions. 27 July 2012. CRC Press. 978-1-4398-6333-6.
38. Book: José Ferreirós. Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. 16 August 2007. Birkhäuser Basel. 978-3-7643-8349-7.
39. Book: Steve Russ. The Mathematical Works of Bernard Bolzano. 9 December 2004. OUP Oxford. 978-0-19-151370-1.
40. Book: William Ewald. William Bragg Ewald. From Kant to Hilbert Volume 1: A Source Book in the Foundations of Mathematics. 1996. OUP Oxford. 978-0-19-850535-8. 249.
41. Book: Paul Rusnock. Jan Sebestík. Bernard Bolzano: His Life and Work. 25 April 2019. OUP Oxford. 978-0-19-255683-7. 430.
42. Beiträge zur Begründung der transfiniten Mengenlehre (1) . Georg Cantor . Mathematische Annalen . 46 . 4 . 481 - 512 . Nov 1895 . German.
43. Book: By an 'aggregate' (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Ganzen) M of definite and separate objects m of our intuition or our thought.. Cantor .
Georg . Jourdain . ((Philip E.B. (Translator))) . 1915 . Contributions to the founding of the theory of transfinite numbers . New York Dover Publications (1954 English translation) . Here: p.85
44. Book: Jose Ferreiros. Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. 1 November 2001. Springer Science & Business Media. 978-3-7643-5749-8.
45. Web site: Raatikainen . Panu . 2022 . Zalta . Edward N. . Gödel’s Incompleteness Theorems . 2024-06-03 . Stanford Encyclopedia of Philosophy . Metaphysics Research Lab, Stanford University.
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Lesson 21
Graphing Linear Inequalities in Two Variables (Part 1)
• Let’s find out how to use graphs to represent solutions to inequalities in two variables.
21.1: Math Talk: Less Than, Equal to, or More Than 12?
Here is an expression: \(2x+3y\).
Decide if the values in each ordered pair, \((x, y)\), make the value of the expression less than, greater than, or equal to 12.
\((0, 5)\)
\((\text-1, \text-1)\)
21.2: Solutions and Not Solutions
Here are four inequalities. Study each inequality assigned to your group and work with your group to:
• Find some coordinate pairs that represent solutions to the inequality and some coordinate pairs that do not represent solutions.
• Plot both sets of points. Either use two different colors or two different symbols like X and O.
• Plot enough points until you start to see the region that contains solutions and the region that contains non-solutions. Look for a pattern describing the region where solutions are plotted.
21.3: Sketching Solutions to Inequalities
1. Here is a graph that represents solutions to the equation \(x-y=5\).
Sketch 4 quick graphs representing the solutions to each of these inequalities:
2. For each graph, write an inequality whose solutions are represented by the shaded part of the graph.
1. The points \((7,3)\) and \((7,5)\) are both in the solution region of the inequality \(x - 2y < 3\).
1. Compute \(x-2y\) for both of these points.
2. Which point comes closest to satisfying the equation \(x-2y=3\)? That is, for which \((x,y)\) pair is \(x-2y\) closest to 3?
2. The points \((3,2)\) and \((5,2)\) are also in the solution region. Which of these points comes closest to satisfying the equation \(x-2y=3\)?
3. Find a point in the solution region that comes even closer to satisfying the equation \(x-2y=3\). What is the value of \(x-2y\)?
4. For the points \((5,2)\) and \((7,3)\), \(x-2y=1\). Find another point in the solution region for which \(x-2y=1\).
5. Find \(x-2y\) for the point \((5,3)\). Then find two other points that give the same answer.
The equation \(x+y = 7\) is an equation in two variables. Its solution is any pair of \(x\) and \(y\) whose sum is 7. The pairs \(x=0, y=7\) and \(x =\text5, y= 2\) are two examples.
We can represent all the solutions to \(x+y = 7\) by graphing the equation on a coordinate plane.
The graph is a line. All the points on the line are solutions to \(x+y = 7\).
The inequality \(x+y \leq 7\) is an inequality in two variables. Its solution is any pair of \(x\) and \(y\) whose sum is 7 or less than 7.
This means it includes all the pairs that are solutions to the equation \(x+y=7\), but also many other pairs of \(x\) and \(y\) that add up to a value less than 7. The pairs \(x=4, y=\text-7\) and \
(x=\text-6, y=0\) are two examples.
On a coordinate plane, the solution to \(x+y \leq 7\) includes the line that represents \(x+y=7\). If we plot a few other \((x,y)\) pairs that make the inequality true, such as \((4, \text-7)\) and \
((\text-6,0)\), we see that these points fall on one side of the line. (In contrast, \((x,y)\) pairs that make the inequality false fall on the other side of the line.)
We can shade that region on one side of the line to indicate that all points in it are solutions.
What about the inequality \(x+y <7\)?
The solution is any pair of \(x\) and \(y\) whose sum is less than 7. This means pairs like \(x=0, y=7\) and \(x =5, y=2\) are not solutions.
On a coordinate plane, the solution does not include points on the line that represent \(x+y=7\) (because those points are \(x\) and \(y\) pairs whose sum is 7).
To exclude points on that boundary line, we can use a dashed line.
All points below that line are \((x,y)\) pairs that make \(x+y<7\) true. The region on that side of the line can be shaded to show that it contains the solutions.
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Limit of x/cosx when x approaches 0 - iMath
The value of the limit of x/cosx is equal to 0 when x approaches to 0. Here we will learn how to find the limit of x/cosx when x tends 0. The formula of x/cosx limit as x goes to 0 is given below:
lim[x→0] $\dfrac{x}{\cos x}$ = 0.
Proof of limit x/cosx is 0 when x→0
We can directly show the limit of x/cosx is equal to zero when x→0. We proceed as follows.
lim[x→0] $\dfrac{x}{\cos x}$
= $\dfrac{\lim\limits_{x \to 0} x}{\lim\limits_{x \to 0} \cos x}$ by the quotient rule of limits.
= $\dfrac{0}{\cos 0}$
= $\dfrac{0}{1}$ as the value of cos0 is 1.
= 0
So the limit of x/cosx is equal to 0 when x approaches to 0.
By the same technique, we can show that
• the limit of x/cos2x is equal to 0 as x tends to 0, that is, lim[x→0] $\dfrac{x}{\cos 2x}$ = 0.
• lim[x→0] $\dfrac{x}{\cos ax}$ = 0 for any number a.
• lim[x→0] $\dfrac{2x}{\cos x}$ = 0.
Epsilon-delta definition of limit
Q1: What is the limit of x/cosx when x approaches 0?
Answer: The limit of x/cosx is equal to 0 when x approaches to 0, that is, lim[x→0] (x/cosx) = 0.
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Transfer function estimate
txy = tfestimate(x,y) finds a transfer function estimate between the input signal x and the output signal y evaluated at a set of frequencies.
• If x and y are both vectors, they must have the same length.
• If one of the signals is a matrix and the other is a vector, then the length of the vector must equal the number of rows in the matrix. The function expands the vector and returns a matrix of
column-by-column transfer function estimates.
• If x and y are matrices with the same number of rows but different numbers of columns, then txy is a multi-input/multi-output (MIMO) transfer function that combines all input and output signals.
txy is a three-dimensional array. If x has m columns and y has n columns, then txy has n columns and m pages. See Transfer Function for more information.
• If x and y are matrices of equal size, then tfestimate operates column-wise: txy(:,n) = tfestimate(x(:,n),y(:,n)). To obtain a MIMO estimate, append 'mimo' to the argument list.
txy = tfestimate(x,y,window) uses window to divide x and y into segments and perform windowing.
txy = tfestimate(x,y,window,noverlap) uses noverlap samples of overlap between adjoining segments.
txy = tfestimate(___,'mimo') computes a MIMO transfer function for matrix inputs. This syntax can include any combination of input arguments from previous syntaxes.
[txy,w] = tfestimate(___) returns a vector of normalized frequencies, w, at which the transfer function is estimated.
[txy,f] = tfestimate(___,fs) returns a vector of frequencies, f, expressed in terms of the sample rate, fs, at which the transfer function is estimated. fs must be the sixth numeric input to
tfestimate. To input a sample rate and still use the default values of the preceding optional arguments, specify these arguments as empty [].
[txy,w] = tfestimate(x,y,window,noverlap,w) returns the transfer function estimate at the normalized frequencies specified in w.
[txy,f] = tfestimate(x,y,window,noverlap,f,fs) returns the transfer function estimate at the frequencies specified in f.
[___] = tfestimate(x,y,___,freqrange) returns the transfer function estimate over the frequency range specified by freqrange. Valid options for freqrange are 'onesided', 'twosided', and 'centered'.
[___] = tfestimate(___,'Estimator',est) estimates transfer functions using the estimator est. Valid options for est are 'H1' and 'H2'.
tfestimate(___) with no output arguments plots the transfer function estimate in the current figure window.
Transfer Function Between Two Sequences
Compute and plot the transfer function estimate between two sequences, x and y. The sequence x consists of white Gaussian noise. y results from filtering x with a 30th-order lowpass filter with
normalized cutoff frequency $0.2\pi$ rad/sample. Use a rectangular window to design the filter. Specify a sample rate of 500 Hz and a Hamming window of length 1024 for the transfer function estimate.
h = fir1(30,0.2,rectwin(31));
x = randn(16384,1);
y = filter(h,1,x);
fs = 500;
Verify that the transfer function approximates the frequency response of the filter.
Obtain the same result by returning the transfer function estimate in a variable and plotting its absolute value in decibels.
[Txy,f] = tfestimate(x,y,1024,[],[],fs);
SISO Transfer Function
Estimate the transfer function for a simple single-input/single-output system and compare it to the definition.
A one-dimensional discrete-time oscillating system consists of a unit mass, $\mathit{m}$ (in kg), attached to a wall by a spring of unit elastic constant. A sensor samples the acceleration, $\mathit
{a}$, of the mass at ${\mathit{F}}_{\mathit{s}}=1$ Hz. A damper impedes the motion of the mass by exerting on it a force proportional to speed, with damping constant $\mathit{b}=0.01$ kg/s.
Generate 2000 time samples. Define the sampling interval $\Delta \mathit{t}=1/{\mathit{F}}_{\mathit{s}}$.
Fs = 1;
dt = 1/Fs;
N = 2000;
t = dt*(0:N-1);
b = 0.01;
The system can be described by the state-space model
$\begin{array}{c}x\left(k+1\right)=Ax\left(k\right)+Bu\left(k\right),\\ y\left(k\right)=Cx\left(k\right)+Du\left(k\right),\end{array}$
where $\mathit{x}={\left[\begin{array}{cc}\mathit{r}& \mathit{v}\end{array}\right]}^{\mathit{T}}$ is the state vector, $\mathit{r}$ and $\mathit{v}$ are respectively the position and velocity of the
mass, $\mathit{u}$ is the driving force, and $\mathit{y}=\mathit{a}$ is the measured output. The state-space matrices are
$A=\mathrm{exp}\left({A}_{c}\Delta t\right),\phantom{\rule{1em}{0ex}}B={A}_{c}^{-1}\left(A-I\right){B}_{c},\phantom{\rule{1em}{0ex}}C=\left[\begin{array}{cc}-1& -b\end{array}\right],\phantom{\rule
$\mathit{I}$ is the $2×2$ identity, and the continuous-time state-space matrices are
${A}_{c}=\left[\begin{array}{cc}0& 1\\ -1& -b\end{array}\right],\phantom{\rule{1em}{0ex}}{B}_{c}=\left[\begin{array}{c}0\\ 1\end{array}\right].$
Ac = [0 1;-1 -b];
A = expm(Ac*dt);
Bc = [0;1];
B = Ac\(A-eye(size(A)))*Bc;
C = [-1 -b];
D = 1;
The mass is driven by random input for half of the measurement interval. Use the state-space model to compute the time evolution of the system starting from an all-zero initial state. Plot the
acceleration of the mass as a function of time.
u = zeros(1,N);
u(1:N/2) = randn(1,N/2);
y = 0;
x = [0;0];
for k = 1:N
y(k) = C*x + D*u(k);
x = A*x + B*u(k);
Estimate the transfer function of the system as a function of frequency. Use 2048 DFT points and specify a Kaiser window with a shape factor of 15. Use the default value of overlap between adjoining
nfs = 2048;
wind = kaiser(N,15);
[txy,ft] = tfestimate(u,y,wind,[],nfs,Fs);
The frequency-response function of a discrete-time system can be expressed as the Z-transform of the time-domain transfer function of the system, evaluated at the unit circle. Verify that the
estimate computed by tfestimate coincides with this definition.
[b,a] = ss2tf(A,B,C,D);
fz = 0:1/nfs:1/2-1/nfs;
z = exp(2j*pi*fz);
frf = polyval(b,z)./polyval(a,z);
hold on
hold off
ylim([-60 40])
Plot the estimate using the built-in functionality of tfestimate.
Transfer Function Estimation of MIMO System
Estimate the transfer function for a multi-input/multi-output (MIMO) system.
Two masses connected to a spring and a damper on each side form an ideal one-dimensional discrete-time oscillating system. The system input array u consists of random driving forces applied to the
masses. The system output array y contains the observed displacements of the masses from their initial reference positions. The system is sampled at a rate Fs of 40 Hz.
Load the data file containing the MIMO system inputs, the system outputs, and the sample rate. The example Frequency-Response Analysis of MIMO System analyzes the system that generated the data used
in this example.
Estimate and plot the frequency-domain transfer functions of the system using the system data and the function tfestimate. Select the "mimo" option to produce all four transfer functions. Use ${2}^
{14}$ sampling points to calculate the discrete Fourier transform, divide the signal into 5000-sample segments, and window each segment with a Hann window. Specify 2500 samples of overlap between
adjoining segments.
wind = hann(5000);
nfs = 2^14;
nov = 2500;
[tXY,ft] = tfestimate(u,y,wind,nov,nfs,Fs,"mimo");
tiledlayout flow
for jk = 1:2
for kj = 1:2
grid on
ylim([-120 0])
title("Input "+jk+", Output "+kj)
xlabel("Frequency (Hz)")
ylabel("Magnitude (dB)")
Input Arguments
x — Input signal
vector | matrix
Input signal, specified as a vector or matrix.
Example: cos(pi/4*(0:159))+randn(1,160) specifies a sinusoid embedded in white Gaussian noise.
Data Types: single | double
Complex Number Support: Yes
y — Output signal
vector | matrix
Output signal, specified as a vector or matrix.
Data Types: single | double
Complex Number Support: Yes
window — Window
integer | vector | []
Window, specified as an integer or as a row or column vector. Use window to divide the signal into segments.
• If window is an integer, then tfestimate divides x and y into segments of length window and windows each segment with a Hamming window of that length.
• If window is a vector, then tfestimate divides x and y into segments of the same length as the vector and windows each segment using window.
If the length of x and y cannot be divided exactly into an integer number of segments with noverlap overlapping samples, then the signals are truncated accordingly.
If you specify window as empty, then tfestimate uses a Hamming window such that x and y are divided into eight segments with noverlap overlapping samples.
For a list of available windows, see Windows.
Example: hann(N+1) and (1-cos(2*pi*(0:N)'/N))/2 both specify a Hann window of length N + 1.
Data Types: single | double
noverlap — Number of overlapped samples
positive integer | []
Number of overlapped samples, specified as a positive integer.
• If window is scalar, then noverlap must be smaller than window.
• If window is a vector, then noverlap must be smaller than the length of window.
If you specify noverlap as empty, then tfestimate uses a number that produces 50% overlap between segments.
Data Types: double | single
nfft — Number of DFT points
positive integer | []
Number of DFT points, specified as a positive integer. If you specify nfft as empty, then tfestimate sets this argument to max(256,2^p), where p = ⌈log[2] N⌉ for input signals of length N and the ⌈ ⌉
symbols denote the ceiling function.
Data Types: single | double
freqrange — Frequency range for transfer function estimate
'onesided' | 'twosided' | 'centered'
Frequency range for the transfer function estimate, specified as a one of 'onesided', 'twosided', or 'centered'. The default is 'onesided' for real-valued signals and 'twosided' for complex-valued
• 'onesided' — Returns the one-sided estimate of the transfer function between two real-valued input signals, x and y. If nfft is even, txy has nfft/2 + 1 rows and is computed over the interval [0,
π] rad/sample. If nfft is odd, txy has (nfft + 1)/2 rows and the interval is [0,π) rad/sample. If you specify fs, the corresponding intervals are [0,fs/2] cycles/unit time for even nfft and [0,fs
/2) cycles/unit time for odd nfft.
• 'twosided' — Returns the two-sided estimate of the transfer function between two real-valued or complex-valued input signals, x and y. In this case, txy has nfft rows and is computed over the
interval [0,2π) rad/sample. If you specify fs, the interval is [0,fs) cycles/unit time.
• 'centered' — Returns the centered two-sided estimate of the transfer function between two real-valued or complex-valued input signals, x and y. In this case, txy has nfft rows and is computed
over the interval (–π,π] rad/sample for even nfft and (–π,π) rad/sample for odd nfft. If you specify fs, the corresponding intervals are (–fs/2, fs/2] cycles/unit time for even nfft and (–fs/2,
fs/2) cycles/unit time for odd nfft.
est — Transfer function estimator
'H1' (default) | 'H2'
Transfer function estimator, specified as 'H1' or 'H2'.
• Use 'H1' when the noise is uncorrelated with the input signals.
• Use 'H2' when the noise is uncorrelated with the output signals. In this case, the number of input signals must equal the number of output signals.
See Transfer Function for more information.
Output Arguments
txy — Transfer function estimate
vector | matrix | three-dimensional array
Transfer function estimate, returned as a vector, matrix, or three-dimensional array.
w — Normalized frequencies
Normalized frequencies, returned as a real-valued column vector.
f — Cyclical frequencies
Cyclical frequencies, returned as a real-valued column vector.
More About
Transfer Function
The relationship between the input x and output y is modeled by the linear, time-invariant transfer function txy. In the frequency domain, Y(f) = H(f)X(f).
• For a single-input/single-output system, the H[1] estimate of the transfer function is given by
where P[yx] is the cross power spectral density of x and y, and P[xx] is the power spectral density of x. This estimate assumes that the noise is not correlated with the system input.
For multi-input/multi-output (MIMO) systems, the H[1] estimator becomes
${H}_{1}\left(f\right)={P}_{YX}\left(f\right){P}_{XX}^{-1}\left(f\right)=\left[\begin{array}{cccc}{P}_{{y}_{1}{x}_{1}}\left(f\right)& {P}_{{y}_{1}{x}_{2}}\left(f\right)& \cdots & {P}_{{y}_{1}{x}_
{m}}\left(f\right)\\ {P}_{{y}_{2}{x}_{1}}\left(f\right)& {P}_{{y}_{2}{x}_{2}}\left(f\right)& \cdots & {P}_{{y}_{2}{x}_{m}}\left(f\right)\\ ⋮& ⋮& \ddots & ⋮\\ {P}_{{y}_{n}{x}_{1}}\left(f\right)&
{P}_{{y}_{n}{x}_{2}}\left(f\right)& \cdots & {P}_{{y}_{n}{x}_{m}}\left(f\right)\end{array}\right]\text{\hspace{0.17em}}{\left[\begin{array}{cccc}{P}_{{x}_{1}{x}_{1}}\left(f\right)& {P}_{{x}_{1}
{x}_{2}}\left(f\right)& \cdots & {P}_{{x}_{1}{x}_{m}}\left(f\right)\\ {P}_{{x}_{2}{x}_{1}}\left(f\right)& {P}_{{x}_{2}{x}_{2}}\left(f\right)& \cdots & {P}_{{x}_{2}{x}_{m}}\left(f\right)\\ ⋮& ⋮& \
ddots & ⋮\\ {P}_{{x}_{m}{x}_{1}}\left(f\right)& {P}_{{x}_{m}{x}_{2}}\left(f\right)& \cdots & {P}_{{x}_{m}{x}_{m}}\left(f\right)\end{array}\right]}^{-1}$
for m inputs and n outputs, where:
□ P[y[i]x[k]] is the cross power spectral density of the kth input and the ith output.
□ P[x[i]x[k]] is the cross power spectral density of the kth and ith inputs.
For two inputs and two outputs, the estimator is the matrix
${H}_{1}\left(f\right)=\frac{\left[\begin{array}{cc}{P}_{{y}_{1}{x}_{1}}\left(f\right){P}_{{x}_{2}{x}_{2}}\left(f\right)-{P}_{{y}_{1}{x}_{2}}\left(f\right){P}_{{x}_{2}{x}_{1}}\left(f\right)& {P}_
{{y}_{1}{x}_{2}}\left(f\right){P}_{{x}_{1}{x}_{1}}\left(f\right)-{P}_{{y}_{1}{x}_{1}}\left(f\right){P}_{{x}_{1}{x}_{2}}\left(f\right)\\ {P}_{{y}_{2}{x}_{1}}\left(f\right){P}_{{x}_{2}{x}_{2}}\left
(f\right)-{P}_{{y}_{2}{x}_{2}}\left(f\right){P}_{{x}_{2}{x}_{1}}\left(f\right)& {P}_{{y}_{2}{x}_{2}}\left(f\right){P}_{{x}_{1}{x}_{1}}\left(f\right)-{P}_{{y}_{2}{x}_{1}}\left(f\right){P}_{{x}_{1}
• For a single-input/single-output system, the H[2] estimate of the transfer function is given by
where P[yy] is the power spectral density of y and P[xy] = P^*[yx] is the complex conjugate of the cross power spectral density of x and y. This estimate assumes that the noise is not correlated
with the system output.
For MIMO systems, the H[2] estimator is well-defined only for equal numbers of inputs and outputs: n = m. The estimator becomes
${H}_{2}\left(f\right)={P}_{YY}\left(f\right){P}_{XY}^{-1}\left(f\right)=\left[\begin{array}{cccc}{P}_{{y}_{1}{y}_{1}}\left(f\right)& {P}_{{y}_{1}{y}_{2}}\left(f\right)& \cdots & {P}_{{y}_{1}{y}_
{n}}\left(f\right)\\ {P}_{{y}_{2}{y}_{1}}\left(f\right)& {P}_{{y}_{2}{y}_{2}}\left(f\right)& \cdots & {P}_{{y}_{2}{y}_{n}}\left(f\right)\\ ⋮& ⋮& \ddots & ⋮\\ {P}_{{y}_{n}{y}_{1}}\left(f\right)&
{P}_{{y}_{n}{y}_{2}}\left(f\right)& \cdots & {P}_{{y}_{n}{y}_{n}}\left(f\right)\end{array}\right]\text{\hspace{0.17em}}{\left[\begin{array}{cccc}{P}_{{x}_{1}{y}_{1}}\left(f\right)& {P}_{{x}_{1}
{y}_{2}}\left(f\right)& \cdots & {P}_{{x}_{1}{y}_{n}}\left(f\right)\\ {P}_{{x}_{2}{y}_{1}}\left(f\right)& {P}_{{x}_{2}{y}_{2}}\left(f\right)& \cdots & {P}_{{x}_{2}{y}_{n}}\left(f\right)\\ ⋮& ⋮& \
ddots & ⋮\\ {P}_{{x}_{n}{y}_{1}}\left(f\right)& {P}_{{x}_{n}{y}_{2}}\left(f\right)& \cdots & {P}_{{x}_{n}{y}_{n}}\left(f\right)\end{array}\right]}^{-1},$
□ P[y[i]y[k]] is the cross power spectral density of the kth and ith outputs.
□ P[x[i]y[k]] is the complex conjugate of the cross power spectral density of the ith input and the kth output.
tfestimate uses Welch's averaged periodogram method. See pwelch for details.
[1] Vold, Håvard, John Crowley, and G. Thomas Rocklin. “New Ways of Estimating Frequency Response Functions.” Sound and Vibration. Vol. 18, November 1984, pp. 34–38.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
• Arguments specified using name value pairs must be compile time constants.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Version History
Introduced before R2006a
R2024a: Code generation support for single-precision variable-size window inputs
The tfestimate function supports single-precision variable-size window inputs for code generation.
R2023b: Use single-precision data and gpuArray objects
The tfestimate function supports single-precision inputs and gpuArray objects. You must have Parallel Computing Toolbox™ to use gpuArray objects.
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Keeping in tune with the order of the day, the School of Mathematics offers courses that span across a variety of areas in Mathematics. Starting off with certain fundamental courses such as Single
Variable Calculus and Group Theory, the school offers mandatory courses as advanced as Functional Analysis and Differential Geometry and elective courses such as Ergodic Theory, Data Sciences and
Machine Learning, as well.
Foundation Courses for BS-MS programme:
Varsha Semester
MATH 111 Single Variable Calculus
IDC 111 Mathematical Tools-I
Vasanth Semester
MAT 121 Introduction to Algebra
IDC 121 Mathematical Tools-II
Varsha Semester
MATH 211 Multi Variable Calculus
Vasanth Semester
MAT 221 Introduction to Probability and Statistics
Major Courses in Mathematics
Varsha Semester
MAT 311 Real Analysis
MAT 312 Abstract Algebra
MAT 313 Linear Algebra
MAT 314 Numerical Analysis
Mat 315 Number Theory and Cryptography
Vasanth Semester
MAT 321 Complex Analysis
MAT 322 Measure Theory and Integration
MAT 323 Galois Theory & Commutative Algebra
MAT 324 Theory of Ordinary Differential Equations
MAT 325 General Topology
Varsha Semester
MAT 411 Functional Analysis
MAT 412 Analysis on Manifolds
MAT 413 Partial Differential Equations
MAT 414 Rings, Modules and Algebras
Vasanth Semester
MAT 421 Probability Theory and Stochastic Process
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Electives available at SoM:
• Algebraic Geometry
• Algebraic Number Theory
• Algebraic Topology
• Lie Groups and Lie Algebras
• Representation Theory
• Matrix Analysis
• Operator Algebras
• C* Algebras
• Graph Theory
• Diophantine Approximations
• Harmonic Analysis
• Stochastic Analysis
• Control Theory
• Mathematical Finance
• Financial Engineering
• Mathematical Fluid Dynamics
• Calculus of Variations
• Operations Research
• Discrete Mathematics
• Programming and Data Structures
• Finite Element Methods
• Operator Theory
• Ergodic Theory
• Category Theory and Applications
• Complex Geometry
• Geometry of Schemes
• Class Field Theory
• Elliptic Curves and Modular Forms
• Fourier Analysis
• Complex Dynamics
• Mathematical Biology
• Wavelets and Frames
• Geometric Measure Theory
• Data Science and Machine Learning
• Several Complex Variables
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Python Integer To Octal With Code Examples
In this article, we will look at how to get the solution for the problem, Python Integer To Octal With Code Examples
How do I get the octal value in Python without 0o?
Method 1: Slicing To skip the prefix, use slicing and start with index 2 on the octal string. For example, to skip the prefix '0o' on the result of x = oct(42) ='0o52' , use the slicing operation x
[2:] that results in just the octal number '52' without the prefix '0o' .
print(str(1)) # convert number to string
print(int("1")) # convert string to int
print(float(1)) # convert int to float
print(list('hello')) # convert string to list
print(tuple('hello')) # convert string to tuple
print(list((1, 2, 3))) # convert tuple to list
print(tuple([1, 2, 3])) # convert list to tuple
print(bool(1)) # convert a number to boolean
print(bool(0)) # convert a number to boolean
print(bool("")) # convert a string to boolean
print(bool("data")) # convert string to boolean
print(bin(10)) # convert an integer to a binary string
print(hex(10)) # convert an integer to a hex string
print(oct(10)) # convert an integer to an octal string
dec = oct(x) #x is your decimal number
We have presented a wealth of illustrative examples to show how the Python Integer To Octal problem can be solved, and we have also explained how to do so.
How can I print integer to octal?
Print value in Decimal, Octal ad Hex using printf() in C %d - to print value in integer format. %o - to print value in octal format.
How do you find the octal of a number in Python?
Python oct() function is used to get an octal value of an integer number. This method takes an argument and returns an integer converted into an octal string. It throws an error TypeError if argument
type is other than an integer.
How set octal value in Python?
In Python, we also have another number data type called octal and hexadecimal numbers. To represent the octal number which has base 8 in Python, add a preceding 0 (zero) so that the Python
interpreter can recognize that we want the value to be in base 8 and not in base 10.
What does 0o mean in Python?
octal number
How do you convert int to decimal in Python?
In Python, you can simply use the bin() function to convert from a decimal value to its corresponding binary value. And similarly, the int() function to convert a binary to its decimal value. The int
() function takes as second argument the base of the number to be converted, which is 2 in case of binary numbers.
How do you convert int to binary in Python?
Use bin() Function to Convert Int to Binary in Python In Python, you can use a built-in function, bin() to convert an integer to binary. The bin() function takes an integer as its parameter and
returns its equivalent binary string prefixed with 0b .
What is octal form in Python?
Octal stands for eight. Right, octal contains eight digits. And instead of bin() or hex(), we use oct() to convert numbers to octal in Python. We prefix octal numbers with a zero followed by a
lowercase o, like '0o'.
How do you convert a decimal number to octal?
In decimal to binary, we divide the number by 2, in decimal to hexadecimal we divide the number by 16. In case of decimal to octal, we divide the number by 8 and write the remainders in the reverse
order to get the equivalent octal number.
What is an octal integer?
Octal is a number system where a number is represented in powers of 8. So all the integers can be represented as an octal number. Also, all the digit in an octal number is between 0 and 7. In java,
we can store octal numbers by just adding 0 while initializing.
How do I get the octal value in Python without 0o?
Method 1: Slicing To skip the prefix, use slicing and start with index 2 on the octal string. For example, to skip the prefix '0o' on the result of x = oct(42) ='0o52' , use the slicing operation x
[2:] that results in just the octal number '52' without the prefix '0o' .
What is octal form in Python?
Octal stands for eight. Right, octal contains eight digits. And instead of bin() or hex(), we use oct() to convert numbers to octal in Python. We prefix octal numbers with a zero followed by a
lowercase o, like '0o'.
How set octal value in Python?
In Python, we also have another number data type called octal and hexadecimal numbers. To represent the octal number which has base 8 in Python, add a preceding 0 (zero) so that the Python
interpreter can recognize that we want the value to be in base 8 and not in base 10.
How do you convert int to binary in Python?
Use bin() Function to Convert Int to Binary in Python In Python, you can use a built-in function, bin() to convert an integer to binary. The bin() function takes an integer as its parameter and
returns its equivalent binary string prefixed with 0b .
What does 0o mean in Python?
octal number
What is an octal integer?
Octal is a number system where a number is represented in powers of 8. So all the integers can be represented as an octal number. Also, all the digit in an octal number is between 0 and 7. In java,
we can store octal numbers by just adding 0 while initializing.
How do you convert a decimal number to octal?
In decimal to binary, we divide the number by 2, in decimal to hexadecimal we divide the number by 16. In case of decimal to octal, we divide the number by 8 and write the remainders in the reverse
order to get the equivalent octal number.
How do you convert int to decimal in Python?
In Python, you can simply use the bin() function to convert from a decimal value to its corresponding binary value. And similarly, the int() function to convert a binary to its decimal value. The int
() function takes as second argument the base of the number to be converted, which is 2 in case of binary numbers.
How do you find the octal of a number in Python?
Python oct() function is used to get an octal value of an integer number. This method takes an argument and returns an integer converted into an octal string. It throws an error TypeError if argument
type is other than an integer.
How can I print integer to octal?
Print value in Decimal, Octal ad Hex using printf() in C %d - to print value in integer format. %o - to print value in octal format.
forEach() method does not call prototype function on array of objects With Code Examples
The forEach() method is a powerful tool for iterating over arrays of objects. It is a method of the Array prototype and is used to execute a given function on each element of the array. However, it
is important to note that the forEach() method does not call the prototype function on the array of objects. The forEach() method is a great way to iterate over an array of objects. It allows you to
execute a given function on each element of the array. This is useful for performing operations on each
How To Open Cmd At Specific Size Using Python With Code Examples
In this article, we will look at how to get the solution for the problem, How To Open Cmd At Specific Size Using Python With Code Examples How do I increase font size in CMD? Right Click the top bar
of the Command Prompt Window (i.e. Right Click the words "Command Prompt" on the top left) and select Properties. There you can change Font, Size, Color and other options. import os os.system('
mode con: cols=100 lines=40') input("Press any key to continue...") What is default window size
Str To Tuple Of Float With Code Examples
In this article, we will look at how to get the solution for the problem, Str To Tuple Of Float With Code Examples Can a tuple have 3 elements? A tuple is a data structure that has a specific number
and sequence of values. The Tuple<T1,T2,T3> class represents a 3-tuple, or triple, which is a tuple that has three components. You can instantiate a Tuple<T1,T2,T3> object by calling either the Tuple
<T1,T2,T3> constructor or the static Tuple. x = str((1, 20, 2.0)) y = tuple(float(s) for s in x.strip
Matplotlib Title Not Fully Visible With Code Examples
In this article, we will look at how to get the solution for the problem, Matplotlib Title Not Fully Visible With Code Examples How do you put a title in a plot? To set title for plot in matplotlib,
call title() function on the matplotlib. pyplot object and pass required title name as argument for the title() function. The following code snippet sets the title of the plot to “Sample Title”. # If
you saving your images just add, bbox_inches='tight' as below: fig.savefig(filename,d
Find The Area Of A Circle In Python With Code Examples
In this article, we will look at how to get the solution for the problem, Find The Area Of A Circle In Python With Code Examples Why is area of a circle π r 2? The usual definition of pi is the ratio
of the circumference of a circle to its diameter, so that the circumference of a circle is pi times the diameter, or 2 pitimes the radius. This give a geometric justification that the area of a
circle really is "pi r squared". #Area Of A Circle in python R = int(input("Enter the radius of the circ
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Maryland College and Career-Ready Standards
AttributesFreeAn attribute describes an object. <br>You use attributes to describe two objects when they are not the same. <br>An attribute can tell you if an object is shorter, taller, longer or
smaller than another object. Read more...iWorksheets: 19Study Guides: 1Vocabulary Sets: 3
PatternsFreeWhat are Patterns? Patterns are all around us. We can see them in nature, clothing, words, and even floor tiles. Read more...iWorksheets: 18Study Guides: 1Vocabulary Sets: 1
Relative PositionWhat is Relative Position? Relative position describes where an object or person is compared to another object or person. The terms used in relative position are: below, up, next to,
left, right, under, over, behind, on front of, far near, down. Read more...iWorksheets: 12Study Guides: 1Vocabulary Sets: 2
SymmetryWhat is Symmetry? Symmetry is when a shape or an object can be folded and both sides of the fold are the same size and shape. The fold line is called the line of symmetry. Not all shapes or
objects have a line of symmetry. Read more...iWorksheets: 3Study Guides: 1Vocabulary Sets: 1
Counting CoinsMoney is what we use to buy the things we want or need. Pennies, nickels, dimes and quarters are all the forms of US money. Read more...iWorksheets: 4Study Guides: 1Vocabulary Sets: 1
Days of the WeekWhat are the days of the week? There are seven days in a week. They are: Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday. Saturday and Sunday are considered weekends.
Monday through Friday are considered weekdays. Read more...iWorksheets: 4Study Guides: 1Vocabulary Sets: 1
Months of the YearFreeThere are twelve months in one year. The months are always in the same order. Read more...iWorksheets: 5Study Guides: 1Vocabulary Sets: 1
MD.MA.1.OA. Operations and Algebraic Thinking (OA)
1.OA.A. Represent and solve problems involving addition and subtraction.
1.OA.A.1. Major Standard: Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in
all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
1.OA.A.1.1. Ability to represent the problem in multiple ways including drawings and or objects/manipulatives (e.g., counters, unifix cubes, Digi-Blocks, number lines, and part-part-whole mats).
Story ProblemsStory problems are a set of sentences that give you the information to a problem that you need to solve. With a story problem, it is your job to figure out whether you will use addition
or subtraction to solve the problem. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Subtraction FactsSubtract means to take away. The meaning of 3-2=1 is that two objects are taken away from a group of three objects and one object remains. Subtraction Facts fun Worksheets and
Printables. Read more...iWorksheets :4Study Guides :1Vocabulary :1
Subtraction is not CommutativeCommutative means you can switch around the numbers you are using without changing the result. Addition is commutative. Subtraction, however, is not commutative. Read
more...iWorksheets :3Study Guides :1Vocabulary :1
Subtraction FactsSubtraction is taking a group of objects and separating them. When you subtract, your answer gets smaller. If you subtract zero from a number, you answer will stay the same. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
Using Number LineWhat is a Number Line? Number lines can be used to help with many different ways. The most common ways are for addition and subtraction. Read more...iWorksheets :4Study Guides :1
Commutative PropertyWhat is the commutative property? It is used in addition. Commutative property is when a number sentence is turned around and it still means the same thing. Read more...i
Worksheets :3Study Guides :1Vocabulary :1
Addition FactsFreeWhen you add, you combine two or more numbers together to get ONE answer… one SUM. A sum is the answer to an addition problem. Read more...iWorksheets :10Study Guides :1Vocabulary
Double Digit Addition without RegroupingSteps to follow when adding a double-digit number:<br> First: Add the two numbers in the ONES place.<br> Second: Add the two numbers in the TENS place. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
Story ProblemsA story problem is a word problem that contains a problem you need to solve by adding, subtracting, multiplying or dividing in order to figure out the answer. Read more...iWorksheets :4
Study Guides :1Vocabulary :1
Addition FactsFreeWhat is Addition? Addition is taking two groups of objects and putting them together. When adding, the answer gets larger. When you add 0, the answer remains the same. <br>How to
Add: The two numbers you are adding together are called addends. Read more...iWorksheets :15Study Guides :1Vocabulary :2
One Less, One MoreWhat is One Less or One More? One less means the number that comes before. One more means the number that comes after. How to figure out one more: If you are given a number, say 2.
You are asked to find the number that is one more. You count on from 2 and the answer is 3. Read more...iWorksheets :3Study Guides :1
1.OA.A.1.2. Ability to take apart and combine numbers in a wide variety of ways.
Subtraction FactsSubtract means to take away. The meaning of 3-2=1 is that two objects are taken away from a group of three objects and one object remains. Subtraction Facts fun Worksheets and
Printables. Read more...iWorksheets :4Study Guides :1Vocabulary :1
Subtraction FactsSubtraction is taking a group of objects and separating them. When you subtract, your answer gets smaller. If you subtract zero from a number, you answer will stay the same. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
Addition FactsFreeWhen you add, you combine two or more numbers together to get ONE answer… one SUM. A sum is the answer to an addition problem. Read more...iWorksheets :10Study Guides :1Vocabulary
Addition FactsFreeWhat is Addition? Addition is taking two groups of objects and putting them together. When adding, the answer gets larger. When you add 0, the answer remains the same. <br>How to
Add: The two numbers you are adding together are called addends. Read more...iWorksheets :15Study Guides :1Vocabulary :2
1.OA.A.1.3. Ability to make sense of quantity and be able to compare numbers.
SequencingSequencing is when when you count, numbers go in a specific order. Read more...iWorksheets :3Study Guides :1Vocabulary :1
Using Number LineWhat is a Number Line? Number lines can be used to help with many different ways. The most common ways are for addition and subtraction. Read more...iWorksheets :4Study Guides :1
Counting to 999When you count, you start with the number 1 and stop counting after you count the last object you happen to be counting. Read more...iWorksheets :3Study Guides :1
Comparing NumbersWhen comparing two numbers, you figure out if one number is GREATER or LESS THAN the other number. You can use SIGNS to show if a number is greater than, less than, or equal to
another number. Read more...iWorksheets :4Study Guides :1Vocabulary :2
Greater Than, Less ThanWhen a number is greater than another number, it means that is is larger. > is the greater than symbol. < is the less than symbol. Read more...iWorksheets :3Study Guides :1
Vocabulary :1
Ordering Numbers and Objects by SizeWhat is Ordering? Ordering is when numbers or objects are in a sequence. They may go from smallest to largest. They may go from largest to smallest. Read more...i
Worksheets :5Study Guides :1
SequencingWhat is Sequencing? Sequencing means in order. When we count, we count in order or in a sequence. We use sequencing in our every day lives. We follow directions and count in sequence. Try
counting by ones. As you say the number, put your finger on the number on the page. 1 2 3 4 5 6 7 8 9 10. Read more...iWorksheets :4Study Guides :1
1.OA.A.1.5. Ability to solve a variety of addition and subtraction word problems (CCSS, Page 88, Table 1).
Story ProblemsStory problems are a set of sentences that give you the information to a problem that you need to solve. With a story problem, it is your job to figure out whether you will use addition
or subtraction to solve the problem. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Story ProblemsA story problem is a word problem that contains a problem you need to solve by adding, subtracting, multiplying or dividing in order to figure out the answer. Read more...iWorksheets :4
Study Guides :1Vocabulary :1
1.OA.A.2. Major Standard: Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for
the unknown number to represent the problem.
1.OA.A.2.1. Ability to add numbers in any order and be able to identify the most efficient way to solve the problem.
Commutative PropertyWhat is the commutative property? It is used in addition. Commutative property is when a number sentence is turned around and it still means the same thing. Read more...i
Worksheets :3Study Guides :1Vocabulary :1
1.OA.A.2.2. Ability to solve a variety of addition and subtraction word problems (CCSS, Page 88, Table 1).
Story ProblemsStory problems are a set of sentences that give you the information to a problem that you need to solve. With a story problem, it is your job to figure out whether you will use addition
or subtraction to solve the problem. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Story ProblemsA story problem is a word problem that contains a problem you need to solve by adding, subtracting, multiplying or dividing in order to figure out the answer. Read more...iWorksheets :4
Study Guides :1Vocabulary :1
1.OA.B. Understand and apply properties of operations and relationship between addition and subtraction.
1.OA.B.3. Major Standard: Apply properties of operations as strategies to add and subtract. (Students need not use formal terms for these properties.) Examples: If 8+3 = 11 is known, then 3+8 = 11 is
also known (Commutative property of addition). To add 2+6+4, the second two numbers can be added to make a ten, so 2+6+4 = 2+10, which equals 12 (Associative property of addition).
1.OA.B.3.1. Knowledge of and ability to use the properties of operations (CCSS, Page 90, Table 3).
Commutative PropertyWhat is the commutative property? It is used in addition. Commutative property is when a number sentence is turned around and it still means the same thing. Read more...i
Worksheets :3Study Guides :1Vocabulary :1
1.OA.B.4. Major Standard: Understand subtraction as an unknown-addend problem. For example, subtract 10–8 by finding the number that makes 10 when added to 8.
1.OA.B.4.2. Ability to apply the strategy to think addition rather than take away: Rather than find 9-6 = ?, ask how many would you add to six to equal nine?
Story ProblemsStory problems are a set of sentences that give you the information to a problem that you need to solve. With a story problem, it is your job to figure out whether you will use addition
or subtraction to solve the problem. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Subtraction FactsSubtract means to take away. The meaning of 3-2=1 is that two objects are taken away from a group of three objects and one object remains. Subtraction Facts fun Worksheets and
Printables. Read more...iWorksheets :4Study Guides :1Vocabulary :1
Subtraction is not CommutativeCommutative means you can switch around the numbers you are using without changing the result. Addition is commutative. Subtraction, however, is not commutative. Read
more...iWorksheets :3Study Guides :1Vocabulary :1
Subtraction FactsSubtraction is taking a group of objects and separating them. When you subtract, your answer gets smaller. If you subtract zero from a number, you answer will stay the same. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
Using Number LineWhat is a Number Line? Number lines can be used to help with many different ways. The most common ways are for addition and subtraction. Read more...iWorksheets :4Study Guides :1
Commutative PropertyWhat is the commutative property? It is used in addition. Commutative property is when a number sentence is turned around and it still means the same thing. Read more...i
Worksheets :3Study Guides :1Vocabulary :1
Addition FactsFreeWhen you add, you combine two or more numbers together to get ONE answer… one SUM. A sum is the answer to an addition problem. Read more...iWorksheets :10Study Guides :1Vocabulary
Double Digit Addition without RegroupingSteps to follow when adding a double-digit number:<br> First: Add the two numbers in the ONES place.<br> Second: Add the two numbers in the TENS place. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
Story ProblemsA story problem is a word problem that contains a problem you need to solve by adding, subtracting, multiplying or dividing in order to figure out the answer. Read more...iWorksheets :4
Study Guides :1Vocabulary :1
Addition FactsFreeWhat is Addition? Addition is taking two groups of objects and putting them together. When adding, the answer gets larger. When you add 0, the answer remains the same. <br>How to
Add: The two numbers you are adding together are called addends. Read more...iWorksheets :15Study Guides :1Vocabulary :2
One Less, One MoreWhat is One Less or One More? One less means the number that comes before. One more means the number that comes after. How to figure out one more: If you are given a number, say 2.
You are asked to find the number that is one more. You count on from 2 and the answer is 3. Read more...iWorksheets :3Study Guides :1
1.OA.B.4.4. Ability to use the open number line to find the unknown.
Using Number LineWhat is a Number Line? Number lines can be used to help with many different ways. The most common ways are for addition and subtraction. Read more...iWorksheets :4Study Guides :1
1.OA.C. Add and subtract within 20.
1.OA.C.5. Major Standard: Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
1.OA.C.5.1. Knowledge of and ability to use addition counting strategies (e.g., Counting All, Counting On, Counting On from the Larger Number) to solve addition problems.
Story ProblemsStory problems are a set of sentences that give you the information to a problem that you need to solve. With a story problem, it is your job to figure out whether you will use addition
or subtraction to solve the problem. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Using Number LineWhat is a Number Line? Number lines can be used to help with many different ways. The most common ways are for addition and subtraction. Read more...iWorksheets :4Study Guides :1
Commutative PropertyWhat is the commutative property? It is used in addition. Commutative property is when a number sentence is turned around and it still means the same thing. Read more...i
Worksheets :3Study Guides :1Vocabulary :1
Addition FactsFreeWhen you add, you combine two or more numbers together to get ONE answer… one SUM. A sum is the answer to an addition problem. Read more...iWorksheets :10Study Guides :1Vocabulary
Double Digit Addition without RegroupingSteps to follow when adding a double-digit number:<br> First: Add the two numbers in the ONES place.<br> Second: Add the two numbers in the TENS place. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
Story ProblemsA story problem is a word problem that contains a problem you need to solve by adding, subtracting, multiplying or dividing in order to figure out the answer. Read more...iWorksheets :4
Study Guides :1Vocabulary :1
Addition FactsFreeWhat is Addition? Addition is taking two groups of objects and putting them together. When adding, the answer gets larger. When you add 0, the answer remains the same. <br>How to
Add: The two numbers you are adding together are called addends. Read more...iWorksheets :15Study Guides :1Vocabulary :2
One Less, One MoreWhat is One Less or One More? One less means the number that comes before. One more means the number that comes after. How to figure out one more: If you are given a number, say 2.
You are asked to find the number that is one more. You count on from 2 and the answer is 3. Read more...iWorksheets :3Study Guides :1
1.OA.C.5.2. Knowledge of and ability to use subtraction counting strategies (Counting Up To, Counting Back From) to solve problems.
Story ProblemsStory problems are a set of sentences that give you the information to a problem that you need to solve. With a story problem, it is your job to figure out whether you will use addition
or subtraction to solve the problem. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Subtraction FactsSubtract means to take away. The meaning of 3-2=1 is that two objects are taken away from a group of three objects and one object remains. Subtraction Facts fun Worksheets and
Printables. Read more...iWorksheets :4Study Guides :1Vocabulary :1
Subtraction FactsSubtraction is taking a group of objects and separating them. When you subtract, your answer gets smaller. If you subtract zero from a number, you answer will stay the same. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
Using Number LineWhat is a Number Line? Number lines can be used to help with many different ways. The most common ways are for addition and subtraction. Read more...iWorksheets :4Study Guides :1
Double Digit Addition without RegroupingSteps to follow when adding a double-digit number:<br> First: Add the two numbers in the ONES place.<br> Second: Add the two numbers in the TENS place. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
Story ProblemsA story problem is a word problem that contains a problem you need to solve by adding, subtracting, multiplying or dividing in order to figure out the answer. Read more...iWorksheets :4
Study Guides :1Vocabulary :1
One Less, One MoreWhat is One Less or One More? One less means the number that comes before. One more means the number that comes after. How to figure out one more: If you are given a number, say 2.
You are asked to find the number that is one more. You count on from 2 and the answer is 3. Read more...iWorksheets :3Study Guides :1
1.OA.C.5.3. Ability to use skip counting to add, understanding when skip counting they are adding groups of, such as when counting by 2s to add 2, understand that a counting by 2s is counting groups
of 2.
Story ProblemsStory problems are a set of sentences that give you the information to a problem that you need to solve. With a story problem, it is your job to figure out whether you will use addition
or subtraction to solve the problem. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Using Number LineWhat is a Number Line? Number lines can be used to help with many different ways. The most common ways are for addition and subtraction. Read more...iWorksheets :4Study Guides :1
Commutative PropertyWhat is the commutative property? It is used in addition. Commutative property is when a number sentence is turned around and it still means the same thing. Read more...i
Worksheets :3Study Guides :1Vocabulary :1
Addition FactsFreeWhen you add, you combine two or more numbers together to get ONE answer… one SUM. A sum is the answer to an addition problem. Read more...iWorksheets :10Study Guides :1Vocabulary
Double Digit Addition without RegroupingSteps to follow when adding a double-digit number:<br> First: Add the two numbers in the ONES place.<br> Second: Add the two numbers in the TENS place. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
Story ProblemsA story problem is a word problem that contains a problem you need to solve by adding, subtracting, multiplying or dividing in order to figure out the answer. Read more...iWorksheets :4
Study Guides :1Vocabulary :1
Addition FactsFreeWhat is Addition? Addition is taking two groups of objects and putting them together. When adding, the answer gets larger. When you add 0, the answer remains the same. <br>How to
Add: The two numbers you are adding together are called addends. Read more...iWorksheets :15Study Guides :1Vocabulary :2
One Less, One MoreWhat is One Less or One More? One less means the number that comes before. One more means the number that comes after. How to figure out one more: If you are given a number, say 2.
You are asked to find the number that is one more. You count on from 2 and the answer is 3. Read more...iWorksheets :3Study Guides :1
1.OA.C.6. Major Standard: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on, making ten (e.g. 8+6 = 8+2+4, which leads to
10+4 = 14); decomposing a number leading to a ten (13–4 = 13–3–1, which leads to 10–1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8+4 = 12, one knows 12–8 = 4);
and creating equivalent but easier or known sums (e.g., adding 6+7 by creating the known equivalent 6+6+1 = 12+1, which equals 13).
1.OA.C.6.1. Ability to use mental math strategies such as counting on, making ten, decomposing a number leading to ten, the relationship between addition and subtraction, and creating equivalent but
easier or know sums to add and subtract within 20, first using visual models and then moving to mental math.
Story ProblemsStory problems are a set of sentences that give you the information to a problem that you need to solve. With a story problem, it is your job to figure out whether you will use addition
or subtraction to solve the problem. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Subtraction FactsSubtract means to take away. The meaning of 3-2=1 is that two objects are taken away from a group of three objects and one object remains. Subtraction Facts fun Worksheets and
Printables. Read more...iWorksheets :4Study Guides :1Vocabulary :1
Subtraction FactsSubtraction is taking a group of objects and separating them. When you subtract, your answer gets smaller. If you subtract zero from a number, you answer will stay the same. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
Using Number LineWhat is a Number Line? Number lines can be used to help with many different ways. The most common ways are for addition and subtraction. Read more...iWorksheets :4Study Guides :1
Commutative PropertyWhat is the commutative property? It is used in addition. Commutative property is when a number sentence is turned around and it still means the same thing. Read more...i
Worksheets :3Study Guides :1Vocabulary :1
Addition FactsFreeWhen you add, you combine two or more numbers together to get ONE answer… one SUM. A sum is the answer to an addition problem. Read more...iWorksheets :10Study Guides :1Vocabulary
Double Digit Addition without RegroupingSteps to follow when adding a double-digit number:<br> First: Add the two numbers in the ONES place.<br> Second: Add the two numbers in the TENS place. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
Story ProblemsA story problem is a word problem that contains a problem you need to solve by adding, subtracting, multiplying or dividing in order to figure out the answer. Read more...iWorksheets :4
Study Guides :1Vocabulary :1
Addition FactsFreeWhat is Addition? Addition is taking two groups of objects and putting them together. When adding, the answer gets larger. When you add 0, the answer remains the same. <br>How to
Add: The two numbers you are adding together are called addends. Read more...iWorksheets :15Study Guides :1Vocabulary :2
One Less, One MoreWhat is One Less or One More? One less means the number that comes before. One more means the number that comes after. How to figure out one more: If you are given a number, say 2.
You are asked to find the number that is one more. You count on from 2 and the answer is 3. Read more...iWorksheets :3Study Guides :1
1.OA.C.6.2. Ability to demonstrate fluency for addition and subtraction within 10, building first on accurate recall of the facts using games, (including technology) and purposeful practice (Tasks
which are timed should not be used unless students have demonstrated accurate recall of the facts).
Story ProblemsStory problems are a set of sentences that give you the information to a problem that you need to solve. With a story problem, it is your job to figure out whether you will use addition
or subtraction to solve the problem. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Subtraction FactsSubtract means to take away. The meaning of 3-2=1 is that two objects are taken away from a group of three objects and one object remains. Subtraction Facts fun Worksheets and
Printables. Read more...iWorksheets :4Study Guides :1Vocabulary :1
Subtraction is not CommutativeCommutative means you can switch around the numbers you are using without changing the result. Addition is commutative. Subtraction, however, is not commutative. Read
more...iWorksheets :3Study Guides :1Vocabulary :1
Subtraction FactsSubtraction is taking a group of objects and separating them. When you subtract, your answer gets smaller. If you subtract zero from a number, you answer will stay the same. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
Using Number LineWhat is a Number Line? Number lines can be used to help with many different ways. The most common ways are for addition and subtraction. Read more...iWorksheets :4Study Guides :1
Commutative PropertyWhat is the commutative property? It is used in addition. Commutative property is when a number sentence is turned around and it still means the same thing. Read more...i
Worksheets :3Study Guides :1Vocabulary :1
Addition FactsFreeWhen you add, you combine two or more numbers together to get ONE answer… one SUM. A sum is the answer to an addition problem. Read more...iWorksheets :10Study Guides :1Vocabulary
Double Digit Addition without RegroupingSteps to follow when adding a double-digit number:<br> First: Add the two numbers in the ONES place.<br> Second: Add the two numbers in the TENS place. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
Story ProblemsA story problem is a word problem that contains a problem you need to solve by adding, subtracting, multiplying or dividing in order to figure out the answer. Read more...iWorksheets :4
Study Guides :1Vocabulary :1
Addition FactsFreeWhat is Addition? Addition is taking two groups of objects and putting them together. When adding, the answer gets larger. When you add 0, the answer remains the same. <br>How to
Add: The two numbers you are adding together are called addends. Read more...iWorksheets :15Study Guides :1Vocabulary :2
One Less, One MoreWhat is One Less or One More? One less means the number that comes before. One more means the number that comes after. How to figure out one more: If you are given a number, say 2.
You are asked to find the number that is one more. You count on from 2 and the answer is 3. Read more...iWorksheets :3Study Guides :1
1.OA.D. Work with addition and subtraction equations.
1.OA.D.7. Major Standard: Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are
true and which are false?: 6 = 6, 7 = 8–1, 5+2 = 2+5, 4+1 = 5+2.
1.OA.D.7.1. Knowledge that an equal sign represents the relationship between two equal quantities.
Using Number LineWhat is a Number Line? Number lines can be used to help with many different ways. The most common ways are for addition and subtraction. Read more...iWorksheets :4Study Guides :1
Comparing NumbersWhen comparing two numbers, you figure out if one number is GREATER or LESS THAN the other number. You can use SIGNS to show if a number is greater than, less than, or equal to
another number. Read more...iWorksheets :4Study Guides :1Vocabulary :2
Greater Than, Less ThanWhen a number is greater than another number, it means that is is larger. > is the greater than symbol. < is the less than symbol. Read more...iWorksheets :3Study Guides :1
Vocabulary :1
Ordering Numbers and Objects by SizeWhat is Ordering? Ordering is when numbers or objects are in a sequence. They may go from smallest to largest. They may go from largest to smallest. Read more...i
Worksheets :5Study Guides :1
1.OA.D.7.3. Understand the equal sign means “is the same as”.
Using Number LineWhat is a Number Line? Number lines can be used to help with many different ways. The most common ways are for addition and subtraction. Read more...iWorksheets :4Study Guides :1
Comparing NumbersWhen comparing two numbers, you figure out if one number is GREATER or LESS THAN the other number. You can use SIGNS to show if a number is greater than, less than, or equal to
another number. Read more...iWorksheets :4Study Guides :1Vocabulary :2
Greater Than, Less ThanWhen a number is greater than another number, it means that is is larger. > is the greater than symbol. < is the less than symbol. Read more...iWorksheets :3Study Guides :1
Vocabulary :1
Ordering Numbers and Objects by SizeWhat is Ordering? Ordering is when numbers or objects are in a sequence. They may go from smallest to largest. They may go from largest to smallest. Read more...i
Worksheets :5Study Guides :1
1.OA.D.8. Major Standard: Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the question
true in each of the equations: 8+? = 11, 5 = ?-3, 6+6 = ?.
1.OA.D.8.1. Ability to represent the problem in multiple ways including drawings and or objects/manipulatives (e.g., counters, connecting cubes, Digi-Blocks, number lines).
Story ProblemsStory problems are a set of sentences that give you the information to a problem that you need to solve. With a story problem, it is your job to figure out whether you will use addition
or subtraction to solve the problem. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Subtraction FactsSubtract means to take away. The meaning of 3-2=1 is that two objects are taken away from a group of three objects and one object remains. Subtraction Facts fun Worksheets and
Printables. Read more...iWorksheets :4Study Guides :1Vocabulary :1
Subtraction is not CommutativeCommutative means you can switch around the numbers you are using without changing the result. Addition is commutative. Subtraction, however, is not commutative. Read
more...iWorksheets :3Study Guides :1Vocabulary :1
Subtraction FactsSubtraction is taking a group of objects and separating them. When you subtract, your answer gets smaller. If you subtract zero from a number, you answer will stay the same. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
Using Number LineWhat is a Number Line? Number lines can be used to help with many different ways. The most common ways are for addition and subtraction. Read more...iWorksheets :4Study Guides :1
Commutative PropertyWhat is the commutative property? It is used in addition. Commutative property is when a number sentence is turned around and it still means the same thing. Read more...i
Worksheets :3Study Guides :1Vocabulary :1
Addition FactsFreeWhen you add, you combine two or more numbers together to get ONE answer… one SUM. A sum is the answer to an addition problem. Read more...iWorksheets :10Study Guides :1Vocabulary
Double Digit Addition without RegroupingSteps to follow when adding a double-digit number:<br> First: Add the two numbers in the ONES place.<br> Second: Add the two numbers in the TENS place. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
Story ProblemsA story problem is a word problem that contains a problem you need to solve by adding, subtracting, multiplying or dividing in order to figure out the answer. Read more...iWorksheets :4
Study Guides :1Vocabulary :1
Addition FactsFreeWhat is Addition? Addition is taking two groups of objects and putting them together. When adding, the answer gets larger. When you add 0, the answer remains the same. <br>How to
Add: The two numbers you are adding together are called addends. Read more...iWorksheets :15Study Guides :1Vocabulary :2
One Less, One MoreWhat is One Less or One More? One less means the number that comes before. One more means the number that comes after. How to figure out one more: If you are given a number, say 2.
You are asked to find the number that is one more. You count on from 2 and the answer is 3. Read more...iWorksheets :3Study Guides :1
1.OA.D.8.2. Ability to take apart and combine numbers in a wide variety of ways.
Subtraction FactsSubtract means to take away. The meaning of 3-2=1 is that two objects are taken away from a group of three objects and one object remains. Subtraction Facts fun Worksheets and
Printables. Read more...iWorksheets :4Study Guides :1Vocabulary :1
Subtraction FactsSubtraction is taking a group of objects and separating them. When you subtract, your answer gets smaller. If you subtract zero from a number, you answer will stay the same. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
Addition FactsFreeWhen you add, you combine two or more numbers together to get ONE answer… one SUM. A sum is the answer to an addition problem. Read more...iWorksheets :10Study Guides :1Vocabulary
Addition FactsFreeWhat is Addition? Addition is taking two groups of objects and putting them together. When adding, the answer gets larger. When you add 0, the answer remains the same. <br>How to
Add: The two numbers you are adding together are called addends. Read more...iWorksheets :15Study Guides :1Vocabulary :2
1.OA.D.8.3. Ability to make sense of quantity and be able to compare numbers.
SequencingSequencing is when when you count, numbers go in a specific order. Read more...iWorksheets :3Study Guides :1Vocabulary :1
Using Number LineWhat is a Number Line? Number lines can be used to help with many different ways. The most common ways are for addition and subtraction. Read more...iWorksheets :4Study Guides :1
Counting to 999When you count, you start with the number 1 and stop counting after you count the last object you happen to be counting. Read more...iWorksheets :3Study Guides :1
Comparing NumbersWhen comparing two numbers, you figure out if one number is GREATER or LESS THAN the other number. You can use SIGNS to show if a number is greater than, less than, or equal to
another number. Read more...iWorksheets :4Study Guides :1Vocabulary :2
Greater Than, Less ThanWhen a number is greater than another number, it means that is is larger. > is the greater than symbol. < is the less than symbol. Read more...iWorksheets :3Study Guides :1
Vocabulary :1
Ordering Numbers and Objects by SizeWhat is Ordering? Ordering is when numbers or objects are in a sequence. They may go from smallest to largest. They may go from largest to smallest. Read more...i
Worksheets :5Study Guides :1
SequencingWhat is Sequencing? Sequencing means in order. When we count, we count in order or in a sequence. We use sequencing in our every day lives. We follow directions and count in sequence. Try
counting by ones. As you say the number, put your finger on the number on the page. 1 2 3 4 5 6 7 8 9 10. Read more...iWorksheets :4Study Guides :1
1.OA.D.8.5. Ability to solve a variety of addition and subtraction word problems (CCSS, Page 88, Table 1).
Story ProblemsStory problems are a set of sentences that give you the information to a problem that you need to solve. With a story problem, it is your job to figure out whether you will use addition
or subtraction to solve the problem. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Story ProblemsA story problem is a word problem that contains a problem you need to solve by adding, subtracting, multiplying or dividing in order to figure out the answer. Read more...iWorksheets :4
Study Guides :1Vocabulary :1
MD.MA.1.NBT. Number and Operations in Base Ten (NBT)
1.NBT.A. Extend the counting sequence.
1.NBT.A.1. Major Standard: Count to 120 starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.
1.NBT.A.1.2. Reading
1.NBT.A.1.2.a. Ability to explore visual representations of numerals, matching a visual representation of a set to a numeral.
Comparing NumbersWhen comparing two numbers, you figure out if one number is GREATER or LESS THAN the other number. You can use SIGNS to show if a number is greater than, less than, or equal to
another number. Read more...iWorksheets :4Study Guides :1Vocabulary :2
1.NBT.A.1.2.b. Ability to read a written numeral.
1.NBT.B. Understand Place Value.
1.NBT.B.2. Major Standard: Understand that the two digits of a two-digit number represent amounts of tens and ones.
1.NBT.B.2.1. Ability to use base ten manipulatives (e.g., base ten blocks, DigiBlocks, connecting cubes, ten frames, interlocking base ten blocks) to represent two-digit numbers.
Place ValueWhat is Place Value? Place value is the AMOUNT that each digit is worth in a number. A number can have MANY place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
Place ValueWhat is place value? Place value is the amount that each digit is worth in a numeral. There are many different place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
1.NBT.B.2.2. Knowledge of the connection between numerals, words, and quantities.
Place ValueWhat is Place Value? Place value is the AMOUNT that each digit is worth in a number. A number can have MANY place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
Place ValueWhat is place value? Place value is the amount that each digit is worth in a numeral. There are many different place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
1.NBT.B.2.3. Knowledge that two-digit numbers are composed of bundles of tens and leftover ones.
Place ValueWhat is Place Value? Place value is the AMOUNT that each digit is worth in a number. A number can have MANY place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
Place ValueWhat is place value? Place value is the amount that each digit is worth in a numeral. There are many different place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
1.NBT.B.2.4. Ability to count by tens and ones.
SequencingSequencing is when when you count, numbers go in a specific order. Read more...iWorksheets :3Study Guides :1Vocabulary :1
Odd and EvenAll numbers are either odd or even. When a number is even, it can be split into two sets without any leftovers. When you split a number into two sets and there is one left over, that
means the number is odd. Read more...iWorksheets :4Study Guides :1
Using Number LineWhat is a Number Line? Number lines can be used to help with many different ways. The most common ways are for addition and subtraction. Read more...iWorksheets :4Study Guides :1
Counting to 999When you count, you start with the number 1 and stop counting after you count the last object you happen to be counting. Read more...iWorksheets :3Study Guides :1
Odd and EvenWhat are Odd and Even Numbers? ODD numbers are numbers that CAN NOT be equally divided in half, by 2. Read more...iWorksheets :5Study Guides :1
Skip CountingSkip counting is when you SKIP a number or numbers when counting. Counting by 2s, 3s, 4s, 5, and 10s. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Double Digit Addition without RegroupingSteps to follow when adding a double-digit number:<br> First: Add the two numbers in the ONES place.<br> Second: Add the two numbers in the TENS place. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
SequencingWhat is Sequencing? Sequencing means in order. When we count, we count in order or in a sequence. We use sequencing in our every day lives. We follow directions and count in sequence. Try
counting by ones. As you say the number, put your finger on the number on the page. 1 2 3 4 5 6 7 8 9 10. Read more...iWorksheets :4Study Guides :1
One Less, One MoreWhat is One Less or One More? One less means the number that comes before. One more means the number that comes after. How to figure out one more: If you are given a number, say 2.
You are asked to find the number that is one more. You count on from 2 and the answer is 3. Read more...iWorksheets :3Study Guides :1
Skip CountingWhat is Skip Counting? Skip counting means you do not say every number as you count. You only count special numbers. There are many different ways to skip count. E.g. when counting by
twos, you only say every second number: 2 4 6 8 10. Read more...iWorksheets :3Study Guides :1
1.NBT.B.2a. Major Standard: Understand that the two digits of a two-digit number represent amounts of tens and ones – Understand the following as a special case: 10 can be thought of as a bundle of
ten ones–called a “ten”.
1.NBT.B.2a.1. Ability to use base ten manipulatives (e.g., base ten blocks, DigiBlocks, connecting cubes, ten frames, interlocking base ten blocks) to build and compare ten ones and ten.
Place ValueWhat is Place Value? Place value is the AMOUNT that each digit is worth in a number. A number can have MANY place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
Place ValueWhat is place value? Place value is the amount that each digit is worth in a numeral. There are many different place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
1.NBT.B.2b. Major Standard: Understand that the two digits of a two-digit number represent amounts of tens and ones – Understand the following as a special case: The numbers from 11 to 19 are
composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
1.NBT.B.2b.1. Ability to use base ten manipulatives (e.g., base ten blocks, Digi-Blocks, connecting cubes, ten frames, interlocking base ten blocks) to build and compare 11 to 19.
Place ValueWhat is Place Value? Place value is the AMOUNT that each digit is worth in a number. A number can have MANY place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
Place ValueWhat is place value? Place value is the amount that each digit is worth in a numeral. There are many different place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
1.NBT.B.2b.2. Ability to match the concrete representations of 11 through 19 with the numerical representations.
Place ValueWhat is Place Value? Place value is the AMOUNT that each digit is worth in a number. A number can have MANY place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
Place ValueWhat is place value? Place value is the amount that each digit is worth in a numeral. There are many different place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
1.NBT.B.2b.3. Ability to understand that numbers 11-19 represent one ten and some more ones.
Place ValueWhat is Place Value? Place value is the AMOUNT that each digit is worth in a number. A number can have MANY place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
Place ValueWhat is place value? Place value is the amount that each digit is worth in a numeral. There are many different place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
1.NBT.B.2c. Major Standard: Understand that the two digits of a two-digit number represent amounts of tens and ones – Understand the following as a special case: The numbers 10, 20, 30, 40, 50, 60,
70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).
1.NBT.B.2c.1. Ability to use base ten manipulatives (e.g., base ten blocks, DigiBlocks, Unifix Cubes, ten frames, interlocking base ten blocks) to build and model counting by tens.
Skip CountingSkip counting is when you SKIP a number or numbers when counting. Counting by 2s, 3s, 4s, 5, and 10s. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Place ValueWhat is Place Value? Place value is the AMOUNT that each digit is worth in a number. A number can have MANY place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
Place ValueWhat is place value? Place value is the amount that each digit is worth in a numeral. There are many different place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
Skip CountingWhat is Skip Counting? Skip counting means you do not say every number as you count. You only count special numbers. There are many different ways to skip count. E.g. when counting by
twos, you only say every second number: 2 4 6 8 10. Read more...iWorksheets :3Study Guides :1
1.NBT.B.2c.2. Ability to skip count by 10s to 100 understanding that each ten counted represents that number of groups of ten.
Skip CountingSkip counting is when you SKIP a number or numbers when counting. Counting by 2s, 3s, 4s, 5, and 10s. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Skip CountingWhat is Skip Counting? Skip counting means you do not say every number as you count. You only count special numbers. There are many different ways to skip count. E.g. when counting by
twos, you only say every second number: 2 4 6 8 10. Read more...iWorksheets :3Study Guides :1
1.NBT.B.3. Major Standard: Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.
1.NBT.B.3.1. Ability to apply their understanding of the value of tens and ones in order to compare the magnitude of two numbers.
Place ValueWhat is Place Value? Place value is the AMOUNT that each digit is worth in a number. A number can have MANY place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
Place ValueWhat is place value? Place value is the amount that each digit is worth in a numeral. There are many different place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
1.NBT.B.3.2. Ability to use base ten manipulatives to represent the numbers and model the comparison of their values.
Place ValueWhat is Place Value? Place value is the AMOUNT that each digit is worth in a number. A number can have MANY place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
Place ValueWhat is place value? Place value is the amount that each digit is worth in a numeral. There are many different place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
1.NBT.B.3.3. Ability to represent their reasoning about the comparison of two two-digit numbers using pictures, numbers, and words.
SequencingSequencing is when when you count, numbers go in a specific order. Read more...iWorksheets :3Study Guides :1Vocabulary :1
Using Number LineWhat is a Number Line? Number lines can be used to help with many different ways. The most common ways are for addition and subtraction. Read more...iWorksheets :4Study Guides :1
Counting to 999When you count, you start with the number 1 and stop counting after you count the last object you happen to be counting. Read more...iWorksheets :3Study Guides :1
Comparing NumbersWhen comparing two numbers, you figure out if one number is GREATER or LESS THAN the other number. You can use SIGNS to show if a number is greater than, less than, or equal to
another number. Read more...iWorksheets :4Study Guides :1Vocabulary :2
Greater Than, Less ThanWhen a number is greater than another number, it means that is is larger. > is the greater than symbol. < is the less than symbol. Read more...iWorksheets :3Study Guides :1
Vocabulary :1
Ordering Numbers and Objects by SizeWhat is Ordering? Ordering is when numbers or objects are in a sequence. They may go from smallest to largest. They may go from largest to smallest. Read more...i
Worksheets :5Study Guides :1
SequencingWhat is Sequencing? Sequencing means in order. When we count, we count in order or in a sequence. We use sequencing in our every day lives. We follow directions and count in sequence. Try
counting by ones. As you say the number, put your finger on the number on the page. 1 2 3 4 5 6 7 8 9 10. Read more...iWorksheets :4Study Guides :1
1.NBT.B.3.5. Ability to use ordinality to compare the placement of the numbers on the number line or 100s chart.
OrdinalsAn ordinal is an object’s position in the order of a group. An ordinal tells whether an object is first or fifth. Read more...iWorksheets :3Study Guides :1Vocabulary :2
OrdinalsOrdinal numbers are numbers that are used to tell what order something is in. Read more...iWorksheets :4Study Guides :1Vocabulary :2
1.NBT.B.3.6. Knowledge of the symbols >, =, < and their meaning.
Using Number LineWhat is a Number Line? Number lines can be used to help with many different ways. The most common ways are for addition and subtraction. Read more...iWorksheets :4Study Guides :1
Comparing NumbersWhen comparing two numbers, you figure out if one number is GREATER or LESS THAN the other number. You can use SIGNS to show if a number is greater than, less than, or equal to
another number. Read more...iWorksheets :4Study Guides :1Vocabulary :2
Greater Than, Less ThanWhen a number is greater than another number, it means that is is larger. > is the greater than symbol. < is the less than symbol. Read more...iWorksheets :3Study Guides :1
Vocabulary :1
Ordering Numbers and Objects by SizeWhat is Ordering? Ordering is when numbers or objects are in a sequence. They may go from smallest to largest. They may go from largest to smallest. Read more...i
Worksheets :5Study Guides :1
1.NBT.C. Use place value understanding and properties of operations to add and subtract.
1.NBT.C.4. Major Standard: Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and
strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand
that in adding two-digit numbers, one adds tens and tens, ones and ones, and sometimes it is necessary to compose a ten.
1.NBT.C.4.1. Knowledge of addition and subtraction fact families.
Subtraction is not CommutativeCommutative means you can switch around the numbers you are using without changing the result. Addition is commutative. Subtraction, however, is not commutative. Read
more...iWorksheets :3Study Guides :1Vocabulary :1
1.NBT.C.4.2. Ability to model addition and subtraction using base ten manipulatives (e.g., base ten blocks, Digi-Blocks, Unifix cubes) and explain the process.
Story ProblemsStory problems are a set of sentences that give you the information to a problem that you need to solve. With a story problem, it is your job to figure out whether you will use addition
or subtraction to solve the problem. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Subtraction FactsSubtract means to take away. The meaning of 3-2=1 is that two objects are taken away from a group of three objects and one object remains. Subtraction Facts fun Worksheets and
Printables. Read more...iWorksheets :4Study Guides :1Vocabulary :1
Subtraction is not CommutativeCommutative means you can switch around the numbers you are using without changing the result. Addition is commutative. Subtraction, however, is not commutative. Read
more...iWorksheets :3Study Guides :1Vocabulary :1
Subtraction FactsSubtraction is taking a group of objects and separating them. When you subtract, your answer gets smaller. If you subtract zero from a number, you answer will stay the same. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
Using Number LineWhat is a Number Line? Number lines can be used to help with many different ways. The most common ways are for addition and subtraction. Read more...iWorksheets :4Study Guides :1
Commutative PropertyWhat is the commutative property? It is used in addition. Commutative property is when a number sentence is turned around and it still means the same thing. Read more...i
Worksheets :3Study Guides :1Vocabulary :1
Addition FactsFreeWhen you add, you combine two or more numbers together to get ONE answer… one SUM. A sum is the answer to an addition problem. Read more...iWorksheets :10Study Guides :1Vocabulary
Double Digit Addition without RegroupingSteps to follow when adding a double-digit number:<br> First: Add the two numbers in the ONES place.<br> Second: Add the two numbers in the TENS place. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
Story ProblemsA story problem is a word problem that contains a problem you need to solve by adding, subtracting, multiplying or dividing in order to figure out the answer. Read more...iWorksheets :4
Study Guides :1Vocabulary :1
Addition FactsFreeWhat is Addition? Addition is taking two groups of objects and putting them together. When adding, the answer gets larger. When you add 0, the answer remains the same. <br>How to
Add: The two numbers you are adding together are called addends. Read more...iWorksheets :15Study Guides :1Vocabulary :2
One Less, One MoreWhat is One Less or One More? One less means the number that comes before. One more means the number that comes after. How to figure out one more: If you are given a number, say 2.
You are asked to find the number that is one more. You count on from 2 and the answer is 3. Read more...iWorksheets :3Study Guides :1
1.NBT.C.4.3. Knowledge of place value.
Place ValueWhat is Place Value? Place value is the AMOUNT that each digit is worth in a number. A number can have MANY place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
Place ValueWhat is place value? Place value is the amount that each digit is worth in a numeral. There are many different place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
1.NBT.C.5. Major Standard: Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.
1.NBT.C.5.1. Ability to use base ten manipulatives, number lines or hundreds charts to model finding 10 more and explain reasoning.
Place ValueWhat is Place Value? Place value is the AMOUNT that each digit is worth in a number. A number can have MANY place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
Place ValueWhat is place value? Place value is the amount that each digit is worth in a numeral. There are many different place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
1.NBT.C.5.2. Knowledge of addition and subtraction fact families.
Subtraction is not CommutativeCommutative means you can switch around the numbers you are using without changing the result. Addition is commutative. Subtraction, however, is not commutative. Read
more...iWorksheets :3Study Guides :1Vocabulary :1
1.NBT.C.5.3. Ability to model addition using base ten manipulatives (e.g., base ten blocks, Digi-Blocks, connecting cubes) and explain the process.
Story ProblemsStory problems are a set of sentences that give you the information to a problem that you need to solve. With a story problem, it is your job to figure out whether you will use addition
or subtraction to solve the problem. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Subtraction is not CommutativeCommutative means you can switch around the numbers you are using without changing the result. Addition is commutative. Subtraction, however, is not commutative. Read
more...iWorksheets :3Study Guides :1Vocabulary :1
Using Number LineWhat is a Number Line? Number lines can be used to help with many different ways. The most common ways are for addition and subtraction. Read more...iWorksheets :4Study Guides :1
Commutative PropertyWhat is the commutative property? It is used in addition. Commutative property is when a number sentence is turned around and it still means the same thing. Read more...i
Worksheets :3Study Guides :1Vocabulary :1
Addition FactsFreeWhen you add, you combine two or more numbers together to get ONE answer… one SUM. A sum is the answer to an addition problem. Read more...iWorksheets :10Study Guides :1Vocabulary
Story ProblemsA story problem is a word problem that contains a problem you need to solve by adding, subtracting, multiplying or dividing in order to figure out the answer. Read more...iWorksheets :4
Study Guides :1Vocabulary :1
Addition FactsFreeWhat is Addition? Addition is taking two groups of objects and putting them together. When adding, the answer gets larger. When you add 0, the answer remains the same. <br>How to
Add: The two numbers you are adding together are called addends. Read more...iWorksheets :15Study Guides :1Vocabulary :2
1.NBT.C.5.4. Knowledge of place value and skip counting by forward 10.
Skip CountingSkip counting is when you SKIP a number or numbers when counting. Counting by 2s, 3s, 4s, 5, and 10s. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Skip CountingWhat is Skip Counting? Skip counting means you do not say every number as you count. You only count special numbers. There are many different ways to skip count. E.g. when counting by
twos, you only say every second number: 2 4 6 8 10. Read more...iWorksheets :3Study Guides :1
1.NBT.C.6. Major Standard: Subtract multiples of 10 in the range of 10-90 from multiples of 10 in the range of 10-90 (positive or zero differences), using concrete models or drawings and strategies
based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
1.NBT.C.6.1. Ability to use base ten manipulatives, number lines or hundreds charts to model finding 10 less and explain reasoning.
Place ValueWhat is Place Value? Place value is the AMOUNT that each digit is worth in a number. A number can have MANY place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
Place ValueWhat is place value? Place value is the amount that each digit is worth in a numeral. There are many different place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
1.NBT.C.6.2. Knowledge of addition and subtraction fact families.
Subtraction is not CommutativeCommutative means you can switch around the numbers you are using without changing the result. Addition is commutative. Subtraction, however, is not commutative. Read
more...iWorksheets :3Study Guides :1Vocabulary :1
1.NBT.C.6.3. Ability to model subtraction using base ten manipulatives (e.g., base ten blocks, Digi-Blocks, Unifix cubes) and explain the process.
Story ProblemsStory problems are a set of sentences that give you the information to a problem that you need to solve. With a story problem, it is your job to figure out whether you will use addition
or subtraction to solve the problem. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Subtraction FactsSubtract means to take away. The meaning of 3-2=1 is that two objects are taken away from a group of three objects and one object remains. Subtraction Facts fun Worksheets and
Printables. Read more...iWorksheets :4Study Guides :1Vocabulary :1
Subtraction is not CommutativeCommutative means you can switch around the numbers you are using without changing the result. Addition is commutative. Subtraction, however, is not commutative. Read
more...iWorksheets :3Study Guides :1Vocabulary :1
Subtraction FactsSubtraction is taking a group of objects and separating them. When you subtract, your answer gets smaller. If you subtract zero from a number, you answer will stay the same. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
Using Number LineWhat is a Number Line? Number lines can be used to help with many different ways. The most common ways are for addition and subtraction. Read more...iWorksheets :4Study Guides :1
Double Digit Addition without RegroupingSteps to follow when adding a double-digit number:<br> First: Add the two numbers in the ONES place.<br> Second: Add the two numbers in the TENS place. Read
more...iWorksheets :5Study Guides :1Vocabulary :1
Story ProblemsA story problem is a word problem that contains a problem you need to solve by adding, subtracting, multiplying or dividing in order to figure out the answer. Read more...iWorksheets :4
Study Guides :1Vocabulary :1
One Less, One MoreWhat is One Less or One More? One less means the number that comes before. One more means the number that comes after. How to figure out one more: If you are given a number, say 2.
You are asked to find the number that is one more. You count on from 2 and the answer is 3. Read more...iWorksheets :3Study Guides :1
1.NBT.C.6.4. Knowledge of place value and skip counting by 10.
Skip CountingSkip counting is when you SKIP a number or numbers when counting. Counting by 2s, 3s, 4s, 5, and 10s. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Place ValueWhat is Place Value? Place value is the AMOUNT that each digit is worth in a number. A number can have MANY place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
Place ValueWhat is place value? Place value is the amount that each digit is worth in a numeral. There are many different place values. Read more...iWorksheets :3Study Guides :1Vocabulary :2
Skip CountingWhat is Skip Counting? Skip counting means you do not say every number as you count. You only count special numbers. There are many different ways to skip count. E.g. when counting by
twos, you only say every second number: 2 4 6 8 10. Read more...iWorksheets :3Study Guides :1
MD.MA.1.MD. Measurement and Data (MD)
1.MD.A. Measure lengths indirectly and by iterating length units.
1.MD.A.2. Major Standard: Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length
measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units
with no gaps or overlaps.
1.MD.A.2.1. Knowledge that length is the distance between the two endpoints of an object.
MeasurementFreeMeasurement in inches, feet, centimeters, meters, cups, pints, quarts, gallons, liters, pounds, grams, and kilograms. Read more...iWorksheets :10Study Guides :1Vocabulary :3
Comparing ObjectsWhen you compare two objects, you identify how the objects are ALIKE and how they are DIFFERENT. Read more...iWorksheets :4Study Guides :1Vocabulary :2
TemperatureTemperature is what we use to measure how hot or cold things are. A thermometer is used to measure temperature. Read more...iWorksheets :3Study Guides :1
MeasurementFreeWhat is measurement? Measurement is used in our everyday lives. We measure to cook or bake, and how far away a place is. There are metric measurements which include liters,
centimeters, grams and kilograms. Read more...iWorksheets :12Study Guides :1Vocabulary :2
1.MD.A.2.2. Ability to identify a unit of measure.
MeasurementFreeMeasurement in inches, feet, centimeters, meters, cups, pints, quarts, gallons, liters, pounds, grams, and kilograms. Read more...iWorksheets :10Study Guides :1Vocabulary :3
Comparing ObjectsWhen you compare two objects, you identify how the objects are ALIKE and how they are DIFFERENT. Read more...iWorksheets :4Study Guides :1Vocabulary :2
TemperatureTemperature is what we use to measure how hot or cold things are. A thermometer is used to measure temperature. Read more...iWorksheets :3Study Guides :1
MeasurementFreeWhat is measurement? Measurement is used in our everyday lives. We measure to cook or bake, and how far away a place is. There are metric measurements which include liters,
centimeters, grams and kilograms. Read more...iWorksheets :12Study Guides :1Vocabulary :2
1.MD.A.2.3. Knowledge of nonstandard (e.g., paper clips, eraser length, toothpicks) as well as standard units of measurement.
MeasurementFreeMeasurement in inches, feet, centimeters, meters, cups, pints, quarts, gallons, liters, pounds, grams, and kilograms. Read more...iWorksheets :10Study Guides :1Vocabulary :3
Comparing ObjectsWhen you compare two objects, you identify how the objects are ALIKE and how they are DIFFERENT. Read more...iWorksheets :4Study Guides :1Vocabulary :2
TemperatureTemperature is what we use to measure how hot or cold things are. A thermometer is used to measure temperature. Read more...iWorksheets :3Study Guides :1
MeasurementFreeWhat is measurement? Measurement is used in our everyday lives. We measure to cook or bake, and how far away a place is. There are metric measurements which include liters,
centimeters, grams and kilograms. Read more...iWorksheets :12Study Guides :1Vocabulary :2
1.MD.B. Tell and write time.
1.MD.B.3. Additional Standard: Tell and write time in hours and half-hours using analog and digital clocks.
1.MD.B.3.1. Ability to apply knowledge of fractional wholes and halves to telling time.
Fractions Greater Than or Less Than 1/2Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and
describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. Read more...iWorksheets :5Study Guides :1Vocabulary :1
FractionsWhat are fractions? When an object is broken into a number of parts, these parts must all be the same size. These equal parts can be counted to become a fraction of that object. Read more...
iWorksheets :8Study Guides :1Vocabulary :2
FractionsA fraction is a part of a whole. Fractions for 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/10, and 1/12 Read more...iWorksheets :7Study Guides :1Vocabulary :2
TimeTell time to the nearest hour, half hour, and quarter hour. Read more...iWorksheets :15Study Guides :1Vocabulary :2
Telling TimeFreeTime is measuring of how long it takes to do different activities like playing a game, doing your Math homework or riding your bike. A clock measures time. It helps us know the time.
Time is measured in hours and minutes. Read more...iWorksheets :10Study Guides :1Vocabulary :1
1.MD.B.3.2. Ability to equate a number line to 12 with the face of a clock.
Using Number LineWhat is a Number Line? Number lines can be used to help with many different ways. The most common ways are for addition and subtraction. Read more...iWorksheets :4Study Guides :1
TimeTell time to the nearest hour, half hour, and quarter hour. Read more...iWorksheets :15Study Guides :1Vocabulary :2
Telling TimeFreeTime is measuring of how long it takes to do different activities like playing a game, doing your Math homework or riding your bike. A clock measures time. It helps us know the time.
Time is measured in hours and minutes. Read more...iWorksheets :10Study Guides :1Vocabulary :1
1.MD.B.3.3. Ability to match time on a digital clock with that on an analog clock.
TimeTell time to the nearest hour, half hour, and quarter hour. Read more...iWorksheets :15Study Guides :1Vocabulary :2
Telling TimeFreeTime is measuring of how long it takes to do different activities like playing a game, doing your Math homework or riding your bike. A clock measures time. It helps us know the time.
Time is measured in hours and minutes. Read more...iWorksheets :10Study Guides :1Vocabulary :1
1.MD.C. Represent and interpret data.
1.MD.C.4. Supporting Standard: Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how
many more or less are in one category than in another.
1.MD.C.4.3. Ability to answer questions about the data such as: ‘Which category has more?’ ‘Which category has less?’ ‘What is the favorite snack of our class?’ ‘How many more stickers does Sam have
than John?’
GraphsGraphs are visual displays of data and information. A bar graph is a graph that uses BARS to show data. Bar graphs are used to compare two or more objects or people. Graphs and charts allow
people to learn information quickly and easily. Read more...iWorksheets :9Study Guides :1
MD.MA.1.G. Geometry (G)
1.G.A. Reason with shapes and their attributes.
1.G.A.1. Additional Standard: Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and
draw shapes to possess defining attributes.
1.G.A.1.1. Ability to sort shapes (e.g., attribute blocks, polygon figures) by shape, number of sides, size or number of angles.
ShapesFreeA shape is the form something takes. Read more...iWorksheets :12Study Guides :1Vocabulary :2
1.G.A.1.2. Ability to use geoboards, toothpicks, straws, paper and pencil, computer games to build shapes that possess the defining attributes.
ShapesFreeA shape is the form something takes. Read more...iWorksheets :12Study Guides :1Vocabulary :2
1.G.A.1.3. Ability to explain how two shapes are alike or how they are different from each other.
ShapesFreeA shape is the form something takes. Read more...iWorksheets :12Study Guides :1Vocabulary :2
1.G.A.3. Additional Standard: Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth
of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.
1.G.A.3.3. Ability to model halves and fourths with concrete materials.
Fractions Greater Than or Less Than 1/2Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and
describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. Read more...iWorksheets :5Study Guides :1Vocabulary :1
FractionsWhat are fractions? When an object is broken into a number of parts, these parts must all be the same size. These equal parts can be counted to become a fraction of that object. Read more...
iWorksheets :8Study Guides :1Vocabulary :2
FractionsA fraction is a part of a whole. Fractions for 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/10, and 1/12 Read more...iWorksheets :7Study Guides :1Vocabulary :2
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Learn Math trough Kid’s Tile-Math
Learn Core Math Through Kid’s Tile-Math
with worksheets here Math Dislike Cured with flexible bundle numbers
Asked ‘How old next time?’, a 3-year-old says ‘Four’ showing four fingers; but objects when seeing them held together two by two: ‘That is not four, that is two twos!’ A child thus sees what exists
in the world, bundles of 2s, and 2 of them. So, adapting to Many, children develop bundle-numbers with units as 2 2s having 1 1s as the unit, i.e. a tile, also occurring as bundle-of-bundles, e.g. 3
3s, 5 5s or ten tens.
Recounting 8 in 2s as 8 = (8/2)x2 gives a recount-formula T = (T/B)xB saying ‘From the total T, T/B times, B can be pushed away’ occurring all over mathematics and science. It solves equations: ux2 =
8 = (8/2)x2, so u = 8/2. And it changes units when adding on-top, or when adding next-to as areas as in calculus, also occurring when adding per-numbers or fractions coming from double-counting in
two units. Finally, double-counting sides in a tile halved by its diagonal leads to trigonometry.
The following papers present close to 50 micro-curricula in Mastering Many inspired by the bundle-numbers children bring to school.
Learn Core Mathematics Through Your Kid’s Tile-Math:
Recounting Bundle-Numbers and Early Trigonometry
This first paper is written for the conference ‘The Research on Outdoor STEM Education in the digiTal Age (ROSETA) Conference’ planned to take place between 16th and 19th June 2020 at Instituto
Superior de Engenharia do Porto in Portugal.
The Power of Bundle- & Per-Numbers Unleashed in Primary School:
Calculus in Grade One – What Else?
This second paper is written for the International Congress for Mathematical Education, ICME 14, planned to be held in Shanghai from July 12th to 19th, 2020, but postponed one year.
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Visual Filters
Note : The filters described here are only supported by Internet Explorer 4.0. Strictly speaking, Visual Filters should be detailed under Style Sheets as they are applied by using Style Sheet
attributes. However, currently, they are a Microsoft Internet Explorer 4.0 specific style sheets extension (in early versions of Internet Explorer 4.0, the filters were implemented as a set of
ActiveX controls, only being implemented as style sheet attributes in the later preview versions of Internet Explorer 4.0
The Visual Filters provide a way for manipulating visual objects (basically, anything on a page) to provide visual effects previously only manageable by using graphics. Also, through scripting, the
applied filters are dynamically changeable, without reloading the document, which gives them a major advantage over images. Most commonly, they would be applied to <IMG> elements, but can be applied
to <DIV> elements, which in turn can contain any HTML, so the visual filters can be applied to virtually any content. Note that if they are applied to text blocks (wrapped in <DIV> elements, then the
<DIV> element must specify width and height style sheet attributes.
Inter-page/site transitions
Internet Explorer 4.0 also supports inter-page and inter-site transitions, using a Visual Filter set in the <META> element. These can be used to set transitions that play when a page is entered (i.e.
first loaded) or exited or when a site (individual sites are determined by a change in the host - i.e. http://www.htmlib.com/ and http://faq.htmlib.com/ would be considered individual sites) is
entered, or exited. Any of the standard Visual Filter effects can be used with either blend or reveal transitions. The syntax is as used for other Visual Filters, setting the filter type in the
CONTENT attribute of the <META> element. For example:
<META HTTP-EQUIV="Page-Enter" CONTENT="filter:RevealTrans(Duration=3.000, Transition=23)">
This would play a random dissolve filter, over 3 seconds when the page is first displayed.
Note : For page-enter, page-exit, site-enter and site-exit transitions to work, the <META> element specifying the filter must be the first element in the <HEAD> section of the document and inter-page
transitions do not appear to work across framed documents.
The Visual filters available are:
Filter effect Description
Alpha Sets a uniform transparency level.
Blur Creates the impression of moving at high speed.
Chroma Makes a specific color transparent.
DropShadow Creates a solid silhouette of the object.
FlipH Creates a horizontal mirror image.
FlipV Creates a vertical mirror image.
Glow Adds radiance around the outside edges of the object.
Gray Drops color information from the image.
Invert Reverses the hue, saturation, and brightness values.
Light Projects a light source onto an object.
Mask Creates a transparent mask from an object.
Shadow Creates an offset solid silhouette.
Wave Creates a sine wave distortion along the X axis.
XRay Shows just the edges of the object.
Internet Explorer also supports two Transition Filters (Reveal and Blend transitions) for creating effects for revealing and blending objects.
The basic syntax for applying visual filters to an object is:
STYLE="filter:filtername(fparameter1, fparameter2...)}
where filtername is the name of the filter (as in the above table) and fparamter1... represents the parameters associated with each different filter type. These are detailed below, by filter name.
The Alpha visual filter can be used to set the opacity of an object, either the whole image, or a gradient region.
STYLE="filter:Alpha(Opacity=opacity, FinishOpacity=finishopacity, Style=style, StartX=startX, StartY=startY, FinishX=finishX, FinishY=finishY)"
Opacity level, 0-100, where 0 is transparent, 100 is fully opaque
Optional finish opacity level, 0-100, 0 is transparent, 100 is fully opaque
values can be 0 (uniform), 1 (linear), 2 (radial) or 3 (rectangular)
X coordinate for start of opacity gradient
Y coordinate for start of opacity gradient
X coordinate for finish of opacity gradient
Y coordinate for finish of opacity gradient
For example:
Pressing the button below the image causes the following filter to be applied:
The Blur filter gives the impression that the object is moving.
STYLE="filter:Blur(Add = add, Direction = direction, Strength = strength)"
Boolean value, any integer value adds the original object to the blurred object, '0' doesn't
0 - 315 in increments of 45 - specifies the direction of the motion blur to be added
An integer value representing the number of pixels of 'depth' for the motion blur
For example:
Pressing the button below the image causes the following filter to be applied:
The Chroma filter makes a specific colour of the object transparent.
STYLE="filter:Chroma(Color = color)"
Any colour (as a #rrggbb triplet). For the Chroma filter to work properly, this must be a colour used in the object
For example:
Pressing the button below the image causes the following filter to be applied:
The dropShadow filter adds a solid silhouette of the object, offset in the specified direction.
STYLE="filter:DropShadow(Color=color, OffX=offX, OffY=offY, Positive=positive)"
A #rrggbb hex triplet, specifying the colour to use for the shadow
Horizontal offset for the shadow
Vertical offset for the shadow
A boolean value. Any nonzero integer (true) creates a shadow for non-transparent pixels in the object, '0' (false) creates a shadow for transparent pixels.
For example:
Pressing the button below the text causes the following filter to be applied:
DropShadow this text!
The FlipH filter flips the object horizontally
For example:
Pressing the button below the text causes the following filter to be applied:
Flip this text!
The FlipV filter flips the object vertically
For example:
Pressing the button below the text causes the following filter to be applied:
Flip this text!
The Glow filter adds radiance around the object, causing it to appear to glow.
STYLE="filter:Glow(Color=color, Strength=strength)"
Any #rrggbb hex triplet for the colour of the glow
Glow intensity, from 0-100
For example:
Pressing the button below the text causes the following filter to be applied:
filter : Glow(Color="#6699CC",Strength="5")
Make me glow
The Gray filter drops colour information from the object, rendering it in gray scales only.
For example:
Pressing the button below the image causes the following filter to be applied:
The Invert filter reverses the hue, saturation and brightness of the object
For example:
Pressing the button below the image causes the following filter to be applied:
The Light filter can be used to make the object appear as if a light source is illuminating it. Initially, Light filters need to be applied, then have the light source specified with one of the
following methods.
Adds an ambient light source to the image. Ambient light is non-directional and lights the entire area. The sun emits ambient light. The syntax is:
call object.style.filters.Light(n).addAmbient(R,G,B,strength)
where R, G and B are values (0-255) to determine the ambient light colour and strength determines the 'amount' of light cast.
Adds a cone light source to the image. Cone light is directional and lights only a defined area. The syntax is:
call object.style.filters.Light(n).addCone(x1,y1,z1,x2,y2,R,G,B,strength,spread)
where x1, y1 represent the location of the source light, x2 and y2 represent the location that the light is targeted towards, R, G and B are values (0-255) to determine the light colour, strength
determines the 'amount' of light cast and spread determines the angle of spread (0-90, in degrees).
Adds an point light source to the image. Point light is emitted by light bulbs. The syntax is:
call object.style.filters.Light(n).addPoint(x,y,z,R,G,B,strength)
where x, y and z represent the point lights coordinates, R, G and B are values (0-255) to determine the ambient light colour and strength determines the 'amount' of light cast.
For example:
Pressing the button below the text causes the following script function to execute:
call document.all.divLight.filters.Light(0).addPoint(10,10,100,255,255,255,1000)
call divLight.filters.Light(0).addAmbient(0,0,255,50)
which adds a white point light and a blue ambient light to the text.
Light me up
The following methods are also available for the Light Visual filter:
ChangeColor(lightnumber, r,g,b, fAbsolute)
The ChangeColor method will change the colour of a light filter applied to an object. Use lightnumber to identify the particular light source whose colour is to be changed (it's position in the
Lights array), r,g and b, represent the new colour to be changed to and fAbsoloute is a boolean flag. If fAbsoloute is true (nonzero), then the referenced Light filter colour is changed to the new
amount specified, if false (i.e. zero), then the referenced Light filter colour is changed by the specified amount.
ChangeStrength(lightnumber, strength, fAbsolute)
ChangeStrength changes the strength of the particular Light filter (referenced by the lightnumber argument) to the strength specified in strength if the fAbsolute flag is true (nonzero), or by the
amount specified if it's false (zero).
The Clear method removes all light sources for the referenced Light filter.
MoveLight(lightnumber, x, y, z, fAbsolute)
The MoveLight method moves the light source (for point lights), or the target location (for cone lights) and has no effect on ambient lights. The x, y and z values represent positions to move the
light to, either absolutely (fAbsoloute=nonzero) or relatively (fAbsolute=false).
The Mask filter takes all the transparent pixels in a visual object, sets them to a certain colour and creates a transparent mask from the nontransparent pixels. The syntax is:
where Color is the colour to be used for the mask.
For example:
Pressing the button below the image causes the following filter to be applied:
filter:Mask (Color="#FFFFE0")
Mask Me
The shadow filter can be used to apply a shadow to the specified object. The syntax is:
filter:Shadow(Color=color, Direction=direction)
A #rrggbb hex triplet that specifies the shadow colour
0-315 in 45 degree increments, specifying the direction that the shadow should be applied for.
For example:
Pressing the button below the image causes the following filter to be applied:
filter:Shadow (Color="#6699CC", Direction="135")
Spooky Shadows
The wave causes sine wave distortion of the referenced object. The syntax is:
filter: Wave(Add=add, Freq=freq, LightStrength=strength, Phase=phase, Strength=strength)
A Boolean value specifying whether the original object is added (true, nonzero) to the filtered object or not (false, zero)
An integer value specifying the number of waves to appear in the distortion
Strength of the light on the wave effect as a percentage value
Specifying the angular offset of the wave, in percentage (i.e. 0/100% = 360 degrees, 25% = 90 degrees
An integer value specifying the intensity of the wave effect
For example:
Pressing the button below the image causes the following filter to be applied:
filter: wave(Add="0", Phase="4", Freq="5", LightStrength="5", Strength="2")
Make me Wavey
The xray filter causes the object to appear as if it had been x-rayed. The syntax is:
For example:
Pressing the button below the image causes the following filter to be applied:
Transition Filters
There are two types of transition filter supported by Internet Explorer 4.0, the Reveal Transition and the Blend Transition. As their names suggest, the Reveal Transition filter allows selective
revealing of any visual object and the blend transition performs a fade in/out of a visual object.
RevealTrans Filter
The RevealTrans filter can be applied to any visual object, to selectively show or hide it, using a variety of different techniques. The basic syntax is:
STYLE="filter: revealtrans(duration=duration, transition=transitionshape)
where Duration is a time value that the transition will take. It accepts values in the format of seconds.milliseconds For example 2.1 = 2 seconds, 100 milliseconds. Transition can be any one of the
Value Description
0 Box in
1 Box out
2 Circle in
3 Circle out
4 Wipe up
5 Wipe down
6 Wipe right
7 Wipe left
8 Vertical blinds
9 Horizontal blinds
10 Checkerboard across
11 Checkerboard down
12 Random dissolve
13 Split vertical in
14 Split vertical out
15 Split horizontal in
16 Split horizontal out
17 Strips left down
18 Strips left up
19 Strips right down
20 Strips right up
21 Random bars horizontal
22 Random bars vertical
23 Random effect (any of the other 23)
Note : The Reveal Transition Filter is most useful when used with the following events:
The Apply method is used to actually apply the filter. Event though it may have been specified in the STYLE attribute of the element, it still needs to be applied via the apply method.
The Play method causes the referenced Reveal Transition filter to start playing. It starts the transition type, for the time specified in the Duration attribute (if no Duration argument is specified
in the method. A Duration can be specified in the Play method, which will override any settings in the elements filter declaration.
The Stop method is used to stop a transition and can be called at any time while the transition is playing. To determine whether the transition is playing, use the status property (described below).
Reveal Transition filters have status and duration properties. The Duration property reflects the current duration set for the filter and status returns a value depending on the current status of the
transition. "0" = transition stopped, "1" = transition applied, "2" = transition playing.
For example. Pressing the button below the (currently hidden) image below starts the random dissolve transition. The image is initially hidden, so that the filter is applied to dissolve the image
into appearance, rather than dissolve it away.
The script function executed when the button is pressed is:
call logo.filters.item(0).Apply()
call logo.filters.item(0).Play()
which applies the filter, then resets the images visibility before playing the dissolve transition.
BlendTrans Filter
The BlendTrans filter can be applied to any visual object, to fade it in or out, over a certain time period. The basic syntax is:
STYLE="filter: blendtrans(duration=duration)"
where Duration is a time value that the transition will take. It accepts values in the format of seconds.milliseconds For example 2.1 = 2 seconds, 100 milliseconds.
For example, below there is an image above the button that's not displayed yet. When the button is clicked, the following script function is executed:
call logo2.filters.item(0).Apply()
call logo2.filters.item(0).Play()
which applies the blendTrans filter on the image, sets it to display as an 'in-line' element, makes it visible and finally, plays the transition (causing it to blend in over a 3 second period).
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Teaching Practice and Teaching Knowledge - Theoretical Background and Relevant Research
2 Theoretical Background and Relevant Research
2.1 Teaching Practice and Teaching Knowledge
Can the skills and characteristics required of an effective teacher be taught? Are some ways of preparing teachers better than others? What is involved in the practice of teaching? Although teaching
in one form or another has existed throughout human history and extensive research on teaching and teacher education has been conducted over the years, these and other questions persist.
Axelrold (1973) described teaching as a didactic or evocative activity. The didactic teaching emphasizes teachers’ responsibility for transmitting pertinent knowledge or instructing others on how to
do something. This teaching model is typically employed by teacher craftsmen (Axelrold, 1973) who have full control of the learning environment, and are solely responsible for the students’ learning
and the direction that the lesson takes in the classroom. In other words, didactic teachers allow learning to occur (Novak, 1998). As a result of this hierarchical and rigid process, learners’ focus
is directed on memorizing facts and prescribed procedures, without seeking to understand the broader context or draw conclusions. The evocative teaching, on the other hand, emphasizes the role of
teachers as teacher artists (Axelrold, 1973) whose aim is to enable learners to take control of their learning process and create evocative situations that promote learning. In this teaching model,
the emphasis is on “inquiry” and
“discovery” due to which lessons are designed to respond to the students’
needs and aspirations with the emphasis on creativity, improvisation, and expressiveness (Gage, 1978).
Describing teaching without considering learning is to understand the work of the teacher only partially. According to Hiebert and Grouws (2007), teaching consists of “classroom interactions among
teachers and
students around content directed toward facilitating students’
achievement of learning goals” (p. 372). This definition encompasses the ways in which multiple features that contribute to defining teachers’
roles impact students’ learning. Therefore, to understand the function of teachers and the effectiveness of teaching, it is necessary to understand the kind of learning goals the teaching is designed
to achieve.
When bringing such a viewpoint to the context of teacher education, both teaching and learning dimensions should be considered.
Pre-service teachers with or without teaching experience are simultaneously aspirant teachers and students. Although they are defined as teachers who had not yet completed a degree course in
teaching, research has shown that pre-service teachers also differ from experienced teachers in terms of the beliefs they hold (Wideen, Mayer-Smith, &
Moon, 1998). Pre-service teachers often begin their education with an intuitive idea about teaching that is established from their previous experience in schools (Barkatsas & Malone, 2005; Wilson,
Cooney, &
Stinson, 2005). During teacher education, pre-service teachers are exposed to many new ideas of teaching as they take theoretical courses and have supervised teaching practice in schools (Lavigne,
2014). After becoming in-service teachers, they are forced to modify their pedagogical beliefs because of contexts and tasks designed (Lavigne, 2014; Sheridan, 2016). The concepts of teaching and
learning take on different shapes as the pre-service teachers become in-service teachers (Ng, Nicholas, & Williams, 2010).
Although teacher education has been studied extensively, the focus is typically given to the cognitive difference between what teachers should learn and what they should be able to do
(Darling-Hammond &
Bransford, 2005). In this context, the works of Shulman (1987), Perrenoud (1993), Freire (1996), and Tardif (2002) are particularly noteworthy. Even though these authors used similar terms to convey
the same meaning, they used different approaches to study teacher preparedness. For example, Perrenoud (1993) and Freire (1996) focused
on the teaching practices and roles of teachers in classrooms, whereas Shulman (1987) and Tardif (2002) direct their attention to the education and professionalization of teachers (Neto & Costa,
2016). As noted by Fernandez (2014), most of Shulman’s work was dedicated to developing
“the body of understanding and skills, and device and values, character and performance that together constitute the ability to teach” (p. 82).
Based on his findings, Shulman (1986) opined:
The teacher needs not only understand that something is so; the teacher must further understand why it is so, on what grounds its warrant can be asserted, and under what circumstances our belief in
its justification can be weakened and even denied.
Moreover, we expect the teacher to understand why a given topic is particularly central to a discipline whereas another may be somewhat peripheral. This will be important in subsequent pedagogical
judgments regarding relative curricular emphasis.
(p. 9)
According to this perspective, the work of teaching entails tasks that teachers must execute to help students to learn (Shulman, 1986).
Teachers must be able to determine the content that is essential to meet students’ learning needs and specificities. This means that once the content’s essence, origins, and the logic-historical
processes that justify the existence of the content are understood, teachers should be able to orient their students’ learning beyond simple facts and predetermined standards, which is a prime
condition for effective teaching (Grossman et al., 2009).
To meet the goals outlined above, Shulman (1986) suggested that teachers needed to possess three categories of knowledge to teach a particular subject effectively: subject matter knowledge,
curricular knowledge, and pedagogical content knowledge. The first category—
subject matter knowledge—refers to “the amount and organization of knowledge per se in the mind of the teacher” (Shulman, 1986, p. 9).
According to Shulman (1986), an effective teacher should know not only the facts and concepts pertinent to the domain, but should also be able to
explicate why the domain is worth knowing and how it relates to other domains. The second category—curricular knowledge—entails awareness of what the curriculum proposes and the norms and principles
of the work setting. In other words, curricular knowledge involves knowledge about the programs of study and curricular materials used to teach a subject, as this allows teachers to make connections
between previously studied material and topics to be introduced later in the learning process, which is an essential aspect of teaching (Brant, 2006).
The third category—pedagogical content knowledge—refers to the knowledge base of teaching at the intersection between content and pedagogy (Shulman, 1986). Such knowledge, according to Shulman
(1986), encompasses “aspects of content most germane to its teachability” (p. 9). It includes the ability to identify and organize concepts presented in class (representations, analogies,
illustrations, examples, explanations, and demonstrations) to make a subject more comprehensible for the students.
Even though Shulman (1986) proposed the aforementioned ideas about pedagogical content knowledge nearly 35 years ago, these conceptualizations have gained momentum in recent investigations about
teacher knowledge. His work has also served as a basis for the recent educational reforms and has influenced research efforts and educational policies in several countries. In recent years,
pedagogical content knowledge is increasingly being taught by teacher educators in teacher educational programs, especially those aimed at primary school education. Given that Shulman (1986)
conceived pedagogical content knowledge in general terms, his ideas have since been expanded to help teachers learn and develop a better sense of the tasks and knowledge demanded for teaching subject
As a part of this research initiative, Shulman’s (1986) ideas have been investigated in the context of pre-service mathematics teacher education. An ample body of frameworks has been produced on this
topic, including the works of Ball et al. (2008), Chevallard’s (2000),
Davis and Simmt (2006), and Rowland, Huckstep, and Thwaites (2005).
Although aligning with Shulman’s (1986) ideas, these frameworks have pursued different ideas and approaches regarding teaching knowledge, including examining associations between mathematical
knowledge and practice (Chevallard, 2000), investigating the complex dynamics of the mathematical knowledge that teachers needed for teaching (Davis &
Simmt, 2006), studying the differences between content knowledge and pedagogical content knowledge and implications for teaching and learning (Baumert et al., 2010; Krauss et al., 2008), and
exploring different aspects of teacher knowledge that contribute to the professional development of pre-service teachers (Rowland et al., 2005).
The framework of Ball et al. (2008), in particular, focuses on representations of the knowledge entailed in the work of mathematics teachers. Such a framework of mathematical knowledge for teaching
comprises the areas that are unique to the role of mathematics teacher by examining how subject matter and pedagogical content knowledge are employed to carry out the tasks of teaching mathematics
(Ball et al., 2008).
Additionally, Ball et al.’s (2008) works focus on the recurrent tasks and problems of teaching mathematics, what teachers do as they teach mathematics, and the mathematical knowledge, skills, and
sensibilities required to manage these tasks. A list of the tasks identified as the tasks entailed in the work teachers do when they are teaching mathematics includes:
x Presenting mathematical ideas,
x Responding to students’ “why” questions,
x Finding an example to make a specific mathematical point, x Recognizing what is involved in using a particular representation, x Linking representations to underlying ideas and other
x Connecting a topic being taught to topics from prior or future years,
x Explaining mathematical goals and purposes to parents,
x Appraising and adapting the mathematical content of textbooks, x Modifying tasks to be either easier or harder,
x Explaining the plausibility of students’ claims (often quickly), x Giving or evaluating mathematical explanations,
x Choosing and developing usable definitions,
x Using mathematical notation and language and critiquing its use, x Asking productive mathematical questions,
x Selecting representations for particular purposes, and x Inspecting equivalencies.
The tasks outlined by the authors are examples of what is required for teachers to carry out to conduct their teaching successfully. They reveal the complexity and dynamic of activities that
regularly occur in the classroom and offer a window into the knowledge entailed in teaching mathematics in broader contexts (Ng, Mosvold, & Fauskanger, 2012; Selling, Garcia, & Ball, 2016).
In analyzing these tasks, Ball et al. (2008) were guided by the empirical evidence supporting the existence of six domains of teaching knowledge needed to carry out the tasks of teaching mathematics
effectively. These domains were typically denoted as common content knowledge (CCK), specialized content knowledge (SCK), horizon content knowledge (HCK), knowledge of content and students (KCS),
knowledge of content and curriculum (KCC), and knowledge of content and teaching (KCT), and their organization into systematic units as presented in Figure 1.
Figure 1. Framework of mathematical knowledge for teaching (Ball et al., 2008, p. 403).
According to Ball et al. (2008), CCK domain refers to the knowledge that is common in a wide variety of settings, rather than pertaining solely to the work of teaching. For example, engineers or
economists use this type of knowledge to solve problems in their daily work. Similarly, using an algorithm to find the answer for a subtraction problem is an example of CCK. In teaching, CCK allows
teachers to appropriately respond to students’ questions and resolve any misunderstandings related to the subject matter (Ndlovu, Amin, &
Samuel, 2017).
SCK, one the other hand, is the knowledge unique to the work of mathematics teaching. It “involves an uncanny kind of unpacking of mathematics that is not needed—or even desirable—in settings other
than teaching” (Ball et al., 2008, p. 400). Some examples of SCK include the knowledge needed to carry out tasks of teaching unique to the work of teaching such as introduce mathematical concepts in
a way that is accessible to the students (Ball et al., 2008). For instance, when introducing students to the notion of numbers, the teacher needs to know
how the students perceive this concept in various real-world contexts. As noted by Worden (2015), this necessitates not only the capacity for
“transforming content knowledge into pedagogical content knowledge but also unpacking one’s content knowledge to make it available for such transformation” (p. 106).
The CCK and SCK domains are interrelated via the HCK domain, which is defined as the mathematical knowledge from a broad perspective (Ball et al., 2008). Thus, HCK entails knowledge of the
discipline, its origins, and the value of curriculum in its multiple dimensions and settings (Jakobsen, Thames, Ribeiro, & Delaney, 2012).
As this necessitates the general knowledge of the previous and forthcoming content, it is often equated with “a peripheral mathematical vision needed in teaching” (Hill, Rowan, & Ball, 2005, p. 70).
In teaching practice, HCK allows teachers to develop a sense of conceptual nexus between the curriculum and a broader perspective of the discipline (Jakobsen et al., 2012).
The KCC domain combines the knowledge of mathematics and the curriculum, as conceived by Shulman (Sleep, 2009). This domain also includes the skills required to effectively use the teaching materials
such as textbooks and didactic materials, teaching instruments such as Blackboard, and technology such as calculators and computers (Koponen, Asikainen, Viholainen, & Hirvonen, 2016).
The KCS domain represents an amalgam of knowledge of content and students (Ball et al., 2008). It implies the capacity for anticipating how students will interpret the taught material and which
aspects they will find difficult to understand. To meet these aims, teachers must be able to hear and respond to students’ arguments and choose instruction approaches that promote student learning.
Consequently, KCS also necessitates the awareness of students’ motivation and aptitude for learning mathematical topics.
Finally, KCT combines knowledge of mathematics and teaching, in recognition of the fact that, in order to teach mathematics effectively,
teachers must be able design lessons appropriately. This includes proper selection of activities, exercises, and representations for different topics.
One crucial characteristic of this knowledge is the teacher’s ability “to recognize situations where teachers should diverge from their original planning, for example, if a student makes a
mathematical discovery”
(Koponen et al., 2016, p. 152).
The six domains presented above imply that the integration of knowledge types is unique to mathematics teachers (Ball & Bass, 2000).
Teaching mathematics includes a core of tasks that teachers must carry out to help students to learn (Ball & Forzani, 2009). Such tasks are complex and reveal qualities that other professions do not
demand. The work of mathematics teachers is a specific activity that differs from casual actions including commonplace showing, telling, or helping (Cohen, 2011; as cited in Ball & Forzani, 2009).
For example, although an engineer possesses high-level mathematics knowledge and at least reasonable science knowledge, the engineer can only provide information or show one another how to do things.
The mathematics teacher, on the other hand, aims at the professional classroom teaching (Ball & Forzani, 2009), an endeavor that includes the creation of opportunities for students to learn and
develop their understanding of the subject matter. In this sense, the teacher’s role is driven by social and moral conduct and a human sense to help students develop their best qualities as human
beings (Jacinto & Cedro, 2012).
Teaching mathematics requires specialized knowledge and skills that go beyond subject matter alone, and Ball et al.’s (2008) Theory of Mathematical Knowledge for Teaching provides the analytical
tools to identify and analyze the kind of knowledge and skills that mathematics teaching actually requires (Ding, 2016; Goos, 2013; Jakobsen et al., 2012; Stephenson, 2018). However, it also has
attracted considerable criticism due to its limited application on how the framework could be useful for guiding teachers to teach mathematics (Mitchell, Charalambous, & Hill, 2014) or provide better
insights into teachers’
views and understandings of the mathematical knowledge for teaching (Mosvold & Fauskanger, 2013). This is a particular challenge for the field of teacher education and teaching knowledge since the
quality of teaching and teaching knowledge depends on the views and understandings of those who actually teach. Therefore, this thesis seeks to provide insights into the understanding that
pre-service teachers develop of the knowledge necessary to teach mathematics.
The following sections aim to contextualize the current study into the research field of teaching knowledge. They focus on relevant research related to teachers’ beliefs of teaching knowledge,
followed by a clarification of the main terms used in this study.
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XLOOKUP with boolean logic
In this video we'll look how to use the XLOOKUP function with Boolean logic to apply multiple criteria.
In this video we'll look at how to use the XLOOKUP function with Boolean logic.
Boolean logic is an elegant way to apply multiple criteria.
In this worksheet we have sample order data in a table called "data".
Let's use the XLOOKUP function to find the first order in March where the color is red.
To make things clear, I'm going to work out the logic in helper columns first. Then, I'll move that logic into the XLOOKUP function, to make an all-in-one formula.
First, we'll test for dates in March with the MONTH function. When we give MONTH the full set of dates in a range, we get back a month number for each date in a dynamic array.
Since we only want dates in March, I simply need to compare this result to the number 3. When I update the formula, we get TRUE for all dates in March, and FALSE for all other dates.
Next, I'll test for the color red. This is just a simple expression that compares values in the color column to the string "red". Again, we get a list of TRUE and FALSE values. Only orders where the
color is Red return TRUE.
Now, since we want Red and March, we need to use AND logic, which means we use multiplication.
When I multiply these two helper columns together, the math operation automatically coerces the TRUE and FALSE values to 1s and 0s.
This will become our lookup array.
Notice this array is dynamic. If I temporarily change a color in March, the results update.
We now have everything we need to configure the XLOOKUP function.
For lookup_value, we use one.
For lookup_array, we use our last helper column.
For return_array, we use the full set of data.
When I enter the formula, we get details for the first order in March where the color is Red.
By default, XLOOKUP will find the first match. In other words, the first 1 in the array.
Now to move this into an all-in-one formula, I'll need to replicate this logic inside the lookup_array argument.
To do this, I need to add parentheses, and use the same expressions we used in columns I and J, multiplied together.
When I enter the formula, we get the same result.
And if I check the LOOKUP array with the F9 key, you can see we have exactly the same array we have in column K.
I can now delete the helper columns, and everything keeps working.
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Calculation 2: Phase line & Stability - TU Delft OCW
Calculation 2: Phase line & Stability
Course subject(s) 1. Introducing Mathematical Modelling
Now that you can find equilibrium solutions of a differential equation, it is time to investigate what kinds of equilibrium solutions can occur. In the next video you will learn about this.
Note: in the video an example of an unstable equilibrium point is given. However, the definition of an unstable equilibrium point is not in the video. That you can find below, in the text.
Phase line & Stability of equilibrium points
Sorry, there don't seem to be any downloads..
Subtitles (captions) in other languages than provided can be viewed at YouTube. Select your language in the CC-button of YouTube.
Autonomous differential equations
In the video you have seen how you can construct a phase line from the direction field of a differential equation. You can only draw a phase line when the differential equation is a so-called
autonomous differential equation.
In an autonomous differential equation, none of the terms depend explicitly on the independent variable.
In our differential equation,
the independent variable t does not occur explicitly: although P and dP/dt are functions of t, the time is not mentioned in the differential equation by itself. Our equation is autonomous.
A differential equation that is not autonomous is for example dy/dt=5y(t)+sin(t). For this equation you cannot draw a phase line.
Different way to draw a phase line
Because our differential equation is autonomous, you can make a graph of dP/dt versus P(t):
In this graph you can see that if P(t) is smaller than the equilibrium value 200.7, the rate change, dPdt is negative, so the value of P(t) will decrease. This means that in the phase line you should
draw an arrow pointing in the decreasing direction for values below the equilibrium. This is the same as you have seen in the video. For values of P(t) larger than the equilibrium, the value of dPdt
is positive, so the arrow in this case must be pointing in the increasing direction.
Drawing a phase line vertical takes up a lot of room, so you could also draw the phase line horizontally. For the figure above, the phase line then becomes:
Stable equilibrium point
The equilibrium solution Pe=20/0.7 is an unstable equilibrium solution. For this we need a definition of what a stable equilibrium is.
We call an equilibrium point stable if any initial value close to the equilibrium point gives solutions that always remain close to the equilibrium point.
For many stable equilibrium points, the solutions that start close by, do not only remain close by, but for t→∞ the solutions even tend to the equilibrium point.
Unstable equilibrium point
Any equilibrium point which is not stable we call unstable, so there is at least one initial value close to the equilibrium which will give a solution that moves away from the equilibrium point.
Another differential equation
In the next exercises we will consider a “slightly” different differential equation:
This differential equation has three equilibrium solutions:
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Understanding Polynomial Regression Model
Hello there, guys! Good day, everyone! Today, we’ll look at Polynomial Regression, a fascinating approach in Machine Learning. For understanding Polynomial Regression Model, we’ll go over several
fundamental terms including Machine Learning, Supervised Learning, and the distinction between regression and classification. Let’s explore polynomial regression model in detail!
This article was published as a part of the Data Science Blogathon
Supervised Machine Learning
In supervised learning, algorithms are trained using labeled datasets, and they learn about each input category. We evaluate the approach using test data (a subset of the training set) and predict
outcomes after completing the training phase. There are two types of supervised machine learning:
• Classification
• Regression
Classification vs Regression
Regression Classification
Predicting continuous variables Categorizing output variables
Continuous Categorical
Weather forecasting, market trends Gender classification, disease diagnosis
Links input and continuous output Categorizes input into classes
Why Do we Need Regression?
Regression analysis is helpful in performing following tasks:
• Forecasting the value of the dependent variable for those who have knowledge of the explanatory components
• Assessing the influence of an explanatory variable on the dependent variable
What is Polynomial Regression?
In polynomial regression, we describe the relationship between the independent variable x and the dependent variable y using an nth-degree polynomial in x. Polynomial regression, denoted as E(y | x),
characterizes fitting a nonlinear relationship between the x value and the conditional mean of y. Typically, this corresponds to the least-squares method. The least-square approach minimizes the
coefficient variance according to the Gauss-Markov Theorem. This represents a type of Linear Regression where the dependent and independent variables exhibit a curvilinear relationship and the
polynomial equation is fitted to the data.
We will delve deeper into this concept later in the article. Machine learning is also a subset of Multiple Linear Regression, achieved by incorporating additional polynomial elements into the
equation, transforming it into a Polynomial Regression equation.
Types of Polynomial Regression
A quadratic equation is a general term for a second-degree polynomial equation. This degree, on the other hand, can go up to nth values. Here is the categorization of Polynomial Regression:
1. Linear – if degree as 1
2. Quadratic – if degree as 2
3. Cubic – if degree as 3 and goes on, on the basis of degree.
Assumption of Polynomial Regression
We cannot process all of the datasets and use polynomial regression machine learning to make a better judgment. We can still do it, but there should be specific constraints for the dataset in order
to get the best polynomial regression results.
• A dependent variable’s behaviour can be described by a linear, or curved, an additive link between the dependent variable and a set of k independent factors.
• The independent variables lack any interrelationship.
• We employ datasets featuring independently distributed errors with a normal distribution, having a mean of zero and a constant variance.
Simple Math to Understand Polynomial Regression
Here we are dealing with mathematics, rather than going deep, just understand the basic structure, we all know the equation of a linear equation will be a straight line, from that if we have many
features then we opt for multiple regression just increasing features part alone, then how about polynomial, it’s not about increasing but changing the structure to a quadratic equation, you can
visually understand from the diagram:
Linear Regression vs Polynomial Regression
Rather than focusing on the distinctions between linear and polynomial regression, we may comprehend the importance of polynomial regression by starting with linear regression. We build our model and
realize that it performs abysmally. We examine the difference between the actual value and the best fit line we predicted, and it appears that the true value has a curve on the graph, but our line is
nowhere near cutting the mean of the points. This is where polynomial regression comes into play; it predicts the best-fit line that matches the pattern of the data (curve).
One important distinction between Linear and Polynomial Regression is that Polynomial Regression does not require a linear relationship between the independent and dependent variables in the data
set. When the Linear Regression Model fails to capture the points in the data and the Linear Regression fails to adequately represent the optimum, then we use Polynomial Regression.
Before delving into the topic, let us first understand why we prefer Polynomial Regression over Linear Regression in some situations, say the non-linear condition of the dataset, by programming and
Python Code
import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
#Let randomly create some data’s for two variables,
x = 2 - 3 * np.random.normal(0, 1, 20)
y = x - 2 * (x ** 2) + 0.5 * (x ** 3) + np.random.normal(-3, 3, 20)
#Visualize the variables spreads for better understanding
plt.scatter(x,y, s=10)
Let’s analyze random data using Regression Analysis:
x = x[:, np.newaxis]
y = y[:, np.newaxis]
model = LinearRegression()
model.fit(x, y)
y_pred = model.predict(x)
plt.scatter(x, y, s=10)
plt.plot(x, y_pred, color='r')
The straight line is unable to capture the patterns in the data. This is an example of under-fitting.
Let’s look at it from a technical standpoint, using measures like Root Mean Square Error (RMSE) and discrimination coefficient (R2). The RMSE indicates how well a regression model can predict the
response variable’s value in absolute terms, whereas the R2 indicates how well a model can predict the response variable’s value in percentage terms.
import sklearn.metrics as metrics
mse = metrics.mean_squared_error(x,y)
rmse = np.sqrt(mse)
r2 = metrics.r2_score(x,y)
print('RMSE value:',rmse)
print('R2 value:',r2)
RMSE value: 93.47170875128153
R2 value: -786.2378753237103
Non-linear data in Polynomial Regression
We need to enhance the model’s complexity to overcome under-fitting. In this sense, we need to make linear analyzes in a non-linear way, statistically by using Polynomial,
Because the weights associated with the features are still linear, this is still a linear model. x2 (x square) is only a function. However, the curve we’re trying to fit is quadratic in nature.
Let’s see visually the above concept for better understanding, a picture speaks louder and stronger than words,
from sklearn.preprocessing import PolynomialFeatures
polynomial_features1 = PolynomialFeatures(degree=2)
x_poly1 = polynomial_features1.fit_transform(x)
model1 = LinearRegression()
model1.fit(x_poly1, y)
y_poly_pred1 = model1.predict(x_poly1)
from sklearn.metrics import mean_squared_error, r2_score
rmse1 = np.sqrt(mean_squared_error(y,y_poly_pred1))
r21 = r2_score(y,y_poly_pred1)
The figure clearly shows that the quadratic curve can better match the data than the linear line.
import operator
plt.scatter(x, y, s=10)
# sort the values of x before line plot
sort_axis = operator.itemgetter(0)
sorted_zip = sorted(zip(x,y_poly_pred), key=sort_axis)
x, y_poly_pred1 = zip(*sorted_zip)
plt.plot(x, y_poly_pred1, color='m')
polynomial_features2= PolynomialFeatures(degree=3)
x_poly2 = polynomial_features2.fit_transform(x)
model2 = LinearRegression()
model2.fit(x_poly2, y)
y_poly_pred2 = model2.predict(x_poly2)
rmse2 = np.sqrt(mean_squared_error(y,y_poly_pred2))
r22 = r2_score(y,y_poly_pred2)
plt.scatter(x, y, s=10)
# sort the values of x before line plot
sort_axis = operator.itemgetter(0)
sorted_zip = sorted(zip(x,y_poly_pred2), key=sort_axis)
x, y_poly_pred2 = zip(*sorted_zip)
plt.plot(x, y_poly_pred2, color='m')
polynomial_features3= PolynomialFeatures(degree=4)
x_poly3 = polynomial_features3.fit_transform(x)
model3 = LinearRegression()
model3.fit(x_poly3, y)
y_poly_pred3 = model3.predict(x_poly3)
rmse3 = np.sqrt(mean_squared_error(y,y_poly_pred3))
r23 = r2_score(y,y_poly_pred3)
plt.scatter(x, y, s=10)
# sort the values of x before line plot
sort_axis = operator.itemgetter(0)
sorted_zip = sorted(zip(x,y_poly_pred3), key=sort_axis)
x, y_poly_pred3 = zip(*sorted_zip)
plt.plot(x, y_poly_pred3, color='m')
In comparison to the linear line, we can observe that RMSE has dropped and R2-score has increased.
Overfitting vs Under-fitting
We keep on increasing the degree, we will see the best result, but there comes the over-fitting problem, if we get r2 value for a particular value shows 100.
When analyzing a dataset linearly, we encounter an under-fitting problem
Polynomial regression can correct this.
However, when fine-tuning the degree parameter to the optimal value, we encounter an over-fitting problem, resulting in a 100 per cent r2 value. The conclusion is that we must avoid both overfitting
and underfitting issues.
Note: To avoid over-fitting, we can increase the number of training samples so that the algorithm does not learn the system’s noise and becomes more generalized.
Bias vs Variance Tradeoff
How do we pick the best model? To address this question, we must first comprehend the trade-off between bias and variance.
The mistake is due to the model’s simple assumptions in fitting the data is referred to as bias. A high bias indicates that the model is unable to capture data patterns, resulting in under-fitting.
The mistake caused by the complicated model trying to match the data is referred to as variance. When a model has a high variance, it passes over the majority of the data points, causing the data to
From the above program, when degree is 1 which means in linear regression, it shows underfitting which means high bias and low variance. And when we get r2 value 100, which means low bias and high
variance, which means overfitting
As the model complexity grows, the bias reduces while the variance increases, and vice versa. A machine learning model should, in theory, have minimal variance and bias. However, having both is
nearly impossible. As a result, a trade-off must be made in order to build a strong model that performs well on both train and unseen data.
Degree – How to Find the Right One?
We need to find the right degree of polynomial parameter, in order to avoid overfitting and underfitting problems:
• Forward selection: increase the degree parameter till you get the optimal result
• Backward selection: decrease degree parameter till you get optimal
Loss and Cost Function – Polynomial Regression
The Cost Function is a function that evaluates a Machine Learning model’s performance for a given set of data. The Cost Function is a single real number that calculates the difference between
anticipated and expected values. Many people dont know the differences between the Cost Function and the Loss Function. To put it another way, the Cost Function is the average of the n-sample error
in the data, whereas the Loss Function is the error for individual data points. To put it another way, the Loss Function refers to a single training example, whereas the Cost Function refers to the
complete training set.
The Mean Squared Error may also be used as the Cost Function of Polynomial regression; however, the equation will vary somewhat.
We now know that the Cost Function’s optimum value is 0 or a close approximation to 0. To get an optimal Cost Function, we may use Gradient Descent, which changes the weight and, as a result, reduces
Gradient Descent – Polynomial Regression
Gradient descent is a method of determining the values of a function’s parameters (coefficients) in order to minimize a cost function (cost). It may decrease the Cost function (minimizing MSE value)
and achieve the best fit line.
The values of slope (m) and slope-intercept (b) will be set to 0 at the start of the function, and the learning rate (α) will be introduced. The learning rate (α) is set to an extremely low number,
perhaps between 0.01 and 0.0001. The learning rate is a tuning parameter in an optimization algorithm that sets the step size at each iteration as it moves toward the cost function’s minimum. The
partial derivative is then determined in terms of m for the cost function equation, as well as derivatives with regard to the b.
With the aid of the following equation, a and b are updated once the derivatives are determined. m and b’s derivatives are derived above and are α.
Gradient indicates the steepest climb of the loss function, but the steepest fall is the inverse of the gradient, which is why the gradient is subtracted from the weights (m and b). The process of
updating the values of m and b continues until the cost function achieves or approaches the ideal value of 0. The current values of m and b will be the best fit line’s optimal value.
Practical Application of Polynomial Regression
We will start with importing the libraries:
#with dataset
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
dataset = pd.read_csv('Position_Salaries.csv')
Segregating the dataset into dependent and independent features,
X = dataset.iloc[:,1:2].values
y = dataset.iloc[:,2].values
Then trying with linear regression,
from sklearn.linear_model import LinearRegression
lin_reg = LinearRegression()
Visually linear regression:
plt.scatter(X,y, color='red')
plt.plot(X, lin_reg.predict(X),color='blue')
plt.title("Truth or Bluff(Linear)")
plt.xlabel('Position level')
from sklearn.preprocessing import PolynomialFeatures
poly_reg = PolynomialFeatures(degree=2)
X_poly = poly_reg.fit_transform(X)
lin_reg2 = LinearRegression()
Application of Polynomial Regression
This equation obtains the results in various experimental techniques. The independent and dependent variables have a well-defined connection.
• Used to figure out what isotopes are present in sediments.
• Utilized to look at the spread of various illnesses across a population
• Research on creation of synthesis.
Advantage of Polynomial Regression
The best approximation of the connection between the dependent and independent variables is a polynomial. It can accommodate a wide range of functions. Polynomial is a type of curve that can
accommodate a wide variety of curvatures.
Disadvantages of Polynomial Regression
One or two outliers in the data might have a significant impact on the nonlinear analysis’ outcomes. These are overly reliant on outliers. Furthermore, there are fewer model validation methods for
detecting outliers in nonlinear regression than there are for linear regression.
Supervised machine learning encompasses classification and regression, with regression crucial for predicting continuous values. Polynomial Regression, a form of regression, captures complex
relationships, requiring careful selection of degree to avoid overfitting or underfitting. Gradient descent optimizes polynomial models, finding practical applications across diverse fields despite
inherent disadvantages.
Responses From Readers
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How Many Inches Are in a Standard Milliliter - Furniture Experts Handyman
How Many Inches Are in a Standard Milliliter
I’d be delighted to assist! The fact remains that 1 milliliter corresponds to an exact measurement of 0.61024 inches, a precise conversion that can be relied upon.
How Do Milliliters and Inches Relate to Each Other in Volume Measurements?
The connection between milliliters and inches in volume measurements is an interesting one. We often think of measuring volumes in terms of liquid quantities, but it’s also important to consider the
physical space those liquids occupy. In practical terms, a single milliliter is equivalent to a volume of 0.0338 fluid ounces, which is roughly the same as the volume of a small paper clip or a
grape. On the other hand, an inch is a unit of length, and in terms of volume measurement, it’s a bit more abstract.
When converting between the two units, it’s helpful to think of the relationship between the volume of a substance and its physical space. For instance, a 1-inch cube of water has a volume of roughly
1.61 milliliters. Conversely, if you had a container filled with 1 liter of water, you could imagine it taking up a space roughly equivalent to the volume of a brick, about 10.7 inches on each side.
The relationship between milliliters and inches is not always intuitive, especially when working with volumes in smaller or larger scales. But understanding the conversion between the two units can
help you better visualize and grasp the concepts of volume and space. By applying this knowledge, you can more easily compare and manipulate volumes in various contexts, making you a whiz with
What is the Conversion Rate for Milliliters to Inches, and How Do I Use It?
Two common units of measurement are milliliters (mL) and inches. If you need to convert between these two units, it’s essential to understand the conversion rate. But before we dive into the
conversion, it’s vital to understand what milliliters and inches represent.
• Milliliters (mL) measure volume in the metric system. It’s commonly used in laboratories, pharmacies, and medical settings to measure small quantities of liquids.
• Inches, on the other hand, measure length or distance in the imperial system. It’s widely used in everyday life, construction, and engineering.
Now, let’s talk about the conversion rate. To convert milliliters to inches, we need to know that 1 milliliter is equivalent to 0.033814 inches. This is the conversion rate we’ll use to change units.
How Many Milliliters Are in a Standard Inch, and Why is This Important in Woodworking?
A standard inch is equivalent to 25.4 millimeters. This conversion is crucial in woodworking due to the precision required in measuring and cutting wood for various projects. Woodworkers rely heavily
on accurate measurements to achieve the desired results, especially when working with thin strips of wood.
To help ensure accuracy, woodworking professionals often use specialized tools, such as calipers, to measure the thickness of the wood. These instruments provide precise readings, allowing
woodworkers to cut the wood with precision and control. By using these tools, woodworkers can ensure that their measurements are accurate, which is vital for producing high-quality pieces.
Another reason accurate measurements are essential in woodworking is that even small variations in the thickness of the wood can significantly impact the quality of the finished product. For example,
if a woodworker is creating a cabinet or table, a slight error in the measurement of the wood can result in a poorly fitting joint or a wobbly table leg.
As a result, many woodworking professionals opt for digital calipers that provide precise measurements in millimeters, making it easier to convert between inches and milliliters. These tools enable
them to work with greater precision, ensuring that their projects meet their exact specifications. By utilizing these specialized tools and conversions, woodworkers can produce high-quality pieces
with confidence.
Can You Convert Cubic Inches to Milliliters, and How Do You Do It?
Cubic inches to milliliters, a conversion that’s as easy as sipping a smoothie on a sunny afternoon. You see, the key is to remember that cubic inches are a unit of volume, while milliliters are a
metric unit of volume. Think of it like comparing apples and oranges, both delicious in their own way, but totally different.
To make this conversion, we need to do a simple calculation. One cubic inch is equivalent to 16.387 cubic centimeters (or cc for short). Now, since we want to convert it to milliliters, we need to
know that one milliliter is equal to 1,000 cubic centimeters. Ah, but wait! We already know that one cubic inch is 16.387 cubic centimeters. So, we can do some math magic and multiply those numbers
together. Ta-da! One cubic inch is equivalent to 16.387 cc, which is roughly 0.016387 liters, or about 16.39 milliliters.
Now, I know what you’re thinking, “Wow, that’s a lot of numbers!” But trust me, it’s easier than it sounds. Just remember that each cubic inch is a small chunk of volume, and by multiplying it by a
bunch of numbers, we get a tasty result – the equivalent volume in milliliters. And there you have it, a conversion that’s as straightforward as making a peanut butter and jelly sandwich. Or, in this
case, a conversion that’s as seamless as sipping that smoothie on a sunny afternoon.
What is the Difference between Milliliters and Millimeters, and How Do They Apply to Volume Measurements?
Both units are used to measure small amounts, but they’re used for different purposes. Let’s break it down.
Milliliters are a unit of volume, used to measure the amount of a liquid or substance. One milliliter is equal to one-thousandth of a liter. In everyday life, you might use milliliters to measure
medicine, juice, or other liquids. For example, a typical medicine bottle might contain 10 mL of liquid.
Millimeters, on the other hand, are a unit of length, used to measure distance or size. One millimeter is equal to one-thousandth of a meter. You might use millimeters to measure the length of a
pencil, the width of a paper, or even the thickness of a coin.
When applying these units to volume measurements, it’s essential to remember that milliliters measure amounts of substance, while millimeters measure distance. For instance, if you’re measuring the
volume of a liquid, you would use milliliters. If you’re measuring the length of a solid object, you would use millimeters.
Practical Examples
Here are a few examples to illustrate the difference:
• A 250 mL bottle of soda contains 250 milliliters of liquid.
• A ruler might measure 15 mm (15 millimeters) in length.
By understanding the difference between milliliters and millimeters, you’ll be better equipped to accurately measure quantities and communicate effectively with others.
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Harold BERJAMIN | PostDoc Position | PhD | University of Galway, Gaillimh | NUI Galway | School of Mathematical and Statistical Sciences | Research profile
Looking for opportunities (Assistant Professor, Lecturer, Researcher, Scientist)
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Learn more
Postdoctoral Researcher in Applied Mathematics at the University of Galway, Ireland. I work on the propagation of mechanical waves in nonlinear materials. In particular, I study the nonlinear dynamic
behavior of viscoelastic and poroelastic solids (theoretical and numerical aspects). This research has various applications in engineering, e.g. in geophysics, biomechanics, nondestructive testing
and materials science.
October 2018 - September 2019
• Applied Mathematics: Numerical Analysis, Probability Theory, Mathematical Statistics (tutorials, practicals).
September 2016 - September 2018
• Graduate Teaching Assistant
• Applied Mathematics: Probability Theory, Mathematical Statistics (tutorials, practicals).
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# 72 Graphs and other ways of displaying data
When you have collected your data and completed your results table, you will generally want to display the data so that anyone looking at them can see any patterns.
1. Line graphs
Line graphs are used when both the independent variable and the dependent variable are continuous. This is the case for the potato strip data on the table below.
The graph can help you to decide if there is a relationship between the independent variable and the dependent variable. This is what a line graph of these data might look like.
• The independent variable goes on the x-axis, and the dependent variable goes on the y-axis.
• Each axis is fully labelled with units. You can just copy the headings from the appropriate columns of your results table.
• The scales on each axis should start at or just below your lowest reading, and go up to or just above your highest reading. Think carefully about whether you need to begin at 0 on either of the
axes, or if there is no real reason to do this.
• The scales use as much of the width and height of the graph paper as possible. If you are given a graph grid on the exam paper, the examiners will have worked out a sensible size for it, so you
should find your scales will fit comfortably. The greater the width and height you use, the easier it is to see any patterns in your data once you have plotted them.
• The scale on each axis goes up in regular steps. Choose something sensible, such as 1s, 2s, 5s or 10s. If you choose anything else, such as 3s, it is practically impossible to read off any
intermediate values. Imagine trying to decide where 7.1 is on a scale going up in 3s...
• Eachpoint is plotted very carefully with a neat cross. Don't usejust a dot, as this may not be visible once you've drawn the line. You could, though, use a dot with a circle round it.
• A smooth best-fit line has been drawn. This is what biologists do when they have good reason to believe there is a smooth relationship between the independent and dependent variables. You know that
your individual points may be a bit off this line (and the fact that the two repeats for each concentration were not always the same strongly supports this view), so you can actually have more faith
in there being a smooth relationship than you do in your plots for each point.
Sometimes in biology (it doesn't often happen in physics or chemistry!) you might have more trust in your individual points than in any possible smooth relationship between them. If that is the case,
then you do not draw a best-fit curve. Instead, join the points with a very carefully drawn straight line, like this:
During your course:
• Get plenty of practice in drawing graphs,so that it becomes second nature always to choose the correct axes. To label them fully and to choose appropriate scales.
In the exam:
• Take time to draw your graph axes and scales - you may need to try out two or even three different scales before finding the best one. • Take time to plot the points - and then go back and check
them. • Use a sharp HB pencil to draw the line, taking great care to touch the centre of each cross if you are joining points with straight lines. If you go wrong, rub the line out completely before
starting again. • If you need to draw two lines on your graph, make sure you label each one clearly.
You may be asked to read off an intermediate value from the graph you have drawn. It is always a good idea to use a ruler to do this - place it vertically to read a value on the x-axis, and
horizontally to do the same on the y-axis. You can draw in faint vertical and horizontal pencil lines along the ruler. This will help you to read the value accurately.
You could also be asked to work out the gradient of a line on a graph. This is explained on
The post #20
During your course:
• Make sure you know how to read off an intermediate value from a graph accurately, and how to calculate a gradient.
In the exam:
• Take time over finding intermediate values on a graph - If you rush it is very easy to read off a value that is not quite correct. 2. Histograms
A histogram is a graph where there is a continuous variable on the x-axis, and a frequency on the y-axis. For example, you might have measured the length of 20 leaves taken from a tree. You could
plot the data like this:
• The numbers on the x-ails scale are written on the lines. The first bar therefore includes all the leaves with a length between 30 and 39 mm. The next bar includes all the leaves with a length
between 40 and 49 mm, and so on.
• The bars are all the same width.
• The bars are all touching - this is important, because the x-axis scale is continuous, without any gaps in it.
3. Bar charts
A bar chart is a graph where the independent variable is made up of a number of
different, discrete categories and the dependent variable is continuous. For example, the independent variable could be type of fruit juice, and the dependent variable could be the concentration of
glucose in the juice.
• The x-axis has an overall heading (type of fruit), and then each bar also has its own heading (orange, apple and so on on).
• The y-axis has a normal scale just as you would use on a line graph.
• The bars are all the same width.
• The bars do not touch.
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What Is Diffie-Hellman Key Exchange in Internet Security?
The Diffie-Hellman algorithm enables two or more parties to create a shared encryption key while communicating over an insecure network. Even though parties exchange plaintext data while generating a
key, the algorithm makes it impossible for eavesdroppers to figure out the chosen encryption key.
This article is a complete guide to the Diffie-Hellman key exchange. Jump in to learn how this algorithm works and see why a 50-year-old cryptographic strategy is still the go-to method for
establishing a secure connection over an insecure channel.
What Is Diffie Hellman Key Exchange?
The Diffie-Hellman key exchange is a protocol that allows devices to establish a shared secret over an insecure medium. Communicating parties use the shared secret to create a unique symmetric key
for encrypting and decrypting messages.
Instead of generating and distributing a key to all participants, the Diffie-Hellman protocol enables each party to create the same custom key individually. At its most basic, this is a three-step
• Two or more parties exchange plaintext info over the network.
• The exchanged info enables participants to compute the same secret number independently.
• Each party inputs the secret number into a key derivation function (KDF) and generates a unique encryption key.
The algorithm never transmits the secret number over the network, which makes the Diffie-Hellman key exchange highly effective at preventing eavesdropping. While intruders can spy on plaintext data
during the key creation process, there is not enough info to determine the key participants plan to use during communication.
Once participants have a unique symmetric key, parties scramble all future messages. Only recipients with the appropriate decryption key can understand the content of the transmitted data.
Each time devices connect again, the Diffie-Hellman algorithm generates a new shared secret (i.e., a new symmetric key). This property aligns the algorithm with perfect forward secrecy (PFS), so
past and future communications stay safe even if a malicious actor determines the key of the current session.
While relatively simple, encryption is the most effective strategy for protecting valuable files. The standard is to encrypt data at rest and in transit, but more organizations are also starting to
implement encryption in use (scrambling data during active processing).
Diffie-Hellman Key Exchange vs. RSA
The Diffie-Hellman key exchange and Rivest-Shamir-Adleman (RSA) are two cryptographic algorithms that serve different purposes.
The Diffie-Hellman enables two parties who don't know each other to generate a shared key without anyone having to send the key to the other party. On the other hand, the RSA enables you to:
• Generate a public/private key pair.
• Publish the public key so anybody can encrypt messages before they send them to you.
• Be the only one capable of decrypting messages since you are the only one with the private key.
Here's a table that outlines the main differences between the Diffie-Hellman and RSA:
Point of comparison Diffie-Hellman algorithm Rivest-Shamir-Adleman (RSA)
Main purpose Enables secure communication over open networks without needing a pre-established secret key. Enables secure messaging via asymmetric encryption.
Key methodology Allows two parties to independently generate a shared symmetric key without directly Asymmetric encryption with a pair of keys: public for encryption, private for
transmitting it over the network. decryption.
Algorithmic Relies on the computational infeasibility of solving discrete logarithm problems. Leverages the difficulty of integer factorization of numbers.
Perfect forward Yes, since participants generate new shared secrets for each session. No, since the key pair is static.
Authentication Does not authenticate communication participants. Authenticates parties involved in communication.
Prevalent use cases Establishing secure connections on insecure networks. Digital signatures, online transactions, and secure communication that require
identity verification.
Diffie-Hellman History
Before the Diffie-Hellman algorithm, all cryptographic systems using symmetric keys had to exchange the plaintext key before they could encrypt traffic. If an eavesdropper intercepted the key, the
intruder could easily decrypt whatever data was moving through the network.
Whitfield Diffie, a researcher at Stanford, and Martin Hellman, a professor at Stanford, began collaborating to address this problem during the early 1970s. In 1976, the two introduced the concept of
public-key cryptography in a paper titled "New Directions in Cryptography."
In this paper, Diffie and Hellman presented the idea of a key exchange protocol that allowed two or more network parties to establish a shared secret without directly exchanging the secret key. The
main idea was to use a pair of mathematically related keys:
• A public key for encryption, which parties can openly share over the network.
• A private key for decryption, which is known only by the recipient.
The proposed algorithm relied on the mathematical difficulty of solving discrete logarithm problems. The algorithm made it computationally infeasible for an eavesdropper to determine the secret key
(i.e., the private key) even if they knew the public parameters (i.e., the public key).
How Diffie Hellman Key Exchange Works
Let's say Dan and Bill want to exchange data over a potentially insecure network. The process starts with both parties publicly defining two numbers:
• The modulus (P), which must be a prime number.
• The base value (G).
In our example, the modulus (P) is 13, while the base (G) is 6. Once Dan and Bill agree on these numbers, both parties randomly generate a secret number (i.e., a private key) they never share with
each other:
• Dan chooses a secret number (a) of 5.
• Bill selects a secret number (b) of 4.
Dan then performs the following calculation to get the number (i.e., the public key) he will send to Bill:
This calculation figures out the remainder after dividing the left side by the right. In our example, Dan calculates:
• A = 6^5 mod 13
• A = 7776 mod 13
• A = 2
Bill does the same calculation, but only for his secret number (b) of 4:
• B = 6^4 mod 13
• B = 1296 mod 13
• B = 9
Dan sends his public number (A) to Bill, while Bill sends his figure (B) to Dan. Dan calculates the shared secret (S) with the following formula:
• S = B^a mod P
• S = 9^5 mod 13
• S = 59049 mod 13
• S = 3
Bill performs the same calculation, but with Dan's public number (A) and his secret number (b):
• S = A^b mod P
• S = 2^4 mod 13
• S = 16 mod 13
• S = 3
Both parties end up with the same number (3), which Dan and Bill use as a basis for an encryption key. The secret number acts as the input to a key derivation function, which then generates a unique
symmetric key.
Remember that the Diffie-Hellman algorithm requires the use of exceptionally large prime numbers (P). We used a small modulus to simplify our example, but a real-life key exchange must use a prime
number that's at least 2048 bits long (the binary equivalent of a decimal number with 512 digits).
Diffie Hellman Algorithm Use Cases
The Diffie-Hellman algorithm has applications in various use cases that require secure key exchanges. The algorithm is valuable in any scenario that involves communication over a potentially unsafe
channel. This key exchange is also vital where pre-shared secret keys are impossible or impractical.
Here are a few common use cases for the Diffie-Hellman algorithm:
• Secure internet communication. Diffie-Hellman is a fundamental component in TLS and SSL protocols. The algorithm enables secure connections between web browsers and servers.
• Wi-Fi security. The Diffie-Hellman key exchange enables secure connections between devices and access points in Wi-Fi networks.
• Remote access protocols. Remote desktop protocols often use Diffie-Hellman to establish encrypted communication channels between remote users and servers.
• Virtual Private Networks (VPNs). VPNs commonly use Diffie-Hellman to establish secure communication channels over the Internet.
• Secure messaging. Many messaging applications, including Signal and WhatsApp, use Diffie-Hellman to protect the privacy of conversations.
• Email protection. Several email security protocols (e.g., Pretty Good Privacy (PGP) or its open standard OpenPGP) use Diffie-Hellman to ensure safe key exchanges.
• Voice over Internet Protocol (VoIP). VoIP services use Diffie-Hellman to establish secure communication channels for voice and video calls.
• Secure file transfers. SSH (Secure Shell) and SFTP (Secure File Transfer Protocol) use Diffie-Hellman for secure key exchanges when establishing a secure channel for data transfers.
Diffie Hellman Algorithm Advantages
The main benefit of the Diffie-Hellman algorithm is that it enables two parties to establish a shared secret key without directly transmitting it over an untrusted channel. The algorithm lowers the
risk of potential eavesdroppers and enables safe use of potentially dangerous networks.
Here are a few other benefits of the Diffie-Hellman key exchange:
• Perfect forward secrecy. The algorithm aligns with PFS, so past and future communications remain secure even if someone compromises a current session's key. PFS enhances overall security posture
and limits the impact of successful breaches.
• High effectiveness. Despite its simplicity, the Diffie-Hellman key exchange is highly effective. The algorithm makes it computationally infeasible for intruders to determine the shared secret
even if they intercept all public parameters (the base value, modulus, and two public keys).
• Interoperability. Diffie-Hellman is widely supported and standardized. The algorithm has high compatibility and interoperability across different systems and platforms, so you can implement the
key exchange in various use cases.
• Simple key management. The Diffie-Hellman algorithm simplifies key management by allowing participants to distribute public keys freely.
Diffie Hellman Algorithm Disadvantages
While the Diffie-Hellman key exchange is highly effective, the algorithm has a few must-know disadvantages. The most notable shortcomings are its lack of authentication and susceptibility to
man-in-the-middle attacks:
• The Diffie-Hellman algorithm establishes a shared secret without checking the identity of involved entities. The process requires additional mechanisms, such as digital signatures or
certificates, to address this limitation.
• Since the algorithm does not authenticate participants, it's relatively simple for someone to intercept and replace public parameters. These man-in-the-middle attacks enable a hacker to connect
with legitimate entities and receive the secret key for the current session.
Here are a few more noteworthy shortcomings of the Diffie-Hellman algorithm:
• Cryptanalysis. Advancements in computing power and cryptanalysis techniques could raise concerns about the algorithm's long-term security. Quantum computing, for example, has the potential to
provide enough resources to crack algorithms that use 2048+ bits long prime numbers.
• Key exchange overhead. The Diffie-Hellman key exchange involves complex mathematical operations. Calculations often result in computational overhead, making the algorithm unideal for
resource-constrained environments.
• Logjam attacks. The logjam attack targets the Diffie-Hellman key exchange with weak or commonly used prime numbers. This attack leverages precomputed tables to perform a fast computation of
discrete logarithms.
If you decide to implement the Diffie-Hellman key exchange, ensure you mandate the use of random and appropriately high prime numbers in your network security policy.
The Diffie-Hellman Algorithm Is as Effective Today as It Was in 1976
Despite being almost 50 years old, the Diffie-Hellman algorithm remains the go-to method for communicating over insecure channels. While this key exchange strategy has a few notable shortcomings,
Diffie-Hellman is a vital enabler for various use cases that require communication over a potentially compromised network.
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Tech Note No.1: MAP sensor recalibration and replacement
Monday, March 13, 2017 - 08:00
Note link to calculator spreadsheet added at bottom of post
The 1.42 bar boost limitation of the stock Land Rover Td5 engine management system has been a long standing issue when increasing boost levels as a performance upgrade.
The standard solution has been to insert a boost box or some other type of "cheater circuit" in line with the MAP sensor to lower the sensor voltage to provide the ECU with false reading of the boost
In the course of disassembling the ECU firmware it became apparent that the sensor parameters could altered with minor adjustments to the settings in the fuel map to accomodate alternative MAP
sensors. The advantage to this approach is that the ECU receives a true reading of the boost pressure rather than a falsified reading that effects pressure readings across the range.
This modification was originally tested on a range of Td5 powered vehicles by members of the Td5Tuning.info forum in January and February 2014.
Under the hood
In order to understand why and how this modification works, it is useful to understand the signal flow from the MAP sensor through the ECU hardware and the processing the ECU firmware applies to the
converted analog voltage.
Sensor Signal Flow
The starting point of the conversion process from manifold pressure to ECU representation of the pressure is MAP/IAT sensor which is mounted on the TD5 inlet manifold. The voltage the MAP sensor
outputs in response to a given manifold pressure is determined by the characteristics of the sensor used. The relationship between pressure and output voltage is often described as the transfer
function of the sensor.
Once the MAP sensor voltage enters the ECU housing it passes through a voltage divider formed by two resistors. The effect of the voltage divider is to reduce the incoming MAP sensor voltage to 90.7%
of the original value.
The reduced MAP sensor voltage is then processed by a 10 bit Analog to Digital Convertor (ADC). The 10 bit range of the ADC allows the sensor voltage to be converted to one of 1024 ( 2^10) values.
The conversion process is referenced to a 5 volt supply within the ECU, meaning the maximum ADC value of 1023 represents 5000mV and the minimum ADC value of 0 represents 0mV.
At this point the signal flow moves from hardware to pure software. The ECU code first checks that the raw value retrieved from the ADC is within the preset range. Both the minimum and maximum values
are related to characteristics of the sensor being used and the range of sensor voltages that are considered within normal range. The maximum value set for the MAP in stock tunes is the equivalent of
a pressure reading of 2.42 bar, which will familiar as the point the ECU limits with over boost. Any value that lies outside the initial check range causes an “out of range” fault to be logged.
In the next stage of processing the range checked ADC value is scaled so the converted value is returned to the required units. In the case of the MAP sensor this is millibar * 100. The scaling and
offset values reverse the transformations applied by the ADC conversion, voltage divider and the sensor transfer curve to give the pressure value the sensor measured.
Reverse engineering from the stock MAP sensor
At this point we have a basic outline of how the MAP sensor voltage progresses to a digital representation of the pressure, and where intervention is required if we wish to replace a MAP sensor.
To illustrate the process of configuring the ECU values for a specific sensor the following section works from first principles using the stock MAP sensor.
While the explanation of the process may seem long winded and overly detailed the aim here is to explore the underlying assumptions, so that the same procedure can be applied to other sensors.
The sensor datasheet
The starting point is to acquire the specification of the MAP sensor as this provides the key information required to accurately configure the ECU. The stock MAP sensor for the Td5 is Bosch part 0
281 002 205, and the key parameters from the data sheet are shown below.
Note that the data table pressure range refers to two points - p1 and p2 The information about these points is given in the plot of the sensor Characteristic Curve.
From the spec sheet we can determine that that this sensor has an output of 400mV at 20kPa and 4650mV at 250 kPa.
Sensor Transfer Curve
Calculating the transfer curve is fairly simple and uses only basic high school algebra.
First we calculate the slope of the sensor curve from the two points given by the datasheet.
p1(mV) = 400
p1(kPa) = 20
p2(mV) = 4650
p2(kPa) = 250
The slope of the sensor curve (m) is the change in voltage divided by the change in pressure.
$ m = \frac{p2(mV) - p1(mV)}{p2(kPa) - p1(kPa)} $
Substituting in the stock MAP sensor values
$$ m = \frac{4650- 400}{250 - 20} = \frac{4250}{230} = 18.478260869565217 mV/kPa $$
The slope m tells us that for 1 kPa change in manifold pressure the output of the sensor increases by 18.4783 mV.
The second piece of information required is the voltage the sensor outputs when the pressure is 0 kPa. This is the sensor offset.
offset = p1(mV) - (m * p1(kPa)
= 400 - (18.4783 * 20)
= 400 - 369.566
= 30.434 mV
The slope (m = 18.4783 mV/kPa) and offset (30.434 mV) provide us with sufficient information about the MAP to configure the ECU.
ECU Hardware
As discussed in the Sensor Signal Flow section the output of the MAP sensor passes through two fixed stages of processing - a voltage divider and the Analog Digital Convertor.
Voltage divider
The Td5 ECU uses a resistor divider arrangement on sensor inputs to provide a form of over voltage protection for the Analog to Voltage Convertor inputs. The divider consists of two resistors:
resistor1 = 121000 ohm
resistor2 = 12400 ohm
$ voltDivider = \frac{121000}{ 121000 + 12400} = 0.9070 $
The divider therefore reduces the sensor voltage to 90.7% of the original value. This means that a sensor output of 5000mV is reduced to 4535mV at the input of the ADC.
ADC Conversion
The 10 bit ADC divides the range of voltages between ground/0mV and the sensor supply voltage/5000mV into 1024 discrete steps.
Dividing the total number of steps by the voltage range gives the step size of 1 millivolt of input voltage. The output of the ADC is a value between 0 and 1023 that represents the measured voltage.
Note that there is debate as to whether n or n-1 steps is correct. Comparing both methods to the ECU curves indicates that n-1 gives the closest match when compared with stock values.
The voltage of each ADC step (or ADC code) is calculated as:
$ step/mV = 1023/5000 = 0.2046 $
Note that it requires a change of at least 4.8876 mV to cause a change of 1 ADC code.
Putting together the hardware scaling factors
The combined effect of the voltage divisor and ADC allows us to calculate the value in "adc codes" the ADC will output for a given sensor voltage at the ECU connector.
hwScale = ADCstep/mV * voltDivider
hwScale = 0.2046 * 0.9070 = 0.18557
Using this hardware scaling factor we can calculate the ADC codes produced by a voltage at the ECU MAP input.
For example if we have 4650mV input...
$ adc codes = 4650 * 0.18557 = 862.9 $
This can be extended to calculate the ADC output for a given pressure.
Using 242kPa as an example...
pressure = 242kPa;
adcCodes = ((pressure * sensor slope ) + sensor offset) * hwScale
adcCodes = ((242 * 18.4783) + 30.434) * 0.18557 $
adcCodes = 4502.18 * 0.18557= 835.47 $
As another illustration lets use the sensor maximum of 250kPa...
pressure = 250kPa;
adcCodes = ((pressure * sensor slope ) + sensor offset) * hwScale
adcCodes = ((250 * 18.4783) + 30.434) * 0.18557
adcCodes = 4650 * 0.18557 = 862.9
Error handling
At this point in the process the ECU does error checking against minimum and maximum values defined for the specific sensor input. These are the values that are available as scalar editors
(ai_limit_min : MAP and ai_limit_max : MAP) in my "donor-ware" Tuner Pro .XDF's.
If the sensor value in ADC codes is below the minimum the ECU logs "below minimum" and "out of range" faults, if the value is above the maximum, "above maximum" and "out of range" faults are logged.
These are the faults the Nanocom reports as "logged low" (1-2,x), "logged high"(3-4,x) and "current" (5-6,x). "Current" simply indicates that there is either a "logged low" or "logged high" fault
Using the stock MAP ADC limit check values as example we can work backwards to find the voltages and pressures that have been set.
min = 93
max = 836
By reversing the transforms performed by the ADC and voltage divisor we can reconstruct the input voltage.
sensorV = ( limits / mvStepADC ) / vDivider
sensorVmin = ( 93 / 0.2046 ) / 0.9070 = 501
sensorVmax = (836 / 0.2046) / 0.9070 = 4505
So it appears the limits are set to a minimum of 500mV and a maximum of 4500mV.
To find the pressure these voltages correspond to we divide the voltage minus the offset by the mV/kPa value m
$ pressureLimits = (sensor voltage - sensor offset) / m $
$ pressureMin = (501 - 30.434) / 18.4783 = 25.46kPa $
$ pressureMax = (4505 - 30.434) / 18.4783 = 242.15 kPa $
25kPa is well below anything you'd encounter while driving on the surface of this planet but higher than the minimum sensor hardware limit. 242kPa matches the well known stock boost limit.
When recalibrating for a new MAP sensor, or using the stock sensor at higher boost it is essential that these limit values are reset to match the new boost levels and sensor curve.
Software: From ADC codes to pressure readings
Working backwards from ADC codes to pressure is a good warmup for the remaining steps of calculating new multiplier, divisor and offset parameters which makes possible substitution of alternative
In effect we are reversing the transformations done by the sensor curve m, sensor offset, the voltage divisor and the steps/mV of the ADC process in the same way the limiter pressures were checked.
The value used internally by the ECU for MAP is kPa*100. Additionally the divisor in the stock configuration is set to 1000. This is done due to the integer math used in processing - the multiplier
is x1000 to give three decimal places of additional precision, and after the multiplication is completed the result is divided by 1000 to the required two places. This means the multiplier should be
multipled by 100000 to bring it to correct units for use in a engine map.
$ multiplier = (1/(m * hwScale ))* 100000 $
$ multiplier = (1/(18.4783 * 0.18557 ))* 100000 $
$ multiplier = (1 / 3.429018) * 100000 = 0.2916287 * 100000 $
$ multiplier = 29163 $
The stock value for this parameter is 29163, so the calculated value is a match for the factory calculations.
Stock divisor parameter is 1000, and reverses the three places precision noted above.
The final parameter is the offset which is the sensor offset calculated in the intial steps converted to ADC codes then scaled by the multiplier and divisor.
$$ offset = \frac{sensor offset * hwScale * multiplier}{divisor} $$
$$ offset = \frac{30.434 * 0.18557 * 29163} {1000} = 164.7 $$
Rounding up to the next highest integer value gives 165.
As noted earlier the offset indicates the voltage the sensor would output at 0kPa pressure. This means that to correct so the sensor curve so the output is 0 mV at 0 kPa you need to subtract the
offset if it is positive and add if it is negative.
The ECU math uses addition for this calculation, so if the offset is positive we need to swap the sign to make the number negative. And if the offset is negative the number added needs to be signed
swapped to make it a positive number.
So in this case the ECU offset parameter should be -165 to remove the positive offset of 165.
In summary, the values calculated from the datasheet information match the stock parameters:
MAP ADC Maximum: 836
MAP Multiplier: 29163
MAP Divisior: 1000
MAP Offset: -165
Current XDF's have a changed naming scheme for scalars which reflects LR documentation.
ai_limit_max_map = 836
ai_anlg_mult_map = 29163
ai_anlg_divisor_map = 1000
ai_anlg_offset_map = -165
1.5 Bar MAP Recalibration
This is a super simple mod to do!
• Change the MAP ADC Max (ai_limit_max : MAP) value from 836 to 863, which raises maximum input to 250kPa.
• Change the Boost Limit (tb_over_pres_enbl) from 14200 to 15000.
• Change the Boost Limiter Recovery (tb_over_pres_disbl) value to 14800.
This mod uses the stock MAP sensor and does not require any hardware changes.
It's a good choice if you are running stock intercooler and turbo.
In the Tuner Pro .XDF's I give as a "thank you" to donors these parameters can be edited using a simple graphical interface.
The parameters can be located by searching for the stock values using a hex editor of course, so it's your choice.
MAP Setting Calculator
There is now a MAP Parameter calculator on Google Spreadsheets.
It's read only so you'll need to download as an XLSX or ODS spreadsheet (or copy to your Google account) from the File menu.
The spreadsheet contains the values required for the VAG 3 Bar and Bosch 3.5 Bar (PN# 0 281 002 244) sensors.
- Copy the values you need across to the area highlighted in yellow or enter for the sensor you want to use.
- Set the boost limit required (pressure from Point 2 or lower *100). Recovery is calculated as 2kPa below this.
- If the calculated multiplier is greater than 32767 you'll need to lower the divisor. Try 750 as a starter.
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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.7
URN: urn:nbn:de:0030-drops-112229
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/11222/
Eden, Alon ; Feige, Uriel ; Feldman, Michal
Max-Min Greedy Matching
A bipartite graph G(U,V;E) that admits a perfect matching is given. One player imposes a permutation pi over V, the other player imposes a permutation sigma over U. In the greedy matching algorithm,
vertices of U arrive in order sigma and each vertex is matched to the highest (under pi) yet unmatched neighbor in V (or left unmatched, if all its neighbors are already matched). The obtained
matching is maximal, thus matches at least a half of the vertices. The max-min greedy matching problem asks: suppose the first (max) player reveals pi, and the second (min) player responds with the
worst possible sigma for pi, does there exist a permutation pi ensuring to match strictly more than a half of the vertices? Can such a permutation be computed in polynomial time?
The main result of this paper is an affirmative answer for these questions: we show that there exists a polytime algorithm to compute pi for which for every sigma at least rho > 0.51 fraction of the
vertices of V are matched. We provide additional lower and upper bounds for special families of graphs, including regular and Hamiltonian graphs. Our solution solves an open problem regarding the
welfare guarantees attainable by pricing in sequential markets with binary unit-demand valuations.
BibTeX - Entry
author = {Alon Eden and Uriel Feige and Michal Feldman},
title = {{Max-Min Greedy Matching}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
pages = {7:1--7:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-125-2},
ISSN = {1868-8969},
year = {2019},
volume = {145},
editor = {Dimitris Achlioptas and L{\'a}szl{\'o} A. V{\'e}gh},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/11222},
URN = {urn:nbn:de:0030-drops-112229},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.7},
annote = {Keywords: Online matching, Pricing mechanism, Markets}
Keywords: Online matching, Pricing mechanism, Markets
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)
Issue Date: 2019
Date of publication: 17.09.2019
DROPS-Home | Fulltext Search | Imprint | Privacy
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susie is painting a wall that is 12 feet tall and 9 feet wid... - Ask Spacebar
susie is painting a wall that is 12 feet tall and 9 feet wide if each can of paint has enough paint to cover 8 square feet how many cans of paint will susie need to cover the whole wall
Views: 0 Asked: 12-27 06:07:25
On this page you can find the answer to the question of the mathematics category, and also ask your own question
Other questions in category
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Unscramble OASTS
How Many Words are in OASTS Unscramble?
By unscrambling letters oasts, our Word Unscrambler aka Scrabble Word Finder easily found 23 playable words in virtually every word scramble game!
Letter / Tile Values for OASTS
Below are the values for each of the letters/tiles in Scrabble. The letters in oasts combine for a total of 7 points (not including bonus squares)
What do the Letters oasts Unscrambled Mean?
The unscrambled words with the most letters from OASTS word or letters are below along with the definitions.
• oast (n.) - A kiln to dry hops or malt; a cockle.
• stoa () - Sorry, we do not have a definition for this word
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Think Nuclear
The “Boy or Girl Paradox” (also called “The Two Child Problem” in addition to other names) is generally phrased as follows:
You know a couple who has two children. At least one of the children is a girl. What is the probability that they have two girls?
This is an ambiguous problem, which leads to different answers depending on the assumptions that are used. Not enough information has been provided to produce a definite answer, and the unstated
assumptions fill in the space needed to complete the logic.
Here I investigate this problem and explain the ambiguity.
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Margins, Multiples, and the Iron Law of Valuation
John Hussman will be speaking at the Wine Country Conference held in Sonoma, CA on May 1^st and 2^nd, 2014. Net proceeds from the conference will go to the Autism Society of America for grant
requests focusing on high-impact programming for individuals on the autism spectrum and their families. More information at www.winecountryconference.com.
The equity market remains valued at nearly double its historical norms on reliable measures of valuation (though numerous unreliable alternatives can be sought if one seeks comfort rather than
reliability). The same measures that indicated that the S&P 500 was priced in 2009 to achieve 10-14% annual total returns over the next decade presently indicate estimated 10-year nominal total
returns of only about 2.7% annually. That’s up from about 2.3% annually last week, which is about the impact that a 4% market decline would be expected to have on 10-year expected returns. I should
note that sentiment remains wildly bullish (55% bulls to 19% bears, record margin debt, heavy IPO issuance, record “covenant lite” debt issuance), and fear as measured by option volatilities is still
quite contained, but “tail risk” as measured by option skew remains elevated. In all, the recent pullback is nowhere near the scale that should be considered material. What’s material is the extent
of present market overvaluation, and the continuing breakdown in market internals we’re observing. Remember – most market tops are not a moment but a process. Plunges and spikes of several percent in
either direction are typically forgettable and irrelevant in the context of the fluctuations that occur over the complete cycle.
The Iron Law of Valuation is that every security is a claim on an expected stream of future cash flows, and given that expected stream of future cash flows, the current price of the security moves
opposite to the expected future return on that security. Particularly at market peaks, investors seem to believe that regardless of the extent of the preceding advance, future returns remain entirely
unaffected. The repeated eagerness of investors to extrapolate returns and ignore the Iron Law of Valuation has been the source of the deepest losses in history.
A corollary to the Iron Law of Valuation is that one can only reliably use a “price/X” multiple to value stocks if “X” is a sufficient statistic for the very long-term stream of cash flows that
stocks are likely to deliver into the hands of investors for decades to come. Not just next year, not just 10 years from now, but as long as the security is likely to exist. Now, X doesn’t have to be
equal to those long-term cash flows – only proportional to them over time (every constant-growth rate valuation model relies on that quality). If X is a sufficient statistic for the stream of future
cash flows, then the price/X ratio becomes informative about future returns. A good way to test a valuation measure is to check whether variations in the price/X multiple are closely related to
actual subsequent returns in the security over a horizon of 7-10 years.
This is very easy to do for bonds, especially those that are default-free. Given the stream of cash flows that the bond will deliver over time, the future return can be calculated by observing the
current price (the only variation from actual returns being the interest rate on reinvested coupon payments). Conversely, the current price can be explicitly calculated for every given
yield-to-maturity. Because the stream of payments is fixed, par value (or any other arbitrary constant for that matter) is a sufficient statistic for that stream of cash flows. One can closely
approximate future returns knowing nothing more than the following “valuation ratio:” price/100. The chart below illustrates this point.
[Geek's Note: the estimate above technically uses logarithms (as doubling the bond price and a halving it are “symmetrical” events). Doing so allows other relevant features of the bond such as the
maturity and the coupon rate to be largely captured as a linear relationship between log(price/100) and yield-to-maturity].
Put simply, every security is a claim on some future expected stream of cash flows. For any given set of expected future cash flows, a higher price implies a lower future investment return, and vice
versa. Given the price, one can estimate the expected future return that is consistent with that price. Given an expected future return, one can calculate the price that is consistent with that
return. A valuation "multiple" like Price/X can be used as a shorthand for more careful and tedious valuation work, but only if X is a sufficient statistic for the long-term stream of future cash
Margins and Multiples
The Iron Law of Valuation is equally important in the stock market, as is the need for representative measures of future cash flows when investors consider questions about valuation. It’s striking
how eager Wall Street analysts become – particularly in already elevated markets – to use current earnings as a sufficient statistic for long-term cash flows. They fall all over themselves to ignore
the level of profit margins (which have always reverted in a cyclical fashion over the course of every economic cycle, including the two cycles in the past decade). They fall all over themselves to
focus on price/earnings multiples alone, without considering whether those earnings are representative. Yet they seem completely surprised when the market cycle is completed by a bear market that
wipes out more than half of the preceding bull market gain (which is the standard, run-of-the-mill outcome).
The latest iteration of this effort is the argument that stock market returns are not closely correlated with profit margins, so concerns about margins can be safely ignored. As it happens, it’s true
that margins aren’t closely correlated with market returns. But to use this as an argument to ignore profit margins is to demonstrate that one has not thought clearly about the problem of valuation.
To see this, suppose that someone tells you that the length of a rectangle is only weakly correlated with the area of a rectangle. A moment’s thought should prompt you to respond, “of course not –
you have to know the height as well.” The fact is that length is not a good sufficient statistic, nor is height, but the product of the two is identical to the area in every case.
Similarly, suppose someone tells you that the size of a tire is only weakly correlated with the number of molecules of air inside. A moment’s thought should make it clear that this statement is
correct, but incomplete. Once you know both the size of the tire and the pressure, you know that the amount of air inside is proportional to the product of the two (Boyle’s Law, and yes, we need to
assume constant temperature and an ideal gas).
The same principle holds remarkably well for equities. What matters is both the multiple and the margin.
Wall Street – You want the truth? You can't handle the truth! The truth is that in the valuation of broad equity market indices, and in the estimation of probable future returns from those indices,
revenues are a better sufficient statistic than year-to-year earnings (whether trailing, forward, or cyclically-adjusted). Don’t misunderstand – what ultimately drives the value of stocks is the
stream of cash that is actually delivered into the hands of investors over time, and that requires earnings. It’s just that profit margins are so variable over the economic cycle, and so
mean-reverting over time, that year-to-year earnings, however defined, are flawed sufficient statistics of the long-term stream of cash flows that determine the value of the stock market at the index
As an example of the interesting combinations that capture this truth, it can be shown that the 10-year total return of the S&P 500 can be reliably estimated by the log-values of two variables: the S
&P 500 price/book ratio and the equity turnover ratio (revenue/book value). Why should these unpopular measures be reliable? Simple. Those two variables – together – capture the valuation metric
that's actually relevant: price/revenue. If you hate math, just glide over any equation you see in what follows – it’s helpful to show how things are derived, but it’s not required to understand the
key points.
price/revenue = (price/book)/(revenue/book)
Taking logarithms and rearranging a bit,
log(price/revenue) = log(price/book) + log(book/revenue)
If price/revenue is the relevant explanatory variable, we should find that in an unconstrained regression of S&P 500 returns on log(price/book) and log(book/revenue), the two explanatory variables
will be assigned nearly the same regression coefficients, indicating that they can be joined without a loss of information. That, in fact, is exactly what we observe.
Similarly, when we look at trailing 12-month (TTM) earnings, the TTM profit margin and P/E ratio of the S&P 500 are all over the map. When profit margins contract, P/E ratios often soar. When profit
margins widen, P/E ratios are suppressed. All of this introduces a terrible amount of useless noise in these indicators. As a result, TTM margins and P/E ratios are notoriously unreliable
individually in explaining subsequent market returns. But use them together, and the estimated S&P 500 return has a 90% correlation with actual 10-year returns. Moreover, the two variables – again –
come in with nearly identical regression coefficients. Why? Because they can be joined without a loss of information, that is, the individual components contain no additional predictive information
on their own. Just like the area of a rectangle and Boyle’s Law:
price/revenue = (earnings/revenue)*(price/earnings)
Again taking logarithms
log(price/revenue) = log(profit margin) + log(P/E ratio)
The chart below shows this general result across a variety of fundamentals. In each case, the fitted regression values have a greater than 90% correlation with actual subsequent 10-year S&P 500 total
returns. Let’s be clear here – I’m not a great fan of this sort of regression, strongly preferring models that have structure and explicit calculations (see for example the models presented in It is
Informed Optimism to Wait for the Rain). The point is that one can’t cry that “profit margins aren’t correlated with subsequent returns” without thinking about the nature of the problem being
addressed. The question is whether P/E multiples, or the Shiller cyclically-adjusted P/E, or the forward operating P/E, or price/book value, or market capitalization/corporate earnings, or a host of
other possibilities can be used as sufficient statistics for stock market valuation. The answer is no.
What we find is that both margins and multiples matter, and they matter with nearly the same regression coefficients – all of which imply that revenue is a better sufficient statistic of the
long-term stream of future index-level cash flows than a host of widely-followed measures. Emphatically, one should not use unadjusted valuation multiples without examining the relationship between
the underlying fundamental and revenues. That is why we care so much about record profit margins here.
Note that in each of these regressions, the coefficients could place a low weight on profit margins and other measures that are connected with revenues, if doing so would improve the fit. They could
place significantly different coefficients on margins and multiples, if doing so would improve the fit. They just don’t, and like the area of a rectangle and Boyle’s Law, this tells you that it is
the product of the two measures that drives the relationship with subsequent market returns.
[Geek’s Note: Gross value added (essentially revenue of U.S. corporations including domestic and foreign operations) is estimated as domestic financial and nonfinancial gross value added, plus
foreign gross value added of U.S. corporations inferred by imputing a 10% profit margin to the difference between total U.S. corporate profits after tax and purely domestic profits. Varying the
assumed foreign profit margin has very little impact on the overall results, but this exercise addresses the primary distinction (h/t Jesse Livermore) between normalizing CPATAX by GDP versus
normalizing by estimated corporate revenues.]
To illustrate these relationships visually, the 3-D scatterplot below shows the TTM profit margin of the S&P 500 along one bottom axis, the TTM price/earnings ratio on the other bottom axis, and the
actual subsequent 10-year annual total return of the S&P 500 on the vertical axis. This tornado of points is not distributed all over the map. Instead, you’ll notice that the worst market returns are
associated with points having two simultaneous features: not only above-average profit margins, but elevated price/earnings multiples as well. This combination is wicked, because it means that
investors are paying a premium price per dollar of earnings, where the earnings themselves are cyclically-elevated and unrepresentative of long-term cash flows. This is the situation we observe at
present. It bears repeating that the S&P 500 price/revenue multiple, the ratio of market capitalization to GDP, and margin-adjusted forward P/E and cyclically-adjusted P/E ratios remain more than
double their pre-bubble historical norms.
[Geek’s Note: On a 3-D chart where the Z variable is determined by the sum or product of X and Y, a quick way to visually identify the relationship is to view the scatter from either {min(X},max(Y)}
or {max(X),min(Y)} as above].
The upshot is that regardless of the metrics used, S&P 500 nominal total returns in the coming decade are likely to be in the very low single digits – from current levels. But remember the Iron Law
of Valuation – for a given stream of long-term expected cash flows, as valuations retreat, prospective returns increase. This should be a cause for optimism about future investment opportunities.
Unfortunately, not present ones.
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1d5 in Tabletop RPGs: Usage Tips
Discover the intriguing world of 1d5 rolls in tabletop RPGs and how this unique dice can enhance your gaming sessions. Whether you're creating characters, generating loot, or resolving actions, a 1d5
offers a fresh twist on traditional gameplay. Here's a quick overview:
• Understanding 1d5 Rolls: Learn how to simulate a 1d5 using a d10 for outcomes ranging from 1 to 5.
• Creative Applications: From character creation and loot generation to resolving actions, explore various ways to incorporate 1d5 rolls.
• Implementing 1d5 Rolls in Popular Systems: See how a 1d5 can fit into games like Dungeons & Dragons 5th Edition, Pathfinder Second Edition, and World of Darkness 5th Edition.
• Step-by-Step Implementation Guide: Get tips on preparing for and executing 1d5 rolls in your games.
• Gameplay Examples: Inspiring scenarios that show the dynamic impact of 1d5 rolls.
Remember, the 'd' in 1d5 stands for 'dice,' and this guide aims to make your RPG sessions more unpredictable and fun with the simple addition of a 1d5.
Definition and Mechanics
When we talk about a 1d5 roll, we mean rolling a die that can land on any number from 1 to 5. But here's the thing: actual 5-sided dice are pretty rare. So, most of the time, players roll a 10-sided
die (d10) and just divide the result by 2, rounding up if needed. This way, numbers 1 to 5 on the d10 represent the same numbers on a d5.
This kind of roll works just like any other dice roll in games. It's a way to bring in some chance. Since it only goes up to 5, it's perfect for when you have a few options and don't need a big range
of numbers. Each number has an equal chance of coming up, which is 20%.
Historical Context
Dice with different numbers of sides have been around since the 1970s, especially with games like Dungeons & Dragons. The usual set includes dice with 4, 6, 8, 10, 12, and 20 sides. The d5 was used
here and there, but it wasn't as popular.
Over time, games have stuck with a standard set of dice to keep things simple for everyone. Most players have d6s (6-sided dice) and d20s (20-sided dice), so those are used the most. Even though the
d5 is pretty unique, it's not used much because it's easier when everyone has the same kinds of dice.
But the d5 has its moments. It's good for when you need a range that's not too big. And even if you don't have a d5, you can still get the same effect with other dice. It's a fun way to add something
different to your game, even if it's a bit unusual!
Creative Applications of 1d5 Rolls
Character Creation
Using a d5 can make creating characters for games like Dungeons & Dragons more fun and random. Here's how:
• Decide on character details. Roll a d5 to see how many flaws or secrets your character has. This makes your character more interesting.
• Figure out character size. Use a d5 to help decide how tall or heavy your character is. This is helpful in games where how much you can carry matters.
• Pick languages. Roll a d5 to see how many languages your character knows. Roll again to pick which ones.
• Find out about your character's past. Roll a d5 to learn how many jobs or adventures they had before the game starts. This adds depth to your character's story.
Using a d5 here adds just the right amount of chance to make each character unique without making things too complicated.
Loot Generation
When players find treasure, a d5 can make it more exciting:
• Decide how rare an item is. First, roll a d5 to set the item's rarity. Then, use another roll or a chart to pick the exact item.
• Figure out how much loot there is. Roll a d5 to see how many items players find. This keeps things interesting.
• Check magic item charges. Use a d5 to see how many uses a magic wand or staff has left. This range keeps it simple.
• Find gold or gems. Multiply your d5 roll by 5, 10, or 20 to decide how much treasure players find. This adds a fun twist to discovering loot.
Using a d5 here makes finding loot less predictable and more fun for everyone.
Resolving Actions
For actions that need a smaller range of outcomes, a d5 is perfect:
• Decide how far something moves. For things like being pushed by wind or a monster, roll a d5 to see how many feet or meters it goes.
• Figure out how long an effect lasts. A d5 can decide how many rounds a potion or spell works for. This keeps things simple.
• Try something tricky. When doing something hard, like fitting through a tight space, roll a d5 to see if you succeed.
• Roll for small injuries. Use a d5 for damage from things like a stubbed toe. It's enough to be a lesson but not too harsh.
The d5 is great for when you need a little bit of randomness without big changes. It adds a nice touch of fun to the game.
Implementing 1d5 Rolls in Popular Systems
Adding a d5 to your game can make things more interesting in lots of different tabletop RPGs. Here's how to do it in some popular games:
Dungeons & Dragons 5th Edition
D&D 5e is super popular and there are many ways to use a d5:
• Initiative order. Players can roll a d5 plus their initiative bonus to decide their turn order. This keeps things random but simple.
• Lair actions. The DM can use a d5 to pick a lair action each round. This makes the game more unpredictable.
• Traps. Use a d5 to decide how much damage a trap does. This method is quick and keeps players on their toes.
• Identifying magic items. Rolling a d5 can determine how many minutes it takes to figure out what an item does. It's fast but adds a bit of suspense.
The d5 adds excitement without making things too complicated.
Pathfinder Second Edition
Pathfinder 2e also works well with a d5:
• Skill feat activation. If a feat activates on a roll of 5 or less, use a d5 to make it happen more often.
• Spell durations. For spells that don't last long, use a d5 to decide how many rounds they work. This adds variety.
• Weapon damage. For weapons like thrown daggers, a d5 can decide the damage. This keeps attacks unpredictable but fair.
• Trap finding. To spot traps, roll a d5 for your Perception check. It makes failing less frustrating.
The d5 brings in a good mix of chance and fairness.
World of Darkness 5th Edition
In story-driven games like WoD, a d5 can add to the drama:
• Complications. When players do something risky, rolling a d5 can add unexpected twists.
• NPC attitudes. Use a d5 to quickly figure out how an NPC feels about the players.
• Willpower. Players can roll a d5 to see how much willpower they get back each game. This keeps things uncertain.
• Health levels. For vampires, roll a d5 to decide damage from attacks. It's quick and keeps the action moving.
With just 5 options, the d5 makes decisions quick and adds surprises to the game.
Step-by-Step Implementation Guide
Before you start using 1d5 in your games, here's what game masters (GMs) should do:
• Review core rules: Make sure you know how dice are used in your game. Look for places where a 1d5 could work instead of other dice.
• Plan integration: Think about when a 1d5 would be useful, like for deciding how much damage a trap does or how long a spell lasts. Plan out when you'll ask for these rolls.
• Adjust difficulty: Sometimes, you might need to change how hard or easy something is so that a 1-5 result makes sense. Make sure the game stays balanced.
• Inform players: Tell your group about using 1d5 rolls. Let them know it's for adding fun and a bit of unpredictability.
Doing these things will help everyone get used to the idea of using a 1d5 in the game.
When it's game time, here's how to use 1d5 rolls:
• Set the scene: Explain why a 1d5 roll is needed this time, so players get the context.
• Have players roll: Ask for a 1d5 roll, reminding them to use a d10 and divide by 2.
• Resolve the outcome: Use the roll to figure out what happens next and describe it.
• Adjust on the fly: If the outcomes feel off, you can make small changes to keep things balanced.
• Gather feedback: After the game, see if the players liked the new twist or found it too random. Be ready to adjust based on what they say.
Keeping things flexible and listening to your players will make using 1d5 rolls smooth.
To fine-tune using 1d5, GMs should:
• Solicit player feedback: Ask your players if they liked the randomness or if the 1d5 was annoying.
• Review session logs: Look back at how 1d5 rolls went. Check if they added fun or slowed things down.
• Tweak mechanics: If the randomness isn't fitting well, you might need to change some things. Maybe use different dice in some places.
• Communicate changes: Let your group know about any changes. The goal is to make the game more fun for everyone.
By making small changes based on how things go, 1d5 rolls can add just the right amount of excitement.
Gameplay Examples
Here are a few ways a 1d5 roll can change the game in fun and unexpected ways:
A Mysterious Stranger
The group runs into an odd old man on the road who says he has important news. The GM uses a d5 to figure out if the man is being honest:
1. He's telling the truth and wants to help the group.
2. He's a thief trying to steal from the group.
3. He's cursed and can only talk in puzzles.
4. He's a spy who's watching adventurers in the area.
5. He's actually a demon in disguise.
The roll leads to very different paths, showing how a simple d5 can make a big difference.
Deadly Traps
While in an old tomb, the group sets off a trap. The GM rolls a d5 to see which player triggers the trap and gets hurt:
1. The fighter in armor gets hit by a poison dart.
2. The quick rogue is caught by spikes from the walls.
3. The wizard steps on a magic rune on the ground.
4. The cleric is grabbed by a moving mummy from a coffin.
5. The ranger finds a bunch of poisonous snakes.
The d5 roll picks who gets hit by the trap.
Mysterious Potions
The group finds some potions without labels. The GM gives each a number from 1 to 5. When someone drinks one, a d5 roll decides what happens:
1. Restore Health - You get some health back.
2. Boost Strength - You do more damage in fights.
3. Enhance Senses - You're better at noticing things.
4. Burst of Speed - You can move twice as fast for a little while.
5. Vile Poison - You get poisoned.
The excitement of not knowing what the d5 roll will do makes trying these potions thrilling!
These examples show that using a 1d5 can add a lot of surprises and fun to tabletop RPGs. Try it out in your next game!
Throwing a 1d5 in tabletop RPGs like Dungeons & Dragons can make your game more exciting by adding a bit of chance. With only 5 outcomes, the 1d5 is a simple way to bring in surprises without making
things too complicated.
Here's what you should remember about using 1d5 rolls:
• A 1d5 adds a little bit of randomness in a simple way, which is great for small choices in the game.
• It can help make characters more unique, find cool treasure, and decide what happens in the game in a fun way.
• 1d5 rolls fit well in many RPGs, including D&D 5e, Pathfinder 2e, and World of Darkness.
• If you're running the game, make sure to explain the new 1d5 rules clearly, keep the game balanced, and ask your players what they think. If something isn't working, you can always tweak it.
• Examples showed how a quick 1d5 roll can change the story in big ways, making the game more fun and exciting.
In short, don't overlook the simple 1d5. It's a great tool for adding fun surprises to your tabletop RPG sessions. When used right, it can make your game even more enjoyable and memorable.
Related Questions
What does the D mean in Dungeons and Dragons?
In games like Dungeons & Dragons, when you see something like 1d4 or 2d6, it's telling you how to roll dice. The 'd' stands for dice. The number before the 'd' tells you how many dice to roll, and
the number after it tells you how many sides those dice have. For example, 1d4 means roll one four-sided die, and 2d6 means roll two six-sided dice.
What does 3d6 mean?
3d6 means you need to roll three six-sided dice and then add up what you get. So, if you roll a 2, a 4, and a 5, your total would be 11. Rolling several dice like this adds randomness, making the
game more unpredictable and fun.
What does 4d6 mean in D&D?
In Dungeons and Dragons, 4d6 means rolling four six-sided dice and adding them up. This method is often used when creating characters to decide their strengths and weaknesses. It's a way to bring in
chance and make each character unique.
How to roll d12 with d6?
To pretend you're rolling a 12-sided die using two 6-sided dice, do this:
1. Roll both dice.
2. The first die gives you a number between 1 and 6.
3. For the second die, if you roll 1-3, don't change the first number. If you roll 4-6, add 6 to the first number.
4. This way, you get numbers from 1 to 12!
It's a bit like a puzzle but it lets you play even if you don't have the exact dice you need.
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Self-consistent Estimates for Hexagonal Symmetry
Next: Appendix B: Crack-influence Decomposition Up: Appendix A: Bounds and Previous: Peselnick-Meister-Watt (PMW) Bounds for
The results obtained for self-consistent estimates can be written in many different ways (Berryman, 2005). We take the self-consistent estimate for bulk modulus to be
where In (35), K^* is determined by (34), depending also on G^*; G^* is determined by the self-consistent expression for the shear modulus to follow, also depending on K^*; and 35). The final result
for G^* = G^*[hex] in polycrystals having grains with hexagonal symmetry is These formulas can be successfully solved by iteration, starting for example by using values corresponding to upper or
lower bounds for the values of K^* and G^*. Some details of the derivation of these formulas can be found in Willis (1977, 1981) and Berryman (2005).
Next: Appendix B: Crack-influence Decomposition Up: Appendix A: Bounds and Previous: Peselnick-Meister-Watt (PMW) Bounds for Stanford Exploration Project
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Calculus I
Here you will find 150+ Calculus I videos that cover every main concept you will see in a typical Calculus I class. Scroll through the gallery below which is sorted in order of topics as they are
typically addressed throughout the semester.
Would you like a free open source Calculus I textbook? Try this no cost Calculus 1 textbook from Openstax.
Topics Covered
*The videos can be viewed in the player on this page, or the hyperlinks go directly to YouTube. Make sure to bookmark this site so you can find additional videos later. Please make sure to LIKE,
SUBSCRIBE, and LEAVE A COMMENT as it greatly helps our channel! Pro tip — Change the ‘playback speed’ to 1.5x in the video’s settings. You can get through more videos faster this way!
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The Last Word Information To The Numpy Package Deal For Scientific Computing In Python – گروه توسعه مدیریت نشا
Just remember that if you use the reshape technique, the array you need to produce must have the identical variety of elements as the original array. If you begin with an array with 12 components,
you’ll need to make sure that your new array also has a complete of 12 elements. We can access the weather in the array utilizing sq. brackets.
If the arrays match in size alongside an axis, then parts will be operated on element-by-element, similar to how the built-in Python perform zip() works. If you simply need to get started with some
examples, comply with together with this tutorial, and start constructing some muscle memory with NumPy, then Repl.it’s a nice choice for in-browser enhancing. You can join and fireplace up a Python
setting in minutes. For this NumPy tutorial, go with the current variations of NumPy and Matplotlib. Since you already know Python, you may be asking your self if you really have to be taught a
complete new paradigm to do knowledge science. Reading and writing CSV files could be done with traditional code.
This compactness is in part as a result of the looping within the vectorized version occurs in the background. There are a quantity of helpful features for sorting array parts. Some of the obtainable
sorting algorithms embrace quicksort, heapsort, mergesort, and timesort. The NumPy array – an n-dimensional information construction – is the central object of the NumPy package. Many readers will
likely be acquainted with the business scientific computing software MATLAB.
Hashes For Numpy-1264-cp39-cp39-manylinux_2_17_x86_64manylinux2014_x86_64whl
If you need to generate a plot on your values, it’s quite simple with Matplotlib. You also can save your array with the NumPy savetxt methodology. You can save a NumPy array as a plain text file like
a .csv or .txt file with np.savetxt.
Since most of your data science and numerical calculations will are probably to involve numbers, they seem like the most effective place to start. There are primarily 4 numerical types in NumPy code,
and each one can take a number of completely different sizes. Because of the particular calculation on this instance, it makes life simpler to have integers within the numbers array. But because the
house between 5 and 50 doesn’t divide evenly by 24, the resulting numbers would be floating-point numbers. You specify a dtype of int to drive the perform to round down and offer you complete
integers. You’ll see a more detailed discussion of information varieties later on.
reserve it as a .npz file utilizing np.savez. You also can save several numpy js arrays into a single file in compressed npz format with savez_compressed.
Cut Up Your Dataset With Scikit-learn’s Train_test_split()
As noted above, NumPy arrays behave a lot like other Python objects, for the sake of comfort. For occasion, they are often indexed like lists; arr[0] accesses the first element of a NumPy array. This
enables you to set or learn individual elements in an array.
In different words, keep only the rows the place the worth in column 1 ends with ‘thirteen’. To do that, we use record comprehension (a pure Python formalism) to generate the masks array to carry out
the indexing. The horizontal counterpart of np.vstack() is np.hstack(), which combines sub-arrays column-wise. For greater dimensional joins, the most common perform is np.concatenate().
NumPy arrays are stored at one steady place in reminiscence in distinction to lists, so processes can access and manipulate them very effectively. To make things more compact, we’ll outline a operate
to index certain rows from the first dataset based mostly on the earlier strategy. To perceive how electrical energy era has changed with time, we’ll need to concentrate to column 1 (date), column 2
(energy generated), and column 4 (description).
Recent Articles On Numpy
The example above shows how important it’s to know not solely what form your information is in but also which knowledge is in which axis. In NumPy arrays, axes are zero-indexed and identify which
dimension is which. For instance, a two-dimensional array has a vertical axis (axis 0) and a horizontal axis (axis 1). Lots of functions and commands in NumPy change their habits based mostly on
which axis you tell them to course of. Here, you use a numpy.ndarray methodology known as .reshape() to kind a 2 × 2 × three block of knowledge.
It is the basic bundle for scientific computing with Python. Besides its obvious scientific uses, Numpy may also be used as an environment friendly multi-dimensional container of generic information.
NumPy (Numerical Python) is an open supply Python library that’s used in almost each subject of science and engineering. It’s the universal commonplace for working with numerical data in Python, and
it’s at the core of the scientific
• NumPy offers a specialised array type that’s optimized to work with machine-native numerical types corresponding to integers or floats.
• traditional Python lists.
• We will learn to cope with nan values in additional element later in this course.
• For this NumPy tutorial, go together with the current variations of NumPy and Matplotlib.
• One is through a typed memoryview, a Cython assemble for quick and bounds-safe access to a NumPy array.
This flexibility has allowed the NumPy array dialect and NumPy ndarray class to turn out to be the de-facto language of multi-dimensional data interchange utilized in Python. Notice that the
matplotlib plotting commands accepted the NumPy arrays as inputs without a downside.
Tasks And Functions With Numpy
We will request that NumPy converts every little thing to a string format earlier than exporting. It is worth noting that it’s easy to keep away from wasting a NumPy array to a text file using the
np.savetxt() perform. Vectorized code may be much less intuitive to those who have no idea tips on how to read it. The skill of understanding how a lot vectorization to use in your code is one thing
that you’ll develop with expertise. The decision will at all times have to be made based on the nature of the appliance in query. In different words, NumPy has broadcast the scalar to a new array of
applicable dimensions to carry out the computation.
Arrays are very incessantly used in data science, where pace and assets are essential. In Python we’ve lists that serve the purpose of arrays, but they’re sluggish to course of. Just for fun, let’s
save our outcomes to a comma-delimited csv file.
You’ll use it in one of the later examples to explore how other https://www.globalcloudteam.com/ libraries make use of NumPy.
Numpy Common Functions (ufuncs)
The Cython library in Python allows you to write Python code and convert it to C for pace, using C sorts for variables. Those variables can include NumPy arrays, so any Cython code you write can work
instantly with NumPy arrays. Another set of features NumPy presents that permit you to do superior computation strategies with out Python loops are known as universal functions, or ufuncs for brief.
Ufuncs take in an array, perform some operation on every factor of the array, and either ship the outcomes to a different array or do the operation in-place. NumPy presents a broad catalog of
built-in routines for manipulating array knowledge. Built-ins for linear algebra, discrete Fourier transforms, and pseudorandom number generators prevent the trouble of having to roll those things
your self, too.
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Missing Blueprint
Professor EHENG is a well known architect, known for his works in modern architecture as well as art. For his latest work, he decided to create a one of a kind staircase for the computer engineering
building. On his way to grab a coffee during a visit to the building, he suddenly realizes that he lost his blueprint for this unique staircase. He can only remember the dimensions of the staircase,
and has access to an e-mail written by one of his students discussing a special property of the blueprint. Can you help Professor EHENG remember this masterpiece?
The staircase can be described as an array \(\mathbf{B}\) with integer elements representing the height of the steps.
\(\mathbf{B}\) has length \(\mathbf{N}\) and height \(\mathbf{H}\) as its dimensions. It is known to have elements between \(\mathbf{1}\) and \(\mathbf{H}\) inclusively. Therefore, the height
dimension \(\mathbf{H}\) both describes the height and the maximum element of \(\mathbf{B}\).
The special property \(\mathbf{SP}\) is given as an array consisting of \(\mathbf{N}\) numbers, and can be built from the blueprint \(\mathbf{B}\) by following the pattern below:
\(\mathbf{SP_i} = \sum_{j=0}^{i-1}(1 \text{ if } \mathbf{B_j} \leq \mathbf{B_i} \text{ else } 0)\)
Informally, it can be said that each index \(i\) of \(\mathbf{SP}\) counts the number of elements in the subarray \(\mathbf{B_0}, \mathbf{B_1}, .. ,\mathbf{B_{i-1}}\) smaller than or equal to \(\
It is guaranteed that with the given dimensions, the staircase will be unique. Thus, \(\mathbf{B}\) cannot be constructed if all \(\mathbf{B_i} < \mathbf{H}\).
The first line contains a single number \(\mathbf{T}\), the number of test cases.
Then for each test case, the following input is given:
For the first line of the current test case, the dimensions of the special property array \(\mathbf{SP}\) is given as two numbers \(\mathbf{N}\) and \(\mathbf{H}\), \(\mathbf{N}\) denoting the length
of both the array \(\mathbf{SP}\) and \(\mathbf{B}\); and \(\mathbf{H}\) denoting the height of the array \(\mathbf{B}\).
The next line of the current test case consists of \(\mathbf{N}\) integers, the elements of \(\mathbf{SP}\).
• \(1 \leq \mathbf{H} \leq \mathbf{SP_i} \leq \mathbf{N} \leq 10^5\)
• \(1 \leq \mathbf{B_i} \leq \mathbf{H}\)
It is guaranteed that the total number of elements in all the test cases won't exceed \(10^5\).
For each test case, print the elements of the staircase array \(\mathbf{B}\) that is unique to the conditions of the test case's \(\mathbf{H}\) and \(\mathbf{SP}\) values.
Input 1:
Output 1:
Input 2:
Output 2:
Input 1: The special property array \(\mathbf{SP}\) can be constructed from the staircase array \(\mathbf{B}\) like so:
\(\mathbf{SP_0}=0\) from \(\mathbf{B}=[\underline{1},-,-,-,-,-]\)
\(\mathbf{SP_1}=1\) from \(\mathbf{B}=[\mathbf{1},\underline{2},-,-,-,-]\)
\(\mathbf{SP_2}=2\) from \(\mathbf{B}=[\mathbf{1},\mathbf{2},\underline{2},-,-,-]\)
\(\mathbf{SP_3}=1\) from \(\mathbf{B}=[\mathbf{1},2,2,\underline{1},-,-]\)
\(\mathbf{SP_4}=2\) from \(\mathbf{B}=[\mathbf{1},2,2,\mathbf{1},\underline{1},-]\)
\(\mathbf{SP_5}=3\) from \(\mathbf{B}=[\mathbf{1},2,2,\mathbf{1},\mathbf{1},\underline{1}]\)
Above, the underline signifies the current index being processed, whereas the bold is used to describe the elements that are smaller than or equal to the element being processed.
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Multiplication in Vedic Mathematics
Tirthaji Maharaj has classified tricks to Multiply Numbers in Vedic Mathematics in Specific and General Methods. Specific Multiplication Methods can be applied when numbers satisfy certain conditions
like both numbers closer to 100 or numbers closer to each other or addition of last digits of both numbers is 10, etc. While General Multiplication Methods can be applied to any types of numbers.
Depending on Specific and General Techniques, Multiplication in Vedic Mathematics are classified in the form of Sutras as below. Lets see the Vedic Mathematics Multiplication techniques.
1. Nikhilam Sutra (Specific Technique)
2. Anurupyena Sutra (Specific Technique)
3. Urdhva Tiryak Sutra and Vinculum Process (General Technique)
4. Ekayunena Purvena (Specific Technique)
5. Antyaordaske’pi (Specific Technique)
Nikhilam Sutra:
This is most simplest trick to multiply numbers using Vedic Mathematics. I personally like this method a lot as multiplication can be done in mind as well.
Using Nikhilam Sutra it is simpler to multiply numbers like 98 & 95, 997 & 987, 102 & 112, 995 & 1008 i.e. the numbers which are closer to power of 10. This Sutra is a Specific method of
Multiplication in Vedic Mathematics which shows shortcuts to multiply numbers which are closer to power of 10 (10, 100, 1000, etc. )
This will generate 3 cases:
• Numbers closer and less than power of 10. Example: 97 * 96, 994 * 992, etc
• Numbers closer and greater than power of 10. Example: 102* 108, 1004 * 1012, etc
• Numbers closer and lying on both sides of power of 10. Example: 102* 95, 1004 * 991, etc
Let’s see few examples on this:
Click Here To Check Process, Types and Examples on Nikhilam Sutra
Anurupyena Sutra:
This is a sub-type of Nikhilam Sutra and another vedic math multiplication trick when numbers are not closer to power of 10 but are closer to themselves. It works on concept of Working Base and then
apply Nikhilam Sutra.
For Example – Multiplication of Numbers like 63 & 67.
1. Working Base(W.B.) concept: As the numbers (63 & 67) are closer to 60, we take working base as 60 (6*10) instead of 100, here factor is 6.
2. Apply concept of Nikhilam as discussed previously i.e. 63 is 3 greater than 60 and 67 is 7 greater than 60
3. Multiply 3 and 7 to get 21 in 2nd compartment. As base is *10, thus we need to have only 1 digit in 2nd compartment and hence need to carry forward 2 to 1st compartment.
4. Like Nikhilam Sutra, Cross Addition of 63 & 7 or 67 & 3 gives 70.
5. In Anurupyena Sutra, before adding carry forward directly to 1st compartment we need to multiply by the factor (6) and then add the carry forward. This Carry Forward (2) is added to 420
6. Final Answer: 4221
Same multiplication 63 and 67 can be solved by considering Working Base of 70 (10 * 7) as below.
Click Here To understand the Process and More Examples of Anurupyena Sutra
Urdhva Tiryak Sutra:
This is another great shortcut method of multiplication using Vedic Mathematics. Urdhva Tiryak is General method of multiplication in Vedic Maths which provides shortcut to multiply any types of
It can be applied very easily to multiply 3 digit numbers, multiply 4 digits numbers and even more than 4 digit numbers.
Lets see an example Multiplication of 3 digit numbers:
Formula Used: (ax^2+bx+c)(dx^2+ex+f) = adx^4 + (ae+bd)x^3 + (af+be+cd)x^2 + (bf+ce)x + cf
Process: (Left –> Right)
1. Vertical Multiplication of 1st digits of 2 numbers.
2. Crosswise Multiplication Addition of 1st 2 digits 2 numbers. (i.e. Crosswise Multiplication of 1st 2 digits and adding them.)
3. Crosswise Multiplication Addition of all 3 digits of both the numbers.
4. Crosswise Multiplication Addition of last 2 digits 2 numbers.
5. Vertical Multiplication of last digits 2 numbers.
6. For all steps, except 1st step, each compartment needs to have ONLY 1 digits. If not then carry forward initial digits to previous compartment (Check below examples to understand).
Click Here to Check Process, Multiplication of 4 & more examples using UrdhvaTiryak Sutra.
Vinculum Process of Multiplication:
Vinculum is a special method of Vedic Maths Multiplication which is used with Urdhva Tiryak whenever we have bigger digits like 6,7,8 and 9.
Vinculum is a process applied when numbers have bigger digits like 6,7,8,9. Carrying out operations like multiplication with bigger digits is time consuming and little tougher as compared to smaller
digits. Hence such digits 6,7,8 and 9 are converted to smaller digits like 4,3,2 and 1 using Vinculum Process.
I highly recommend to go through the concept of Vinculum Process.
Ekayunena Purvena Sutra:
This sutra is applicable whenever multiplier has only 9’s as digits.
Click Here => To understand the Process and more examples on Ekayunena Purvena Sutra.
This sutra has another great multiplication trick in Vedic Mathematics which can be applied when last digits of both numbers totals as 10.
1. Check if addition of last digits of the numbers is 10.
2. If yes, multiply them and write in 2nd compartment.
3. Apply Ekadhikena Purvena for the remaining digits i.e. Add 1 to the remaining digits.
4. Eg: In case of 34 x 36, Apply Ekadhikena Purvena on 3 so we have 4. Now multiply 3 and 4 and write in the 1st compartment.
Click Here => Multiplication Shortcuts of Antyaordasake’pi Sutra in Vedic Mathematics.
1. Celia says
I don’t usually comment but I gotta tell regards for the post
oon this grdeat one :D.
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Pcb factory: introduction of varistor characteristics and related physical quantities
Building 6, Zone 3, Yuekang Road,Bao'an District, Shenzhen, China
Pcb factory: introduction of varistor characteristics and related physical quantities
Pcb factory: introduction of varistor characteristics and related physical quantities
The circuit board manufacturer and circuit board designer explain the characteristics of varistor and related physical quantities
Characteristics of varistor
The voltage and current of varistors do not obey Ohm's law, so they have a special nonlinear relationship. When the voltage applied at both ends is lower than the nominal rated voltage, the
resistance of the varistor is close to infinity, and there is almost no current flowing through the varistor; When the voltage applied at both ends is slightly higher than the nominal rated voltage,
the varistor will break down and conduct quickly, and change from high resistance state to low resistance state, and the working current will also increase sharply; When the applied voltage at both
ends is lower than the nominal rated voltage, the varistor will return to the high resistance state; When the voltage applied at both ends exceeds the maximum limit voltage value, the varistor will
be completely broken down and damaged, and can no longer recover by itself.
Key parameters of varistor
Voltage sensitive voltage: voltage sensitive is breakdown voltage or threshold voltage. It is generally believed that when the temperature is 20 degrees and there is 1mA current flowing through the
varistor, the corresponding voltage value is applied to both ends of the varistor. The varistor voltage is the nonlinear starting voltage at the inflection point of the I-U curve of the varistor, and
it is the nonlinear voltage that determines the rated voltage of the varistor. In order to ensure that the circuit is within the normal working range and the varistor works normally, the voltage
value of the varistor must be greater than the maximum rated working voltage of the protected circuit.
Maximum limiting voltage: the maximum limiting voltage refers to the maximum voltage that can be borne by both ends of the varistor. The popular explanation is that when the surge voltage exceeds the
voltage sensitive voltage, the highest peak voltage measured at both ends of the varistor is also called the maximum clamping voltage. In order to ensure that the protected circuit will not be
damaged, when selecting the varistor, the maximum limiting voltage of the varistor must be less than the rated maximum working voltage of the circuit (if multi-level protection is adopted, it can be
considered separately).
Through current capacity: the through current capacity, also known as through current, refers to the maximum pulse (peak) current value allowed to pass through the varistor under specified conditions
(applying standard impulse current at specified time interval and times).
Usually, the flow rate given by the product is the maximum current that the product can withstand when conducting pulse test according to the waveform, impact times and gap time given in the product
standard. The number of shocks that the product can withstand is a function of the waveform, amplitude and gap time. The number of shocks can be doubled when the current waveform amplitude decreases
by 50%. Therefore, in practical applications, the surge current absorbed by the varistor should be greater than the maximum flow of the product.
The surge current amplitude absorbed by the varistor shall be less than the maximum flow of the product given in the manual. However, from the perspective of protection effect, it is required to
select a larger flow rate. In many cases, the actual flow is difficult to calculate accurately, so 2-20kA products are selected. If the flow capacity of the product at hand cannot meet the use
requirements, several individual varistors can be used in parallel. The voltage of the varistor after parallel connection will not change, and the flow capacity is the sum of the values of each
individual varistor. It is required that the voltage current characteristics of the parallel varistors should be the same as much as possible, otherwise it is easy to cause uneven shunt and damage
the varistors.
Voltage ratio: the voltage ratio refers to the ratio between the voltage value generated when the current of the varistor is 1mA and the voltage value generated when the current of the varistor is
Residual voltage ratio: when the current flowing through the varistor is a certain value, the voltage generated at its two ends is called the residual voltage. Residual voltage ratio is the ratio of
residual voltage to nominal voltage.
Leakage current: also known as waiting current, leakage current refers to the current flowing through the varistor at the specified temperature and maximum DC voltage. The smaller the leakage
current, the better. For the leakage current, it should be emphasized that it must be stable, and it is not allowed to increase automatically during operation. Once the leakage current automatically
increases, it should be eliminated immediately, because the instability of the leakage current is the direct reason for accelerating the aging of the lightning protector and the explosion of the
lightning protector. Therefore, when selecting the leakage current parameter, the smaller the better cannot be blindly pursued. As long as the leakage current value is within the allowable range of
the power grid, the lightning arrester with a relatively large leakage current value is selected, which is more stable.
The circuit board manufacturer and circuit board designer explain the characteristics of varistor and related physical quantities
Just upload Gerber files, BOM files and design files, and the KINGFORD team will provide a complete quotation within 24h.
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We were discussing “Rankine cycle” in our previous post, where we have seen the various components of Rankine cycle and its basic operations also. We have also discussed the “Rankine cycle with
Today we will see here the basic concept of regeneration in Rankine cycle process with the help of this post. First we will see here the few basic concepts and terms and after that we will see the
very important concept i.e. regeneration in Rankine cycle in this post.
First we will draw here a simple Rankine cycle and after that we will try to find here the method to increase the efficiency of a Rankine cycle. Let us see here the following figure, where a simple
Rankine cycle is displayed.
As we can see here that heat energy will be added during the process 4 to 1 and heat energy will be rejected during the process 2 to 3. So let us see the efficiency of simple rankine cycle.
Efficiency of a simple Rankine cycle will be calculated by following formula.
η= 1-[Temperature of heat rejection / Temperature of heat addition]
Temperature of heat rejection i.e. T[2] will be constant because either heat energy will be rejected in to atmosphere or heat energy will be rejected at temperature lower than the atmospheric
temperature but temperature of heat rejection will be constant throught out the heat rejection process.
We can also think to increase the efficiency of the rankine cycle by decreasing the temperature of heat rejection but we must note it here that temperature of heat rejection i.e. T[2] could not be
decreased below a specific level.
Let us see the temperature of heat addition, as we can see that heat energy will be added during the process 4-1 and therefore only partial heat energy will be added at constant temperature and rest
heat energy will be added at varying temperature. So we will have one term known as mean temperature of heat addition i.e. T[m1].
Mean temperature of heat addition i.e. T[m1] will be termed as a constant temperature, located between T[1] and T[4], at which if same quantity of heat energy will be added then we will have same
change in entropy as we were having changes in entropy during the process 4-1.
Therefore , Efficiency of a simple rankine cycle will be calculated by following formula.
η= 1-[T[2] / T[m1]]
Now in order to increase the efficiency of the Rankine cycle , we will have to increase the value of mean temperature of heat addition i.e. T[m1]. We can increase the value of of mean temperature of
heat addition i.e. T[m] by increasing the maximum temperature of the Rankine cycle i.e. T[1].
We must note it here that temperature of heat addition could be increased up to a limit only as it will be restricted by various practical parameters such as material properties of turbine blades.
Turbine blade material will not work satisfactory if we increase the maximum temperature of rankine cycle i.e. T1 above a certain level and that level of temperature will be termed as maximum
allowable temperature and we could not increase the temperature T[1] of the Rankine cycle beyond this maximum allowable temperature.
We can also increase the boiler pressure in order to increase the efficiency of the Rankine cycle because range of temperature for heat energy addition will be increased.
Concept of regeneration in rankine cycle
So in order to increase the efficiency of the rankine cycle, we will require to increase the mean temperature of heat addition and it could be increased by increasing the amount of heat energy
addition at higher temperature.
As we can see the above Rankine cycle, considerable quantity of heat energy will be added to the working fluid during its liquid phase or during sensible heating or during subcooled region. Only less
part of heat energy addition will be added at maximum temperature i.e. at T[1].
If we want to increase the efficiency of the cycle, we must be aware that all heat energy must be supplied at maximum temperature of the cycle i.e. at temperature T[1] in this cycle and hence we will
have to think the method by which we can permit the feed water to enter in to the boiler at state 5 so that all heat energy supplied by boiler to the working fluid will be carried out at maximum
temperature of the rankine cycle i.e. T[1] in this case.
So if we can use the heat energy of the high temperature steam, which is flowing through the turbine during the expansion process 1-2, to heat the feed water from 4 to 5 then all heat energy supplied
by the boiler will be done at maximum temperature of the cycle.
Hence mean temperature of heat addition will be T[1] itself because all heat energy supplied by the boiler to the working fluid will be completed at maximum temperature T[1] during process 5 to 1.
Therefore, feed water leaving the feed pump will be circulated around the casing of the turbine in opposite direction of the direction of steam flow during expansion process 1-2 in the turbine.
The basic concept of regenration in rankine cycle is that we will have to make an arrangement to heat the feed water leaving the feed pump by the high temperature steam flowing through the turbine
during the process 1-2.
So if feed water will reach to dry saturated liquid state, by receiving heat energy from the hot steam flowing through turbine, before entering to the boiler then all heat energy supplied by the
boiler to the working fluid will be done at maximum temperature of the cycle and hence mean temperature of heat addition will be T[1] and therefore in that situation we will have higher efficiency of
the rankine cycle.
If feed water could not reach to dry saturated liquid state but also it is reaching at a point between 4 and 5, by receiving heat energy from the hot steam flowing through turbine, before entering to
the boiler then all heat energy supplied by the boiler to the working fluid will not be done at maximum temperature of the cycle.
But in that situation also we will have higher efficiency of the rankine cycle because we will have higher mean temperature of heat addition as compared to the simple rankine cycle without
regeneration concept.
So this is the concept of regeneration in Rankine cycle, we will see another topic i.e. rankine cycle with regeneration
Do you have any suggestions? Please write in comment box.
Engineering thermodynamics by P. K. Nag
Engineering thermodynamics by Prof S. K. Som
Image courtesy: Google
Also read
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Possible Topics for 2020 O'level 4047 Additional Math Paper 2
Here are the possible topics to be tested for AMath paper 2 on 26th October
1. Sketch of quadratic functions and Discriminant
2. Surds --> can be real world context or finding unknowns like a, b --> rationalisation of denominator
3. Binomial theorem --> specific term formula, expansion formula, nCr formula
4. Sketching of power functions, parabola, log graphs, exponential graphs, question involving find another straight line equation to be drawn
5. Logarithm equations and simplifying involving log laws
6. Coordinate Geometry --> midpoint formula, m1m2 =-1, length formula, area formula, find equation of line and simultaneous equation, gradient Involving angle.
7. Linear Law - plotting of graph, and/or non-graph question finding m and c, solving unknown.
8. Trigonometry graphs, proving involving double angle, R-formula, Trigo in real world context
9. Differentiation: tangent and normal, rate of change, chain rule or product rule of x functions, ln, functions, trigo functions
10. Integration: reverse of differentiation, finding equation of curve, evaluation of definite integral, x function,1/f(x) functions
11. Simultaneous eqn -- can be problem sum
ALL THE BEST!
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teaching replication
Fear of rejection, part I
To replicate a study, you need information. Probably information that is not fully disclosed in a 6-12,000 word journal article. Except for a recent trend, information such as data and analytical
procedure are not going to be available publicly. This means you should, or must in case the data are not retrievable from some other source, contact the original author. Be prepared for rejection.
One study demonstrated that among the top sociology journals, less than 30% of replication materials were available (even though as many as 75% claimed otherwise). Political science was only
marginally better at around 50% as of 2015. Professors are likely to ignore emails asking for their data and code. One group of sociology students contacted 53 different authors asking for
replication materials and only 15 provided them (28%). Ten never responded to the requests at all, despite several follow up emails. So don’t take it personally, social scientists are not known for
their forthcomingness in this area.
Verification is not affirmation
Imagine being a student who tries to verify the results of a prolific, senior scholar and cannot. If it were me, I would be anxious that I made a mistake. But the only real mistake would be to assume
my lack of verification is a refutation of my own skills. Of course, it’s good to double check everything. Have a colleague look at your work if you are unsure, a teacher or supervisor if you are a
student. Un-verifiable results are common, no need for self-doubt. Things like reverse coding biological sex so that women appear less supportive of welfare state policies or accidentally analyzing
values of 88 (a missing code) as a relevant value of coital frequency leading to a surprising rate for older persons are actually a normal part of social science.
When replicating a study just assume there will be at least one mistake. Like a treasure hunt.
Verification comes down to the availability of the materials. If the data and code are not fully available, it really is a treasure hunt because you will be unsure what you are going to find or
learn. On the other hand, if the data and code are available and in good order, then it is more like cooking than hunting. This often comes down to the difference between teaching replication – the
recipe approach, where students should come to the same results every time when following the exact same steps, and replication as a form of social research – the treasure hunt approach, where
researchers (i.e., students) may not have a coherent recipe from the original ‘chef’. But make no mistake(!) even fully transparent studies often come with mistakes in the code or data.
Fear of mistakes
If I am not making mistakes, I am not doing research. You will make mistakes and there is nothing to fear. There are all kinds of reasons that replication results will diverge, not all of them are
mistakes. Recently a well-known and well-respected sociologist retracted his own paper after someone trying to replicate the study identified coding errors. One journal started checking that data and
code produced results in accepted papers, and almost none were verifiable on the first attempt. In a crowdsourced replication, mostly PhD students, postdocs and a few professors came to an exact
verification of the original study only 82% of the time, despite having the original code!
Fear of the unknown
Designing statistical models using a software is like learning a new language. Student replications often involve methods unfamiliar to the them. This is a great didactic tool – learning by doing.
There is nothing to fear here. Professors’ original studies often involve methods that they are not experts in. One extremely famous scholar and his colleague ran a regression with an interaction
term in it and botched the interpretation of the effects, the results were basically the opposite of what they reported.
Science is a process of exploring the unknown. Replications use what is known as a tool for finding what is unknown.
Fear of rejection, Part II
Students may be interested in publishing their replications, they should be, because how else will others put their knowledge into practical use? Get prepared again for rejection. Journals and
reviewers across the social sciences are not very excited about replications. A pair of researchers studied the instructions and aims of 1,151 psychology journals in 2016 and discovered that only 3%
explicitly accepted replications. One sociologist pointed out not so long ago that replication is just not the norm in sociology, and another one recently came to the same conclusion. The good news
is that we don’t need journals anymore to make useful science, at least in theory. Students can immediately publish their results as preprints and share data and code in a public repository. If a
student elects to use Open Science Framework preprint servers, their work will be immediately found in scholarly search engines.
Fear of ego
Scientists tend to overestimate the impact of a negative replication on their reputations. Ego-alert. Assume a scientist worried about a replication is a professor. This is a person who is most like
tenured, certainly the highly cited professors are. This is also a person who “professes” knowledge on a topic, meaning that they should be an expert and engage in teaching students, policymakers,
the public and really anyone interested about this topic. If any of this professor’s results were shown to be unreliable or false, this would be a critical piece of information if that professor’s
goal was to actually profess knowledge on that topic. Unfortunately, professors regularly suffer from some kind of ‘rock-star syndrome’ or ego-mania where they are doing science as a means to get
recognition and fame. This leads them to react aggressively against anything that contradicts them. This is very bad for science. If a student replicator can help deplete a runaway professor ego
through replication, then that student is doing a great service to science.
Fear of not addressing fear
In a typical primary or secondary school chemistry class, students repeat the basic experiments of chemical reactions that have been done for hundreds of years. These students are learning through
replication. They are gaining knowledge in a way that cannot be simply taught in a lecture or by reading a book. They are also affirming the act of science, thus developing a faith that science
works. In social science especially, we face a reliability crisis if not a public image crisis. Students should be reassured that there is a repetitive and reliable nature to doing social science,
whether they will continue as a social scientist or (in the most likely case) not. Part of this reliability can be a lack of reliability. Science is simply a process of trying to understand the
unknown, and even quantify this unknown. I fear that without more student replications, we are diminishing the value of social science and contributing to the perception that social science is
Good social science should be reliably able to identify unreliability, and this is best taught through conducting replications.
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Activity report
RNSR: 202023616M
In partnership with:
Team name:
Stochastic Approaches for Complex Flows and Environment
Applied Mathematics, Computation and Simulation
Stochastic approaches
Creation of the Project-Team: 2020 November 01
• A6.1. Methods in mathematical modeling
• A6.1.1. Continuous Modeling (PDE, ODE)
• A6.1.2. Stochastic Modeling
• A6.2. Scientific computing, Numerical Analysis & Optimization
• A6.2.1. Numerical analysis of PDE and ODE
• A6.2.2. Numerical probability
• A6.2.3. Probabilistic methods
• A6.2.4. Statistical methods
• A6.2.7. High performance computing
• A6.3. Computation-data interaction
• A6.3.5. Uncertainty Quantification
• A6.4.1. Deterministic control
• A6.5. Mathematical modeling for physical sciences
• A6.5.2. Fluid mechanics
• B1.1.8. Mathematical biology
• B3.2. Climate and meteorology
• B3.3.2. Water: sea & ocean, lake & river
• B3.3.4. Atmosphere
• B4.3.2. Hydro-energy
• B4.3.3. Wind energy
• B9.5.2. Mathematics
• B9.5.3. Physics
1 Team members, visitors, external collaborators
Research Scientists
• Mireille Bossy [Team leader, Inria, Senior Researcher, HDR]
• Jeremie Bec [CNRS, Senior Researcher, HDR]
• Laetitia Giraldi [Inria, Researcher]
• Christophe Henry [Inria, ISFP, (SRP until 10/2021)]
Post-Doctoral Fellows
• Aurore Dupre [Inria, January 2021]
• Kerlyns Martínez Rodríguez [Univ Côte d'Azur, until February 2021]
PhD Students
• Sofia Allende Contador [Univ Côte d'Azur, until March 2021]
• Luca Berti [Univ de Strasbourg]
• Lorenzo Campana [Univ Côte d'Azur]
• Zakarya El Khiyati [Univ Côte d'Azur, from Oct 2021]
• Robin Vallée [CEMEF, Mines ParisTech, until Marsh 2021]
Interns and Apprentices
• Raphael Chesneaux [Inria, from May to Aug 2021]
• Zakarya El Khiyati [Inria, from Apr to Oct 2021]
• Thomas Ponthieu [Inria, from Apr to Sep 2021]
External Collaborators
• Areski Cousin [Univ de Strasbourg, IRMA]
• Nadia Maïzi [École Nationale Supérieure des Mines de Paris ]
• Simon Thalabard [Univ Côte d'Azur]
2 Overall objectives
Turbulence modeling and particle dynamics are at play in numerous situations in which inertial particles are transported by turbulent flows. These particles can interact with each other, form
aggregates which can fragment later on, and deposit on filters or solid walls. In turn, this deposition phenomenon includes many aspects, from the formation of monolayer deposits to heavy fouling
that can clog flow passage sections. Taking into account the potentially complex morphology of these particles then requires to develop new approaches to predict the resulting statistical quantities
(turbulent dispersion, formation of aggregates, nature of formed deposits, etc.).
The variety of situations (deposition, resuspension, turbulent mixing, droplet/matter agglomeration, thermal effect) involves specific models that need to be improved. Yet, one of the key
difficulties lies in the fact that the relevant phenomena are highly multi-scale in space and time (from chemical reactions acting at the microscopic level to fluid motion at macroscopic scales), and
that consistent and coherent models need to be developed together. This raises many challenges related both to physical sciences (i.e. fluid dynamics, chemistry or material sciences) and to numerical
Through the unique synergy between team members from various disciplines, Calisto is developing Stochastic Approaches for complex Flows and Environment to address the following challenges:
• produce original answers (methodological and numerical) for challenging environmental simulation models, with applications to renewable energy, filtration/deposition technology in industry
(cooling of thermal or nuclear power plants) and filtration/deposition, dispersion of materials or active agents (such as biological organisms, micro-robots);
• design new mathematical tools to analyze the fundamental physics of turbulence;
• develop numerical methods to analyze the displacement of micro-swimmers into a range of fluids such as water, non-Newtonian bodily fluids, etc.;
• optimize and control the displacement of artificial micro-swimmers;
• develop stochastic modeling approaches and approximation methods, in the rich context of particle-particle and fluid-particle interactions in complex flows;
• contribute to the field of numerical probability, with new simulation methods for complex stochastic differential equations (SDEs) arising from multi-scale Lagrangian modeling for the dynamics of
material/fluid particle dynamics with interaction.
3 Research program
Calisto is structuring its research according to five interacting axes.
• Axis A
Complex flows: from fundamental science to applied models.
• Axis B
Particles and flows near boundaries: specific Lagrangian approaches for large-scale simulations.
• Axis C
Active agents in a fluid flow.
• Axis D
Mathematical and numerical analysis of stochastic systems.
• Axis E
Variability and uncertainty in flows and environment.
3.1 Axis A Complex flows: from fundamental science to applied models
This axis aims at promoting significant advances in the understanding and modeling of realistic dispersed, multiphase turbulent flows. In situations where basic mechanisms are still not fully
apprehended, the proposed research aims at bringing out the underlying physics by identifying novel effects and quantifying their impacts. These results will then be used to foster new macroscopic
models that are expected to be computationally sufficiently undemanding. These models should also be adaptable to open the way to systematic studies of turbulent suspensions as a function of
settings, parameters, system geometry. Such aspects are essential in exploratory researches aimed at optimizing combustion processes, heat transfers, phase changes, or the design of energy-efficient
hydraulic or aerodynamic processes.
Accurate modeling of the location, attributes, and effects of particles transported by turbulent flows is key to optimize the design and performance of several processes in industry, in particular in
power production. Yet, current macroscopic approaches often oversimplify physical phenomena related to small-scale physics and fail to capture various effects, such as heterogeneous distributions of
sizes and shapes, particle deformation, agglomeration, as well as their interactions with boundaries. Improving models remains a huge challenge that requires monitoring spatial and temporal
correlations through particle relative dynamics.
Our overall objective here is to design, validate and apply new efficient modeling and simulation tools for fluid-particle systems that account for relative particle motions, two-particle
interactions and complex flow geometries. Our methodology consists in simultaneously (i) building up a comprehensive microscopic description, (ii) developing efficient macroscopic models, and (iii)
applying these two approaches to study practical situations to compare and validate them.
Continuous exchanges between these two viewpoints make it possible to quickly identify pitfalls in models. Furthermore, fine-scale descriptions will progressively provide suggestions for
This research axis is currently investigating the following distinct topics
• Models for polydisperse, complex-shaped, deformable particles;
• Particle interactions and size evolution;
• Transfers between the dispersed phase and its environment.
3.2 Axis B - Particles and flows near boundaries: specific Lagrangian approaches for large scale simulations
This research axis aims at developing Lagrangian macroscopic models for single phase and particle-laden turbulent flow simulations. This activity addresses important situations of environmental
flows, such as atmospheric boundary layer (ABL), and pollutants, pollen, micro-plastic dispersion and resupension in the atmosphere or river/marine systems. These are situations where boundaries
bring additional complexity, in terms of turbulent description, and in terms of the interaction between wall and particles.
In the hierarchy of turbulent models, the Lagrangian stochastic approach (or probability density function (PDF) approach) is distinguished by several important features, mainly: (i) it is a
stochastic method that resolves the probability density function of some physical relevant variables, needed to provide sufficient statistical information. For example, in the case of single-phase
turbulent flows, this method provides the velocity distribution, compatible with the imposed momentum turbulent closure of the considered model. In particular, it delivers the whole tensor of
correlations between the flow velocity components in adequacy with the given closure; (ii) thanks to its Lagrangian formulation, this approach allows to develop a fully coherent model of a turbulent
flow, of particles embedded in it, as well as their interactions.
For two-phase turbulent flows, the combination of fluid-particle approaches with discrete particle approaches –called here Lagrange-Lagrange approaches– appears to be particularly interesting for
near boundary flows where interactions with surface boundaries are coming into the problem. Until now, this Lagrangian-Lagrangian modelling approach has never really been explored. The Calisto
in-house SDM software, as a mature fluid-particle Lagrangian simulation code, offers an exciting opportunity to investigate this direction.
This research axis is currently investigating the following distinct topics.
3.2.1 Stand-alone Lagrangian simulations in atmospheric boundary layer (ABL)
The turbulent nature of the atmospheric boundary layer (ABL) contributes to the uncertainty of the wind energy estimation. This has to be taken into account in the modeling approach when assessing
the wind power production. The purpose of the Stochastic Downscaling Model (SDM) is to compute the wind at a refined scale in the ABL, from a coarse wind computation obtained with a mesoscale
meteorological solver. The main features of SDM reside in the choice of a fully Lagrangian viewpoint for the turbulent flow modeling. This is allowed by stochastic Lagrangian modeling approaches that
adopt the viewpoint of a fluid-particle dynamics in a flow. Such methods are computationally inexpensive when one needs to refine the spatial scale. This is a main advantage of the SDM approach, as
particles methods are free of numerical constraints (such as the Courant Friedrichs Lewy condition that imposes a limit to the size of the time step for the convergence of many explicit time-marching
numerical methods).
A particular attention is now focused on improving stand-alone Lagrangian numerical models in the ABL (such as additional buoyancy model, canopy models). Furthermore, the coupling of fluid particle
modeling with phase particle models is of crucial interest for some of our applications.
3.2.2 Advanced stochastic models for discrete particle dispersion and resuspension
As a particle nears a surface, deposition can occur depending on the interactions between the two objects. Deposits formed on a surface can then be resuspended, i.e. detached from the surface and
brought back in the bulk of the fluid. Resuspension results from a subtle coupling between forces acting to move a particle (including hydrodynamic forces) and forces preventing its motion (such as
adhesive forces, gravity). In the last decades, significant progresses have been achieved in the understanding and modeling of these processes within the multiphase flow community. Despite these
recent progresses, particle resuspension is still often studied in a specific context and cross-sectoral or cross-disciplinary exchange are scarce. Indeed, resuspension depends on a number of
processes making it very difficult to come up with a general formulation that takes all these processes into account.
Our goal here is to improve deposition law and resuspension law for more complex deposits in turbulent flows, especially towards multilayered deposits. For that purpose, we are improving existing
Lagrangian stochastic models while resorting to meta-modeling to develop tailored resuspension law from experimental measurements and fine-scale numerical simulations. We are targeting practical
applications such as pollutants in the atmosphere and plastic in marine systems.
3.2.3 Coherent descriptions for fluid and particle phases
Various particles are present in the ABL, such as pollutant, fog or pollen. This surface layer is characterized by various complex terrains (as urban cities or forests), forming the so-called canopy.
This canopy strongly affects the near-wall turbulent motion as well as the radiative and thermal transfers.
Simulations of two-phase flows requires to couple solvers for the fluid and particle phases. Numerical Weather Prediction (NWP) software usually rely on a Eulerian solver to solve Navier-Stokes
equations. Solid particles are often treated using a Lagrangian point of view, i.e. their motion is explicitly tracked by solving Newton's equation of motion, the key difficulty being then to couple
these intrinsically different approaches together. In line with the models and numerical methods developed in Sections 3.2.1 and 3.2.2, as an alternative to Eulerian-Lagragian approaches, Calisto is
developing a new Lagrange-Lagrange formulation that remains tractable to perform simulations for two-phase turbulent flows. We are particularly interested in Lagrange-Lagrange models for interactions
with surfaces, as turbulence and collisions with surfaces can significantly affect the concentration of particles in the near-wall region.
3.2.4 Active particles near boundary
Surface effects can lead to the trapping of micro-swimmers near boundaries, as the presence of a boundary breaks both the symmetry of the fluid (leading to strong anisotropy) and the symmetry of the
fluid-swimmer system. The better understanding of fluid-particle interactions near boundaries are expected here to help in the design of new control actuation for driving artificial swimmers in
confined environments (developed in Axis C).
3.3 Axis C - Active agents in a fluid flow
Active agents are entities immersed into a fluid, capable of converting stored or ambient free energy (for instance through deformation) into systematic movement. Active agents, also called swimmers,
can interact with each other as well as with the surrounding medium.
This research axis is devoted to new mathematical modeling approaches to simulate the displacement of swimmers, to get results on control and optimal control associated with them, to study the
presence of an additional stochastic effect for driving a swarm of such micro-swimmers.
Modeling approach
The equations of motion of the swimmer derive from its hydrodynamical interactions with the fluid through Newton laws. At a high level of description, this can be described by coupling the
Navier-Stokes equations with the hyper-elastic equations describing the swimmer's deformation (in the case of elastic body). In the case of artificial magnetic swimmers, additional contribution
representing the action of an external magnetic field on the swimmer needs to be added in the equations of motion. Solving the resulting system of PDEs is a challenging task, since it combines a set
of equations deemed to be numerically difficult to solve even when they are decoupled. To overcome these difficulties, Calisto considers various types of models, ranging from simpler but rough models
to more realistic but complex models.
Control and optimal control for swimmers displacement
Calisto investigates the controllability issues and the optimal control problems related in particular to two situations: the displacement of (i) real self-propelled swimmer by assuming that the
control is the deformation of its body (ii) artificial bio-inspired swimmers that are able to swim using an external magnetic field.
Another line of research concerns optimal path planning in turbulent flow. As a microswimmer swims towards a target in a dynamically evolving turbulent fluid, it is buffeted by the flow or it gets
trapped in whirlpools. The general question we want to address is whether such a microswimmer can develop an optimal strategy that reduces the average time or energy it needs to reach a target at a
fixed distance.
Stochastic effect on artificial swimmers
Calisto investigates also the effect of the presence of noise in the response of a micro-robot (to the external magnetic field for instance) by developing new model and related numerical simulation
of such systems.
3.4 Axis D - Mathematics and numerical analysis of stochastic systems
This research axis is devoted to fundamental aspects of our models or objects though their mathematical analysis.
Mathematics for fundamental aspects of turbulence and turbulence transport
This research line has the scope of providing a unified description of turbulent flows in the limit of large Reynolds numbers and thus will be applicable to a large range of physical applications. It
is conjectured since Kolmogorov and Onsager that the flow develops a sufficiently singular structure to provide a finite dissipation of kinetic energy when the viscosity vanishes. This dissipative
anomaly gives a consistent framework to select physically acceptable solutions of the limiting inviscid dynamics. However, recent mathematical constructions of weak dissipative solutions face the
problem of non-uniqueness, raising new questions on the relevance to turbulence and on the notion of physical admissibility.
On the one hand, the conservation of kinetic energy is actually not the only symmetry that is broken by turbulence. Various experimental and numerical measurements show significant deviations from
simple scaling, time-irreversible fluctuations along fluid elements trajectories, and possibly other broken inviscid symmetries, such as circulation. Still, these anomalies may have a universal
nature and, as such, provide new constraints for the design of physically admissible solutions. On the other hand, non-uniqueness could be an intrinsic feature of turbulence. Singular solutions to
non-linear problems have an explosive sensitivity leading to spontaneously stochastic behaviors, thus questioning the pertinence of uniqueness and providing a framework to interpret solutions at a
probabilistic level. To address such issues and provide unified appreciation, we simultaneously develop three strongly interrelated viewpoints: a) numerical approach, exploiting relevant and
efficient fully-resolved simulations; b) new theoretical approaches based on the statistical physics of turbulent flow; c) mathematical construction of "very weak" flows, such as measure-valued
solutions to the Euler equations.
Interacting Stochastic Systems, and Mean Field Interactions
A birds flock, a school of fish, a group of fireflies, a crowd in the street, or even the neurons of our brain, are all examples of interacting entities that can suddenly start to behave collectively
in a more complex and richer way than their constitutive elements. The mathematical modeling of such phenomena started mainly motivated by biological systems, but lately has gained a lot of attention
due to new applications in economics, finance, robotics and even opinion formation in human behavior. Calisto considers examples of particle systems in interaction, possibly under mean field
interaction, with the overall goal of analyzing the effect of stochasticity in such system. In particular, we aim to detect and analyze conditions for the emergence of collective behaviors such as
collective motions, synchronization and organization with or without the notion of leaders.
Another important example of complex interacting system is given by collisioning particle system under Langevin dynamics. If the case of collisioning systems in the context of gas dynamics –where
particles experiment free path between two collision events– and in the context of overdamped Brownian dynamics have been largely studied, until now, situation of a finite number of particles
collisioning under a Langevin dynamics is poorly addressed. This last case, describing particles in turbulent flow, is of great interest for Calisto from both numerical and theoretical view points.
3.5 Axis E - Variability and uncertainty in flows and environment
Variability in wind/hydro simulation at small scale: application to wind/hydro energy
The turbulent nature of the atmospheric boundary layer (ABL) contributes to the uncertainty of the wind energy estimation. This has to be taken into account in the modeling approach when assessing
the wind power production. The stochastic nature of the SDM approach developed in Axis B offers some rich perspectives to asses variability and uncertainty quantification issues in the particular
context of environmental flows and power extraction evaluation. In particular, as a PDF method, SDM delivers a probability distribution field of the computed entities. Merging such numerical strategy
with Sensitivity Analysis (SA)/Uncertainty Quantification (UQ) are potentially fruitful in terms of computational efficiency.
Metamodeling and uncertainty
While building and using computational fluid dynamics (CFD) simulation models, sensitivity analysis and uncertainty quantification methods allow to study how the uncertainty in the output of a model
can be apportioned to different sources of uncertainty in the model input. UQ approaches allow to model verification and factor prioritization. It is a precious aid in the validation of a computer
code, guidance research efforts, and in terms of system design safety in dedicated application. As CFD code users, we aim at applying UQ tools in our dedicated modeling and workflow simulation. As
Stochastic Lagrangian CFD developers, we aim at developing dedicated SA and UQ tools as Stochastic solvers have the ability to support cross Monte Carlo strategy at the basis of SA methodology.
Another goal is to address some control and optimization problems associated with the displacement of swimmers through metamodeling, such as Gaussian process regression model, proved to be efficient
for solving optimization of PDEs systems in other contexts.
Anomalies modeling through machine-learned dataset of meteorological observations and forecasts
Stochastic modeling approaches are known to be able to describe the intrinsic variability of a phenomenon, preserving the spatial coherence of variability and interacting with the dynamics of the
physical processes involved. The machine learning (or meta model) approach is recognized for these prediction capabilities. It is nowadays everywhere in the forecast data delivered, using past events
to de-bias/select the future, when the physical dynamic model becomes too heavy to handle. We aim at intersecting the two approaches to develop methodologies for selecting/enriching future scenarios,
starting from the observation that we can not calibrate the model of variability to be associated with a future forecast (distribution of extremes accounts for climatic changes), the same way one
calibrates the variability model to be associated with observables.
4 Application domains
Environmental challenges: predictive tools for particle transport and dispersion
Particles are omnipresent in the environment:
• formation of clouds and rain results from the coalescence of tiny droplets in suspension in the atmosphere;
• fog corresponds to the presence of droplets in the vicinity of the Earth's surface, reducing the visibility to below 1 km 25;
• pollution corresponds to the presence of particulate matter in the air. Due to their impact on human health 33, the dispersion of fine particulate matter is of primary concern: PM2.5 and PM10
(particles smaller than 2.5 or 10 $\mu$ m) and Ultra Fine Particles (UFP) are particularly harmful for human respiratory systems while pollen can trigger severe allergies;
• the dispersion of radioactive particles following their release in nuclear incidents has drawn a great deal of attention to deepen our understanding and ability to model these phenomena 39;
• the dispersion/deposition of ash and soots and their consequences for the environment and health have been highlighted by recent events in France and abroad;
• plastic contamination in oceans impacts marine habitats and human health 28;
• suspension of real micro-swimmers 20 such as sperm cell, bacteria, and in environmental issues with animal flocks attracted intrinsic biological interest 30;
• accretion of dusts is responsible for the formation of planetesimals in astrophysics 29.
These selected examples show that the presence of particles affects a wide range of situations and has implications in public, industrial and academic sectors.
Each of these situations (deposition, resuspension, turbulent mixing, droplet/matter agglomeration, thermal effect) involves specific models that need to be improved. Yet, one of the key difficulties
lies in the fact that the relevant phenomena are highly multi-scale in space and time (from chemical reactions acting at the microscopic level to fluid motion at macroscopic scales), and that
consistent and coherent models need to be developed together. This raises many issues related both to physical sciences (i.e. fluid dynamics, chemistry or material sciences) and to numerical
Next generation of predictive models for complex flows
Many processes in power production involve circulating fluids that contain inclusions, such as bubbles, droplets, debris, sediments, dust, powders, micro-swimmers or other kinds of materials. These
particles can either be inherent components of the process, for instance liquid drops in sprays and soot formed by incomplete combustion, or external foul impurities, such as debris filtered at water
intakes or sediments that can obstruct pipes. Active particles, seen as artificial micro-swimmers, have attracted particular attention for medical applications since they can be used as vehicles for
the transport of therapeutics or as tools for limited invasive surgery. In these cases, optimization and control requires monitoring the evolution of their characteristics, their trajectories (with/
without driving), and their effects on the fluid with a sufficiently high level of accuracy. These are very challenging tasks given a numerical complexity of the numerical models.
These challenges represent critical technological locks and power companies are devoting significant design efforts to deal with these issues, increasingly relying on the use of macroscopic numerical
models. This framework is broadly referred to as “Computational Fluid Dynamics”. However, such large-scale approaches tend to oversimplify small-scale physics, which limits their suitability and
precision 21. Particles encountered in industrial situations are generally difficult to model: they are polydisperse, not exactly spherical but of any shape, and deform; they have complex
interactions, collide and can agglomerate; they usually deposit or stick to the walls and can even modify the very nature of the flow (e.g. polymeric flows). Extending present models to these complex
situations is thus key to improve their applicability, fidelity, and performance.
Models operating in industry generally incorporate rather minimalist descriptions of suspended inclusions. They rely on statistical closures for single-time, single-particle probability
distributions, as is the case for the particle-tracking module in the open-source CFD software Code_Saturne developed and exploited by EDF R&D. The underlying mean-field simplifications do not
accurately reproduce complex features of the involved physics that require higher-order correlation descriptions and modeling. Indeed, predicting the orientation and deformation of particles requires
suitable models of the fluid velocity gradient along their trajectories 40 while concentration fluctuations and clustering depend on relative particle dispersion 36, 26. Estimates of collision and
aggregation rates should also be fed by two-particle dynamics 34, while wall deposition is highly affected by local flow structures 37. Improving existing approaches is thus key to obtain better
prediction tools for multiphase flows.
New simulation approach for renewable energy and meteorological/climate forecast
A major challenge of sustainable power systems is to integrate climate and meteorological variability into operational processes, as well as into medium/long term planning processes 24. Wind, solar,
marine/rivers energies are of growing importance, and the demand for forecasts goes hand in hand with it 23, 19. Numerous methods exist for different forecast horizons 22. One of the main
difficulties is to address refined spatial description. In the case of wind energy, wind production forecasts are submitted to the presence of turbulence in the near wall atmospheric boundary layer.
Turbulence increases the variability of wind flows interacting with mill structures (turbine, mast, nacelle), as well as neighboring structures, terrain elevation and surface roughness. Although some
computational fluid dynamics models and software are already established in this sector of activity 3532, the question of how to enrich and refine wind simulations (from meteorological forecast, or
from larger scale information, eventually combined with local measurements) remains largely open.
Though hydro turbine farms are of a less assertive technological maturity than wind farms, simulating hydro turbines farms in rivers and sea channels submitted to tidal effect present similar
features and challenges. Moreover in the marine energy context, measures are technically more difficult and more costly, and the demand in weather forecast concerns also the safety in maintenance
At the time scale of climate change, the need for uncertainty evaluation of predictions used in long-term planning systems is increasing. For managers and decision makers in the field of hydrological
forecasts, assessing hydropower predictions taking into account their associated uncertainties is a major research issue, as shown by the recent results of the European QUICS project 38. The term
uncertainty here refers to the overall error of the output of a generic model 31. Translating time series of meteorological forecast into time series of run-of-river hydropower generation
necessitates to capture the complex relationship between the availability of water and the generation of electricity. The water flow is itself a nonlinear function of the physical characteristics of
the river basins and of the weather variables whose impact on the river flow may occur with a delay.
5 New results
5.1 Axis A – Complex flows: from fundamental science to applied models
5.1.1 Lagrangian stochastic model for the orientation of non-spherical particles in turbulent flow: an efficient numerical method for CFD approach
Participants: Lorenzo Campana, Mireille Bossy, Christophe Henry, Jérémie Bec.
Suspension of anisotropic particles can be found in various industrial applications. Microscopic ellipsoidal bodies suspended in a turbulent fluid flow rotate in response to the velocity gradient of
the flow. Understanding their orientation is important since it can affect the optical or rheological properties of the suspension. The equations of motion for the orientation of microscopic
ellipsoidal particles were obtained by Jeffery 27. But so far, this description has always been investigated in the framework of direct numerical simulations (DNS) and experimental measurements. In
particular, inertia-free particles, with sizes smaller than the Kolmogorov length, follow the fluid motion with an orientation generally defined by the local turbulent velocity gradient.
In this work, our focus is to characterize the dynamics of these objects in turbulence by means of a stochastic Lagrangian approach. The development of a model that can be used as predictive
computational tool in industrial computational fluid dynamics (CFD) codes is highly valuable for practical applications. Models that reach an acceptable compromise between simplicity and accuracy are
needed for progressing in the field of medical, environmental and industrial processes.
Firstly, the formulation of a stochastic orientation model is studied in two-dimensional turbulent flow with homogeneous shear, where results are compared with direct numerical simulations (DNS). We
address several issues, i.e finding analytical results, the model, scrutinizing the effect of the anisotropies when they are included in the model, and extending the notion of rotational dynamics in
the stochastic framework. Analytical results give a reasonable qualitative response, even if the diffusion model is not designed to reproduce the non-Gaussian characteristics of the DNS experiments.
A further extension to the three-dimensional case shows that the implementation of efficient numerical schemes in 3D models is far from straightforward. A numerical scheme has been devised, able to
preserve the dynamical features at reasonable computational costs for such highly nonlinear SDEs. The convergence is analyzed, obtaining a strong mean-square convergence of order 1/2 and a weak
convergence of order 1.
Eventually, the model and the numerical scheme have been implemented in the open-source CFD Code_Saturne software. The model was used to study the orientational and rotational behavior of anisotropic
inertia-free particles in an applicative prototype of inhomogeneous turbulence in a channel flow. This application faces two different modeling issues: the first concerns whether and to which extent
the model is able to reproduce the DNS experiments in a channel flow; the second is about its numerical implementation within a fully stochastic Lagrangian framework provided by the Lagrangian module
of Code_Saturne. In this context, the stochastic Lagrangian model for the orientation reproduces with some limits the orientation and rotation statistics of the DNS.
Three related publications are in preparation.
5.1.2 Dynamics and statistics of inertial spheroidal particles in turbulence
Participants: Sofia Allende, Jérémie Bec.
Many industrial processes involve the transport of material inclusions (dust, debris) by a turbulent fluid. Quantifying properties of such particles is essential to optimize the design and
performance of these systems. Despite these challenges, the classical approaches used in industry oversimplify the physics at small scale and fail to capture various effects, especially in the case
of non-spherical and deformable particles. The improvement of macroscopic models remains to this day a real challenge. In continuation to the collaboration developed between Inria and EDF R&D on
models for the transport of non-ideal particles in turbulent flows, we have developed direct numerical simulation tools to provide a microscopic description of the dynamical and statistical
properties of inertial non-spherical particles
In this framework we have performed several numerical experiments of rigid ellipsoidal particles (described by the Jeffery equation) passively transported by an incompressible 3D homogeneous
isotropic turbulent flow. The idea was to understand the effects of non-sphericity on the statistics of particles velocity, acceleration, rotation and concentration properties. Our results seem to
indicate that the translational dynamics of particles solely depends on an angle-averaged Stokes number. Everything happens as if the orientation of the particles is not correlated with its
translational dynamics. An article on this topic has been submitted to the Journal of Fluid Mechanics.
5.1.3 Turbophoresis of heavy inertial particles in statistically homogeneous flow
Participants: Jérémie Bec, Robin Vallée.
Dispersed particles suspended in turbulent flows are widely encountered in nature or industry under the form of droplets, dust, or sediments. When they are heavier than the fluid, such particles
possess inertia and are ejected by centrifugal forces from the most violent vortical structures of the carrier phase. Once cumulated along particle paths, this small-scale mechanism produces an
effective large-scale drift where particles leave the excited turbulent zones and converge to calmer regions to form uneven spatial distributions. This fundamental phenomenon, called turbophoresis,
has been extensively used to explain why particles transported by non-homogeneous flows concentrate near the minima of the turbulent kinetic energy.
We have shown that turbophoretic effects are just as crucial in statistically homogeneous and isotropic flows. Instantaneous spatial fluctuations of the turbulent activity, despite their uniform
average, trigger local fluxes that play a key role in the emergence of inertial-range inhomogeneities in the particle distribution. Direct numerical simulations have been used to thoroughly probe and
depict the statistics of particle accelerations and in particular their scale-averaged properties conditioned on the local turbulent activity. They confirm the relevance of the local energy
dissipation to describe instantaneous spatial fluctuations of turbulence. This analysis yields an effective coarse-grained dynamics, in which particles detachment from the fluid and their ejection
from excited regions are accounted for by a space and time-dependent non-Fickian diffusion.
Such considerations led us to cast inertial-range fluctuations in the particles distributions in terms of a local Péclet number Pe, which measures the relative importance of turbulent advection
compared to turbophoresis induced by inertia. Numerical simulations confirm the relevance of this dimensionless parameter to characterize how particle concentration recovers homogeneity at large
scales. This approach also explains the presence of voids with inertial-range sizes, and in particular that their volumes have a non-trivial distribution with a power-law tail whose exponent depends
on the particle response time. These results are gathered in an article that will be submitted to the Journal of Fluid Mechanics in the coming months.
5.1.4 Modeling of the formation and maturation of soot particle aggregates
Participant: Christophe Henry.
Studying the agglomeration of small nanoparticles (a few nanometers in size) or atomic clusters has remarkable importance for the synthesis of nanoparticles at industrial scale. However, this is a
challenge since different physical phenomena have to be considered for instance atomic clusters can experience coalescence upon collisions while larger nanoparticles may experience a rebound after
collisions. This means that a sticking probability has to be taken into account. This sticking probability is currently poorly understood especially for nanoparticles formed in flames where changes
in agglomeration and flow regimes occur simultaneously.
This study focuses on the aggregation of nascent soot particles, which are very important to predict well soot particle size distribution and morphology in flames. Such nascent soot particles may
grow in the reaction-limited aggregation regime (sticking probability $\ll$ 1). However, it is currently unknown how fast would be the transition towards diffusion/ballistic-limited aggregation
regimes as observed for mature soot (sticking probability close to 1). In this collaborative work, we intend to fill this gap by focusing on numerically simulated soot particles formed in a laminar
premixed flame. To this end, a recent fast and accurate Monte Carlo discrete element code called MCAC (developed at CORIA) is used. In these simulations the individual trajectories of particles are
integrated in time. The MCAC has been adapted to non-unitary collision and sticking probability considering three different outcomes for interacting aggregates: no collision, sticking or rebound.
Using such fine-scale simulations, we have shown that assuming a unitary sticking and collision probability produces no big changes in the aggregation kinetics, particle size distribution, and
aggregate morphology. Meanwhile, the soot particles bulk density was found to affect the aggregation kinetics and particle size distribution. This is an important result for macroscopic models: such
effects should be considered in future simulations relying on Population Balance Equations (PBE).
These results have been realized in collaboration with José Moran and Jérôme Yon from CORIA in Rouen. The results were published in Carbon 8 and were presented by José Moran at the French Conference
on Aerosol in January 2021, at the Cambridge Particle Meeting in June 2021 and at the European Aerosol Conference in August 2021 13, 18.
5.2 Axis B – Particles and flows near boundaries: specific Lagrangian approaches for large-scale simulations
5.2.1 New spatial decomposition method for accurate, mesh-independent agglomeration predictions in particle-laden flows
Participants: Mireille Bossy, Christophe Henry, Kerlyns Martínez Rodríguez.
Computational fluid dynamics simulations in practical industrial/environmental cases often involve non-homogeneous concentrations of particles. In Euler-Lagrange simulations, this can induce the
propagation of numerical error when the number of collision/agglomeration events is computed using mean-field approaches. In fact, mean-field statistical collision models allow to sample the number
of collision events using a priori information on the frequency of collisions (the collision kernel). Yet, since such methods often rely on the mesh used for the Eulerian simulation of the fluid
phase, the particle number concentration within a given cell might not be homogeneous, leading to numerical errors. In this article, we apply the data-driven spatial decomposition (D2SD) algorithm,
recently proposed in a previous work reported in 7, to control such error in simulations of particle agglomeration. This D2SD algorithm provides a spatial splitting according to the spatial
distribution of particles. More precisely, the D2SD algorithm uses as an input data only the information on the location of the center of gravity of each particle. One of the many advantages of the
D2SD algorithm is that the parameters leading to the optimal domain decomposition are automatically tuned through the statistical information coming from the data (position of particles). Thus, there
is no bias coming from the choice of arbitrary parameter.
Significant improvements are made to design a fast D2SD version, minimizing the additional computational cost by developing re-meshing criteria. Several options are assessed, introducing a criterion
to avoid applying the full version of the D2SD algorithm every time step, or simplifying uniformity tests. The main difficulty is to ensure that the adapted algorithm keeps an appropriate balance
between its accuracy and its computational costs.
Through the application to some practical simulation cases, we show the importance of splitting the domain when computing agglomeration events in Euler/Lagrange simulations, so that there is a
spatially uniform distribution of particles within each elementary cell. The algorithm is coupled to 3D simulations of particle agglomeration in practical cases with a two-fold objective: first, we
assess the accuracy and efficiency of the method in a validation case; second, we illustrate how the D2SD can be applied in a practical case that is representative of situations of interest in the
multiphase flow community.
This study is detailed in 6, published in the International Journal of Multiphase Flow.
5.2.2 Evidence of collision-induced resuspension of microscopic particles from a monolayer deposit and new models
Participants: Mireille Bossy, Christophe Henry.
This study aims at bridging the gap between the understanding and modeling of particle resuspension in monolayer deposits and multilayer deposits. More precisely, modeling resuspension is indeed a
challenging task owing to its complexity and multiscality. In practice, numerical concepts describing the resuspension at the particle scale, that is in the micron to millimeter size, exist. However,
such models have been designed to treat two limit cases: monolayer or multilayer deposits. In the monolayer case, the inter-particle distance $L$ is implicitly assumed to be much greater than the
particle diameter ${D}_{p}$ ($L\gg {D}_{p}$), so that each resuspension event can be treated independently. In the multilayer case where particles sit on top of one another ($L\ll {D}_{p}$),
resuspension events involve either single particles or clusters of particles depending on the local deposit structure and inter-particle cohesion forces. Yet, a unified description of particle
resuspension from monolayer to multilayer deposits is still missing.
The present work bridges the gap by addressing the very special case where the inter-particle distance becomes comparable to the particle diameter ($L\sim {D}_{p}$). Experimental investigations
performed by co-authors at Technische Universität Dresden (Germany) have revealed two distinct detachment mechanisms. At relatively low flow velocities, few loosely adhering particles move on the
wall to eventually collide with neighboring particles resulting in a clustered resuspension. At higher fluid velocities, mostly individual particles resuspend due to their interaction with the
turbulent flow.
In line with these new observations, the existing model for particle resuspension from monolayered deposits has been extended to account for the effect of inter-particle collision. Despite its
simplicity, this extended model confirms the role played by inter-particle collisions even at relatively low surface coverage while highlighting the importance of initial clustering (which can
significantly increase the probability of collision between particles at the local scale).
These results were published in Physical Review Fluids 1 and presented at the Dispersed Two-Phase Flow Conference in October 2021. Another publication is under preparation to further explore the role
of adhesive forces.
5.2.3 Effective accretion rates of small inertial particles by a large towed sphere
Participants: Jérémie Bec, Robin Vallée.
The capture of small suspended particles by a streamlined or bluff body is an important process in many natural systems (wind pollination, collection of phytoplancton by passive suspension-feeding
invertebrates, planet formation, growth of raindrops by accretion of cloud droplets, riming of supercooled droplets by ice crystals, scavenging of aerosols during wet deposition). Achieving precise
predictions requires, on the one hand, elucidating mesoscopic fluid-dynamical effects that determine whether or not impaction occurs, and on the other hand, specifying the microphysical features and
processes that affect the outcome of such collisions and a possible capture by the collector.
In collaboration with Christoph Siewert (Deutscher Wetterdienst, Germany), we have studied the collision efficiency of small particles by a large sphere. We found that the rate at which small
inertial particles collide with a moderate-Reynolds-number body is strongly affected when these particles are also settling under the effect of gravity. The sedimentation of small particles indeed
changes the critical Stokes number above which collisions occur. We explain this by the presence of a shielding effect caused by the unstable manifolds of a stagnation-saddle point of an effective
velocity field perceived by the small particles. We also found that there exists a secondary critical Stokes number above which no collisions occur. This is due to the fact that large-Stokes number
particles settle faster, making it more difficult for the larger one to catch them up. Still, in this regime, the flow disturbances create a complicated particle distribution in the wake of the
collector, sometimes allowing for collisions from the back. We demonstrated that this effect can lead to collision efficiencies higher than unity at large values of the Froude number. An article on
this topic has been submitted to Physical Review Fluids.
5.3 Axis C – Active agents in a fluid flow
5.3.1 Finite Element Methods for simulate displacement of flagellated micro-swimmers
Participants: Laetitia Giraldi, Luca Berti.
In collaboration with Vincent Chabannes (IRMA, Strasbourg) and Christophe Prud'Homme (IRMA, Strasbourg), in 2, we propose a numerical method for the finite element simulation of micro-swimmers
displacement with a prescribed stroke. We focus on swimmers composed of several rigid bodies in relative motion. Three distinct formulations are proposed to impose the relative velocities between the
rigid bodies. We validate our model on the three-sphere swimmer, for which analytical results are available.
This paper was published in Comptes Rendus – Mathématiques.
5.3.2 Reinforcement learning with function approximation for 3-spheres swimmer
Participants: Luca Berti, Zakarya El-khyiati, Laetitia Giraldi.
In collaboration with Christophe Prud'Homme (IRMA, Strasbourg) and Youssef Essoussy (IRMA, Strasbourg), the paper 14 investigates the swimming strategies that maximize the speed of the three-sphere
swimmer using reinforcement learning methods. First of all, we ensure that for a simple model with few actions, the Q-learning method converges. However, this latter method does not fit a more
complex framework (for instance the presence of boundary) where states or actions have to be continuous to obtain all directions in the swimmer's reachable set. To overcome this issue, we investigate
another method from reinforcement learning which uses function approximation, and benchmarks its results in absence of walls.
This work was initiated with the internship of Youssef Essoussy. We have also been supported by UCA Fox 2021 School1 which allows some participant to see each other.
5.3.3 Necessary conditions for local controllability of a particular class of systems with two scalar controls
Participant: Laetitia Giraldi.
In this paper 17 in collaboration with Pierre Lissy (Ceremade, Paris), Jean-Baptiste Pomet (Inria, McTAO) and Clement Moreau (RIMS, Kyoto, Japan), we consider control-affine systems with two scalar
controls, such that one control vector field vanishes at an equilibrium state. We state two necessary conditions for local controllability around this equilibrium, involving the iterated Lie brackets
of the system vector fields, with controls that are either bounded, small in ${\mathrm{L}}^{\infty }$ or small in ${\mathrm{W}}^{1,\infty }$. These results were deduced by the behavior of the
magnetic flagellated swimmers and they are illustrated with several examples.
The paper is submitted. It was also a chapter of the PhD thesis of Clement Moreau.
5.3.4 Reinforcement learning for the locomotion and navigation of undulatory micro-swimmers in chaotic flow
Participants: Raphaël Chesneaux, Zakarya El Khyiati, Jérémie Bec, Laetitia Giraldi.
We developed a framework to study the motion of vermiform micro-swimmers, self-propelling by undulating their body. Such deformable swimmers have a high potential because of their aptness to carry
out a broad set of swimming strategies and to select the most efficient one according to the biological media where they evolve. Many questions are still open on how these micro-swimmers optimize
their displacement, in particular when they are embedded in a complex environment. In practice the swimmers navigate in a fluctuating medium comprising walls and obstacles, a fluid flow possibly with
non-Newtonian properties or containing other swimmers. In this framework, optimizing their navigation requires dealing with a strongly nonlinear and chaotic high-dimensional dynamics.
Using machine-learning tools, we have developed new methods to tackle this optimization problem where swimming and navigation are tightly bonded. Techniques borrowed from partially-observable Markov
decision processes were found to be particularly promising. Combining an efficient locomotion strategy with optimal navigation and path-planning is particularly novel in the field. An article
demonstrating the efficiency of genetic reinforcement learning for the displacement of undulatory swimmers in two-dimensional flow is currently in preparation and will be submitted in the coming
months to Physical Review Letters.
5.4 Axis D – Mathematics and numerical analysis of stochastic systems
5.4.1 Anomalous fluctuations for the Lyapunov exponents of tracers in developed turbulent flow
Participants: Jérémie Bec, Simon Thalabard.
The infinitesimal separation between tracers transported by a turbulent flow is generally characterized in terms of stretching rates and Lyapunov exponents obtained from the integration of the
tangent system to the dynamics. We have shown that turbulent intermittency is responsible for long-range correlations in the Lagrangian fluid velocity gradient. This behavior, which does not question
the existence of a law of large numbers and of Lyapunov exponents, seriously questions large-deviation approaches that are usually used to characterize the fluctuations of finite-time stretching
rates and thus to quantify small-scale turbulent mixing. We propose alternative manners to qualify fluctuations based on generalizations of the central-limit theorem to sums of correlated variables.
These results were obtained in the framework of the ANR TILT project and are the subject of a manuscript that will be soon submitted to Physical Review Letters.
These results suggest to introduce new Lagrangian stochastic models for small-scale turbulent mixing that extend traditional diffusive approach to noises with long-range time correlations. Fractional
Brownian motion seems a promising candidate.
5.5 Axis E – Variability and uncertainty in flows and environment
5.5.1 Instantaneous turbulent kinetic energy modeling based on Lagrangian stochastic approach in CFD and application to wind energy
Participants: Mireille Bossy, Kerlyns Martínez Rodríguez.
The need of statistical information on the wind, at a given location and on large time period, is of major importance in many applications such as the structural safety of large construction projects
or the economy of a wind farm, whether it concerns an investment project, a wind farm operation or its repowering. The evaluation of the local wind is expressed on different time scales: monthly,
annually or over several decades for resource assessment, daily, hourly or even less for dynamical forecasting (these scales being addressed with an increasing panel of methodologies). In the
literature, wind forecasting models are generally classified into physical models (numerical weather prediction models), statistical approaches (time-series models, machine learning models, and more
recently deep learning methods), and hybrid physical and statistical models. At a given site and height in the atmospheric boundary layer, measuring instruments record time series of characteristics
of the wind, such as wind speed characterizing load conditions, wind direction, kinetic energy and possibly power production. Such observations should feed into forecasting, but also uncertainty
modeling. In this context, probabilistic or statistical approaches are widely used, helping to characterize uncertainty through quantile indicators.
In this work, we construct an original stochastic model for the instantaneous turbulent kinetic energy at a given point of a flow, and we validate estimator methods on this model with observational
data examples. Motivated by the need for wind energy industry of acquiring relevant statistical information of air motion at a local place, we adopt the Lagrangian stochastic description of fluid
flows to derive, from the 3D+time equations of the physics, a 0D+time-stochastic model for the time series of the instantaneous turbulent kinetic energy at a given position. First, we derive a family
of mean-field dynamics featuring the square norm of the turbulent velocity. By approximating at equilibrium the characteristic nonlinear terms of the dynamics, we recover the so called
Cox-Ingersoll-Ross process, which was previously suggested in the literature for modeling wind speed. We then propose a calibration procedure for the parameters employing both direct methods
(motivating partially the numerical analysis in 3 by the same authors) and Bayesian inference. In particular, we show the consistency of the estimators and validate the model through the
quantification of uncertainty, with respect to the range of values given in the literature for some physical constants of turbulence modeling.
This work 15, in collaboration with Jean-Francois Jabir from National Research University HSE Moscow, is now accepted in Journal of Computational Physics. It was also presented (12) during the annual
meeting of the European Meteorological Society 2021.
5.5.2 Methodology to quantify uncertainties in droplet dispersion in the air
Participants: Christophe Henry, Kerlyns Martínez Rodríguez, Mireille Bossy, Jérémie Bec.
In this work, we resorted to standard uncertainty quantification (UQ) and sensitivity analysis (SA) tools that are available in the open-source software OpenTurns. The present methodology relies on
variance-based methods (such as the “Sobol indices” or “variance-based sensitivity indices”) to analyze the variability of the numerical results with respect to a number of input parameters (e.g.
droplet size, droplet emission velocity, wind velocity). This methodology has been validated on a demonstration case that consisted in a simulation of droplet dispersion in a quiescent flow without
evaporation/condensation models. We are currently working on setting up more realistic simulations of droplet dispersion in the air.
This research is described in a short communication in ERCIM News 5, which was done in collaboration with Hervé Guillard from Team Castor as well as Nicolas Rutard and Angelo Murrone from ONERA. This
research has actually been carried out through the Inria's Covid Mission Spreading_Factor project 2020, which aimed at setting up a methodology to help quantify the relative importance between the
input physical parameters and their impact on droplet dispersion as well as to quantify uncertainties on the output results. The results were also presented at the French Aerosol Conference in
January 2021 11.
5.5.3 Methodology to quantify uncertainties in dispersed two-phase flows
Participants: Aurore Dupré, Christophe Henry, Mireille Bossy.
A similar methodology has been applied to study dispersed two-phase flows. This methodology has actually been developed within the framework of the VIMMP EU project (Virtual Materials Market Place).
The objective is to set up a methodology to analyze the sensitivity and then quantify uncertainty in numerical simulations of multiphase flows to a number of input variables. For that purpose, we
focused on the case of a point-source dispersion of particles in a turbulent pipe flow. Numerical simulations were performed by coupling a CFD simulation of the turbulent pipe flow (using standard
turbulence models) to a particle-tracking simulation (using a stochastic Lagrangian model). The simulations were performed in Code_Saturne CFD software. The simulation workflow is launched using
tools from the Salome platform, which allows to handle the coupling of the fluid phase simulation and the particle-phase simulation. The results obtained are then analyzed using existing tools within
OpenTurns. For that purpose, a dataset is obtained by running the workflow with a range of input variables (e.g. the fluid velocity, number of particles injected, size of particles) and accounting
for the intrinsic stochasticity of each run. Sensitivity analysis techniques (here the Sobol sensitivity indices) were used to identify the key parameters affecting the observed results.
These results were presented at the OpenTurns User Days held in June 2021 10. A paper is also in preparation with other partners involved in the VIMMP project (Pascale Noyret, Eric Fayolle and
Jean-Pierre Minier from EDF R&D).
5.5.4 Analyzing the Applicability of Random Forest-Based Models for the Forecast of Run-of-River Hydropower Generation
Analyzing the impact of climate variables into the operational planning processes is essential for the robust implementation of a sustainable power system. The work, published in 9, deals with the
modeling of the run-of-river hydropower production based on climate variables on the European scale. A better understanding of future run-of-river generation patterns has important implications for
power systems with increasing shares of solar and wind power. Run-of-river plants are less intermittent than solar or wind but also less dispatchable than dams with storage capacity. However,
translating time series of climate data (precipitation and air temperature) into time series of run-of-river-based hydropower generation is not an easy task as it is necessary to capture the complex
relationship between the availability of water and the generation of electricity. This task is also more complex when performed for a large interconnected area. In 9, in collaboration with Valentina
Sessa and Edi Assoumou from CMA Mines ParisTech, and Sofia G Simões from Laboratório Nacional de Energia e Geologia in Portugal, a model is built for several European countries by using machine
learning techniques. In particular, we compare the accuracy of models based on the Random Forest algorithm and show that a more accurate model is obtained when a finer spatial resolution of climate
data is introduced. We then discuss the practical applicability of a machine learning model for the medium term forecasts and show that some very context specific but influential events are hard to
5.6 Other
5.6.1 Selection of microalgue
Participant: Laetitia Giraldi.
The papers 4, 16, in collaboration with Walid Djema and Olivier Bernard (Inria, Biocore) and Sofya Maslovskaya (Paderborn University, Germany), proposes a strategy to separate two strains of
microalgae in minimal time. The control is the dilution rate of the continuous photobioreactor. The microalgae dynamics is described by the Droop's model, taking into account the internal quota
storage of the cells. Using Pontryagin’s principle, we develop a dilution-based control strategy that leads to the most efficient species separation in minimal time. A numerical optimal synthesis
–based on direct optimization methods– is performed throughout the paper, in order to determine the structure of the optimal feedback-control law, which is bang-singular. Our numerical study reveals
that singular arcs play a key role in the optimization problem since they allow the optimal solution to be close to an associated static optimal control problem. A resulting turnpike-like behavior,
which characterizes the optimal solution, is highlighted throughout this work.
6 Bilateral contracts and grants with industry
6.1 Bilateral grants with industry
aVENTage – Towards a very high resolution wind forecast chain on the sailing basin of Marseilles.
Participants: Mireille Bossy, Thomas Ponthieu.
aVENTage is an industrial partnership project with the two French startups SportRizer and RiskWeatherTech. Starting at the end of 2020, the genesis of this project was motivated by the next Paris
2024 Olympic Games, where the sailing events will take place in the Marseilles sailing basin. The reading of the wind is one of the major stakes in the search for performance for the sailing
Olympics. However, the exhaustive knowledge of the wind of a body of water is not yet resolved.
aVENTage aims to complete the knowledge database of the different local effects in the Marseilles sailing basin, thus facilitating the exploitation of the water body.
A high resolution wind forecast allows to reduce the margin of error in the decision making. This is a determining factor for progress, accelerating learning and supporting the material and technical
development underlying performance. To reach a very high resolution of 50 m horizontally, aVENTage relies on two distinct and successive downscaling processes to produce its results.
• 1.
An operational processing chain from large-scale weather forecasts (GFS 50 km) up to 1 km resolution. Each day, the SportRIZER & RiskWeatherTech operational chain downloads the 0h00 GFS forecast
data for the 0h00+1h to 0h00+48h time frames and performs a downscaling simulation with the WRF model down to 1 km resolution over the Marseilles area.
• 2.
A specific downscaling from the previous operation. To refine the wind simulation up to a resolution of 50 m, this second step relies on the SDM-WindPoS model.
Preliminary results and case studies, as well as detailed methodologies are available on the SDM-WindPos software webpage.
7 Partnerships and cooperations
7.1 International initiatives
7.1.1 STIC/MATH/CLIMAT AmSud project
Participant: Mireille Bossy.
Calisto was involved in the MATH-AmSUD project Fantastic, ended in 2021, on statistical inference and sensitivity analysis for models described by stochastic differential equations. In particular
Calisto was collaborated with Universidad de Valparaíso on the diffusive limit of system of piecewise deterministic Markov processes under mean field interaction.
7.1.2 Participation in other International Programs
Participant: Jérémie Bec.
The team participates in the CNRS IRL IFCAM (Indo-French Center for Applied Mathematics, see website) that provides support for recurrent collaborations with teams at the Indian Institute of Science
and the International Center for Theoretical Science in Bangalore. The pandemic however prevented from planning any visit between the French and Indian teams during the year 2021.
7.2 International research visitors
7.2.1 Visits of international scientists
International visits to the team
• In October 2021, Mara Chiricotto came for a 5-day visit in Calisto. Mara Chiricotto is a Post-Doctoral fellow at the University of Manchester and has an expertise in Molecular Dynamics
simulations. Her visit took place in the framework of the VIMMP project. The specific goal was to assist Mireille Bossy and Christophe Henry in the development of a scientific workflow to
quantify uncertainty in nano-particle agglomeration using Molecular Dynamics tools. This work is still in progress.
7.2.2 Visits to international teams
Research stays abroad
Jérémie Bec
• Visited institution:
Göteborg University
• Country:
• Dates:
Nov. 28 – Dec. 4, 2021
• Context of the visit:
Preparation of a review article on statistical models for turbulent aerosols
• Mobility program/type of mobility:
research stay
7.3 European initiatives
7.3.1 FP7 & H2020 projects
Participants: Aurore Dupré, Mireille Bossy, Christophe Henry.
VIMMP (Virtual Materials Market Place) is a EU H2020 project (started in 2018) in the program Industrial Leadership project in Advanced materials. VIMMP is a four-year development for a software
platform and simulation market place on the topic of complex multiscale CFD simulations.
As a VIMMP partner, Calisto is co-working with EDF R&D at designing complex workflows through the EDF's cross-platform Salome, involving Lagrangian aggregations, fragmentation with Code_Saturne.
Calisto also addresses some typical workflow design for uncertainty quantification, and experiments with them in two-phase flow simulation situation. Precisely, we are designing a workflow case of
particle dispersion in a turbulent pipe flow, with a selection of physical and numerical inputs as well as observable output. We have performed some sensitivity analysis (based on the Sobol indices
method) and meta-modeling (based on polynomial chaos) to asses some main features in term of workflow run in a simulation platform, identifying also the relative HPC needs, and expert supervision
needs. This workflow case also served as demonstration case for the development of a common data model (CDM) led by EDR R&D.
7.4 National initiatives
7.4.1 ANR PACE
Participant: Christophe Henry.
Christophe Henry was the coordinator of the PACE project, a MRSEI project funded by the ANR to help prepare European projects. As for PAIRE, the project aims at creating new international and
cross-sector collaborations to foster innovative solutions for particle contamination in the environment. This is achieved by bringing together partners in a consortium to submit a research proposal.
Submissions have been made to the European MSCA-RISE-2019 and MSCA-RISE-2020 calls. Members of the consortium are now considering the option to submit a research project MSCA-DN (doctoral network) in
7.4.2 ANR TILT
Participant: Jérémie Bec.
The ANR PRC project TILT (Time Irreversibility in Lagrangian Turbulence) started on Jan. 1, 2021. It is devoted to the study and modeling of the fine structure of fluid turbulence, as it is observed
in experiments and numerical simulations. In particular, recall that the finite amount of dissipation of kinetic energy in turbulent fluid, where viscosity seemingly plays a vanishing role, is one of
the main properties of turbulence, known as the dissipative anomaly. This property rests on the singular nature and deep irreversibility of turbulent flows, and is the source of difficulties in
applying concepts developed in equilibrium statistical mechanics. The TILT project aims at exploring the influence of irreversibility on the motion of tracers transported by the flow. The consortium
consists of 3 groups with complementary numerical and theoretical expertise, in statistical mechanics and fluid turbulence. They are located in Saclay, at CEA (Bérengère Dubrulle), in Lyon, at ENSL
(Laurent Chevillard, Alain Pumir), and in Sophia Antipolis (Jérémie Bec). A postdoc will be hired by the team on this contract in fall 2022.
7.4.3 ANR NEMO
Participant: Laetitia Giraldi.
The JCJC project NEMO (controlliNg a magnEtic Micro-swimmer in cOnfined and complex environments) was selected by ANR in 2021, and started on Jan. 1, 2022 for four years. NEMO team's is composed of
Laetitia Giraldi, Mickael Binois (Inria, Acumes) and Laurent Monasse (Inria, Coffee).
NEMO aims to develop numerical methods to control a micro-robot swimmer in the arteries of the human body. These robots could deliver drugs specifically to cancer cells before they form new tumors,
thus avoiding metastasis and the traditional chemotherapy side effects.
NEMO will focus on micro-robots, called Magnetozoons, composed of a magnetic head and an elastic tail immersed into a laminar fluid possibly non-Newtonian. These robots imitate the propulsion of
spermatozoa by propagating a wave along their tail. Their movement is controlled by an external magnetic field that produces a torque on the head of the robot, producing a deformation of the tail.
The tail then pushes the surrounding fluid and the robot moves forward. The advantage of such a deformable swimmer is its aptness to carry out a large set of swimming strategies, which could be
selected according to the geometry or the rheology of the biological media where the swimmer evolves (blood, eye retina, or other body tissues).
Although the control of a such micro-robots has mostly focused on simple unconfined environment, the main challenge is today to design external magnetic fields that allow them to navigate efficiently
in complex realistic environments.
NEMO aims to elaborate efficient controls, which will be designed by tuning the external magnetic field, through a combination of Bayesian optimization and accurate simulations of the swimmer's
dynamics with Newtonian or non-Newtonian fluids. Then, the resulting magnetic fields will be validated experimentally in a range of confined environments. In such an intricate situation, where the
surrounding fluid is bounded laminar and possibly non-Newtonian, optimization of a strongly nonlinear, and possibly chaotic, high-dimensional dynamical system will lead to new paradigms.
7.5 Regional initiatives
Participant: Laetitia Giraldi.
Laetitia Giraldi was the investigator of a project Reboost-2021, from the Academy of excellence "Complex Systems" of the IDEX Université Côte d'Azur, on ”Locomotion and optimal navigation of
micro-swimmers in complex environements”. The project aimed to support the internships of Zakarya El-Khiyati (Inria, Calisto) and Raphaël Chesneaux (Inria, Calisto).
7.6 Others
The Calisto team members are involved in the GdR (CNRS Research network) Turbulence, in the GdR Mascot-NUM on stochastic methods for the analysis of numerical codes, and in the GdR Théorie et Climat.
8 Dissemination
Participants: Jérémie Bec, Mireille Bossy, Laetitia Giraldi, Christophe Henry.
8.1 Promoting scientific activities
8.1.1 Scientific events: organisation
Member of the organizing committees
• Jérémie Bec was member of the organizing committee of the conference “Dynamics Days Europe XL” held in Nice in August 2021 (link here).
• Jérémie Bec and Laetitia Giraldi were members of the organizing committee of the first edition of UCA Fall program on Complex Systems devoted to “Mobility, self-organization and swimming
strategies” in October 2021 (link here).
• Mireille Bossy was member of the Committee of the “Prix Pierre Lafitte 2021”. She is also member of the Steering Committee of the GdR MascotNum.
• Christophe Henry was the organizer of a workshop on “Microplastics in the atmosphere” in November 2021 (details on the program on Calisto website).
Scientific seminars of the Team
• Since November 2020, the team is organizing a regular seminar every 4 weeks. In 2021, the following researchers were invited to give a presentation (mostly online due to the sanitary situation):
Aurore Dupré (Calisto), Florence Marcotte (Inria, Castor), Grégory Lécrivain (HZDR, Germany), Jérôme Yon and José Moran (CORIA, Rouen), Agnese Seminara (InPhyNi, Nice), Rudy Valette (CEMEF, Mines
ParisTech, Sophia Antipolis), Mickael Binois (Inria, Acumes), Areski Cousin (IRMA, Strasbourg and external collaborator in Calisto), Christophe Brouzet (InPhyNi, UCA, Nice), Angelica Bianco
(LaMP, UCA, Clermont), Simon Thalabard (InPhyNi, UCA, Nice).
8.1.2 Scientific events: selection
Member of the conference program committees
• Jérémie Bec was member of the scientific committee of the conference “Fluids & Complexity” held in Nice in November 2021 (link here).
8.1.3 Journal
Member of the editorial boards
• Jérémie Bec acted as a guest editor for a special issue of the Philosophical Transactions of the Royal Society A entitled “Scaling the turbulence edifice” and gathering 25 contributions.
Reviewer - reviewing activities
• Jérémie Bec acted as a reviewer for International Journal of Multiphase flow, Journal of Fluid Mechanics, Journal of Mathematical Physics, Physical Review Fluids.
• Mireille Bossy reviewed project propositions form the generic ANR AAP 2021 and from ANRT. She also acted in 2021 as a reviewer for the following international journals: Annals of Applied
Probability, Journal of Computational and Applied Mathematics, IMA Journal of Numerical Analysis, Stochastics and Partial Differential Equations: Analysis and Computations, and Stochastics.
• Laetitia Giraldi reviewed several papers as for instance for Physical Review Fluids, Journal of Fluids Mechanics, IEEE Transactions on Automatic Control.
• Christophe Henry reviewed papers for the following journals in 2021: Talanta (February 2021), Aerosol and Air Quality Research (May 2021), Atmospheric Pollution Research (May 2021) Journal of
Aerosol Science (November 2021).
8.1.4 Invited talks
• Mireille Bossy was invited to give a presentation at the 33ème séminaire CEA/GAMNI de mécanique des fluides numérique, January 25-26,2021. She also gave a plenary talk at the 13th International
Conference on Monte Carlo Methods and Applications (MCM 2021, from 16.8 to 20.8.2021). She was an invited speaker at the Conference of Numerical Probability (in honor of Gilles Pagès' 60th
birthday) 26-28 May 2021 Paris (France).
• Christophe Henry was invited to give presentations at the OpenTurns User Days (June 2021) and at Helmholtz Zentrum Dresden Rossendorf (in October 2021).
• Laetitia Giraldi was invited to give a presentation at SMAI Congres (June 2021) near Montpellier.
8.1.5 Leadership within the scientific community
• Jérémie Bec is in charge of the Academy of excellence "Complex Systems" of the IDEX Université Côte d'Azur (Decision-making role for funding; Coordination and animation of federative actions;
Participation in the IDEX evaluation).
• Mireille Bossy is Chairing of the Scientific Council of the Academy of excellence "Complex Systems" of the IDEX Université Côte d'Azur.
8.1.6 Scientific expertise
• Jérémie Bec was a member of the selection committee for a Professor position in Physics at Université Côte d'Azur.
• Mireille Bossy was the Chair of the selection committee for CRCN and ISFP position at Inria Bordeaux Sud Ouest.
8.1.7 Research administration
• Jérémie Bec is a member of Inria's Comité NICE and of the scientific council of the CNRS GDR “Theoretical challenges for climate sciences”.
• Laetitia Giraldi is a member of Inria’s Comité NICE, Comité de Suivi Doctoral et du Comité du centre.
8.2 Teaching - Supervision - Juries
8.2.1 Teaching
• Fluid dynamics and turbulence (Jérémie Bec, 6h, Doctoral courses, Mines Paris).
• The physics of turbulent flow (Jérémie Bec, 4h, 2nd-year courses, Mines Paris).
• “Research Trimester” project supervision (Jérémie Bec and Laetitia Giraldi, research project of 2 months followed by 2nd-year students of Mines Paris).
• Microswimming (Laetitia Giraldi, 6h, course, Master 2 cell physics, Université de Strasbourg).
• Khôlle en classes préparatoire MPSI, MP* (Laetitia Giraldi, 2h par semaine scolaire par niveau, Centre International de Valbonne).
• Advanced modeling (Christophe Henry, 50h, Master of Hydrology, Polytech Nice Sophia Université Côte d'Azur).
8.2.2 Supervision
• PhD in progress: Lorenzo Campana, “Stochastic modeling of non-spherical particles in turbulence”; Defense is announced for March 29, 2022; supervised by Mireille Bossy.
• PhD in progress: Zakarya El Khiyati, “Reinforcement learning for the optimal locomotion of micro-swimmers in a complex chaotic environment” started in October 2021; supervised by Jérémie Bec and
Laetitia Giraldi.
• PhD in progress: Fabiola Gerosa, “Turbulent fluid-particles coupling and applications to planet formation” started in October 2021; supervised by Jérémie Bec and Héloïse Méheut (Lagrange,
Observatoire de la Côte d'Azur).
• PhD defended in March 4, 2021: Sofia Allende Contador, “Dynamics and statistics of elongated and flexible particles in turbulent flows”; supervised by Jérémie Bec.
• PhD defended in March 30, 2021: Robin Vallée, “Suspensions of inertial particles in turbulent flows”; supervised by Jérémie Bec.
• PhD defended in December 13, 2021: Luca Berti, “Mathematical modeling and simulation of magnetic micro-Swimmers”; co-supervised by Laetitia Giraldi and Christophe Prud'Homme (IRMA, Strasbourg).
• M2 Internship: Thomas Ponthieu, “Very high resolution numerical wind simulation. Assessment of the SDM-WindPoS model for use in sports sailing”; March to September 2021; supervised by Mireille
• M2 Internship: Zakarya El Khiyati, “Smart strategies for the collective motion of deformable micro-swimmers”, April 2021 to September 2021, supervised by Jérémie Bec and Laetitia Giraldi.
• M2 Internship: Youssef Essoussy (IRMA, Strasbourg),“The locomotion optimization for micro-swimmers using machine learning”, April 2021 to September 2021, supervised by Luca Berti, Laetitia
Giraldi and Christophe Prud'Homme (IRMA, Strasbourg).
• M1 Internship: Raphael Chesneaux, “Steering undulatory microswimmers in a moving fluid through machine learning”, December 2020 to February 2021 and June 2021 to August 2021, supervised by
Jérémie Bec and Laetitia Giraldi.
8.2.3 Juries
• Jérémie Bec was referee for the Habilitation thesis of Gautier Verhille, Deformable Objects in Turbulence, at IRPHE, Aix-Marseille University, June 2021.
• Mireille Bossy served as a referee for the Ph.D. thesis of Arthur Macherey, Approximation and model reduction for partial differential equations with probabilistic interpretation, at École
Centrale Nante, June 2021.
• Jérémie Bec was examiner for the Ph.D. theses of Pierre Azam (Université Côte d'Azur, September 2021) and Luca Berti (Université de Strasbourg, December 2021).
• Mireille Bossy served as an examiner for the Ph.D. theses of Sofia Allende Contador at Université Côte d'Azur, March 2021, and Camille Choma at Université Le Havre Normandie, July 2021.
• Laetitia Giraldi served as an examiner for the Ph.D. thesis of Maxime Etiévant at Université de Besançon, July 2021.
8.3 Popularization
8.3.1 Interventions
• Christophe Henry was involved in the following popularization events:
□ Café In at Inria Sophia Antipolis Méditerranée in May 2021 (to present the results of Inria's Mission Covid Spreading Factors);
□ Interview of his research activities at Interstice (link here).
9 Scientific production
9.1 Publications of the year
International journals
• 1 articleEvidence of collision-induced resuspension of microscopic particles from a monolayer deposit.Physical Review Fluids68August 2021
• 2 articleModeling and finite element simulation of multi-sphere swimmers.Comptes Rendus. Mathématique3599November 2021, 1119-1127
• 3 articleOn the weak convergence rate of an exponential Euler scheme for SDEs governed by coefficients with superlinear growth.Bernoulli2712021, 312-347
• 4 articleTurnpike features in optimal selection of species represented by quota models.Automatica2021
• 5 articleSocial Distancing: The Sensitivity of Numerical Simulations.ERCIM News20211242021
• 6 articleParticle agglomeration in flows: fast data-driven spatial decomposition algorithm for CFD simulations.International Journal of Multiphase FlowJanuary 2022
• 7 articleNew spatial decomposition method for accurate, mesh-independent agglomeration predictions in particle-laden flows.Applied Mathematical Modelling902021, 582-614
• 8 articleImpact of the maturation process on soot particle aggregation kinetics and morphology.Carbon182September 2021, 837-846
• 9 articleAnalyzing the Applicability of Random Forest-Based Models for the Forecast of Run-of-River Hydropower Generation.Clean Technologies34December 2021, 858-880
Conferences without proceedings
• 10 inproceedingsSensitivity analysis and uncertainty in CFD simulations of multiphase flow.OpenTurns User day 14 (2021)Paris, FranceJune 2021
• 11 inproceedingsSensitivity of droplet dispersion to emission and ambient air properties.CFA2021 - 34ème Congrès Français sur les AérosolsParis, FranceJanuary 2021
• 12 inproceedingsLocal turbulent kinetic energy modelling based on Lagrangian stochastic approach in CFD and application to wind energy.EMS Annual MeetingVirtual format, GermanySeptember 2021
• 13 inproceedingsImpact of the maturation process on soot particle aggregation kinetics and morphology.Cambridge Particle MMeetingVirtual Conference, United KingdomJune 2021
Reports & preprints
• 14 miscReinforcement learning with function approximation for 3-spheres swimmer.January 2022
• 15 miscInstantaneous turbulent kinetic energy modelling based on Lagrangian stochastic approach in CFD and application to wind energy.January 2021
• 16 reportTurnpike Features in Optimal Selection of Species Represented by Quota Models: Extended Proofs.RR-9399Inria - Sophia AntipolisJune 2021, 29
• 17 miscNecessary conditions for local controllability of a particular class of systems with two scalar controls.August 2021
Other scientific publications
• 18 inproceedingsImpact of the maturation process on soot particle aggregation kinetics and morphology.European Aerosol Conferenceonline presentation, United KingdomAugust 2021
9.2 Cited publications
• 19 miscAdaptation challenges and opportunities for the European energy system Building a climate‑resilient low‑carbon energy system.2019
• 20 articleSelf-propulsion of slender micro-swimmers by curvature control: N-link swimmers.International Journal of Non-Linear Mechanics562013, 132--141
• 21 articleTurbulent Dispersed Multiphase Flow.Annual Review of Fluid Mechanics4212010, 111-133
• 22 articleThe quiet revolution of numerical weather prediction.Nature52575672015, 47--55
• 23 misc10 Trends Reshaping Climate and Energy.2018
• 24 inbookWhat Does the Energy Industry Require from Meteorology?Weather & Climate Services for the Energy IndustryA.A. TroccoliChamSpringer International Publishing2018, 41--63
• 25 techreportFog, Glossary of Meteorology.American Meteorological Society2017
• 26 articleStatistical models for spatial patterns of heavy particles in turbulence.Advances in Physics6512016, 1-57
• 27 articleThe motion of ellipsoidal particles immersed in a viscous fluid.Proc. Royal Soc. Lond. A1027151922, 161--179
• 28 articleSources, fate and effects of microplastics in the marine environment: part 2 of a global assessment.Reports and studies-IMO/FAO/Unesco-IOC/WMO/IAEA/UN/UNEP Joint Group of Experts on the
Scientific Aspects of Marine Environmental Protection (GESAMP) eng no. 932015
• 29 articlePlanet formation.Annual review of astronomy and astrophysics3111993, 129--172
• 30 articleHydrodynamics of soft active matter.Reviews of Modern Physics8532013, 1143
• 31 articleWhat do we mean by ‘uncertainty’? The need for a consistent wording about uncertainty assessment in hydrology.Hydrological Processes: An International Journal2162007, 841--845
• 32 articleAnalytical Modeling of Wind Farms: A New Approach for Power Prediction.Energies99, 7412016
• 33 articleAmbient air pollution: A global assessment of exposure and burden of disease.2016
• 34 articleCollisional Aggregation Due to Turbulence.Annual Review of Condensed Matter Physics712016, 141-170
• 35 articleAerodynamic Aspects of Wind Energy Conversion.Annual Review of Fluid Mechanics4312011, 427-448
• 36 articleTURBULENT RELATIVE DISPERSION.Annual Review of Fluid Mechanics3312001, 289-317
• 37 articlePhysics and modelling of turbulent particle deposition and entrainment: Review of a systematic study.International Journal of Multiphase Flow359Special Issue: Point-Particle Model for
Disperse Turbulent Flows2009, 827 - 839
• 38 techreportQUICS - D.1.1 Report on uncertainty frameworks; QUICS - D.4.2 Report on application of uncertainty frameworks, potential improvements.USFD and TUD2017
• 39 articleModel of particle resuspension in turbulent flows.Nuclear Engineering and Design238112008, 2943--2959
• 40 articleAnisotropic Particles in Turbulence.Annual Review of Fluid Mechanics4912017, 249-276
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ML Aggarwal Class 7 Solutions for ICSE Maths Chapter 14 Symmetry Ex 14.1
ML Aggarwal Class 7 Solutions Chapter 14 Symmetry Ex 14.1 for ICSE Understanding Mathematics acts as the best resource during your learning and helps you score well in your exams.
ML Aggarwal Class 7 Solutions for ICSE Maths Chapter 14 Symmetry Ex 14.1
Question 1.
Draw all lines of symmetry, if any, in each of the following figures:
Question 2.
Copy the figures with a punched hole(s) and draw all the axes of symmetry in each of the following:
Question 3.
In the following figure, mark the missing hole(s) in order to make them symmetrical about the dotted line:
Question 4.
In the following figures, the mirror line (line of symmetry) is given as dotted line. Complete each figure by performing reflection in the mirror (dotted) line and name the figure you complete:
Question 5.
Copy the adjoining figure.
Take any one diagonal as a line of symmetry and shade a few more squares to make the figure symmetric about a diagonal. Is there more than one way to do that? Will the figure be symmetric about both
the diagonals?
Question 6.
Draw the reflection of the following figures/letter in the given mirror line shown dotted:
Question 7.
What other names can you give to the line of symmetry of
(i) an isosceles triangle
(ii) rhombus
(iii) circle?
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Frederic Marazzato - Teaching
Linear Algebra and Machine Learning
A project class taught as "flip the classroom" which introduces the students to Machine Learning through the prism of linear algebra.
Reference: P. Wolenski (LSU)
Analysis and Scientific Computing
Class for senior undergrads that contained basics of distributions, Lebesgue and Sobolev spaces, linear PDEs, finite elements and numerical methods for ODEs.
References: G. Stolz (Ecole des Ponts)
I taught calculus III at LSU, calculus I at U of A and its equivalent at Universite Paris Dauphine (France).
Reference: C. Delzell (LSU)
Undergraduate research projects held during the summer. The objective is to get students to work on real life problems and get some experience before applying for jobs.
Reference: P. Wolenski (LSU)
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Processor Comprising Three-Dimensional Memory (3D-M) Array
Patent application title: Processor Comprising Three-Dimensional Memory (3D-M) Array
IPC8 Class: AH03K19177FI
USPC Class: 708503
Class name: Arithmetical operation floating point multiplication
Publication date: 2017-08-17
Patent application number: 20170237440
The present invention discloses a processor comprising three-dimensional memory (3D-M) array (3D-processor). Instead of logic-based computation (LBC), the 3D-processor uses memory-based computation
(MBC). It comprises an array of computing elements, with each computing element comprising an arithmetic logic circuit (ALC) and a 3D-M-based look-up table (3DM-LUT). The ALC performs arithmetic
operations on the LUT data, while the 3DM-LUT is stored in at least one 3D-M array.
A three-dimensional processor (3D-processor), comprising: a semiconductor substrate including transistors thereon; at least a computing element formed on said semiconductor substrate, said computing
element comprising an arithmetic logic circuit (ALC) and a three-dimensional memory (3D-M)-based look-up table (3DM-LUT), wherein said ALC is formed on said semiconductor substrate and configured to
perform at least one arithmetic operation on data from said 3DM-LUT; said 3DM-LUT is stored in at least a 3D-M array, said 3D-M array being stacked above said ALC; said 3D-M array and said ALC are
communicatively coupled by a plurality of contact vias.
The processor according to claim 1, wherein the data stored in said 3DM-LUT is associated with a mathematical function.
The processor according to claim 2, wherein the data stored in said 3DM-LUT includes function values of said mathematical function.
The processor according to claim 2, wherein the data stored in said 3DM-LUT includes derivative values of said mathematical function.
The processor according to claim 2, wherein said mathematical function includes at least a composite function.
The processor according to claim 2, wherein said mathematical function includes at least a special function.
The processor according to claim 1, wherein the data stored in said 3DM-LUT is associated with a mathematical model.
The processor according to claim 7, wherein the data stored in said 3DM-LUT is associated with raw measurement data.
The processor according to claim 7, wherein the data stored in said 3DM-LUT is associated with smoothed measurement data.
The processor according to claim 1, wherein said ALC comprises at least a pre-processing circuit and/or a post-processing circuit.
The processor according to claim 1, wherein said ALC comprises at least an adder, a multiplier, or a multiplier accumulator (MAC).
The processor according to claim 1, wherein said ALC carries out operations on integer numbers, fixed-point numbers, and/or floating-point numbers.
The processor according to claim 1, wherein said 3D-M array is three-dimensional writable memory (3D-W).
The processor according to claim 1, wherein said 3D-M array is a three-dimensional printed memory (3D-P).
The processor according to claim 1, wherein the memory cells in said 3D-M array comprises diodes or diode-like devices.
The processor according to claim 1, wherein the memory cells in said 3D-M array comprises transistors or transistor-like devices.
The processor according to claim 1, wherein said 3D-M array at least partially covers said ALC.
The processor according to claim 1, wherein said 3DM-LUT is stored in first and second 3D-M arrays, wherein said first and second 3D-M arrays are stacked above said semiconductor substrate and at
least partially covers said ALC.
The processor according to claim 18, wherein said second 3D-M array is formed on a same memory level as said first 3D-M array.
The processor according to claim 18, wherein said second 3D-M array is vertically stacked above said first 3D-M array.
This application claims priority from Chinese Patent Application 201610083747.7, filed on Feb. 13, 2016; Chinese Patent Application 201610260845.3, filed on Apr. 22, 2016; Chinese Patent Application
201610289592.2, filed on May 2, 2016; Chinese Patent Application 201710237780.5, filed on Apr. 12, 2017, in the State Intellectual Property Office of the People's Republic of China (CN), the
disclosure of which are incorporated herein by references in their entireties.
BACKGROUND [0002]
1. Technical Field of the Invention
The present invention relates to the field of integrated circuit, and more particularly to processors.
2. Prior Art
Conventional processors use logic-based computation (LBC), which carries out computation primarily with logic circuits (e.g. XOR circuit). Logic circuits are suitable for arithmetic operations (i.e.
addition, subtraction and multiplication), but not for non-arithmetic functions (e.g. elementary functions, special functions). Non-arithmetic functions are computationally hard. Rapid and efficient
realization thereof has been a major challenge.
For the conventional processors, only few basic non-arithmetic functions (e.g. basic algebraic functions and basic transcendental functions) are implemented by hardware and they are referred to as
built-in functions. These built-in functions are realized by a combination of logic circuits and look-up tables (LUT). For example, U.S. Pat. No. 5,954,787 issued to Eun on Sep. 21, 1999 taught a
method for generating sine/cosine functions using LUTs; U.S. Pat. No. 9,207,910 issued to Azadet et al. on Dec. 8, 2015 taught a method for calculating a power function using LUTs.
Realization of built-in functions is further illustrated in FIG. 1A. A conventional processor 300 generally comprises a logic circuit 380 and a memory circuit 370. The logic circuit 380 comprises an
arithmetic logic unit (ALU) for performing arithmetic operations, while the memory circuit 370 stores an LUT for the built-in function. To obtain a desired precision, the built-in function is
approximated to a polynomial of a sufficiently high order. The LUT 370 stores the coefficients of the polynomial; and the ALU 380 calculates the polynomial. Because the ALU 380 and the LUT 370 are
formed side-by-side on a semiconductor substrate 0, this type of horizontal integration is referred to as two-dimensional (2-D) integration.
Computation has been developed along the directions of computational density and computational complexity. The computational density is a figure of merit for parallel computation and it refers to the
computational power (e.g. the number of floating-point operations per second) per die area. The computational complexity is a figure of merit for scientific computation and it refers to the total
number of built-in functions supported by a processor. The 2-D integration severely limits computational density and computational complexity.
For the 2-D integration, inclusion of the LUT 370 increases the die size of the conventional processor 300 and lowers its computational density. This has an adverse effect on parallel computation.
Moreover, because the ALU 380 is the primary component of the conventional processor 300 and occupies a large die area, the LUT 370 is left with a small die area and only supports few built-in
functions. FIG. 1B lists all built-in transcendental functions supported by an Intel Itanium (IA-64) processor (referring to Harrison et al. "The Computation of Transcendental Functions on the IA-64
Architecture", Intel Technical journal, Q4 1999, hereinafter Harrison). The IA-64 processor supports a total of 7 built-in transcendental functions, each using a relatively small LUT (from 0 to 24
kb) in conjunction with a relatively high-degree Taylor-series calculation (from 5 to 22).
This small set of built-in functions (.about.10 types, including arithmetic operations) is the foundation of scientific computation. Scientific computation uses advanced computing capabilities to
advance human understandings and solve engineering problems. It has wide applications in computational mathematics, computational physics, computational chemistry, computational biology,
computational engineering, computational economics, computational finance and other computational fields. The prevailing framework of scientific computation comprises three layers: a foundation
layer, a function layer and a modeling layer. The foundation layer includes built-in functions that can be implemented by hardware. The function layer includes mathematical functions that cannot be
implemented by hardware (e.g. non-basic non-arithmetic functions). The modeling layer includes mathematical models, which are the mathematical descriptions of the input-output characteristics of a
system component.
The mathematical functions in the function layer and the mathematical models in the modeling layer are implemented by software. The function layer involves one software-decomposition step:
mathematical functions are decomposed into combinations of built-in functions by software, before these built-in functions and the associated arithmetic operations are calculated by hardware. The
modeling layer involves two software-decomposition steps: the mathematical models are first decomposed into combinations of mathematical functions; then the mathematical functions are further
decomposed into combinations of built-in functions. Apparently, the software-implemented functions (e.g. mathematical functions, mathematical models) run much slower and less efficient than the
hardware-implemented functions (i.e. built-in functions), and extra software-decomposition steps (e.g. for mathematical models) would make these performance gaps even more pronounced.
To illustrate how computationally intensive a mathematical model could be, FIGS. 2A-2B disclose a simple example--the simulation of an amplifier circuit 20. The amplifier circuit 20 comprises a
transistor 24 and a resistor 22 (FIG. 2A). All transistor models (e.g. MOS3, BSIM3 V3.2, BSIM4 V3.0, PSP of FIG. 2B) model the transistor behaviors based on the small set of built-in functions
provided by the conventional processor 300. Due to the limited choice of the built-in functions, calculating even a single current-voltage (I-V) point for the transistor 24 requires a large amount of
computation (FIG. 2B). As an example, the BSIM4 V3.0 transistor model needs 222 additions, 286 multiplications, 85 divisions, 16 square-root operations, 24 exponential operations, and 19 logarithmic
operations. This large amount of computation makes simulation extremely slow and inefficient.
Objects and Advantages [0013]
It is a principle object of the present invention to provide a paradigm shift for scientific computation.
It is a further object of the present invention to provide a processor with improved computational complexity.
It is a further object of the present invention to provide a processor with a large set of built-in functions.
It is a further object of the present invention to realize non-arithmetic functions rapidly and efficiently.
It is a further object of the present invention to realize rapid and efficient modeling and simulation.
It is a further object of the present invention to provide a processor with improved computational density.
In accordance with these and other objects of the present invention, the present invention discloses a processor comprising three-dimensional memory (3D-M) arrays (3D-processor). Instead of
logic-based computation (LBC), the 3D-processor uses memory-based computation (MBC).
SUMMARY OF THE INVENTION [0020]
The present invention discloses a processor comprising three-dimensional memory (3D-M) array (3D-processor). It comprises an array of computing elements formed on a semiconductor substrate, with each
computing element comprising an arithmetic logic circuit (ALC) and a look-up table (LUT) based on 3D-M (3DM-LUT). The ALC is formed on the substrate and it performs arithmetic operations on the
3DM-LUT data. The 3DM-LUT is stored in at least a 3D-M array. The 3D-M array is stacked above the ALC and at least partially covers the ALC. The 3D-M array is further communicatively coupled with the
ALC with the contact vias. These contact vias are collectively referred to as 3-D interconnects.
The present invention further discloses a memory-based computation (MBC), which carries out computation primarily with the 3DM-LUT. Compared with the conventional logic-based computation (LBC), the
3DM-LUT used by the MBC has a much larger capacity than the conventional LUT. Although arithmetic operations are still performed for most MBCs, using a larger LUT as a starting point, the MBC only
needs to calculate a polynomial to a smaller order. For the MBC, the fraction of computation done by the 3DM-LUT could be more than the ALC.
Because the 3DM-LUT is stacked above the ALC, this type of vertical integration is referred to as three-dimensional (3-D) integration. The 3-D integration has a profound effect on the computational
density. Because the 3D-M array does not occupy any substrate area, the footprint of the computing element is roughly equal to that of the ALC. However, the footprint of a conventional processor is
roughly equal to the sum of the footprints of the LUT and the ALU. By moving the LUT from aside to above, the computing element becomes smaller. The 3D-processor would contain more computing
elements, become more computationally powerful and support massive parallelism.
The 3-D integration also has a profound effect on the computational complexity of the 3D-processor. For a conventional processor, the total LUT capacity is less than 100 kb. In contrast, the total
3DM-LUT capacity for a 3D-processor could reach 100 Gb (for example, a 3D-XPoint die has a storage capacity of 128 Gb). Consequently, a single 3D-processor die could support as many as 10,000
built-in functions, which are three orders of magnitude more than the conventional processor.
Significantly more built-in functions shall flatten the prevailing framework of scientific computation (including the foundation, function and modeling layers). The hardware-implemented functions,
which were only available to the foundation layer, now become available to the function and modeling layers. Not only mathematical functions in the function layer can be directly realized by
hardware, but also mathematical models in the modeling layer can be directly described by hardware. In the function layer, mathematical functions can be realized by a function-by-LUT method, i.e. the
function values are calculated by reading the 3DM-LUT plus polynomial interpolation. In the modeling layer, mathematical models can be described by a model-by-LUT method, i.e. the input-output
characteristics of a system component are modeled by reading the 3DM-LUT plus polynomial interpolation. Rapid and efficient computation would lead to a paradigm shift for scientific computation.
Accordingly, the present invention discloses a three-dimensional processor (3D-processor), comprising: a semiconductor substrate including transistors thereon; at least a computing element formed on
said semiconductor substrate, said computing element comprising an arithmetic logic circuit (ALC) and a three-dimensional memory (3D-M)-based look-up table (3DM-LUT), wherein said ALC is formed on
said semiconductor substrate and configured to perform at least one arithmetic operation on data from said 3DM-LUT; said 3DM-LUT is stored in at least a 3D-M array, said 3D-M array being stacked
above said ALC; said 3D-M array and said ALC are communicatively coupled by a plurality of contact vias.
BRIEF DESCRIPTION OF THE DRAWINGS [0026]
FIG. 1A is a schematic view of a conventional processor (prior art); FIG. 1B lists all transcendental functions supported by an Intel Itanium (IA-64) processor (prior art);
FIG. 2A is a circuit block diagram of an amplifier circuit; FIG. 2B lists number of operations to calculate a current-voltage (I-V) point for various transistor models (prior art);
FIG. 3A is a block diagram of a preferred 3D-processor; FIG. 3B is a block diagram of a preferred computing element;
FIGS. 4A-4C are the block diagrams of three preferred ALC;
FIG. 5A is a cross-sectional view of a preferred computing element comprising at least a three-dimensional writable memory (3D-W) array; FIG. 5B is a cross-sectional view of a preferred computing
element comprising at least a three-dimensional printed memory (3D-P) array; FIG. 5C is a perspective view of a preferred computing element;
FIG. 6A is a schematic view of a 3D-M cell comprising a diode or a diode-like device; FIG. 6B is a schematic view of a 3D-M cell comprising a transistor or a transistor-like device;
FIGS. 7A-7C are the substrate layout views of three preferred 3D-processors;
FIG. 8A is a block diagram of a first preferred computing element; FIG. 8B is its substrate layout view; FIG. 8C is a detailed circuit diagram of the first preferred computing element;
FIG. 9A is a block diagram of a second preferred computing element; FIG. 9B is its substrate-circuit layout view;
FIG. 10A is a block diagram of a third preferred computing element; FIG. 10B is its substrate-circuit layout view.
It should be noted that all the drawings are schematic and not drawn to scale. Relative dimensions and proportions of parts of the device structures in the figures have been shown exaggerated or
reduced in size for the sake of clarity and convenience in the drawings. The same reference symbols are generally used to refer to corresponding or similar features in the different embodiments. The
symbol "/" means a relationship of "and" or "or".
Throughout the present invention, the phrase "memory" is used in its broadest sense to mean any semiconductor-based holding place for information, either permanent or temporary; the phrase
"permanent" is used in its broadest sense to mean any long-term storage; the phrase "communicatively coupled" is used in its broadest sense to mean any coupling whereby information may be passed from
one element to another element; the phrase "on the substrate" means the active elements of a circuit (e.g. transistors) are formed on the surface of the substrate, although the interconnects between
these active elements are formed above the substrate and do not touch the substrate; the phrase "above the substrate" means the active elements (e.g. memory cells) are formed above the substrate and
do not touch the substrate.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS [0038]
Those of ordinary skills in the art will realize that the following description of the present invention is illustrative only and is not intended to be in any way limiting. Other embodiments of the
invention will readily suggest themselves to such skilled persons from an examination of the within disclosure.
Referring now to FIG. 3A-3B, a preferred processor 100 comprising a three-dimensional memory (3D-M) array (3D-processor) is disclosed. The preferred 3D-processor 100 comprises an array of computing
elements 110-1, 110-2 . . . 110-i . . . 110-N (FIG. 3A). The computing elements 110-1 . . . 110-N could realize a same function or different functions. Each computing element 110-i could have one or
more input variables 150, and one or more output variables 190 (FIG. 3B). Each computing element 110-i comprises an arithmetic logic circuit (ALC) 180 and a look-up table (LUT) based on 3D-M
(3DM-LTU) 170. The ALC 180 performs arithmetic operations on the 3DM-LUT data, while the 3DM-LUT 170 is stored in at least a 3D-M array. The 3DM-LUT may possess a two-dimensional structure (e.g. the
function represented by the 3DM-LUT has one input variable and one output value), or a multi-dimensional structure (e.g. the function represented by the 3DM-LUT has two input variables and one output
value). The ALC 180 and the 3DM-LUT 170 are communicatively coupled by 3D-interconnects 160. Because the 3D-M array 170 is formed on a different level than the ALC 180 (shown in FIGS. 5A-5C), it is
represented by dotted line in this and following figures.
The 3D-processor 100 uses memory-based computation (MBC), which carries out computation primarily with the 3DM-LUT 170. Compared with the conventional logic-based computation (LBC), the 3DM-LUT 170
used by the MBC has a much larger capacity than the conventional LUT 370. Although arithmetic operations are still performed for most MBCs, using a larger LUT as a starting point, the MBC only needs
to calculate a polynomial to a smaller order. For the MBC, the fraction of computation done by the 3DM-LUT 170 could be more than the ALC 180.
FIGS. 4A-4C are the block diagrams of three preferred ALC 180. The first preferred ALC 180 comprises an adder 180A, the second preferred ALC 180 comprises a multiplier 180M, with the third preferred
ALC 180 comprising a multiplier-accumulator (MAC), which includes an adder 180A and a multiplier 180M. The preferred ALC 180 could perform integer arithmetic operations, fixed-point arithmetic
operations, or floating-point arithmetic operations. To those skilled in the art, besides the above arithmetic circuits, the preferred ALC 180 may also comprise memory circuits, e.g. registers,
flip-flops, buffer RAMs.
Referring now to FIGS. 5A-5C, the computing element 110-i comprising different types of 3D-M are disclosed. 3D-M was disclosed in U.S. Pat. No. 5,835,396 issued to Zhang on Nov. 10, 1998. It
comprises a plurality of vertically stacked memory levels formed on a semiconductor substrate, with each memory level comprising a plurality of 3D-M arrays. Each 3D-M array is a collection of 3D-M
cells in a memory level that share at least one address-line.
3D-M can be categorized into 3D-RAM (random access memory) and 3D-ROM (read-only memory). As used herein, the phrase "RAM" is used in its broadest sense to mean any memory for temporarily holding
information, including but not limited to registers, SRAM, and DRAM; the phrase "ROM" is used in its broadest sense to mean any memory for permanently holding information, wherein the information
being held could be either electrically alterable or un-alterable. Most common 3D-M is 3D-ROM. The 3D-ROM is further categorized into 3-D writable memory (3D-W) and 3-D printed memory (3D-P).
For the 3D-W, data can be electrically written (or, programmable). Based on the number of programmings allowed, a 3D-W can be categorized into three-dimensional one-time-programmable memory (3D-OTP)
and three-dimensional multiple-time-programmable memory (3D-MTP). The 3D-OTP can be written once, while the 3D-MTP is electrically re-programmable. An exemplary 3D-MTP is 3D-XPoint. Other types of
3D-MTP include memristor, resistive random-access memory (RRAM or ReRAM), phase-change memory, programmable metallization cell (PMC), conductive-bridging random-access memory (CBRAM), and the like.
For the 3D-W, the 3DM-LUT 170 can be configured in the field. This becomes even better when the 3D-MTP is used, as the 3DM-LUT 170 would become re-configured.
For the 3D-P, data are recorded thereto using a printing method during manufacturing. These data are fixedly recorded and cannot be changed after manufacturing. The printing methods include
photo-lithography, nano-imprint, e-beam lithography, DUV lithography, and laser-programming, etc. An exemplary 3D-P is three-dimensional mask-programmed read-only memory (3D-MPROM), whose data are
recorded by photo-lithography. Because electrical programming is not required, a memory cell in the 3D-P can be biased at a larger voltage during read than the 3D-W and therefore, the 3D-P is faster
than the 3D-W.
FIG. 5A discloses a preferred computing element 110-i comprising at least a 3D-W array. It comprises a substrate circuit 0K formed on the substrate 0. The ALC 180 is a portion of the substrate
circuit 0K. A first memory level 16A is stacked above the substrate circuit 0K, with a second memory level 16B stacked above the first memory level 16A. The substrate circuit 0K includes the
peripheral circuits of the memory levels 16A, 16B. It comprises transistors 0t and the associated interconnect 0M. Each of the memory levels (e.g. 16A, 16B) comprises a plurality of first
address-lines (i.e. y-lines, e.g. 2a, 4a), a plurality of second address-lines (i.e. x-lines, e.g. 1a, 3a) and a plurality of 3D-W cells (e.g. 6aa). The first and second memory levels 16A, 16B are
coupled to the ALC 180 through contact vias 1 av, 3av, respectively. The LUTs stored in all 3D-M arrays coupled to the ALC 180 are collectively referred to as the 3DM-LUT 170. Coupling the 3DM-LUT
170 with the ALC 180, the contact vias 1 av, 3av are collectively referred to as 3D-interconnects 160.
The 3D-W cell 5aa comprises a programmable layer 12 and a diode layer 14. The programmable layer 12 could be an antifuse layer (which can be programmed once and is used for the 3D-OTP) or a
re-programmable layer (which is used for the 3D-MTP). The diode layer 14 is broadly interpreted as any layer whose resistance at the read voltage is substantially lower than when the applied voltage
has a magnitude smaller than or polarity opposite to that of the read voltage. The diode could be a semiconductor diode (e.g. p-i-n silicon diode), or a metal-oxide (e.g. TiO.sub.2) diode.
FIG. 5B discloses a preferred computing element 110-i comprising at least a 3D-P array. It has a structure similar to that of FIG. 5A except for the memory cells. 3D-P has at least two types of
memory cells: a high-resistance 3D-P cell 5aa, and a low-resistance 3D-P cell 5ac. The low-resistance 3D-P cell 5ac comprises a diode layer 14, while the high-resistance 3D-P cell 5aa comprises at
least a high-resistance layer 13. The diode layer 14 is similar to that in the 3D-W. The high-resistance layer 13, on the other hand, could simply be a layer of insulating dielectric (e.g. silicon
oxide, or silicon nitride). It is physically removed at the location of the low-resistance 3D-P cell 5ac during manufacturing.
FIG. 5C is a perspective view of the preferred computing element 110-i. The ALC 180 is formed on the substrate 0. The 3DM-LUT 170 is vertically stacked above and at least partially covers the ALC
180. The 3-D integration moves the 3DM-LUT 170 physically close to the ALC 180. Because the contact vias 1 av, 3av coupling them are short (on the order of an um in length) and numerous (thousands at
least), the 3D-interconnects 160 have a much larger bandwidth than the conventional processor 300. As the 2-D integration places the ALU 380 and the LUT 370 side-by-side on the substrate 0, the
interconnects coupling them are much longer (hundreds of ums in length) and fewer (hundreds at most).
FIGS. 6A-6B show two types of the preferred 3D-M cell 5ab. In the preferred embodiment of FIG. 6A, the 3D-M cell 5ab comprises a variable resistor 12 and a diode (or a diode-like device) 14. The
variable resistor 12 is realized by the programmable layer of FIG. 5A. It can be varied during manufacturing or after manufacturing. The diode (or diode-like device) 14 is realized by the diode layer
of FIG. 5A. It is broadly interpreted as any two-terminal device whose resistance at the read voltage is substantially lower than when the applied voltage has a magnitude smaller than or polarity
opposite to that of the read voltage.
In the preferred embodiment of FIG. 6B, the 3D-M cell 5ab comprises a transistor or a transistor-like device 16. The transistor or transistor-like device 16 is broadly interpreted as any three- (or,
more-) terminal device whose resistance between the first and second terminals can be modulated by an electrical signal on a third terminal. In this preferred embodiment, the device 16 further
comprises a floating gate 18 for storing electrical charge which represents the digital information stored in the 3D-M cell 5ab. To those skilled in the art, the devices 16 can be organized into
NOR-arrays or NAND-arrays. Depending on the direction of the current flow between the first and second terminals in the devices 16, the 3D-M could be categorized into horizontal 3D-M (e.g. 3D-XPoint)
and vertical 3D-M (e.g. 3D-NAND).
Referring now to FIGS. 7A-7C, the substrate layout views of three preferred computing elements 110-i are shown. In the embodiment of FIG. 7A, the ALC 180 is only coupled with a single 3D-M array 170o
and processes the 3DM-LUT data therefrom. The 3DM-LUT 170 is stored in the 3D-M array 170o. The ALC 180 is covered by the 3D-M array 170. The 3D-M array 170o has four peripheral circuits, including
X-decoders 15o, 15o' and Y-decoders 17o, 17o'. The ALC 180 is bound by these four peripheral circuits. As the 3D-M array is stacked above the substrate circuit 0K and does not occupy any substrate
area, its projection on the substrate 0 is shown by dotted lines in this and following figures.
In the embodiment of FIG. 7B, the ALC 180 is coupled with four 3D-M arrays 170a-170d and processes the 3DM-LUT data therefrom. The 3DM-LUT 170 is stored in four 3D-M arrays 170a-170d. Different from
FIG. 7A, each 3D-M array (e.g. 170a) has two peripheral circuits (e.g. X-decoder 15a and Y-decoder 17a). The ALC 180 is bound by eight peripheral circuits (including X-decoders 15a-15d and Y-decoders
17a-17d) and located below four 3D-M arrays 170a-170d. Apparently, the ALC 180 of FIG. 7B could be four times as large as that of FIG. 7A.
In the embodiment of FIG. 7C, the ALC 180 is coupled with eight 3D-M arrays 170a-170d, 170w-170z and processes the 3DM-LUT data therefrom. The 3DM-LUT 170 is stored in eight 3D-M arrays 170a-170d,
170w-170z. These 3D-M arrays are divided into two sets: a first set 150a includes four 3D-M arrays 170a-170d, and a second set 150b includes four 3D-M arrays 170w-170z. Below the four 3D-M arrays
170a-170d of the first set 150a, a first component 180a of the ALC 180 is formed. Similarly, below the four 3D-M array 170w-170z of the second set 150b, a second component 180b of the ALC 180 is
formed. In this embodiment, adjacent peripheral circuits (e.g. adjacent x-decoders 15a, 15c, or, adjacent y-decoders 17a, 17b) are separated by physical gaps G. These physical gaps allow the
formation of the routing channel 182, 184, 186, which provide coupling between different components 180a, 180b, or between different ALCs 180a, 180b. Apparently, the ALC 180 of FIG. 7C could be eight
times as large as that of FIG. 7A.
Because the 3DM-LUT 170 is stacked above the ALC 180, this type of vertical integration is referred to as three-dimensional (3-D) integration. The 3-D integration has a profound effect on the
computational density of the 3D-processor 100. Because the 3DM-LUT 170 does not occupy any substrate area 0, the footprint of the computing element 110-i is roughly equal to that of the ALC 180. This
is much smaller than a conventional processor 300, whose footprint is roughly equal to the sum of the footprints of the LUT 370 and the ALC 380. By moving the LUT from aside to above, the computing
element becomes smaller. The 3D-processor 100 would contain more computing elements 110-1, become more computationally powerful and support massive parallelism.
The 3-D integration also has a profound effect on the computational complexity of the 3D-processor 100. For a conventional processor 300, the total LUT capacity is less than 100 kb. In contrast, the
total 3DM-LUT capacity for a 3D-processor 100 could reach 100 Gb (for example, a 3D-XPoint die has a storage capacity of 128 Gb). Consequently, a single 3D-processor die 100 could support as many as
10,000 built-in functions, which are three orders of magnitude more than the conventional processor 300.
Significantly more built-in functions shall flatten the prevailing framework of scientific computation (including the foundation, function and modeling layers). The hardware-implemented built-in
functions, which were only available to the foundation layer, now become available to the function and modeling layers. Not only mathematical functions in the function layer can be directly realized
by hardware (FIGS. 8A-9B), but also mathematical models in the modeling layer can be directly described by hardware (FIGS. 10A-10B). In the function layer, mathematical functions can be realized by a
function-by-LUT method, i.e. the function values are calculated by reading the 3DM-LUT plus polynomial interpolation. In the modeling layer, mathematical models can be described by a model-by-LUT
method, i.e. the input-output characteristics of a system component are modeled by reading the 3DM-LUT plus polynomial interpolation. Rapid and efficient computation would lead to a paradigm shift
for scientific computation.
Referring now to FIGS. 8A-8C, a first preferred computing element 110-i implementing a built-in function Y=f(X) is disclosed. It uses the function-by-LUT method. FIG. 8A is its circuit block diagram.
The ALC 180 comprises a pre-processing circuit 180R, a 3DM-LUT 170P, and a post-processing circuit 180T. The pre-processing circuit 180R converts the input variable (X) 150 into an address (A) of the
3DM-LUT 170P. After the data (D) at the address (A) is read out from the 3DM-LUT 170P, the post-processing circuit 180T converts it into the function value (Y) 190. A residue (R) of the input
variable (X) is fed into the post-processing circuit 180T to improve the calculation precision.
FIG. 8B is its substrate-circuit layout view. The 3D-M storing the 3DM-LUT 170P comprises at least a 3D-M array 170p, as well as its X-decoder 15p and Y-decoder 17p. The 3D-M array 170p covers the
pre-processing circuit 180R and the post-processing circuit 180T. Although a single 3D-M array 170p is shown in this figure, the preferred embodiment could use multiple 3D-M arrays, as those shown in
FIGS. 7B-7C. Because the 3DM-LUT 170 does not occupy any substrate area, the 3-D integration between the 3DM-LUT 170 and the ALC 180 (including the pre-processing circuit 180R and the post-processing
circuit 180T) leads to a smaller footprint for the computing element 110-i.
FIG. 8C discloses the first preferred computing element 110-i which realizes a single-precision built-in function Y=f(X). The input variable X 150 has 32 bits (x.sub.31 . . . x.sub.0). The
pre-processing circuit 180R extracts the higher 16 bits (x.sub.31 . . . x.sub.16) thereof and sends it as a 16-bit address A to the 3DM-LUT 170P. The pre-processing circuit 180R further extracts the
lower 16 bits (x.sub.15 . . . x.sub.0) and sends it as a 16-bit residue R to the post-processing circuit 180T. The 3DM-LUT 170P comprises two 3DM-LUTs 170Q, 170R. Both 3DM-LUTs 170Q, 170R have 2 Mb
capacities (16-bit input and 32-bit output): the 3DM-LUT 170Q stores the function value D1=f(A), while the 3DM-LUT 170R stores the first-order derivative value D2=f'(A). The post-processing circuit
180T comprises a multiplier 180M and an adder 180A. The output value (Y) 190 has 32 bits and is calculated from polynomial interpolation. In this case, the polynomial interpolation is a first-order
Taylor series: Y(X)=D1+D2*R==f(A)+f'(A)*R. To those skilled in the art, higher-order polynomial interpolation (e.g. higher-order Taylor series) can be used to improve the calculation precision.
When calculating a built-in function, combining the LUT with polynomial interpolation can achieve a high precision without using an excessively large LUT. For example, if only LUT (without any
polynomial interpolation) is used to realize a single-precision function (32-bit input and 32-bit output), it would have a capacity of 2.sup.32*32=128 Gb, which is impractical. By including
polynomial interpolation, significantly smaller LUTs can be used. In the above embodiment, a single-precision function can be realized using a total of 4 Mb LUT (2 Mb for function values, and 2 Mb
for first-derivative values) in conjunction with a first-order Taylor series calculation. This is significantly less than the LUT-only approach (4 Mb vs. 128 Gb).
Besides elementary functions, the preferred embodiment of FIG. 8C can be used to implement non-elementary functions such as special functions. Special functions can be defined by means of power
series, generating functions, infinite products, repeated differentiation, integral representation, differential difference, integral, and functional equations, trigonometric series, or other series
in orthogonal functions. Important examples of special functions are gamma function, beta function, hyper-geometric functions, confluent hyper-geometric functions, Bessel functions, Legrendre
functions, parabolic cylinder functions, integral sine, integral cosine, incomplete gamma function, incomplete beta function, probability integrals, various classes of orthogonal polynomials,
elliptic functions, elliptic integrals, Lame functions, Mathieu functions, Riemann zeta function, automorphic functions, and others. The 3D-processor will simplify the calculation of special
functions and promote their applications in scientific computation.
Referring now to FIGS. 9A-9B, a second preferred computing element 110-i implementing a composite function Y=exp[K*log(X)]=X.sup.K is disclosed. It uses the function-by-LUT method. FIG. 9A is its
schematic circuit block diagram. The preferred computing element 110-i comprises two 3DM-LUTs 170S, 170T and a multiplier 180M. The 3DM-LUT 170S stores the Log( ) values, while the 3DM-LUT 170T
stores the Exp( ) values. The input variable X is used as an address 150 for the 3DM-LUT 170S. The output Log(X) 160a from the 3DM-LUT 170S is multiplied by an exponent parameter K at the multiplier
180M. The multiplication result K*Log(X) is used as an address 160b for the 3DM-LUT 170T, whose output 190 is Y=X.sup.K.
FIG. 9B is its substrate-circuit layout view. The substrate circuit 0K comprises the X-decoders 15s, 15t and the Y-decoders 17s, 17t for the 3D-M arrays 170s, 170t, as well as a multiplier 180M.
Placed side-by-side, both 3D-M arrays 170s, 170t partially cover the multiplier 180M. Note that both embodiments in FIG. 8C and FIG. 9A comprise two 3DM-LUTs. These 3DM-LUTs could be stored in a
single 3D-M array 170p (as in FIG. 8B), in two 3D-M arrays 170s, 170t placed side-by-side (as in FIG. 9B), or in two vertically stacked 3D-M arrays (i.e. on different memory levels 16A, 16B, as in
FIGS. 5A-5C). Apparently, the 3DM-LUT can be stored in more 3D-M arrays.
Referring now to FIGS. 10A-10B, a third preferred computing element 110-i to simulate the amplifier circuit 20 of FIG. 2A is disclosed. It uses the model-by-LUT method. FIG. 10A is its schematic
circuit block diagram. The preferred computing element 110-i comprises a 3DM-LUT 170U, an adder 180A and a multiplier 180M. The 3DM-LUT 170U stores the data associated with the behaviors (e.g.
input-output characteristics) of the transistor 24. By using the input voltage value (V.sub.IN) as an address 150 for the 3DM-LUT 170U, the readout 160 of the 3DM-LUT 170U is the drain-current value
(I.sub.D). After the I.sub.D value is multiplied with the minus resistance value (-R) of the resistor 22 by the multiplier 180M, the multiplication result (-R*I.sub.D) is added to the V.sub.DD value
by the adder 180A to generate the output voltage value (V.sub.OUT) 190.
The 3DM-LUT 170U stores different forms of mathematical models. In one case, the mathematical model data stored in the 3DM-LUT 170U is raw measurement data, i.e. the measured input-output
characteristics of the transistor 24. One example is the measured drain current vs. the applied gate-source voltage (I.sub.D-V.sub.GS) characteristics. In another case, the mathematical model data
stored in the 3DM-LUT 170U is the smoothed measurement data. The raw measurement data could be smoothed using a purely mathematical method (e.g. a best-fit model). Or, this smoothing process can be
aided by a physical transistor model (e.g. a BSIM4 V3.0 transistor model). In a third case, the mathematical data stored in the 3DM-LUT include not only the measured data, but also its derivative
values. For example, the 3DM-LUT data include not only the drain-current values of the transistor 24 (e.g. the I.sub.D-V.sub.GS characteristics), but also its transconductance values (e.g. the
G.sub.m-V.sub.GS characteristics). With derivative values, polynomial interpolation can be used to improve the modeling precision using a reasonable-size 3DM-LUT, as in the case of FIG. 8C.
FIG. 10B is its substrate-circuit layout view. The substrate circuit 0K comprises the X-decoder 15u and the Y-decoder 17u for the 3D-M array 170u, as well as the multiplier 180M and the adder 180A.
The 3D-M array 170u covers the multiplier 180M and the adder 180A. Although a single 3D-M array 170u is shown in this figure, the preferred embodiment could use multiple 3D-M arrays 170u, as those
shown in FIGS. 7B-7C.
Model-by-LUT offers many advantages. By skipping two software-decomposition steps (from mathematical models to mathematical functions, and from mathematical functions to built-in functions), it saves
substantial modeling time and energy. Model-by-LUT may need less LUT than function-by-LUT. Because a transistor model (e.g. BSIM4 V3.0) has hundreds of model parameters, calculating the intermediate
functions of the transistor model requires extremely large LUTs. However, if we skip function-by-LUT (namely, skipping the transistor models and the associated intermediate functions), the transistor
behaviors can be described using only three parameters (including the gate-source voltage V.sub.GS, the drain-source voltage V.sub.DS, and the body-source voltage V.sub.BS). Describing the
mathematical models of the transistor 24 requires relatively small LUTs.
While illustrative embodiments have been shown and described, it would be apparent to those skilled in the art that many more modifications than that have been mentioned above are possible without
departing from the inventive concepts set forth therein. For example, the processor could be a micro-controller, a central processing unit (CPU), a digital signal processor (DSP), a graphic
processing unit (GPU), a network-security processor, an encryption/decryption processor, an encoding/decoding processor, a neural-network processor, or an artificial intelligence (AI) processor.
These processors can be found in consumer electronic devices (e.g. personal computers, video game machines, smart phones) as well as engineering and scientific workstations and server machines. The
invention, therefore, is not to be limited except in the spirit of the appended claims.
User Contributions:
Comment about this patent or add new information about this topic:
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Covariate Matching
CovMatch {DSWE} R Documentation
Covariate Matching
The function aims to take list of two data sets and returns the after matched data sets using user specified covariates and threshold
CovMatch(data, xCol, xCol.circ, thrs, priority)
data a list, consisting of data sets to match, also each of the individual data set can be dataframe or a matrix
xCol a vector stating the column position of covariates used
xCol.circ a vector stating the column position of circular variables
a numerical or a vector of threshold values for each covariates, against which matching happens It should be a single value or a vector of values representing threshold for each of the
thrs covariate
priority a boolean, default value False, otherwise computes the sequence of matching
a list containing :
• originalData - The data sets provided for matching
• matchedData - The data sets after matching
• MinMaxOriginal - The minimum and maximum value in original data for each covariate used in matching
• MinMaxMatched - The minimum and maximum value in matched data for each covariates used in matching
Ding, Y. (2019). Data Science for Wind Energy. Chapman & Hall, Boca Raton, FL.
data1 = data1[1:100, ]
data2 = data2[1:100, ]
data = list(data1, data2)
xCol = 2
xCol.circ = NULL
thrs = 0.1
priority = FALSE
matched_data = CovMatch(data, xCol, xCol.circ, thrs, priority)
version 1.8.2
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Still confused about Reaper XP [Archive] - Dungeons & Dragons Online Forums
02-06-2022, 10:00 AM
I've read every post I can find and referenced the ddowiki, but I'm still not getting the basic math here.
I'm only dealing with Epic level quests here trying to understand the equations involved.
The basic formula 50 + (3 x Base Level of Quest x Number of Skulls) works just fine for predicting the Base XP listed in my XP window. No problems there.
But how does that number get converted into the final Total RXP listed at the bottom of the XP window?
Prior to any kill/breakable/etc. bonuses, prior to any deaths or optional completions, just and only just after the first 10 things in a dungeons are killed and the RXP shows up.......what is the
math that leads from point A to Point B?
A VIP Level 26 character enters a Base Level 22 Long Quest on Reaper 1, kills 10 things and has a base RXP of 116 and a total of 443. 10% tome bonus showing
The same character and another VIP Level 28 character enter the same quest. Base RXP of 116 and a total of 444 on each. 15% tome bonus showing on the second character
So.....tome bonuses don't figure in - at least at this level of the math.
And no level spread penalties (at least in Epics).
One character recalls. Total drops to 443. That would be the VIP 1% grouping bonus. So that's in there, but I'm having trouble making any other combination of bonuses work to get the base up to the
Any insight/information out there folks?
Edit: It was a level 22 Base level quest, fixed.
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Geodetector method
Spatial stratified heterogeneity (SSH), referring to the within strata are more similar than the between strata, such as landuse types and climate zones, is ubiquitous in spatial data. SSH instead of
random is a set of information, which has been being a window for humans to understand the nature since Aristotle time. In another aspect, a model with global parameters would be confounded if input
data is SSH, the problem dissolves if SSH is identified so simple models can be applied to each stratum separately. Note that the “spatial” here can be either geospatial or the space in mathematical
Geodetector is a novel tool to investigate SSH: (1) measure and find SSH of a variable Y ; (2) test the power of determinant X of a dependent variable Y according to the consistency between their
spatial distributions; and (3) investigate the interaction between two explanatory variables X[1] and X[2] to a dependent variable Y. All of the tasks are implementable by the geographical detector q
-statistic: \[$$q=1- \frac{1}{N\sigma^2}\sum_{h=1}^{L}N_h\sigma_h^2$$\]
where N and σ^2 stand for the number of units and the variance of Y in study area, respectively; the population Y is composed of L strata (h = 1, 2, …, L), N[h] and σ[h]^2 stand for the number of
units and the variance of Y in stratum h, respectively. The strata of Y (red polygons in Figure 1) are a partition of Y, either by itself ( h(Y) in Figure 1) or by an explanatory variable X which is
a categorical variable ( h(Y) in Figure 1). X should be stratified if it is a numerical variable, the number of strata L might be 2-10 or more, according to prior knowledge or a classification
(Notation: Yi stands for the value of a variable Y at a sample unit i ; h(Y) represents a partition of Y ; h(X) represents a partition of an explanatory variable X. In geodetector, the terms
“stratification”, “classification” and “partition” are equivalent.)
Interpretation of q value (please refer to Fig.1). The value of q ∈ [0, 1].
If Y is stratified by itself h(Y), then q = 0 indicates that Y is not SSH; q = 1 indicates that Y is SSH perfectly; the value of q indicates that the degree of SSH of Y is q.
If Y is stratified by an explanatory variable h(X), then q = 0 indicates that there is no association between Y and X ; q = 1 indicates that Y is completely determined by X ; the value of q-statistic
indicates that X explains 100q% of Y. Please notice that the q-statistic measures the association between X and Y, both linearly and nonlinearly.
For more detail of Geodetector method, please refer:
[1] Wang JF, Li XH, Christakos G, Liao YL, Zhang T, Gu X, Zheng XY. Geographical detectors-based health risk assessment and its application in the neural tube defects study of the Heshun Region,
China. International Journal of Geographical Information Science, 2010, 24(1): 107-127.
[2] Wang JF, Zhang TL, Fu BJ. A measure of spatial stratified heterogeneity. Ecological Indicators,2016, 67(2016): 250-256.
[3] Wang JF, Xu CD. Geodetector:Principle and prospective. Geographica Sinica,2017,72(1):116-134.
R package for geodetector
geodetector package includes five functions: factor_detector, interaction_detector, risk_detector, ecological_detector and geodetector. The first four functions implementing the calcution of factor
detector, interaction detector, risk detector and ecological detector, which can be calculated using table data, e.g. csv format(Table 1). The last function geodetector is an auxiliary function,
which can be used to implement the calculation for shapefile format map data(Figure 2).
Table 1.
Demo data
in table
7.20 2 3 6
7.01 2 3 6
6.79 2 3 6
6.73 4 3 6
6.77 4 3 1
6.74 4 3 6
geodetector package is available for data.frame. Please check the data type in advance.
As a demo, neural-tube birth defects (NTD) Y and suspected risk factors or their proxies Xs in villages are provided, including data for the health effect GIS layers and environmental factor GIS
layers, “elevation”, “soil type”, and “watershed”.
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International Conferences
E1. Çevik, A.S., “The p-Cockcroft Property of Central Extensions of Groups”, I. Antalya Algebra Days, 1999.
E2. Çevik, A.S., “The Efficiency of Standard Wreath Product”, IV. Antalya Algebra Days, 2002.
E3. Çevik, A.S., “The p-Cockcroft property of Semi-Direct Products of Monoids”, V. Antalya Algebra Days, 2003.
E4. Çevik, A.S., “Some Minimal but Inefficient Monoid Presentations”, VI. Antalya Algebra Days, 2004.
E5. Çevik, A.S., “Minimal but Inefficient Presentations for Semidirect products of Finite Cyclic Monoids”, VIII. Antalya Algebra Days, 2006.
E6. Çevik, A.S., “Some Minimal Monoid Presentations”, Mathematical Methods in Applied Sciences, Bursa, 2008.
E7. Çevik, A.S., “Geometric Approximations to Minimality of Monoids”, 20 th International Congress of Jangjeon Mathematical Society, 21-23 August, Bursa, 2008.
E8. Çevik, A.S. & Karpuz G.E., “Decision Problems over Semigroups”, 77th Workshop on General Algebra (24th. Conference for Young Algebraist), University of Potsdam, Germany, 20-22 March 2009.
E9. Çevik, A.S., “A Relationship between Subgroup Separability and Efficiency”, 5th Asian Mathematical Conference, 22-26 June, Kuala Lumpur, 2009.
E10. Çevik, A.S. & Karpuz G.E., “The Word and Generalized Word Problem for Semigroups under Wreath Products”, 79th Workshop on General Algebra, Olomouc, February 12-14, 2010.
E11. Çevik, A.S. (joint work with Gungor A.D.), “Some Bounds for Extreme Singular Values of a Complex Matrix”, International Congress in Honour of Professor H. M. Srivastava on his 70th Birth
Anniversary, Uludag University, Bursa, Turkey, 18-21 August 2010.
E12. Çevik, A.S. (joint work with Karpuz E.G., Gungor A.D., Ates F., Cangul I.N.), “One dimension higher of the word problem for monoids”, International Congress in Honour of Professor H. M.
Srivastava on his 70th Birth Anniversary, Uludag University, Bursa, Turkey, 18-21 August 2010.
E13. Çevik, A.S. (joint work with Gungor A.D., Ates, F., Karpuz, E.G., Cangul I.N.), “A New Example of Deficiency One Groups”, “Generating functions of special numbers and polynomials and their
applications” with in International Conference of Numerical Analysis and Applied Mathematics 2010 (ICNAAM 2010)”, Rhodes, Grecee, 19-26 September 2010.
E14. Çevik, A.S. (joint work with Gungor A.D., Ates, F., Karpuz, E.G., Cangul I.N.), “On the Efficiency of Semi-Direct Products of Finite Cyclic Monoids by One-Relator Monoids”, “Generating
functions of special numbers and polynomials and their applications” with in International Conference of Numerical Analysis and Applied Mathematics 2010 (ICNAAM 2010)”, Rhodes, Grecee, 19-26
September 2010.
E15. Çevik, A.S. (joint work with Gungor A.D., Ates, F., Karpuz, E.G., Cangul I.N.), “Generalization for Estrada Index”, “Generating functions of special numbers and polynomials and their
applications” with in International Conference of Numerical Analysis and Applied Mathematics 2010 (ICNAAM 2010)”, Rhodes, Grecee, 19-26 September 2010.
E16. Çevik, A.S. (joint work with Gungor A.D., Ates, F., Karpuz, E.G., Cangul I.N.), “On the Norms of Toeplitz and Henkel Matrices with Pell Numbers”, “Generating functions of special numbers and
polynomials and their applications” with in International Conference of Numerical Analysis and Applied Mathematics 2010 (ICNAAM 2010)”, Rhodes, Grecee, 19-26 September 2010.
E17. Çevik, A.S. (joint work with Gungor A.D., Namli, D., Tekcan, A., Cangul I.N.), “Primes in ${\mathbb Z}[exp \frac{2i\pi}{3}]$”, “Generating functions of special numbers and polynomials and
their applications” with in International Conference of Numerical Analysis and Applied Mathematics 2010 (ICNAAM 2010)”, Rhodes, Grecee, 19-26 September 2010.
E18. Çevik, A.S., “Deficiencies on Groups and Monoids”, 81st Workshop on General Algebra, Salzburg University, Salzburg, Austria, 03-06 February 2011.
E19. Çevik, A.S., “Some Minimality Results on Monoids”, 82nd Workshop on General Algebra, University of Potsdam, Potsdam, Germany, 24-26 June 2011.
E20. Çevik, A.S., 24th The Conference of International Jangjeon Mathematical Society, 20-23 July 2011 Konya, Türkiye. (Head of Organisation).
E21. Çevik, A.S., (with Firat Ates, Eylem G. Karpuz), “The efficiency of the semi-direct products of free abelian monoid with rank n by the infinite cyclic monoid”, “Generating functions of special
numbers and polynomials and their applications” with in International Conference of Numerical Analysis and Applied Mathematics 2011 (ICNAAM 2011)”, Halkidiki, Grecee, 19-26 September 2011.
E22. Çevik, A.S., (with Firat Ates, Eylem G. Karpuz), “Conjugacy for free groups under split extensions”, “Generating functions of special numbers and polynomials and their applications” with in
International Conference of Numerical Analysis and Applied Mathematics 2011 (ICNAAM 2011)”, Halkidiki, Grecee, 19-26 September 2011.
E23. Çevik, A.S., “Grobner-Shirshov Bases of Bruck-Reilly *-Extensions of Monoids”, 25th The Conference of International Jangjeon Mathematical Society, 23-27 Temmuz 2012 Seul, Korea.
E24. Çevik, A.S., (with Nihat Akgüneş and Kinkar Chandra Das) “On a Graph of Monogenic Semigroups”, International Congress in Honour of Professor Hari M. Srivastava at The Auditorium at the Campus of
Uludag University Bursa-TURKEY August 23-26, 2012.
E25. Çevik, A.S., (with Kinkar Chandra Das and I. Naci Cangül) “The Number of Spanning Trees of a Graph”, International Congress in Honour of Professor Hari M. Srivastava at The Auditorium at the
Campus of Uludag University Bursa-TURKEY August 23-26, 2012.
E26. Çevik, A.S., (with Kinkar Chandra Das, Aysun Yurttaş, Müge Togan and I. Naci Cangül) “Multiplicative Zagrep Indices of some Graph Operations of Complete Graphs”, International Congress in
Honour of Professor Hari M. Srivastava at The Auditorium at the Campus of Uludag University Bursa-TURKEY August 23-26, 2012.
E27. Çevik, A.S., (with Kinkar Chandra Das, Ayse Dilek Maden, Ismail Naci Cangul, Betul Acar) “The Kirchhoff matrix, new Kirchhoff indices and the Kirchhoff Energy”, International Conference on
the Theory, Methods and Applications of Nonlinear Equations. Texas A&M University – Kingsville, 17-21 December 2012, USA.
E28. Çevik, A.S., (with Kinkar Chandra Das, Aysun Yurttaş, Müge Togan and I. Naci Cangül) “Bounds over Multiplicative Zagrep Indices”, International Conference on the Theory, Methods and
Applications of Nonlinear Equations. Texas A&M University – Kingsville, 17-21 December 2012, USA.
E29. Çevik, A.S., (with I. Naci Cangül, Aysun Yurttaş, Muge Togan),“Some formulae for the Zagreb indices of graphs”, Conference Information: 10th International Conference of Numerical Analysis and
Applied Mathematics 2012 (ICNAAM 2012), SEP 19-25 2012, Kos, GREECE Source: GENERATING FUNCTIONS OF SPECIAL NUMBERS AND POLYNOMIALS AND THEIR APPLICATIONS Book Series: AIP Conference Proceedings,
(Edt. Theodore E. Simos, George Psihoyios, Ch. Tsitouras, Zacharias Anastassi), Volume 1479, pg. 365-367, 2012. (Web of Science)
E30. Çevik, A.S., (with I. Naci Cangül and Yilmaz Simsek) “A new approach to connect algebra with analysis: Relationships and applications between presentations and generating functions”,
International Workshop “Questions, Algorithms, and Computations in Abstract Group Theory”. Braunschweigh, 21-24 May 2013 Germany.
E31. Çevik, A.S., (with N. Akgüneş) “Some indices on a special graph”, 4th International Conference on Matrix Analysis and Applications, Konya, 02-05 July 2013 Turkey.
E32. Çevik, A.S., (with I. Naci Cangül and Yilmaz Simsek) “Relationships between presentations and generating functions”, 26th The Conference of International Jangjeon Mathematical Society, 01-05
Agust 2013, Bangalore, India.
E33. Çevik, A.S., International Conference on Algebra in honour of Patrick Smith’s and John Clark’s 70th Birthdays, http://ica.balikesir.edu.tr/ 12-15 August 2013, Balikesir, Turkey. (Member of
the Organisation Committee).
E34. Çevik, A.S., International Congress in honour of Ravi Agarwal, June 23-26, 2014, Bursa Turkey (Member of the Organisation and Scientific Committees).
E35. Çevik, A.S., (with I. Naci Cangül and Yilmaz Simsek) “Relationships between presentations and generating functions”, 27th International Conference of the Jangjeon Mathematical Society,
p-Analysis, Umbral Algebra and their Applications, 02-04 August 2014, Daejeon, Korea (Speaker - Member of the Scientific Committee).
E36. Çevik, A.S., “Some new indices on special graphs”, International Conference on Recent Advances in Pure and Applied Mathematics ICRAPAM 2014, 06-09 November 2014, Antalya, Turkey (Speaker -
Member of the Scientific Committee).
E37. Çevik, A.S. (with M. Togan, A. Yurttas, I.N. Cangul) “Minimal polynomials corresponding to spectral sets of some graphs”, 28. International Conference on the Jangjeon Mathematical Society
ICJMS 2015, 15-19 May 2015, Antalya, Turkey (Joint worker - Member of the Scientific Committee).
E38. Çevik, A.S. (with F. Ates, E.G. Karpuz, I.N. Cangul) “A presentation and some finiteness conditions for a new version of the Shützenberger product of monoids”, 28. International Conference
on the Jangjeon Mathematical Society ICJMS 2015, 15-19 May 2015, Antalya, Turkey (Speaker - Member of the Scientific Committee).
E39. Çevik, A.S. (with E.G. Karpuz, N. Urlu) “Gröbner-Shirshov basis of an exceptional Braid groups”, 28. International Conference on the Jangjeon Mathematical Society ICJMS 2015, 15-19 May 2015,
Antalya, Turkey (Joint worker - Member of the Scientific Committee).
E40. Çevik, A.S. (with F. Ates, E.G. Karpuz, I.N. Cangul) “The New Type of Shützenberger Products of Monoids”, 2nd International Conference on Recent Advances in Pure and Applied Mathematics ICRAPAM
2015, 03-06 June 2015, Istanbul, Turkey (Speaker - Member of the Scientific Committee).
E41. Çevik, A.S. (with S. Topkaya) “A new graph over semi-direct products of groups”, 2nd International Conference on Recent Advances in Pure and Applied Mathematics ICRAPAM 2015, 03-06 June 2015,
Istanbul, Turkey (Joint worker - Member of the Scientific Committee).
E42. Çevik, A.S. (with I.N. Cangul, M. Togan, A. Yurttas) “Some Zagreb Indices of Double Graphs”, 2nd International Conference on Recent Advances in Pure and Applied Mathematics ICRAPAM 2015, 03-06
June 2015, Istanbul, Turkey (Joint worker - Member of the Scientific Committee).
E43. Çevik, A.S. (with E. Kangal, E.G. Karpuz) “The Word problem on Special Cases”, 2nd International Conference on Recent Advances in Pure and Applied Mathematics ICRAPAM 2015, 03-06 June 2015,
Istanbul, Turkey (Joint worker - Member of the Scientific Committee).
E44. Cangul, I. N., Yurttas, A., Togan, M., Cevik, A. S., “Graph Theoretical Indices and Some Applications”, International Conference on Applied Mathematics and Numerical Methods, 14-16.04.2016,
University of Craiova, Romania (Joint worker).
E45. Cangul, I. N., Yurttas, A., Togan, M., Cevik, A. S., “Chemical Applications of Graph Indices”, 3rd International Conference on Recent Advances in Pure and Applied Mathematics (ICRAPAM 2016),
19-23.05.2016, Bodrum-Mugla-Turkey (Joint worker).
E46. Cangul, I. N., Yurttas, A., Togan, M., Cevik, A. S., “Results and Applications regarding the Topological Graph Indices”, 11^th Ankara Mathematics Days, Ankara University, 26-27.05.2016,
Ankara-Turkey (Joint worker).
E47. Cangul, I. N., Yurttas, A., Togan, M., Cevik, A. S., “Topological Descriptors of Some Graphs”, 3rd Istanbul Design Theory, Graph Theory and Combinatorics Workshop, Koc University, 13-17.06.2016,
Istanbul-Turkey (Joint worker).
E48. Cangul, I. N., Yurttas, A., Togan, M., Cevik, A. S., “Edge Operations in Graphs and Zagreb Indices”, Interational Conference on Analysis and its Applications, Ahi Evran University,
12-15.07.2016, Kırşehir-Turkey (Joint worker).
E49. Cangul, I. N., Yurttas, A., Togan, M., Cevik, A. S., “Some Inequalities with Zagreb Indices”, Distance in Graph 2016, 18-22.07.2016, Bali-Indonesia (Joint worker).
E50. Cangul, I. N., Yurttas, A., Togan, M., Cevik, A. S., “Graph Descriptors and Chemical Applications”, 5th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2016),
16-19.08.2016, Belgrade-Serbia (Joint worker).
E51. Cangul, I. N., Yurttas, A., Togan, M., Cevik, A. S., “Applications of Topological Graph Indices”, 2nd International Conference on Algebra in Honour of Leonid BOKUT and Surender K. JAIN,
26-29.08.2016, Burhaniye-Balikesir-Turkey (Joint worker – Head of Organization).
E52. Cangul, I. N., Yurttas, A., Togan, M., Cevik, A. S., “New Formulae for Zagreb Indices”, 14^th International Conference of Numerical Analysis and Applied Mathematics, ICNAAM 2016, 19-25 September
2016, Rodos Palace Hotel, Rhodes, Greece (Joint worker).
E53. Cevik, A. S., Wazzan, S. A., Ates, F., “New constructions on a special type of General products over monoids”, The 2^nd Mediterranean International Conference of Pure & Applied Mathematics and
Related Areas (MICOPAM 2019), 28-31 August 2019, Paris, France (Speaker - Member of the Scientific Committee).
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Core Research Area
Core Research Area D: Differential geometry and supersymmetry
Leading Researchers
For many years, there has been a fruitful interplay between supersymmetry, string theory and differential geometry [C]. In this Core Research Area (CRA) we continue to explore the the geometry of
deformation spaces of geometrical structures related to the scalar field spaces of supergravity and string theory. Of particular interest are four-dimensional N = 2 supergravities whose scalar field
space locally is a product of a special Kähler and a quaternionic Kähler manifold. The special Kähler manifold arising in the low energy effective action of string theories is reasonably well
understood, while comparatively little is known about the quaternionic Kähler component as it generically receives perturbative and non-perturbative quantum corrections. It arises in the
hypermultiplet sector of N=2 type II and heterotic compactification and thus is related to properties of K3 and/or Calabi-Yau threefolds.
The following projects are current and future topics in the CRA:
• Study of the hypermultiplet metric in string compactifications arising from Calabi-Yau manifolds with vanishing Euler number. Building on earlier work [CLST] [DLMST] [HLS] [LST10] it was shown in
[KMT] that they also admit a non-integrable SU(2)-structure corresponding to a phase of spontaneously broken N = 4 -> N = 2 supergravity which in turn constrains the possible quantum corrections.
• Extending the study of supersymmetric AdS_4 backgrounds, their moduli spaces and their holographically dual superconformal field theory along the lines of [DLMTW] [LST12] [LT] to other dimensions
and to consistent truncations of 10/11-dimensional supergravity of the form AdS_d x Sasaki-Einstein spaces.
• Dimensional reduction of supergravity theories and perturbative quantum corrections are methods inspired by string theory which, based on research in our group [CHM] [CDL] [CNS] [D], can be
effectively used to construct many new complete quaternionic Kähler manifolds of negative scalar curvature. We are systematically studying the geometric properties of these constructions.
• It would be nice to have a mathematical description of non-perturbative quantum corrections to quaternionic Kähler metrics, such as the metric on the hypermultiplet sector of type II string
theory compactified on a Calabi-Yau three-fold. In particular, it is not known in which situations non-perturbative corrections lead to complete metrics. Perturbative corrections of the
hypermultiplet moduli space can be described using a one-parameter deformation of a c-map metric induced by the HK/QK-correspondence [APP] [ACM] [ACDM]. The method can be also applied in the case
of other interesting moduli spaces such as the Hitchin system.
[C] D.V. Alekseevsky, V. Cortés, M. Dyckmanns and T. Mohaupt, Quaternionic Kähler metrics associated with special Kähler manifolds, J. Geom. Phys. 92 (2015), 271-287. arXiv:1305.3549[math.DG].
[ACM] D.V. Alekseevsky, V. Cortés and T. Mohaupt, Conification of Kähler and hyper-Kähler manifolds, Commun. Math. Phys. 324 (2013) 637-655. arxiv:1205.2964[math.DG].
[APP] S. Alexandrov, D. Persson and B. Pioline, Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence, JHEP 1112 (2011) 027. arXiv:1110.0466 [hep-th].
[C] V. Cortés (ed.), Handbook of pseudo-Riemannian geometry and supersymmetry, IRMA Lectures in Mathematics and Theoretical Physics 16 (2010), 964 pages.
[CDL] V. Cortés, M. Dyckmanns and D. Lindemann, Classification of complete projective special real surfaces, Proc. London Math. Soc. 109 (2014), no. 2, 353-381. arXiv:1302.4570[math.DG].
[CHM] V.Cortés,X.Han and T.Mohaupt, Completeness in supergravity constructions , Comm. Math. Phys. 311 (2012) 191-213. arXiv:1101.5103[hep-th].
[CLST] V. Cortés, J. Louis, P. Smyth and H. Triendl, On certain Kähler quotients of quaternionic Kähler manifolds, Commun. Math. Phys. 317 (2013), no. 3, 787-816. [arXiv:1111.0679 [math.DG]].
[CNS] V. Cortés, M. Nardmann and S. Suhr, Completeness of hyperbolic centroaffine hypersurfaces, Comm. Anal. Geom. (accepted March 4, 2015), arXiv:1407.3251[math.DG].
[DLMST] T. Danckaert, J. Louis, D. Martinez-Pedrera, B. Spanjaard and H. Triendl, The N = 4 effective action of type IIA supergravity compactified on SU(2)- structure manifolds, JHEP 1108 (2011)
024. arXiv:1104.5174 [hep-th].
[DLMTW] S. de Alwis, J. Louis, L. McAllister, H. Triendl and A. Westphal, Moduli spaces in AdS4 supergravity, JHEP 1405 (2014) 102. arXiv:1312.5659 [hep-th].
[D] M. Dyckmanns, The hyper-Kähler/quaternionic Kähler correspondence and the geometry of the c-map, PhD thesis, University of Hamburg, 2015.
[HLS] C. Horst, J. Louis and P. Smyth, Electrically gauged N=4 supergravities in D=4 with N=2 vacua, JHEP 1303(2013) 144. arXiv:1212.4707 [hep-th].
[KMT] A. K. Kashani-Poor, R. Minasian and H. Triendl, Enhanced supersymmetry from vanishing Euler number, JHEP 1304 (2013) 058. arXiv:1301.5031 [hep-th].
[LST12] J. Louis, P. Smyth and H. Triendl, Supersymmetric Vacua in N=2 Supergravity, JHEP 1208 (2012) 039. arXiv:1204.3893 [hep-th].
[LST10] J. Louis, P. Smyth and H. Triendl, Spontaneous N=2 to N=1 Supersymmetry Breaking in Supergravity and Type II String Theory, JHEP 1002 (2010) 103. arXiv:0911.5077 [hep-th].
[LSV] J. Louis, M. Schasny and R. Valandro, Effective Action of Heterotic Compactification on K3 with Nontrivial Gauge Bundles, JHEP 1204 (2012) 028. arXiv:1112.5106 [hep-th].
[LT] J. Louis and H. Triendl, Maximally Supersymmetric AdS4 Vacua in N=4 Supergravity, JHEP 1410 (2014) 7. arXiv:1406.3363 [hep-th].
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Algebraic Geometry Seminar | Ben Bakker (UIC) 10/15/2024
When: Tuesday, October 15, 2024
3:00 PM - 4:00 PM CT
Where: Lunt Hall, 104, 2033 Sheridan Road, Evanston, IL 60208 map it
Audience: Faculty/Staff - Student - Public - Post Docs/Docs - Graduate Students
Contact: Yuchen Liu (847) 491-5553
Group: Department of Mathematics: Algebraic Geometry Seminar
Category: Lectures & Meetings
Title: Algebraicity of Shafarevich maps and the Shafarevich conjecture
Abstract: For a normal complex algebraic variety X equipped with a complex representation V of its fundamental group, a Shafarevich map f:X->Y is a map which contracts precisely those algebraic
subvarieties on which V has finite monodromy. Such maps were constructed for projective X by Eyssidieux, and recently have been constructed analytically in the quasiprojective case by Brunebarbe and
Deng--Yamanoi, in both cases using techniques from non-abelian Hodge theory. In joint work with Y. Brunebarbe and J. Tsimerman, we show that these maps are algebraic. This is a generalization of the
Griffiths conjecture on the algebraicity of images of period maps, and the proof critically uses o-minimal GAGA. We will also explain how the same techniques can be used to prove the Shafarevich
conjecture in the "linear case", which puts strong restrictions on the complex analytic varieties that arise as universal covers of algebraic varieties admitting linear representations of their
fundamental groups.
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Exploring pi123: The Constant Redefining Precision in Science and Tech
Pi is one of the most well-known mathematical constants. It represents the ratio of a circle’s circumference to its diameter, approximately 3.14159. While traditional pi has been studied for
centuries, a lesser-known variation, pi123, has recently gained attention.
Pi123 is an adaptation of pi with unique characteristics. Unlike the traditional value of pi, this version provides a different perspective on calculations involving circles and geometry. It serves
specialized applications in mathematics, data science, engineering, and technology.
Understanding this constant requires knowledge of its derivation and the ways it differs from pi. Its distinct properties make it useful for certain types of mathematical computations. Many
scientists and researchers are exploring this constant’s potential for solving complex problems.
This constant also has applications beyond basic mathematics. In fields like data science, it can optimize certain algorithms and calculations. Engineers also use it when working on specific
technical projects, especially those involving precise measurements.
One exciting area where it is applied is in cryptography. It supports the creation of more complex encryption algorithms, which enhances data security. This version is also significant in physics,
helping scientists with calculations for circular or oscillatory motion.
As research continues, more applications for it may emerge. This constant not only broadens our mathematical toolkit but also offers new possibilities for technological advancements. In the following
sections, we will explore its properties, applications, and potential for future innovations.
The Mathematical Significance
Derivation and Calculation
This adaptation is derived by extending the concept of pi with a unique approach to its calculation. Unlike the original, which has an infinite, non-repeating decimal expansion, this constant uses a
more structured formula. This results in a number with distinct properties that differ from traditional pi.
Mathematical Properties
It holds unique characteristics that set it apart in mathematical studies. Its structure makes it suitable for certain computations where precision is crucial. These properties allow it to simplify
specific types of calculations, especially in advanced mathematics.
Applications in Advanced Mathematics
This constant is valuable in fields like number theory and calculus. It offers new ways to solve equations that involve complex circular measurements. Many researchers are investigating how it could
be applied to broaden our understanding of mathematical constants.
Applications in Various Fields
Data Science and Analysis
In data science, this constant can optimize certain algorithms and statistical models. Its structured properties make it a useful constant for computations that require high precision. Researchers
are exploring how it can enhance the accuracy of machine learning models and data processing tasks.
Engineering and Technology
Engineers use this constant for specific technical projects that involve precise measurements. For example, it is helpful in calculating volumes and areas in advanced engineering designs. Its
properties make it useful in fields such as robotics, material science, and aerospace engineering.
Cryptography and Cybersecurity
This constant also plays a role in cryptography, where it supports the development of complex encryption algorithms. Its unique structure helps create secure data transmission channels, which is
critical in modern cybersecurity. This constant adds an extra layer of security by enabling more sophisticated encryption methods.
Physics and Astronomy
In physics, this constant assists with calculations related to circular motion and oscillations. Physicists use it in the study of planetary orbits and wave behavior. It is also valuable in astronomy
for modeling the paths of celestial objects with high accuracy.
Computer Science
In computer science, this constant is used in simulations, computer graphics, and artificial intelligence. It allows for efficient calculations that are essential in graphic rendering and complex
simulations. Additionally, this constant is used to improve the performance of algorithms in AI and computational tasks.
This constant’s versatility makes it a powerful tool in these various fields. Its potential for innovation continues to attract interest from scientists, engineers, and technologists. Next, we will
compare it with other mathematical constants to further understand its unique role in science and technology.
Comparing with Other Mathematical Constants
Comparison with Traditional Pi (π)
This variation and traditional pi (π) have distinct purposes. While pi is commonly used to measure the ratio between a circle’s circumference and diameter, this constant serves specialized
applications. Situations that require precise computational efficiency may benefit from using this constant instead of pi.
Comparison with Other Constants like Euler’s Number (e) and the Golden Ratio (φ)
In addition to pi, other mathematical constants, such as Euler’s number (e) and the golden ratio (φ), play significant roles. Each constant has unique properties that make it suitable for specific
mathematical and scientific fields. This constant joins this group, providing a new option for applications that require different properties than e or φ.
Practical Uses versus Other Constants
This constant is particularly useful in calculations involving precision in circular measurements. For example, it may be used over pi when accuracy and computational efficiency are paramount. In
contrast, Euler’s number is commonly used in growth and decay models, and the golden ratio appears in geometry and art.
This constant provides unique advantages in areas where other constants might fall short. Its specialized role has led to new insights and techniques in fields like engineering and computer science.
Next, we will explore theoretical implications and future possibilities for this constant in mathematical research.
Theoretical Exploration
Potential Implications in Mathematics
This constant has the potential to reshape certain areas of mathematics. Its unique properties allow it to simplify complex calculations, particularly those involving circular and spherical
measurements. Researchers are studying how it can contribute to new methods in calculus and number theory.
Research Developments and Innovations
Scientists and mathematicians are continuously exploring new applications for this constant. Recent studies show promise for it in fields requiring high precision, such as quantum mechanics and
advanced engineering. Universities and research institutions are beginning to examine its role in computational mathematics.
Future Exploration in Academia
As interest in this constant grows, academic institutions are likely to incorporate it into their mathematics and engineering programs. It could provide a fresh perspective for students and
researchers working with mathematical constants. It may also become a focus of research papers and theoretical studies in the coming years.
This constant’s theoretical applications continue to intrigue the scientific community. Its potential for innovation spans various areas, from basic mathematics to complex physics. In the following
section, we will debunk common myths about this constant and clarify its accurate uses.
Debunking Common Myths
Misconceptions and Confusions
Some believe this constant is simply another version of traditional pi (π). However, it is distinct and has specific properties suited to different applications. Understanding the differences helps
clarify when and how to use it effectively.
Clearing Up Misinterpretations
Another common misconception is that this constant replaces traditional pi in all calculations. In reality, it is designed for specific scientific and mathematical tasks that need higher precision.
It is not a substitute for pi but an additional tool for unique scenarios.
Educating on the Correct Use
This constant should be used in contexts where its unique characteristics can be fully utilized. For example, it is beneficial in technical fields that require advanced measurements, such as
engineering and cryptography. Educators and researchers emphasize the importance of knowing when this constant can enhance calculations and when traditional pi is more suitable.
By debunking these myths, we gain a better understanding of this constant’s specialized role. This clarity supports its effective application in fields like data science, physics, and engineering.
Next, we will explore practical examples and case studies to illustrate the use of this constant in real-world situations.
Pi123 offers a new way to approach mathematical and scientific calculations that require high precision. Its unique properties make it a valuable tool in fields like data science, engineering,
cryptography, and physics. By providing a different structure compared to traditional pi, it allows for more specialized applications and greater efficiency in certain contexts.
As research on pi123 grows, its potential for future innovations becomes even more exciting. Scientists, mathematicians, and engineers are continually finding new ways to leverage it, and educational
institutions are beginning to take notice. Whether it’s enhancing data security, advancing quantum mechanics, or optimizing algorithms, this constant shows that mathematics is still full of
What is pi123?
Pi123 is a mathematical constant similar to traditional pi (π) but with unique properties that make it suitable for specialized applications. It offers a different approach to calculations involving
circles, helping to increase precision and efficiency in certain fields.
How does pi123 differ from traditional pi (π)?
While both constants relate to circles, pi123 has distinct structural properties that differentiate it from traditional pi. Pi123 is tailored for more specific tasks, especially in data science,
engineering, and cryptography, where higher precision is needed.
Where is pi123 used?
Pi123 is used in data science, engineering, cryptography, and physics. It is especially useful in calculations involving circular and spherical measurements, as well as in developing secure
encryption methods and optimizing algorithms for AI and simulations.
Why is pi123 important in cryptography?
In cryptography, pi123 supports the development of more complex and secure encryption algorithms. Its unique properties help create encrypted data transmission channels, which are critical for
protecting sensitive information.
Can pi123 replace traditional pi (π)?
No, pi123 is not meant to replace traditional pi. It serves as an additional tool for specific calculations that benefit from its unique characteristics. Traditional pi is still widely used for
general purposes, especially in geometry and trigonometry.
Is pi123 used in academic research?
Yes, pi123 is gaining interest in academic research. Many universities and research institutions are exploring its applications in mathematics, physics, and engineering. It is becoming a focus of
studies related to computational mathematics and theoretical physics.
How is pi123 calculated?
Pi123 is calculated using a structured formula that differs from traditional pi. This formula provides a precise numerical value that is useful for computations requiring high accuracy, though exact
details of the formula may vary depending on the specific application.
What are the future possibilities for pi123?
As research advances, pi123 may unlock new possibilities in technology and science. Its potential applications include more efficient algorithms in artificial intelligence, improved modeling in
physics and astronomy, and further innovations in data security.
Is pi123 recognized as an official mathematical constant?
Pi123 is recognized as a mathematical constant within specialized fields, though it is not as universally established as traditional pi. Its use is generally limited to applications that require its
specific properties rather than general mathematical calculations.
Where can I learn more about pi123?
You can learn more about pi123 by exploring academic papers, mathematical journals, and online resources focused on advanced mathematics and data science. Universities with strong mathematics and
engineering departments may also offer resources and research papers on pi123.
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Modeling Capacitive Discharge
Here, we address how to model the discharging of a capacitor that is connected to a set of electrical components, which can be modeled either with full geometric fidelity or in combination with a set
of lumped components.
It is possible to model the discharge of the electric energy stored within a capacitor using the Electromagnetic Waves, Transient interface. The initial stored electric energy can either be computed
using the Electrostatics interface, which solves for the electric fields within the structure of the capacitor, or alternatively, the capacitor can be modeled using the Electrical Circuits interface,
where a lumped capacitor with an initial charge defines the initial stored electric energy. The objective of these models is to compute the electromagnetic fields and the losses. The electric and
magnetic energy are computed, as well as the conversion into thermal energy and the radiated energy.
The structure being modeled. An explicitly modeled capacitor is connected to a transformer, which is then connected to a Lumped Element model of a capacitor equivalent. Supporting structures are
omitted under the assumption that they are not electromagnetically relevant. The surrounding region of free space and a ground plane are modeled.
Modeling Approach
Discharge modeling involves two steps: first, setting up an electrostatics model that computes the electric fields around a charged capacitor and then using those fields as initial conditions in a
transient electromagnetic model. You can follow along using the MPH-file attached to this article.
The Electrostatics Model
To model the initial charge, the modeling domain is partitioned to consider only the dielectric and a small volume of space around the capacitor where there will be significant electric fields.
Within this domain, the boundary conditions are set to Ground on one of the capacitor plates, and a fixed Electric Potential on the other plate. The interior of the connecting wires is not modeled.
All other boundaries are set to Zero Charge. The solution from this electrostatic model is used as the initial state for the transient electromagnetic problem, where the wires will be explicitly
A close-up view of the Model Builder with the Stationary node highlighted and the corresponding Settings window.
A separate Stationary study is used to solve for the electrostatic fields in the dielectrics around the capacitor plates. Within this study, only the Electrostatics interface is solved for.
The Transient Electromagnetic Model
To model the transient behavior, the Electromagnetic Waves, Transient interface is solved on all domains with the exception of a domain representing the lumped Electrical Circuit elements. This
cylindrical domain bridges a gap in the conductive wires. The Electrical Circuit adds additional impedance to the system across this gap and is connected via the Lumped Port feature, of type Via. The
Lumped Port feature is valid to use under the assumption that the electric field is uniform and parallel to the wire around its perimeter. The cross-sectional boundaries of the wire on either end of
the Via are Perfect Electric Conductor, implying an equipotential condition across these surfaces.
The Perfect Electric Conductor boundary condition is applied on the bottom boundary of the model, representing a lossless ground plane. The remaining outside boundaries of the domain are Scattering
Boundary Conditions, which approximate an open boundary to free space. Electromagnetic waves will pass through these boundaries with minimal reflections.
The Initial Values feature defines the computed electrostatic fields as the initial value for the first time derivative of the Magnetic vector potential field.
A close-up view of the Model Builder with the Time Dependent node highlighted and the corresponding Settings window.
The study is set up to first solve for the initial electrostatic fields, then compute the electromagnetic fields, the lumped circuit, and a set of global equations for the power and energy. The
initial values used to compute the electromagnetic fields are taken from the electrostatic initialization. It is also possible to save results only on some selections to reduce the amount of data
A close-up view of the Model Builder with the Time-Dependent Solver node highlighted and the corresponding Settings window.
The Time-Dependent Solver settings are adjusted based on the maximum frequency of interest and the element size. Consistent initialization is on.
A Time Dependent study is used to solve for the electromagnetic fields over time. Based on the maximum frequency of interest, it is possible to manually specify the time step, which reduces the
computational cost. Since the global equations are used to store all integrated quantities, it is possible to reduce the amount of data that is stored in the model by only saving results on a few
selected domains, or none at all.
Results and Discussion
It is useful to examine the plot of energy as well as the relative losses. Note that:
• The total energy of the system is nearly constant over time. In the limit of mesh and time-step refinement, this can be improved further.
• The frequency content is initially high but reduces over time.
• The fraction of total thermal losses in the conductors is relatively small. It is possible to ignore losses in the conductors altogether by omitting these domains from the analysis and modeling
the boundaries of the conductors as Perfect Electric Conductor boundary conditions.
• The model can instead be run with the lumped capacitor having an initial potential, and discharging into the modeled domains.
A 1D plot showing the magnetic, electric, dissipated, radiated, and total energy over time.
Plot of the electric, magnetic, thermal, and radiated energy over time. The sum stays nearly constant.
A 1D plot showing the total thermal losses over time.
The thermal losses as a fraction of total losses.
Further Learning
To learn more about the techniques introduced here and explore new ones, check out the following resources:
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Chapter 4. Bayesian Modeling – Basic Models
After learning how to represent graphical models, how to compute posterior distributions, how to use parameters with maximum likelihood estimation, and even how to learn the same models when data is
missing and variables are hidden, we are going to delve into the problem of modeling using the Bayesian paradigm. In this chapter, we will see that some simple problems are not easy to model and
compute and will necessitate specific solutions. First of all, inference is a difficult problem and the junction tree algorithm only solves specific problems. Second, the representation of the models
has so far been based on discrete variables.
In this chapter we will introduce simple, yet powerful, Bayesian models, and show how to represent them as probabilistic graphical models. We will see how their parameters can be learned efficiently,
by using different techniques, and also how to perform inference on those models in the most efficient...
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To determine the angle of emergence i' for varours angle of incidence i and to draw the i-i' curve.
Spectrometer, equilateral prism, sodium light etc.
The graph connecting the angles i and i' is shown in fig(1). The bisector, meets the curve at P. At the point P, i and i' are equal.
OB = OC = i
The angle of deviation s is given by
Where A is teh angle of the prism.
When i =i' , the deviation is minimum (D).
The regractive index of the prism,
Where θ[2 ]is difference between the reflected ray and direct ray.
Spectrometer i-i' curve
A graph is drawn with angle of incidence i along X-axis and the angle of emergence i' along Y-axis. The graph is a rectangular parabola. from it, angle of incidence corresponding to minimum deviation
is calculated.
Fig: 1
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2 Digit By 2 Digit Multiplication Using Area Model Worksheets
Math, especially multiplication, forms the foundation of numerous academic self-controls and real-world applications. Yet, for numerous learners, mastering multiplication can position a difficulty.
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Intro to 2 Digit By 2 Digit Multiplication Using Area Model Worksheets
2 Digit By 2 Digit Multiplication Using Area Model Worksheets
2 Digit By 2 Digit Multiplication Using Area Model Worksheets -
Welcome to The Multiplying 2 Digit by 2 Digit Numbers A Math Worksheet from the Long Multiplication Worksheets Page at Math Drills This math worksheet was created or last revised on 2021 02 17 and
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CIE March 2022 9709 Pure Maths Paper 1
CIE March 2022 9709 Pure Maths Paper 1 (pdf)
1. A curve with equation y = f(x) is such that …
2. A curve has equation y = x^2 + 2cx + 4 and a straight line has equation y = 4x + c, where c is a constant.
Find the set of values of c for which the curve and line intersect at two distinct points.
3. Find the term independent of x in each of the following expansions.
4. The first term of a geometric progression and the first term of an arithmetic progression are both equal to a.
The third term of the geometric progression is equal to the second term of the arithmetic progression.
The fifth term of the geometric progression is equal to the sixth term of the arithmetic progression.
Given that the terms are all positive and not all equal, find the sum of the first twenty terms of the arithmetic progression in terms of a.
5. (a) Express 2x^2 − 8x + 14 in the form 2[(x − a)^2 + b]
(b) Describe fully a sequence of transformations that maps the graph of y = f(x) onto the graph of y = g(x), making clear the order in which the transformations are applied.
6. The circle with equation (x + 1)^2 + (y − 2)^2 = 85 and the straight line with equation y = 3x − 20 are shown in the diagram. The line intersects the circle at A and B, and the centre of the
circle is at C.
(a) Find, by calculation, the coordinates of A and B
(b) Find an equation of the circle which has its centre at C and for which the line with equation y = 3x − 20 is a tangent to the circle.
7. (a) Show that
8. The diagram shows the circle with equation
9. Functions f, g and h are defined as follows:
10. The diagram shows a circle with centre A of radius 5 cm and a circle with centre B of radius 8 cm. The circles touch at the point C so that ACB is a straight line. The tangent at the point D on
the smaller circle intersects the larger circle at E and passes through B.
(a) Find the perimeter of the shaded region.
11. It is given that a curve has equation
Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
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What is: Q-Factor
What is Q-Factor?
The Q-Factor, often referred to in the context of statistics and data analysis, is a quantitative measure that helps in understanding the quality of a dataset or a statistical model. It serves as an
indicator of how well a particular model or dataset can predict outcomes based on the input variables. The Q-Factor is particularly significant in fields such as data science, where the accuracy and
reliability of predictive models are paramount. By evaluating the Q-Factor, data analysts can ascertain the effectiveness of their models and make informed decisions about further data collection or
model refinement.
Understanding the Components of Q-Factor
The Q-Factor is derived from several key components that contribute to its overall value. These components typically include the model’s predictive accuracy, the complexity of the model, and the
amount of data used for training. Predictive accuracy refers to how closely the model’s predictions align with actual outcomes, while model complexity pertains to the number of parameters or features
included in the model. The amount of training data is crucial as well; models trained on larger datasets tend to have a higher Q-Factor due to their ability to generalize better to unseen data.
Understanding these components is essential for data scientists aiming to optimize their models.
Applications of Q-Factor in Data Science
In data science, the Q-Factor is utilized in various applications, including machine learning, statistical modeling, and data mining. For instance, in machine learning, the Q-Factor can help in
selecting the best model among several candidates by providing a quantitative measure of each model’s performance. This is particularly useful in scenarios where multiple algorithms are tested, as it
allows data scientists to choose the model that not only performs well on training data but also generalizes effectively to new data. Additionally, the Q-Factor can be employed in feature selection
processes, guiding analysts in identifying which variables contribute most significantly to the model’s predictive power.
Calculating the Q-Factor
Calculating the Q-Factor involves a systematic approach that typically includes evaluating the model’s performance metrics, such as accuracy, precision, recall, and F1 score. These metrics provide
insights into how well the model is performing and can be combined to derive a single Q-Factor score. The formula for calculating the Q-Factor may vary depending on the specific context and the type
of model being evaluated. However, it generally incorporates elements that reflect both the model’s predictive capabilities and its complexity, ensuring a comprehensive assessment of its overall
Q-Factor in Predictive Analytics
In the realm of predictive analytics, the Q-Factor plays a crucial role in determining the reliability of forecasts generated by statistical models. Predictive analytics relies heavily on the ability
to make accurate predictions based on historical data, and the Q-Factor serves as a benchmark for assessing the effectiveness of these predictions. By analyzing the Q-Factor, analysts can identify
potential weaknesses in their models, such as overfitting or underfitting, and take corrective measures to enhance predictive accuracy. This iterative process of evaluation and refinement is
essential for developing robust predictive models that can withstand real-world applications.
Q-Factor and Model Validation
Model validation is a critical step in the data analysis process, and the Q-Factor is integral to this phase. Validation techniques, such as cross-validation and bootstrapping, often incorporate the
Q-Factor to assess the stability and reliability of a model’s predictions. By applying these techniques, data scientists can evaluate how well their models perform on different subsets of data,
thereby gaining insights into their generalizability. A high Q-Factor during validation indicates that the model is likely to perform well on unseen data, which is a key requirement for any
predictive modeling task.
Limitations of Q-Factor
Despite its usefulness, the Q-Factor is not without limitations. One significant drawback is that it may not fully capture the nuances of model performance in all scenarios. For instance, a model
with a high Q-Factor may still exhibit poor performance in specific contexts or datasets. Additionally, the Q-Factor can be influenced by the choice of evaluation metrics, which may lead to varying
interpretations of a model’s effectiveness. Therefore, while the Q-Factor is a valuable tool for assessing model quality, it should be used in conjunction with other evaluation methods to obtain a
more comprehensive understanding of model performance.
Improving Q-Factor Scores
Improving the Q-Factor score of a model involves several strategies that focus on enhancing predictive accuracy and reducing model complexity. One effective approach is to conduct feature
engineering, which involves creating new features or modifying existing ones to better capture the underlying patterns in the data. Additionally, employing ensemble methods, such as bagging and
boosting, can help improve the Q-Factor by combining the strengths of multiple models. Regularly updating the model with new data and retraining it can also lead to better Q-Factor scores, as it
ensures that the model remains relevant and accurate in a changing data landscape.
Future Trends in Q-Factor Analysis
As the fields of statistics, data analysis, and data science continue to evolve, the concept of the Q-Factor is likely to undergo significant advancements. Emerging technologies, such as artificial
intelligence and deep learning, may introduce new methodologies for calculating and interpreting the Q-Factor. Furthermore, the integration of big data analytics could enhance the Q-Factor’s
applicability across diverse datasets and industries. Researchers and practitioners will need to stay abreast of these developments to effectively leverage the Q-Factor in their analytical endeavors,
ensuring that they maintain a competitive edge in the rapidly changing landscape of data science.
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Non Parametric Analysis Homework Solution
When to perform non-parametric analysis
To achieve the best results from a
non-parametric test
, we should know the kind of situations in which such tests are appropriate. Here are the most common:
• If the data being tested does not meet parametric assumptions: Basically, parametric analysis requires that the data being studied meets a given set of assumptions. For instance, the data should
be normally distributed and the variance of the population should be homogenous. But some data samples may display skewness, rendering parametric tests less powerful. On the other hand,
non-parametric tests work perfectly with skewed distributions and will come in handy in this situation.
• The size of the population sample is too small: The size of the sample is an essential aspect when it comes to selecting the most suitable statistical method. If the sample size is relatively
large, a parametric test can be applied. If the size is too small, however, validating the distribution of data can be difficult, and the only way around it is to use non-parametric analysis.
• The data being analyzed is nominal or ordinal: Unlike parametric analysis that only works with continuous data, non-parametric analysis can be performed on other data types like nominal or
ordinal data. For such data types, a non-parametric test is the only suitable solution.
To learn more about when to use non-parametric tests,
our-parametric analysis online tutors.
Types of non-parametric tests
When we come across the word “parametric analysis” the first thing that we think of is an analysis of variance or t-test. These two tests assume that the data being observed is distributed normally.
non-parametric analysis
does not assume that the data has a normal distribution and the only non-parametric analysis you are likely to perform in a statistics class is the
chi-square test
. But there are many other tests that can be carried out when performing non-parametric analysis and these include:
• 1 sample sign test: Used to determine the median of a data set and comparing it to a target value or a reference value.
• Wilcoxon signed-rank test: Like 1 sample sign test, the Wilcoxon test allows you to make an approximation of a data set’s median and compare it to a target or reference value. Nevertheless, the
test makes an assumption that the data has been obtained from a symmetric distribution like a uniform distribution or Cauchy distribution.
• Friedman test: Used to determine the difference between various groups with ordinal dependent variables. The Friedman test can also be performed on continuous data if some assumptions have been
violated, for instance, if one-way analysis of variance is inappropriate.
• Goodman Kruska’s Gamma: Used to test the relationship between ranked variables
• Kruskal Wallis test: Used in place of a one-way analysis of variance to determine whether multiple medians are different. In this test, the calculations use the ranks of data points instead of
the data points themselves.
• Mann Kendall trend test: Used to identify trends in time series data
• Mann Whitney test: Used to check the differences between two independent groups of data sets when the dependent variables are either continuous or ordinal.
• Mood’s median test: Used in place of the 1 sample sign test when the data being analyzed has two independent variables
• Spearman rank correlation: Used to determine a correlation between two data sets
Advantages and disadvantages of non-parametric analysis
• Fewer assumptions
• Acceptance of smaller sample sizes
• More powerful when the parametric test assumptions have been violated
• Can be used on almost all types of data including interval variables, nominal variables, or data that has been measured inaccurately or that has outliers
However, like most data manipulation methods, non-parametric analysis has its drawbacks. Here are the most notable ones:
• More labor-intensive and time consuming when calculated manually
• Essential value tables for many tests are not incorporated into many statistical software packages
To further understand the different types of non-parametric tests as well as their advantages and disadvantages, connect with our non-parametric analysis homework help experts.
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Quantum Computers, a sword of Damocles above the digital economy
Photo by sebastiaan stam on Unsplash
Fortunately, faced with the risks of these new uses, we have two major advantages. We have at our disposal two main families of cryptographic algorithms that provide us on the one hand with the
guarantee of identity and on the other hand with the confidentiality of exchanges. The algorithms of the first family are based on asymmetric keys. In an exchange, each participant has a private key
and a public key that he provides to his interlocutors. The private key allows him to sign his messages and recipients can ensure his integrity by verifying his signature with the public key.
The link between the public key and the private key also allows the recipient to verify the identity of the sender. They can also send an encrypted response with the public key and only the person in
possession of the corresponding private key can decrypt it.
Very often, these keys are
RSA keys
, initials of the name of their inventors (Ronald Rivest, Adi Shamir and Leonard Adleman). Their principle has been known since 1977. It is based on the difficulty of finding each of its prime
numbers from the product of two large prime numbers. If the number of bits required to represent this product is less than 256 bits, the key can be broken in a few minutes on any PC. The recommended
length today is 2048 bits, which puts factoring out of reach of today's computers for many years to come. It is thanks to these keys that the vast majority of digital transactions are secured, from
credit cards to health cards, the foundation of our society's digital transformation.
However, the appearance of quantum computers is shaking up all this; indeed, the
mathematician Peter Williston Shor showed in 1994
that they could in theory be used to quickly factor large numbers, rendering inoperative not only the RSA keys, but also those using elliptic curve mechanisms. Last year, an
article published in the journal Nature
announced the factorization of the numbers 15, 143, 59989, and 376289 using 4, 12, 59, and 94 logical qubits. The qubit is the elementary unit that these quantum computers handle; unlike the bits of
conventional computers, they do not correspond to a single possible state but to a set of states. The number of qubits is one measure of the power of quantum computers. These results were achieved
using a D-WAWE 2000Q computer. As its name suggests, this computer consists of 2000 qubits, but these qubits are of a particular type that could not be used for the factorization problem. The quantum
computers that can be used to run the Shor algorithm on them are currently much less powerful, with less than 80 qubits, because they are more complex to build. This is therefore a significant step.
If there is still a long way to go for 2048-bit number factorization, since the largest of the first factorized numbers is 18 bits, there is now a Damocles sword above the security of RSA keys.
It is therefore important to start now to focus on finding new algorithms that will resist the advent of quantum computing. That is why the National Institute of Standards and Technology (NIST)
launched a competition at the end of 2016 to designate the successor to the RSA algorithm. We are now in the 2nd round and the
candidates in the running are known
. Many French teams answered the call. The validation of the proposals, and the choice of one of them, will be a long process whose completion is not actually expected until 2024.
The other algorithm, which we have not yet discussed, ensures the confidentiality of the exchanges. This is the Advanced Encryption Standard (AES), which is also the result of a NIST competition that
began in 1997. It’s vulnerable too, but to a lesser extent, to advances in quantum computation. The Grover algorithm would halve the security related to key length. Today the minimum length of a key
considered secure is 128-bits, which will no longer be the case using the Grover algorithm. However, the standard length used nowadays is 256-bits and would therefore be sufficient.
In order to ensure a smooth transition, it is important to prepare for these changes as soon as possible. It is very difficult to predict when the first powerful enough quantum computers will be
built or even if they will exist one day, but by the time they are there, it will be too late to act.
About Serge Adda - CTO WALLIX
Serge Adda has more than twenty years of experience in the world of IT and software publishing. For 15 years, he was Vice President of Research and Development at Infovista. He has been Product Vice
President of WALLIX GROUP since 2012, in charge of research and development, roadmaps and product life cycle. Serge Adda graduated from the Ecole Nationale des Mines de Saint-Etienne in France.
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Reply To: Difference in Meas Q between calculations of flow WinRiverI ? vs WinRiverII/Qrev
I think that Jeff describes it well. A bit more detail is provided in https://hydroacoustics.usgs.gov/memos/OSW2009-02.pdf. I have pasted in the pertinent wording below.
For each ensemble, WinRiver calculates a depth below which velocities measured by the ADCP are not used due to possible errors caused by side-lobe interference. This depth is referred to as the
side-lobe cutoff. In previous versions of WinRiver (2.02 or earlier) the sidelobe cutoff was calculated as 6% of the depth, computed as the mean of all valid beam depths for an ensemble (figure 1a).
The new side-lobe cutoff is calculated as 6% of the shallowest beam depth in an ensemble (figure 1b). For example, the side lobe cutoff computed for ensemble 78 in figure 1 for prior versions of
WinRiver II is 2.4 ft. The beam depths measured for this ensemble were 1.9, 3.5, 3.3, and 1.7 ft. The new side lobe cutoff, computed using WinRiver II version 2.03 and 2.04 is 1.6 ft. The change in
the side-lobe cutoff may result in fewer valid depth cells in ensembles near sloping banks and in cross sections with irregular or rough streambeds. As a consequence, the middle (measured portion of
the cross-section) discharge will decrease and the bottom (unmeasured or extrapolated portion of the cross-section) discharge will increase.
OSW has used the new version of WinRiver II to reprocess discharge measurements from Oberg and Mueller (2007) as well as other available measurements. Based on the discharge measurements reviewed to
date, the new side-lobe cutoff has resulted in a median difference in discharge of +0.5 percent. The typical change in discharge was less than 1 percent. However, for about 4 percent of the discharge
measurements reviewed the discharge changed by more than 5 percent. The largest changes were found for measurements made in shallow uneven cross sections that would typically be considered poor
measurement sections. These cross sections are characterized by mean differences of beam depths for individual ensembles ranging from 17 to 32 percent.
Measurements collected in the 2009 Water Year with a mean difference of beam depths in individual ensembles exceeding 15% or a mean number of valid depth cells per ensemble less than 4 must be
reprocessed using version 2.04.
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On a map 1 inch represents 1000 miles If the area of a country is actually 16 million square miles what is the area of the countrys representation on the map
On a map, 1 inch represents 1,000 miles. If the area of a country is actually 16 million
square miles, what is the area of the country’s representation on the map?
Answer :
1 inch x 1 inch=1 squared inch
1000 miles x 1000 miles=1 million squared miles
16 million/(1 million squared miles per squared inch)= 16 squared icnhes
Answer Link
Answer: 16 square inches .
Step-by-step explanation:
Given : On a map, 1 inch represents 1,000 miles.
That means , [tex]\text{1 mile =}\dfrac{1}{1000}\text{ inch on map.}[/tex]
[tex]\text{1 square mile =}\dfrac{1}{1000}\times\dfrac{1}{1000}\text{ square inch on map.}[/tex]
If the area of a country is actually 16 million square miles, then the area of the country’s representation on the map is given by :-
[tex]16,000,000\times\dfrac{1}{1000}\times\dfrac{1}{1000}=16\text{ square inches on map.}[/tex]
Answer Link
Other Questions
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Minimal strings and topological strings
In http://arxiv.org/abs/hep-th/0206255 Dijkgraaf and Vafa showed that the closed string partition function of the topological B-model on a Calabi-Yau of the form $uv-H(x,y)=0$ coincides with the free
energy of a certain matrix model.
Then, after taking the double-scaling limit, they get an identification between the B-model partition function and the minimal string partition function. The latter is a minimal model coupled to the
Liouville theory, and the equation $H(x,y)=0$ corresponds to what is known as the minimal string Riemann surface (see http://arxiv.org/abs/hep-th/0312170). For the $(p,q)$ minimal model (without any
insertions) one gets $H(x,y)=y^p+x^q$.
There are two kinds of branes in the Liouville theory: FZZT and ZZ, where the FZZT branes are parametrized (semiclassically) by the points on the Riemann surface $H(x,y)=0$.
What are the equivalents of the FZZT and ZZ open string partition functions in the B-model?
This post has been migrated from (A51.SE)
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"Physics and Astronomy Colloquium-Raphael Bousso" - Department of Physics and Astronomy
UNC-CH Physics and Astronomy Colloquium
Raphael Bousso, University of California, Berkeley
“Black Holes, Quantum Information, and Unification”
The study of black holes has revealed a deep connection between quantum information and spacetime geometry. Its origin must lie in a quantum theory of gravity, so it offers a valuable hint in our
search for a unified theory. Precise formulations of this relation recently led to new insights in Quantum Field Theory, some of which have been rigorously proven. An important example is our
discovery of the first universal lower bound on the local energy density. The energy near a point can be negative, but it is bounded below by a quantity related to the information flowing past the
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Power System Formulas for GATE EE Exam | GATE Notes and Videos for Electrical Engineering - Electrical Engineering (EE) PDF Download
FAQs on Power System Formulas for GATE EE Exam - GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)
1. What are some important power system formulas that I should know for the GATE EE exam in Electrical Engineering?
Ans. Some important power system formulas that you should know for the GATE EE exam in Electrical Engineering include: 1. Power formula: P = VI, where P is the power in watts, V is the voltage in
volts, and I is the current in amperes. 2. Ohm's Law: V = IR, where V is the voltage in volts, I is the current in amperes, and R is the resistance in ohms. 3. Power factor formula: PF = P/VI, where
PF is the power factor, P is the active power in watts, V is the voltage in volts, and I is the current in amperes. 4. Apparent power formula: S = VI, where S is the apparent power in volt-amperes, V
is the voltage in volts, and I is the current in amperes. 5. Reactive power formula: Q = √(S^2 - P^2), where Q is the reactive power in volt-amperes reactive (VAR), S is the apparent power in
volt-amperes, and P is the active power in watts.
2. How can I use the power factor formula to calculate the power factor of a given electrical system?
Ans. To calculate the power factor of a given electrical system using the power factor formula (PF = P/VI), you need to know the active power (P), voltage (V), and current (I) of the system. 1.
Calculate the active power (P) by multiplying the voltage (V) and current (I). 2. Calculate the apparent power (S) by multiplying the voltage (V) and current (I). 3. Substitute the values of P, V,
and I into the power factor formula (PF = P/VI). 4. Divide the active power (P) by the apparent power (S) to get the power factor (PF). The power factor is a dimensionless quantity ranging from 0 to
1, where a higher value indicates a more efficient use of electrical power.
3. How do I calculate the reactive power of an electrical system using the reactive power formula?
Ans. To calculate the reactive power of an electrical system using the reactive power formula (Q = √(S^2 - P^2)), you need to know the apparent power (S) and active power (P) of the system. 1.
Calculate the apparent power (S) by multiplying the voltage (V) and current (I). 2. Calculate the active power (P) by multiplying the voltage (V), current (I), and power factor (PF). 3. Substitute
the values of S and P into the reactive power formula (Q = √(S^2 - P^2)). 4. Square the apparent power (S) and subtract the square of the active power (P^2) to get the value inside the square root.
5. Take the square root of the result to get the reactive power (Q). The reactive power is measured in volt-amperes reactive (VAR) and represents the power consumed or generated by reactive elements
such as inductors and capacitors in the electrical system.
4. How can I apply Ohm's Law to solve power system problems in the GATE EE exam?
Ans. Ohm's Law (V = IR) can be applied to solve power system problems by using the relationship between voltage (V), current (I), and resistance (R) in an electrical circuit. 1. If you know the
values of two out of the three variables (V, I, R), you can use Ohm's Law to calculate the unknown variable. 2. If you know the voltage (V) and resistance (R), you can calculate the current (I) by
dividing the voltage by the resistance (I = V/R). 3. If you know the current (I) and resistance (R), you can calculate the voltage (V) by multiplying the current by the resistance (V = I*R). 4. If
you know the voltage (V) and current (I), you can calculate the resistance (R) by dividing the voltage by the current (R = V/I). By applying Ohm's Law correctly, you can solve various power system
problems related to voltage, current, and resistance in electrical circuits.
5. How can I use the power formula to calculate the power consumed by an electrical device?
Ans. To calculate the power consumed by an electrical device using the power formula (P = VI), you need to know the voltage (V) and current (I) flowing through the device. 1. Measure the voltage (V)
across the device using a voltmeter. 2. Measure the current (I) flowing through the device using an ammeter. 3. Multiply the voltage (V) and current (I) to calculate the power (P) consumed by the
device. The power consumed by an electrical device is measured in watts (W) and represents the rate at which the device uses electrical energy.
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commit dc4efd2b5bff43a5a040e73e6e534a145a6b1fb2
parent 998a5036fe2c9aa1ceb047663be6ad9a2063b88d
Author: Sebastiano Tronto <sebastiano.tronto@gmail.com>
Date: Tue, 28 Dec 2021 12:47:01 +0100
Changed time estimate for optimal solving (after running a test on 100 scrambles).
M README.md | 2 +-
M doc/nissy.1 | 2 +-
2 files changed, 2 insertions(+), 2 deletions(-)
diff --git a/README.md b/README.md
@@ -5,7 +5,7 @@ For optimal HTM solving Nissy uses techniquest from Herbert Kociemba's
[Cube Explorer](http://kociemba.org/cube.htm) and Tomas Rokicki's
With 4 cores at 2.5GHz and using less than 3Gb of RAM, Nissy can find an
-optimal solution in less than a minute (18 moves or less) to a few minutes.
+optimal solution in about a minute on average.
Nissy can also solve many different substeps of Thistlethwaite's algorithm
(DR/HTR), and can use NISS (Normal-Inverse Scramble Switch).
diff --git a/doc/nissy.1 b/doc/nissy.1
@@ -17,7 +17,7 @@ is a Rubik's Cube solver.
It uses techniques from Herbert Kociemba's Cube Explorer and
Tomas Rokicki's nxopt. With 4 cores at 2.5GHz and using less than 3Gb
of RAM, Nissy can find the optimal solution for a random Rubik's cube position
-in less than a minute (18 moves or less) to a few minutes.
+in about a minute on average.
Nissy can also solve different substeps of the Thistlethwaite's algorithm and more.
When run without any argument an interactive shell is launched, otherwise
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A 2^{n/2}-Time Algorithm for \sqrt{n}-SVP and \sqrt{n}-Hermite SVP, and an Improved Time-Approximation Tradeoff for (H)SVP
A 2^{n/2}-Time Algorithm for \sqrt{n}-SVP and \sqrt{n}-Hermite SVP, and an Improved Time-Approximation Tradeoff for (H)SVP
Divesh Aggarwal , National University of Singapore
Authors: Zeyong Li , National University of Singapore
Noah Stephens-Davidowitz , Cornell University
Download: DOI: 10.1007/978-3-030-77870-5_17 (login may be required)
Conference: EUROCRYPT 2021
We show a 2^{n/2+o(n)}-time algorithm that, given as input a basis of a lattice $\lat \subset \R^n$, finds a (non-zero) vector in whose length is at most $\widetilde{O}(\sqrt{n})\cdot \
min\{\lambda_1(\lat), \det(\lat)^{1/n}\}$, where $\lambda_1(\lat)$ is the length of a shortest non-zero lattice vector and $\det(\lat)$ is the lattice determinant. Minkowski showed that $
\lambda_1(\lat) \leq \sqrt{n} \det(\lat)^{1/n}$ and that there exist lattices with $\lambda_1(\lat) \geq \Omega(\sqrt{n}) \cdot \det(\lat)^{1/n}$, so that our algorithm finds vectors that
are as short as possible relative to the determinant (up to a polylogarithmic factor). The main technical contribution behind this result is new analysis of (a simpler variant of) a 2^{n/
Abstract: 2 + o(n)}-time algorithm from [ADRS15], which was only previously known to solve less useful problems. To achieve this, we rely crucially on the ``reverse Minkowski theorem'' (conjectured
by Dadush [DR16] and proven by [RS17]), which can be thought of as a partial converse to the fact that $\lambda_1(\lat) \leq \sqrt{n} \det(\lat)^{1/n}$. Previously, the fastest known
algorithm for finding such a vector was the 2^{0.802n + o(n)}-time algorithm due to [LWXZ11], which actually found a non-zero lattice vector with length $O(1) \cdot \lambda_1(\lat)$.
Though we do not show how to find lattice vectors with this length in time $2^{n/2+o(n)}$, we do show that our algorithm suffices for the most important application of such algorithms:
basis reduction. In particular, we show a modified version of Gama and Nguyen's slide-reduction algorithm [GN08], which can be combined with the algorithm above to improve the time-length
tradeoff for shortest-vector algorithms in nearly all regimes---including the regimes relevant to cryptography.
Video from EUROCRYPT 2021
title={A 2^{n/2}-Time Algorithm for \sqrt{n}-SVP and \sqrt{n}-Hermite SVP, and an Improved Time-Approximation Tradeoff for (H)SVP},
author={Divesh Aggarwal and Zeyong Li and Noah Stephens-Davidowitz},
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Expressing one number as a percentage of another | Quizalize
Expressing one number as a percentage of another
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• $\frac\left\{7\right\}\left\{20\right\}$ as a percentage is ______%.
• Q1
$\frac\left\{7\right\}\left\{20\right\}$ as a percentage is ______%.
• Q2
Which decimal is equivalent to 1%?
• Q3
$\frac\left\{6\right\}\left\{25\right\}$ as a percentage is ______%.
• Q4
Match each percentage to its decimal equivalent.
• Q5
$\frac\left\{3\right\}\left\{8\right\}$ as a decimal is ______.
• Q6
Using short division or otherwise, write $\frac\left\{7\right\}\left\{9\right\}$ as a percentage.
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Functions in real life
Good morning!
Today, we are going to talk about functions and how they can be applied in real life situation. But first, what are functions? Do you have any idea about what functions are? And now, I am going to
start giving you a definition. Are you ready? Let's go!
A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
It is also a bunch of ordered pairs of things (in your case, the things will be numbers, but they can be otherwise), with the property that the first members of the pairs are all different from one
IN OTHER WORDS, a function is a mathematical relationship between two variables, where every input variable has one output variable.
In functions, the x-variable is known as the input or independent variable, because its value can be choosen freely. The calculated y-variable is known as the output or dependent variable, because
its value depends on the chosen input value.
A shorthand used to write sets, often sets with an infinite number of elements.
Constant function is a linear function of the form y=b, where b is a constant. It is also written as f(x)=b. The graph is a horizontal line.
Identity function: It can be written in the form f(x)=x. It's graph is a straight line passing through the origin.
Linear function has one independent variable and one dependent variable. The independent variable is x and the dependent variable is y. It is written in the form f(x)=mx+b. It's graph is a straight.
Radical function contain functions involving roots. Most examples deal with square roots.
Piecewise function is a function that is defined as a sequence of intervals.
Quadratic function is one of the form f(x)=ax2+bx+c, where a, b and c are numbers with a not equal to zero. The graph of a quadratic function is a curve called a parabola.
The domain of a function is the set of all independents x-values for which there is one dependent y-value according to that function.
The range of a function is the set of all dependent y-values which can be obtained using an independent x-value.
Functions are mathematical building blocks for designing machines, predicting natural disasters, curing diseases, understanding world economies and for keeping aeroplanes in the air. Functions can
take input from many variables, but always give the same output, unique to that function.
Money as a function of time. You never have more than one amount of money at any time because you can always add everything to give one total amount. By understanding how your money changes over
time, you can plan to spend your money sensibly.
Temperature as a function of various factors. Temperature is a very complicated function because it has so many inputs, including: the time of day, the season, the amount of clouds in the sky, the
strength of the wind, where you are and many more. But the important thing is that there is only one temperature output when you measure ir in a specific place.
Location as a function of time. You can never be in two places at the same time. If you were to plot the graphs of where two people are as a function of time, the place where the lines cross means
that the two people meet each other at that time. This idea is used in logistics, an area of mathematics that tries to plan where people and items are for businesses.
Now we are going to learn how quadratic functions can be applied in real life situations.
The throw ends when the shot hits the ground. The height y at that point is 0, so set the equal to zero.
This equation is difficult to factor or to complete the square, so well solve by applying the quadratic formula.
Simplify or find both roots. x=46.4 or -4,9.
Do the roots make sense? The parabola described by the quadratic function has two x-intercepts, but the shot only traveled along part of that curve.
One solution, -4,9, cannot be the distance traveled because it is a negative number.
The other solution, 46,4 feet, must give the distance of the throw.
Now that we've studied different types of function and how a quadratic function can be applied in real life, we now say that everything can related in real life and that these can be solved through
these mathematical equations learned in school.
I hope you had a nice time watching this video, thank you!
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Prediction from Partial Information and Hindsight, with Application to Circuit Lower Bounds
Consider a random sequence of n bits that has entropy at least n−k, where k≪ n. A commonly used observation is that an average coordinate of this random sequence is close to being uniformly
distributed, that is, the coordinate “looks random.” In this work, we prove a stronger result that says, roughly, that the average coordinate looks random to an adversary that is allowed to query ≈nk
other coordinates of the sequence, even if the adversary is non-deterministic. This implies corresponding results for decision trees and certificates for Boolean functions. As an application of this
result, we prove a new result on depth-3 circuits, which recovers as a direct corollary the known lower bounds for the parity and majority functions, as well as a lower bound on sensitive functions
due to Boppana (Circuits Inf Process Lett 63(5):257–261, 1997). An interesting feature of this proof is that it works in the framework of Karchmer and Wigderson (SIAM J Discrete Math 3(2):255–265,
1990), and, in particular, it is a “top-down” proof (Håstad et al. in Computat Complex 5(2):99–112, 1995). Finally, it yields a new kind of a random restriction lemma for non-product distributions,
which may be of independent interest.
Bibliographical note
Publisher Copyright:
© 2019, Springer Nature Switzerland AG.
• 68Q15
• Certificate complexity
• Circuit complexity
• Circuit complexity lower bounds
• Decision tree complexity
• Information theoretic
• Query complexity
• Sensitivity
ASJC Scopus subject areas
• Theoretical Computer Science
• General Mathematics
• Computational Theory and Mathematics
• Computational Mathematics
Dive into the research topics of 'Prediction from Partial Information and Hindsight, with Application to Circuit Lower Bounds'. Together they form a unique fingerprint.
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