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Iterative In-order B Tree Traversal
Efficient sorted collections are extremely important. There are many situations where a developer may choose a sorted collection with logarithmic access times over a hashtable with constant access
times because the benefit of maintaining the data in sorted order is either a requirement or highly advantageous. B Trees in particular have held an established role in filesystems and database
management systems practically since their inception. More recently, B Tree's have also been seeing something of a rebirth as in-memory data structures. In some instances even rivaling traditional
in-memory structures like Red/Black Trees - which themselves are an abstraction of B Trees implemented over binary search trees. This is possible because B Trees maintain "the search tree property"
which dictates that values in the subtree to the left of a key are less than that key, and values in the subtree to the right of that key are greater.
It is because of this constraint placed on the ordering of the keys that allow an in-order traversal to yield the keys in ascending order. For both binary search trees and B Tree's an in order
traversal can easily be written recursively. For many applications, such as implementing an iterator, a non-recursive traversal is not only preferable: it's necessary. As we shall see shortly, those
easy recursive traversals hide quite a bit of details behind the scenes.
//BST traversal
void traverse(link node) {
if (node != nullptr) {
cout<<h->info.key()<<" ";
//B Tree traversal
void traverse(link node) {
if (node != nullptr) {
int i = 0;
while (i < node->n) {
cout<<node->data[i++].key()<<" ";
In todays post I'm going to show you how to replace the recursion in the above algorithm with an explicit stack. Iterative implementations allow us to more easily "pause" the traversal so that it can
be performed step-wise. I've covered
building in-memory B Tree's
in a previous post, which I will use as the basis for this post.
Warm Up: Iterative Binary Search Tree Traversal
As both Binary Search Trees and B Trees maintain the search tree property their traversal algorithms follow the same general strategy. As such, it makes sense to review the iterative traversal of
binary search tree's as it will lay a good foundation for us to start from. Starting from the root of the BST, an in order traversal begins by traversing the current nodes left subtree, then
processing the current node, and then doing the same to the right subtree. This is sometimes written as (LNR) for Left, Node, Right. In the recursive variant it is the call stack which stores the
paths as we iterate over the tree, which makes finding the next node possible. For us to perform the traversal iteratively we must then maintain that stack ourselves.
void traverse(node* h) {
stack<node*> sf;
node* x = h;
while (x != nullptr) {
x = x->left;
while (!sf.empty()) {
node* x = sf.top();
if (x != nullptr) {
cout<<x->info<<" ";
x = x->right;
while (x != nullptr) {
x = x->left;
There are two important pieces of the following code I would like to highlight. The first piece is the "priming of the stack" before we enter the main loop. To prime the stack we store the left most
path of the tree in the stack from root to leaf. Now when we pop the first value off of the stack for processing, it is the min node and we can immediately display it. We will perform this "primeing
of the stack" for B Tree's as well. The second piece I want to highlight is the jump to the right branch after displaying the nodes contents. In a binary search tree it is simple as we only have one
choice of where to go as binary trees only store one value, but for B Tree's this will require a bit of thought, as the way we traverse the internal nodes is a bit more complicated.
We can now adapt this algorithm to work with the m-ary nodes of B Trees.
The main event: B Tree Traversal
The key to iterartively traversing a B Tree is to remember not only the node we came from, but also the index of the node we were previously processing. To do this, we "tag" the nodes we add to the
stack with the index to proceed from when we remove the node from the stack. To do this tagging, we use an std::pair<int, node*> for the stack item. The traversal once again begins with a priming of
the left most branch, and then jumping into the main loop.
const int M = 8;
typedef bnode<int,int,M>* link;
void savePathToLeaf(stack<pair<int,link>>& sf, link node) {
link x = node;
while (x != nullptr) {
sf.push(make_pair(0, x));
x = x->next[0];
When we remove the top item from the stack, we have the current node and the index of the element we in that node which we are processing. We begin by checking if the current node is a leaf, as we
process leaf nodes differently than internal nodes.
void inorder(link root) {
stack<pair<int, link>> sf;
savePathToLeaf(sf, root);
while (!sf.empty()) {
int idx = sf.top().first;
link x = sf.top().second;
if (x->isleaf) {
} else {
handleInternalNode(sf, x, idx);
As leaf nodes have no children, they have no subtrees to process and as such we simply iterate over the leafs entire array of values without adding any nodes to the stack.
void printLeaf(link x) {
for (int j = 0; j < x->n; j++)
cout<<x->data[j].key()<<" ";
Internal nodes are a bit trickier. We first need to check if the index we are to process is valid. If so, we display the data at that index, saving the next index to the stack. With that done, we
next check if the child node at index+1 is a valid node, and if so we move to that node, marking it as current: this is analagous to moving to the right branch during BST traversal. From here we once
again proceed to the next leaf node by following the left most path, saving the path to the stack as we go.
void handleInternalNode(stack<pair<int,link>>& sf, link x, int idx) {
//print current position in internal node and save where to resume on stack.
if (idx < x->n) {
cout<<x->data[idx].key()<<" ";
if (idx+1 < x->n) {
sf.push(make_pair(idx+1, x));
if (idx+1 < M) { //valid index?
savePathToLeaf(sf, x->next[idx+1]);
Et voila, Iterative in-order traversal.
Thats what I've got for you today. Until next time, Happy Hacking! | {"url":"http://maxgcoding.com/iterative-btree-traversal","timestamp":"2024-11-03T06:00:42Z","content_type":"text/html","content_length":"15597","record_id":"<urn:uuid:f1a9f8dc-7263-4576-8079-c16ba8c26380>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00103.warc.gz"} |
Applications with Radical Equations
Learning Outcome
• Solve application problems involving kinetic energy, volume, and free fall
Kinetic Energy
One way to measure the amount of energy that a moving object (such as a car or roller coaster) possesses is by finding its kinetic energy. The kinetic energy ([latex]E_{k}[/latex], measured in
Joules) of an object depends on the object’s mass (m, measured in kg) and velocity (v, measured in meters per second) and can be written as [latex] v=\sqrt{\frac{2{{E}_{k}}}{m}}[/latex].
What is the kinetic energy of an object with a mass of [latex]1,000[/latex] kilograms that is traveling at [latex]30[/latex] meters per second?
Show Solution
Here is another example of finding the kinetic energy of an object in motion.
Harvester ants found in the southwest of the U.S. create a vast interlocking network of tunnels for their nests. As a result of all this excavation, a very common above-ground hallmark of a harvester
ant nest is a conical mound of small gravel or sand ^[1]
We will use this equation in the next example.
A mound of gravel is in the shape of a cone with the height equal to twice the radius. Calculate the volume of such a mound of gravel whose radius is [latex]3.63[/latex] ft. Use [latex]\pi =3.14[/
Show Solution
Here is another example of finding volume given the radius of a cone.
When you drop an object from a height, the only force acting on it is gravity (and some air friction) and it is said to be in free-fall. We can use math to describe the height of an object in free
fall after a given time because we know how to quantify the force of Earth pulling on us – the force of gravity.
An object dropped from a height of 600 feet has a height, h, in feet after t seconds have elapsed, such that [latex]h=600 - 16{t}^{2}[/latex]. In our next example we will find the time at which the
object is at a given height by first solving for t.
Find the time it takes to reach a height of 400 feet by first finding an expression for t.
Show Solution
Analysis of the Solution
We have made a point of restricting the radicand of radical expressions to non-negative numbers. In the previous example, we divided by a negative number then took the square root to solve for t. In
this example, is it possible to get a negative number in the radicand? In other words, for what values for height would we have an issue where we may be taking the square root of a negative number?
We can use algebra to answer this question.
Let us translate our question into an inequality. Again, for what values of h would we get a negative quantity under the radical?
The radicand is [latex]\frac{h-600}{-16}[/latex], so if we set up an inequality we can solve it for h:
We can interpret this as “when the height is greater than [latex]600[/latex] ft. the radicand will be negative and therefore not a real number.” If you re-read the question, you will see that heights
greater than [latex]600[/latex] do not even make sense, because the object starts at a height of 600 feet and is falling toward the ground, so height is decreasing. Understanding what domain our
variables have is important in application problems so we can get answers that make sense.
Try It
1. Taber, Stephen Welton. The World of the Harvester Ants. College Station: Texas A & M University Press, [latex]1998[/latex]. ↵ | {"url":"https://courses.lumenlearning.com/wm-developmentalemporium/chapter/read-applications-with-radical-equations/","timestamp":"2024-11-09T23:48:00Z","content_type":"text/html","content_length":"56598","record_id":"<urn:uuid:05736726-2090-46e6-9516-34369ac1ab8e>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00266.warc.gz"} |
Combinatorics Calculators | List of Combinatorics Calculators
List of Combinatorics Calculators
Combinatorics calculators give you a list of online Combinatorics calculators. A tool perform calculations on the concepts and applications for Combinatorics calculations.
These calculators will be useful for everyone and save time with the complex procedure involved to obtain the calculation results. You can also download, share as well as print the list of
Combinatorics calculators with all the formulas. | {"url":"https://www.calculatoratoz.com/en/combinatorics-Calculators/CalcList-11118","timestamp":"2024-11-03T21:51:43Z","content_type":"application/xhtml+xml","content_length":"84532","record_id":"<urn:uuid:0c0eb298-8038-4a0c-923c-487e98105c0f>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00353.warc.gz"} |
What's New in Math.NET Numerics 2.6
Math.NET Numerics v2.6, released in July 2013, is focused on filling some gaps around the very basic numerical problems of fitting a curve to data and finding solutions of nonlinear equations. As
usual you'll find a full listing of all changes in the release notes. However, I'd like to take the chance to highlight some important changes, show some code samples and explain the reasoning behind
the changes.
A lot of high quality code contributions made this release possible. Just like last release, I've tried to attribute them directly in the release notes. Thanks again!
Please let me know if these "What's New" articles are useful in this format and whether I should continue to put the together for future releases. See also what's new in the previous version 2.5.
Linear Curve Fitting
Fitting a linear-parametric curve to a set of samples such that the squared errors are minimal has always been possible with the linear algebra toolkit, but it was somewhat complicated to do and
required understanding of the algorithm. See Linear Regression with Math.NET Numerics for an introduction and some examples.
Note: if you need to have the curve go exactly through all your data points, use our Interpolation routines instead.
We now finally provide a shortcut with a few common functions to fit to data, but also a method to fit a linear combination of arbitrary functions. For fitting a simple line it uses an efficient
direct algorithm:
1: var x = new [] { 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 };
2: var y = new [] { 4.986, 2.347, 2.061, -2.995, 2.352, -5.782 }
4: C#: var p = Fit.Line(x, y);
5: var offset = p[0]; // = 7.01013
6: var slope = p[1]; // = -2.08551
8: F#: let offset, slope = Fit.line x y
Otherwise it usually applies an ordinary least squares regression to find the best parameters using a thin QR decomposition (leveraging a native provider like Intel MKL if enabled). This also works
with arbitrary functions, like sine and cosine:
1: F#: let p = (x, y) ||> Fit.linear [(fun _ -> 1.0); (Math.Sin); (Math.Cos)]
2: C#: var p = Fit.LinearCombination(x, y, z => 1.0, Math.Sin, Math.Cos);
4: // p = [ -0.287, 4.02, -1.46 ], hence f: x -> -0.287 + 4.02*sin(x) - 1.46*cos(x)
The intention is to add more special cases for common curves like the logistic function in the future. Like the line they may have more appropriate direct implementations. For now there is one other
special case, for fitting to a polynomial. It returns the best parameters, in ascending order (coefficient for power k has index k) compatible to the Evaluate.Polynomial routine:
1: C#: var coeff = Fit.Polynomial(x, y, 2); // order 2
2: Evaluate.Polynomial(1.2, coeff); // ...
3: F#: let coeff = Fit.polynomial 2 x y
In practice your x values are not always just real numbers. Maybe you need multi-dimensional fitting where the x values are actually arrays, or even full data structures. For such cases we provide a
version that is generic in x and where you can provide a list of functions that accept such x directly without the need to convert to an intermediate double vector first:
1: C#: var p = Fit.LinearMultiDim(xarrays, y, f1, f2, f3, ...);
2: var p = Fit.LinearGeneric(xstructs, y, f1, f2, f3, ...);
3: F#: let p = Fit.linear [f1; f2; f3; ...] xgeneric y
Often after evaluating the best fitting linear parameters you'd actually want to evaluate the function with those parameters. For this scenario we provide a shortcut as well: For each of these
methods there is also a version with a "Func" suffix ("F" in F#) which, instead of the parameters, returns the composed function:
1: F#: let f = Fit.lineF x y
2: [1.0..0.1..2.0] |> List.map f
4: C#: var f = Fit.LinearCombinationFunc(x, y, z => z*z, Math.Sin, SpecialFunctions.Gamma);
5: Enumerable.Range(0,11).Select(x => f(x/10.0))
Root Finding
We now provide basic root finding algorithms. A root of a function x -> f(x) is a solution of the equation f(x)=0. Root-finding algorithms can thus help finding numerical real solutions of arbitrary
equations, provided f is reasonably well-behaved and we already have an idea about an interval [a,b] where we expect a root. As usual, there is a facade class FindRoots for simple scenarios:
The routines usually expect a lower and upper boundary as parameters, and then optionally the accuracy we try to achieve and the maximum number of iterations.
1: C#: FindRoots.OfFunction(x => x*x - 4, -5, 5) // -2.00000000046908
2: C#: FindRoots.OfFunction(x => x*x - 4, -5, 5, accuracy: 1e-14) // -2 (exact)
3: C#: FindRoots.OfFunctionDerivative(x => x*x - 4, x => 2*x, -5, 5) // -2 (exact)
5: F#: FindRoots.ofFunction -5.0 5.0 (fun x -> x*x - 4.0)
6: F#: FindRoots.ofFunctionDerivative -5.0 5.0 (fun x -> x*x - 4.0) (fun x -> 2.0*x)
A NonConvergenceException is thrown if no root can be found by the algorithm.
In practice you'd often want to use a specific well-known algorithm. You'll find them in the RootFinding namespace. Each of these algorithm provides a FindRoot method with similar arguments as those
above. However, the algorithms may sometimes fail to find a root or the function may not actually have a root within the provided interval. Failing to find a root is thus not exactly exceptional.
That's why the algorithms also provide an exception-free TryFindRoot code path with the common Try-pattern as in TryParse.
A simple and robust yet rather slow algorithm, implemented in the Bisection class.
Example: Find the real roots of the cubic polynomial 2x^3 + 4x^2 - 50x + 6:
1: Func<double, double> f = x => Evaluate.Polynomial(x, 6, -50, 4, 2);
2: Bisection.FindRoot(f, -6.5, -5.5, 1e-8, 100); // -6.14665621970684
3: Bisection.FindRoot(f, -0.5, 0.5, 1e-8, 100); // 0.121247371972135
4: Bisection.FindRoot(f, 3.5, 4.5, 1e-8, 100); // 4.02540884774855
6: F#: f |> FindRoots.bisection 100 1e-8 3.5 4.5 // Some(4.0254..)
Note that the F# function returns a float option. Instead of throwing an exception it will simply return None if it fails.
Brent's Method
We use Brent's method as default algorithm, implemented in the Brent class. Brent's method is faster than bisection, but falls back to something close to bisection if the faster approaches
(essentially the secant method and inverse quadratic interpolation) fails and is therefore almost as reliable.
The same example as above, but using Brent's method:
1: Func<double, double> f = x => Evaluate.Polynomial(x, 6, -50, 4, 2);
2: Brent.FindRoot(f, -6.5, -5.5, 1e-8, 100); // -6.14665621970684
3: Brent.FindRoot(f, -0.5, 0.5, 1e-8, 100); // 0.121247371972135
4: Brent.FindRoot(f, 3.5, 4.5, 1e-8, 100); // 4.02540884774855
6: F#: f |> FindRoots.brent 100 1e-8 3.5 4.5 // Some(4.0254..)
Note that there are better algorithms for finding all roots of a polynomial. We plan to add specific polynomial root finding algorithms later on.
The Newton-Raphson method leverages the function's first derivative to converge much faster, but can also fail completely. The pure Newton-Raphson algorithm is implemented in the NewtonRaphson class.
However, we also provide a modified algorithm that tries to recover (instead of just failing) when overshooting, converging too slowly or even when loosing bracketing in the presence of a pole. This
modified algorithms is available in the RobustNewtonRaphson class.
Example: Assume we want to find solutions of x+1/(x-2) == -2, hence x -> f(x) = 1/(x-2)+x+2 with a pole at x==2:
1: Func<double, double> f = x => 1/(x - 2) + x + 2;
2: Func<double, double> df = x => -1/(x*x - 4*x + 4) + 1;
3: RobustNewtonRaphson.FindRoot(f, df, -2, -1, 1e-14, 100, 20); // -1.73205080756888
4: RobustNewtonRaphson.FindRoot(f, df, 1, 1.99, 1e-14, 100, 20); // 1.73205080756888
5: RobustNewtonRaphson.FindRoot(f, df, -1.5, 1.99, 1e-14, 100, 20); // 1.73205080756888
6: RobustNewtonRaphson.FindRoot(f, df, 1, 6, 1e-14, 100, 20); // 1.73205080756888
8: F#: FindRoots.newtonRaphsonRobust 100 20 1e-14 1.0 6.0 f df
9: F#: (f, df) ||> FindRoots.newtonRaphsonRobust 100 20 1e-14 1.0 6.0
Broyden's Method
The quasi-newton method by Broyden, implemented in the Broyden class, may help you to find roots in multi-dimensional problems.
Linear Algebra
As usual there have been quite a few improvements around linear algebra, see the release notes for the complete list. If you've enabled our Intel MKL native linear algebra provider, then eigenvalue
decompositions should be much faster now. Matrices now also support the new F# 3.1 array slicing syntax.
Note that we're phasing out the MathNet.Numerics.IO library and namespace and plan to drop it entirely in v3. We've already replaced it with two new separate NuGet packages and obsoleted all members
of the old library. The new approach with separate libraries makes it possible to introduce specific dependencies e.g. to read and write Excel files, without forcing these dependencies on all of
Math.NET Numerics. We recommend to switch over to the new packages as soon as possible.
We've had a Pearson correlation coefficient routine for a while, but no Covariance routine. In addition to a new Spearman ranked correlation routine, this release finally also adds sample and
population Covariance functions for arrays and IEnumerables.
1: ArrayStatistics.Covariance(new[] {1.2, 1.3, 2.4}, new[] {2.2, 2.3, -4.5})
Multiple items
module List
from Microsoft.FSharp.Collections
type List<'T> =
| ( [] )
| ( :: ) of Head: 'T * Tail: 'T list
interface IEnumerable
interface IEnumerable<'T>
member GetSlice : startIndex:int option * endIndex:int option -> 'T list
member Head : 'T
member IsEmpty : bool
member Item : index:int -> 'T with get
member Length : int
member Tail : 'T list
static member Cons : head:'T * tail:'T list -> 'T list
static member Empty : 'T list
Full name: Microsoft.FSharp.Collections.List<_>
val map : mapping:('T -> 'U) -> list:'T list -> 'U list
Full name: Microsoft.FSharp.Collections.List.map
Multiple items
val double : value:'T -> double (requires member op_Explicit)
Full name: Microsoft.FSharp.Core.ExtraTopLevelOperators.double
type double = System.Double
Full name: Microsoft.FSharp.Core.double | {"url":"https://christoph.ruegg.name/blog/new-in-mathnet-numerics-2-6","timestamp":"2024-11-09T06:08:13Z","content_type":"text/html","content_length":"36667","record_id":"<urn:uuid:c7356c25-3d37-46c9-ab1d-0c59523b4e6e>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00690.warc.gz"} |
How do you integrate int (2-sqrtx)^5/sqrtx using substitution? | HIX Tutor
How do you integrate #int (2-sqrtx)^5/sqrtx# using substitution?
Answer 1
$\int {\left(2 - \sqrt{x}\right)}^{5} / \sqrt{x} \mathrm{dx} = - {\left(2 - \sqrt{x}\right)}^{6} / 3 + C$
#t = 2-sqrtx#
#dt = -dx/(2sqrtx)#
#int (2-sqrtx)^5/sqrtx dx = -2 int t^5dt =-t^6/3 + C#
and undoing the substitution:
#int (2-sqrtx)^5/sqrtx dx = - (2-sqrtx)^6/3+C#
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Answer 2
To integrate (\int (2-\sqrt{x})^5/\sqrt{x}) using substitution, let (u = 2 - \sqrt{x}). Then, (x = (2 - u)^2 = 4 - 4u + u^2), and (du = -\frac{1}{2\sqrt{x}} dx). Substituting these into the integral,
we get: (\int (2 - \sqrt{x})^5/\sqrt{x} dx = -2 \int u^5 du). Integrate (u^5) to get (-\frac{1}{6}u^6 + C). Substitute (u = 2 - \sqrt{x}) back in to get the final answer: (-\frac{1}{6}(2 - \sqrt{x})^
6 + C).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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A right triangle has sides A, B, and C. Side A is the hypotenuse and side B is also a side of a rectangle. Sides A, C, and the side of the rectangle adjacent to side B have lengths of 11 , 4 , and 12 , respectively. What is the rectangle's area? | Socratic
A right triangle has sides A, B, and C. Side A is the hypotenuse and side B is also a side of a rectangle. Sides A, C, and the side of the rectangle adjacent to side B have lengths of #11 #, #4 #,
and #12 #, respectively. What is the rectangle's area?
1 Answer
${\text{Area}}_{\square} \approx 122.96$ sq.units.
Here is my understanding of the arrangement:
By Pythagorean Theorem:
$\textcolor{w h i t e}{\text{XXX}} B = \sqrt{{11}^{2} - {4}^{2}} = \sqrt{105} \approx 10.247$
And the area of the rectangle is
$\textcolor{w h i t e}{\text{XXX}} B \times 12 \approx 10.247 \times 12 \approx 122.96$
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Transposition -- from Wolfram MathWorld
An exchange of two elements of an ordered list with all others staying the same. A transposition is therefore a permutation of two elements. For example, the swapping of 2 and 5 to take the list
123456 to 153426 is a transposition. The permutation symbol permutation. | {"url":"https://mathworld.wolfram.com/Transposition.html","timestamp":"2024-11-13T08:53:58Z","content_type":"text/html","content_length":"51954","record_id":"<urn:uuid:bdd29037-40a2-4498-a23b-e1dedb7e898e>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00708.warc.gz"} |
Optimization of a multi-objective function
Hello, everyone,
Please I am looking for the minimum of the multi-objective function given in the joined file.
If an analytical solution is possible, please, provide it. if it is not possible please say why, and use a numerical method(weighted sum method, Pareto optimality...). f1 is more important than f2
and f3. f2 and f3 are at the same level of importance.
Please give all the calculations and method details.
Answers can only be viewed under the following conditions:
1. The questioner was satisfied with and accepted the answer, or
2. The answer was evaluated as being 100% correct by the judge.
View the answer
1 Attachment
• Leave a comment if you have any questions.
• Good job
• Does the function still convex if we don't neglect h ?
• For the gradient decent method, you do not need convexity. The algorithm will converge. But I don't think putting the h there will make a sensible difference.
• thanks very much
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• answered
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When using a machine learning model to make important decisions, like in healthcare, finance, or engineering, we not only need accurate predictions but also want to know how sure the model is about
its answers [1-3]. CP offers a practical solution for generating certified “error bars”—certified ranges of uncertainty—by post-processing the outputs of a fixed, pre-trained base predictor. This is
crucial for safety and reliability. At the upcoming ISIT 2024 conference, we will present our research work, which aims to bridge the generalization properties of the base predictor with the expected
size of the set predictions, also known as informativeness, produced by CP. Understanding the informativeness of CP is particularly relevant as it can usually only be assessed at test time.
Conformal prediction
The most practical form of CP, known as inductive CP, divides the available data into a training set and a calibration set [4]. We use the training data to train a base model, and the calibration
data to determine the prediction sets around the decisions made by the base model. As shown in Figure 1, a more accurate base predictor, which generalizes better outside the training set, tends to
produce more informative sets when CP is applied.
Our work’s main contribution is a high probability bound on the expected size of the predicted sets. The bound relates the informativeness of CP to the generalization properties of the base model and
the amount of available training and calibration data. As illustrated in Figure 2, our bound predicts that by increasing the amount of calibration data CP’s efficiency converges rapidly to a quantity
influenced by the coverage level, the size of the training set, and the predictor’s generalization performance. However, for finite amount of calibration data, the bound is also influenced by the
discrepancy between the target and empirical reliability measured over the training data set. Overall, the bound justifies a common practice: allocating more data to train the base model compared to
the data used to calibrate it.
Since what really proves the worth of a theory is how well it holds up in real-world testing, we also compare our theoretical findings with numerical evaluations. In our study, we looked at two
classification and regression tasks. We ran CP with various splits of calibration and training data, then measured the average efficiency. As shown in the Figure 3, the empirical results from our
experiments matched up nicely with what our theory predicted in Figure 2.
[1] A. L. Beam and I. S. Kohane, “Big data and machine learning in health care,” JAMA, vol. 319, no. 13, pp. 1317–1318, 2018.
[2] J.. W. Goodell, S. Kumar, W. M. Lim, and D. Pattnaik, “Artificial intelligence and machine learning in finance: Identifying foundations, themes, and research clusters from bibliometric analysis,”
Journal of Behavioral and Experimental Finance, vol. 32, p. 100577, 2021.
[3] L. Hewing, K. P. Wabersich, M. Menner, and M. N. Zeilinger, “Learning-based model predictive control: Toward safe learning in control,” Annual Review of Control, Robotics, and Autonomous Systems,
vol. 3, pp. 269–296, 2020.
[4] V. Vovk, A. Gammerman, and G. Shafer, Algorithmic learning in a random world, vol. 29. Springer, 2005.
With the advent of 5G, cellular systems are expected to play an increasing role in enabling Internet of Things (IoT). This is partly due to the introduction of NarrowBand IoT (NB-IoT), a
cellular-based radio technology allowing low-cost and long-battery life connections, in addition to other IoT protocols that operate in the unlicensed band such as LoRa. However, these protocols
allow for a successful transmission only when a radio resource is used by a single IoT device. Therefore, generally, the amount of resources needed scales with the number of active devices. This
poses a serious challenge in enabling massive connectivity in future cellular systems. In our recent IEEE Transactions on Wireless Communications paper, we tackle this issue.
Suggested Solution
In our new paper, we propose an information-centric radio access technique where IoT devices making (roughly) the same observation of a given monitored quantity, e.g., temperature, transmit using the
same radio resource, i.e., in a non-orthogonal fashion. Thus, the number of radio resources needed scales with the number of possible relevant values observed, e.g., high or low temperature and not
with the number of devices.
Cellular networks are evolving toward Fog-Radio architectures, as shown in Figure 1. In these systems, instead of the entire processing happening at the edge node, radio access related
functionalities can be distributed between the cloud and the edge. We propose that detection in the IoT system under study be implemented at either cloud or edge depending on backhaul conditions and
on the statistics of the observations.
Some Results
One of the important findings of this work is that cloud detection is able to leverage inter-cell interference in order to improve detection performance, as shown in the figure below. This is mainly
due to the fact that devices transmitting the same values in different cells are non-orthogonally superposed and thus, the cloud can detect these values with higher confidence.
More details and results can be found in the complete version of the paper here.
Quantifying the causal flow of information between different components of a system is an important task for many natural and engineered systems, such as neural, genetic, transportation and social
networks. A well-established metric of the information flow between two time sequences Transfer Entropy (TE). The TE equals the mutual information between the past of sequence intrinsic, or
exclusive, information flow from sequence in this paper, the TE captures not only the amount of information on in addition to that already present in the past of
In the same paper, the authors propose to decompose the TE as the sum of an Intrinsic TE (ITE) and a Synergistic TE (STE), and introduce a measure of the ITE based on cryptography. The idea is to
measure the ITE as the size (in bits) of a secret key that can be generated by two parties, one holding the past of sequence
The computation of ITE is generally intractable. To estimate ITE, in recent work, we proposed an estimator, referred to as ITE Neural Estimator (ITENE), of the ITE that is based on variational bound
on the KL divergence, two-sample neural network classifiers, and the pathwise estimator of Monte Carlo gradients.
Some Results
We first apply the proposed estimator to the following toy example. The joint processes
for some threshold λ, where variables
information about
For a real-world example, we apply the estimators at hand to historic data of the values of the Hang Seng Index (HSI) and of the Dow Jones Index (DJIA) between 1990 and 2011 (see Fig. 2). As
illustrated in Fig. 3, both the TE and ITE from the DJIA to the HSI are much larger than in the reverse direction, implying that the DJIA influenced the HSI more significantly than the other way
around for the
given time range. Furthermore, we observe that not all the information flow is estimated to be intrinsic, and hence the joint observation of the history of the DJIA and of the HSI is partly
responsible for the predictability of the HSI from the DJIA.
The full paper will be presented at 2020 International Zurich Seminar on Information and Communication and can be found here. | {"url":"https://blogs.kcl.ac.uk/kclip/category/information-theory/","timestamp":"2024-11-13T18:27:38Z","content_type":"text/html","content_length":"59861","record_id":"<urn:uuid:433c7d05-5fd3-4205-9207-6b1dba68270e>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00816.warc.gz"} |
Puzzle and Possible Formula Solution Inquiry?
I need to create a formula to pull information from a Smartsheet to another with two different columns. For example:
Year Criteria Geography
2023 Yellow North
2022 Orange East
Let's say there are 50 entries that meet these criteria. I need a formula that will pull all 2023 Yellow North data to another Smartsheet. Then, pull all 2022 Orange East data.
I am running into a situation where the "Countif" functionality is only counting in one category.
Has anyone run across this situation in Smartsheet? If so, how did you resolve it? Thanks!
• Are you trying to pull multiple rows over or generate a count? If the COUNTIF is not providing the flexibility or detail you need because it only allows for a single range/criteria set, try the
COUNTIFS function which allows for multiple range/criteria sets to be included.
• I need to reference three columns into a single count. Using the example above, 2023 Yellow North - 50, 2022 Orange East - 50,..etc.
The Countifs function is giving me issues since the columns are not located next to each other. Now, I need to copy a Smartsheet, hide the unnecessary columns, then...I can't figure out the
formula without getting an unparseable error.
• They don't have to be next to each other. You just reference them one at a time.
=COUNTIFS([Column A]:[Column A], @cell = "This", [Column B]:[Column B], @cell = "That")
• Don't I need to put the report in the formula to pull from that sheet?
For example: =Countifs({Report Name}),[Column A]:[Company A],"Color",[Year]:[Year],"2021"
Does that formula make sense? I keep getting an unparsable error so I'm doing something wrong here...
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/108467/puzzle-and-possible-formula-solution-inquiry","timestamp":"2024-11-05T03:44:34Z","content_type":"text/html","content_length":"418253","record_id":"<urn:uuid:db5e75db-7b00-4a21-bf7d-f3f6afb61f63>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00258.warc.gz"} |
Difference Between Sine and Arcsine | Compare the Difference Between Similar Terms
Sine vs Arcsine
Sine is one of the basic trigonometric ratios. It is an inevitable mathematical entity you find in any mathematical theory from the high school level onwards. Just as the Sine gives a value for a
given angle, the angle for a given value can also be calculated. Arcsin or Inverse Sin is that process.
More about Sine
Sin can be defined basically in the context of a right angled triangle. In its basic form as a ratio, it is defined as the length of the side opposite the angle considered (α) divided by the length
of the hypotenuse. sin α= (length of the opposite side )/(length of the hypotenuse).
In a much broader sense, the sin can be defined as a function of an angle, where the magnitude of the angle is given in radians. It is the length of the vertical orthogonal projection of the radius
of a unit circle. In modern mathematics, it is also defined using Taylor series, or as solutions to certain differential equations.
The sine function has a domain ranging from negative infinity to positive infinity of real numbers, with the set of real numbers as the codomain too. But the range of the sine function is between -1
and +1. Mathematically, for all α belonging to real numbers, sin α belongs to the interval [-1,+1];{ ∀ α∈R ,sin α ∈[-1,+1] . That is, sin: R→ [-1,+1]
Following identities hold for the sine function;
Sin (nπ±α) = ± sin α ; When n∈Z and sin (nπ±α) = ± cos α when n∈ 1/2 ,3/2 ,5/2 ,7/2 …… (Odd multiples of 1/2). The reciprocal of the sine function is defined cosecant, with the domain R-{0} and range
More about Arcsine (Inverse Sine)
Inverse sine is known as the arcsine. In the inverse sine function, the angle is calculated for a given real number. In the inverse function, the relationship between the domain and the codomain is
mapped backwards. The domain of the sine acts as the codomain for the arcsine, and the codomain for the sine acts as the domain. It’s a mapping of a real number from [-1,+1] to R
However, one problem with the inverse trigonometric functions is that their inverse is not valid for the whole domain of the considered original function. (Because it violates the definition of a
function). Therefore, the range of the inverse sin is restricted to [-π,+π] so the elements in the domain are not mapped into multiple elements in the codomain. So sin^-1: [-1,+1]→ [-π,+π]
What is the difference between Sine and Inverse Sine (Arcsine)?
• Sine is a basic trigonometric function, and the arcsine is the inverse function of the sine.
• Sine function maps any real number/ angle in radians into a value between -1 and +1, whereas the arcsine maps a real number in [-1,+1] To [-π,+π]
Leave a Reply Cancel reply | {"url":"https://www.differencebetween.com/difference-between-sine-and-vs-arcsine/","timestamp":"2024-11-13T05:00:35Z","content_type":"text/html","content_length":"54319","record_id":"<urn:uuid:32bbfc38-990d-4430-a698-6210f73e4e7f>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00127.warc.gz"} |
Elastic Collision Between Different Masses
Consider two disks of equal size but different mass sliding on a smooth surface. Disk A has mass 250 grams and initial velocity of 2 m/s. Disk B has mass 175 grams and is initially at rest. The disks
collide directly, with the velocity of Disk A along the line connecting their centers. The collision is totally elastic, with a restitution coefficient of 1.0. Determine the velocity of each disk
immediately after the collision and show that the total kinetic energy is the same before and after the collision.
The coefficient of restitution, Cr, is the ratio of the velocity of separation to the velocity of approach in a collision. In the case of an elastic collision it is 1.0. For two moving objects, A and
B, Cr = (vA-vB)/(vBi-vAi) where the i indicates initial conditions. Here is a good opportunity to get the signs messed up. The velocity of approach carries the opposite sign from the velocity of
separation so the order of the terms in numerator and denominator must be reversed. In this case (vBi-vAi) = (vA-vB) since Cr=1.0. We also know that momentum is conserved so mA*vAi + mB*vBi = mA*vA +
mB*vB. This gives us two equations in the two unknown post collision velocities, vA and vB.
Two solve this set of equations let's use the Cr relationship to get vA in terms of vB. We know that (0 - 2 m/s)=(vA-vB) or vA = vB-2. Plugging this into the momentum equation we get mA*vAi + mB*vBi
= mA*(vB-2) + mB*vB = (mA + mB)*vB -2*mA. So vB = (mA*vAi+mB*vBi+2*mA)/(mA+mB). Putting in the known quantities we get vB = (.25*2+0+2*.25)/(.250+.175) = 1/.425 =
2.35294117647058823529411764705882 m/s.
Pardon the absurd number of decimal places but we are trying to make a point here.
Since vA=vB-2, vA=0.35294117647058823529411764705882/s
The kinetic energy of the system before the collision was all in disk A. KE=1/2*.25*2^2 Joules = 0.5 Joules. After the collision the kinetic energy is shared between the disks. The disk A kinetic
energy, KEA is
1/2*.25*0.35294117647058823529411764705882^2 Joules =
0.0155709342560553633217993079584775 Joules.
KEB is 1/2*.175*2.35294117647058823529411764705882^2 Joules =
0.484429065743944636678200692041522 Joules.
The total KE after the collision is
0.499999999999999999999999999999522 Joules,
about as close to the original value as we might expect.
This information is brought to you by M. Casco Associates, a company dedicated to helping humankind reach the stars through understanding how the universe works. My name is James D. Jones. If I can
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Science:Math Exam Resources/Courses/MATH103/April 2016/Question 07 (b) (ii)
MATH103 April 2016
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q3 • Q4 (a) • Q4 (b) • Q5 (a) • Q5 (b) • Q6 (a) (i) • Q6 (a) (ii) • Q6 (a) (iii) • Q6 (a) (iv) • Q6 (a) (v) • Q6 (b)
(i) • Q6 (b) (ii) • Q6 (b) (iii) • Q7 (a) (i) • Q7 (a) (ii) • Q7 (b) (i) • Q7 (b) (ii) • Q8 (a) • Q8 (b) • Q8 (c) • Q8 (d) • Q9 (a) • Q9 (b) • Q9 (c) • Q9 (d) •
Question 07 (b) (ii)
Consider the power series: ${\displaystyle \sum _{n=1}^{\infty }{\frac {2^{-n}}{\sqrt {n}}}x^{n}}$.
(ii) Find all values of ${\displaystyle x}$ such that the power series converges.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is
correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still
stuck, go for the next hint.
Hint 1
When ${\displaystyle x=2,}$ consider the p-series test.
Hint 2
When ${\displaystyle x=-2,}$ apply the alternating series test.
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
• If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you
are stuck or if you want to check your work.
• If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem
and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.
Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies.
From part (i), we already know that the given series converges when ${\displaystyle |x|<2}$ and diverges when ${\displaystyle |x|>2.}$
Therefore, it is enough to consider its convergence/divergence when ${\displaystyle |x|=2,}$ i.e., when ${\displaystyle x=\pm 2.}$
When ${\displaystyle x=2,}$ the given series diverges since
${\displaystyle \sum _{n=1}^{\infty }{\frac {2^{-n}}{\sqrt {n}}}\cdot 2^{n}=\sum _{n=1}^{\infty }{\frac {1}{\sqrt {n}}}=+\infty ,}$
where the last equality follows from the p-series test (with p = 1/2).
On the other hand, when ${\displaystyle x=-2,}$ the given series can be written as an alternating series:
${\displaystyle \sum _{n=1}^{\infty }{\frac {2^{-n}}{\sqrt {n}}}\cdot (-2)^{n}=\sum _{n=1}^{\infty }{\frac {2^{-n}}{\sqrt {n}}}(-1)^{n}\cdot 2^{n}=\sum _{n=1}^{\infty }(-1)^{n}{\frac {1}{\sqrt
Then, since ${\displaystyle b_{n}={\frac {1}{\sqrt {n}}}}$ is a decreasing sequence converging to 0, by the alternating series test, the series converges.
Therefore, the given series converges when ${\displaystyle \color {blue}x\in [-2,2).}$ | {"url":"https://wiki.ubc.ca/Science:Math_Exam_Resources/Courses/MATH103/April_2016/Question_07_(b)_(ii)","timestamp":"2024-11-05T09:01:23Z","content_type":"text/html","content_length":"57402","record_id":"<urn:uuid:c7763e16-1d6a-48de-bf05-c4776126a6d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00801.warc.gz"} |
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Definition of inductive reasoning
Inductive reasoning is a reasoning method that recognizes patterns and evidence from specific occurrences to reach a general conclusion. The general unproven conclusion we reach using inductive
reasoning is called a conjecture or hypothesis.
With inductive reasoning, the conjecture is supported by truth but is made from observations about specific situations. So, the statements may not always be true in all cases when making the
conjecture. Inductive reasoning is often used to predict future outcomes. Conversely, deductive reasoning is more certain and can be used to draw conclusions about specific circumstances using
generalized information or patterns.
Deductive reasoning is a reasoning method that makes conclusions based on multiple logical premises which are known to be true.
The difference between inductive reasoning and deductive reasoning is that, if the observation is true, then the conclusion will be true when using deductive reasoning. However, when using inductive
reasoning, even though the statement is true, the conclusion won’t necessarily be true. Often inductive reasoning is referred to as the "Bottom-Up" approach as it uses evidence from specific
scenarios to give generalized conclusions. Whereas, deductive reasoning is called the "Top-Down" approach as its draws conclusions about specific information based on the generalized statement.
Let’s understand it by taking an example.
Deductive Reasoning
Consider the true statements – Numbers ending with 0 and 5 are divisible by 5. Number 20 ends with 0.
Conjecture – Number 20 must be divisible by 5.
Here, our statements are true, which leads to true conjecture.
Inductive Reasoning
True statement – My dog is brown. My neighbor’s dog is also brown.
Conjecture – All dogs are brown.
Here, the statements are true, but the conjecture made from it is false.
Caution: It is not always the case that the conjecture is true. We should always validate it, as it may have more than one hypothesis that fits the sample set. Example: ${x}^{2}>x$ . This is correct
for all integers except 0 and 1.
Examples of inductive reasoning
Here are some examples of inductive reasoning that show how a conjecture is formed.
Find the next number in the sequence $1,2,4,7,11$ by inductive reasoning.
Observe: We see the sequence is increasing.
Here the number increases by $1,2,3,4$ respectively.
Conjecture: The next number will be 16, because $11+5=16.$
Types of inductive reasoning
The different types of inductive reasonings are categorized as follows:
This form of reasoning gives the conclusion of a broader population from a small sample.
Example: All doves I have seen are white. So, most of the doves are probably white.
Here, the conclusion is drawn based on a statistical representation of the sample set.
Example: 7 doves out of 10 I have seen are white. So, about 70% of doves are white.
This is similar to statistical induction, but additional information is added with the intention of making the hypothesis more accurate.
Example: 7 doves out of 10 in the U.S. are white. So about 70% of doves in the U.S. are white.
This type of reasoning forms a causal connection between evidence and hypothesis.
Example: I have always seen doves during winter; so, I will probably see doves this winter.
This inductive method draws conjecture from similar qualities or features of two events.
Example: I have seen white doves in the park. I also have seen white geese there. So, doves and geese are both of the same species.
This inductive reasoning predicts a future outcome based on past occurrence(s).
Example: There are always white doves in the park. So, the next dove which comes will also be white.
Methods of inductive reasoning
Inductive reasoning consists of the following steps:
1. Observe the sample set and identify the patterns.
2. Make a conjecture based on the pattern.
3. Verify the conjecture.
How to make and test conjectures?
To find the true conjecture from provided information, we first should learn how to make a conjecture. Also, to prove the newly formed conjecture true in all similar circumstances, we need to test it
for other similar evidence.
Let us understand it by taking an example.
Derive a conjecture for three consecutive numbers and test the conjecture.
Remember: Consecutive numbers are numbers that come after another in increasing order.
Consider groups of three consecutive numbers. Here these numbers are integers.
To make a conjecture, we first find a pattern.
Pattern: $1+2+3=6⇒6=2×3$
As we can see this pattern for the given type of numbers, let’s make a conjecture.
Conjecture: The sum of three consecutive numbers is equal to three times the middle number of the given sum.
Now we test this conjecture on another sequence to consider if the derived conclusion is in fact true for all consecutive numbers.
Test: We take three consecutive numbers $50,51,52.$
A conjecture is said to be true if it is true for all the cases and observations. So if any one of the cases is false, the conjecture is considered false. The case which shows the conjecture is false
is called the counterexample for that conjecture.
It is sufficient to show only one counterexample to prove the conjecture false.
The difference between two numbers is always less than its sum. Find the counterexample to prove this conjecture false.
Let us consider two integer numbers say -2 and -3.
Sum: $\left(-2\right)+\left(-3\right)=-5$
Difference: $\left(-2\right)-\left(-3\right)=-2+3=1\phantom{\rule{0ex}{0ex}}\therefore 1less -5$
Here the difference between two numbers –2 and –3 is greater than its sum. So, the given conjecture is false.
Examples of making and testing conjectures
Let’s once again take a look at what we learned through examples.
Make a conjecture about a given pattern and find the next one in the sequence.
Observation: From the given pattern, we can see that every quadrant of a circle turns black one by one.
Conjecture: All quadrants of a circle are being filled with color in a clockwise direction.
Next step: The next pattern in this sequence will be:
Make and test conjecture for the sum of two even numbers.
Consider the following group of small even numbers.
Step 1: Find the pattern between these groups.
From the above, we can observe that the answer of all the sums is always an even number.
Step 2: Make a conjecture from step 2.
Conjecture: The sum of even numbers is an even number.
Step 3: Test the conjecture for a particular set.
Consider some even numbers, say, $68,102.$
The answer to the above sum is an even number. So the conjecture is true for this given set.
To prove this conjecture true for all even numbers, let’s take a general example for all even numbers.
Step 4: Test conjecture for all even numbers.
Consider two even numbers in the form: $x=2m,y=2n$, where $x,y$ are even numbers and $m,n$ are integers.
Hence, it is an even number, as it is a multiple of 2 and $m+n$ is an integer.
So our conjecture is true for all even numbers.
Show a counterexample for the given case to prove its conjecture false.
Two numbers are always positive if the product of both those numbers is positive.
Let us first identify the observation and hypothesis for this case.
Observation: The product of the two numbers is positive.
Hypothesis: Both numbers taken must be positive.
Here, we have to consider only one counterexample to show this hypothesis false.
Let us take into consideration the integer numbers. Consider –2 and –5.
Here, the product of both the numbers is 10, which is positive. But the chosen numbers –2 and –5 are not positive. Hence, the conjecture is false.
Advantages and limitations of inductive reasoning
Let's take a look at some of the advantages and limitations of inductive reasoning.
• Inductive reasoning allows the prediction of future outcomes.
• This reasoning gives a chance to explore the hypothesis in a wider field.
• This also has the advantage of working with various options to make a conjecture true.
• Inductive reasoning is considered to be predictive rather than certain.
• This reasoning has limited scope and, at times, provides inaccurate inferences.
Application of inductive reasoning
Inductive reasoning has different uses in different aspects of life. Some of the uses are mentioned below:
• Inductive reasoning is the main type of reasoning in academic studies.
• This reasoning is also used in scientific research by proving or contradicting a hypothesis.
• For building our understanding of the world, inductive reasoning is used in day-to-day life.
Inductive Reasoning — Key takeaways
• Inductive reasoning is a reasoning method that recognizes patterns and evidence to reach a general conclusion.
• The general unproven conclusion we reach using inductive reasoning is called a conjecture or hypothesis.
• A hypothesis is formed by observing the given sample and finding the pattern between observations.
• A conjecture is said to be true if it is true for all the cases and observations.
• The case which shows the conjecture is false is called a counterexample for that conjecture.
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Frequently Asked Questions about Inductive Reasoning
What is inductive reasoning in math?
Inductive reasoning is a reasoning method that recognizes patterns and evidence to reach a general conclusion.
What is an advantage of using inductive reasoning?
Inductive reasoning allows the prediction of future outcomes.
What is inductive reasoning in geometry?
Inductive reasoning in geometry observes geometric hypotheses to prove results.
Which area is inductive reasoning applicable?
Inductive reasoning is used in academic studies, scientific research, and also in daily life.
What are the disadvantages of applying inductive reasoning?
Inductive reasoning is considered to be predictive rather than certain. So not all predicted conclusions can be true.
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What Every Engineer Should Know About Data-Driven Analytics
Book description
What Every Engineer Should Know About Data-Driven Analytics provides a comprehensive introduction to the machine learning theoretical concepts and approaches that are used in predictive data
analytics through practical applications and case studies.
Product information
• Title: What Every Engineer Should Know About Data-Driven Analytics
• Author(s):
• Release date: April 2023
• Publisher(s): CRC Press
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2 Digit By 2 Digit Multiplication Worksheets Printable
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Realized Volatility |Realized Volatility vs Implied Volatility
Realized volatility (RV) in options refers to the actual volatility of an underlying asset’s price movement over a specific period of time, as opposed to implied volatility (IV), which is the
market’s expectation of the future volatility of the underlying asset.
• RV is often used in the pricing and valuation of options. It is calculated by measuring the standard deviation of the asset’s returns over a given period, usually using historical price data. This
information can be used to estimate the future volatility of the asset and can be incorporated into option pricing models, such as the Black-Scholes model.
• In options trading, RV can also be used to assess the accuracy of an option’s IV. If the RV differs significantly from the IV used in pricing the option, it can create opportunities for traders to
take advantage of mispricing in the market.
• It’s worth noting that RV is retrospective and is therefore unable to predict future volatility with certainty. However, it can provide valuable information about the volatility of the underlying
asset and can be a useful tool for options traders and investors.
How is Realized Volatility calculated
Here are the general steps to calculate realized volatility in options:
• Obtain historical price data of the underlying asset: This can be obtained from a financial data provider or through a trading platform.
• Calculate daily returns: Calculate the percentage change in the price of the underlying asset for each trading day in the historical data. This can be done using the following formula:• Daily
Return = (Price Today – Price Yesterday) / Price Yesterday
• Calculate the standard deviation of daily returns: The standard deviation measures the dispersion of the daily returns around their average value. This can be calculated using a statistical
software or Excel function such as STDEV.
• Annualize the standard deviation: Multiply the standard deviation by the square root of the number of trading days in a year. This assumes that volatility is constant over time and that there are
252 trading days in a year.
• Annualized Volatility = Standard Deviation of Daily Returns * Sqrt(252)
• Use the annualized volatility to calculate the implied volatility of an options contract: Compare the calculated annualized volatility to the IV of an options contract to determine whether the
options contract is overpriced or underpriced relative to historical volatility.
Note that there are different methods to calculate RV in options and the specific approach may depend on the data and the analysis being conducted.
Relation between Realized Volatility and Implied Volatility
IV and RV are two different measures of volatility used in options trading.
• IV is a forward-looking measure that is based on the market price of the option. It represents the expected volatility of the underlying asset over the life of the option, as implied by the market
participants. IV is an important input parameter in option pricing models such as the Black-Scholes model.
• RV, on the other hand, is a historical measure that is based on the actual price movements of the underlying asset over a specific time period. It measures the actual volatility that occurred in
the past.
• The relationship between IV and RV is important in options trading because it can provide insight into the market’s expectations for future price movements of the underlying asset.
• If IV is higher than RV, it suggests that the market is expecting higher volatility in the future than what has occurred in the past. This could indicate that the option is overpriced, or that
there is a higher level of uncertainty about the future price movements of the underlying asset.
• Conversely, if IV is lower than RV, it suggests that the market is expecting lower volatility in the future than what has occurred in the past. This could indicate that the option is underpriced,
or that there is a lower level of uncertainty about the future price movements of the underlying asset.
Concisely, the relationship between IV and RV can provide important information to traders and investors in making informed decisions about option pricing and risk management.
Key difference between Realized Volatility and Implied Volatility
• The main difference between implied and realized volatility is that IV is based on market expectations for future price movements of the underlying asset, while RV is based on past price movements
of the underlying asset.
• Another difference is that IV can vary depending on the strike price and expiration date of the option, while RV is calculated over a specific time period.
• IV is generally higher than RV because market participants tend to be more cautious and price in more uncertainty about future price movements. However, in some cases, realized volatility can be
higher than implied volatility, indicating that the market underestimated the potential price movements of the underlying asset.
Overall, both implied and realized volatility are important measures in options trading, and understanding the relationship between them can help traders make informed decisions about option pricing
and risk management.
Read More:
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The Stacks project
Definition 66.12.5. Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. Let $Z \subset |X|$ be a closed subset. An algebraic space structure on $Z$ is given by a closed subspace $Z'$ of
$X$ with $|Z'|$ equal to $Z$. The reduced induced algebraic space structure on $Z$ is the one constructed in Lemma 66.12.3. The reduction $X_{red}$ of $X$ is the reduced induced algebraic space
structure on $|X|$.
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ISTQB Foundation Level Exam Crash Course Part-9 - Software Testing Genius
ISTQB Foundation Level Exam Crash Course Part-9
This is Part 9 of 35 containing 5 Questions (Q. 41 to 45) with detailed explanation as expected in ISTQB Foundation Level Exam Latest Syllabus updated in 2011
Deep study of these 175 questions shall be of great help in getting success in ISTQB Foundation Level Exam
Q. 41: What is the purpose of Equivalence partitioning the specification-based test case design technique
Equivalence partitioning is based on a very simple idea: it is that in many cases the inputs to a program can be �chunked� into groups of similar inputs.
For example, a program that accepts integer values can accept as valid any input that is an integer (i.e. a whole number) and should reject anything
else (such as a real number or a character). The range of integers is infinite, though the computer will limit this to some finite value in both the negative and positive directions (simply because
it can only handle numbers of a certain size; it is a finite machine). Let us take an example, that the program accepts any value between −10,000 and +10,000 (computers actually represent numbers in
binary form, which makes the numbers look much less like the ones we are familiar with, but we will stick to a familiar representation).If we imagine a program that separates numbers into two groups
according to whether they are positive or negative the total range of integers could be split into three �partitions�: the values that are less than zero; zero; and the values that are greater than
zero. Each of these is known as an �equivalence partition� because every value inside the partition is exactly equivalent to any other value as far as our program is concerned. So if the computer
accepts −2,905 as a valid negative integer we would expect it also to accept −3. Similarly, if it accepts 100 it should also accept 2,345 as a positive integer.
Note that we are treating zero as a special case. We could, if we chose to, include zero with the positive integers, but my rudimentary specification did not specify that clearly, so it is really
left as an undefined value (and it is not untypical to find such ambiguities or undefined areas in specifications). It often suits us to treat zero as a special case for testing where ranges of
numbers are involved; we treat it as an equivalence partition with only one member. So we have three valid equivalence partitions in this case.
The equivalence partitioning technique takes advantage of the properties of equivalence partitions to reduce the number of test cases we need to write. Since all the values in an equivalence
partition are handled in exactly the same way by a given program, we need only test one of them as a representative of the partition. In the example given, then, we need any positive integer, any
negative integer and zero. We generally select values somewhere near the middle of each partition, so we might choose, say, −5,000, 0 and 5,000 as our representatives. These three test inputs would
exercise all three partitions and the theory tells us that if the program treats these three values correctly it is very likely to treat all of the other values, all 19,998 of them in this case,
The partitions we have identified now are called valid equivalence partitions because they partition the collection of valid inputs, but there are other possible inputs to this program that would not
be valid – real numbers, for example. We also have two input partitions of integers that are not valid: integers less than −10,000 and integers greater than 10,000. We should test that the program
does not accept these, which is just as important as the program accepting valid inputs.
Non-valid partitions are also important to test. If you think about the example we have been using you will soon recognize that there are far more possible non-valid inputs than valid ones, since all
the real numbers (e.g. numbers containing decimals) and all characters are non-valid in this case. It is generally the case that there are far more ways to provide incorrect input than there are to
provide correct input; as a result, we need to ensure that we have tested the program against the possible non-valid inputs. Here again equivalence partitioning comes to our aid: all real numbers are
equally non-valid, as are all alphabetic characters. These represent two non-valid partitions that we should test, using values such as 9.45 and �r� respectively. There will be many other possible
non-valid input partitions, so we may have to limit the test cases to the ones that are most likely to crop up in a real situation.
<<<<<< =================== >>>>>>
Q. 42: Describe some example of Equivalence Partitions
Valid input: integers in the range 100 to 999.
# Valid partition: 100 to 999 inclusive.
# Non-valid partitions: less than 100, more than 999, real (decimal) numbers and non-numeric characters.
Valid input: names with up to 20 alphabetic characters.
# Valid partition: strings of up to 20 alphabetic characters.
# Non-valid partitions: strings of more than 20 alphabetic characters, strings containing non-alphabetic characters.
<<<<<< =================== >>>>>>
Q. 43: What is the purpose of Boundary Value Analysis the specification-based test case design technique
There is a common mistakes that programmers make is that errors tend to cluster around boundaries. For example, if a program should accept a sequence of numbers between 1 and 10, the most likely
fault will be that values just outside this range are incorrectly accepted or that values just inside the range are incorrectly rejected. In the programming world these faults coincide with
particular programming structures such as the number of times a program loop is executed or the exact point at which a loop should stop executing.
This works well with our equivalence partitioning idea because partitions must have boundaries. A partition of integers between 1 and 99, for instance, has a lowest value, 1, and a highest value, 99.
These are called boundary values. Actually they are called valid boundary values because they are the boundaries on the inside of a valid partition. What about the values on the outside? Yes, they
have boundaries too. So the boundary of the non-valid values at the lower end will be zero because it is the first value you come to when you step outside the partition at the bottom end. (You can
also think of this as the highest value inside the non-valid partition of integers that are less than one, of course.) At the top end of the range we also have a non-valid boundary value, 100.
This is the boundary value technique, more or less. For most practical purposes the boundary value analysis technique needs to identify just two values at each boundary. For reasons that need not
detain us here there is an alternative version of the technique that uses three values at each boundary.
For this variant, which is the one documented in BS 7925-2, we include one more value at each boundary when we use boundary value analysis: the rule is that we use the boundary value itself and one
value (as close as you can get) either side of the boundary.
So, in this case lower boundary values will be 0, 1, 2 and upper boundary values will be 98, 99, 100. What does �as close as we can get� mean? It means take the next value in sequence using the
precision that has been applied to the partition. If the numbers are to a precision of 0.01, for example, the lower boundary values would be 0.99, 1.00, 1.01 and the upper boundary values would be
98.99, 99.00, 99.01.
<<<<<< =================== >>>>>>
Q. 44: Describe some example of Boundary Values
1) The boiling point of water: The boundary is at 100 degrees Celsius, so for the 3 Value Boundary approach the boundary values will be 99 degrees, 100 degrees, 101 degrees�unless you have a very
accurate digital thermometer, in which case they could be 99.9 degrees, 100.0 degrees, 100.1 degrees. For the 2 value approach the corresponding values would be 100 and 101.
2) Exam pass: If an exam has a pass boundary at 40 per cent, merit at 60 per cent and distinction at 80 per cent the 3 value boundaries would be 39, 40, 41 for pass, 59, 60, 61 for merit, 79, 80, 81
for distinction. It is unlikely that marks would be recorded at any greater precision than whole numbers. The 2 value equivalents would be 39 and 40, 59 and 60, and 79 and 80 respectively.
<<<<<< =================== >>>>>>
Q. 45: What is the purpose of Decision Table Testing the specification-based test case design technique
Specifications generally contain business rules to define the functions of the system and the conditions under which each function operates. Individual decisions are usually simple, but the overall
effect of these logical conditions can become quite complex.
As testers we need to be able to assure ourselves that every combination of these conditions that might occur has been tested, so we need to capture all the decisions in a way that enables us to
explore their combinations. The mechanism usually used to capture the logical decisions is called a decision table.
A decision table lists all the input conditions that can occur and all the actions that can arise from them. These are structured into a table as rows, with the conditions at the top of the table and
the possible actions at the bottom.
Business rules, which involve combinations of conditions to produce some combination of actions, are arranged across the top. Each column therefore represents a single business rule (or just �rule�)
and shows how input conditions combine to produce actions. Thus each column represents a possible test case, since it identifies both inputs and expected outputs. The schematic structure of a
decision table is shown in the following table.
│ │ Business rule 1 │ Business rule 2 │ Business rule 3 │
│ Condition 1 │ T │ F │ T │
│ Condition 2 │ T │ T │ T │
│ Condition 3 │ T │ “-“ │ F │
│ Action 1 │ Y │ N │ Y │
│ Action 2 │ N │ Y │ Y │
Business rule 1 requires all conditions to be true to generate action 1. Business rule 2 results in action 2 if condition 1 is false and condition 2 is true but does not depend on condition 3.
Business rule 3 requires conditions 1 and 2 to be true and condition 3 to be false.
In reality the number of conditions and actions can be quite large, but usually the number of combinations producing specific actions is relatively small. For this reason we do not enter every
possible combination of conditions into our decision table, but restrict it to those combinations that correspond to business rules – this is called a limited entry decision table to distinguish it
from a decision table with all combinations of inputs identified.
Part – 10 of the Crash Course – ISTQB Foundation Exam
Access The Full Database of Crash Course Questions for ISTQB Foundation Level Certification
An expert on R&D, Online Training and Publishing. He is M.Tech. (Honours) and is a part of the STG team since inception.
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Worksheet #11 More Verbal Problems Involving Trigonometry
Worksheet #11 – More Verbal Problems Involving Trigonometry
Directions: On a separate sheet of paper, answer all of these questions. SHOW ALL WORK!
Part 1: More Verbal Problems
For 1-20, answer the following questions in a sentence.
1. A fireman rests his ladder against a building, making a 57° angle with the ground. The bottom of the ladder is 28 feet from the base of the building. How long is the ladder?
1. A six-meter-long ladder leans against a building. If the ladder makes an angle of 60° with the ground, how far up the wall does the ladder reach? How far from the wall is the base of the ladder?
1. A five-meter-long ladder leans against a wall, with the top of the ladder being four meters above the ground. What is the approximate angle that the ladder makes with the ground?
1. A ramp has an angle of inclination of 20. It has a vertical height of 1.8 m. What is the length, L meters, of the ramp?
1. A damaged tree is supported by a guy wire 10.0 m long. The wire makes an angle of 61 with the ground. Calculate the height at which the guy wire is attached to the tree.
1. A helicopter is hovering above a road at an altitude of 24 m. At a certain time, the distance between the helicopter and a car on the road is 45.0 m. Calculate the angle of elevation of the
helicopter from the car
1. A tree that is 8.5 m tall casts a shadow 6m long. At what angle are the sun’s rays hitting the ground?
1. John wants to measure the height of a tree. He walks exactly 100 feet from the base of the tree and looks up. The angle from the ground to the top of the tree is 33º. How tall is the tree?
1. A tree casts a shadow on the ground. If the shadow is 95 feet long and the angle of the sun is 32º. How tall is the tree?
1. A 16 foot ladder leans against a building. If the bottom of the ladder is 3 feet from the building, what angle does the ladder make with the ground?
1. A pilot of an airplane in flight looks down at a point on the ground that is some distance away. The angle of depression is 28°, and the plane's altitude is 1200 meters. What is the distance from
the pilot to the point on the ground?
1. Suppose a tree 50 feet in height casts a shadow of length 60 feet. What is the angle of elevation from the end of the shadow to the top of the tree with respect to the ground?
1. A building is 50 feet high. At a distance away from the building, an observer notices that the angle of elevation to the top of the building is 41º. How far is the observer from the base of the
1. An airplane is flying at a height of 2 miles above the ground. The distance along the ground from the airplane to the airport is 5 miles. What is the angle of depression from the airplane to the
1. A bird sits on top of a lamppost. The angle of depression from the bird to the feet of an observer standing away from the lamppost is 35º. The distance from the bird to the observer is 25 meters.
How tall is the lamppost?
1. If your distance from the foot of the tower is 20 m and the angle of elevation is 40, find the height of the tower.
1. A ladder must reach the top of a building. The base of the ladder will be 25′ from the base of the building. The angle of elevation from the base of the ladder to the top of the building is 64°.
Find the height of the building (h) and the length of the ladder ( m).
1. From a point on the ground 25 feet from the foot of a tree, the angle of elevation of the top of the tree is 32º. Find to the nearest foot, the height of the tree.
1. From the top of a barn 25 feet tall, you see a cat on the ground. The angle of depression of the cat is 40º. How many feet, to the nearest foot, must the cat walk to reach the barn?
1. A ladder 6 feet long leans against a wall and makes an angle of 71º with the ground. Find to the nearest tenth of a foot how high up the wall the ladder will reach.
Part 2: Angle of Elevation (Find the missing ANGLE)
1. Find the angle of elevation of the sun when a tree that is 10 yards tall casts a shadow 14 yards long.
2. A large totem pole in Alaska is 160 feet tall. On a particular day at noon, it casts a 222 ft shadow. What is the sun’s angle of elevation at that time?
Part 3: Angle of Elevation (Find the missing SIDE)
1. From a point on level ground 25 feet from the base of a tower, the angle of elevation to the top of the tower is 78 degrees, as shown in the accompanying diagram. Find the height of the tower, to
the nearest tenth of a foot.
1. A tree casts a shadow that is 20 feet long. The angle of elevation from the end of the shadow to the top of the tree is 66 degrees. Determine the height of the tree to the nearest foot.
1. To find the height of a pole, a surveyor moves 80 feet away from the base of the pole and measures the angle of elevation to the top of the pole to be 57 degrees. What is the height of the pole?
Round to the nearest foot.
Part 4: Angles of Depression (Find the missing ANGLE)
1. A spotlight is mounted on a wall 7.4 feet above a security desk in an office building. It is used to light an entrance door 9.3 feet from the desk. Find the angle of depression from the spotlight
to the entrance door.
Part 5: Angles of Depression (Find the missing SIDE)
1. A person measures the angle of depression from the top of a wall to a point on the ground. The point is located on level ground 62 feet from the base of the wall and the angle of depression is 52
degrees. How high is the wall, to the nearest tenth of a foot?
1. A lookout spots a fire from a 32 m high tower. The angle of depression from the tower to the fire is 13 degrees. To the nearest meter, how far is the fire from the base of the tower?
Part 6: Angles of Elevation and Depression (Mixed)
1. An airplane that is 500 m in the air spots a boat at an 11 degree angle of depression. If the plane was to drop down a ladder to the water, how far would a man have to swim from the dropped
ladder to the boat?
1. If a tree 28 meters tall casts a shadow 32 meters long, what is the angle of elevation of the sun to the nearest degree?
1. A ship on the ocean surface detects a sunken ship on the ocean floor at an angle of depression of 50 degrees. The distance between the ship on the surface and the sunken ship on the ocean floor
is 200 meters. If the ocean floor is level in this area, how far above the ocean floor, to the nearest meter, is the ship on the surface?
1. The angle of elevation from a point 25 feet from the base of a tree on level ground to the top of the tree is 30 degrees. Which equation can be used to find the height of the tree?
a) tan 30\
b) tan 25 = ,30-?.
c) sin x = ,
25d) tan 30 = ,?-25.
Part 7: Angles of Elevation and Depression (Challenge)
1. Suppose the angle of elevation from a swimmer to the top of a cliff is 45 degrees. The swimmer is x feet from the bottom of the cliff and the cliff is y feet high. Find two possible values for x
and y.
1. A lighthouse is built on the edge of a cliff near the ocean, as shown in the accompanying diagram. From a boat located 200 feet from the base of the cliff, the angle of elevation to the top of
the cliff is 18 degrees and the angle of elevation to the top of the lighthouse is 28 degrees. What is the height of the lighthouse, x, to the nearest tenth of a foot?
Part 8: Mixed Review
For 35-46,
answer the following questions
1. Simplify:
1. Simplify:
1. Simplify:
1. Find the missing side.
1. Find the missing side.
1. Find the missing angle.
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Exact Joint Probabilities for Low n
exactbin {crossrun} R Documentation
Exact Joint Probabilities for Low n
Exact joint probabilities, for low n, of the number of crossings C and the longest run L in n independent Bernoulli observations with success probability p. Probabilites are multiplied by 2^{n-1}.
exactbin(n, p = 0.5, prec = 120)
n number, length of seqience, at most 6.
p success probability.
prec precision in mpfr calculations. Default 120.
mpfr array
exactbin(n=5, p=0.6)
version 0.1.1 | {"url":"https://search.r-project.org/CRAN/refmans/crossrun/html/exactbin.html","timestamp":"2024-11-02T20:31:20Z","content_type":"text/html","content_length":"2344","record_id":"<urn:uuid:8cf7b2be-e313-4f86-9252-2bc1cd8e7936>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00072.warc.gz"} |
• Choice K
A square root of a number a is a number y such that y^2 = a; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is a. $$\sqrt{a} = 36 \\ ⇒a = 36^2
= 1296$$
Skills you are tested for:
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mp_arc 00-220
00-220 Simon B.
Resonances in one dimension and Fredholm determinants (53K, LaTeX) May 10, 00
Abstract , Paper (src), View paper (auto. generated ps), Index of related papers
Abstract. We discuss resonances for Schrodinger operators in whole- and half-line problems. One of our goals is to connect the Fredholm determinant approach of Froese to the Fourier transform
approach of Zworski. Another is to prove a result on the number of antibound states --- namely, in a half-line problem there are an odd number of antibound states between any two bound states.
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Voltages sag and swell mitigation using DPFC for Multibus system
Volume 02, Issue 10 (October 2013)
Voltages sag and swell mitigation using DPFC for Multibus system
DOI : 10.17577/IJERTV2IS100884
Download Full-Text PDF Cite this Publication
Shanker.N, B.Sampath Kumar, 2013, Voltages sag and swell mitigation using DPFC for Multibus system, INTERNATIONAL JOURNAL OF ENGINEERING RESEARCH & TECHNOLOGY (IJERT) Volume 02, Issue 10 (October
• Open Access
• Total Downloads : 475
• Authors : Shanker.N, B.Sampath Kumar
• Paper ID : IJERTV2IS100884
• Volume & Issue : Volume 02, Issue 10 (October 2013)
• Published (First Online): 24-10-2013
• ISSN (Online) : 2278-0181
• Publisher Name : IJERT
• License: This work is licensed under a Creative Commons Attribution 4.0 International License
Text Only Version
Voltages sag and swell mitigation using DPFC for Multibus system
Shanker.N 1, B.Sampath Kumar 2
1PG scholar, Dept. of EEE, Teegala Krishna Reddy Egg College, Meerpet, Hyd, A.P., India.
2Associate Professor & HOD of EEE, Teegala Krishna Reddy Egg College, Meerpet, Hyd, A.P., India.
ABSTRACT: Due to increasing in demand of electricity and nonlinear loads increases, power quality problems will be occurs in the interconnected power system network. Power quality problems are
voltage sag and swell can be eliminated by using distributed power flow controller then power quality can be improved. The switching level model is constructed using three phase six pulse shunt
converter and single phase four pulse series converters. Both the converters are modelled as back to back voltage source inverters connected without the D.C link are controlled by pulse width
modulation scheme. This model is implemented in single-machine infinite bus power system including two parallel transmission linesystems. The detailed DPFC simulation in switching level model is
performed in Matlab/Simulink environment.
1. INTRODUCTION
In recent years, power quality disturbances become most issue which makes many researchers interested to find the best solutions to solve it. Power quality in the power system is the important
issue for industrial, commercial and residential applications today. The voltage problem is mainly considered from under-voltage (voltage sag) condition over current caused by short circuit or
fault somewhere in the system. In customer opinion a power problem is deviation in voltage, current and frequency that results in failure.
To overcome the voltage sag and swell problems fast acting power electronics based FACTS (flexible A.C transmission system) devices are introduced. The flexible ac-transmission system (FACTS)
that is defined by IEEE as a power- electronic based system and other static equipment that provide control of one or more ac-transmission system parameters to enhance controllability and
increase power-transfer capability and can be
Utilized for power-flow control. Currently, the distributed power-flow controller (DPFC) shown in Fig. 1
Fig.1. DPFC configuration
This paper introduces a new concept, called distributed power-flow controller (DPFC) that is derived from the UPFC. The same as the UPFC, the DPFC is able to control all system parameters. The
DPFC eliminates the common dc link between the shunt and series converters. The active power exchange between the shunt and the series converter is through the transmission line at the
third-harmonic frequency. The series converter of the DPFC employs the distributed FACTS (D- FACTS) concept.
Fig. 2. The DPFC Structure
Comparing with the UPFC, the DPFC have two major advantages: 1) low cost because of the low-voltage isolation and the low component rating of the series converter and 2) high reliability because
of the redundancy of the series converters. This paper begins with presenting the principle of
the DPFC, followed by its steady-state analysis. After a short introduction of the DPFC control, the paper ends with the experimental results of the DPFC.
In this paper, a distributed power flow controller, introduced as a new FACTS device, is used to mitigate voltage and current waveform deviation and improve power quality in a matter of seconds.
The DPFC structure is derived from the UPFC structure that is included one shunt converter and several small independent series converters, as shown in Fig. 2. The DPFC has capability to balance
the line parameters, i.e., line impedance, transmission angle, and bus voltage magnitude.
2. PROPOSED MODEL OF DPFC
An infinite bus is a source of constant frequency and voltage either in magnitude or angle. Single Machine Infinite Bus System (SMIB) equipped with a DPFC is connected to the remote system
through a transformer and a parallel Transmission line having 2 section models as shown in Fig.3. A DPFC is placed in the transmission line at point m (between middle of two line sections m-n) to
improve the dynamic behaviour of the system. The DPFC consists of shunt and series converters controlled by pulse width modulation (PWM) controller.
Fig.3.Single line diagram of UPFC with parallel transmission lines
Within the framework of traditional power transmission concepts, the UPFC is able to control, simultaneously or selectively, all the parameter affecting power flow in the transmission line (i.e.,
voltage, impendence, and phase angle). Alternatively, it can independently control both the real and reactive power flow in the line.
The DPFC Advantages
(A)High Control Capability
The DPFC can simultaneously control all the parameters of the power system: the line
impedance, the transmission angle, and the bus voltage. The elimination of the common dc link enables separated installation of the DPFC converters. The shunt and series converters can be placed
at the most effectively location. Due to the high control capability, the DPFC can also be used to improve the power quality and system stability, such as low-frequency power oscillation damping,
voltage sag restoration.
(B)High Reliability
The redundancy of the series converter gives an improved reliability. In addition, the shunt and series converters are independent, and the failure at one place will not influence the other
converters. When a failure occurs in the series converter, the converter will be short-circuited by bypass protection, thereby having little influence to the network. In the case of the shunt
converter failure, the shunt converter will trip and the series converter will stop providing active compensation and will act as the D-FACTS controller.
(C)Low Cost
The single-phase series converters rating are lower than one three-phase converter. Furthermore, the series converters do not need any high voltage isolation in transmission line connecting;
single- turn transformers can be used to hang the series converters.
(D)Eliminate DC Link
Within the DPFC, there is a common connection between the ac terminals of the shunt and the series converters, which is the transmission line. Therefore, it is possible to exchange the active
power through the ac terminals of the converters. The method is based on the power theory of non- sinusoidal components. According to the Fourier analysis, a non-sinusoidal voltage and current
can be expressed by the sum of sinusoidal functions in different frequencies with different amplitudes.
1. Control Scheme of Shunt Converter
The objective of the shunt control is to inject a constant third harmonic current into the line to provide active power for the series converters. The third-harmonic current is locked with
the bus voltage at the fundamental frequency.
A PLL is used to capture the bus-voltage frequency, and the output phase signal of the PLL is multiplied by three to create a virtual rotation reference frame for the third-harmonic
The central control generates the reference signals for both the shunt and series converters of the DPFC. It is focused on the DPFC tasks at the power-system level, such as power-flow
control, low-frequency power oscillation damping, and balancing of asymmetrical components.
Fig.4.Control Scheme of Shunt Converter
The shunt converters fundamental frequency
From2 VabcS
Vabc1 From1
Vdref Goto1
control aims to inject a controllable reactivecurrent to grid and to keep the capacitor dc voltage at a
Vabc(pu) wt Sin_Cos
Trigonometric Function2
VqrefPI Controller4
constant level. The control for the fundamental
frequency components consists of two cascaded controllers. The current control is the inner control loop, which is to modulate the shunt current at the
3-phase PLL
Trigonometric Function3
Discrete PI Controller
fundamental frequency. The q-component of the reference signal of the shunt converter is obtained from the central controller, and d-component is generated by the dc control.
2. Control scheme for series converter Each single-phase converter has its own series control through the line. The controller inputs are series capacitor voltages, line current, and series
voltage reference in the dq frame. The third- harmonic frequency control is the major control loop with the DPFC series converter control. The principle of the
Fig.5. Control scheme for series converter
vector control is used here for the dc-voltage control. The third-harmonic current through the line is selected as the rotation reference frame for the single-phase park transformation, because
it is easy to be captured by the phase-locked loop(PLL) in the series converter.
Fig.6. Central Control Scheme
According to the system requirement, the central control gives corresponding voltage-reference signals for the series converters and reactive current signal for the shunt converter. All the
reference signals generated by the central control are at the fundamental frequency.
3. MODELING OF DPFC
This modelling is done with Simulink block set and simulationis carried out in MATLAB environment as shown in Fig.7. The system is modelled with a three phase source connected to a load. The
source is connected to load through parallel transmission lines. The transmission line consists of transmission line I, transmission line II .These two transmission lines are connected in
parallel. Each parallel transmission line has equal length.
The DPFC is incorporated between transmission line I & II. The inductive and capacitive loads are connected for dynamic performance analysis. To obtain the transient analysis the fault can be
connected near the load. The system circuit parameters are given in appendix. The simulation model of DPFC is modelled with a three phase voltage source inverter connectedto different
loads. Each transmission line has the busmeasurement block to measure the real power, reactive power, voltage and the current. The shunt and series device of DPFC consists of three phase IGBT
converter with PWM controller. The shunt converter is connected to the transmission line in parallel through a three phase transformer. The series converter is connected to the transmission line
in series through three independent single phase transformers. The IGBT firing pulses are generated for shunt & series converters as described earlier in section (2).Three leg, six pulse bridges
are used for the model of converters.
4. IMPLEMENTATION OF FLC IN DPFC
FLC are formed by simple rule based on If x and y then z. These rules are defined by taking help from persons experience and knowledge about the system behaviour. The performance of the system is
improved by the correct combinations of these rules. Each of the rules defines one membership which is the function of FLC. More sensitivity is provided in the control mechanism of FLC by
increasing the numbers of membership functions. In this study, the inputs of the fuzzy system are assigned by using 7 membership functions and the fuzzy system to be formed in 49 rules. Hence,
the sensitivity in the control mechanism is increased.
The basic if-then rule is defined as If (error is very small and error rate is very small) then output. The signals errorand error rate are described as linguistic variables in the FLCsuch as
large negative (LN), medium negative (MN), small negative (SN), very small (VS), small positive (SP), mediumpositive (MP) and large positive (LP). These are shown in Fig.5. In the same way, the
input values of the fuzzy controller areconnected to the output values by the if-then rules. The relationship between the input and the output values can be achieved easily by using Takagi-
Sugeno type inference method. The output values are characterized by memberships and named as linguistic variables such as negative big (NB), negative medium (NM), negative small (NS), zero (Z),
positive small (PS), positive medium (PM) and positive big(PB). The membership functions of output variables and the decision tables for FLC rules are seen in Table I
FUZZY DECISION TABLE
Error rate\error LP MP SP VS SN MN LN
LP PB 1 PB2 PB 3 PM4 PM5 PM6 Z7
MP PB 8 PB 9 PM 10 PM 11 PS 12 Z 13 NS 14
SP PB 15 PM 16 PM 17 PS 18 Z 19 NS 20 NM 21
VS PM 22 PM 23 PS 24 Z 24 NS 26 NM 27 NM 28
SN PM 29 PS 30 Z 31 NS 32 NM 33 NM 34 NB 35
MN PS 36 Z 37 NS 38 NM 39 NM 40 NB 41 NM 42
LN Z 43 NS 44 NS 45 NM 46 NB 47 NB 48 NB 49
Fig.8.Error and error rate of fuzzy membership functions
5. RESULT ANALYSI
Case (i)
In this case the impact of sag can be analysed by creating the three phase fault on the network system as shown in Fig.9. The time duration for fault is 0.05s (0.05-0.1 s).The length between the
feeders will determine the severity of dropped voltage. In this case, the SMIB is connected with capacitive load. The DPFC can compensate 100% of drop voltage in theSystem as shown in Fig.10&
Case (ii)
In this case, the impact of swell can be analysed by effect of line current due to three phase fault on the
network system as shown in Fig.12.The time duration of the fault is 0.05 seconds.
In this simulation study after implementation of DPFC the magnitude of line current is comparatively reduced. The mitigation of swell for this simulation can be observed from
the Fig.13& fig.14.
due to the effect three phase fault on the network system the real and reactive power magnitude will be distorted in the time duration of 0.05s to 0.1s as shown in fig.15. In this simulation
study after implementation of DPFC the magnitude of real and reactive power is comparatively increased as shown in fig.16 & fig.17
Fig.9.Without DPFC Load voltage sag wave form
Fig.10.Mitigation of load voltage sag with DPFC(PI)
Fig.11.Mitigation of load voltage sag with DPFC (Fuzzy)
Fig.12.Without DPFC swell wave form of load current
Fig.13.mitigation of load current with DPFC(PI)
Fig.14.mitigation of load current with DPFC(fuzzy)
Fig.15.without DPFC realand reactive power wave form
Fig.16.with DPFC real and reactive power wave form(PI)
Fig.17.with DPFC real and reactive power wave form(Fuzzy)
Fig.18.without DPFC THD Fig.20.with DPFC THD (fuzzy)
Fig.19.with DPFC THD (pi)
The load voltage harmonic analysis without presence of DPFC is illustrated in Fig.18. It can be seen, after DPFC implementation in system, the even harmonics is eliminated, the odd harmonicsare
reduced within acceptable limits, and total harmonic distortion (THD) of load voltage is minimized from 18.48 to 0.32percentage (Fig.19 & 20).
6. CONCLUSION
To improve power quality in the power transmission system, there are some effective methods. In this paper, the voltage sag and swell mitigation, using a new FACTS device called distributed power
flow controller (DPFC) is presented. The DPFC has a control capability to balance the line parameters, i.e., line impedance, transmission angle, and bus voltage magnitude. However, the DPFC offers
some advantages, such as high control capability, high reliability, and low cost. The DPFC is modelled and three control loops, i.e., central controller, series control, and shunt control are
designed. The system under study is a single machine infinite-bus system, with and without DPFC. To simulate the dynamic performance, a three-phase fault is considered near the load. It is shown that
the DPFC gives an acceptable performance in power quality mitigation and power flow control and simulation results values are shown in table II.
Rated voltage (kv) Load voltage(kv) Load current( A) Real power(MW) Reactive power(MVAR) Apparent power(MV A)
Without DPFC 230 190 1200 75 45 88 18.48
With DPFC using PI controller 230 210 1300 48 125 134 0.78
With DPFC using Fuzzy controller 230 210 1300 48 125 134 0.32
Table.II. Simulation results
TABLE III. Simulation System Parameters
Parameters values
Three phase source
Rated voltage 230 kV
Rated power/Frequency 100MW/60 HZ
X/R 3
Short circuit capacity 11000MW
Transmission line
Resistance 0.012 pu/km
Inductance/ Capacitance reactance 0.12/0.12pu/k m
Length of transmission line 100 km
Shunt Converter 3-phase
Nominal power 60 MVAR
DC link capacitor 600 F
Continue of Table I :
Coupling transformer (shunt)
Nominal power 100 MVA
Rated voltage 230/15 kV
Series Converters
Rated voltage 6 kV
Nominal power 6 MVAR
Three-phase fault
Type ABC-G
Ground resistance 0.01ohm
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(NAPS), pp.1 7, 2011
2. I Nita R. Patne, Krishna L. Thakre Factor Affecting Characteristics Of Voltage Sag Due to Fault in the Power System Serbian Journal OfElectrical engineering. vol. 5, no.1, May2008, pp. 171-182.
3. J. R. Enslin, Unified approach to power quality mitigation, in Proc.IEEE Int. Symp. Industrial Electronics (ISIE 98), vol. 1, 1998, pp. 820.
1. B. Singh, K. Al-Haddad, and A. Chandra, A review of active filters for power quality improvement, IEEE Trans. Ind. Electron. vol. 46, no. 5, pp. 960971, 1999.
2. A. L. Olimpo and E. Acha, Modeling and analysis of custom power systems by PSCAD/EMTDC, IEEE Trans. Power Delivery, vol. 17, no.1, pp. 266272, Jan. 2002.
3. P. Pohjanheimo and E. Lakervi, Steady state modeling of custom power components in power distribution networks, in Proc. IEEEPower Engineering Society Winter Meeting, vol. 4, Jan. 2000,
4. Zhihui Yuan, Sjoerd W.H de Haan, Braham Frreira and DaliborCevoric A FACTS Device: Distributed Power Flow Controller (DPFC) IEEE Transaction on Power Electronics, vol.25, no.10, October 2010.
5. Ahmad Jamshidi,S.MasoudBarakati and Mohammad MoradiGhahderijani Power quality improvement and mitigation case study using Distributed power flow controller
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Whole cell biophysical modeling of codon-tRNA competition reveals novel insights related to translation dynamics
The importance of mRNA translation models has been demonstrated across many fields of science and biotechnology. However, a whole cell model with codon resolution and biophysical dynamics is still
lacking. We describe a whole cell model of translation for E. coli. The model simulates all major translation components in the cell: ribosomes, mRNAs and tRNAs. It also includes, for the first time,
fundamental aspects of translation, such as competition for ribosomes and tRNAs at a codon resolution while considering tRNAs wobble interactions and tRNA recycling. The model uses parameters that
are tightly inferred from large scale measurements of translation. Furthermore, we demonstrate a robust modelling approach which relies on state-of-the-art practices of translation modelling and also
provides a framework for easy generalizations. This novel approach allows simulation of thousands of mRNAs that undergo translation in the same cell with common resources such as ribosomes and tRNAs
in feasible time. Based on this model, we demonstrate, for the first time, the direct importance of competition for resources on translation and its accurate modelling. An effective supply-demand
ratio (ESDR) measure, which is related to translation factors such as tRNAs, has been devised and utilized to show superior predictive power in complex scenarios of heterologous gene expression. The
devised model is not only more accurate than the existing models, but, more importantly, provides a framework for analyzing complex whole cell translation problems and variables that haven't been
explored before, making it important in various biomedical fields.
Author summary
mRNA translation is a fundamental process in all living organisms and the importance of its modeling has been demonstrated across many fields of science and biotechnology. Specifically, modeling a
whole cell context with a high resolution has been a great challenge in the field, making many important problems un-addressable. In this study we devised a novel model, which allows, for the first
time, simultaneous simulation of thousands of mRNAs, along with various bio-physical aspects that affect translation (such as codon-resolution dynamics and shared resources pool of both ribosomes and
tRNAs). We demonstrated (using experimental data) that this model is more accurate than existing ones, and, more importantly, provides a framework for addressing complex translation problems (such as
heterologous expression) at whole cell scale and in reasonable time. We demonstrated the model using E. coli data, but the model can be easily tailored to other organisms as well. Our model addresses
an urgent unmet need for biophysically accurate whole cell translation model with resources coupling and has potential applications in many fields, including medicine and biotechnology.
Citation: Levin D, Tuller T (2020) Whole cell biophysical modeling of codon-tRNA competition reveals novel insights related to translation dynamics. PLoS Comput Biol 16(7): e1008038. https://doi.org/
Editor: Christos A. Ouzounis, CPERI, GREECE
Received: January 16, 2020; Accepted: June 10, 2020; Published: July 10, 2020
Copyright: © 2020 Levin, Tuller. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction
in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the manuscript and its Supporting Information files.
Funding: This work was partially supported by a grant from the Ela Kodesz institute for medical physics and engineering and by a research grant from the U.S.-Israel Binational Science Foundation
(BSF), the Israeli Ministry of Science, Technology and Space, the Koret-UC Berkeley-Tel Aviv University Initiative in Computational Biology and Bioinformatics. The funders had no role in study
design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Dozens of studies in recent years have demonstrated the advantages of using computational models of mRNA translation in basic science and biotechnology (see, for example, [1–11]). Whole cell
modelling of translation is a fundamental component of synthetic and systems biology, with implications in various fields, such as biotechnology and biomedical engineering. Despite its undoubted
importance, the selection of models and tools is quite limited. According to a recent review [11], several computational and mathematical models related to competition for finite resources during
translation have been suggested [12–18]. However, these models commonly address only a single resource type (ribosomes) and do not include aspects such as tRNA availability. Other studies, e.g. [19],
emphasize the importance of incorporating competition for tRNAs, but usually (e.g. due to computational limitations) simulate a single mRNA or a small group of mRNAs, thus inaccurately modeling a
whole cell scenario, what limits their applicability. In other cases, thorough mathematical descriptions are provided, but the models are not tailored to real data and thus are not validated and are
not predictive. In other words, a comprehensive model that includes all major aspects of translation (such as discrete ribosomal dynamics, various limited resources, including tRNA and ribosomes, and
the affinity of tRNA-codon interactions) on a cellular level (with thousands of simultaneously translating mRNAs) currently does not exist. All the mentioned aspects are required to properly model
complex scenarios related to mutations and heterologous expression, which are fundamental both for answering basic questions in cell biophysics and evolution and for various medical and
biotechnological applications. In this study we aimed to bridge this gap by providing such model which is efficient enough to simulate real whole cell scenarios.
We focused on incorporation of the following two aspects in a whole cell model of translation: 1) accurate elongation dynamics at a codon resolution; 2) finite resources pools of various types. The
first aspect is important because many phenomena can be analyzed and observed only when addressing the dynamics and interactions within a polysome. Furthermore, numerous biomedical implications of
translation are related to mutations (such as SNPs) along the ORF (Open Reading Frame), which often cannot be addressed well with models that treat mRNA as a bulk (e.g. mean field approximations [20
]). The second aspect is particularly important when considering significant changes from the native state of the cell, like in the case of heterologous expression. Such scenarios alter the resources
competition profile inside the cell, leading to a change in translation resources allocation and consequently may influence the vitality of the cell [21–23]. This aspect is also important when
studying the evolution of genomes as it many times induces strong evolutionary constraints.
Currently, there are no models available that fully utilize both aspects discussed above in a way that allows feasible whole cell simulations. To bridge this gap, we have developed and implemented
MP-SMTM (Multiple Pool State Machine Translation Model): a novel model, based on well-established practices of translation elongation modeling and several new concepts, including the finite pools of
various translation resources. We demonstrated the model on E. coli and showed that the model has not only proved to be more accurate than existing models, but, more importantly, allowed us to study
codon-resolution questions on a whole cell level, what was previously impossible.
Whole cell translation elongation model
The proposed single-cell model operates on three levels:
Cell level.
The modelled cell has a transcriptome that consists of N mRNA molecules, a finite ribosome pool G[tot] and a finite tRNA pool H[tot] (Fig 1A). Both pools are assumed to be constant in time, while
their occupancy and availability change with time t. The amount of every mRNA corresponds to the actual mRNA level inside a real cell. There can be several independent copies of a given mRNA or a
partial mRNA (S1 File), which accounts for cases with average levels smaller than one. The ribosome pool is consumed by initiation events and refilled upon termination events. At time t (on a
timeline with resolution ΔT), the number of ribosomes on mRNA i is denoted by g[i](t), while the free ribosome pool is G(t). Thus, . The tRNA pool consists of all available tRNA types in the cell
(specific to the modelled organism), with amounts corresponding to actual tRNA levels in the cell. Suppose that there are N[tRNA] unique tRNA types. The total amount of tRNA of type j, j = 1,…,N[tRNA
], is denoted by H[tot,j], while the free amount is H[j](t) and the amount on an mRNA i is . Thus, the total amount of tRNA molecules is . We assume that upon release, tRNAs undergo aminoacylation
and return to the available pool (this is also related to the aspect of tRNA recycling, which is defined in sub-section Effective supply-demand ratio).
Schematic visualization of the simulated elements at the cell level (A), at the mRNA level (B) and a simplified view of the state machines at the codon level (C), for initiation and elongation (the
detailed state machines appear in F-G Figs in S1 File). At the cell level, a transcriptome of mRNA molecules with a finite pool of ribosomes and tRNAs is simulated, leading to a competition for these
resources. At the mRNA level, we utilized a novel generalized deterministic TASEP model that incorporates both accurate dynamics at the codon level and dependence on cellular resources, which
influences these dynamics. On the codon level, for each codon the dynamics are dictated by a state machine, an object that holds the state (e.g. “this codon is anticipating a tRNA”) and the rules of
state transition order and conditions (see sub-section Generalized deterministic TASEP and state machines).
mRNA level.
A single mRNA molecule is modeled by a generalized version of deterministic TASEP (Totally Asymmetric Exclusion Process), described in more details in sub-section Generalized deterministic TASEP and
state machines. Each mRNA is modeled as an array of codons, on which unidirectional movement (5’ to 3’) of ribosomes takes place. Ribosomes have physical size (of s codons) and they are not allowed
to overlap or overtake one another, resulting in possible “traffic jams”. The movement of the ribosomes is modeled similarly to the current understating of the elongation cycle, including the
incorporation of a suitable tRNA at the A-site, translocation and release of a non-charged tRNA from the E-site (Fig 1B). Dynamics are driven by timers in the following manner: when a certain step is
required, the expected waiting time is calculated (according to the available resources) and a timer is assigned. In other words, a timer of a translation step is a discrete count-down of the time
required for its completion. Only when the timer nullifies, the required step is performed, given all the requirements are fulfilled (otherwise, the timer is delayed). As a result, the availability
of resources controls the timers, which dictate the dynamics of the system. In order to avoid resources allocation bias, at each iteration we randomized the order in which mRNAs are analyzed,
resulting in uniform exposure of the mRNAs to the resources.
tRNA-codon interactions and the interaction coefficient
We used an exact, E. coli specific, codon recognition scheme [24] (Table A in S1 File), which allows wobble interactions. In order to account for the affinity of a tRNA-codon interaction (i.e. to
consider the fact that a Watson-Crick interaction is more likely to happen than wobble interaction under similar conditions) we introduced an interaction coefficient α(c,j) between codon c and tRNA j
(defined in the range (0,1]). We used similar approach to the one used in the definition of the tAI (tRNA Adaptation Index [2]), which utilizes similar coefficients to evaluate the extent to which a
codon is adapted to the tRNA pool. However, we used real tRNA levels from [24] rather than tRNA gene copy number. We then defined an optimization problem and found the interaction coefficients that
gave the best overall correlation between tAI and both PA (Protein Abundance) and TDR (Typical Decoding Rate, a useful translational efficiency index) [25]. Detailed steps appear in supplementary
Figs A-D and Table B in S1 File.
Effective supply-demand ratio
A fundamental aspect of translation we aimed to model is the competition for resources (such as ribosomes and tRNAs). It is known that such competition drives co-evolutionary adaptations of the
organism and affects its genome (e.g. by creating codon usage bias) [1,26,27]. Furthermore, when modelling a cellular infection (e.g. with a bacteriophage) or a heterologous gene expression, the
allocation of resources is expected to change, leading to a change in translational dynamics. To incorporate competition into our model, we have devised two terms (defined for each point t in time):
RSDR (Ribosomal Supply-Demand Ratio).
The ratio between the free ribosome pool G(t) (supply) and the number of mRNAs that are waiting for an initiation complex at time t (demand). Higher RSDR values imply lower initiation times (i.e.
higher initiation rates) and vice versa.
ESDR (Effective Supply-Demand Ratio).
For each codon c, ESDR(c) represents the ratio between the number of available tRNAs that recognize c (supply) and the competition for these resources, namely all the codons that are recognized by
these tRNAs and require a tRNA at time t (demand). To make ESDR as realistic as possible, we incorporated weightening based on codon-tRNA interaction coefficients and accounted for an aspect of tRNA
recycling [26,28,29]; we specifically assumed that the effective demand of codon types which tend to be close to each other is lower due to recycling. ESDR was defined as: (1) where RT(c) is the
group of tRNAs that recognize the codon c (i.e. tRNAs j that satisfy α(c,j)>0), D[j](t) is the effective competition for tRNA j, d[c] is the distance score of codon c (related to tRNA recycling) and
w is the associated normalization factor. All arguments in the definition above are defined and described in detail in S1 File. Higher ESDR values imply lower elongation times (or higher elongation
rates) and vice versa.
Both RSDR and ESDR are used continuously throughout a MP-SMTM simulation to determine the expected waiting time for a resource: when a state machine (described in sub-section Generalized
deterministic TASEP and state machines) dictates that a timer needs to be defined (either for initiation or for elongation), the corresponding supply-demand ratio is calculated and the timer is
defined accordingly. By doing so we assume that both supply and demand affect the waiting time. The role of the supply is more obvious. However, to justify incorporation of demand, consider, for
example, a given pool of resources Y and one specific demanding object X (e.g. an initiation site X, “waiting” for ribosomes, denoted as Y). The amount of competitors (demand) can directly influence
the availability of Y for X, since it affects the chance that Y will bind to a competitor, rather than to X.
In order for the model to be accurate enough to fit measured data, the estimated waiting times must be small enough, so that no significant changes will happen during this waiting time. Indeed, the
error is expected to be small (up to ~3%, see S1 File). Indeed, at the regime of nominal resources, we validated that the prediction of the model in terms of protein synthesis (see sub-section
Prediction of protein synthesis rate and the Results section) corresponds to the available empirical protein abundance data of E. coli. It should be noted, however, that the regime of severe
resources depletion is expected to be less accurate (see S1 File). The functional relations between the timers and the supply-demand ratios are given in the S1 File.
Generalized deterministic TASEP and state machines
TASEP traditionally incorporates a one-dimensional lattice of consecutive sites and a statistical description of transition of an object (ribosome, in our case) from one site (codon, in our case) to
the next one. TASEP models commonly utilize the Gillespie approach [30] for stochastic continuous-time dynamics modeling. However, since we model a large number (thousands) of TASEP-like objects
coupled to a complex resources pool (that affect many of the parameters of translation), even the most efficient stochastic implementations of TASEP results in an extremely computationally intensive
task. For example, stochastic TASEP simulation is estimated to take several hours and can take 6 times longer (depending in the specific parameters and CPUs) than deterministic implementation for a
system with no resources coupling [11]. This ratio is expected to be much higher with pool coupling (close to two orders of magnitudes higher), since regardless of the implementation, additional
factor need to be taken into account in every iteration. To avoid unreasonable running times, we took a deterministic approach. Furthermore, to incorporate accurate biophysical behavior into the
model, we integrated the idea of codon state machines. Below we briefly describe the approach and provide a simple example.
In the proposed approach, each codon holds a symbolic state. The duration of each state is defined by a timer which is initiated upon state transition, according to predefined rules and up-to-date
system parameters. State transition is associated with an action (e.g. ribosomal movement) and/or parameters update (e.g. ribosomal free pool increment by 1). At each simulation’s iteration, the
timer is reduced by ΔT and only when it reaches zero, the state machine switches to the next state and the associated action takes place if possible (e.g. a ribosome is allowed to proceed only if all
bio-physical conditions are fulfilled, namely the downstream ribosome is distant enough and the required resources are available). Every codon on every mRNA may have an individual timer and a state.
Transition between states is monitored, and allows efficient tracking of the number of codons that require a certain resource. Detailed schemes of the state machines for both initiation and
elongation are provided in Figs F-G in S1 File. A simplified version is presented in Fig 1C.
Consider, for example, the following scenario: The E site of a ribosome has translocated to a UGG codon, that now awaits a tRNA charged with Tryptophan. First, the state machine sets the state of the
codon to “NeedtRNA” (the counter of “number of UGG codons that currently need tRNA” is increased by 1). Next, the state machine may require to define a timer based on the current ESDR(UGG) value.
When that timer nullifies, the tRNA is allocated (if H[Tryptophan]>0), an amino acid bond is formed, the ribosome progresses, the tRNA is released, etc. In cases where wobble interactions may take
place, the interaction coefficient α (sub-section tRNA-codon interactions and the interaction coefficient) is used to statistically choose the candidate tRNA.
To conclude, we kept the biophysical qualities of the TASEP model to account for realistic ribosomal movement. We used deterministic timers rather than stochastic approach for computational
efficiency. The timers are defined ad-hoc according to up-to-date supply and demand metric (ESDR and RSDR). The whole process is controlled by a codon-specific state machines, which hold the state of
each codon at every point in time and are the de-facto implementation of the elongation cycle.
System parameters
The proposed model is intended for the study of various cellular conditions (e.g. viral infection) and for design and engineering modifications of existing and even new organisms. Thus, it is
particularly important to simulate conditions that are as realistic as data availability permits. All prediction reported here were performed on E. coli data (K-12 MG1655 strain, downloaded from
Ensembl Bacteria ASM584v2.31). Below we briefly describe how some of the parameters were obtained. Detailed description of these and other parameters, as well as their estimation methods, is provided
in Figs H-I in S1 File.
Local initiation times.
The initiation time of a given mRNA was determined by a local, mRNA-specific, value and a factor related to the global supply of initiation-related resources. Local initiation times were estimated
based on ribosomal sequencing data [31]. These values where then normalized to obtain simulated ribosomal density that is in the expected range.
mRNAs, ribosomes and tRNAs.
We maintained the relation between the number of ribosomes and number of tRNAs according to reported values [32]. We chose a representative value of 5,100 mRNA molecules, distributed according to
reported mRNA levels [31]. We than performed optimization to find the size of pools that resulted in 80% ribosomal activity [32,33]. The key parameters chosen for the simulation are: N = 5,100; G[tot
] = 70,000; H[tot] = 650,407; Average initiation time: 0.95sec. We have performed comprehensive sensitivity tests for these and other simulation parameters to make sure that the behavior of the
simulation is robust and exhibits expected trends (Figs J-K in S1 File).
Prediction of protein synthesis rate
One of the desired outcomes of a translation model is a prediction of protein synthesis rate, which is also a measure of translational efficiency. The number of termination events is a proxy to the
actual number of proteins synthesized, as long as protein degradation is negligible, which is what we assume. To avoid the dependence on a specific time frame, we commonly refer to synthesis rate
rather than the total number of synthesized proteins. Throughout the manuscript we use the terms termination rate and protein synthesis/production rate interchangeably. To estimate the total number
of proteins synthesized, we count the total number of termination events, while taking into account the number of mRNAs of each type. We only considered termination since steady state, defined in our
context as the time in which the consumption of the ribosomal pool stabilizes. PA (protein abundance) is commonly used as a proxy for the translation efficiency, and here we took the same approach,
i.e. used correlations between empirical PA and simulated synthesis rate as a way to assess the predictivity of the model. We used protein abundance data from PAXdb id: 511145 (integrated dataset).
Heterologous expression
One important property of a translation model is its usefulness in explaining a heterologous expression outcome, in part, due to its vast industrial implications. In order to asses this property, we
have performed several whole cell simulations in which a heterologous gene was introduced into the E. coli cell. We used a GFP (Green Fluorescent Protein) gene from the U57608.1 cloning vector, in
order to support future experimental verification. The mRNA of the GFP was inserted with a high copy-number (20% of the total mRNA molecules) and with a high initiation rate (see S1 File). The
simulated cell contains both the native mRNA transcriptome and the heterologous GFPs to allow the complex interaction of pool sharing between the native and heterologous mRNAs.
Several mutated variants were simulation and compared to the original variant. Variants were based on synonymous mutations (while preserving the amino acid content) and were generated using two
methods: 1) single representing codon for each amino acid; and 2) single representing codon for only one amino acid. In the first method, single codon was chosen for each amino acid based on a given
(highest) score. We used the following scoring systems: TDR, mean ESDR and inverse mean occupancy. The two latter were calculated based on a simulation without a GFP gene. These scores are correlated
with codon translational efficiency, thus suit as candidates for naively choosing optimal codons. The second method of variant generation is based on choosing a single codon and substituting all
synonymous codons with the chosen one (leaving all other amino acids unchanged), resulting in 61 possible variants.
A landmark study of heterologous expression in the context of synonymous variants was published by Kudla et. al [34], in which they examined ~150 GFP variants (created by inducing random synonymous
mutations) and their expression in E. coli. Here we aimed to show that the predictive power (in terms of protein synthesis and cell growth rate) has improved when using cell-level supply-demand
considerations and specifically when using MP-SMTM. We performed a calibration procedure to properly integrate the variants into our model. Briefly, we assumed that initiation time is τ[h] exp(−FE/c[
h]), where FE is the folding energy at the beginning of the ORF, c[h] is a calibration factor (deduced by a calibration procedure that maximizes correlation with PA, see Fig L in S1 File) and τ[h] is
some baseline value (see supplementary methods in S1 File). We then performed an independent whole cell simulation for each one of the variants, using the corresponding initiation rate and mRNA
amount to study the relation between simulation predictions and empirical measurements. Finally, we used these relations to assess the prediction power of our model.
Regressor features selection
In order to estimate the predictive power of our model, we built linear least squares multivariable regressors. In order to predict OD (Optical Density), which is related to cellular growth and PA
and is an obvious prediction objective, we used features that are known to relate to translational efficiency, such as CAI (Codon Adaptation Index [35]), FE (Folding Energy) and features based on
ESDR. We aimed to show that supply-demand-based features are important for accurate predictions. We took the following approach for feature selection: given a set of features x[1],…,x[M] and an
objective y∈{OD,PA}, we performed N[rep] = 100 repetitions of a train-test, in every one of which we iteratively chose features with decreasing importance. The training, i.e. the parameters
estimation, was performed on ~67% of the variants (randomly chosen in each repetition) and tested on the other ~33%. The first feature at repetition j, denoted by , was selected to be the one that
maximized R^2 (the coefficient of determination). At iteration i, we used features and chose the next feature from the set so that it maximized the R^2 change. To prioritize the features for the
final model, a score has been assigned as follows: if a feature f[k] was the i[j]-th feature to be selected at repetition j, the score was defined as . The features were then sorted ascendingly, with
the lowest score being the best: .
Additional terminology
The term occupancy of a codon stands for the number of simulation iterations for which a ribosome was located on the codon. Similarly, occupancy profile simply refers to the location dependent
occupancies of a series of codons, such as an ORF. Ribo-seq stands for ribosome sequencing or ribosome profiling, which is an experimental method of estimation of ribosomes location at the codon
level [36]. In processed ribo-seq results, each codon on each mRNA hold a read count value, which is related to the chance that a ribosome is located in this codon and to the mRNA level (among other
factors). CUB stands for Codon Usage Bias, namely the extent to which the distribution of synonymous codons differs from an expected uniform distribution. When a quantitative representation of CUB
was needed, we used CAI [35] (see supplementary methods in S1 File for the calculated values for E. coli codons).
We have performed several analyses with the aim of showing the usefulness and accuracy of MP-SMTM. In short, we showed high correlations between simulated and empirical values (sub-section High
correlation between model predictions and experimental measurements), demonstrated the effect of codon order on translation efficiency (sub-section Estimation of the effect of codon composition and
order on translation efficiency) and provided a thorough case study of heterologous gene expression, while showing the importance of competition for resources (sub-section Competition for tRNAs
explains variations in heterologous translation). In all cases we used empirical E. coli data.
High correlation between model predictions and experimental measurements
As a first validation step, we correlated the predictions of our model with various experimental gene expression measurements. We observed a high Spearman correlation of r = 0.733, p[value] = 5.62×10
^−256 between the simulated termination rate and the empirical protein abundance (Fig 2A). This value is higher than previous state-of-the-art PA predictors [2,20,37]. Due to the fact that
translation tends to be an initiation rate limited process (leading to a high correlation between initiation rate and termination rate), one may argue that the correlation observed in Fig 2A is
explained solely by initiation rate. This is, however, not the case, since the correlation between the local initiation rate and the protein abundance is only r = 0.605, p[value] = 6.52×10^−152 (Fig
2B). Thus, the initiation rate explains ~37% of the protein level variance, while the complete model (initiation + elongation) explains ~54% of the variance. We conclude that incorporating aspects of
elongation into the model indeed improves its predictions.
(A) Termination rate in steady state (predicted by the model) and protein abundance (empirical data); (B) Local initiation rate (as estimated for the model) and protein abundance (empirical data);
(C) Mean ribo-seq read count and mean simulated occupancy. (D) Ribosomal density profiles for both simulation (average occupancy per codon) and ribo-seq (average read count per codon). In (A), (B)
and (D) mRNAs with level lower than 0.2 were omitted from this analysis to avoid discretization errors of the simulation. In (A), (B) and (C) each point represents a single mRNA type. (All terms in
this figure are defined in the Methods section, sub-sections System parameters, Prediction of protein synthesis rate and Additional terminology).
We then compared the mean occupancies to the mean ribo-seq read count (obtained from [31]) and obtained an exceptionally high correlation of r = 0.974, p[value]<10^−300 (Fig 2C), even though the
correlation between the read count and the corresponding estimated local initiation rate is r = 0.554, p[value] = 1.81×10^−281 (Fig M in S1 File). Lastly, we qualitatively compared the simulated
average ribosomal profile with ribo-seq results. Specifically, we calculated the average occupancy (for the simulation) and the average read count (for the ribosomal sequencing), per codon, across
the entire transcriptome. In both cases (Fig 2D) it is evident that the first 50–100 codons exhibit higher ribosomal density, which is with agreement with previous reports [27].
Estimation of the effect of codon composition and order on translation efficiency
It was suggested that codon order and composition are strongly related to the translation efficiency [26,38,39] as it expected to affect ribosomal dynamics and thus protein synthesis. Translation
efficiency can be assessed using the total rate of protein synthesis, which is expected to be related to growth rate. However, the previous studies in the field failed to show that indeed there is a
direct effect of codon usage and order on growth rate and did not provide any evaluation of the magnitude of this effect.
To provide initial answers to these questions we have performed 4 types of random modifications to the genome (Fig 3A), with different combinations of CUB and amino acid preservations within each ORF
or across the entire genome. Each type was randomized 10 independent times. The total protein synthesis rate was calculated for each case and compared to the un-randomized baseline using a
single-sample two-sided t-test. We observed a significant change in translation rate both for the local and the global randomizations (Fig 3B). Naturally, in the cases where the CUB was maintained at
the ORF level, a small decrease in the total termination rate was observed. In the cases where CUB was only maintained at the genome level, the total termination rate has been reduced significantly
(A) Schematic illustration of the different randomization types. (B) Box plot for total simulated termination rate (i.e. the sum of termination rates of all mRNAs) distributions (10 values each)
compared to the un-randomized scenario. The p-values shown are the result of a single-sample two-sided t-test.
Competition for tRNAs explains variations in heterologous translation
We have examined the total protein synthesis rates (of E. coli and GFP) for the two variant generation methods (as described in the methods sub-section Heterologous expression) (Fig 4A). It can be
seen that the values for 61 variants of method 2 are distributed around the values of the original variant, showing the optimization potential. Not surprisingly, the more rational variants of method
1 (which is based on scoring codons per amino acid) led to a higher GFP synthesis rate. Let us discuss two specific variants and demonstrate how ESDR provides a useful analysis framework. Two
variants (generated using method 2) are marked with an arrow: a variant in which Glycine is represented solely by GGU (variant I) and a variant in which the same amino acid is represented by GGA
(variant II). Interestingly, these two variants exhibit very distinct results. We can observe the mean ESDR values of all codons for these two variants and compare them, as shown in Fig 4B. Most of
the ESDR values are similar in both variants, but those that don’t demonstrate a possible reason for the different behavior. Relative to variant I, the amount of free tRNA-UCC in variant II is lower,
resulting in lower ESDR of GGA. Since tRNA-UCC is also recognized by GGG, its ESDR also drops in variant II relative to variant I. On the other side, codon-GGU is only recognized by tRNA-GCC. This
tRNA is expected to be more abundant in variant II (since it is no longer demanded by GGU codons in the GFP mRNAs), what indeed increases the ESDR of codon GGU and codon-GGC, the other codon that
this tRNA recognizes. Furthermore, we can see an increase in the ESDR of UUG, which cannot be easily explained with the reasoning just provided. Such analysis shows what resources are important for
the native E. coli genes and for the GFP gene. For example, the availability of tRNA-GCC is important for the E. coli genes, since its reduction resulted in decreased protein production.
(A) Total E. coli and GFP genes termination rates for various GFP variants as heterologous genes. Two method were used for variant generation, as described in the sub section Heterologous expression
in the Methods. The original variant is shown in orange. Smaller blue dots are related to variants in which one codon was chosen as substituted all synonymous codons (61 such variants in total).
Other dots represent variants in which a single representing codon was chosen for each amino acid, according to some optimality score (either TDR, ESDR or inverse occupancy). (B) Two variants (marked
with arrows in (A)) are compared in terms of ESDR. Codons for which the ESDR is different represent changes in supply and demand of associated tRNAs and allow to understand the results in (A). (C)
Comparison of three optimization variants in terms of their ESDR, per codon, relative to the unoptimized variant (red represents a higher ESDR value). Some codons (such as CCC) exhibit an increase in
ESDR in all variants, indicating that increasing the supply/demand for these codons can improve overall translational efficiency.
We have compared the three variants generated by method 1 in Fig 4C, which depicts relative changes in ESDR for all codons, grouped by amino acid. Interestingly, all variants indicate that the ESDR
of CCC, GCC, CCU (and to a lower extent GGG, AGC and AGU) should be increased in order to increase the overall GFP synthesis. On the other side, the ESDR of UCA and UCG should be decreased.
We then performed model validation using results published by Kudla et. al [40]. In order to show that the model provides better predictivity of both PA and OD relative to other common approaches, we
have calculated several correlations and built various relevant regressors. We have considered FE, N[mRNA] (the number of mRNAs, as reported by Kudla et. al), CAI and two results of the simulation:
TR (translation rate of the GFP mRNAs) and %AR (percent of active, i.e. translating, ribosomes). We have considered only variants with complete data availability. The TR was found to be highly
correlated to the PA (r[Spear] = 0.66, p[value] = 4.28×10^−11 and r[Pears] = 0.66, p[value] = 2.77×10^−11, Fig N in S1 File). The FE, which was claimed to be highly correlated with PA as well,
exhibited a slightly lower correlation of r[Spear] = 0.63, p[value] = 4.14×10^−10 and r[Pears] = 0.63, p[value] = 5.83×10^−10.
When linear regression was used for PA prediction, the FE underperformed the proposed model. Indeed, linear regressor based on FE gave r[Spear] = 0.62, p[value] = 2.36×10^−9 and r[Pears] = 0.62, p[
value] = 1.6×10^−9 while TR gave r[Spear] = 0.64, p[value] = 2.44×10^−10 and r[Pears] = 0.65, p[value] = 1.76×10^−10 (in all cases, the correlation is between the prediction objective and the
prediction). A multivariable linear regressor with both FE and TR gave r[Spear] = 0.73, p[value] = 3.14×10^−14 and r[Pears] = 0.73, p[value] = 5.71×10^−14, i.e. 37% improvement in the explained
variance relative to FE regressor. CAI failed to predict PA (r[Spear] = 0.13, r[Pears] = 0.08). Details for all the regressors are provided in the S1 File.
In terms of OD prediction, FE, as expected, did not perform well (r[Spear] = −0.01,r[Pears] = 0.06). CAI predicted OD better than TR (r[Spear] = 0.6, p[value] = 8.15×10^−9 and r[Pears] = 0.53, p[
value] = 6.96×10^−7 for CAI). However, the model provided additional valuable information, since the best multivariable regressor turned to be based on CAI, TR and %AR, with r[Spear] = 0.64, p[value]
= 3.28×10^−10 and r[Pears] = 0.73, p[value] = 7.55×10^−14. The relation between %AR and OD is not surprising, since higher % of free ribosomes means more polysomes in the cell cytoplasm available for
translating the endogenous genes, leading to increased growth rate.
We hypothesized that the higher accuracy of MP-SMTM is related to the supply and demand of resources, or more specifically, to the ESDR. The multitude of variants was sufficient to observe how ESDR
correlates positively with PA (or OD) for some codons, but negatively for others. To isolate the codon-specific effect, we simply calculated the correlation between the PA (or OD) and the mean ESDR
values across all variants, for each codon. To ensure that the effect is not the result of CUB or the FE, we controlled for these two variables. In the PA case, 8 codons exhibited negative
correlation of −0.40<r[Spear]<−0.19 and 4 codons exhibited positive correlation of 0.1<r[Spear]<0.3 (while , i.e. 5% with a strict Bonferonni correction). The OD case showed similar numbers (all
results are available in S1 File). The asymmetry between positive and negative correlations is not surprising, since increase in ESDR of a codon (resulted in an increased average decoding rate) is
more likely to cause a ribosomal traffic jam and decrease translational efficiency than the opposite scenario.
The fact that ESDR can explain PA and OD at the codon level suggests that some codons can serve as regressor features for PA and OD predictions. We defined 64 features (ESDR of 61 codons and the
start codon, FE and CAI) and performed feature selection using the approach described in the Methods sub-section Regressor features selection, resulting in a prioritized list of features . The result
of such process is demonstrated in Fig 5A. For evaluation, we have compared the results to an alternative model, resulted from an alternative set of features that can be calculated per codon per
variant–namely the number of occurrences of a codon at a given variant. The comparison was done using the entire data set, with the features for increasing number of k for each model. As seen in Fig
5B, for both PA and OD, the ESDR-based model has outperformed the alternative model, achieving an impressive correlation (the lists of chosen features for all regressors are given in Figs O-R in S1
(A) Pearson correlation between model prediction and the measured PA values of the GFP variants, as function of the number of feature selected for the train and the test sets. This graph demonstrates
the approach taken for feature selection: For 100 times, a train (~67%) and test (~33%) sets were randomly selected. Each time, the next best feature was selected to be the one the increases R^2 the
most in the test set (for more details, see sub-section 'Regressor features selection' in the Methods section). This result suggests that a model with more than ~10 features will show poor
predictivity due to over-fitting. (B) After choosing the best features and sorting them, this figure shows the correlation between the predicted PA (blue) and OD (red) values and the measured ones,
for increasing number of features. A features' set based on ESDR (continues line) was compared to a simpler metric of codon-count (dashed lines). In both cases (PA and OD) the ESDR-based model
performed better and reached impressive correlation with empirical data, demonstrating the importance of our model (Data for this model was taken from Kudla et. al [34]).
We have presented, for the first time, a framework that allows not only accurate translation elongation modeling at the codon level, but also incorporation of various whole cell aspects such as
competition for resources (both ribosomes and tRNA). The main challenge was to build a model that not only captures all the mentioned aspects, but is also efficient enough to allow real whole cell
simulations with real parameters, such as thousands of mRNAs undergoing translation at the same time (a task which various other studies failed to accomplish). Moreover, the formulation of this
framework as a system of generalized deterministic TASEP objects allows relatively easy generalization: it is possible to incorporate additional factors to further refine the model when additional
experimental data will be available.
The chosen model organism, E. coli, proved to be useful to establish the fundamentals of the model and simulation as there are many available relevant gene expression measurements for this organism
that can be used for inferring the model parameters. However, similar framework can be utilized to simulate Eukaryotic cells and even multi-cellular organisms. Such modification will require
formulation of different state machines (that may have additional signals as an inputs) and a relevant codon-tRNA recognition table (to account for the organism-specific interactions that can also
include non-standard amino acids), but the basic principle of a system of generalized TASEP objects remains.
We have demonstrated that the model can be used for solving fundamental problems such as predicting the outcome of heterologous gene expression, both in terms of protein synthesis rate and cellular
growth. We have shown that observing ESDR at the codon level is extremely beneficial in such prediction problems. Specifically, we have shown that a multivariable regressors based on codon-specific
ESDR features can reach Spearman correlation of as high as 0.8 with protein abundance (and 0.7 with optical density), outperforming models with traditional features such as folding energy and codon
adaptation index. We have also demonstrated various case studies in which the model can be harnessed to gain a deeper understanding of the complex supply and demand dynamics in the cell.
The importance of studying supply and demand of tRNA has already been demonstrated before [1,10,19]. The relation between tRNA abundance and codon decoding rates was previously modeled [41,42] and
various aspects of tRNA aminoacylation and recycling dynamics [19,43–45]. Despite all these advances, our model is currently the first that considers the biophysics of tRNA supply and demand at a
cellular level with thousands of mRNA molecules performing translation simultaneously, while accounting for wobble interactions, finite pool of tRNAs and ribosomes, ribosomal dynamics, and tRNA
recycling while maintaining correspondence to real empirical data.
The devised model has several additional advantages. The codon resolution allows using this model for complex medical and biomedical engineering problems. For example, various types of cancers are
associated with mutations that affect protein synthesis rate. This model allows examining the effect of possible therapeutic mechanisms and their effect on the behavior of the cell. Furthermore, the
model allows planning and predicting the outcome of complex experimental setups, allowing significant time optimization when such assays are performed.
Nevertheless, as in any model, several limitations should be discussed. First, all simulations presented in this paper assumed constant resources pools. However, it is known that these may vary due
to various cellular conditions, division, external stimuli or stress. For simulating non-constant resources all relevant parameters (e.g. the values of tRNA, mRNAs, and ribosomes over time, DNA
replication rates, etc) are needed. When the right parameters are can be estimated, modeling a time-variable pool is possible within the suggested framework of the current model, and should be
thoroughly considered; this can be done for example via assumption and simulation of pseudo steady state of the system. Similarly, translation is known to be coupled to additional aspects such as DNA
replications and cell growth that can be added to our model when high resolution measurements of these processes are available.
Additionally, the model is not expected to behave accurately is scenarios with severe resources depletion. Since the discussed organism was E. coli, it worth mentioning that co-transcriptional
translation is known to occur [46], but we did not incorporate this aspect into the model. Additional aspects that were not taken into account (mainly due to high complexity and/or lack of reliable
experimental data) are mRNA/tRNA degradation and cell-division (which is, by itself, a very challenging process to model, involving DNA replication and various steps of the cell cycle). With that
said, a previous study showed that for a simple degradation model, the protein levels should be correlated with predicted translation rate even when omitting degradation [20]. Finally, the model is
in-part deterministic and does not include all the stochastic aspects of translation. These aspects should be addressed in a future study.
To conclude, we believe that MP-SMTM is a meaningful step towards accurate whole cell modeling of translation. This work opens the door to many more advances in the field, and allows further progress
in the utilization of such modelling and analyzing important problems in medicine, synthetic biology, biotechnology and biomedical engineering.
We would like to thank Yoram Zarai and Hadas Zur for helpful discussions and comments and the Koret-UC Berkeley-Tel Aviv University Initiative in Computational Biology and Bioinformatics. | {"url":"https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1008038","timestamp":"2024-11-14T07:45:14Z","content_type":"text/html","content_length":"222130","record_id":"<urn:uuid:e2fc5f98-2fc0-4699-8dc1-7df697ce2818>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00241.warc.gz"} |
Str8ts (also known as "Straights") is a logic puzzle, invented by Jeff Widderich (Canada). It is a grid, partially divided by black cells into compartments. Each compartment, vertically or
horizontally, must contain a straight - a set of consecutive numbers, but in any order (for example: 2-1-3-4). The aim is to fill all white cells with the numbers from 1 to N (where N is the size of
the grid). No single number can repeat in any row or column. Clues in black cells remove that number as an option in that row and column, and are not part of any straight.
Cross+A can solve puzzles from 3 x 3 to 9 x 9. | {"url":"https://cross-plus-a.com/html/cros7s8t.htm","timestamp":"2024-11-07T01:19:06Z","content_type":"text/html","content_length":"1510","record_id":"<urn:uuid:a8aa4066-cbbd-4198-8926-d9016d6ef9ae>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00689.warc.gz"} |
Topics: Causality
In General > s.a. causality in quantum theory [including information causality and indefinite causal structures]; causality violation; determinism; locality.
* Idea: Explanations are given in terms of efficient/physical cause, as opposed to final cause (teleology); Often associated with predictability; The dominant paradigm is the "machine", a
deterministically predictive one, despite setbacks from thermodynamics, special relativity, and quantum mechanics (Prigogine); Should be modified, according to him, to allow for self-organization and
creation of order in non-linear dissipative systems and non-equilibrium thermodynamics.
* Rem: Causality implies conservation of identity, a far from simple notion; It imposes strong demands on the universalizing power of theories concerned, often met by the introduction of a metalevel
which encompasses the notions of 'system' and 'lawful behaviour'; In classical mechanics, the division between universal and particular leaves its traces in the separate treatment of kinematics and
dynamics; > s.a. Dynamics [synthesis between kinematics and dynamics].
* History: Francis Bacon considered causality as a mechanical relationship, as opposed to an abstract one.
@ General references: Fermi RMP(32); Margenau PhSc(34)apr; Bohm 57; Svechnikov 71; Jones AJP(96)mar [RL]; Hunter et al ed-98; Dowe & Noordhof 04; Hájíček phy/06 [and liberty]; Ross & Spurrett BJPS
(07) [notions of cause and Russell]; Butterfield BJPS-a0708 [stochastic Einstein locality]; Janzing a0708 [asymmetry between cause and effect, Occam's razor, and thermodynamics]; Hájíček GRG(09)-
a0803 [and freedom of choice]; Pearl 09; Coecke & Lal a1010-wd [time asymmetry, quantum information processing]; Ellis a1212-FQXi [top-down causation]; Coecke et al a1711 [the time-reverse of any
causal theory is eternal noise]; Chvykov & Hoel Ent(21)-a2010 [geometric approach to information in causal relationships].
@ Causation: Ma FdP(00)qp/99-proc; Dowe PhSc(04)dec [conserved quantity theory]; Corry PhSc(06)jul [revision avoiding Bertrand Russell's arguments]; Kistler 06.
@ Other conceptual, philosophical: Mehlberg IJTP(69) [vs determinism]; Salmon PhSc(94)jun, PhSc(97)sep; Eckhardt SHPMP(06) [and irreversibility]; Cat PhSc(06)jan [fuzzy]; Smith BJPS(07) [relationship
between causal dependence and causal laws]; Frisch BJPS(09) [role of causality]; Norton BJPS(09), reply Frisch BJPS(09); Verelst a1203 [analysis, application to theories of Newton and Leibniz]; Rédei
& San Pedro SHPMP(12)-a1204 [inequivalent causality principles]; Vidunas Axiom(18)-a1707 [delegated causality in complex systems]; > s.a. Explanation; paradigms in physics.
> Four causes: see Efficient (Moving) Cause; Final Cause; Formal Cause; Material Cause; Marc Cohen's page.
> Online resources: see Wikipedia page.
In Classical Theories > s.a. causal structures; causality conditions; dispersion [Kramers-Kronig relations]; geometry; spacetime subsets.
* Classical field theory: Expressed by the support of Green functions or the Kramers-Kronig dispersion relations, or v[f].
* General relativity: For matter propagation, built in by the requirement that spacetime satisfy a causality condition.
@ General references: de Souza ht/97, BJP(02)ht/00, Bergqvist & Senovilla CQG(99)gq [field theory]; Patricot ht/04 [and symmetries]; Triacca PLA(07) [Granger causality for stochastic processes];
Yuffa & Scales EJP(12) [electrodynamics, and linear response]; Ajaib a1302 [physical vs numerical causality, and the Courant-Friedrichs-Lewy condition]; Baumeler & Wolf NJP(16)-a1507 [classical
processes without causal order]; Weaver a2011 [causation and Hamiltonian mechanics]; > s.a. field theory.
@ k-essence: Bruneton PRD(07) [and MOND and other modified theories]; Babichev et al JHEP(08)-a0708.
@ In relativity and gravity: Jacobson in(91) [general relativity]; Rohrlich AJP(02)apr [and electromagnetism]; Bertolami & Lobo NQ-a0902; Kochiras SHPSA(09) [Newton's causal and substance counting
problems]; Reall a2101 [gravitational theories].
@ And dispersion relations: Wigner ed-64; Nussenzveig 71; Fearn & Gibb qp/03.
@ Wave propagation: Bonilla & Senovilla PRL(97) [gravity in vacuum]; Mitchell & Chiao AJP(98)jan [v[g] < 0]; Smolyaninov JO(13)-a1210 [metamaterial model of causality]; > s.a. electromagnetism.
> Related topics: see gauge choice [causality and gauge in electromagnetism]; gravitating matter [and the speed of sound].
Related Topics > s.a. Retrocausation; tachyons; time; velocity.
* Causality vs correlations: Statistical and causal information are related, but causal information goes beyond correlations.
* Principle of common cause: The idea that simultaneous correlated events must have prior common causes, first made precise by Hans Reichenbach in 1956; It can be used to infer the existence of
unobserved and unobservable events, and to infer causal relations from statistical relations; It may not be universally valid (there is no agreement as to the circumstances in which it is valid), and
its validity is questionable in quantum mechanics (because quantum statistics violates Bell's inequalities, variables serving as common causes cannot exist); If there is a separate common cause for
both, the correlation disappears when probabilities are conditioned to the common cause; > s.a. Stanford Encyclopedia of Philosophy page.
* Emergent causality: In one proposal, information is seen as a more fundamental concept than the laws of physics, which leads to a different understanding of spacetime where causality emerges from
correlations between random variables representing physical quantities.
* Other related concepts: Arguments by design (> see cosmology).
@ Causality vs correlations: news pt(14)nov [information flow, and application].
@ Principle of common cause: Reichenbach 56; Henson SHPMP(05)qp/04, reply to comment SHPMP(13) [quantum mechanics]; Mazzola FP(12); Hofer-Szabó et al 13 [r CP(14)#3]; Cavalcanti & Lal JPA(14)-a1311
[modifications in light of Bell's theorem]; Mazzola & Evans FP(17)-a1703 [existence of Reichenbachian common cause systems]; > s.a. causality in quantum theory.
@ Emergent causality: Baumeler & Wolf ISIT(14)-a1312, a1602 [intrinsic definition of randomness, and complexity]; Rossi & Souza a1901 [in quantum mechanics].
@ Probabilistic causality: Price BJPS(91); Twardy & Korb PhSc(04)jul; Dzhafarov & Kujala JMPsy-a1110; Zaopo a1209 [causal relations as observer-dependent]; Zhang BJPS(13) [connection between
causality and probability].
@ Other generalizations: Choudhury & Mondal TMP(13) [almost causality, reflecting and distinguishing spacetimes]; Minguzzi RVMP(18)-a1709 [causality with non-round cones]; Milburn & Shrapnel a2009
[causal perspectivalism]; > s.a. causality conditions.
main page – abbreviations – journals – comments – other sites – acknowledgements
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Facts about Zero (0)
Zero (0) is most probably the most mysterious number. The ancient mathematicians were far ahead in Maths but they did not realized about Zero (0). The thought that number how low is it there must
have a number. There is no number, how it comes?
The feeling of Zero (0) first came from our subcontinent (Indian subcontinent) and it was a great feel in the field of mathematics. And so we can be proud of our mathematicians.
If any number is multiplied by Zero (0) then the answer is Zero (0). But what if we divide any number by Zero (0)? Is it infinity or undefined?
If you think it is infinity then you are wrong. The answer is undefined. Are you getting confused or thinking that I am wrong, then wait I will prove it to you.
Mathematics if broken when you attempt to divide by zero. Consider the limit of 2/x as x approaches zero.
2/2 = 1
2/1 = 2
2/.5 = 4
2/.1 = 20
2/.01 = 200
2/.0001 = 2000
Now imagine 2/.0000000000000000000000…01. It would be infinitely large. So you might think 2/0 is infinity… but now approach it from below: 2/(-1) = -2 2/(-.01) = -200 etc. So you might be temped to
think it is negative infinity. Since it cannot be both negative infinity and positive infinity, it is undefined.
Still not satisfied or need more clarification, then wait.
We all know
And if
A/0= ∞
∞x0 should be equal to A
But we know ∞x0=0
So any thing divided by Zero (0) is undefined.
If we look at everyday life example then it will be clearer to us. Think of you need to divide 100 taka to Zero (0) man, then how much will one get? The answer is undefined. Same is in the case of
Now let’s think another fact. What is 0^0 (zero to the power zero). Some mathematicians say it is 1 and some say it is 0. Here nothing is need to be said.
2 Comments
1. Have you yet read it? If not then go.
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The Department of Earth & Planetary Sciences
Yale University scientists have answered a 40-year-old question about Arctic ice thickness by treating the ice floes of the frozen seas like colliding molecules in a fluid or gas.
Although today’s highly precise satellites do a fine job of measuring the area of sea ice, measuring the volume has always been a tricky business. The volume is reflected through the distribution of
sea ice thickness — which is subject to a number of complex processes, such as growth, melting, ridging, rafting, and the formation of open water.
For decades, scientists have been guided by a 1975 theory (by Thorndike et al.) that could not be completely tested, due to the unwieldy nature of sea ice thickness distribution. The theory relied
upon a term that could not be related to the others, which represented the mechanical redistribution of ice thickness. As a result, the complete theory could not be mathematically tested.
Enter Yale professor John Wettlaufer, inspired by the staff and students at the Geophysical Fluid Dynamics Summer Study Program at the Woods Hole Oceanographic Institution, in Massachusetts. Over the
course of the summer, Wettlaufer and Yale graduate student Srikanth Toppaladoddi developed and articulated a new way of thinking about the space-time evolution of sea ice thickness.
The resulting paper appears in the Sept. 17 edition of the journal Physical Review Letters.
“The Arctic is a bellwether of the global climate, which is our focus. What we have done in our paper is to translate concepts used in the microscopic world into terms appropriate to this problem
essential to climate,” said Wettlaufer, who is the A.M. Bateman Professor of Geophysics, Mathematics and Physics at Yale.
Wettlaufer and co-author Toppaladoddi recast the old theory into an equation similar to a Fokker-Planck equation, a partial differential equation used in statistical mechanics to predict the
probability of finding microscopic particles in a given position under the influence of random forces. By doing this, the equation could capture the dynamic and thermodynamic forces at work within
polar sea ice.
“We transformed the intransigent term into something tractable and — poof — solved it,” Wettlaufer said.
The researchers said their equation opens up the study of this aspect of climate science to a variety of methods normally used in nonequilibrium statistical mechanics. | {"url":"https://earth.yale.edu/news/solving-problem-sea-ice-thickness-distribution-using-molecular-concepts","timestamp":"2024-11-01T22:55:09Z","content_type":"text/html","content_length":"34990","record_id":"<urn:uuid:b459a90d-c58c-48ee-a5cd-21c36e644186>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00388.warc.gz"} |
Strength of Materials(21-40)
21.The weakest section in a diamond riveting as shown in Fig. 10.1 is
22. According to Unwin's formula, the diameter of rivet in mm to suit the t mm
thickness of plate is given by
23.A flat carrying a pull of 690 kN is con nected to a gusset plate using rivets. If the
pulls required to shear the rivet, to crush the rivet and to tear the plate per pitch
length are 68.5 kN, 46 kN and 69 kN respectively, then the number of rivets
24.If the rivet value is 16.8 kN and force in the member is 16.3 kN, then the number
of rivets required for the connection of the member to a gusset plate is 2
25.In the riveted connection shown in Fig.10.2, which of the rivets will be subjected
26.When a member is subjected to axial ten sile load, the greatest normal stress is
equal to twice the maximum shear stress
27.At a point in a strained body carrying two unequal unlike principal stresses pi and p2 (Pi > P2), the maximum shear stress is given by (p, + p2)/2
28. If a point in a strained material is subjected.to equal normal and tangential
stresses, then the angle of obliquity is 45°
29. If the principal stresses at a point in a strained body are pi and p2 (pi > P2), then the resultant stress on a plane carrying the maximum shear stress is equal to
30. The plane of maximum obliquity is inclined to the major principal plane at an
where Φ max is the angle of maximum obliquity.
31.A point in a strained body is subjected to a tensile stress of 100 MPa on one plane and a tensile stress of 50 MPa on a plane at right angle to it. If these planes are carrying shear stresses of
50 MPa, then the principal stresses are inclined to the larger normal stress at an angle
32.If a prismatic member with area of cross section A is subjected to a tensile load P, then the maximum shear stress and its inclination with the direction of load
respectively are P/2A and 45°
33.The sum of normal stresses is constant
34.The radius of Mohr's circle for two equal unlike principal stresses of magnitude p is p
35.Shear stress on principal planes is zero
36.The state of pure shear stress is produced by tension in one direction and equal
compression in perpendicular direction
37.A prismatic bar is carrying only an axial force. The two planes on which normal
and shearing stresses are equal are inclined to the axial force at angles
38.According to Rankine's hypothesis, the criterion of failure of a brittle material is
39.Maximum bending moment in a beam occurs where shear force changes sign
40.A simply supported beam of span l carries over its full span a load varying linearly from zero at either end to w/unit length at midspan. The maximum bending moment occurs at | {"url":"https://www.sudhanshucivil2010.com/post/strength_2","timestamp":"2024-11-04T02:23:24Z","content_type":"text/html","content_length":"1050497","record_id":"<urn:uuid:4f5ce132-0763-4542-b5f1-ddd011efcb25>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00061.warc.gz"} |
Mastering Algebra - Coursepig.com
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Learn through carefully explained and annotated examples
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Is there an open oriented superstring?
Type I superstring theory is unoriented, and it seems that it needs to be so in order to exist.
Now, we always have open-closed duality, that connects at least the ultraviolet sector of a theory with the infrared sector of another, so in principle we should have some oriented open strings
coming from it, by duality with a closed oriented theory.
And we have all the D-brane stuff, that surely produces another kinds of open string theories with the restriction of terminating there in the branes.
¿Is some of these D-brane examples an oriented theory? ¿Is there some canonical, well known, example of a consistent oriented theory with open strings?
This post imported from StackExchange Physics at 2014-03-07 13:44 (UCT), posted by SE-user arivero
No. There is no theory of open, oriented strings. Any string theory must contain closed strings, while the open strings are optional. If there is a string theory which contains oriented open strings,
then it has the problem that the oriented open strings cannot couple to the oriented closed strings. Why?
This is my understanding of the explanation given by the by Thomas Mohaupt in Lecture notes "Introduction to String theory":
In the closed string spectrum, there is an $\mathcal N = 2A$ algebra and and an $\mathcal N = 2B$ algebra which lead to different string theories. Both have 32 supercharges. In each of these,, there
are 2 gravitinoes, dilationoes, 1 in the Ramond Neveu-Schwarz Sector and 1 more in the Neveu-Schwarz Ramond Sector. These 2 gravitinos need 2 different supercurrents to couple to. But the $\mathcal N
=1$ supersymmetric algebra with only 16 supercharges clearly cannot allow this!
Thus, the open oriented strings would not couple with the closed oriented strings. The solution is to have open unoriented strings instead. This along with the unoriented closed strings IS the Type I
string theory. The "only unoriented closed strings" theory is also inconsistent because of other reasons.
This post imported from StackExchange Physics at 2014-03-07 13:44 (UCT), posted by SE-user Dimensio1n0
What about the strings between D-branes? Are they unoriented too, even if they have Chan-Paton labels?
This post imported from StackExchange Physics at 2014-03-07 13:44 (UCT), posted by SE-user arivero
They are unoriented. Orientation is a property of the 2d worldsheet, not the 1d string. Unoriented just means that crosscaps and the like can appear in the worldsheet.
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Re: Finding the set of recursive calls
"Hans Aberg" <haberg@matematik.su.se>
14 Aug 2002 02:16:36 -0400
From comp.compilers
| List of all articles for this month |
From: "Hans Aberg" <haberg@matematik.su.se>
Newsgroups: comp.compilers
Date: 14 Aug 2002 02:16:36 -0400
Organization: Mathematics
References: 02-08-007 02-08-040
Keywords: analysis
Posted-Date: 14 Aug 2002 02:16:36 EDT
"VBDis" <vbdis@aol.com> wrote:
>In a discussion with Prof. Eppstein it turned out, that a vertex can
>form a strongly connected component, even if no explicit path exists
>from the vertex back to itself. This is due to the reflexive property
>of equivalence relations. Thus the set of recursive procedures is
>only a subset of the strongly connected components of an graph.
>So in the case of the set of recursive calls I'm not sure, which
>algorithms really are applicable without modificiations. The set of
>strongly connected components can be used as a base, from which all
>single-vertex components must be removed, which have no edges to
>(I knew that there was a problem... ;-)
At least the C++ variation I sent you in the mail puts the components on a
list before writing them out.
So it seems you should merely write out the components of size > 1 plus
the singletons of functions that call themselves, if that is included in
your definition of "recursive".
It should be easy to construct a directed graph putting label on each
vertex of a function that calls itself.
Hans Aberg * Anti-spam: remove "remove." from email address.
* Email: Hans Aberg <remove.haberg@member.ams.org>
* Home Page: <http://www.matematik.su.se/~haberg/>
* AMS member listing: <http://www.ams.org/cml/>
Post a followup to this message
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Convex Optimization for CP, Part 2: How do we solve Convex Optimization tasks? - Codeforces
Before we begin: This blog is my attempt to Codeforces Month of Blog Posts. Simple rule: Write an interesting CodeForces blog post until February 15th and win $300.
In this blog, I will be describing the concept of Convex Optimization, and techniques that can be used to solve Convex Optimization tasks. The blog will be divided into two parts — "What is Convex
Optimization", and "How do we solve Convex Optimization tasks". This part of the blog is for "How do we solve Convex Optimization tasks". (Part 1 is here)
There are just too many ways to solve them
One thing is for sure — There are too many ways to solve convex optimization tasks. By "too many", I mean, at least a dozen. We can't cover all of them here! So, I will be covering the methods that
can be used in a majority of convex optimization tasks in this blog. Those methods would be Gradient Descent (including the Subgradient method), Coordinate descent, and Ternary Search.
Gradient Descent
Gradient Descent is a technique where we repeatedly move closer to the optimal point. To do this, we must find the gradient of the objective function at the current point, and move based on the
gradient. If we know the gradient, we know one direction on which a more optimal point exists, so we move based on the gradient. There are two major flaws of the usual Gradient Descent, though. One
flaw is that if we start at a point where the gradient is too small, the convergence will be very slow. There are ways to solve this (Backtracking Line Search is an example), but we will not cover
them too technically in this article. Just remember that the methods are done by tweaking the movements, and we would be good to go. The greater issue is that, our objective function may or may not
have a gradient. Not always are the objective functions differentiable, so if the objective functions are not differentiable, we cannot use the traditional Gradient Descent.
Luckily, we can still use a similar way, called the Subgradient method. For every convex function at every point, a "subgradient" exists. If a gradient exists, the gradient is a subgradient. If it
doesn't, we define any gradient-like vector, where the plane/line defined by the vector underestimates the function in any point, as the subgradient. We can use this subgradient just like the
gradient in Gradient Descent. The only thing to remember is that we may not converge at all if the step size is fixed, so we must reduce the step size at every iteration. This method is very
convenient if we have a way to easily determine a subgradient.
Coordinate Descent
Coordinate Descent is probably the simplest of the methods used for Convex Optimization. Not many tasks allow the usage of Coordinate Descent, but if we can use it, it makes solving the task much
easier. Basically, we move along the axes on the euclidean plane. We define a variable "delta" initially, and in each step, we seek towards four directions (+x,-x,+y,-y) and see if moving towards any
direction by delta reduces the value. If moving towards any direction reduces the value, we move towards that direction. If delta is constant, then we will clearly be stuck in one point at some point
in time (likely the second step), so we decrease delta at every step. Coordinate Descent's flaw is that it can be stuck at situations where there are more than one direction having greatest
reduction. Still, this flaw does not exist on one dimension, so you can simply treat it as a "Lazy man's Ternary Search" in one dimension.
Ternary Search
At this point you may say, "Wait, isn't ternary search limited to one dimension?" and you may be right. But think about it, $$$\mathbb{R}^2$$$ is just $$$\mathbb{R}$$$ in two axes. Instead of just
one dimension, we can think of using the result from ternary search (the minimized/maximized value) as the function to minimize/maximize in another ternary search. This is one very simple way to
solve convex optimization tasks, and is guaranteed to work very often. (If time complexity isn't an issue, that is!) Again, coordinate descent always works in one dimension, so you can replace one
ternary search with coordinate descent if you want to.
Practice Tasks
Easy Difficulty
• Almost any ternary search task. Try to prove whether the objective function is convex before you solve it!
Medium Difficulty
• NCNA 2019 — "Weird Flecks, but OK": Usage of the Smallest Enclosing Circle task. (Kattis) (Baekjoon)
• Waterloo's local Programming Contests — "A Star not a Tree?": The Geometric Median task. (Baekjoon)
Medium-Hard Difficulty
• JAG Summer Camp 2019 — "All your base are belong to us": if K=1, this is the Smallest Enclosing Circle. if K=N, this is the Geometric Median. Is our objective function convex even for any
arbitrary N? Time to find out. (Baekjoon)
• 2013 Japan Domestic Contest — "Anchored Balloon": Designing the objective function may be tricky for this task. Bruteforcing is one solution for the task, but using convex optimization techniques
to solve it is a very good practice. (Baekjoon)
Hard Difficulty
• Asia Regional Contest 2007 in Tokyo — "Most Distant Point from the Sea": Some people reading the task may know this as a half-plane intersection task. And you're not wrong. Still, This task can
be solved as a Convex Optimization task well enough in the TL! (Baekjoon)
• SWERC 2021-2022 "Pandemic Restrictions": The intended solution was based on convex optimization. For proof, please check the official editorial of the mirror contest. (CF Gym)
UPD: Solutions here! | {"url":"https://mirror.codeforces.com/blog/entry/111048","timestamp":"2024-11-04T07:49:56Z","content_type":"text/html","content_length":"91327","record_id":"<urn:uuid:86fbe02f-1fb7-4b43-bc85-90b12687ede0>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00053.warc.gz"} |
How do you convert kN m2 to T m2?
How do you convert kN m2 to T m2?
How do you convert kN m2 to T m2?
How many kn/m2 in 1 ton/square metre? The answer is 9.80665. We assume you are converting between kilonewton/square metre and ton/square metre.
How do you convert PSF to kN M?
How to Convert PSF to Kilonewton/Square Meter (psf to kN/m2) By using our PSF to Kilonewton/Square Meter conversion tool, you know that one PSF is equivalent to 0.047880258977532 Kilonewton/Square
Meter. Hence, to convert PSF to Kilonewton/Square Meter, we just need to multiply the number by 0.047880258977532.
How many kiloNewtons are in a ton?
Kilonewton to Ton-force (metric) Conversion Table
Kilonewton [kN] Ton-force (metric) [tf]
1 kN 0.1019716213 tf
2 kN 0.2039432426 tf
3 kN 0.3059148639 tf
5 kN 0.5098581065 tf
What is 5kn m2 in tonnes?
Kilonewton Meter to Ton-force (metric) Meter Conversion Table
Kilonewton Meter [kN*m] Ton-force (metric) Meter
2 kN*m 0.2039432426 ton-force (metric) meter
3 kN*m 0.3059148639 ton-force (metric) meter
5 kN*m 0.5098581065 ton-force (metric) meter
10 kN*m 1.019716213 ton-force (metric) meter
How do you work out KN m2?
How do you calculate kN? Multiply the load per unit area or length by the total area or length. For the rectangle, you compute 10 kN per square meter multiplied by 24 square meters to get 240 kN. For
the beam, you calculate 10 kN per meter multiplied by 5 meters to get 50 kN.
What is a KN m2?
kN/m² – Kilonewton Per Square Meter.
What unit is psf?
Pounds per Square Foot
Pounds or Pounds Force per Square Foot is a British (Imperial) and American pressure unit which is directly related to the psi pressure unit by a factor of 144 (1 sq ft = 12 in x 12 in = 144 sq in).
1 pound per square foot equals 47.8803 pascals.
What is relation between ton and kN?
How many Kilonewtons are in a Tonne? The answer is one Tonne is equal to 9.81 Kilonewtons.
How many kg is 1kN?
One kilonewton, 1 kN, is equivalent to 102.0 kgf, or about 100 kg of load under Earth gravity. 1 kN = 102 kg × 9.81 m/s2. So for example, a platform that shows it is rated at 321 kilonewtons (72,000
lbf) will safely support a 32,100-kilogram (70,800 lb) load. | {"url":"https://poletoparis.com/how-do-you-convert-kn-m2-to-t-m2/","timestamp":"2024-11-02T17:41:57Z","content_type":"text/html","content_length":"43392","record_id":"<urn:uuid:1a99df27-92fd-4a03-992f-d6c7b04af3b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00182.warc.gz"} |
Curve Fitting for the Core Loss Model
The core loss of magnetic components plays an important role in engineering practices. Particularly at high-frequency conditions, the loss of magnetic components accounts for a large proportion of
the energy loss of the whole unit. Most manufacturers provide the core loss curve in the product manuals. The core loss also has corresponding theoretical models and coefficients. Some core material
manufacturers give these coefficients, but some manufacturers do not provide it. In this scenario, the user needs to perform curve-fitting on the core loss curves to obtain the parameters by
themselves. The core loss is also a common factor in electromagnetic simulation, especially in the post-processing analysis of the electromagnetic results, the fitted coefficients of core loss are
commonly used.
Core loss is closely related to magnetic material characteristics and operating frequency. Equally, the core loss is also related to working temperature. There are valid methods for calculating core
loss at different temperatures. This article focuses on the curve fitting of the core loss P-B test data. The effect of temperature on energy loss will be described in a future article.
Models and numerical methods of core loss curve fitting
Common core loss models include the classical model, Steinmetz model, and modified Steinmetz models, etc. Since these models are similar, this article will only focus on the theoretical and numerical
methods of the classical and Steinmetz models.
Computation of electrical steel core loss from loss curves
The iron-core loss without DC flux bias is expressed as the following:
where Bm is the amplitude of the AC flux component, f is the frequency, Kh is the hysteresis core loss coefficient, Kc is the eddy-current core loss coefficient, and Ke is the excess core loss
coefficient. Hysteresis loss (Ph) is the energy loss caused by the magnetic domain during the magnetization process of the magnetic domain to overcome the friction between the magnetic domains. This
part of the loss eventually heats the component and dissipated. The energy lost per unit volume of the core is proportional to the area enclosed by the hysteresis loop. Each magnetization cycle
consumes energy proportional to the area enclosed by the hysteresis loop, so the smaller the hysteresis curve area is, the smaller the hysteresis loss is; the higher the frequency, the greater the
power loss. Eddy current loss (Pc) is due to the resistivity of the magnetic core material is finite, and has a certain resistance. At high frequencies, the eddy current in the magnetic core will be
caused by the exciting magnetic field and lead to losses. Residual loss (Pe) is a loss due to the magnetization relaxation effect or magnetic hysteresis effect. The so-called relaxation means that
during the process of magnetization or demagnetization, the magnetization state does not immediately change to its final state with the change of the magnetization intensity, but requires a process.
This ‘time effect’ is the cause of excess loss.
For the given P-B test data, as long as the quadratic form below is minimized, the curve parameters Kf, Kc, and Ke can be obtained.
where m is the number of loss curves, n_i is the number of data points for the i-th loss curve, and p_ij is a two-dimensional look-up table for loss curves.
Computation of power ferrite core loss (Steinmetz model)
Although the classical method gives a reasonable explanation of magnetic core loss, it is inconvenient to calculate certain magnets in the practices. The mathematician and electrical engineer
Steinmetz summed up an empirical formula suitable for engineering calculation of core loss. The expression is as follows:
where p_v is the average power loss density, f is the excitation frequency, and Bm is the peak magnetic flux density. This formula shows that the loss per unit volume p_v is an exponential function
of the repetitive magnetization frequency and magnetic flux density. Cm, x, and y are empirical parameters. Both exponents can be non-integer, generally 1<x<3 and 2<y<3. For different materials,
manufacturers generally give a corresponding set of parameters. This formula is commonly used to represent the state of magnetization under sinusoidal excitation and cannot be used for non-sinusoidal
magnetic excitations such as square waves. The Steinmetz model has only three parameters and has proved to be a useful tool for calculating core loss. For the sinusoidal waveform, it is more
convenient and accurate to calculate the core loss using this formula.
To linearize the equation for curve fitting, we used base-10 logarithms. The equation above can be rewritten to
where c=log(Cm)。
By minimizing its quadratic form below, its parameters c, x, and y can be calculated.
where m is the number of loss curves, n_i is the number of points of the i-th loss curve, and p_vij is a two-dimensional look-up table for loss curves. Then coefficient Cm is calculated from the
equation c=log(Cm).
The numerical algorithms for function minimization involve more content, which we will discuss in future articles.
Procedures of core loss curve fitting
The following describes how to use MatEditor to fit the parameters of the core loss curves. MatEditor is a free engineering material data editing software. The same module is also included in the
finite element analysis software WelSim.
Add P-B Test Data material property, and input the data into the table, these data are often from the manufacturer’s manual of magnetic products. You can also enter table data by loading a text or
Excel file. After inputting, click the frequency of each line, and the chart window will display the related P-B curves. Clicking on the header of the frequency column will display all P-B curves
2) Add Core Loss Model material property and set Electrical Steel or Power Ferrite in the Model Type.
3) Right-click the Core Loss Model property pop-up menu and add Curve Fitting sub-property.
4) After adding the curve fitting sub-property, right-click, and select Solve Curve Fit from the pop-up context menu to solve the curve fitting.
5) If the solving succeeds, the calculated coefficients will be automatically displayed in the table window. As the picture shows,
6) Right-click on the curve fitting property and select Copy Calculated Values to Property. The calculated coefficients will be set to the coefficient properties.
7) (Optional) For the magnetic core material, we often need to display the curves in the logarithmic axis. Click the logarithmic axis button to display the curves in the logarithmic axis.
This case uses the Power Ferrite (Steinmetz) model as an example. The curve fitting procedures for the Electrical Steel model are the same.
A tutorial video is attached below for your reference. | {"url":"https://welsim.com/2021/06/30/curve-fitting-for-the-core-loss-model.html","timestamp":"2024-11-12T13:43:39Z","content_type":"text/html","content_length":"26439","record_id":"<urn:uuid:3214418c-6cd1-40aa-818b-39b13b3bb18b>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00236.warc.gz"} |
Tails You Win, One Tail You Lose
I presented the math for this at the #cosyne19 diversity lunch today.
Success rates for first authors with known gender:
Female: 83/264 accepted = 31.4%
Male: 255/677 accepted = 37.7%
37.7/31.4 = a 20% higher success rate for men https://t.co/u2sF5WHHmy
— Megan Carey (@meganinlisbon) March 2, 2019
Controversy over hypothesis testing methodology encountered in the wild a second time! At this year’s Computational and Systems Neuroscience conference, CoSyNE 2019, there was disagreement over
whether the acceptance rates indicated bias against women authors. As it turns out, part of the disupute turned over which statistical test to run!
Controversial Data
CoSyNe is an annual conference where COmputational and SYstems NEuroscientists to get together. As a conference in the intersection of two male-dominated fields, concerns about gender bias abound.
Further, the conference uses single-blind review, i.e. reviewers but not submitters are anonymous, which could be expected to increase bias against women, though effects might be small.
During the welcome talk, the slide below was posted (thanks to Twitter user @neuroecology for sharing their image of the slide; they have a nice write-up data mining other CoSyNe author data) to
support the claim that bias was “not too bad”, since the ratio of male first authors to female first authors was about the same between submitted and accepted posters.
However, this method of viewing the data has some problems: the real metric for bias isn’t the final gender composition of the conference, it’s the difference in acceptance rate across genders. A
subtle effect there would be hard to see in data plotted as above.
And so Twitter user @meganinlisbon got hold of the raw data and computed the acceptance rates and their ratio in the following tweet:
I presented the math for this at the #cosyne19 diversity lunch today.
Success rates for first authors with known gender:
Female: 83/264 accepted = 31.4%
Male: 255/677 accepted = 37.7%
37.7/31.4 = a 20% higher success rate for men https://t.co/u2sF5WHHmy
— Megan Carey (@meganinlisbon) March 2, 2019
Phrased as “20% higher for men”, the gender bias seems staggeringly high!
It seems like it’s time for statistics to come and give us a definitive answer. Surely math can clear everything up!
Controversial Statistics
Shortly afterwards, several other Twitter users, including @mjaztwit and @alexpiet attempted to apply null hypothesis significance testing to determine whether the observed gender bias was likely to
be observed in the case that there was, in fact, no bias. Such a result is called significant, and the degree of evidence for significance is quantified by a value \(p\). For historical reasons, a
value of \(0.05\) is taken as a threshold for a binary choice about significance.
And they got different answers! One found that the observation was not significant, with \(p \approx 0.07\), while the other found the the observation was significant, with \(p \approx 0.03\). What
There were some slight differences in low-level, quantitative approach: one was parametric, the other non-parametric. But they weren’t big enough to change the \(p\) value. The biggest difference was
a choice made at a very high level: namely, are we testing whether there was any gender bias in CoSyNe acceptance, or are we testing whether there was more specifically gender bias against women.
The former is called a two-tailed test and is more standard. Especially in sciences like biology and psychology, we don’t know enough about our data to completely discount the possibility that
there’s an effect opposite to what we might expect.
Because we consider extreme events “in both directions”, the typical effect of switching from a two to a one-tailed test is to cut the \(p\)-value in half. And indeed, we \(0.03\) is approximately
half of \(0.07\).
But is it reasonable to run a two-tailed test for this question? The claims and concerns of most of the individuals concerned about bias was framed specifically in terms of female-identifying authors
(to my recollection, choices for gender identification were male, female, and prefer not to answer, making it impossible to talk about non-binary authors with this data). And given the other evidence
for misogynist bias in this field (the undeniably lower rate of female submissions, the near-absence of female PIs, the still-greater sparsity of women among top PIs) it would be a surprising result
indeed if there were bias that favored women in just this one aspect. Suprising enough that only very strong evidence would be sufficient, which is approximately what a two-tailed test does.
Even putting this question aside, is putting this much stock in a single number like the \(p\) value sensible? After all, the \(p\) value is calculated from our data, and it can fluctuate from sample
to sample. If just two more female-led projects had been accepted or rejected, the two tests would agree on which side of \(0.05\) the \(p\) value lay!
Indeed, the CoSyNe review process includes a specific mechanism for randomness, namely that papers on the margin of acceptance due to a scoring criterion have their acceptance or rejection determined
by the output of a random number generator.
And the effect size expected by most is probably not too much larger than what is reported, since the presumption is that the effect is mostly implicit bias from many reviewers or explicit bias from
a small cohort. In that case, adhering to a strict \(p\) cutoff is electing to have your conclusions from this test determined almost entirely by an explicitly random mechanism. This is surely
It would seem to me that the more reasonable conclusion is that there is moderately strong evidence of a gender bias in the 2019 CoSyNe review process, but that the number of submissions is
insufficient to make a definitive determination possible based off of a single year’s data. This data is unfortunately not available for previous years.
At the end of the conference, the Executive Committee announced that they had heard the complaints of conference-goers around this bit of gender bias and others and would be taking concrete steps to
address them. First, they would be adding chairs for Diversity and Inclusion to the committee. Second, they would move to a system of double-blind review, in which the authors of submissions are also
anonymous to the reviewers. Given the absence of any evidence that such a system is biased against men and the evidence that such a system reduces biases in general, this is an unambiguously good
move, regardless of the precise \(p\) value of the data for gender bias this year. | {"url":"https://charlesfrye.github.io/stats/2019/03/06/cosyne19-gender-bias.html","timestamp":"2024-11-07T12:18:27Z","content_type":"text/html","content_length":"15134","record_id":"<urn:uuid:2aa1586e-bc47-4481-ad4a-fade65bfca8b>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00810.warc.gz"} |
Caught Looking: How Responsible Are Pitchers For Called Strikes?
This article was written by Connelly Doan
This article was published in Spring 2022 Baseball Research Journal
The strikeout is one of the few outcomes ascribed to a pitcher’s control, as indicated by metrics such as Fielding Independent Pitching (FIP) and Skill-Interactive Earned Run Average (SIERA).
However, overall pitcher success is typically attributed to generating swinging strikes1—without considering the ability to induce called strikes. Although discussions around called strikes from the
pitcher’s perspective are infrequent, discussions about other actors’ perspectives, including the umpire and catcher, are abundant within the baseball community. Technology has been working its way
into baseball over recent years, and one feature that has been debated and tested has been introducing an electronic strike zone to reduce human error in called balls and strikes.2 Further, general
conversations and specific metrics have advanced around called strikes from the catcher’s perspective in the form of pitch framing, implying that the catcher holds an important role in inducing a
called strike.3,4
While a large proportion of called-strike analyses have been from perspectives other than the pitcher, there have been several useful developments directed towards understanding pitchers’
contributions. A Called Strike Above Average (CSAA) metric now exists on Baseball Prospectus, which identifies the additional called strikes created by a particular pitcher in relation to the average
among all pitchers.5 Additionally, Alex Fast and Nick Pollack of Pitcher List developed a Called Strikes + Whiffs (CSW) metric in an attempt to understand a pitcher’s value beyond the proportion of
swinging strikes they generate.6
These more nuanced metrics can help us understand called strikes from beyond the outcome seen from a singular, non-pitcher perspective. Relatively little has been done in terms of attempting to
isolate how much each actor (pitcher, catcher, umpire) contributes to a called strike.7 This paper will add to these analyses by studying pitch-level data from 2015 through 2019. Using logistic
regression, the effect of the pitcher, catcher, and umpire on a given taken pitch will be measured in relation to the probability of that pitch being called a strike. Not only will this analysis help
create a more complete understanding of how each party influences a called strike, but it will also shed light specifically on how a pitcher affects the generation of a called strike.
The raw pitch-by-pitch data for this article were scraped from MLB’s BaseballSavant.com using RStudio code and packages based on Bill Petti’s BaseballR.8 Specifically, the raw data comprised
pitch-level events from every game of the 2015 season through the 2019 season where the pitch outcome was either a ball or a called strike. Each pitch outcome also included the pitcher who threw the
pitch, the catcher behind the plate, and the home-plate umpire, along with other descriptive features. Pitcher and catcher CSAA data were taken from Baseball Prospectus.9
The data were then filtered for adequate sample sizes. Data were only included for: catchers with at least 500 framing chances in a given season, umpires with at least 500 called-strike chances in a
given season, and pitchers who pitched at least 100 innings in a given season. The decision to limit the data to pitchers who threw at least 100 innings in a given season was determined considering
the varying overall strike rates for starters and relievers; the standard for a strong strike rate for a starting pitcher is different from that of a relief pitcher.10 The role and definition of a
starting pitcher has changed over recent years with the introduction of openers and nominal starters being used as long-relief pitchers,11 hence the decision to use 100 innings pitched as the cutoff,
rather than a slightly higher mark.
Because the goal of this analysis was to understand how each actor individually impacts the probability of a called strike, a fixed-effects logistic regression model was implemented. The dependent
variable was measured as a dichotomous variable, whether or not the taken pitch was called a strike. Several control variables were also included. The count of the particular pitch was included, as
research has shown that the general probability of called strikes varies in different counts.12 Same pitcher-batter handedness was included to account for potential relative obscurity of the umpire’s
view of a pitch. An umpire may be able to see a pitch more clearly depending on the handedness of the pitcher in relation to the batter, which could give the umpire more confidence in calling the
pitch a strike.13
The pitcher’s and catcher’s individual impact was measured using Baseball Prospectus’s respective CSAA metrics. Practically, the catcher CSAA metric captures a catcher’s pitch-framing ability, while
the pitcher CSAA metric captures a pitcher’s level of command, or the ability to precisely locate pitches, in or out of the strike zone, with the goal of keeping pitches out of the middle of the
plate.14 These CSAA metrics were calculated using a mixed-effects model in order to isolate the most likely individual contributions of each actor, while controlling for their relative effects on
each other.15 The home-plate umpire’s individual impact was measured using each umpire’s called-strike percentage on eligible pitches in a particular season. If the umpire’s called-strike percentage
fell above the third quartile of data, they were labeled a generous umpire; if the umpire’s called-strike percentage fell below the first quartile of data, they were labeled a tough umpire.
The model used can be represented by the formula:
Y= β0 +σ βkXk+ β1X1 + β2X2 + β3U1 + β4U2
where Y is the probability of an eligible pitch being called a strike. β0 is the constant intercept, in this case the log odds of an eligible pitch being called a strike in a different-handed
pitcher-batter matchup in a 0-0 count with a league-average pitcher, catcher, and home-plate umpire. Effects of k control variables are denoted by σ βk. β1 is the coefficient of a pitcher with a CSAA
of X1, and β2 is the coefficient of a catcher with a CSAA of X2. β3 is the coefficient of having a tough umpire behind the plate, and β4 is the coefficient of having a generous umpire behind the
Descriptive Statistics
Tables 1 and 2 present descriptive statistics for the categorical and continuous variables in this analysis.
(Click images to enlarge)
Approximately one-third of the 1.2 million pitches in this dataset were called strikes, with 41.5% of those pitches occurring in same-handed pitcher-batter matchups. Approximately one-third of the
pitches occurred in a 0-0 count, with an overall larger percentage of called-strike-eligible pitches occurring in earlier counts. Umpire called-strike data were normally distributed; as such, roughly
one-quarter of the pitches occurred with a generous home-plate umpire, one-quarter with tough umpires, and half with average umpires.
Both pitcher and catcher CSAA showed similar distributions; the average pitcher CSAA was 0.003196 with a standard deviation of 0.010191, while the average catcher CSAA was 0.003028 with a standard
deviation of 0.012168.
Model Results
Table 3 presents the intercepts, standard errors, Z scores, and P values for the logistic regression model. Logistic regression centers around log odds, as using log-odds results in symmetry around
(Click image to enlarge)
Table 4 shows some conversions of probabilities to log odds for reference.16
(Click image to enlarge)
The constant intercept value of -0.244 is the log odds of an eligible pitch being called a strike in a different-handed pitcher-batter matchup in a 0-0 count with a league-average pitcher, catcher,
and home-plate umpire. All counts except 2-0 had statistically significant log-odd coefficients; indicating that, controlling for all other variables, all counts other than 2-0 affected the
probability of a taken pitch being called a strike. The count with the largest impact on increasing the log odds of a called strike was 3-0. Counts with the largest impact on decreasing the log odds
of a called strike were 0-2, 1-2, and 2-2. While other research has been done on the size of the strike zone in different counts or for particular types of pitches, this article focused on the impact
of each actor in the generation of a called strike; the impact of the count simply groups the recorded outcome of each pitch in general (ball or called strike).17
The log-odds coefficient for same-handedness (0.152) was statistically significant, indicating that, controlling for all other variables, having a same-handed pitcher-batter matchup slightly
increased the probability of a taken pitch being called a strike.
The log-odds coefficients for pitchers (2.151) and catchers (3.811) were both statistically significant. This indicates that, controlling for all other variables, each one-unit increase in pitcher
CSAA increased the log-odds of a taken pitch being called a strike by 2.151. Likewise, controlling for all other variables, each one-unit increase in catcher CSAA increased the log-odds of a taken
pitch being called a strike by 3.811.
The log-odds coefficients for having a tough umpire behind the plate (-0.0751) and a generous umpire behind home plate (0.0908) were both statistically significant. This indicates that, controlling
for all other variables, having a tough umpire slightly decreased the probability of a taken pitch being called a strike. Likewise, controlling for all other variables, having a generous umpire
slightly increased the probability of a taken pitch being called a strike.
A logistic regression model was run using data from roughly 1.2 million taken pitches from the 2015 through 2019 seasons to understand how various factors and actors impact the probability of a pitch
being called a strike. All counts except 2-0 had a statistically significant impact on the probability of a taken pitch being called a strike. The probability of a called strike increased in 3-0
counts, while all other counts exhibited a decrease in the probability. These results match previous research on the changing of called strike zones in different counts.18 Additionally, taken pitches
in same-handed pitcher-batter circumstances had a higher probability of being called a strike compared to different-handed matchups. This suggests that umpires may be more comfortable viewing the
delivery of a pitch if the pitcher is releasing the ball on the same side of the plate as the batter. The result aligns with research that suggests a bias is held towards calling outside pitches
strikes given how umpires set up behind the plate, particularly in same-handed pitcher-batter matchups.19
Regarding the actors involved, catchers had the most influence over the probability of a taken pitch being called a strike. Specifically, the log-odds coefficient for catcher CSAA was roughly 75%
greater than that of pitcher CSAA. However, pitchers also had a sizable influence over the probability of a taken pitch being called a strike. While this result may not be surprising to some, it
supports the assertion that pitch framing alone is not the sole factor in generating called strikes; a pitcher’s command plays a non-trivial role as well. Further investigation and discussion around
pitchers’ contributions to generating called strikes should continue. As for umpires, having either a tough umpire or generous umpire (based on called-strike percentages) behind the plate affected
the probability of a taken pitch being called a strike, with tough umpires decreasing the probability and generous umpires increasing the probability. The relative effect of generous umpires was
slightly more than tough umpires; meaning that the increase in probability of a called strike with a generous umpire was greater than the decrease in probability of a called strike with a tough
Several limitations exist in this analysis. First, batters were not considered in this model. It is clear that the batter is an actor in the generation (or not) of a called strike, and some research
has been conducted in terms of showing the effect the batter may have.20 However, data capturing a batter’s independent impact on a taken pitch being called a strike are not currently publicly
available. Further, a clear explanation for how a batter impacts such an outcome is not as obvious as that of a pitcher, catcher, or umpire. The pitcher’s impact can be attributed to their command,
the catcher to their pitch framing, and the umpire to the fact that they are in charge of making the call. A number of possible explanations could be attributed to the batter, such as stance or
location to home plate, their height, individual swing mechanics, general reputation as a player, and plate discipline, etc.21 It would make sense to attempt to capture the batter’s effect in a model
only if said effect could be fully understood and explained.
The second limitation was the level of nuance of the metric used to represent the effect of the umpire. Ideally, an umpire CSAA metric (similar to the pitcher CSAA and catcher CSAA metrics used)22
would be publicly available for usage. However, such a metric is not present on Baseball Prospectus. That being said, the effect of the umpire was still captured adequately in the model for the
purposes of this analysis.
This article provides support that pitchers are an important actor in the generation of called strikes, and that further investigation, much like that of understanding pitch framing for catchers,
would provide valuable insight. In that vein, there are several logical avenues of study that could follow this article.
The first could be a deep dive into better understanding why pitchers have the level of command that they do. In other words, it would make sense to attempt to understand what characteristics
pitchers with high CSAA possess and the same for pitchers with low CSAA. Characteristics of interest could include velocity, pitch spin rate, handedness, and arm angle or delivery mechanics.23
Analyzing which aspects of pitching a pitcher can improve upon to maximize their effectiveness—beyond swinging strikes or “raw stuff”24—would allow for better theoretical understanding and practical
The second could be the investigation of the relationship between a pitcher’s ability to generate swinging strikes and their ability to generate called strikes. Little has been done in terms of
analyzing a pitcher’s overall value in terms of generating both called and swinging strikes, other than the recent introduction of the CSW metric.25 This is not surprising, as pitchers who have the
ability to make batters miss with high velocity or devastating breaking pitches (such as Randy Johnson, Clayton Kershaw, Aroldis Chapman, etc.) make a greater impression than those who have the
ability to pitch around batters with strong command (such as Jamie Moyer, Kyle Hendricks, etc.). The relative lack of analyses on pitchers impacting called strikes was the motivation behind this
article, so it would make sense to attempt to understand how valuable that aspect of a pitcher’s game is in relation to other aspects.
CONNELLY DOAN, MA is a data analyst in Las Vegas who has applied his professional skills to the game of baseball, both personally and for RotoBaller.com. He has been a SABR member since 2018. He can
be reached on Twitter (@ConnellyDoan) and through email (doanco01@gmail.com).
1. Jarad Evans, “Sabermetrics Glossary: Swinging Strike Rate,” FantasyPros, February 4, 2020, accessed February 27, 2022. https://www.fantasypros.com/2020/02/
2. Mark T. Williams, “MLB Umpires Missed 34,294 Ball-Strike Calls in 2018: Bring on Robo-umps?” BU Today, April 8, 2019, accessed February 27, 2022. https://www.bu.edu/articles/2019/
mlb-umpires-strike-zone-accuracy; and Mhatter106, “Out of the Frame: The Effect of an Electronic Strike Zone on Catching,” Crawfish Boxes, SB Nation, November 19, 2019, accessed February 27, 2022.
3. Harry Pavlidis, “You Got Framed: A New Metric Reveals Just How Valuable an Artful Catcher Who Can Turn a Borderline Pitch into a Called Strike Can Be,” ESPN: The Magazine, June 27, 2014, accessed
Februray 27, 2022. http://www.espn.com/espn/feature/story/_/id/11127248/how-catcher-framing-becoming-great-skill-smart-teams-new-york-yankees-espn-magazine; and Mike Axisa, “Pitch-Framing Might Soon
be a Los MLB Art: But it Could Still Matter for These Pitchers and New Battery Mates,” CBS Sports, May 20, 2020, accessed February 27, 2022. https://www.cbssports.com/mlb/news/
4. Jared Cross, “FanGraphs Pitch Framing,” FanGraphs, March 20, 2019, accessed February 27, 2022. https://blogs.fangraphs.com/fangraphs-pitch-framing/; and Jonathan Judge, Harry Pavlidis, and Dan
Brooks, “Moving Beyond WOWY: A Mixed Approach to Measuring Catcher Framing,” Baseball Prospectus, February 5, 2015, accessed February 27, 2022. https://www.baseballprospectus.com/news/article/25514/
5. Harry Pavlidis, Bret Sayre, Jonathan Judge, and Jeff Long, “Prospectus Feature: Command and Control,” Baseball Prospectus, January 23, 2017, accessed February 27, 2022. https://
6. Alex Fast, “CSW Rate: An Intro to an Important New Metric,” PitcherList, April 16, 2019, accessed February 27, 2022. https://www.pitcherlist.com/csw-rate-an-intro-to-an-important-new-metric/; and
“Glossary,” PitcherList, accessed February 18, 2021. https://www.pitcherlist.com/glossary.
7. Joe Rosales and Scott Spratt, “Who is Responsible for a Called Strike?” MIT Sloan Sports Analytics Conference, February 27, 2015, accessed February 27, 2022. https://global-uploads.webflow.com/
8. “Baseballr: A Package for the R Programming Language,” baseballr, last modified May 13, 2020, accessed February 27, 2022. http://billpetti.github.io/baseballr.
9. “Custom Statistic Report: Catcher Stats—Full Season,” Baseball Prospectus, last modified September, 30, 2019, accessed February 27, 2022. https://legacy.baseballprospectus.com/sortable/index.php?
cid=2519121; and “Custom Statistic Report: Pitcher Season,” Baseball Prospectus, last modified September, 30, 2019, accessed February 27, 2022. https://legacy.baseballprospectus.com/sortable/
10. Fast.
11. J.J. Cooper, “Are ERA Qualifying Standards Becoming Obsolete?” Baseball America, March 5, 2018, accessed February 27, 2022. https://www.baseballamerica.com/stories/
12. Max Marchi and Jim Albert, Analyzing Baseball Data With R (Florida: Taylor and Francis Group, 2013), 181-84; and John Walsh, “The Compassionate Umpire,” The Hardball Times, April 7, 2010,
accessed February 27, 2022. https://tht.fangraphs.com/the-compassionate-umpire.
13. TC Zencka, “Which Pitchers Should Fear Robot Umpires?,” MLB Trade Rumors, April 18, 2020, accessed February 27, 2022. https://www.mlbtraderumors.com/2020/04/
14. Pavlidis, Sayre, Judge, and Long.
15. Judge, Pavlidis, and Brooks.
16. James Jaccard, Interaction Effects in Logistic Regression (California, Sage Publications, 2001).
17. Matthew Carruth, “The Size of the Strike Zone by Count,” Fangraphs.com, December 18, 2012: https://blogs.fangraphs.com/the-size-of-the-strike-zone-by-count/; Bill Petti, “Getting Strikes on the
Edge,” Fangraphs.com, July 15, 2013: https://blogs.fangraphs.com/getting-strikes-on-the-edge.
18. Walsh; and Marchi and Albert.
19. Camden Kay, “The Inherent Umpire Bias,” Balk It Off, July 31, 2020, accessed February 27, 2022. https://medium.com/balk-it-off/umpire-bias-8e5b776ae1b0.
20. Judge, Pavlidis, and Brooks; and Rosales and Spratt.
21. Regarding player height, see Travis Sawchik, “Aaron Judge is Hitting Even Better with an Even Worse Strike Zone,” Fangraphs.com, May 17, 2018: https://blogs.fangraphs.com/
22. Judge, Pavlidis, and Brooks.
23. Zencka.
24. Pavlidis, Sayre, Judge, and Long.
25. Fast.
https://sabr.org/wp-content/uploads/2020/03/research-collection4_350x300.jpg 300 350 sabr /wp-content/uploads/2020/02/sabr_logo.png sabr2022-05-18 05:54:512022-05-18 20:25:44Caught Looking: How
Responsible Are Pitchers For Called Strikes? | {"url":"https://sabr.org/journal/article/caught-looking-how-responsible-are-pitchers-for-called-strikes/","timestamp":"2024-11-09T20:30:18Z","content_type":"text/html","content_length":"118828","record_id":"<urn:uuid:80552bff-3df2-4581-8f44-f7d86eeb0c9c>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00435.warc.gz"} |
Exploring the Fascinating World of Multiples of 3 – From Basics to Advanced Concepts
I. Introduction
When it comes to mathematics, multiples of 3 play a crucial role. In this blog post, we will delve into the world of multiples of 3, exploring their basics, patterns, properties, and even advanced
concepts. By the end of this article, you will have a comprehensive understanding of all the multiples of 3 and their significance within various mathematical domains.
II. Basics of Multiples of 3
Before we dive into the fascinating world of multiples of 3, let’s start with the definition. Multiples of 3 are numbers that can be obtained by multiplying 3 by a whole number. For example, the
multiples of 3 include 3, 6, 9, 12, and so on. Finding multiples of 3 is relatively easy, especially with a few examples.
A. Examples of Finding Multiples of 3
To get a better grasp of multiples of 3, let’s look at some examples. Let’s start with multiples of 3 up to 10:
Now, let’s explore multiples of 3 up to 100:
• 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99
B. Properties of Multiples of 3
Multiples of 3 have interesting properties worth exploring. Let’s delve into two key properties:
1. Relationships with Odd and Even Numbers
Notably, multiples of 3 are closely linked with odd and even numbers. Every alternate multiple of 3 is even, while the ones in between are odd. For example, 6 (even) is followed by 9 (odd), 12
(even), 15 (odd), and so on.
2. Relationship with Divisibility by 3
Multiples of 3 are related to divisibility by 3. If a number is divisible by 3, it is considered a multiple of 3. For instance, 9 is divisible by 3, making it a multiple of 3.
III. Patterns and Properties of Multiples of 3
Now that we have covered the basics, it’s time to delve deeper into the patterns and properties of multiples of 3. Understanding these patterns can provide further insights into the nature of
multiples of 3 and their relationships.
A. Alternative Ways to Find Multiples of 3
There are alternative methods for finding multiples of 3 that can be convenient in different scenarios. Let’s explore two such methods:
1. Using the Sum of Digits Method
One method to find multiples of 3 is by examining the sum of the digits of a given number. If the sum is divisible by 3, then the number itself is a multiple of 3. For example, consider the number
123. The sum of its digits is 1 + 2 + 3 = 6, which is divisible by 3. Therefore, 123 is a multiple of 3.
2. Using Modular Arithmetic
Modular arithmetic offers another approach to finding multiples of 3. By taking the modulus of a number with 3, we can determine if it is a multiple of 3. If the remainder is 0, then the number is
indeed a multiple of 3. For instance, 21 % 3 equals 0, indicating that 21 is a multiple of 3.
B. Understanding the Cyclical Pattern of Multiples of 3
Multiples of 3 exhibit a cyclical pattern when observing the last digits. By exploring this pattern, we gain further insights into how multiples of 3 are formed.
1. Relationship between the Last Digit and Divisibility by 3
The last digit of a number determines its divisibility by 3. If the last digit is either 0, 3, 6, or 9, then the number is divisible by 3. This relationship helps identify multiples of 3 more
2. Exploring the Repeating Pattern of the Last Two Digits
When examining the last two digits of multiples of 3, we notice a repeating pattern. The sequence of the last two digits repeats after every three numbers. For example, the last two digits of
multiples of 3 are 00, 03, 06, 09, 12, 15, and so on.
C. Curious Properties of Multiples of 3
The world of multiples of 3 holds some fascinating properties that are worth exploring. Let’s dive into two curious properties:
1. Sum of Digits of Multiples of 3
If we sum the digits of any multiple of 3, the result is also a multiple of 3. For instance, let’s take the number 18. The sum of its digits, 1 + 8, is 9, which is divisible by 3. This property holds
true for all multiples of 3.
2. Relationship with Other Numeral Systems
Not only in base 10, but multiples of 3 also exhibit interesting properties in other numeral systems. For instance, in binary (base 2), multiples of 3 have a repeating pattern of 11. This
demonstrates the universality of the concept of multiples of 3 across number systems.
IV. Advanced Concepts Related to Multiples of 3
Now that we have covered the basics, patterns, and fascinating properties of multiples of 3, it’s time to explore some advanced concepts and applications.
A. Multiples of 3 in a Number System Other than Base 10
In different number systems, multiples of 3 exhibit unique properties and patterns. Exploring multiples of 3 in number systems other than base 10 can provide valuable insights into the connections
between numbers and their representations.
B. Applications of Multiples of 3 in Practical Scenarios
Multiples of 3 find practical applications in various scenarios. For example, they are essential in determining time intervals, such as seconds, minutes, and hours. Additionally, multiples of 3 play
a role in scheduling and organizing repetitive events.
C. Exploring Multiples of 3 within Larger Mathematical Domains
To truly appreciate the significance of multiples of 3, we can extend our exploration into larger mathematical domains, such as number theory and algebra.
1. Multiples of 3 in Number Theory
In number theory, multiples of 3 play a crucial role. They are closely connected to prime numbers, divisibility rules, and prime factorization. Understanding these connections can deepen our
understanding of number theory concepts.
2. Multiples of 3 in Algebra and Beyond
In Algebra, multiples of 3 serve as a foundation for solving equations and expressing patterns. They help establish relationships between variables and contribute to the broader field of algebraic
V. Conclusion
In conclusion, multiples of 3 are not just another mathematical concept but a fundamental building block with numerous applications and intriguing properties. In this blog post, we have explored the
basics, patterns, and advanced concepts related to multiples of 3. By understanding the nature of multiples of 3, we gain valuable insights into various branches of mathematics and their practical
applications. Now, armed with this knowledge, it’s time to further explore and apply the power of multiples of 3 in our mathematical endeavors. | {"url":"https://skillapp.co/blog/exploring-the-fascinating-world-of-multiples-of-3-from-basics-to-advanced-concepts/","timestamp":"2024-11-11T08:18:32Z","content_type":"text/html","content_length":"112112","record_id":"<urn:uuid:e8fa5580-6925-48f8-ab71-a5c79d91f964>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00444.warc.gz"} |
Number Theory Concepts
Podcast Beta
Play an AI-generated podcast conversation about this lesson
What is the definition of a prime number?
A positive integer that is greater than 0 and has multiple divisors
Any positive integer greater than 1 that is divisible by only itself
Any positive integer greater than 1 that is divisible by 1 and itself only (correct)
A positive integer that has factors other than 1 and itself
Which statement correctly defines the Greatest Common Divisor (GCD)?
It is always equal to the smaller of the two numbers.
It is the smallest number that divides both numbers without a remainder.
It is the largest number that divides both numbers exactly. (correct)
It is the sum of the two numbers divided by two.
Which of these is true according to Euclid's Lemma?
If p is prime and p divides ab, then p divides a or b. (correct)
If p is prime, then it cannot divide any product of two integers.
If p is prime and p divides ab, then p divides both a and b.
If p is prime and p divides a, then p divides b.
What characterizes a quadratic equation?
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Which method is employed to solve quadratic equations?
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What does the Fundamental Theorem of Arithmetic state?
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Which of the following represents a linear equation?
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In the context of number theory, what is a composite number?
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What is the purpose of the Euclidean algorithm?
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Which of the following statements about algebraic equations is NOT true?
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Study Notes
Number Theory
• Definition: Number theory is the branch of mathematics that deals with the properties and behavior of integers and other whole numbers.
• Key concepts:
□ Divisibility: a | b if a divides b exactly without leaving a remainder
□ Prime numbers: positive integers greater than 1 that are divisible only by 1 and themselves
□ Composite numbers: positive integers greater than 1 that are not prime
□ Greatest Common Divisor (GCD): the largest number that divides both a and b exactly
□ Euclidean algorithm: a method for finding the GCD of two numbers
• Theorems:
□ Fundamental Theorem of Arithmetic: every positive integer can be expressed as a product of prime numbers in a unique way
□ Euclid's Lemma: if p is prime and p | ab, then p | a or p | b
Algebraic Equations
• Definition: An algebraic equation is an equation that involves variables and constants, and can be expressed using only addition, subtraction, multiplication, and division, and roots (such as
square roots or cube roots).
• Types of equations:
□ Linear equations: equations of the form ax + by = c, where a, b, and c are constants
□ Quadratic equations: equations of the form ax^2 + bx + c = 0, where a, b, and c are constants
□ Polynomial equations: equations of the form a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0 = 0, where a_n, ..., a_1, a_0 are constants
• Methods for solving equations:
□ Factoring: expressing an equation as a product of simpler equations
□ Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a, for solving quadratic equations
□ Synthetic division: a method for dividing a polynomial by another polynomial
• Applications:
□ Solving systems of equations
□ Finding roots of polynomials
□ Modeling real-world problems
Number Theory
• Branch of mathematics focused on integers and their properties
• Divisibility: Indicated as a | b; means a divides b without remainder
• Prime Numbers: Greater than 1, only divisible by 1 and itself; examples include 2, 3, 5, 7
• Composite Numbers: Greater than 1 and not prime; includes numbers like 4, 6, 8
• Greatest Common Divisor (GCD): Largest integer that divides two numbers without leaving a remainder
• Euclidean Algorithm: A systematic method for calculating the GCD of two integers
• Fundamental Theorem of Arithmetic: States every positive integer can be uniquely expressed as a product of prime numbers
• Euclid's Lemma: If a prime number p divides the product of two integers ab, then p must divide at least one of a or b
Algebraic Equations
• Definition: Involves variables and constants, expressible through basic arithmetic operations and roots
• Types:
□ Linear Equations: Form ax + by = c; a, b, and c are constants, represents a straight line on a graph
□ Quadratic Equations: Form ax² + bx + c = 0; represents a parabolic curve, solutions can be found using the quadratic formula
□ Polynomial Equations: General form a_n x^n + a_(n-1) x^(n-1) +...+ a_1 x + a_0 = 0; includes multiple degrees of x
• Methods for Solving Equations:
□ Factoring: Process of breaking down an expression into simpler components that multiply to the original equation
□ Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a; used for finding roots of quadratic equations
□ Synthetic Division: Simplifies the process of dividing a polynomial by a linear factor
• Applications:
□ Solving systems of equations to find variable values that satisfy multiple conditions
□ Finding roots of polynomials, relevant in calculus and function analysis
□ Modeling real-world problems in fields like physics, finance, and engineering through algebraic relationships
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Test your understanding of key concepts in number theory, including divisibility, prime numbers, composite numbers, and greatest common divisors. | {"url":"https://quizgecko.com/learn/number-theory-concepts-ixmbrr","timestamp":"2024-11-09T15:53:59Z","content_type":"text/html","content_length":"315317","record_id":"<urn:uuid:41d9d061-f31b-4d39-b357-588b30a5ea8f>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00323.warc.gz"} |
Problem with Satorra-Bentler Chi-Squared in WLS
The OpenMx users and the OpenMx Development Team have found a problem with the current implementation of the Satorra-Bentler scaled chi-squared difference test which is used for weighted least
squares (WLS) models. The problem is isolated to models that use WLS, and further isolated to the use of mxCompare() with WLS. The statistics reported by summary() are unaffected. The anova() and
mxCompareMatrix() functions are wrappers around mxCompare() and thus show the same problem.
The values of the 'SBchisq' column in mxCompare() do not behave how they theoretically should. The 'SBchisq' values are frequently preposterously large and sometimes negative, the latter of which is
theoretically impossible. We recommend that users ignore all 'SBchisq' values until the problem is resolved. In the source code, we have added a warning to mxCompare(), mxCompareMatrix(), and anova()
that advises users to not use 'SBchisq', and instead use the robust chi-squared value for chi-squared difference testing.
The procedure for chi-squared difference testing is quite simple: pchisq(diffchisq, df=diffdf, lower.tail=FALSE) where 'diffchisq' is the difference in the chi-squared values of models you are
testing, 'diffdf' is the difference in the degrees of freedom of the models you are testing, and the output gives you the *p*-value for the test.
We anticipate resolving this issue in the next few weeks, but in the meantime want to protect users from inaccurate information. | {"url":"https://openmx.ssri.psu.edu/news/problem-satorra-bentler-chi-squared-wls","timestamp":"2024-11-03T02:51:24Z","content_type":"text/html","content_length":"25368","record_id":"<urn:uuid:a167b859-707b-4bbf-9b14-adca03bd63ce>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00462.warc.gz"} |
Amps to Volts Converter - Electrical Calculation Made Easy
In the realm of electronics and electrical engineering, understanding the relationship between amps and volts is crucial. Whether you're a professional electrician, an electronics hobbyist, or just
someone curious about how electricity works, the ability to convert between amps and volts accurately is invaluable. This is where an "Amps to Volts Calculator" becomes an indispensable tool.
Importance of Converting Amps to Volts
Converting amps to volts, and vice versa, is crucial for various reasons. Whether you're designing electrical circuits, troubleshooting issues, or simply trying to understand the power requirements
of a device, accurate conversions between these units are indispensable.
Amps: Amps, short for amperes, represent the unit of electrical current. In simpler terms, it measures the rate of flow of electric charge. Understanding amperes is essential for assessing the amount
of electricity flowing through a circuit or an electrical appliance.
Volts: Volts, denoted by the symbol "V", signify the unit of electrical potential difference or electromotive force. Essentially, volts measure the pressure that drives electric current. It indicates
the strength of the electrical force pushing the current through a circuit.
How does an Amps to Volts Calculator Work?
An Amps to Volts Calculator operates based on the fundamental principles of electricity, specifically Ohm's Law. Ohm's Law states that the current passing through a conductor between two points is
directly proportional to the voltage across the two points and inversely proportional to the resistance between them.
Application of Ohm's Law: Once the current and resistance values are inputted, the calculator applies Ohm's Law equation, which states:
Voltage (V) = Current (I) × Resistance (R)
By multiplying the current (in amperes) with the resistance (in ohms), the calculator computes the voltage (in volts) across the circuit.
After the calculation is performed, the calculator displays the resulting voltage value. This value represents the electromotive force driving the electric current through the circuit.
If the current is 5 amps and the resistance is 10 ohms, the voltage would be 50 volts.
If the current is 8 amps and the resistance is 20 ohms, the voltage would be 160 volts.
Speed and Efficiency: Calculations that might take significant time manually can be done instantaneously.
Accuracy: Online calculators ensure precise results, minimizing the risk of human error.
Convenience: Accessible anytime, anywhere, without the need for complex formulas or equations.
Dependence on Input Accuracy: The accuracy of results depends on the accuracy of the input data.
Complexity: Calculators may not account for every variable in intricate electrical systems.
Not a Substitute for Understanding: They shouldn't replace a fundamental understanding of electrical principles.
Frequently Asked Questions
Amps (amperes) and volts are two fundamental units used in electricity. Amps measure the rate of flow of electric charge, representing current. Volts measure electrical potential difference or
electromotive force, indicating the pressure driving the current.
In simpler terms, amps quantify the amount of electricity flowing through a circuit, while volts measure the force pushing the electricity through the circuit.
Converting between amps and volts is essential for various reasons. Understanding the relationship between these units helps in designing electrical circuits, determining power requirements for
appliances, and troubleshooting electrical issues.
For example, knowing the voltage requirements of a device allows you to ensure that it receives the appropriate amount of electrical power without risking damage.
Yes, amps to volts calculators can be utilized for a wide range of applications, including household appliances.
Whether you're checking the power requirements of a refrigerator, microwave, or television, an amps to volts calculator can assist in determining the voltage needed to operate the appliance safely
and efficiently.
DWhile online amps to volts calculators are generally safe to use, it's essential to ensure that you're using a reputable and reliable source.
Inaccurate calculations could potentially lead to incorrect voltage determinations, which may pose risks, especially in critical applications. Therefore, it's advisable to double-check inputs and
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3.49e10 watts to gigawatts: Unlocking the Unseen Power Shift in Energy Conversion
In an era where energy efficiency and power management are paramount, understanding units of power becomes essential. When we delve into electrical power, we often encounter various units such as
watts, kilowatts, and gigawatts. This article aims to elucidate the conversion of 3.49e10 watts to gigawatts, shedding light on the significance of this conversion in both practical and theoretical
What Are Watts and Gigawatts?
Before we proceed with the conversion, let’s define the units involved. A watt (W) is a basic unit of power in the International System of Units (SI), defined as one joule per second. It quantifies
the rate of energy transfer. For instance, a typical light bulb consumes about 60 watts.
On the other hand, a gigawatt (GW) is a much larger unit, equal to one billion watts (1 GW = 1,000,000,000 W). Gigawatts are commonly used to express the output of large power plants or the total
energy consumption of cities. Given the vast differences in scale, converting 3.49e10 watts to gigawatts is not just a numerical exercise; it provides insights into energy production and consumption
on a grand scale.
The Conversion Formula
To convert watts to gigawatts, one must use the following formula:
Gigawatts=Watts1,000,000,000\text{Gigawatts} = \frac{\Text{Watts}}{1,000,000,000}Gigawatts=1,000,000,000Watts
Using this formula, we can calculate:
Gigawatts=3.49e10 W1,000,000,000=34.9 GW\text{Gigawatts} = \frac{3.49e10 \Text{ W}}{1,000,000,000} = 34.9 \Text{ GW}Gigawatts=1,000,000,0003.49e10 W=34.9 GW
Thus, 3.49e10 watts converts to approximately 34.9 gigawatts. This conversion highlights the figure’s immense power, making it an excellent context for discussions about energy generation and
Practical Implications of Gigawatt Measurements
Understanding the scale of power generation and consumption is crucial for various sectors, including:
1. Energy Production
Large-scale power plants, such as nuclear or hydroelectric plants, often produce power measured in gigawatts. For example, a nuclear power plant can generate between 1 and 3 gigawatts. Therefore,
converting 3.49e10 watts to gigawatts places this power within the context of substantial energy production facilities.
2. Urban Energy Consumption
Measurements in gigawatts provide a clearer picture when assessing a city’s energy consumption. Cities can consume several gigawatts of power at peak times, especially during summer when air
conditioning demands rise. The conversion can help planners and policymakers understand and predict energy needs.
3. Renewable Energy
Understanding how to convert power measurements becomes essential as we transition to renewable energy sources. For instance, solar and wind farms are often measured in megawatts or gigawatts. The
ability to convert 3.49e10 watts to gigawatts allows energy analysts to compare various energy sources effectively.
4. Energy Efficiency
In the quest for energy efficiency, businesses and homeowners often look for ways to reduce their power consumption. Knowing how much power they use in gigawatts can aid in making informed decisions
about energy use and efficiency improvements.
The Role of Technology in Power Measurement
Modern technology plays a crucial role in accurately measuring and converting power units. Smart meters, for example, allow consumers to monitor their energy consumption in real time, providing
insights that can help reduce energy waste.
Energy Monitoring Systems
These systems aggregate data on energy usage, converting it into understandable metrics. Using these systems to convert 3.49e10 watts to gigawatts helps users assess their consumption and compare it
with the grid or their energy-saving goals.
Innovations in Energy Storage
As energy storage technologies evolve, understanding these conversions aids in evaluating the capacity of storage solutions. For instance, an extensive battery storage system might have a capacity of
several gigawatts, enabling it to store energy generated from renewable sources for later use.
Environmental Considerations
As we convert and analyze figures like 3.49e10 watts to gigawatts, we must also consider the sources of that energy. Renewable energy sources, such as solar and wind, provide cleaner alternatives to
fossil fuels, and understanding these conversions can help promote cleaner energy policies.
Future Trends in Power Consumption
The future of energy is evolving rapidly, with technology playing a pivotal role. As more devices become interconnected through the Internet of Things (IoT), the power demand will increase,
necessitating a solid grasp of managing and converting power measurements effectively.
Smart Grids
The development of intelligent grids allows for more efficient distribution and consumption of electricity. By understanding how to convert 3.49e10 watts to gigawatts, utilities can better manage the
load and ensure a reliable power supply.
Energy as a Service
The “Energy as a Service” concept is gaining traction, allowing consumers to pay for energy usage without owning the infrastructure. This model requires accurate power measurement and conversion,
making knowledge of figures like 3.49e10 watts to gigawatts increasingly vital.
In summary, converting 3.49e10 watts to gigawatts is more than a straightforward mathematical exercise; it encapsulates significant power that can influence various sectors, from energy production to
urban planning and environmental policy. Understanding these conversions will become even more essential for individuals and organizations as the world moves toward greater energy efficiency and
renewable sources.
1. What is the conversion of 3.49e10 watts to gigawatts?
The conversion of 3.49e10 watts to gigawatts is approximately 34.9 gigawatts.
2. How can I convert 3.49e10 watts to gigawatts myself?
To convert 3.49e10 watts to gigawatts, divide 3.49e10 by 1,000,000,000, resulting in about 34.9 gigawatts.
3. Why is the conversion from 3.49e10 watts to gigawatts necessary?
Understanding the conversion from 3.49e10 watts to gigawatts is essential for comprehending the scale of energy generation, especially in large power plants.
4. Can I express other watt values similar to 3.49e10 watts to gigawatts?
Yes, any watt value can be converted to gigawatts using the same method as 3.49e10 watts to gigawatts by dividing the watt value by 1 billion.
5. Where can I find tools to convert 3.49e10 watts to gigawatts easily?
Online conversion tools and calculators can quickly convert 3.49e10 watts to gigawatts and other watt values.
6. What are some practical applications of knowing the conversion of 3.49e10 watts to gigawatts?
Knowing the conversion of 3.49e10 watts to gigawatts can help in energy planning, assessing the output of renewable energy sources, and understanding large-scale energy consumption.
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Gimbal distance and Speed Range Estimates using Lines of Bearing and/or DCS
Venus can be seen in the daytime for sure if you know exactly where to look. You don't need a telescope.
I took this photo of Venus at 11:43am on January 1, 2019 with a Canon D77and my zoom lens at 300mm (iso 100, f7.1, 1/500sec)
the brightness of Venus is higher than the crescent moon. I saw the two earlier morning when the sky was dark and that helped me find it later in the day and take this shot.
View attachment 46194
Venus can be seen in the daytime for sure if you know exactly where to look. You don't need a telescope.
I took this photo of Venus at 11:43am on January 1, 2019 with a Canon D77and my zoom lens at 300mm (iso 100, f7.1, 1/500sec)
the brightness of Venus is higher than the crescent moon. I saw the two earlier morning when the sky was dark and that helped me find it later in the day and take this shot.
When you took that (really nice) photo, Venus was at the edge of its orbit, when it is the brightest, due to its phase and distance to the Earth (0.64 AU, being 1 AU the distance Earth - Sun)
For the gimbal video, AFAIK, it was some day in January 2015. By that time, Venus was at the far side of the orbit, "behind" the sun. Even if its phase was full, it was a 1.5 AU of distance. Also, as
the gimbal object is near the horizon, it means it had to be either before 9 am in the morning (with the sun already up in the sky, making it more difficult to see venus), or near 7 pm, when the sun
already set, it's darker and it may be easier to spot venus in IR (exact hours depends on the time zone and place, but anyway, it's just a quick look.)
Anyway, the apparent size of Venus is so small that even if it could be seen in MWIR (which I doubt), in a 640 pixel FPA with 0.7º FoV it would be as big as ...2 pixels. (NAR 2X FoV is a digital
zoom, it doesn chage the number of pixels actually illuminated by the object).
Anyway, the apparent size of Venus is so small that even if it could be seen in MWIR (which I doubt), in a 640 pixel FPA with 0.7º FoV it would be as big as ...2 pixels. (NAR 2X FoV is a digital
zoom, it doesn chage the number of pixels actually illuminated by the object).
Venus is in a similar position in its orbit now as it was in Jan '15.
This is a cropped photo I took with a full spectrum camera and R720 IR filter a couple evenings ago.
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Venus is in a similar position in its orbit now as it was in Jan '15.
This is a cropped photo I took with a full spectrum camera and R720 IR filter a couple evenings ago.
Nice pic. But that's in the near IR, limited to what a Silicon device (CCD) is sensible to (max. 1.1 um wavelength). Also, that's the reflection of the sun in Venus, which is still a lot of energy.
Compared to the Mid-IR, its about 100 times more energy than you have in MWIR (3.7-5.0 um wavelength), where an InSb FPA is sensible.
(blue: Visible+near IR. Red: MWIR)
Nice pic. But that's in the near IR, limited to what a Silicon device (CCD) is sensible to (max. 1.1 um wavelength). Also, that's the reflection of the sun in Venus, which is still a lot of
energy. Compared to the Mid-IR, its about 100 times more energy than you have in MWIR (3.7-5.0 um wavelength), where an InSb FPA is sensible.
The IR filter knocks down the brightness and improves contrast, which according to the article I linked in a previous post, helps to see Venus during the day.
The other trick is focus. One time I saw Venus before sunset naked eye, and what really helped was it was near a crescent moon, like in Amber Robot's photo above. I could only see Venus after my eyes
adjusted to seeing the distant moon. Same goes for the camera. without something to focus on, Venus disappears against the bright sky. My photo above was taken after sunset when Venus was clearly
No idea what can be seen with an InSb FPA chip. The FLIR pod has a much larger aperture (12" ?) than my camera lens, so collects far more light.
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Searched the internet for mid/far infrared planet images and found but one:
Infrared image of Jupiter from SOFIA's First Light flight composed of individual images at wavelengths of 5.4 (blue), 24 (green) and 37 microns (red) made by Cornell University's FORCAST camera.
Ground-based infrared observations are impossible at 5.4 and 37 microns and normally very difficult at 24 microns even from high mountain-top observatories such as Mauna Kea due to absorption by
water and other molecules in Earth's atmosphere.
I think this puts to bed any chance a FLIR camera can capture celestial objects or man made satellites outside earths atmosphere, especially an object on the horizon like in the GIMBAL video.
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The 3-5 micron range is a transmission window in atmosphere for IR, with very low absorbance. That's one of the reasons(*) why it is used by ATFLIR.
But being able to image a planet I guess has more to do with very low radiation from the source, using large apertures and long exposures, and struggling with scattered light from other sources.
Maybe with a dedicated equipment you can get it, but it's not ATFLIR purpouse and I would't expect it to do it.
(*)Another one is that the range includes the emission band of CO2, one of the by-products of combustion of a jet motor.
I'm a little confused reading through this thread. Was the airspeed of the Gimbal object ever estimated/calculated?
It seems the airspeed of the jet was calculated from what I can see. But I'm more interested in what speed the Gimbal object is doing
I'm a little confused reading through this thread. Was the airspeed of the Gimbal object ever estimated/calculated?
It seems the airspeed of the jet was calculated from what I can see. But I'm more interested in what speed the Gimbal object is doing
As are we but that can't be worked out without knowing the distance to it which this thread tried to work out from lines of bearing using realistic turning rates of the f/18. But the conclusions are
that minor changes in the angle rate estimate change the calculations vastly the angle changes shown on the FLIR are not actually accurate enough.
Staff member
As are we but that can't be worked out without knowing the distance to it which this thread tried to work out from lines of bearing using realistic turning rates of the f/18. But the conclusions
are that minor changes in the angle rate estimate change the calculations vastly the angle changes shown on the FLIR are not actually accurate enough.
Yes, with the more detailed calculations it was shown that the lines of sight could vary from converging on a relatively near region, to being essentially parallel (meaning it could be 100s of miles
Yes, with the more detailed calculations it was shown that the lines of sight could vary from converging on a relatively near region, to being essentially parallel (meaning it could be 100s of
miles away).
So the claim that the Gimbal object is stationary in the air, could be true ?
So the claim that the Gimbal object is stationary in the air, could be true ?
Or more probably, that it was moving along the F-18 line of sight, I guess.
I'm a little confused reading through this thread. Was the airspeed of the Gimbal object ever estimated/calculated?
It seems the airspeed of the jet was calculated from what I can see. But I'm more interested in what speed the Gimbal object is doing
That's something that we have overlooked in my opinion. I have some new results I'd like to share. Here is my line of reasoning.
1) The movement of Gimbal, relative to the background (clouds), that we see, is
due to Gimbal motion. Parallax is secondary here because the fighter is behind Gimbal, and Gimbal goes away from the fighter. This is a very different situation than GoFast, for which the parallax
effect is very important as GoFast is on the side, coming towards the fighter.
2) Because the motion we see is primarily from Gimbal, we can, very roughly, but meaningfully I think, estimate its speed from the time it takes to cross the field of view (FOV). We know it's 0.7
deg. Looking at the cloud features, a rough estimate is that it takes 2 seconds for a cloud "peak" to cross the FOV. Therefore, let's say Gimbal crosses the FOV in 2 seconds. It's not crossing it
perpendicularly, but sideway, because it's seen from the back/right. From previous reconstructions, geometrical and also from flight simulations, that angle is at minimum 45deg. Even if it's less,
this is only a factor 1.4 (sqrt 2) in the following calculations.
3) In GeoGebra, it's easy to create a cone with a radius that corresponds to the FOV, in function of the distance. For a given distance, the diameter of the cone, divided by the time Gimbal takes to
cross it (~2sec), multipled by sqrt(2) (to account for the angle of crossing) gives us a rough speed estimate.
I've made the model here :
The position of Gimbal can be moved, the corresponding distance to the fighter is indicated in Nautical Miles (NM).
The corresponding minimum speed (i.e. perpendicular trajectory) for Gimbal is given.
You'll see that at greater distances, the speed quickly become unrealistic for a plane. At 90 NM (~100 miles), the "distant plane" would have to go at ~5000 km/h.
"Sane" speeds (500-1000 km/h) are only found between 5 and 15Nm, which is consistent with previous estimates for the distance. A larger FOV makes for even greater speeds. I think the angle of
crossing of 45deg is an underestimation, is is probably larger than that (see for example the DCS simulation of
in the previous page). So if something the speed is likely underestimated.
Those are rough estimates, but even considering uncertainties along the way, the number I get at large distances are way beyond what is possible for a plane. I see people discussing a rocket in the
other thread, would that be a plausible candidate? What is the speed of a rocket ?
Again, if it's a plane, how come its features cannot be seen for such a relatively close distance from the fighter ?
I'll be happy to correct some mistakes I may have made, I'm simply trying to fuel the discussion in a new direction and see if we can learn from it.
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That's something that we have overlooked in my opinion. I have some new results I'd like to share. Here is my line of reasoning.
1) The movement of Gimbal, relative to the background (clouds), that we see, is
due to Gimbal motion. Parallax is secondary here because the fighter is behind Gimbal, and Gimbal goes away from the fighter. This is a very different situation than GoFast, for which the
parallax effect is very important as GoFast is on the side, coming towards the fighter.
2) Because the motion we see is primarily from Gimbal, we can, very roughly, but meaningfully I think, estimate its speed from the time it takes to cross the field of view (FOV). We know it's 0.7
deg. Looking at the cloud features, a rough estimate is that it takes 2 seconds for a cloud "peak" to cross the FOV. Therefore, let's say Gimbal crosses the FOV in 2 seconds. It's not crossing it
perpendicularly, but sideway, because it's seen from the back/right. From previous reconstructions, geometrical and also from flight simulations, that angle is at minimum 45deg. Even if it's
less, this is a factor 1.7 (sqrt 2) in the following calculations.
3) In GeoGebra, it's easy to create a cone with a radius that corresponds to the FOV, in function of the distance. For a given distance, the diameter of the cone, divided by the time Gimbal takes
to cross it (~2sec), multipled by sqrt(2) (to account for the angle of crossing) gives us a rough speed estimate.
I've made the model here :
The position of Gimbal can be moved, the corresponding distance to the fighter is indicated in Nautical Miles (NM).
The corresponding minimum speed (i.e. perpendicular trajectory) for Gimbal is given.
You'll see that at greater distances, the speed quickly become unrealistic for a plane. At 90 NM (~100 miles), the "distant plane" would have to go at ~5000 km/h.
"Sane" speeds (500-1000 km/h) are only found between 5 and 15Nm, which is consistent with previous estimates for the distance. A larger FOV makes for even greater speeds. I think the angle of
crossing of 45deg is an underestimation, is is probably larger than that (see for example the DCS simulation of
in the previous page).
Those are rough estimates, but even considering uncertainties along the way, the number I get at large distances are way beyond what is possible for a plane. I see people discussing a rocket in
the other thread, would that be a plausible candidate? What is the speed of a rocket ?
Again, if it's a plane, how come its features cannot be seen for such a relatively close distance from the fighter ?
I'll be happy to correct some mistakes I may have made, I'm simply trying to fuel the discussion in a new direction and see if we can learn from it.
There have been some developments lately about the geometry of the Gimbal video, you could check
Could The Gimbal Video Show an Atlas V Launch?
, starting from post #106 if you're in a hurry.
That's something that we have overlooked in my opinion. I have some new results I'd like to share. Here is my line of reasoning.
1) The movement of Gimbal, relative to the background (clouds), that we see, is
due to Gimbal motion. Parallax is secondary here because the fighter is behind Gimbal, and Gimbal goes away from the fighter. This is a very different situation than GoFast, for which the
parallax effect is very important as GoFast is on the side, coming towards the fighter.
2) Because the motion we see is primarily from Gimbal, we can, very roughly, but meaningfully I think, estimate its speed from the time it takes to cross the field of view (FOV). We know it's 0.7
deg. Looking at the cloud features, a rough estimate is that it takes 2 seconds for a cloud "peak" to cross the FOV. Therefore, let's say Gimbal crosses the FOV in 2 seconds. It's not crossing it
perpendicularly, but sideway, because it's seen from the back/right. From previous reconstructions, geometrical and also from flight simulations, that angle is at minimum 45deg. Even if it's
less, this is only a factor 1.7 (sqrt 2) in the following calculations.
3) In GeoGebra, it's easy to create a cone with a radius that corresponds to the FOV, in function of the distance. For a given distance, the diameter of the cone, divided by the time Gimbal takes
to cross it (~2sec), multipled by sqrt(2) (to account for the angle of crossing) gives us a rough speed estimate.
I've made the model here :
The position of Gimbal can be moved, the corresponding distance to the fighter is indicated in Nautical Miles (NM).
The corresponding minimum speed (i.e. perpendicular trajectory) for Gimbal is given.
You'll see that at greater distances, the speed quickly become unrealistic for a plane. At 90 NM (~100 miles), the "distant plane" would have to go at ~5000 km/h.
"Sane" speeds (500-1000 km/h) are only found between 5 and 15Nm, which is consistent with previous estimates for the distance. A larger FOV makes for even greater speeds. I think the angle of
crossing of 45deg is an underestimation, is is probably larger than that (see for example the DCS simulation of
in the previous page). So if something the speed is likely underestimated.
Those are rough estimates, but even considering uncertainties along the way, the number I get at large distances are way beyond what is possible for a plane. I see people discussing a rocket in
the other thread, would that be a plausible candidate? What is the speed of a rocket ?
Again, if it's a plane, how come its features cannot be seen for such a relatively close distance from the fighter ?
I'll be happy to correct some mistakes I may have made, I'm simply trying to fuel the discussion in a new direction and see if we can learn from it.
We already have a thread for this, the theory is that the features of a plane cannot be seen because the IR glare of the heat source is larger than the physical size of the object and obscures it or
it's so far away that all we can see is the glare from the heat source.
: That's exactly the point of my post, it cannot be very far because its speed would be unphysical for a plane.
To me there is no question anymore, it's either a relatively close object, or a supersonic object very far, although I have a hard time following the arguments for the rocket from the other
: That's exactly the point of my post, it cannot be very far because its speed would be unphysical for a plane.
To me there is no question anymore, it's either a relatively close object, or a supersonic object very far, although I have a hard time following the arguments for the rocket from the other
Hint: it's a supersonic object very far (call it rocket if you wish).
Note that I used a FOV of 0.7, and because the ATFLIR is in Mode 2, it may be 0.35 deg (is it for sure?). Regardless of this, the speed at long distances are too fast for a plane.
Note that I used a FOV of 0.7, and because the ATFLIR is in Mode 2, it may be 0.35 deg (is it for sure?). Regardless of this, the speed at long distances are too fast for a plane.
I assume these steps:
NAR zoom1=0,7°
NAR zoom2=0,35°
Do you think Is It correct my consideration?
WFOV = 6.0
MFOV = 2.8 MFOV zoom = 1.4
NAR = 0.7 NAR zoom = 0.35
There's no zoom2
WFOV = 6.0
MFOV = 2.8 MFOV zoom = 1.4
NAR = 0.7 NAR zoom = 0.35
There's no zoom2
But gimbal video Is NAR zoom 2.0. Is It correct?
But gimbal video Is NAR zoom 2.0. Is It correct?
Yes, at least the Gimbal video shows Z 2.0 in the display, it should be the (electronic) zoom level.
the 2.0x is zoomed once if that makes sense
There isn't NAR then NAR Zoom level 1 and then NAR Zoom level 2
There is NAR and then NAR 2x
the 2.0x is zoomed once if that makes sense
There isn't NAR then NAR Zoom level 1 and then NAR Zoom level 2
There is NAR and then NAR 2x
I think we can all agree on this, there is NAR and then NAR 2x.
I think we can all agree on this, there is NAR and then NAR 2x.
I was correcting this which says there is 3 NAR states, which then was later said to be what we agreed on, it isn't.
I assume these steps:
NAR zoom1=0,7°
NAR zoom2=0,35°
Do you think Is It correct my consideration?
Reply Report
WFOV = 6.0
MFOV = 2.8 MFOV zoom = 1.4
NAR = 0.7 NAR zoom = 0.35
There's no zoom2
Thank you.
NAR=>NAR Z1.0
The FOV in Gimbal is 0.35°.
The movement of Gimbal, relative to the background (clouds), that we see, is mostly due to Gimbal motion.
Have you considered that the clouds may be moving?
Have you considered that the clouds may be moving?
I have not. If the clouds have a speed due to difference in wind relative to the altitude of Gimbal and the fighter, I think this is secondary compared to the speed of Gimbal, and/or apparent speed
due to the F18 speed (parallax effect, important at close distance, less at far distance).
While I'm at it, I want to copy what I posted in another thread because it belong more here.
This is relevant to what
proposed in the previous page, as a potential steady trajectory with parallel line of sights. I think it's not possible for the following reasons :
If we look at the video, the clouds move from left to right, i.e. they are scanned by the camera from right to left (or in another words we see more and more of the clouds to the left of the FOV).
Let's put this information into the equation, and consider parallel line of sights with a simple schematic. That gives us something like this :
Parallel (or at least non-crossing) line of sights
See the problem ? If we are in this configuration, the camera should scan the clouds from left to right, i.e. we should see more and more clouds to the right of the FOV.
This doesn't work.
Now, if the lines of sighting cross, it's possible to reconcile what we see with the clouds.
Crossing line of sights
In that configuration we see more and more clouds to the left of the FOV. This is the configuration we see in the video. With crossing lines of sight, it becomes very difficult to build a steady
trajectory for a distant plane that would not deviate too much from the lines of sight we have.
I think if
had background clouds in his simulation, we would see the clouds moving in the opposite direction to the video.
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How valid is the assumption that those are background clouds and not foreground clouds?
How valid is the assumption that those are background clouds and not foreground clouds?
This is a really good question. In my opinion one of the most probable clues that the clouds are in front of the object is that they are rather detailed, we can distinguish the various cumulus clouds
and above all the various transversal bands typical of stratiform formations. If they had been further away it would have been less distinct.
If the lines of sight were parallel, clouds in the forefront would still move from left to right, wouldn't they? It doesn't change my point that the lines of sight have to cross.
1s to 10s is 1.4737°
10s to 20s is 1.1454°
20s to 30s is 0.6653°
Based on a 0.35° Field of View
We were on an interesting path before being distracted by the object possibly being Venus. I suggest we keep going based on what we've learnt so far.
The LoS angles that CassiO retrieved using frames (post #179 and before) are very interesting because they can help refine the geometrical reconstruction based on the ATFLIR angles and F-18
trajectory. I've included them in my model, to refine the rate of turns so they match the "frame analysis" angles. This gives something like this :
Unless somebody comes up with good arguments against it, I consider this is a good estimate of the situation. This matches the retrieval of the lines of sight using two different methods, and it is
consistent with the movement of the background clouds (opening to the left).
There may be an influence of the movement of the clouds because they are not at the same altitude of the F-18, and there is probably some wind shear so they are not in the same frame of reference.
But even considering a very high wind shear (120 Knots), the clouds would only cover the distance marked by the reference blue segment in the figure above. I consider this of secondary importance
compared to the speed of the F-18, and the speed of a potential plane (plus the wind shear is very likely less than that).
If Gimbal was only moving against the wind (120 Knots to the west) and thus being stationary in the F-18 frame of reference, it should be in the 20-30 Nm range (23Nm in the position marked above).
This very well aligns with
@Edward Current
estimate that the object gets 9% closer over the course of the video (see post #190), because the distance at mark 0'01 is 23.76 Nm, at mark 0'31 it is 21.2 Nm. That's ~ 10% closer.
This is of course inconsistent with a plane. For a plane to be the answer here, it has to go from one line of sight to the other behind the interception point, and cover a reasonable distance given
the speed of a plane at this altitude (see reference segment for a plane flying at 500 Knots). This is not impossible, but the plane has to turn because it has to cover a same distance between each
line of sight (for it to be a steady trajectory). That is what makes me doubt about the distant plane hypothesis, as we should see some kind of change in the IR glare if the engine's angle of sight
changes. And it would be a coincidence that the plane turns right when the F-18 tracks it, far away from any airport. Finaly, a far away plane does not match the 9% growth in the size of the object,
aka post #190 :
To me this starts to be too much evidence against the distant, or not so distant in fact, plane.
I've been posting a lot lately as I had time to go back to this, I will stop now and keep others share their thoughts. Thanks for reading.
EDIT :
just to add that if Gimbal is between 20-30Nm from the F-18, it's 11-17m long (based on the 0.35° FOV)
EDIT 2 :
the latest version of the model I use
Last edited:
Unless somebody comes up with good arguments against it,
The clouds are still moving while the UAP is bearing 0⁰, they slow down but don't quite stop until the end when the UAP is bearing right. And the F-18 is still turning left at the time.
With a stationary UAP, the clouds should stop shifting at 0⁰ and then reverse direction.
Please explain how the end of the video works.
Source: https://m.youtube.com/watch?v=QKHg-vnTFsM
Excellent insight! While the LOS remains fixed, the object and the clouds being aligned, the azimuth continues to increase to the right.
Do you have your own hypothesis?
The clouds are still moving while the UAP is bearing 0⁰, they slow down but don't quite stop until the end when the UAP is bearing right. And the F-18 is still turning left at the time.
With a stationary UAP, the clouds should stop shifting at 0⁰ and then reverse direction.
Yes I noticed that too, I have two hypotheses to explain it :
- the object is not stationary, it's moving but then stops at the end of the video. I think when Ryan Graves mentions the object being stationary, he may mean that the object was flying at a low
speed impossible to achieve for a regular aircraft without stalling. Or maybe he only talks about the very end of the video. So the object could be a bit before or after the point of intersection,
and have a relatively slow motion during most of the video that contributes to its speed relative to the clouds (i.e. the motion of Gimbal is not only an apparent motion due to parallax). The
parallax effect must stop at bearing 0⁰, but it doesn't, because Gimbal is moving. But then when Gimbal stops, the angle between the F-18 and Gimbal lines of bearing is too low to induce any
significant parallax.
- Gimbal is stationary, or close, but this is an effect of clouds motion at that point, when their displacement is not secondary anymore compared to the apparent speed due to parallax.
Or maybe a bit of both.
I'll be happy to hear other's takes on this.
Graves, from 20:25 in the video below:
This object [gimbal] was proceeding behind the larger group and essentially it approached and then it stopped for a bit and then as the formation turned back 180 degrees it just stopped from that
position and then immediately went in the other direction more or less.
My take... The plane is still banked left, but it's not turning anymore. That is to say, the turn was coordinated until the target was more or less front of the plane. | {"url":"https://www.metabunk.org/threads/gimbal-distance-and-speed-range-estimates-using-lines-of-bearing-and-or-dcs.11836/page-7","timestamp":"2024-11-06T08:53:01Z","content_type":"text/html","content_length":"358565","record_id":"<urn:uuid:1d8a4dcf-b963-4d09-8e95-62e73b39eac1>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00339.warc.gz"} |
Compatibility of temporal spectra with Kolmogorov (1941): the Taylor hypothesis.
Earlier this year I received an enquiry from Alex Liberzon, who was puzzled by the fact that some people plot temporal frequency spectra with a
Consider a turbulent velocity field
If we now consider the turbulence to be convected by a uniform velocity
The dimensional consistency of the two forms is obvious from inspection.
Next let us examine the dimensions of the temporal and spatial spectra. We will use the angular frequency
Evidently the dimensions are given by:
or velocity squared.
Then we introduce Taylor’s hypothesis in the form:
and hence:
The Kolmogorov wavenumber spectrum (in the one-dimensional form that is usually measured) is given by:
We should note that
which is easily shown to have the correct dimensions of velocity squared.
After seeing this analysis, Alex came back with: but what about when the field is homogeneous and isotropic, with
I intend to return to this, but not necessarily next week!
[1] W. D. McComb. The Physics of Fluid Turbulence. Oxford University Press, 1990.
[2] A. N. Kolmogorov. The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. C. R. Acad. Sci. URSS, 30:301, 1941.
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://blogs.ed.ac.uk/physics-of-turbulence/2021/02/25/compatibility-of-temporal-spectra-with-kolmogorov-1941-the-taylor-hypothesis/","timestamp":"2024-11-07T07:39:37Z","content_type":"text/html","content_length":"71566","record_id":"<urn:uuid:ab534817-7588-4977-8b9a-10c81c2e0818>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00777.warc.gz"} |
Interval orders, semiorders and ordered groups
We prove that the order of an ordered group is an interval order if and only if it is a semiorder. Next, we prove that every semiorder is isomorphic to a collection $\mathcal J$ of intervals of some
totally ordered abelian group, these intervals being of the form $[x, x+ \alpha[$ for some positive $\alpha$. We describe ordered groups such that the ordering is a semiorder and we introduce
threshold groups generalizing totally ordered groups. We show that the free group on finitely many generators and the Thompson group $\mathbb F$ can be equipped with a compatible semiorder which is
not a weak order. On another hand, a group introduced by Clifford cannot.
arXiv e-prints
Pub Date:
June 2017
□ Mathematics - Combinatorics;
□ 06A05;
□ 06A06;
□ 06F15;
□ 06F20
32 pages, 2 figures | {"url":"https://ui.adsabs.harvard.edu/abs/2017arXiv170603276P/abstract","timestamp":"2024-11-10T19:46:35Z","content_type":"text/html","content_length":"36398","record_id":"<urn:uuid:6ac7e059-6429-4310-b3eb-4779037bc398>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00315.warc.gz"} |
Conditional Distribution - (Engineering Applications of Statistics) - Vocab, Definition, Explanations | Fiveable
Conditional Distribution
from class:
Engineering Applications of Statistics
Conditional distribution refers to the probability distribution of a random variable given that another random variable takes a specific value. This concept is crucial for understanding relationships
between variables, especially when dealing with joint probability distributions, as it helps to analyze how the outcome of one variable influences the outcome of another. It also allows for the
examination of dependencies and can reveal insights into how probabilities change under certain conditions.
congrats on reading the definition of Conditional Distribution. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Conditional distributions are derived from joint distributions by fixing one variable and observing how the other behaves.
2. The notation for conditional probability is usually written as P(X|Y), representing the probability of X given Y.
3. The sum or integral of a conditional distribution over all possible values of the conditioning variable equals 1, indicating that it is a valid probability distribution.
4. Conditional distributions help in understanding how two variables are related, and can indicate whether they are dependent or independent.
5. They are essential in Bayesian statistics, where updating beliefs about one variable based on information from another is often necessary.
Review Questions
• How does conditional distribution help in understanding the relationship between two random variables?
□ Conditional distribution allows us to analyze how one random variable behaves when another is held at a specific value. By focusing on P(X|Y), we can see how changes in Y affect the
probabilities associated with X. This understanding helps determine if there's a dependency between the two variables, which can be critical for data analysis and interpretation.
• In what way does conditional distribution differ from marginal distribution, and why is this distinction important?
□ While marginal distribution provides probabilities of individual random variables without considering others, conditional distribution examines probabilities within the context of specific
values of another variable. This distinction is important because it reveals how one variable influences another, providing deeper insights into relationships and dependencies that marginal
distributions alone cannot convey.
• Evaluate how conditional distributions can be applied in real-world situations such as risk assessment or decision-making.
□ In risk assessment, conditional distributions allow analysts to evaluate potential outcomes based on certain risk factors being present. For instance, if assessing health risks, understanding
P(Disease|Age) enables better predictions about disease prevalence in specific age groups. In decision-making, organizations use conditional distributions to inform choices based on various
scenarios, improving strategic planning and resource allocation by considering the impact of different influencing factors.
© 2024 Fiveable Inc. All rights reserved.
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Subsituting Variables For Numbers Worksheet
Subsituting Variables For Numbers Worksheet function as foundational tools in the world of maths, supplying a structured yet versatile system for students to discover and understand mathematical
principles. These worksheets offer an organized method to comprehending numbers, supporting a strong foundation upon which mathematical proficiency flourishes. From the simplest counting workouts to
the details of sophisticated estimations, Subsituting Variables For Numbers Worksheet satisfy learners of varied ages and ability levels.
Unveiling the Essence of Subsituting Variables For Numbers Worksheet
Subsituting Variables For Numbers Worksheet
Subsituting Variables For Numbers Worksheet -
Simplifying Exponents of Numbers Worksheet Simplifying Exponents of Variables Lessons Simplifying Exponents of Variables Worksheet Simplifying Expressions and Equations Simplifying Fractions With
Negative Exponents Lesson Negative Exponents in Fractions Worksheet Simplifying Multiple Positive or Negative Signs Lessons
Substitution into Algebraic Expressions Worksheet Directions Please answer the following in your binders scribblers Do not do the work on the sheet there is not enough room If a 2 b 5 and c 7
evaluate the following by substituting these values into the following 3b f 2b 3 k a b p 3a 2b b 6a g 3a 1
At their core, Subsituting Variables For Numbers Worksheet are vehicles for theoretical understanding. They envelop a myriad of mathematical concepts, directing students through the maze of numbers
with a series of engaging and purposeful workouts. These worksheets transcend the boundaries of standard rote learning, motivating energetic engagement and fostering an intuitive understanding of
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Nurturing Number Sense and Reasoning
Solving Systems Of Equations By Substitution Worksheets Math Monks
Solving Systems Of Equations By Substitution Worksheets Math Monks
Using the method of substitution in algebra a variable such as x or y is replaced with its value The expression can then be simplified even further In this problem we replace the variables b and c
since their values are given Everywhere in the problem where the variable b is present it is substituted
s i2 h0Z1Q2F QKDu0tuab WSGoDf1t6w ba UrGeo uL iL zC h i J TA2lDlh Crvi gQhjt 4sX vr zeis jedr0vfeqdE x 3 MMa7dLeP dwRict Nhu lI xnhf5i6n0i PtPeD cPZrNeD KA Vlmg5eXbJrDaT P Worksheet by Kuta Software
LLC 13 zy 4y use y 5 and z 2 30 14 b a b a use a 9 and b 4 61 15 p2 4 m use m 3 and p 4 1 16 x y
The heart of Subsituting Variables For Numbers Worksheet lies in growing number sense-- a deep understanding of numbers' meanings and interconnections. They urge exploration, inviting students to
explore math procedures, understand patterns, and unlock the enigmas of series. With provocative difficulties and sensible puzzles, these worksheets become portals to sharpening reasoning abilities,
nurturing the logical minds of budding mathematicians.
From Theory to Real-World Application
Substituting Variables Worksheet
Substituting Variables Worksheet
Three Substituting Numbers for Variables Worksheets w Answer Keys by Worksheet Central 4 5 2 1 60 PDF Activity You will receive one 12 question worksheet in which students will be required to solve
Substituting Numbers for Variables problems THIS IS A GOOGLE FORM VERSION In addition the answers have been put in and each
Click here for Answers substituting into expressions Practice Questions Previous Algebraic Notation Practice Questions Next Algebraic Fractions Practice Questions The Corbettmaths Practice Questions
and Answers to Substitution
Subsituting Variables For Numbers Worksheet work as avenues linking theoretical abstractions with the apparent truths of day-to-day life. By instilling functional scenarios into mathematical
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Versatility is inherent in Subsituting Variables For Numbers Worksheet, using a collection of instructional devices to deal with diverse knowing styles. Visual help such as number lines,
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In a significantly diverse globe, Subsituting Variables For Numbers Worksheet welcome inclusivity. They transcend social limits, integrating instances and issues that reverberate with learners from
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Crafting a Path to Mathematical Mastery
Subsituting Variables For Numbers Worksheet chart a course towards mathematical fluency. They instill willpower, crucial reasoning, and analytical skills, vital features not only in maths yet in
different elements of life. These worksheets equip students to navigate the complex terrain of numbers, nurturing an extensive gratitude for the sophistication and logic inherent in maths.
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option, at a time.
Substiution Worksheets With Answers Cazoom Maths Worksheets Math Expressions Algebra
KS3 KS4 Substitution Maths Worksheet Substitution Maths GCSE
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Substituting Values Of Variables Concept CW Worksheet
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8 1 Substituting Values Into Algebraic Expressions
Substitution into Algebraic Expressions Worksheet Directions Please answer the following in your binders scribblers Do not do the work on the sheet there is not enough room If a 2 b 5 and c 7
evaluate the following by substituting these values into the following 3b f 2b 3 k a b p 3a 2b b 6a g 3a 1
Free Worksheets For Evaluating Expressions With Variables
With this worksheet generator you can make printable worksheets for evaluating simple variable expressions when the value of the variable s is given There are three levels the first level only
including one operation For example the student might find the value of the expression 2 t 5 when t has the value 6
Substitution into Algebraic Expressions Worksheet Directions Please answer the following in your binders scribblers Do not do the work on the sheet there is not enough room If a 2 b 5 and c 7
evaluate the following by substituting these values into the following 3b f 2b 3 k a b p 3a 2b b 6a g 3a 1
With this worksheet generator you can make printable worksheets for evaluating simple variable expressions when the value of the variable s is given There are three levels the first level only
including one operation For example the student might find the value of the expression 2 t 5 when t has the value 6
Substitution Of Variables In Equations Worksheet Learning With Mrs Du Preez
Substitution Method Worksheet Answer Key
Algebra Substitution Worksheets Grade 7
Year 7 Algebra Substitution Worksheet Thekidsworksheet
Unknown Variables In Equations Subtraction Range 1 To 20 Any Position A
Introduction To Substitution Animated PowerPoint Worksheet Functional Skills L2 GCSE
Introduction To Substitution Animated PowerPoint Worksheet Functional Skills L2 GCSE
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Colour in the Square
Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?
Can you put the $25$ coloured tiles into the $5\times 5$ square below so that no column, no row and no diagonal line have the same colour in them?
Use the interactivity below to try out your ideas.
NRICH Roadshow
Getting Started
Try starting with just one colour, then fill in a colour at a time.
Student Solutions
Liam and Joanne from Moorfield Junior School sent us this solution:
We worked out that every new line we started had to have each colour two spaces away from the same colour on the line above.
Do you agree with them? Did you find any of the other ways to solve this problem?
Teachers' Resources
Why do this problem?
This problem
is one that requires working systematically. It is a good activity for promoting discussion between learners working together and also for giving encouragement to those whose spatial ability is
better than their numerical achievements.
Key questions
Which row and which column have none of that colour in them?
Have you checked the diagonals as well as the rows and columns?
Possible extension
Learners could try other-sized squares such as $4\times 4$ and $6\times 6$. With some squares it is possible to place one colour correctly but no more. Of which sized squares is this true?
Possible support
You could suggest starting with just one colour, then fitting in the other colours, one at a time. | {"url":"http://nrich.maths.org/problems/colour-square-0","timestamp":"2024-11-07T12:45:43Z","content_type":"text/html","content_length":"39648","record_id":"<urn:uuid:d442fa1d-7c2d-48e4-95d8-8ef8a637d0ed>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00073.warc.gz"} |
A cylinder has a height of 6 meters and a diameter that is 6 times the measure of the height. Using 3.14 for pi, which of the following can be used to calculate the volume of the cylinder? (6 points)A.(3.14)(36m)2(6m)B.(3.14)(18m)2(6m)c.(3.14)(6m)2(36m)d.(3.14)(6m)2(18m)which letter is the answer?
1. Home
2. General
3. A cylinder has a height of 6 meters and a diameter that is 6 times the measure of the height. Using... | {"url":"https://thibaultlanxade.com/general/a-cylinder-has-a-height-of-6-meters-and-a-diameter-that-is-6-times-the-measure-of-the-height-using-3-14-for-pi-which-of-the-following-can-be-used-to-calculate-the-volume-of-the-cylinder-6-points-a-3-14-36m-2-6m-b-3-14-18m-2-6m-c-3-14-6m-2-3","timestamp":"2024-11-04T14:17:26Z","content_type":"text/html","content_length":"30469","record_id":"<urn:uuid:8f563841-aa04-4adf-b52f-86542ff75c07>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00128.warc.gz"} |
Genspace --- Introduction ---
This module is an exercise of linear algebra, on the generation of the vector space ${\mathrm{ℝ}}^{n}$ by a given set of vectors.
The exercise first gives you the vectors (randomly generated), and automatically computes the rank of the matrix formed by these vectors. You have to determine, according to these data, whether the
vectors generate the whole space, and give the reason for your reply.
A second question will be asked if the exercise is in an advanced mode. This second question depends on the first reply: if you think that the vectors generate the space, you will be presented
another vector, and should express it as a linear combination of the given generator vectors. Otherwise, you should find a vector in the space which is not in the subspace generated by the given
You may go to work on the exercise, by choosing the level of difficulty:
The most recent version
This page is not in its usual appearance because WIMS is unable to recognize your web browser.
Please take note that WIMS pages are interactively generated; they are not ordinary HTML files. They must be used interactively ONLINE. It is useless for you to gather them through a robot program.
• Description: does a given set of vectors generate the whole vector space?. interactive exercises, online calculators and plotters, mathematical recreation and games, Pôle Formation CFAI-CENTRE
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Georges Lemaître and the Big Bang - When a priest corrected Einstein's worldview | Lars JaegerGeorges Lemaître and the Big Bang - When a priest corrected Einstein's worldview | Lars Jaeger
Georges Lemaître and the Big Bang – When a priest corrected Einstein’s worldview
The excitement was great when in February this year researchers at the international LIGO Virgo Collaboration announced that for the first time they had directly measured gravitational waves. Now
they managed to do so for the second time. And again praises are dedicated to Albert Einstein, who in his greatest stroke of genius formulated the general theory of relativity, a theory that
describesgravity not as the effect a spatially acting and temporally independent force, but as an action on the very structure of space-time itself, which allows wavelike distortions of space-time,
namely those gravitational waves.
After the completion of his general theory of relativity in November 1915 Einstein quickly sensed that his theory not only represents a new theory of gravity, but also provides an insight into our
universe as a whole. Can one possibly deduct from an assumed cosmic mass distribution the space-time structure of the entire universe? And would this not pave the way to a first unicosmological model
along with its respective dynamics (i.e. its temporal evolution), and ultimately its beginning? Almost immediately after publishing his theory, Einstein took on this question. On theoretical grounds,
he searched especially for solutions that describe a finite universe. But to his irritation, he had to recognize that nosuch universe could be found as a solution to his equations, which at the same
time served a second condition that was as central to Einstein: His universe was supposed to be static, thatis not subject to any evolutionary development or changein time. He only found solutions of
his equations that described contracting or expanding universes (which is essentially due to the fact that the gravitational forces actsonly as an attracting and never as a repelling force, as e.g.
the electric force does). A global cosmos which inflates like a balloon or shrinks in itself, however, was to Einstein’s greatest disliking. The insolubility of his equations for a finite and at the
same timestatic universe was such an essential problem for him that he found himself desperate enough to do something he was deeply reluctant to do: He added ad hoc an additional term to his
equations such that it allowed a solution describinga finite and static universe (this term corresponds to a gravitational force of opposite, i.e. repelling nature).
But others had meanwhile begun to engage in the consideration of the universe as a whole on the basis of Einstein’s equations. Already in 1917 the Dutch mathematician Willem de Sitter had found a
strange solution in which the universe expands (which, however, implied a universe without any matter), and in 1922 the Russian physicist Alexander Friedmann found solutions of the Einstein equations
for a universe(with masses), which was not limited by Einstein’s condition to be static. A static „sphere like“ universe, as Friedmann recognized was extremely unlikely since it would be very fragile
with respect to its boundary conditions.Smallest deviations from an ideal mass distribution would lead it to (depending on the sign of that perturbation) collapse or expand. Friedmann’s universe in
contrast did expand and contract periodically. Einstein himself violently fought Friedmann’s calculation and even accused him of erroneous mathematics, but ultimately had to admitthe calculation was
correct (sadly Friedmann passed away shortly thereafter, so that they could not continue their dispute).
From 1925 on, a young Belgian priests became interested in the Einstein equations. Georges Lemaître was a PhD in mathematics and graduate of the priest seminary (who however never took a post as
chaplain and emphasized a strict separation between research and questions of faith). Lemaître was toresume Friedmann’s work (however apparently without knowing of it) and thus equallyfound himself
inconfrontation with Albert Einstein. As Friedmann hefound solutions of Einstein’s equations in which the universe expands (plus he was able to correct an error inde Sitter’s calculations).
Furthermore, Lemaître deduced directly from the general theory of relativity that the universe’s expansion follows a simple law which could be put to an empirical test: v = H⋅d. The velocity v with
which two galaxies are moving away from each other by the expansion of space, is all the greater the greater the distance d between them is (H is a constant of proportionality, which was later called
„Hubble constant“, see below). However, at first no one took notice ofhis publication from 1927 which appearedin the second-rate French-language journal Annales de la Société Scientifique de
Bruxelles. In the same year Lemaître turned directly to Albert Einstein. As with Friedmann Einstein had nothing to counter the mathematical calculation of Lemaître. But for his belief system it was
totally unacceptable. „Your calculations are correct, but your physics is awful!“, said Einstein in ending the conversation.
But Lemaître was persistent, not least because he had the impression that Einstein’s knowledge of the latest results in astronomy was rather limited. Because astronomers had actually already found
preliminary evidence that some galaxies drift away from our Milky Way. As early as in 1915 Vesto Slipher had found that in 11 of the examined 15 galaxies so called „spectral redshifts“ (an indication
of a movement away from the observer) can beidentified. But the data was generally not yet sufficient to draw definite conclusions. In 1929 came finally the breakthrough: The American astronomer
Edwin Hubble found clear evidence that the galaxies move away from each other. Hubble’s observations were based on distance measurements of pulsating stars (so-called „Cepheid“) in galaxies outside
the Milky Way for which he observed that the redshifts of the stars increase proportionally to their distance. This is exactly the relationship Lemaître had previously deducted! It is now called
„Hubble’s Law“ (it is however only true up to a certain redshift or equivalently a certain distance). Hubble was not a theoretician and initially did not expressan opinionabout the meaning of the
relationship he found. Ironically, for long he did not evenbelieve in an expanding universe, but sought the reasons for his law elsewhere.
When he heard about Hubble’s findings, Einstein called the introduction of the additional term in his equations and his insistence on a static universe „the greatest blunder of my life“. In 1931, he
finally took it out and returned to the field equations in their original form. Calculating the observed expansion of the universe backwards one can conclude that at an earlier time the universe must
have taken a much smaller space, comparable to a balloon before inflation. Was it not obvious to suppose that at its beginning the universe was concentrated in a single point, to be born in a violent
explosion? This was precisely what Lemaître had indicated in his 1927work: a moment of cosmic origin in which space-time itself arose. The priest became the first champion of a physical theory of a
cosmic beginning. Einstein himself in 1933 called this theory „the most beautiful and satisfactory explanation of the creation that I have ever heard“, and thereby he took back his criticism of
Friedmann and Lemaître and declared himself an ardent advocate of Lemaître’stheory.
But the theory was initially still very controversial (to some degree probably because Lemaître was a priest). One of its most eccentric critics, the English physicist Fred Hoyle in a BBC program
called once it ironically “big bang „. To Hoyle’s annoyance this term stuck firmly in both popular as well as scientific parlance. The „Big Bang theory“ is now the scientifically widely accepted
theory of the origin of our universe. And on his sick bed, shortly before his death, a friend of Lemaître’s brought the issue of the journal Astrophysical Journal, in which the discovery of the
cosmic background radiation was announced, which was the last empirical puzzle piece for the recognition of Lemaître’s theory. Georges Lemaître, the priest, who disproved Einstein’s cosmology and the
first scientist proposing the Big Bang theory died 50 years ago, on 20 June 1966.He deserves far more fame and recognition than what is bestowed to him today. | {"url":"https://larsjaeger.ch/georges-lemaitre-and-the-big-bang-when-a-priest-corrected-einsteins-worldview/","timestamp":"2024-11-12T21:39:59Z","content_type":"text/html","content_length":"88416","record_id":"<urn:uuid:b0dbd16e-dee0-48a2-99dc-757ddffa679a>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00521.warc.gz"} |
Nevertheless, analysis of hCD20 expression during B cell advancement uncovered that hCD20 expression in these mice begins only on the immature stage (IgM+), where about 40% from the cells within this people, mostly later immature (simply because revealed simply by high expression of IgM), exhibit hCD20 (Figure ?(Figure2A)
Nevertheless, analysis of hCD20 expression during B cell advancement uncovered that hCD20 expression in these mice begins only on the immature stage (IgM+), where about 40% from the cells within this
people, mostly later immature (simply because revealed simply by high expression of IgM), exhibit hCD20 (Figure ?(Figure2A)
Nevertheless, analysis of hCD20 expression during B cell advancement uncovered that hCD20 expression in these mice begins only on the immature stage (IgM+), where about 40% from the cells within this
people, mostly later immature (simply because revealed simply by high expression of IgM), exhibit hCD20 (Figure ?(Figure2A).2A). as defined (28). Briefly, mice were injected with two dosages each day
of 2 intraperitoneally?mg/mouse of BrdU. On times 2, 4, and 7, BM and spleen cells were isolated and stained with fluorescently labeled antibodies for surface markers, as detailed in the previous
section. Subsequently, cells were stained Mcl1-IN-9 with a FITC-conjugated anti-BrdU antibody using the BrdU Flow Kit (BD Biosciences), according to the manufacturers protocol. Analysis of Ki67
Expression B cell proliferation was estimated in control and B cell-depleted mice 21?days after B cell depletion by flow cytometry using an intracellular antibody against Ki67, the nuclear protein
expressed in proliferating cells, as described in Ref. (37). For determination of Ki67-positive cells, cells were first stained for surface B cells markers: B220, AA4.1, and IgM, followed by
intracellular staining for Ki67 as follows. Mcl1-IN-9 Cells were fixed and permeabilized in Cytofix/Cytoperm solution (BD Biosciences) for 20?min at 4C and then incubated with Ki67
Alexa647-conjugated monoclonal antibody (Santa Cruz Biotechnology). Analysis of Apoptosis by Annexin V The extent of apoptosis in developing B cells was determined by Annexin V staining in control
and B cell-depleted mice 21?days after B cell depletion, as described in Ref. (36). BM cells were stained for B220, AA4.1, and IgM, followed by Annexin V (Biolegend, catalog number 640920) according
to the manufacturers protocol. Cells were then analyzed by flow cytometry. Statistical Analysis of Experimental Data We first tested whether there are significant differences in BrdU labeling
kinetics between the control Mcl1-IN-9 and the depleted mice using generalized linear model (GLM) repeated measures, a method based on analysis of variance (ANOVA). Repeated measures ANOVA is the
equivalent of the one-way ANOVA, but is used for related rather than independent measurements, and is the extension of the dependent the carrying capacity would be interpreted as competition by other
cells for survival niches. Additionally, since the labeling data did not include pro- and pre-B cell subpopulations, a regulation of the source of pro-/pre-B cells by peripheral B cells or by other
cells in response to the depletion was not explicitly examined (see Discussion). Immature B cells either differentiate to BM mature cells at rate i_re, or emigrate from the BM to the spleen and
differentiate HBEGF to transitional B cells at rate i_t (Eqs 2C4). Transitional B cells differentiate to splenic mature B cells at rate t (Eqs 4 and 5). After their maturation, splenic mature B cells
can go back to the PB and then to the mature recirculating population in the BM. The flow of mature B cells from the spleen to the mature (recirculating) population in the BM is represented by the
parameter ?S. The flow in the opposite direction is represented by the parameter ?BM (Eqs 3 and 5). The death rates are denoted by i, t, and rec for equal probability intervals, and each interval is
sampled exactly once, and thus, values are tested for each parameter. parameter combinations are set by assigning a value for each parameter, selected from a random bin. Using LHS allows us to run
the simulation for a magnitude smaller number of combinations Mcl1-IN-9 (41). We used maximum likelihood parameter estimation (MLE) to determine the parameter values that maximize the probability
[likelihood Mcl1-IN-9 (L)] of the data.. | {"url":"http://www.mycareerpeer.com/2022/09/06/%EF%BB%BFnevertheless-analysis-of-hcd20-expression-during-b-cell-advancement-uncovered-that-hcd20-expression-in-these-mice-begins-only-on-the-immature-stage-igm-where-about-40-from-the-cells-with/","timestamp":"2024-11-02T04:31:49Z","content_type":"text/html","content_length":"22594","record_id":"<urn:uuid:c79bd827-4909-49a9-b376-5c8c821572bf>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00311.warc.gz"} |
Relational algebra - Complex systems and AI
Relational algebra
The basic operators of algebra relational are unary or binary operators applied to relationships. The application of each operation produces a new relationship as a result. We distinguish the
following operators: restriction, projection, join, union, difference, intersection, division.
Definitions of basic operators
(1) Restriction of a relationship:
The restriction is a unary operation that selects a set of lines (n-tuples) of a relation, based on a selection criterion (predicate or logical expression of predicates). The result of a restriction
is a relationship with the same schema as the initial relationship.
(2) Projection of a relationship:
Projection is a unary operation that selects a set of columns from a relation. The result of a projection is a relation having as many lines as the initial relation. However, at the end of a
projection, the result relationship may contain identical lines called “doubles”. According to theory, a relationship cannot have identical lines, but most DBMS allow, at the programmer's choice, to
keep or delete them.
(3) Combination of restriction and projection operators:
The restriction and projection operations can be combined to carry out more elaborate processing on relationships. The frequency of these combinations often leads to the realization of a single
operator called selection.
(4) Joining two relationships:
The join is a binary operation which, applied to two relations R1 and R2, produces a restriction of the Cartesian product of these two relations. When the restriction criterion is equality, we speak
of equi-join, otherwise we speak of q-join.
(4a) Natural join
it is an equi-join which is done on the key of one relation and the reference to this key in the other relation.
(4b) Semi-join
A semi-join is a join in which we only keep the attributes of one of the two joined relations.
(4c) Outer join
An outer join is a join that includes in the result relation the tuples of one or other of the operand relations even if they do not satisfy the join condition. These tuples are completed by zero
values in the result relation.
We speak of a left outer join when we take all the tuples of the left operand and of a right outer join when we take all the tuples of the right relation.
(5) Set operators:
Set operators correspond to the usual operators of set theory, defined on tables of the same schemas, considered as sets of tuples.
(6) Division:
The result of dividing a relation R(X,Y) by a relation S(Y) is a relation Q(X) defined by: (i) the schema of Q made up of all the attributes of R not belonging to S, let X, (ii) the tuples qj of Q
such that, whatever the tuple si of S, the tuple (qj,si) is a tuple of R (i.e. S).
The symbol * preceding the name of the relations in outer joins indicates the side with respect to which the operation is carried out. | {"url":"https://complex-systems-ai.com/en/analyse-logicielle/relational-algebra/","timestamp":"2024-11-05T06:15:17Z","content_type":"text/html","content_length":"124851","record_id":"<urn:uuid:d5a640b5-9ded-45d0-909e-08349ca4d6b8>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00087.warc.gz"} |
How to Find the Volume of a Complicated Shape with the Meat-Slicer Method of Integration - dummies
In geometry, you learned how to figure the volumes of simple solids like boxes, cylinders, and spheres. Integration enables you to calculate the volumes of an endless variety of much more complicated
The meat-slicer metaphor is actually quite accurate. Picture a hunk of meat being cut into very thin slices on one of those deli meat slicers. That’s the basic idea here. You slice up a
three-dimensional shape, then add up the volumes of the slices to determine the total volume.
Here’s a problem: What’s the volume of the solid whose length runs along the x-axis from 0 to ð and whose cross sections perpendicular to the x-axis are equilateral triangles such that the midpoints
of their bases lie on the x-axis and their top vertices are on the curve y = sin(x)? Is that a mouthful or what? This problem is almost harder to describe and to picture than it is to do. Take a look
at this thing in the following figure.
So what’s the volume?
1. Determine the area of any old cross section.
Each cross section is an equilateral triangle with a height of sin(x).
2. Find the volume of a representative slice.
The volume of a slice is just its cross-sectional area times its infinitesimal thickness, dx. So you’ve got the volume:
3. Add up the volumes of the slices from 0 to pi by integrating.
If the following seems a bit difficult, well, heck, you better get used to it. This is calculus after all. (Actually, it’s not really that bad if you go through it patiently, step by step.)
It’s a piece o’ cake slice o’ meat.
About This Article
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About Me | misitio
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The study of the mind is one of the most fascinating and multifaceted concerns of mankind. So, in order to obtain effective and useful models explaining and describing its essential features a fully
interdisciplinary approach is needed. Besides, mathematics is, among many others, the language in which the laws of nature seem to be written with maximum precision. Therefore, a strong and mature
formation in pure and applied mathematics represents a huge advantage for starting this enhancing scientific journey through global laws of the mind.
My current research ranges from commutative algebra, algebraic geometry, model theory, Artificial Mathematical Intelligence (AMI), i.e., the theoretical and practical foundations of software able to
solve mathematical conjectures with a human-style output;
the interdisciplinary study on the foundations of mathematics, quantum mechanics (and its relation with ZFC), the philosophical and formal foundations of an experimental science of natural
consciousness, the design of global intelligence tests, to the design and formalization of a general theory of mind with a sound mathematical framework.
Moreover, the development of solid and highly effective learning/teaching techniques with a sound cognitive and multidisciplinary background.
Finally, I love to to multi-, trans-, inter- and intradisciplinary science involving and blending topics in pure and applied mathematics, cognitive science, artificial (mathematical) intelligence,
computer science, philosophy of mind and AI. I enjoy as well to propose new solutions to fundamental problems in philosophy from a multidisciplinary manner.
2024 - present
Professor and Multidisciplinary Scientist
Area of Fundamental Sciences
EAFIT University
Medellín, Colombia.
2019 - 2020; 2022 - 2023
Professor and Scientific Advisor
Research and Innovation Park Parque Tech
University Institution Pascual Bravo
Medellín, Colombia.
2021 - 2022
General Leader and Researcher
Research and Innovation Park Parque Tech
University Institution Pascual Bravo
Medellín, Colombia.
General Leader and Researcher
Research and Innovation Park Parque I
University Institution ITM
Medellín, Colombia.
2017 - 2019
Associated Researcher
Researc18h Groups
Computational Logic and Algebra
Vienna University of Technology
Vienna, Austria.
Commutative Algebra (e.g. The Homological Conjectures, Closure Operations and Forcing Algebras), Algebraic Geometry and its connection with Model theory
Number Theory from an Intra-, Inter and Multidisciplinary Perspective
Cognitively-inspired Foundations for Mathematics
Quantum Mechanics (and its Connections with Zermelo-Fraenkel Set Theory with Choice)
The Interdisciplinary Study of a Global and Mathematically-sound Theory of Mind
General Taxonomy of the Fundamental Cognitive Mechanisms used in Scientific Invention
General Foundations of an Experimental Science of Natural Conciousness
2013 - 2017
Associated Researcher
Artificial Intelligence Group
Institute of Cognitive Sciences
University of Osnabrueck.
Member of the Consorsium
COINVENT (Concept Invention Theory)
European Research Project
2010 - 2013
Affiliated Researcher
Institute of Mathematics
University of Osnabrueck
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Multiple Quantifiers :: CIS 301 Textbook
Multiple Quantifiers
Translations that involve more than one quantifier (which often happens when some of the predicates have more than one parameter) are more challenging. We will divide these translations into two
• Translations that involve several of the same quantifier (multiple universal quantifiers or multiple existential quantifiers)
• Translations that mix quantifiers
In many of the sections, we will using the predicates below (which are over the domain of shapes):
• isCircle(x) - whether shape x is a circle
• isSquare(x) - whether shape x is a square
• isRectangle(x) - whether shape x is a rectangle
• biggerThan(x, y) - whether shape x is bigger than shape y
Several of the same quantifier
First, we consider translations that involve several of the same quantifier. There are two ways we can translate such statements – either using prenex form (quantifiers out front) or Aristotelian
form (quantifiers nested).
Prenex form
The prenex form of a predicate logic translation lists all the quantifiers at the beginning of the statement. This is only recommended when all the quantifiers are the same type – either all
universal or all existential.
Prenex example 1
Suppose we wished to translate, Some circle is bigger than some square. Here, we are making three claims:
• There exists a shape that is a circle
• There exists a shape that is a square
• The shape that is a circle is bigger than the shape that is a square
With that in mind, we can see that we will use two existential quantifiers. We can translate the statement as follows:
∃ x ∃ y (isCircle(x) ∧ isSquare(y) ∧ biggerThan(x, y))
Which reads: There are two shapes, x and y, where x is a circle, y is a square, and x (which is a circle) is bigger than y (which is a square).
Equivalently, we could have written:
∃ x ∃ y (isCircle(y) ∧ isSquare(x) ∧ biggerThan(y, x))
Which reads: There are two shapes, x and y, where y is a circle, x is a square, and y (which is a circle) is bigger than x (which is a square).
Prenex example 2
Next, suppose we wished to translate: Every circle is bigger than all squares. Again, we are quantifying two things – ALL circles and also ALL squares. We can see that we will need to use two
universal quantifiers. We can translate the statement as follows:
∀ x ∀ y ((isCircle(x) ∧ isSquare(y)) → biggerThan(x, y))
Which reads: For each combination (x, y) of shapes, if x is a circle and y is a square, then x (which is a circle) is bigger than y (which is a square).
Aristotelian form
The Aristotelian form of a predicate logic translation embeds the quantifiers within the translation. This format is possible for any kind of translation – whether the quantifiers are all the same
type or mixed types.
Aristotelian form example 1
Suppose we wished to translate, Some circle is bigger than some square using Aristotelian form. We know that we will still need two existential quantifiers, but we will only introduce each quantifier
just before the corresponding variable is needed in a predicate.
We can translate the statement using Aristotelian form as follows:
∃ x (isCircle(x) ∧ (∃ y (isSquare(y) ∧ biggerThan(x, y)))
Which reads as: There exists a shape x that is a circle and there exists a shape y that is a square, and x (which is a circle) is bigger than y (which is a square).
Aristotelian form example 2
Let’s repeat our translation for, Every circle is bigger than all rectangles using Aristotelian form. We know that we will still need two existential quantifiers, but we will only introduce each
quantifier just before the corresponding variable is needed in a predicate.
We can translate the statement using Aristotelian form as follows:
∀ x (isCircle(x) → (∀ y (isSquare(y) → biggerThan(x, y))))
Which reads as: For every shape x, if x is a circle, then for every shape y, if y is s square, then x (which is a circle) is bigger than y (which is a square).
Mixed quantifiers
Now, we will turn to examples that mix universal and existential quantifiers. We will see below that quantifier order matters in this case, so it is safest to translate such statements using embedded
quantifiers. The embedded form can be tricky to write, so we will see a way to systematically translate any statement that needs multiple quantifiers into predicate logic (using Aristotelian form).
Systematic translation
Suppose we wish to translate, Every circle is bigger than at least one square. We see that we are first making a claim about all circles. Without worrying about the rest of the statement, we know
that for all circles, we are saying something. So we write:
For all circles, SOMETHING
Trying to formalize a bit more, we assign a variable to the current circle we are describing (x). For each circle x, we are saying something about that circle. So we express SOMETHING(x) as some
claim about our current circle, and write:
For each circle x, SOMETHING(x)
We see that we will need a universal quantifier since we are talking about ALL circles, and we also follow the guide of using an implies statement to work with a for-all statement:
∀ x (isCircle(x) → SOMETHING(x))
Next, we describe what SOMETHING(x) means for a particular circle, x:
SOMETHING(x): x is bigger than at least one square
Trying to formalize a bit more about the square, we write:
SOMETHING(x): There exists a square y, and x is bigger than y
Now we can use an existential quantifier to describe our square, and plug in our isSquare and biggerThan predicates to have a translation for SOMETHING(x):
SOMETHING(x): ∃ y (isSquare(y) ∧ biggerThan(x, y))
Now, we can plug SOMETHING(x) into our first partial translation, ∀ x (isCircle(x) → SOMETHING(x)). The complete translation for Every circle is bigger than at least one square is:
∀ x (isCircle(x) → (∃ y (isSquare(y) ∧ biggerThan(x, y))))
Follow-up examples
In these examples, suppose our domain is animals and that we have the following predicates:
• El(x): whether animal x is an elephant
• Hi(x): whether animal x is a hippo
• W(x, y): whether animal x weighs more than animal y
Suppose we wish to translate: There is exactly one hippo. We might first try saying: ∃ x Hi(x). But this proposition would be true even if we had 100 hippos, so we need something more restricted.
What we are really trying to say is:
• There exists a hippo
• AND, any other hippo is the same one
Let’s use our systematic approach, streamlining a few of the steps:
• There exists an animal x that is a hippo, and SOMETHING(x)
• ∃ x (Hi(x) ∧ SOMETHING(x))
To translate SOMETHING(x), the claim we are making about our hippo x:
• SOMETHING(x): any other hippo is the same as x
• SOMETHING(x): for each hippo y, x is the same as y
• SOMETHING(x): `∀ y (Hi(y) → (x == y))
Now we can put everything together to get a complete translation:
∃ x (Hi(x) ∧ (∀ y (Hi(y) → (x == y)))
Here are a few more translations from English to predicate logic. Think about what the following statements mean, and click to reveal each answer:
• Every elephant is heavier than some hippo.
□ Click here for solution
∀ x (El(x) -> (∃ y (Hi(y) ^ W(x, y))))
• There is an elephant that is heavier than all hippos.
□ Click here for solution
∃ x (El(x) ^ (∀ y (Hi(y) -> W(x, y))))
• No hippo is heavier than every elephant.
□ Click here for solution
¬(∃ x (Hi(x) ^ (∀ y (El(y) -> W(x, y)))))
Order matters!
We have learned that when dealing with mixed quantifiers, it is safest to embed them within the translation. If we put the mixed quantifiers out front of a translation, then we can accidentally
include them in the wrong order and end up with an incorrect translation.
Suppose we have this predicate, over the domain of people:
likes(x, y): whether person x likes person y
Further suppose that liking a person is not necessarily symmetric: that just because person x likes person y does not mean that person y necessarily likes person x.
Consider these pairs of propositions:
∀ x ∀ y likes(x, y) vs. ∀ y ∀ x likes(x, y)
∃ x ∃ y likes(x, y) vs. ∃ y ∃ x likes(x, y)
Is there any difference between each one? No! The two versions of the first proposition both say that every person likes every other person, and the two versions of the second proposition both say
that there is a person who likes another person.
But what about:
∀ x ∃ y likes(x, y) vs. ∃ y ∀ x likes(x, y)
Suppose our domain is made up of the following people:
Bob: likes Alice and James
Alice: likes Bob
James: likes Alice
The first proposition, ∀ x ∃ y likes(x, y), says that all people have some person (not necessarily the same person) that they like. This would certainly be true for our domain, as every person has at
least one person that they like. The second proposition, ∃ y ∀ x likes(x, y) is saying that there is a person (the SAME person) that everyone likes. This proposition would be false for our domain, as
there is no one person that is liked by everyone.
Precedence with quantifiers
In section 2.2, we discussed operator precedence for propositional logic statements. The same operator precedence holds for predicate logic statements, except that our two quantifiers (∀ and ∃) have
the same precedence as the NOT operator. If we have a proposition with multiple quantifiers, then the quantifiers are resolved from right to left. For example, ∃ y ∀ x likes(x, y) should be
interpreted as ∃ y (∀ x likes(x, y)).
Here is an updated list of operator precedence, from most important (do first) to least important (do last):
1. Parentheses
2. Not operator (¬), universal quantifier (∀), existential quantifier (∃)
3. And operator, ∧
4. Or operator, ∨
5. Implies operator, →
And here is our updated list of how to resolve multiple operators with the same precedence:
1. Multiple parentheses - the innermost parentheses are resolved first, working from inside out.
2. Multiple not ( ¬ ) operators – the rightmost ¬ is resolved first, working from right to left. For example, ¬¬p is equivalent to ¬(¬p).
3. Multiple and ( ∧ ) operators – the leftmost ∧ is resolved first, working from left to right. For example, p ∧ q ∧ r is equivalent to (p ∧ q) ∧ r.
4. Multiple or ( ∨ ) operators – the leftmost ∨ is resolved first, working from left to right. For example, p ∨ q ∨ r is equivalent to (p ∨ q) ∨ r.
5. Multiple implies ( → ) operators – the rightmost → is resolved first, working from right to left. For example, p → q → r is equivalent to p → (q → r).
6. Multiple quantifiers – the rightmost quantifier is resolved first, working from right to left. For example, ∃ y ∀ x likes(x, y) should be interpreted as ∃ y (∀ x likes(x, y)).
When we get to predicate logic proofs in Chapter 6, we will see that Logika uses a different precedence for quantifiers – there, quantifiers have the LOWEST precedence (done last) of any operator.
This ends up being more forgiving than confusing, as it will accept propositions as correct that are really missing parentheses. For example, Logika will accept: ∃ x isMouse(x) ∧ inHouse(x).
Technically, this proposition should be incorrect – if we correctly treat quantifiers as having a higher precedence than ∧, then inHouse(x) would no longer know about the variable x or its
We should use parentheses with quantifiers to express our intended meaning, and so we should write ∃ x (isMouse(x) ∧ inHouse(x)) instead. But if we forget the parentheses, then Logika will forgive | {"url":"https://textbooks.cs.ksu.edu/cis301/5-chapter/5_4-multquant/tele.html","timestamp":"2024-11-11T23:10:42Z","content_type":"text/html","content_length":"24905","record_id":"<urn:uuid:597552c8-cd72-4ecd-9e8d-87a083963f9d>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00317.warc.gz"} |
Using graphical, numerical, and algebraic approaches, Pre-Calculus covers floor, power, linear, rational, circular, exponential, logarithmic, polynomial, and absolute-value functions. The course also
explores the quadratic functions forming circles, ellipses, parabolas, and hyperbolas. The course highlights where calculus uses these elements and functions, and concludes by introducing students to
limits, numerical derivatives, and numerical integrals. If time allows, students explore parametric equations, polar coordinates, sequences and series, the binomial theorem, and vectors in the
two-dimensional plane. | {"url":"https://catalog.wmcc.edu/mathematics/math180w","timestamp":"2024-11-04T16:45:15Z","content_type":"text/html","content_length":"12710","record_id":"<urn:uuid:593f23c1-64d2-49d9-a5d1-1c1fd8055cdc>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00515.warc.gz"} |
st: xtabond2 "too many instruments" query
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st: xtabond2 "too many instruments" query
From "S M Ali Abbas" <[email protected]>
To <[email protected]>
Subject st: xtabond2 "too many instruments" query
Date Thu, 6 Apr 2006 13:59:53 +0100
Hi folks
I am trying to estimate a panel growth model as follows:
gY = aY_1 + bX +cE + d(i.period) +error (d is the parameter vector for
period dummies)
where Y is income, X is a set of exogenous regressors (excl time dummies), E
is endogenous or predetermined variables and TIME is the time dummies.
The problem is when I try to put all the E variables in gmm( ) and put
iv(i.period, eq(level)) I get the warning: "Number of instruments may be
large relative to number of observations." Also the computation takes a long
time (expected) and the Hansen Chi-Squared is 1.000.
Six "very brief" questions related to this. Please answer whatever you
comfortably can. Many many thanks in advance.
My total observations are 642 and no. of instruments is 101. [This is
"after" I have reduced the lags to (3 3).] Is this a serious warning (I read
on statalist that if the number of instruments is large relative to number
of observations, there can be serious small sample biases)? At what ratio of
obs/instruments does this cease/begin to be a problem.
I have also noticed that if I remove the iv(i.period, eq(level)) and most of
the E variables in gmm(.), the Hansen-Ch-Squared p-value becomes less than 1
(as it healthily should) and the WARNING disappears. However, if I plug back
one additional E variable in gmm(.) the WARNING comes back, and the Hansen
p-value climbes (although not up to 1). My question, therefore, is: what is
the relationship ship between the Hansen Chi-squared and the WARNING, and is
it permissible to ignore the warning as long as Hansen p-value is below 1?
I was a bit concerned that perhaps by removing iv(i.period, eq(level)) from
the command line, I was not actually removing the i.period dummies as
instruments, rather only removing the information that they were time
dummies. If this fear of mine is correct, then this is serious. Please let
me know if this fear is well-founded or have I actually removed the time
dummies from the instrument list.
A more general question is: is it acceptable to remove time dummies from the
instrument set? What are the conditions when it is, and when it is not?
Also I was not clear if my X exogenous regressors are included as
instruments; I was assuming xtabond2 would automatically include them noting
that they are in the regression equation but not in the gmm(.) list. If my
assumption is incorrect, how can I include say variable X with say a lag
structure of (2 2)?
Finally, my E set contains predetermined variables say Ep and other
endogenous variabels Ed. I also have, in addition the Y_1 lagged dependent
variable. I was not sure how I should write my command so that it
distinguishes between these three categories of endogenous variables. I
understand that the lag length used (a b) should differ across the three
categories. But I am not sure what the synatx would be, or whether I would
put Y_1 in the gmm(.) or just Y but starting with a deeper lag? Also I
wasn't sure what the treatment would be for endogenous variables that I have
included with lags in the regression. For e.g., you can think of EDU
affecting growth with a lag so my E set contains EDU_1. Should I write this
as EDU in gmm(.) with a deeper lag (say 3 3), or just EDU_1 with (2 2). Any
guidance on this would be highly appreciated.
Ali Abbas
Oxford University
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"https://www.stata.com/statalist/archive/2006-04/msg00198.html","timestamp":"2024-11-03T13:54:40Z","content_type":"text/html","content_length":"11020","record_id":"<urn:uuid:3a1bafc8-3fb3-47bf-9575-b43629e5c6e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00221.warc.gz"} |
023.) Point Lights - OpenGL 4 - Tutorials - Megabyte Softworks
023.) Point Lights
<< back to OpenGL 4 series
Hello again and I welcome you to my 23rd tutorial of my OpenGL4 tutorial series! In this one we will explore point lights - simply a light with some position and some strength, like a light bulb. How
to program it and what we need - you can find all this out if you keep reading this article
Point light is a type of light that shines from a single spot (point) equally in all directions, so unlike directional light, it doesn't have any specific direction. The typical example of point
light is a light bulb. Here is a picture of light bulb, so that there are some pictures in this article
As you can see, the bulb emits light to all directions. And the common sense tells us, that the further any point from the light is, the less it will be illuminated. Sounds easy, right? And that's
exactly what we're going to program! Let's have a look at the point light shader structure, that holds all its data:
struct PointLight
vec3 position;
vec3 color;
float ambientFactor;
float constantAttenuation;
float linearAttenuation;
float exponentialAttenuation;
bool isOn;
Let's examine what they're about:
• position - represents position of the point light
• color - color of the point light
• ambientFactor - even though it's a point light, its light might be already scattered in the whole world, so this is exactly like a constant contribution to the global illumination. It's not
obligatory to have this attribute, it's just my implementation of point light (in fact, there is no "one and only correct" implementation, you can customize your code as much as you want as long
as you achieve the desired effect
• constantAttenuation - represents constant attenuation of the light (attenuation is simply how fast light weakens with rising distance), but because this is constant attenuation, distance has no
effect on it
• linearAttenuation - represents linear attenuation of the light, so it rises in a linear fashion with rising distance
• exponentialAttenuation - represents exponential attenuation of the light so it rises in an exponential fashion with rising distance
• isOn - simple boolean saying if light is on or off (this is again not something completely necessary, but sometimes it might be easier to just turn light off rather than remove it completely)
I really hope you got the point of those attributes, now let's have a look at math behind the point lights
Looking at the parameters described above, we will need two things for the equation - directional vector of how light shines on the fragment and the distance from the illuminated fragment to the
light. With those two values, we can calculate diffuse factor and the total attenuation of the light depending on the distance. Below is the shader code fragment, that does the whole calculation:
vec3 getPointLightColor(const PointLight pointLight, const vec3 worldPosition, const vec3 normal)
if(!pointLight.isOn) {
return vec3(0.0);
vec3 positionToLightVector = worldPosition - pointLight.position;
float distance = length(positionToLightVector);
positionToLightVector = normalize(positionToLightVector);
float diffuseFactor = max(0.0, dot(normal, -positionToLightVector));
float totalAttenuation = pointLight.constantAttenuation
+ pointLight.linearAttenuation * distance
+ pointLight.exponentialAttenuation * pow(distance, 2.0);
return pointLight.color * (pointLight.ambientFactor + diffuseFactor) / totalAttenuation;
As you can see, first if condition is just checking if the light is on. If it's not, simply return completely black color (so this light does not contribute to the final illumination at all). Second
step is calculation of the position to the light vector. This vector serves two purposes - first, we can get the distance of the fragment to the light, second we can calculate the diffuse factor of
the point light (here applies same logic as explained in the tutorial 14.) Normals and Diffuse Lighting - the more directly the light shines on the fragment, the more it gets illuminated).
The last step is to calculate the total accumulated attenuation of the light depending on its distance from the fragment. The equation for the attenuation is following:
aC is constant attenuation, aL is linear attenuation, aQ is exponential attenuation (quadratic) and d is our calculated distance of the fragment to the point light. This equation corresponds to the
variable totalAttenuation in the shader fragment code. Now that we have all the needed values, the final color of the fragment is simply the point light color multiplied by the ambient and diffuse
factor summed up and divided by the total attenuation
I was actually thinking if the ambientFactor should be divided by attenuation too, but I came to conclusion that it's just fine. Because even if point light contributes to the ambient illumination, I
think it should fade with the distance. One night lamp in a city simply won't illuminate the whole city, just the particular street it is on
The final step is to use the getPointLightColor function in the main shader - here is the most important excerpt of the code:
// ...
uniform AmbientLight ambientLight;
uniform DiffuseLight diffuseLight;
uniform Material material;
uniform PointLight pointLightA;
uniform PointLight pointLightB;
uniform vec3 eyePosition;
void main()
vec3 normal = normalize(ioVertexNormal);
vec4 textureColor = texture(sampler, ioVertexTexCoord);
vec4 objectColor = textureColor*color;
vec3 ambientColor = getAmbientLightColor(ambientLight);
vec3 diffuseColor = getDiffuseLightColor(diffuseLight, normal);
vec3 specularHighlightColor = getSpecularHighlightColor(ioWorldPosition.xyz, normal, eyePosition, material, diffuseLight);
vec3 pointLightColorA = getPointLightColor(pointLightA, ioWorldPosition.xyz, normal);
vec3 pointLightColorB = getPointLightColor(pointLightB, ioWorldPosition.xyz, normal);
vec3 lightColor = ambientColor + diffuseColor + specularHighlightColor + pointLightColorA + pointLightColorB;
outputColor = objectColor * vec4(lightColor, 1.0);
In this tutorial, we have two point lights - point light A and point light B. Of course this is not very dynamic, but it's sufficient to demonstrate the effect
In the C++ code, I have created new shader struct that holds the point light data for easy access. Check the renderScene method to see the whole setup of the point lights.
This is the result that we have achieved with all that I've explained above:
I think it looks very nice
So that's it for today! I really hope that you've enjoyed this tutorial and that you have learned a thing or two today
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[Solved] Find the coordinates of the foci and the vertices, the... | Filo
Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola
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The given equation is
It can be written as
On comparing equation (1) with the standard equation of hyperbola, i.e., , we obtain and b=2
We know that,
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Question Text Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola
Updated On Nov 10, 2023
Topic Conic Sections
Subject Mathematics
Class Class 11
Answer Type Text solution:1 Video solution: 10
Upvotes 976
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1.1 Angles and Lines II
1.1 Angles and Lines II
Identifying Parallel, Transversals, Corresponding Angles, Alternate Angles and Interior Angles
(A) Parallel lines
Parallel lines are lines with the same direction. They remain the same distance apart and never meet.
(B) Transversal lines
(C) Alternate angles
(D) Corresponding angles
(E) Interior angles
PT3 Smart TIP
Alternate angles are easily identified by tracing out the pattern “Z” as shown.
Corresponding angles are easily identified by the pattern “F” as shown.
Interior angles are easily identified by the pattern “C” as shown.
Example 1:
Construct a line parallel to PQ and passing through W.
a = 40^o and b = 50^o← (Alternate angles)
y = a + b
= 40^o+ 50^o
= 90^o
Example 2:
In Diagram below, PSQ and STU are straight lines. Find the value of x.
$\begin{array}{l}\angle WSQ={180}^{o}-{150}^{o}\\ \text{}={30}^{o}←\overline{)\text{Supplementary angle}}\\ \angle XTU=\angle WSQ+\angle x←\overline{)\text{Corresponding angle}}\\ \text{}{75}^{o}=
{30}^{o}+\angle x\\ \text{}\angle x={75}^{o}-{30}^{o}\\ \text{}\angle x={45}^{o}\\ \text{}x=45\end{array}$ | {"url":"http://content.myhometuition.com/2017/12/01/11-angles-and-lines-ii/","timestamp":"2024-11-06T14:18:57Z","content_type":"text/html","content_length":"33785","record_id":"<urn:uuid:ff9acfc4-acb0-4964-a4f7-028165cae824>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00437.warc.gz"} |
NOT So Powerful
Note: Thanks to Sasho and Badih Ghazi for pointing out that I had misread the Tardos paper. Approximating the Shannon graph capacity is an open problem. Grötschel, Lovász and Schrijver approximate a
related function, the
Lovász Theta function
, which also
has the properties
we need to get an exponential separation of monotone and non-monotone circuits.
Also since I wrote this post, Norbert Blum has
retracted his proof
Below is the original post.
A monotone circuit has only AND and OR gates, no NOT gates. Monotone circuits can only produce monotone functions like clique or perfect matching, where adding an edge only makes a clique or matching
more likely. Razborov in a famous 1985 paper showed that the clique problem does not have polynomial-size monotone circuits.
I choose Razborov's monotone bound for clique as one of my
Favorite Ten Complexity Theorems
(1985-1994 edition). In that section I wrote
Initially, many thought that perhaps we could extend these [monotone] techniques into the general case. Now it seems that Razborov's theorem says much more about the weakness of monotone models
than about the hardness of NP problems. Razborov showed that matching also does not have polynomial-size monotone circuits. However, we know that matching does have a polynomial-time algorithm
and thus polynomial-size nonmonotone circuits. Tardos exhibited a monotone problem that has an exponential gap between its monotone and nonmonotone circuit complexity.
I have to confess I never actually read Éva Tardos' short
at the time but since it serves as
Exhibit A
against Norbert Blum's recent
P ≠ NP paper
, I thought I would take a look. The paper relies on the notion of the
Shannon graph capacity
. If you have a k-letter alphabet you can express k
many words of length n. Suppose some pairs of letters were indistinguishable due to transmission issues. Consider an undirected graph G with edges between pairs of indistinguishable letters. The
Shannon graph capacity is the value of c such that you can produce c
distinguishable words of length n for large n. The Shannon capacity of a 5-cycle
turns out
to be the square root of 5.
Grötschel, Lovász, Schrijver
use the ellipsoid method to approximate the Shannon capacity in polynomial time.
The Shannon capacity is anti-monotone, it can only decrease or stay the same if we add edges to G. If G has an independent set of size k you can get k
distinguishable words just by using the letters of the independent set. If G is a union of k cliques, then the Shannon capacity is k by choosing one representation from each clique, since all letters
in a clique are indistinguishable from each other.
So we have the largest independent set is at most the Shannon capacity is at most the smallest clique cover.
Let G' be the complement of a graph G, i.e. {u,v} is an edge of G' iff {u,v} is not an edge of G. Tardos' insight is to look at the function f(G) = the Shannon capacity of G'. Now f is monotone in G.
f(G) is at least the largest independent set of G' which is the same as the largest clique in G. Likewise f(G) is bounded above by the smallest partition into independent sets which is the same as
the chromatic number of G since all the nodes with the same color form an independent set. We can only approximate f(G) but by careful rounding we can get a monotone polynomial-time computable
function (and thus polynomial-size AND-OR-NOT circuits) that sits between the clique size and the chromatic number.
Finally Tardos notes that the techniques of Razborov and
show that any monotone function that sits between clique and chromatic number must have exponential-size monotone (AND-OR) circuits. The NOT gate is truly powerful, bringing the complexity down
2 comments:
1. The Shannon capacity is interesting but I feel this post is misleading in several ways by trying very hard to avoid talking about SDPs and the Lovasz theta function. Tardos's function is in fact
the Lovasz theta function, appropriately rounded, and I don't think the theta function approximates Shannon capacity to any useful factor in general. So I don't think it's true that we know how
to use the ellipsoid method to approximate Shannon capacity. However, we know how to solve SDPs efficiently, and the theta function is the solution to an SDP. It's also monotone and distinguishes
k-cliques from complete (k+1)-partite graphs, which is all that's needed.
2. More details about Tardos' function can be found here. | {"url":"https://blog.computationalcomplexity.org/2017/08/not-so-powerful.html","timestamp":"2024-11-05T12:41:03Z","content_type":"application/xhtml+xml","content_length":"178387","record_id":"<urn:uuid:a45c0eff-005b-4471-a151-c1a25ae02f3e>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00351.warc.gz"} |
Weekly Homework X
This is the homework my students use for typing up their weekly homework in LaTeX.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Welcome to writeLaTeX --- just edit your LaTeX on the left, % and we'll compile it for you on the right. If you give % someone the
link to this page, they can edit at the same % time. See the help menu above for more info. Enjoy! % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
-------------------------------------------------------------- % This is all preamble stuff that you don't have to worry about. % Head down to where it says "Start here" %
-------------------------------------------------------------- \documentclass[12pt]{article} \usepackage[margin=1in]{geometry} \usepackage{amsmath,amsthm,amssymb} \newcommand{\N}{\mathbb{N}} \
newcommand{\Z}{\mathbb{Z}} \newenvironment{theorem}[2][Theorem]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}} \newenvironment{lemma}[2]
[Lemma]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}} \newenvironment{exercise}[2][Exercise]{\begin{trivlist} \item[\hskip \labelsep {\
bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}} \newenvironment{problem}[2][Problem]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end
{trivlist}} \newenvironment{question}[2][Question]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}} \newenvironment{corollary}[2][Corollary]{\
begin{trivlist} \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}} \newenvironment{solution}{\begin{proof}[Solution]}{\end{proof}} \begin{document} %
-------------------------------------------------------------- % Start here % -------------------------------------------------------------- \title{Weekly Homework X}%replace X with the appropriate
number \author{Tony Stark\\ %replace with your name Foundations of Mathematics} %if necessary, replace with your course title \maketitle \begin{theorem}{x.yz} %You can use theorem, exercise, problem,
or question here. Modify x.yz to be whatever number you are proving Delete this text and write theorem statement here. \end{theorem} \begin{proof} %You can also use solution in place of proof. Blah,
blah, blah. Here is an example of the \texttt{align} environment: %Note 1: The * tells LaTeX not to number the lines. If you remove the *, be sure to remove it below, too. %Note 2: Inside the align
environment, you do not want to use $-signs. The reason for this is that this is already a math environment. This is why we have to include \text{} around any text inside the align environment. \
begin{align*} \sum_{i=1}^{k+1}i & = \left(\sum_{i=1}^{k}i\right) +(k+1)\\ & = \frac{k(k+1)}{2}+k+1 & (\text{by inductive hypothesis})\\ & = \frac{k(k+1)+2(k+1)}{2}\\ & = \frac{(k+1)(k+2)}{2}\\ & = \
frac{(k+1)((k+1)+1)}{2}. \end{align*} \end{proof} \begin{theorem}{x.yz} Let $n\in \Z$. Then yada yada. \end{theorem} \begin{proof} Blah, blah, blah. I'm so smart. \end{proof} %
-------------------------------------------------------------- % You don't have to mess with anything below this line. % -------------------------------------------------------------- \end{document} | {"url":"https://www.overleaf.com/latex/templates/weekly-homework-x/cbpdxbqknrvq","timestamp":"2024-11-13T09:46:55Z","content_type":"text/html","content_length":"38880","record_id":"<urn:uuid:5bf36516-c7bb-4e3c-adac-2defb039845e>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00680.warc.gz"} |
P&L Calculations (USDT Contract)
2024-11-02 18:12:21
Regardless of any trades, it is important to understand how P&L is calculated before entering one. In sequential order, traders need to understand the following variables in order to accurately
calculate their P&L.
1) Average Entry Price of position
2) Unrealized P&L and unrealized P&L% of the position
1) Average Entry Price (AEP) of position
In Bybit, whenever traders add on to their position via new orders, AEP will change.
For example: Trader A holds an existing BTCUSDT open buy position of 0.5 qty with an entry price of USD 5,000. After an hour, Trader A decided to increase his buy position by opening an additional
0.3 qty with an entry price of USD 6,000.
Below shows how the formula for AEP and the computation steps:
Average entry price = Total contract value in USDT/Total quantity of contracts
Total contract value in USDT = ( (Quantity1 x Price1) + (Quantity2 x Price2)...)
By using the figures above:
Total contract value in USDT
= ( (Quantity1 x Price1) + (Quantity2 x Price2) )
= ( (0.5 x 5,000) + (0.3 x 6,000) )
= 4300
Total quantity of contracts
= 0.5 + 0.3
= 0.8 BTC
Average Entry Price
= 4,300 / 0.8
= 5,375
2) Unrealized P&L
Once an order is successfully executed, an open position and its real-time unrealized P&L will be shown inside the positions tab.
Depending on which side of the trade you are in, the formula used to calculate the unrealized P&L will differ.
For long position:
For example:
Trader B holds an existing BTCUSDT open buy position of 0.2 qty with an entry price of USD 7,000. When the Last Traded Price inside the order book is showing USD 7,500, the unrealized P&L shown will
be 100 USDT.
Unrealized P&L = Contract Qty x (Last Traded Price - Entry Price)
= 0.2 x (7,500 - 7,000)
= 100 USDT
For short position
For example: Trader C holds an existing BTCUSDT open sell position of 0.4 qty with an entry price of USD 6,000. When the Last Traded Price inside the order book is showing USD 5,000, the unrealized P
&L shown will be 400 USDT.
Unrealized P&L = Contract Qty x (Entry Price - Last Traded Price)
= 0.4 x ( 6,000 - 5,000)
= 400 USDT
a) In USDT contracts, your P&L is also settled in USDT. This is opposite to inverse contracts where P&L is settled depending on the coin being traded (ex. BTCUSD inverse is settled in BTC).
b) When the price movement is by a certain price (example USD 1,000) in the profitable or non-profitable direction, assuming the position size of 1 BTC, this means that a trader will gain or lose USD
1,000 respectively.
c) Increasing leverage does not directly multiply the profits/losses. Instead, profits and losses are determined by the position size and price movement. In short,
• The higher the leverage, the lower the margin collateral needed to open your position
• The larger the contract quantity, the bigger the profits/losses
• The larger the price movement relative to the entry price, the bigger the profits/losses
d) The default unrealized P&L is shown based on the Last Traded Price. When hovering a mouse cursor on top of the figure, the unrealized P&L will change and show an unrealized P&L based on the Mark
e) Last but not least, unrealized P&L does not factor in any trading or funding fees which traders may have received/paid out in the process of opening and holding the position.
2A) Unrealized P&L%
Unrealized P&L% basically shows the Return on Investment (ROI) of the position in its percentage form. Similar to Unrealized P&L, the figure shows changes depending on the movement of the Last Traded
Price. As such, the Unrealized PNL% or ROI formula is below.
Unrealized P&L% = [ Position's unrealized P&L / Position Margin ] x 100%
Position Margin = Initial margin + Fee to close
Using Trader B as an example, Trader B holds an existing BTCUSDT open buy position of 0.2 qty with an entry price of USD 7,000. When the Last Traded Price inside the order book is showing USD 7,500,
the unrealized P&L shown will be 100 USDT. Assuming the leverage used is 10x.
Based on our earlier calculation, the position's unrealized P&L = 100 USDT
Initial margin = (Qty x Entry price) / leverage = (0.2 x 7000) /10 = 140 USDT
Fee to close = Bankruptcy price x Qty x 0.055% = 6,300 x 0.2 x 0.055% = 0.693 USDT
Unrealized P&L% = [ 100 USDT / ( 140 USDT + 0.693 USDT ) ] x 100% = 71.07%
a) Some traders may have misunderstood this but adjustments to increase leverage do not increase your unrealized profits. Instead, traders will see an increase in unrealized P&L% due to a reduction
in your position margin and not because of an increase in actual profits. Using Trader B as an example again, notice that regardless if leverage is 10x, 5x or 20x, the unrealized P&L remains the
• If Trader B uses the same 10x leverage, his unrealized P&L = 100 USDT, unrealized P&L% = 71.07%
• If Trader B reduces the leverage to 5x, his unrealized P&L = 100 USDT, unrealized P&L% = 35.62%
• If Trader B increases the leverage to 20x his unrealized P&L = 100 USDT, unrealized P&L% = 141.45%
b) For cross margin mode, the position margin will always be calculated using the maximum leverage allowed under the current risk limit level for the particular coin (Example BTCUSDT = 100x).
3) Closed P&L
When traders finally close their position, the P&L becomes realized and is recorded inside the Closed P&L tab within the Assets page. Unlike unrealized P&L, there are some major differences in the
calculation. Below summarizes the differences between the unrealized P&L and closed P&L.
Therefore, assuming full closing of the entire position, the formula for calculating Closed P&L is as follows:
Closed P&L = Position P&L - Fee to open - Fee to close - Sum of all funding fees paid/received
Position P&L =
Long: Position P&L = Contract Qty x (Exit Price - Entry Price)
Short: Position P&L = Contract Qty x (Entry Price - Exit Price)
Using Trader C as an example, Trader C holds an existing BTCUSDT open sell position of 0.4 qty with an entry price of USD 6,000. When the Last Traded Price inside the order book is showing USD 5,000,
trader C decided to close the entire position via the Close by Market function.
Assuming that Trader C also opened the position via a market order and funding fees totaling 2.10 USDT were paid out while holding the position.
Fee to open = Qty x Entry price x 0.055% = 1.32 USDT paid out
Fee to close = Qty x Exit price x 0.055% = 1.1 USDT paid out
Sum of all funding fees paid/received = 2.10 USDT paid out
Closed P&L = 400 - 1.32 - 1.1 - 2.10 = 395.48 USDT
a) The above example only applies when the entire position is opened and closed via a single order in both directions.
b) For partial closing of positions, Closed P&L will prorate all fees (fee to open and funding fee(s)) according to the percentage of the position partially closed and use the pro-rated figure to
compute the Closed P&L
c) Traders can view their Closed P&L history from here.
4) Realized P&L
Realized P&L = Sum of realized position P&L - Trading fees - Funding fees over the period of position opening
Realized P&L can be found on the position tab and it shows the sum of realized P&L of the position over the period. This includes all the trading fees, funding fees, and any position P&L realized
from partial closing (same formula as unrealized P&L).
We can use Trader C as an example. Assuming Trader C did not fully close the 0.4 qty short position, but only 0.3 qty with an exit price of $5,000.
Position's P&L = 0.3 x [ 6,000 - 5,000 ] = 300 USDT
Fee to open = 0.4 x 6,000 x 0.055% = 1.32 USDT
Fee to close = 0.3 x 5,000 x 0.055% = 0.825 USDT
Sum of funding fees paid = 1.5 USDT
Realized P&L of the position = 300 - 1.32 - 0.825 - 1.5 = 296.355 USDT
Now, Trader C is left with 0.1 qty of short position. He then opened another 0.2 qty of short position with an entry price of USD 5,500, the realized P&L for the position is as follows:
Realized P&L carried forward = 296.355 USDT
Fee to open = 0.2 x 5,500 x 0.055% = 0.605 USDT
Realized P&L (up-to-date) = 296.355 - 0.605 = 295.75 USDT
Outstanding Open Position = 0.3 Qty of Short Position
The difference between the realized P&L and closed P&L is that for Closed P&L, in the event of partial closing of positions, it will prorate all fees (fee to open and funding fee(s)) according to the
percentage of the position partially closed and use the pro-rated figure to compute the Closed P&L while the realized P&L will update in real time and accumulate until the respective direction of
position is fully closed.
If Trader C places a 0.5 qty long order, the 0.3 qty short position will be closed and a new 0.2 qty long position will be opened. The realized P&L will recalculate and show the realized P&L of the
0.2 qty long position.
Note: The feature will be supported on July 13, 2022. Hence, any realized P&L of the position that was opened before and yet to be closed after July 13, 2022 will not be captured and included. | {"url":"https://www.bybit.com/en/help-center/article/Profit-Loss-calculations-USDT-ContractUSDT_Perpetual_Contract","timestamp":"2024-11-05T00:44:16Z","content_type":"text/html","content_length":"196794","record_id":"<urn:uuid:1b8f1f67-52ff-428a-bdf2-35634c31861a>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00701.warc.gz"} |
Predicting JEE cut-off
The cut-off is a popular word during the admission season, especially in India. For us at Questionbang, the cut-off is something often asked by mock-set-plus users – Has my score is good enough to
qualify Joint Entrance Examination (JEE)? Hence, we decided to predict the cut-off for the next season and have it in result analysis.
Most of us have used basic regression analysis during our class 12th maths, e.g, time series and forecasting. In reality, the outcomes of such predictions are going to be dependent on various
factors. The use of simple regression may not be sufficient in such cases.
Let us consider our requirement – predicting cut-off marks for JEE. The table below shows cut-off scores for the last 6 years.
Table 1. Cut-off scores for the last 6 years.
Let us try a simple curve fitting approach (Figure 1); this is giving a prediction of 69 for the year 2019. However, we cannot relate these points to any reasoning. As we can see, the cut-off (Y) was
113 for the year 2013 and became 74 in 2018 (Table 1). Surely many factors influence the cut-off.
Figure 1. Scatter plot diagram for table 1.
Let us assume those cut-offs (Table 1) are a measure of competition and hence, are a function of the following variables:
• Number of seats available (),
• Difficulty level (),
• Number of applicants ().
How these individual variables influence the outcome is something to be predicted.
Choosing a regression method
What is the regression analysis?
In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships among variables. It includes many techniques for modeling and analyzing several
variables when the focus is on the relationship between a dependent variable and one or more independent variables [wiki].
There are many different types of regression techniques. They are mainly of two categories – linear regression and non-linear regression.
In our case, the cut-off score (predictor) is dependent on three independent variables (, , ) as discussed before. Hence, this is going to be a multiple linear regression scenario.
The table below (Table 2) is an extension of Table 1 (Cut-off scores for the last 6 years) to include the number of seats available, difficulty level and the number of applicants.
Table 2. Extended table to include other variables.
The difficulty level is a categorical variable having level 1 (high) or 0 (moderate).
About data
The data – cut-off scores, seat availability, number of applicants and difficulty level have been gathered from online news portals. The JEE format changed a few times during the past 20+ years. It
has a two-phase (Mains & Advanced) format since 2013. We will use data from 2013 to 2018.
Revisiting the basics of least square regression method – a single independent variable condition
Assume a single independent variable condition and set of values as below (Table 3). Let us call these observations.
Table 3. Observations.
– Independent variable,
– Actual dependent variable.
Following is an equation for simple linear regression:
Let us compute predictions , using the the above values (Table 3):
In generic form, the equation for predicted value will become:
Table 4. Observations and predictions.
– Predicted value.
Let us verify the accuracy of our prediction (Table 5),
Table 5. Observations, predictions and errors.
= error term.
As you can see, we subtracted the predictions () from the actual observations () to compute errors (). The next objective would be to refit the line so that the error () is minimized.
From (2) and (3),
Eq (4) is a squared error function; we need to find coefficients a & b to achieve minimum (zero) error. Take the partial derivative of eq (4) with respect to a and b:
From (5) and (6),
Computing JEE Cut-off – three independent variable condition
In our case, we have three independent variables – , , and coefficients – , , , . The regression equation becomes,
Intercept a is:
, and are means of respective variable values.
And the normal equations are,
Writing the above equations in matrix form and solving using Cramer’s rule:
PRACTICE FREE JEE TEST SERIES >>
Let us calculate the coefficients a, b, c and d using Microsoft Excel.
(Courtesy: https://onlinecourses.science.psu.edu/stat501/node/380/)
Figure 2: Excel showing observations.
Using data analysis toolkit in MS Excel:
a = 26.0402,
b = -0.00225,
c = -6.4645,
d = 0.000119.
After substituting the above coefficients, eq (9) becomes,
We can use the above eq (17) to compute the JEE cut-off.
We will assume the following values for the year 2019:
Number of seats () = 36500,
Difficulty level () = 1 or 0,
Number of applicants () = 11 lakh.
A) Using eq (10), high difficulty ()
B) Using eq (10), moderate difficulty ()
Cut-off score range: – .
The above prediction may not be accurate as it is based on a very limited set of data. It is to be noted that, this prediction is not relevant for the year 2019 (onwards), as the cut-off is going to
be in percentile not in the score.
Questionbang users can find these values in the mock-set-plus result analysis section.
We value your feedback and welcome any comments to help us serve you better. | {"url":"https://www.questionbang.com/blog/2018/12/19/predicting-jee-cut-off/","timestamp":"2024-11-10T00:02:19Z","content_type":"text/html","content_length":"288538","record_id":"<urn:uuid:a2e117a8-de31-4373-ab5c-55d4bccf1425>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00537.warc.gz"} |
Excel 2024: New LAMBDA Helper Functions
October 09, 2024 - by Bill Jelen
About a year after LAMBDA was introduced, Microsoft realized that we needed some helper functions. They gave us MAP, REDUCE, SCAN, MAKEARRAY, BYROW, BYCOL, and ISOMITTED.
• The MAP function runs a LAMBDA on each cell in an array or range and returns an identical-sized array or range.
• The REDUCE function runs a LAMBDA on each cell in an array or range but uses an accumulator variable to return one single answer.
• The SCAN function is sort of a combination of the two. It runs a LAMBDA on each cell of an array or range and returns an array the same size as the input range, showing the accumulator value
after each step.
• The MAKEARRAY function will create an array of any size that you specify. You provide a LAMBDA to calculate each cell in the new array.
• The BYROW function forces a LAMBDA to calculate on each row in a range instead of the entire range.
• The BYCOL function forces a LAMBDA to calculate on each column in a range.
• LAMBDAs now support optional arguments. You can test if an optional argument was skipped using the new ISOMITTED function.
Evaluating a LAMBDA for Each Cell in a Range or Ranges
The MAP function will perform a LAMBDA calculation for each cell in a range. In the example below, you are passing two ranges to MAP. Because there are two incoming ranges, your LAMBDA needs two
incoming variables A and B. Notice that each of the incoming arrays are 5 rows by 3 columns and the result from MAP is also 5 rows by 3 columns.
Note that MAP can accept multiple incoming arrays. This is not true for REDUCE, discussed next.
Accumulating a LAMBDA for Each Cell Using REDUCE
With REDUCE, a LAMBDA will be evaluated for each cell in an incoming array or range. On each pass through the logic, the result of the LAMBDA can be added to an accumulator variable. At the end of
the calculation, the formula returns the final value of the accumulator variable.
In this image, a REDUCE formula in B8 calculates the total bonus pool after several shifts. The initial value is set to 0. The incoming array is each cell in B2:D6. Inside the LAMBDA, the first two
arguments are the variable for the accumulator and for the cell from the incoming array. The last argument in the LAMBDA is the logic. Notice how the logic is adding the previous value of the
accumulator to some calculation from this cell of the incoming range.
Seeing the Results From Each Step of REDUCE with SCAN
The SCAN function performs the same calculation as REDUCE shown on the previous page. However, instead of returning a single value, it shows each intermediate value along the way.
In the image below, the Monday morning shift with sales of $1533 did not qualify for a bonus, so B8 shows 0. The Monday afternoon shift qualified for a $100 bonus, so C8 shows the total bonus earned
so far is $100. The Monday evening shift earned another $100 for the bonus pool, so the total bonus as of the end of Monday is $200 shown in D8. Notice how the $3100 in sales for Tuesday evening
kicked the bonus pool up from $200 to $800, with the $800 being shown in D9.
Evaluate a LAMBDA for Each Row or Column
Say that you asked for the MAX(A5:D11). You would get one single number that was the largest value in the range. Sometimes, though, it would be good to have MAX run on a column-by-column basis or a
row-by-row basis and return the results as a spillable array. The BYCOL and BYROW functions allow you to do this.
Note that the MAX in the above formulas is an Eta-Lambda introduced in November 2023. Before the Eta-Lambdas were introduced, you would use LAMBDA(A,Max(A)).
Make an Array of Any Size
The MAKEARRAY function lets you specify a number of rows and columns for the new array. The third argument is a LAMBDA function with three arguments. The first is the row number. The second is the
column number. The third argument is the logic to apply to this cell of the array.
Thanks to Chris Gross and his team in Redmond for these great new LAMBDA helper functions.
This article is an excerpt from MrExcel 2024 Igniting Excel
Title photo by Matthew Waring on Unsplash | {"url":"https://www.mrexcel.com/excel-tips/excel-2024-new-lambda-helper-functions/","timestamp":"2024-11-04T15:17:32Z","content_type":"text/html","content_length":"40890","record_id":"<urn:uuid:81cd0d41-06b0-4346-934c-e4b45d6afd26>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00653.warc.gz"} |
[Solved] The first national bank pays a 4% interes | SolutionInn
The first national bank pays a 4% interest rate compound continuously. The effective annual rate paid by
The first national bank pays a 4% interest rate compound continuously. The effective annual rate paid by the bank is __________.
a. 4.16%
b. 4.20%
c. 4.08%
d. 4.12%
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The CELL Function | SumProduct are experts in Excel Training: Financial Modelling, Strategic Data Modelling, Model Auditing, Planning & Strategy, Training Courses, Tips & Online Knowledgebase
A to Z of Excel Functions: The CELL Function
Welcome back to our regular A to Z of Excel Functions blog. Today we look at the CELL function.
The CELL function
With our A t Z of Excel Functions series, you could argue we have been trying to make a soft CELL (get it?). We are well and truly in the high C’s now and the puns do not get any better. Ah well…
The CELL function returns information about the formatting, location, or contents of a cell. For example, if you want to verify that a cell contains a numeric value instead of text before you perform
a calculation on it, you can use the following formula:
=IF(CELL("type", A1) = "v", A1 * 2, 0)
This formula calculates A1*2 only if cell A1 contains a numeric value, and returns 0 if A1 contains text or is blank. The CELL function employs the following syntax to operate:
CELL(info_type, [reference])
The CELL function has the following arguments:
• Info_type: this is required. This is a text value that specifies what type of cell information you want to return. The following list shows the possible values of the Info_type argument and the
corresponding results:
• reference: this is optional. This is the cell that you want information about. If this argument is omitted, the information specified in the Info_type argument is returned for the last cell that
was changed. If the reference argument is a range of cells, the CELL function returns the information for only the upper left cell of the range.
CELL Format Codes
The following list describes the text values that the CELL function returns when the Info_type argument is "format" and the reference argument is a cell that is formatted with a built-in number
Please see my example below:
There’s lots you can do with CELL. You may recall in one of our Thought articles we used it to automate the file name using the formula
You can read up on how this formula works by visiting the relevant article here. | {"url":"https://www.sumproduct.com/blog/article/a-to-z-of-excel-functions/the-cell-function","timestamp":"2024-11-10T14:06:14Z","content_type":"text/html","content_length":"25935","record_id":"<urn:uuid:7281d032-1a6a-4f5e-aa29-d2231bd351c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00607.warc.gz"} |
American Mathematical Society
$T$-faithful subcategories and localization
HTML articles powered by AMS MathViewer
Trans. Amer. Math. Soc. 195 (1974), 61-79
DOI: https://doi.org/10.1090/S0002-9947-1974-0364322-4
PDF | Request permission
For any additive functor from a category of modules into an abelian category there is a largest Giraud subcategory for which the functor acts faithfully on homomorphisms into the subcategory. It is
the largest Giraud subcategory into which the functor reflects exact sequences, and under certain conditions it is just the largest Giraud subcategory on which the functor acts faithfully. If the
functor is exact and has a right adjoint, then the subcategory is equivalent to the quotient category determined by the kernel of the functor. In certain cases, the construction can be applied to a
Morita context in order to obtain a recent theorem of Mueller. Similarly, the functor defines a certain reflective subcategory and an associated radical, which is a torsion radical in case the
functor preserves monomorphisms. Certain results concerning this radical, when defined by an adjoint functor, can be applied to obtain two theorems of Morita on balanced modules. References
• Gorô Azumaya, Some properties of $\textrm {TTF}$-classes, Proceedings of the Conference on Orders, Group Rings and Related Topics (Ohio State Univ., Columbus, Ohio, 1972) Lecture Notes in Math.,
Vol. 353, Springer, Berlin., 1973, pp. 72–83. MR 0338073
• John A. Beachy, Generating and cogenerating structures, Trans. Amer. Math. Soc. 158 (1971), 75–92. MR 288160, DOI 10.1090/S0002-9947-1971-0288160-3
• John A. Beachy, Cotorsion radicals and projective modules, Bull. Austral. Math. Soc. 5 (1971), 241–253. MR 292879, DOI 10.1017/S0004972700047122
• Kent R. Fuller, Density and equivalence, J. Algebra 29 (1974), 528–550. MR 374192, DOI 10.1016/0021-8693(74)90088-X
• A. G. Heinicke, Triples and localizations, Canad. Math. Bull. 14 (1971), 333–339. MR 318229, DOI 10.4153/CMB-1971-061-2
• J. P. Jans, Some aspects of torsion, Pacific J. Math. 15 (1965), 1249–1259. MR 191936
• Joachim Lambek, Torsion theories, additive semantics, and rings of quotients, Lecture Notes in Mathematics, Vol. 177, Springer-Verlag, Berlin-New York, 1971. With an appendix by H. H. Storrer on
torsion theories and dominant dimensions. MR 0284459
• Joachim Lambek, Localization and completion, J. Pure Appl. Algebra 2 (1972), 343–370. MR 320047, DOI 10.1016/0022-4049(72)90011-4
• Barry Mitchell, Theory of categories, Pure and Applied Mathematics, Vol. XVII, Academic Press, New York-London, 1965. MR 0202787
• Kiiti Morita, Localizations in categories of modules. I, Math. Z. 114 (1970), 121–144. MR 263858, DOI 10.1007/BF01110321
• Kiiti Morita, Flat modules, injective modules and quotient rings, Math. Z. 120 (1971), 25–40. MR 286833, DOI 10.1007/BF01109715 Bruno J. Mueller, The quotient category of a Morita context, (1972)
• Bo Stenström, Rings and modules of quotients, Lecture Notes in Mathematics, Vol. 237, Springer-Verlag, Berlin-New York, 1971. MR 0325663
• R. G. Swan, Algebraic $K$-theory, Lecture Notes in Mathematics, No. 76, Springer-Verlag, Berlin-New York, 1968. MR 0245634
• Hiroyuki Tachikawa, On splitting of module categories, Math. Z. 111 (1969), 145–150. MR 246923, DOI 10.1007/BF01111195
Similar Articles
• Retrieve articles in Transactions of the American Mathematical Society with MSC: 16A08
• Retrieve articles in all journals with MSC: 16A08
Bibliographic Information
• © Copyright 1974 American Mathematical Society
• Journal: Trans. Amer. Math. Soc. 195 (1974), 61-79
• MSC: Primary 16A08
• DOI: https://doi.org/10.1090/S0002-9947-1974-0364322-4
• MathSciNet review: 0364322 | {"url":"https://www.ams.org/journals/tran/1974-195-00/S0002-9947-1974-0364322-4/?active=current","timestamp":"2024-11-03T16:47:38Z","content_type":"text/html","content_length":"64221","record_id":"<urn:uuid:4ee8f922-5c99-4d95-8d3d-67d7e5eb4744>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00672.warc.gz"} |
The mp-expectation command calculates the expectation value of an operator with finite support. The wavefunction can be finite, infinite, or infinite-boundary.
mp-expectation [options] <psi> <operator> [psi2]
calculates {$\bigbraket{\mbox{psi}}{\mbox{operator}}{\mbox{psi2}}$}
The operator must be defined over a finite range of lattice sites, it is not possible to use mp-iexpectation to evaluate TriangularMPO's or ProductMPO's as these have infinite support. For infinite
wavefunctions, it is not possible to calculate mixed expectation values where psi2 is different to psi1.
show help message
-r, --real
display the real part of the result
-i, --imag
display the imaginary part of the result
If none of --real or --imag is specified, the default is to show both the real and imaginary parts.
1. local magnetization at site 5
mp-expectation psi lat:"Sz(5)"
2. correlations of an SU(2) spin chain
mp-expectation psi lat:"inner(S(0),S(1))"
See also
• To calculate expectation values of TriangularMPO's, see MpIMoments
• To calculate expectation values of ProductMPO's, see MpIOverlap | {"url":"https://mptoolkit.qusim.net/Tools/MpExpectation","timestamp":"2024-11-05T14:14:03Z","content_type":"application/xhtml+xml","content_length":"11756","record_id":"<urn:uuid:f0115da6-0add-402e-b4f5-9464ed8c00fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00401.warc.gz"} |
The Problem of the Cosmological Constant
Astoundingly, Einstein recanted his cosmological constant Λ when Hubble’s 1929 red shift calculations showed that the cosmos is not a static biblical “firmament” as was the prevailing proto-religious
ideology in 1916, but was rather dynamic and expanding, and therefore Einstein’s Λ was no longer required to contort the lovely calculus of the GR field equations into Fred Hoyle’s static, spatially
closed universe.
In 1922 Alexander Friedman—mentor to Sheldon Glashow—derived the Friedman Equation indicating that Einstein’s original GR field equations, before his insertion of Λ indicated that the universe was
expanding, thus a contrived Λ theorem that precluded this expansion was wrong-headed. Even before Einstein’s 1931 formal renunciation of Λ, both relativistic cosmologist Arthur Eddington (The
Expanding Universe, 1933), and Georges Lemaitre (Ann. Soc. Sci. Brux. 47:49 1927) retained it. Both agreed with Friedman that Einstein’s static universe (Λ>0) is unstable, and untenable and that Λ is
indeed the necessary basis for a consistent cosmology.
Later Einstein referred to his insertion of the Λ term into the field equations as his “greatest blunder”. Why? Had he trusted the geometry of his wondrous field equations he would have predicted the
expansion of the universe, and dark energy to boot, 13 years before Hubble’s great 1929 discovery. As Steven Weinberg might have said, he “did not take his mathematics seriously enough”.
Just so, the same could be said for the four marvelous equations of Maxwell that unified electricity and magnetism; and later of Dirac’s mathematical masterpiece that unified the quantum theory with
Special Relativity (discovering antimatter in the process) to give us Relativistic Quantum Field Theory, the very ground of Feynman’s QED. It was perhaps Dirac’s cognitive reticence to trust his
equations regarding antimatter that permitted later genius to steal his theoretical thunder.
Alas for the great mind that was Einstein, his hastily added, then retracted cosmological constant Λ, or something very like it is now back in the cosmic game as a ploy to make sense of “dark
energy”. Dark energy is necessary to explain the recent discovery that the space of the universe is not only expanding, but accelerating exponentially. Platonic irony?
So it is, Einstein’s “greatest blunder”, the cosmological constant Λ has again arisen, phoenix-like, into the cosmological chess game. In 1980 Λ was proffered as the physical cause of the repulsive
force of that great expansion—a trillionth of a second, give or take a trillionth, after a Big Bang singularity—that we now know and love as chaotic “inflation”. Again, the 1998 discovery of the
repulsive dark energy that is hypothesized as the physical cause of the exponentially accelerating expanding cosmos has, as well, been attributed to Λ.
As we have seen, Einstein’s GR tells us that gravity is the curvature of spacetime. This curvature of space is the same everywhere, and the rate at which it expands throughout the expanding universe
indicates its energy density. What is the topology of this curvature? We have three options. It may be negative, like a saddle, positive, like a sphere, or zero, flat. The current best guess based
upon interpretations of the cosmic microwave background radiation (CMB)—the primordial energy remnants of the Big Bang—suggest that the actual curvature is approximately zero. The energy density of
the universe then, the energy present in any volume of space, on the accord of GR, is a function of this curvature of space and its rate of expansion, probably infinite, ending in the high entropy
“heat death” that is the frosty “Big Chill”.
So for Einstein’s GR the rate of expansion of the universe is relative to its overall energy density. It was the 1998 data from type Ia supernovae explosions that revealed this surprising
acceleration of space, along with all of its galactic contents; which by the by, rescues us from the compactified fate of a contracting universal “Big Crunch” at the end of what may seem to be
“Empty space”—the vacuum of space—contains a small bit of fundamental energy. This tiny energy value is the cosmological constant Λ, usually known as the vacuum energy. As energy and matter are
related by E=mc², GR predicts that Λ will have gravitational effects. Λ has a negative pressure that is equivalent to its energy density resulting in an accelerated expansion of the cosmos. Hence,
the current Standard Model cosmological theory is known as the Lambda-CDM Model (cold dark matter) where lambda or Λ is the basal form of dark energy, and for now anyway, includes as its primary
theorem Einstein’s cosmological constant Λ, a scalar field which comprises the energy density of a flat universe as a vacuum energy. Such current theory is supported by temperature anisotropy data
from WMAP, and SDSS surveys of the redshift of distant galaxies (2002 to 2007).
Alternative explanations of mysterious dark energy include 1) several theories of “modified gravity” wherein Einstein’s GR gravity (the equivalence principal) is tweaked; and 2) the quintessence
field. Quintessence is a hypothetical dynamical field, vis-á-vis the constant vacuum energy field, of a universal gradually changing energy density evolution. Thus quintessence differs from Λ for it
is not constant but dynamic in space and time.
Non-baryonic dark matter by hypothesis is about 70 percent of the mass-energy density (remember E=mc²) of the cosmos. Dark matter—a neutral, uncharged non-interacting, or weakly interacting massive
particle (wimp), not yet known to humanity—constitutes 25 percent; and fully 5 percent is baryonic (good old protons and neutrons) ordinary matter. One might well refer to such a Panglossian
explanation of our wondrous cosmos as the Substandard Model of particles and forces, but that would be disrespectful.
So what, in heaven and earth, is the diabolical “cosmological constant problem”, first described by Steven Weinberg in 1989—later exclaimed by Leonard Susskind to be “the worst prediction ever….the
mother of all physics problems”?
As seen above, the cosmological constant Λ, was introduced into the GR field equations by Einstein in 1917 (and later retracted) in order to defend his belief that the universe is static, while we
now know that it is not only expanding, but accelerating exponentially. Λ is generally viewed as the zero point energy density of the quantum vacuum of space, the energy of space empty of all but
virtual matter particles. This density was assumed to be zero (Λ=0).
The cosmological constant of 1998 is considered by cosmologists to be the current best physical explanation for dark energy, the repulsive force that is the cause for the expanding, accelerating
universe. The bad news: theory seems to demonstrate that the value of this zero point energy is 120 orders of magnitude greater than what is actually observed! Such a value would inflate the universe
at a rate that would preclude the formation of galaxies, and thus conscious observers to ponder the equation. Therefore this calculation must be incorrect. Hence the cosmological constant problem.
This is indeed a physics sticky wicket.
What to do? Must we wait for a consistent quantum gravity theory? This requires profound changes to both of the “perfect theories” that are General Relativity and Relativistic Quantum Field Theory.
We are led therefore to the multiverse.
Surely, a perfectly just and rational creator God, desirous of people to praise Him, would have no choice than to fine-tune a cosmological constant with a negative value, so that we might have an
earth upon which to evolve and return to the light. | {"url":"https://davidpaulboaz.org/philosophy-of-physics-and-cosmology/the-problem-of-the-cosmological-constant/","timestamp":"2024-11-03T20:09:32Z","content_type":"text/html","content_length":"39454","record_id":"<urn:uuid:a3aae340-4f95-4d34-a162-0e72db00f88f>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00172.warc.gz"} |
- 99Worksheets
5th Grade Math Worksheets
All You Need to Know About 5th Grade Math Worksheets
If you are teaching a bunch of 5th graders about math, then you certainly need some 5th grade math worksheets that will help you do it properly. Math is one of the most crucial subjects that you can
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Sadly, a lot of children do not like math. Some even say that they hate the subject. Well, hate is a strong word to use. But in reality, a lot of children feel like that. This is exactly the reason
why teachers have to step up their game when it comes to teaching math. But how are they supposed to do that? Is there a magical way to be able to overcome this challenging task?
Teaching Math with the Help of Worksheets
This might be your lucky day because we have the best answer for you. And that answer is the only math worksheets. If you are a teacher, then you are probably familiar with worksheets. They are
perfect to use when it comes to testing your students’ abilities on a subject, including, yours truly, math.
Think about it. Using length word problem worksheets to teach kids how to do math is way easier than trying to explain theories in math. It is like what they say, you will be able to learn something
better if you are doing it. Also, practice makes perfect, which is exactly the purpose of using math worksheets.
What Math Problems Do 5th Graders Have to Learn?
There are so many types of math worksheets that you can use out there. There are exponent worksheets, price value worksheets, ordering numbers worksheets, and so many more. However, not all of these
worksheets are appropriate for 5th graders. Do you want to give out simple addition worksheets to 5th graders? No way. That would be way too easy for them. That is why this next section is important
for you to understand. Here are some math problems that 5th graders need to master.
Multiple-Digit Whole Numbers
The first problem that 5th graders have to know about is to add, subtract, multiply, and divide multi-digit whole numbers. That is why one of the best 5th grade math worksheets are some good old
multiply numbers near 100 worksheets.
5th graders also have to be able to divide up to four-digit numbers by two-digit numbers, which follows the divisibility rules in math. Once they can master these math skills, then you can start
giving them more advanced worksheets.
Place Value
The second math problem that you have to teach 5th graders is place value. They have to understand that in multiple digit numbers, a digit will represent one-tenth of what the digit on the left of
that digit represents. Just like you can teach prime numbers using prime numbers worksheets, you can teach 5th graders to learn place value by using place value worksheets.
Another math problem that your 5th-grade students need to be able to solve is various fraction problems. Your students have to be able to multiply and divide fractions effortlessly. Some simple
completing whole numbers worksheets can be used for fractions as well. Trust us when we say your students will master fractions in no time with the help of some great worksheets.
Conversion Problems
The next math problem that we would like to tell you all about is the conversion problems. Yes, teaching conversion problems to children is not a walk in the park. But you have to know how to do it
properly if you want your students to thrive.
5th graders need to be able to solve multiple steps word problems by using conversions of measurement units. You can see an easy example of this type of worksheet by looking at some convert lengths
worksheets that will help you immensely when you are teaching math to children. Solving conversion problems will be so easy with the help of these amazing worksheets.
What Kind of Worksheets that are Appropriate for 5th Grade Students?
Now that you know some of the many math problems that 5th graders have to master, let’s talk about some math worksheets that you can use for 5th graders. Here are a few examples of the helpful
worksheets that make math feel like a breeze.
Capacity Word Problem Worksheet
Are you trying to help your 5th-grade students to solve capacity word problems? Then some useful capacity word problems worksheets are the thing for you. This worksheet will help your students to put
their thinking caps on and challenge their abilities.
Missing Factor Worksheet
Although some missing factor problems worksheets sound like simple worksheets that are too easy for 5th graders, that is not always the case. You can spice up this kind of worksheet by adding using
fractions, which is something that 5th graders have to master. Ask your 5th-grade students to either multiply or divide the fractions and fill in the missing factors on the worksheet. Solving math
problems has never sounded so fun.
Common Denominator Worksheet
The last type of worksheet that we are going to tell you about is the common denominator worksheet. Finding a common denominator is one of the math problems that 5th-grade students have to be able to
master. That is why you should leave the classify angles worksheets to lower grade students and give your 5th graders some challenging and fun common denominator worksheets.
Final Thoughts
Teaching math to children is not an easy task to do. This is especially true if you do not even know what 5th graders are supposed to know when it comes to math. However, once you have gotten the
idea of what they are supposed to master, all you need to do is to find some great math worksheets that can help them practice their skills. These 5th grade math worksheets will turn math into
everyone’s favorite subject. | {"url":"https://www.99worksheets.com/5th-grade/math-5th-grade/","timestamp":"2024-11-05T19:57:39Z","content_type":"text/html","content_length":"90405","record_id":"<urn:uuid:0271deb2-6c50-453f-8cb5-a8244f9d2983>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00773.warc.gz"} |
Sensitivity function at specified point using slLinearizer or slTuner interface
linsys = getSensitivity(s,pt) returns the sensitivity function at the specified analysis point for the model associated with the slLinearizer or slTuner interface, s.
The software enforces all the permanent openings specified for s when it calculates linsys. If you configured either s.Parameters, or s.OperatingPoints, or both, getSensitivity performs multiple
linearizations and returns an array of sensitivity functions.
linsys = getSensitivity(s,pt,temp_opening) considers additional, temporary, openings at the point specified by temp_opening. Use an opening, for example, to calculate the sensitivity function of an
inner loop, with the outer loop open.
linsys = getSensitivity(___,mdl_index) returns a subset of the batch linearization results. mdl_index specifies the index of the linearizations of interest, in addition to any of the input arguments
in previous syntaxes.
Use this syntax for efficient linearization, when you want to obtain the sensitivity function for only a subset of the batch linearization results.
[linsys,info] = getSensitivity(___) returns additional linearization information.
Sensitivity Function at Analysis Point
For the ex_scd_simple_fdbk model, obtain the sensitivity at the plant input, u.
Open the ex_scd_simple_fdbk model.
mdl = 'ex_scd_simple_fdbk';
In this model:
Create an slLinearizer interface for the model.
sllin = slLinearizer(mdl);
To obtain the sensitivity at the plant input, u, add u as an analysis point to sllin.
Obtain the sensitivity at the plant input, u.
sys = getSensitivity(sllin,'u');
ans =
From input "u" to output "u":
s + 5
s + 8
Continuous-time transfer function.
The software uses a linearization input, du, and linearization output u to compute sys.
sys is the transfer function from du to u, which is equal to .
Specify Temporary Loop Opening for Sensitivity Function Calculation
For the scdcascade model, obtain the inner-loop sensitivity at the output of G2, with the outer loop open.
Open the scdcascade model.
mdl = 'scdcascade';
Create an slLinearizer interface for the model.
sllin = slLinearizer(mdl);
To calculate the sensitivity at the output of G2, use the y2 signal as the analysis point. To eliminate the effects of the outer loop, break the outer loop at y1m. Add both these points to sllin.
Obtain the sensitivity at y2 with the outer loop open.
sys = getSensitivity(sllin,'y2','y1m');
Here, 'y1m', the third input argument, specifies a temporary opening of the outer loop.
Obtain Sensitivity Function for Specific Parameter Combination
Suppose you batch linearize the scdcascade model for multiple transfer functions. For most linearizations, you vary the proportional (Kp2) and integral gain (Ki2) of the C2 controller in the 10%
range. For this example, obtain the sensitivity at the output of G2, with the outer loop open, for the maximum values of Kp2 and Ki2.
Open the scdcascade model.
mdl = 'scdcascade';
Create an slLinearizer interface for the model.
sllin = slLinearizer(mdl);
Vary the proportional (Kp2) and integral gain (Ki2) of the C2 controller in the 10% range.
Kp2_range = linspace(0.9*Kp2,1.1*Kp2,3);
Ki2_range = linspace(0.9*Ki2,1.1*Ki2,5);
[Kp2_grid,Ki2_grid] = ndgrid(Kp2_range,Ki2_range);
params(1).Name = 'Kp2';
params(1).Value = Kp2_grid;
params(2).Name = 'Ki2';
params(2).Value = Ki2_grid;
sllin.Parameters = params;
To calculate the sensitivity at the output of G2, use the y2 signal as the analysis point. To eliminate the effects of the outer loop, break the outer loop at y1m. Add both these points to sllin as
analysis points.
Determine the index for the maximum values of Ki2 and Kp2.
mdl_index = params(1).Value == max(Kp2_range) & params(2).Value == max(Ki2_range);
Obtain the sensitivity at the output of G2 for the specified parameter combination.
sys = getSensitivity(sllin,'y2','y1m',mdl_index);
Obtain Offsets from Sensitivity Function
Open Simulink® model.
mdl = 'watertank';
Create a linearization option set, and set the StoreOffsets option.
opt = linearizeOptions('StoreOffsets',true);
Create slLinearizer interface.
sllin = slLinearizer(mdl,opt);
Add an analysis point at the tank output port.
addPoint(sllin,'watertank/Water-Tank System');
Calculate the sensitivity function at the analysis point, and obtain the corresponding linearization offsets.
[sys,info] = getSensitivity(sllin,'watertank/Water-Tank System');
View offsets.
ans =
struct with fields:
x: [2x1 double]
dx: [2x1 double]
u: 1
y: 1
StateName: {2x1 cell}
InputName: {'watertank/Water-Tank System'}
OutputName: {'watertank/Water-Tank System'}
Ts: 0
Input Arguments
pt — Analysis point signal name
character vector | string | cell array of character vectors | string array
Analysis point signal name, specified as:
• Character vector or string — Analysis point signal name.
To determine the signal name associated with an analysis point, type s. The software displays the contents of s in the MATLAB^® command window, including the analysis point signal names, block
names, and port numbers. Suppose that an analysis point does not have a signal name, but only a block name and port number. You can specify pt as the block name. To use a point not in the list of
analysis points for s, first add the point using addPoint.
You can specify pt as a uniquely matching portion of the full signal name or block name. Suppose that the full signal name of an analysis point is 'LoadTorque'. You can specify pt as 'Torque' as
long as 'Torque' is not a portion of the signal name for any other analysis point of s.
For example, pt = 'y1m'.
• Cell array of character vectors or string array — Specifies multiple analysis point names. For example, pt = {'y1m','y2m'}.
To calculate linsys, the software adds a linearization input, followed by a linearization output at pt.
Consider the following model:
Specify pt as 'u':
The software computes linsys as the transfer function from du to u.
If you specify pt as multiple signals, for example pt = {'u','y'}, the software adds a linearization input, followed by a linearization output at each point.
du and dy are linearization inputs, and, u and y are linearization outputs. The software computes linsys as a MIMO transfer function with a transfer function from each linearization input to each
linearization output.
Output Arguments
linsys — Sensitivity function
state-space model
Sensitivity function, returned as described in the following:
• If you did not configure s.Parameters and s.OperatingPoints, the software calculates linsys using the default model parameter values. The software uses the model initial conditions as the
linearization operating point. linsys is returned as a state-space model.
• If you configured s.Parameters only, the software computes a linearization for each parameter grid point. linsys is returned as a state-space model array of the same size as the parameter grid.
• If you configured s.OperatingPoints only, the software computes a linearization for each specified operating point. linsys is returned as a state-space model array of the same size as
• If you configured s.Parameters and specified s.OperatingPoints as a single operating point, the software computes a linearization for each parameter grid point. The software uses the specified
operating point as the linearization operating point. linsys is returned as a state-space model array of the same size as the parameter grid.
• If you configured s.Parameters and specified s.OperatingPoints as multiple operating point objects, the software computes a linearization for each parameter grid point. The software requires that
s.OperatingPoints is the same size as the parameter grid specified by s.Parameters. The software computes each linearization using corresponding operating points and parameter grid points. linsys
is returned as a state-space model array of the same size as the parameter grid.
• If you configured s.Parameters and specified s.OperatingPoints as multiple simulation snapshot times, the software simulates and linearizes the model for each snapshot time and parameter grid
point combination. Suppose that you specify a parameter grid of size p and N snapshot times. linsys is returned as a state-space model array of size N-by-p.
For most models, linsys is returned as an ss object or an array of ss objects. However, if your model contains one of the following blocks in the linearization path defined by pt, then linsys returns
the specified type of state-space model.
Block linsys Type
Block with a substitution specified as a genss object or tunable model object genss
Block with a substitution specified as an uncertain model, such as uss uss (Robust Control Toolbox)
Sparse Second Order block mechss
Descriptor State-Space block configured to linearize to a sparse model sparss
More About
Sensitivity Function
The sensitivity function, also referred to simply as sensitivity, measures how sensitive a signal is to an added disturbance. Sensitivity is a closed-loop measure. Feedback reduces the sensitivity in
the frequency band where the open-loop gain is greater than 1.
To compute the sensitivity at an analysis point, x, the software injects a disturbance signal, dx, at the point. Then, the software computes the transfer function from dx to x, which is equal to the
sensitivity function at x.
Analysis Point in Simulink Model How getSensitivity Interprets Analysis Point Sensitivity Function
Transfer function from dx to x
For example, consider the following model where you compute the sensitivity function at u:
Here, the software injects a disturbance signal (du) at u. The sensitivity at u, S[u], is the transfer function from du to u. The software calculates S[u] as follows:
$\begin{array}{l}u=du-KGu\\ \to \left(I+KG\right)u=du\\ \to u={\left(}^{I}du\\ \therefore {S}_{u}={\left(}^{I}\end{array}$
Here, I is an identity matrix of the same size as KG.
Similarly, to compute the sensitivity at y, the software injects a disturbance signal (dy) at y. The software computes the sensitivity function as the transfer function from dy to y. This transfer
function is equal to (I+GK)^-1, where I is an identity matrix of the same size as GK.
The software does not modify the Simulink model when it computes the sensitivity transfer function.
Analysis Point
Analysis points, used by the slLinearizer and slTuner interfaces, identify locations within a model that are relevant for linear analysis and control system tuning. You use analysis points as inputs
to the linearization commands, such as getIOTransfer, getLoopTransfer, getSensitivity, and getCompSensitivity. As inputs to the linearization commands, analysis points can specify any open-loop or
closed-loop transfer function in a model. You can also use analysis points to specify design requirements when tuning control systems using commands such as systune.
Location refers to a specific block output port within a model or to a bus element in such an output port. For convenience, you can use the name of the signal that originates from this port to refer
to an analysis point.
You can add analysis points to an slLinearizer or slTuner interface, s, when you create the interface. For example:
s = slLinearizer('scdcascade',{'u1','y1'});
Alternatively, you can use the addPoint command.
To view all the analysis points of s, type s at the command prompt to display the interface contents. For each analysis point of s, the display includes the block name and port number and the name of
the signal that originates at this point. You can also programmatically obtain a list of all the analysis points using getPoints.
For more information about how you can use analysis points, see Mark Signals of Interest for Control System Analysis and Design and Mark Signals of Interest for Batch Linearization.
Permanent Openings
Permanent openings, used by the slLinearizer and slTuner interfaces, identify locations within a model where the software breaks the signal flow. The software enforces these openings for
linearization and tuning. Use permanent openings to isolate a specific model component. Suppose that you have a large-scale model capturing aircraft dynamics and you want to perform linear analysis
on the airframe only. You can use permanent openings to exclude all other components of the model. Another example is when you have cascaded loops within your model and you want to analyze a specific
Location refers to a specific block output port within a model. For convenience, you can use the name of the signal that originates from this port to refer to an opening.
You can add permanent openings to an slLinearizer or slTuner interface, s, when you create the interface or by using the addOpening command. To remove a location from the list of permanent openings,
use the removeOpening command.
To view all the openings of s, type s at the command prompt to display the interface contents. For each permanent opening of s, the display includes the block name and port number and the name of the
signal that originates at this location. You can also programmatically obtain a list of all the permanent loop openings using getOpenings.
Version History
Introduced in R2013b
R2016b: Compute operating point offsets for model inputs, outputs, states, and state derivatives during linearization
You can compute operating point offsets for model inputs, outputs, states, and state derivatives when linearizing Simulink models. Thee offsets streamline the creation of linear parameter-varying
(LPV) systems.
To obtain operating point offsets, first create a linearizeOptions or slTunerOptions object and set the StoreOffsets option to true. Then, create an slLinearizer or slTuner interface for the model.
You can extract the offsets from the info output argument of getSensitivity and convert them into the required format for the LPV System block using the getOffsetsForLPV function. | {"url":"https://nl.mathworks.com/help/slcontrol/ug/sllinearizer.getsensitivity.html","timestamp":"2024-11-04T07:09:29Z","content_type":"text/html","content_length":"142513","record_id":"<urn:uuid:d62a68eb-d5c5-41de-b45b-887f8d97592f>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00596.warc.gz"} |
what is critical speed of ball mill
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WEBOct 19, 2015 · Power draw is related directly to mill length, and, empirically to the diameter to the power Theoretically, this exponent should be (Bond, 1961). Power draw is directly related
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WEBBall mill, most of the size reduction is done by impact. Critical Speed of a Ball Mill (ƞc): 𝜂𝑐 = 1 2𝜋 ∗ √𝑔 √𝑅−𝑟 (1) Where, ηc is a critical rotational speed, 'R' is radius of the ball mill
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WEBJul 1, 2002 · This critical speed would, therefore, be a key condition in milling for designing suitable and optimum mechanical milling performance. Introduction A planetary ball mill is known
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WEBWhat is the critical rotation speed for a ball mill of m diameter charged with 80 mm diameter balls? a. rps b. rpm OC rpm d. rps e. rpm
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WEBMar 20, 2020 · Ball mill： Ball mills are the most widely used one. Rod mill： The rod mill has the highest efficiency when the feed size is <30mm and the discharge size is about 3mm with
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WEBMay 16, 2024 · Nc is the critical speed of the mill (in revolutions per minute, rpm). g is the acceleration due to gravity (approximately m/s²). R is the radius of the mill (in meters). The
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WEBApr 13, 2023 · Speed of disc and grinding media also affect the milling energy which leads to variation in the time required for amorphization. For the maximum collision energy, speed of the
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WEBStepbystep explanation. At critical speed particle is just about to leave the ball mill surface . So centrifugal force try that particle will in contact with surface and gravity force try that
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WEBJun 1, 2012 · u 2 − fresh ore feed rate, u 3 − mill critical speed fraction, u 4 − sump dilution water . ... Ball mills can grind a wide range of materials, including metals, ceramics, and
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WEBWhat is the critical rotation speed for a ball mill of m diameter charged with 70 mm diameter balls? Q1
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WEBJan 5, 2016 · There could be a loss in power with rubber particularly if the mill speed is faster than about 72% of critical speed, and the ball size is larger than 75 mm. Because of the
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WEBOct 27, 2023 · For a ball mill to operate, critical speed needs to be achieved. The enclosed ball begins to rotate along the inner walls of the ball mill. If it fails to reach critical speed,
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WEBApr 28, 2017 · N1 = critical speed of mill in rev. per min. Vb = relative velocity of particle at od, ft. per sec. w = weight of portion of charge, lb. ... The proper operating speed of a
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WEBWhether you're a seasoned engineer or just starting out in the field, understanding the critical speed of a ball mill is essential knowledge. So, let's dive in and explore what exactly a ball
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WEBJan 1, 2016 · abrasive and impact wear due to their large. (75 – 100 mm) dia meters. Ball mill balls. experience a greater number of impacts, but at. lower magnitude than SAG mill balls, due t
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WEBJan 1, 2022 · The filling levels M* was taken as 30%, 40% and 50% of full mill and the mill speed N* was selected as, and of the critical speed. The critical speed is the speed at which a mill
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WEBAnalysis of ball and pulp flow in ball mills indies that three factors may become critical with increasing mill diameters: ball size, fraction critical speed, and average pulp flow velocities.
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WEBJan 28, 2024 · 1. Loading and Rotation: Loading the material to be ground into the cylindrical shell, the mill is then rotated at a critical speed, where the centrifugal force equals the
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Mastering Formulas In Excel: What Is The Formula For Finding
Mastering formulas in Excel is a crucial skill for anyone looking to excel in data analysis and reporting. These formulas allow you to perform complex calculations, manipulate data, and automate
processes, saving you valuable time and effort. In this blog post, we will explore one of the fundamental formulas in Excel and discuss its application in real-world scenarios.
Here's what you can expect from this post:
• A brief overview of the importance of mastering formulas in Excel
• An in-depth look at a specific formula and its functionality
• Practical examples to help you understand how to apply the formula in your own work
Key Takeaways
• Mastering formulas in Excel is crucial for data analysis and reporting.
• Formulas in Excel allow for complex calculations, data manipulation, and process automation.
• This blog post will cover basic arithmetic formulas, commonly used Excel functions, advanced formulas for data analysis, percentage and standard deviation formulas.
• Practical examples will be provided to help understand how to apply different formulas in real-world scenarios.
• It is important to practice and explore more complex formulas to further enhance Excel skills.
Understanding Basic Formulas
When it comes to mastering formulas in Excel, it's essential to start with the basics. Understanding basic arithmetic formulas is the foundation for more complex functions and can greatly improve
your efficiency and accuracy when working with data.
A. Introduction to basic arithmetic formulas
Basic arithmetic formulas include addition, subtraction, multiplication, and division. These formulas are the building blocks of more advanced calculations in Excel and are widely used in everyday
spreadsheet tasks.
B. Example of how to use basic formulas in Excel
Let's take a simple example of using basic formulas in Excel. Suppose you have a column of numbers representing sales figures, and you want to calculate the total sales for the month. In this case,
you can use the SUM formula to add up all the numbers in the column and get the total sales value.
Commonly Used Excel Functions
When it comes to mastering formulas in Excel, understanding the commonly used functions is essential. In this chapter, we will delve into the explanation of SUM, AVERAGE, and MAX functions, and
provide a demonstration of how to apply these functions in Excel.
Explanation of SUM, AVERAGE, and MAX functions
SUM function: The SUM function in Excel is used to add up the values in a range of cells. It is a versatile function that can be used to quickly calculate the total of a series of numbers.
AVERAGE function: The AVERAGE function calculates the average of a group of numbers. This is useful for obtaining the mean value of a dataset without having to manually add up all the numbers and
divide by the count.
MAX function: The MAX function returns the largest value in a set of numbers. This can be helpful when you need to identify the highest value in a range of cells.
Demonstration of how to apply these functions in Excel
Let's take a look at how these functions can be applied in Excel using a simple example:
• First, input a series of numbers into a column in an Excel spreadsheet.
• Next, select a cell where you want the total, average, or maximum value to appear.
• For the SUM function, type "=SUM(" followed by the range of cells you want to add up, and then close the parentheses.
• For the AVERAGE function, type "=AVERAGE(" followed by the range of cells, and close the parentheses.
• For the MAX function, type "=MAX(" followed by the range of cells, and close the parentheses.
• Press Enter, and the result will be calculated and displayed in the selected cell.
By understanding and applying these functions, you can streamline your data analysis and make more efficient use of Excel's capabilities.
Advanced Formulas for Data Analysis
When it comes to analyzing data in Excel, mastering advanced formulas such as VLOOKUP and IF functions can significantly enhance your efficiency and accuracy. In this chapter, we will delve into the
use of these powerful tools for data analysis.
Introduction to VLOOKUP and IF functions
VLOOKUP: VLOOKUP is a versatile function that allows you to search for a value in the first column of a table and return a value in the same row from another column. This function is particularly
useful for looking up data from a large dataset and retrieving specific information.
IF function: The IF function allows you to perform a logical test and return one value if the test is true, and another value if the test is false. It is commonly used to make decisions or perform
calculations based on certain conditions.
Walkthrough of how to use VLOOKUP and IF functions for data analysis
• Using VLOOKUP: To use the VLOOKUP function for data analysis, you first need to specify the lookup value, table array, column index number, and range lookup. This will enable you to easily
retrieve relevant data from a table based on a specific criterion.
• Applying IF function: The IF function can be applied to various scenarios in data analysis. For instance, you can use it to categorize data, calculate bonuses based on performance, or highlight
exceptions in a dataset. By setting up logical tests and defining the values to return, you can effectively streamline your data analysis process.
Formula for Finding Percentage
Calculating percentages in Excel is a common task for many professionals, and it can be a powerful tool for analyzing data and making informed decisions. The formula for finding a percentage in Excel
is straightforward and can be used in various scenarios.
Explanation of how to calculate percentage in Excel
To calculate a percentage in Excel, you can use the following formula:
= (Part / Total) * 100
Where Part is the number you want to find the percentage of, and Total is the total number or amount. This formula will give you the percentage of the part in relation to the total.
Example of using the percentage formula in a practical scenario
Let's say you have a sales report with the total sales for the month and you want to calculate the percentage contribution of each salesperson. You can use the percentage formula in Excel to easily
calculate this. For example, if Salesperson A's total sales are $10,000 and the total sales for the month are $50,000, you can use the formula: = (10000 / 50000) * 100 to find out that Salesperson A
contributed 20% to the total sales for the month.
Formula for Finding Standard Deviation
When working with data in Excel, it's important to be able to calculate the standard deviation to understand the variability of the data. The formula for finding standard deviation in Excel is a
crucial tool for data analysis and decision-making.
A. Explanation of the formula for standard deviation in Excel
The standard deviation formula in Excel is used to measure the amount of variation or dispersion of a set of values. It shows how much individual values differ from the mean (average) of the set. The
formula for the sample standard deviation in Excel is:
=STDEV.S(number1, [number2][number2],…: These are the numerical values for which you want to calculate the standard deviation. You can separate up to 255 individual numbers or cell references with
B. Step-by-step guide on calculating standard deviation using the formula
Calculating standard deviation in Excel involves a few simple steps. Here's a step-by-step guide on how to use the formula:
• First, open a new or existing Excel spreadsheet containing the data for which you want to calculate the standard deviation.
• Select an empty cell where you want the standard deviation result to appear.
• Enter the =STDEV.S( formula into the selected cell.
• Highlight the range of cells or manually input the individual numbers for which you want to calculate the standard deviation inside the parentheses.
• Close the parentheses and press Enter.
• The standard deviation for the provided data will be calculated and displayed in the selected cell.
By mastering the formula for finding standard deviation in Excel, you can gain valuable insights into the variability of your data and make informed decisions based on the analysis.
It is crucial to master formulas in Excel as they are the backbone of data analysis and manipulation. Formulas enable users to quickly and accurately perform complex calculations, saving time and
increasing productivity. As you continue to work with Excel, I encourage you to practice and explore more complex formulas. The more familiar you become with various formulas, the more efficient and
effective you will be at using Excel for your data needs.
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3 Ways to Increase High School Students’ Engagement in Math - Corwin Connect
Has this ever happened to you?
“My lecture went perfectly! I worked the sample problems. I could tell the students really understood the material.”
The next day: “I don’t know what went wrong! I thought all my students understood what I went over yesterday.”
If we’ve been there, we don’t want to go back there. The teacher left something out of the equation when talking about the class – the students. There are a few key ideas we can take from this
scenario to help us engage each and every student.
“My lecture went perfectly!”
There is a place for direct instruction, but mathematics learning requires doing mathematics. Your Mathematics Standards Companion, High School: What They Mean and How to Teach Them has a set of
suggestions about “What the teacher does” for each standard. Part of “What the teacher does” is having good problems/tasks prepared for the students to consider and having good questions related to
the tasks that anticipate student thinking and that prod students along a productive path to deeper understanding of the mathematical learning goals. For example, instead of telling students what the
definition of a function is, the teacher can provide samples of functions and nonfunctions and then challenge the students to determine what defines a function. When students have agency in their
learning and are uncovering mathematics through exploring and discovering, they are engaged students.
“I worked the sample problems.”
Teachers need to use tasks that address the mathematics learning goals of the lesson. The tasks must have accessibility for each student. Sometimes, students need an entry into a problem. Stepping
back and asking the students to state things they noticed about a problem is an open-ended, low-stress way to get students involved in thinking about a problem. Then students are invited to start
reasoning about a task. Tasks can be of different types, but students should be encouraged to think about different solution paths and, as part of the whole class, about connections among and between
the solution paths. All students are part of the discussion of solutions, and all students are learning from each other, so that each and every student is engaged in doing mathematics.
“I could tell the students understood the material.”
How are you gaining information about what the students understand? Having students do exit slips based on the learning goals for the day, writing what they learned from the lesson, asking one
question about something they aren’t sure about are all ways to find out what students know and to engage the students in monitoring their own understanding. Eliciting information from students about
their understanding engages them in the lesson.
Students can be engaged by making them a part of discovering and generalizing concepts, by being part of discourse about different solution paths based on problems and good tasks with accessibility
for the diverse learners in the class, and by reflecting on their learning and sharing their understandings and questions. The focus of the classroom is moved from the teacher (lecture, doing
examples, looking for cues) to the engaged students. | {"url":"https://corwin-connect.com/2018/05/3-ways-to-increase-high-school-students-engagement-in-math/","timestamp":"2024-11-11T13:01:50Z","content_type":"text/html","content_length":"166087","record_id":"<urn:uuid:a1b808d9-2d6f-458d-bf4e-41574d54dbbd>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00197.warc.gz"} |
PENGARUH PENGGUNAAN MODEL PEMBELAJARAN PROBLEM BASED LEARNING TERHADAP HASIL BELAJAR SISWA KELAS V SDN 107103 LANTASAN BARU PADA BIDANG STUDI IPAS T.P 2023/2024
PENGARUH PENGGUNAAN MODEL PEMBELAJARAN PROBLEM BASED LEARNING TERHADAP HASIL BELAJAR SISWA KELAS V SDN 107103 LANTASAN BARU PADA BIDANG STUDI IPAS T.P 2023/2024
Kata Kunci:
Learning Outcomes, Problem based learning, Sciences, Model
This research was conducted with the aim of knowing the effect of student learning outcomes in the Science subject on temperature and changes in the shape of objects using the problem based learning
and Conventional Learning models in class V SDN 107103 Lantasan baru FY 2023/2024.The location of this research was carried out at SDN 107103 Lantasan baru FY 2023/2024 on March 29-30 January 2024 as
the subjects were students in class V-A and V-B with the number of students in class V-A there were 22 people and class V- B there are 21 people. This type of research is a quasi-experimental with an
essay test instrument of 5 questions that have been validated by the validator.The average initial test result for class V-A was 44,28 and the average initial test result for class V-B was 45,9.
After learning was carried out in class V-A using the Direct Instruction learning model, the average student learning outcomes 74,25 and in class V-B using Conventional Learning to obtain an average
learning result of 60,9.Based on the calculation of the hypothesis test with a two-factor independent test, students who were taught by the Direct Instruction model and students who were taught by
Conventional Learning obtained the results obtained Xcount (X2) = 4,76 > Xtable = 2,01. SoH_0 is rejected and H_1 is accepted, meaning that there is an influence from the problem based learning model
on the science learning outcomes of fifth grade students at SDN 107103 Lantasan baru FY 2023/2024. | {"url":"http://jurnal.semnaspssh.com/index.php/pssh/article/view/662","timestamp":"2024-11-03T09:19:01Z","content_type":"text/html","content_length":"16286","record_id":"<urn:uuid:1a30a2b2-babe-4358-84e5-fc2c59056215>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00741.warc.gz"} |
Solving Word Problems Involving Average Velocity
Question Video: Solving Word Problems Involving Average Velocity Mathematics • Second Year of Secondary School
A person is late for an appointment at an office that is at the other end of a long, straight road to his home. He leaves his house and runs towards his destination for a time of 45 seconds before
realizing that he has to return home to pick up some documents that he will need for his appointment. He runs back home at the same speed he ran at before and spends 185 seconds looking for the
documents, and then he runs towards his appointment again. This time, he runs at 5.5 m/s for 260 seconds and then arrives at the office. How much time passes between the person first leaving his
house and arriving at his appointment? What is the distance between the person’s house and his office? What is the person’s average velocity between first leaving his house and finally arriving at
his office? Give your answer to two decimal places.
Video Transcript
A person is late for an appointment at an office that is at the other end of a long, straight road to his home. He leaves his house and runs towards his destination for a time of 45 seconds before
realizing that he has to return home to pick up some documents that he will need for his appointment. He runs back home at the same speed he ran at before and spends 185 seconds looking for the
documents, and then he runs towards his appointment again. This time, he runs at 5.5 meters per second for 260 seconds and then arrives at the office.
We’re then asked three different questions. So let’s begin with the first one. How much time passes between the person first leaving his house and arriving at his appointment?
It might be helpful to begin by visualizing what happens at each stage of this person’s journey. The journey begins with running for 45 seconds towards the office. The person then realizes that
they’ve forgotten something they need, so they go home at the same speed as they traveled before. So that means that the time will also be 45 seconds. They then spend 185 seconds looking for these
documents but not traveling anywhere. And then, finally, he runs towards the office at 5.5 meters per second for 260 seconds.
We’re then asked for the time that passes between first leaving and then arriving at the appointment. So that means that we just add up the four time periods: 45 seconds, 45 seconds, 185 seconds, and
260 seconds. And when we work that out, we get 535 seconds. And that’s the answer for the first part of this question.
The next question asks us, what is the distance between the person’s house and his office?
In order to find the distance, we can use this information on the last stage of the journey, when we’re given the speed and the time taken. We can remember that distance is equal to speed multiplied
by time. So we can fill in the values then. The speed is 5.5, and the time is 260. It is always worthwhile making sure that we do have the same equivalent units. In each case, the time unit is given
in seconds, so we can simply multiply these values. When we work this out, we get a value of 1430. And the units here will be the distance units of meters. And that’s the second part of this question
The third part of this question asks, what is the person’s average velocity between first leaving his house and finally arriving at his office? Give your answer to two decimal places.
We can recall the formula that average velocity is equal to net displacement over total time. In this problem, the net displacement will simply be the direct distance between the man’s home and the
office. We have already calculated this distance in the second part of the question. It’s 1430. And the total time taken in the whole journey was 535 seconds. This gives us 2.672 and so on. And when
we round that to two decimal places, we have a value of 2.67 meters per second. So if the positive direction is from home towards the office, then the person’s average velocity can be given as 2.67
meters per second.
It’s worth noting that if we’d been asked for the average speed instead, we would’ve needed to know the distances in the first two parts of the journey along with the distance in the final part of
the journey. In this case, average speed would have been calculated by the total distance divided by the total time. However, since average velocity uses displacement, then we have the value of 2.67
meters per second. | {"url":"https://www.nagwa.com/en/videos/726162578142/","timestamp":"2024-11-05T06:14:51Z","content_type":"text/html","content_length":"257054","record_id":"<urn:uuid:15318202-62a1-404b-b04f-d335cdf76bdd>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00546.warc.gz"} |
Rolle's theorem – Serlo – Wikibooks, Sammlung freier Lehr-, Sach- und Fachbücher
Rolle's theorem – illustration and explanation (YouTube video in German, by the channel "MJ Education")
We already know from the extreme value theorem that a continuous function ${\displaystyle f}$ attains a maximum and a minimum on a closed interval ${\displaystyle [a,b]}$:
The function f is bounded and hence attains a maximum and a minimum
This is of course also true, if ${\displaystyle f(a)=f(b)}$. In this case (if the function is not constant) there must be a maximum or minimum inside the domain of definition. In the following
figure, both the maximum and the minimum are inside ${\displaystyle [a,b]}$, i.e. within the open interval ${\displaystyle (a,b)}$:
Special case of the extreme value theorem
Let us now additionally assume that ${\displaystyle f}$ is differentiable on ${\displaystyle (a,b)}$. Let ${\displaystyle \xi }$ be a maximum or minimum. If ${\displaystyle \xi }$ is inside the
domain of definition, i.e. if ${\displaystyle \xi is\in (a,b)}$, then ${\displaystyle f'(\xi )=0}$ according to the main criterion for extremes values of a differentiable function. This means that
the tangent to ${\displaystyle f}$ at ${\displaystyle \xi }$ is horizontal. This is exactly what Rolle's theorem says: For every continuous function ${\displaystyle f:[a,b]\to \mathbb {R} }$ with ${\
displaystyle f(a)=f(b)}$, which is differentiable at ${\displaystyle (a,b)}$, there is an argument ${\displaystyle \xi \in (a,b)}$ with ${\displaystyle f'(\xi )=0}$.
• The derivative at the maximum of ${\displaystyle f}$ is zero.
• The derivative at the minimum of ${\displaystyle f}$ is zero.
Of course, ${\displaystyle f}$ can also assume several (partly local) maxima and minima on ${\displaystyle (a,b)}$ . Furthermore, it is possible that ${\displaystyle f}$ attains only one maximum (and
no minimum) or one minimum (and no maximum) on ${\displaystyle (a,b)}$:
• The function ${\displaystyle f}$ attains one maximum and no minimum within its domain of definition. At that point, the derivative is zero.
• The function ${\displaystyle f}$ attains one minimum and no maximum within its domain of definition. At that point, the derivative is zero.
A special case is ${\displaystyle f}$ being constant on ${\displaystyle [a,b]}$. In this case there is ${\displaystyle f'(x)=0}$ for all ${\displaystyle x\in (a,b)}$:
Sketch of the special case of the extreme value theorem.
This may also happen on a finite sub-interval of ${\displaystyle [a,b]}$, i.e. on a "horizontal plateau".
No matter which case we looked at, there was always at least one point inside the domain of definition where the derivative of the function is zero.
Rolle's theorem. (YouTube video (in German) by the channel Quatematik)
The theorem named after Michel Rolle (1652-1719) represents a special case of the mean value theorem of differential calculus and reads as follows:
Theorem (Rolle's theorem)
Let ${\displaystyle f:[a,b]\to \mathbb {R} }$ be a continuous function with ${\displaystyle a<b}$ and ${\displaystyle f(a)=f(b)}$. Furthermore, ${\displaystyle f}$ is assumed to be differentiable on
the open interval ${\displaystyle (a,b)}$. Then there exists a ${\displaystyle \xi \in (a,b)}$ with ${\displaystyle f'(\xi )=0}$.
If ${\displaystyle f}$ is differentiable on ${\displaystyle (a,b)}$, then ${\displaystyle f}$ is continuous on ${\displaystyle (a,b)}$. Therefore, it is sufficient to prove the continuity of ${\
displaystyle f}$ at the boundary points ${\displaystyle a}$ and ${\displaystyle b}$ in order to check the requirements.
Example (Rolle's theorem)
Let us consider the function ${\displaystyle f:[0,2]\to \mathbb {R} }$ with ${\displaystyle f(x)=x^{2}-2x-3}$. There is
• ${\displaystyle f}$ continuous as a polynomial on ${\displaystyle [0,2]}$
• ${\displaystyle f(0)=-3=f(2)}$
• ${\displaystyle f}$ continuous as a polynomial on ${\displaystyle (0,2)}$
Rolle's theorem now asserts: there is at least one ${\displaystyle \xi \in (0,2)}$ with ${\displaystyle f'(\xi )=0}$.
Question: What is a value ${\displaystyle \xi }$ where the derivative of ${\displaystyle f}$ in the above example is zero?
Graph of ${\displaystyle f}$ and its derivative
The derivative of ${\displaystyle f}$ is ${\displaystyle f'(x)=2x-2}$. We set ${\displaystyle f}$ equal to 0 and get:
{\displaystyle {\begin{aligned}&f'(\xi )=2\xi -2=0\\\iff \ &2\xi =2\\\iff \ &\xi =1\end{aligned}}}
At the position ${\displaystyle \xi =1}$ the derivative of ${\displaystyle f}$ is zero. This value lies within the domain of definition ${\displaystyle [0,2]}$ of ${\displaystyle f}$ and is the only
zero of the derivative. Thus ${\displaystyle \xi =1}$ is the value sought.
There are several necessary requirements in Rolle's theorem. We will show now that if we drop any one of them, the theorem is no longer true.
Condition 1: ${\displaystyle f}$ is continuous on ${\displaystyle [a,b]}$
Exercise (Condition: continuity)
Find a function ${\displaystyle f:[a,b]\to \mathbb {R} }$, which is differentiable only on ${\displaystyle (a,b)}$ and for which ${\displaystyle f(a)=f(b)}$ holds, but for which the implication of
Rolle's theorem does not hold.
The searched function ${\displaystyle f}$ fulfils all requirements of the set of roles except continuity in the complete domain of definition. If we drop continuity on the endpoints, we run into
trouble! The reason is that ${\displaystyle f}$ may then diverge to infinity at the end points.
Solution (Condition: continuity)
Graph of the function ${\displaystyle f}$
${\displaystyle f:[0,1]\to \mathbb {R} }$ with
${\displaystyle f(x)={\begin{cases}x&{\text{ if }}xeq 1,\\0&{\text{ if }}x=1,\end{cases}}}$
is differentiable on ${\displaystyle (0,1)}$ and there is ${\displaystyle f(0)=0=f(1)}$. But since ${\displaystyle f'(x)=1}$ for all ${\displaystyle x\in (0,1)}$, there is no ${\displaystyle \xi \in
(0,1)}$ with ${\displaystyle f'(\xi )=0}$.
Condition 2: ${\displaystyle f(a)=f(b)}$:
Exercise (Equality of function values)
Find a continuous function ${\displaystyle f:[a,b]\to \mathbb {R} }$ which is differentiable on ${\displaystyle (a,b)}$ for which the implication of Rolle's theorem does not hold.
This task shows that the condition ${\displaystyle f(a)=f(b)}$ is necessary for Rolle's theorem. Otherwise, we may "build a slight slope" between the end points, which has no maximum or minimum.
Solution (Equality of function values)
The identity function ${\displaystyle f}$ defined on ${\displaystyle [0,1]}$ with ${\displaystyle f(x)=x}$
Such a function is for example ${\displaystyle f:[0,1]\to \mathbb {R} }$ with ${\displaystyle f(x)=x}$. This function is continuous at ${\displaystyle [0,1]}$ and also differentiable at ${\
displaystyle (0,1)}$. There is however ${\displaystyle f(0)=0eq 1=f(1)}$.
For this function, there is ${\displaystyle f'(x)=1}$ for all ${\displaystyle x\in (0,1)}$. So there is no ${\displaystyle \xi \in (0,1)}$ with ${\displaystyle f'(\xi )=0}$.
Condition 3: ${\displaystyle f}$ is differentiable on ${\displaystyle (a,b)}$:
Exercise (Condition: differentiability)
Find a continuous function ${\displaystyle f:[a,b]\to \mathbb {R} }$ with ${\displaystyle f(a)=f(b)}$, for which the implication of Rolle's theorem does not hold.
Solution (Condition: differentiability)
Plot of the function ${\displaystyle f}$
The function ${\displaystyle f:[0,1]\to \mathbb {R} }$ with
${\displaystyle f(x)={\begin{cases}x&{\text{ if }}x\leq {\tfrac {1}{2}}\\1-x&{\text{ if }}x>{\tfrac {1}{2}}\end{cases}}}$
is continuous and there is ${\displaystyle f(0)=0=f(1)}$. This function is only differentiable on the intervals ${\displaystyle \left[0,{\tfrac {1}{2}}\right)}$ and ${\displaystyle \left({\tfrac {1}
{2}},1\right]}$. The derivative function ${\displaystyle g:[a,b]\to \mathbb {R} }$ has the assignment rule:
${\displaystyle g(x)={\begin{cases}1&x<{\tfrac {1}{2}}\\-1&x>{\tfrac {1}{2}}\end{cases}}}$
Hence, there is no ${\displaystyle \xi \in (0,1)}$ with ${\displaystyle f'(\xi )=0}$.
Summary of proof (Rolle's theorem)
We first consider the special case that ${\displaystyle f}$ is a constant function. Here the derivative is zero everywhere. If ${\displaystyle f}$ is not constant, we use the extreme value theorem to
find a maximum or minimum within the domain of definition. At this extremum, the derivative vanishes according to the criterion for the existence of an extremum.
Proof (Rolle's theorem)
Let ${\displaystyle f:[a,b]\to \mathbb {R} }$ be continuous function with ${\displaystyle a<b}$, which is differentiable on ${\displaystyle (a,b)}$ . Let further ${\displaystyle f(a)=f(b)}$.
Fall 1: ${\displaystyle f}$ is constant.
Let ${\displaystyle f}$ be constant. Then, there is ${\displaystyle f'(\xi )=0}$ for all ${\displaystyle \xi \in (a,b)}$. So there is at least one ${\displaystyle \xi \in (a,b)}$ with ${\displaystyle
f'(\xi )=0}$ (any ${\displaystyle \xi }$ can be chosen from ${\displaystyle (a,b)}$). Role's theorem is fulfilled in that simple case.
Fall 2: ${\displaystyle f}$ is not constant.
Let ${\displaystyle f}$ now be non-constant. By the extreme value theorem, ${\displaystyle f}$ attains both maximum and minimum on the compact interval ${\displaystyle [a,b]}$. The maximum or minimum
of ${\displaystyle f}$ must be different from ${\displaystyle f(a)=f(b)}$, otherwise ${\displaystyle f}$ would be constant. Thus (at least) one extremum is attained at a position ${\displaystyle \xi
\in (a,b)}$.
Since ${\displaystyle f}$ is differentiable at ${\displaystyle (a,b)}$, ${\displaystyle f}$ is also differentiable at the extremum ${\displaystyle \xi }$. Here, according to the necessary criterion
for extrema, there is ${\displaystyle f'(\xi )=0}$. Thus there exists at least one ${\displaystyle \xi \in (a,b)}$ where the derivative is zero. So the theorem of Rolle also gives the right
implication in this case.
Exercise (Exercise)
Let ${\displaystyle k\in \mathbb {N} }$. Show with Rolle's theorem that the derivative function ${\displaystyle f'(x)}$ of the function ${\displaystyle f:[0,k\pi ]\to [-1,1]}$ with ${\displaystyle f
(x)=\sin(x)}$ has at least ${\displaystyle k}$ zeros.
Solution (Exercise)
The sine function is differentiable on all of ${\displaystyle \mathbb {R} }$, i.e. also continuous. Furthermore there is ${\displaystyle \sin(l\pi )=0}$ for all ${\displaystyle l\in \mathbb {Z} }$.
By Rolle's theorem, there is a ${\displaystyle \xi \in (l\pi ,(l+1)\pi )}$ with ${\displaystyle f'(\xi )=0}$. For every ${\displaystyle l\in \mathbb {N} }$ with ${\displaystyle 0\leq l<k}$ we find a
${\displaystyle \xi }$ where the derivative is zero. Since there are ${\displaystyle k}$ different natural numbers for ${\displaystyle l}$ with ${\displaystyle 0\leq l<k}$, we can also find ${\
displaystyle k}$ different zeroes ${\displaystyle \xi }$ of the derivative function. The derivative of ${\displaystyle f}$ must therefore have at least ${\displaystyle k}$ distinct zeros.
Rolle's theorem can also be used in proofs of existence of zeros. And it can be used to show that a function has at most one zero on an interval. On the other hand, the intermediate value theorem can
be used to show that a function has at least one zero on an interval. Together the existence of exactly one zero can be implied.
Example (Zeros of a polynomial)
Let us consider the polynomial ${\displaystyle p(x)=x^{3}+x+1}$ on the interval ${\displaystyle [-1,0]}$. Now,
• ${\displaystyle p}$ is continuous on ${\displaystyle [-1,0]}$. Furthermore, ${\displaystyle p(-1)=-1<0}$ and ${\displaystyle p(0)=1>0}$. According to the intermediate value theorem, the
polynomial has at least one zero at ${\displaystyle [-1,0]}$.
• ${\displaystyle p}$ is differentiable at ${\displaystyle (-1,0)}$ with ${\displaystyle p'(x)=3x^{2}+1}$. We now assume that ${\displaystyle p}$ had two zeros on ${\displaystyle [-1,0]}$ ${\
displaystyle x_{1}}$ and ${\displaystyle x_{2}}$. Let without loss of generality be ${\displaystyle x_{1}<x_{2}}$. There is also ${\displaystyle p(x_{1})=0=p(x_{2})}$. Since ${\displaystyle p}$
is continuous on ${\displaystyle [x_{1},x_{2}]\subseteq [-1,0]}$ and differentiable on ${\displaystyle (x_{1},x_{2})\subseteq (-1,0)}$, Rolle's theorem can be applied. Hence, there is a ${\
displaystyle \xi \in (x_{1},x_{2})}$ with ${\displaystyle p'(\xi )=3\xi ^{2}+1=0}$. But now ${\displaystyle p'}$ has no zeros because of ${\displaystyle p'(x)=\underbrace {3x^{2}} _{\geq 0}+1\geq
1}$. So on ${\displaystyle [-1,0]}$, the polynomial ${\displaystyle p}$ cannot have more than one zero.
From both points we get that ${\displaystyle p}$ has exactly one zero on ${\displaystyle [-1,0]}$.
Exercise (Finding a unique zero)
Show that
${\displaystyle f:[1,2]\to \mathbb {R} :x\mapsto {\frac {2}{x^{4}}}-x+1}$
has exactly one zero.
Summary of proof (Finding a unique zero)
First, use the intermediate value theorem to show that ${\displaystyle f}$ has at least one zero. Then we show by Rolle's theorem that ${\displaystyle f}$ has at most one zero. From both steps the
assertion follows.
Proof (Finding a unique zero)
Proof step: ${\displaystyle f}$ has at least one zero.
${\displaystyle f}$ is continuous as a composition of continuous functions. Further, there is ${\displaystyle f(1)={\tfrac {2}{1}}-1+1=2>0}$ and ${\displaystyle f(2)={\tfrac {2}{16}}-2+1=-{\tfrac {7}
{8}}<0}$. By the intermediate value theorem the function has therefore at least one zero.
Proof step: ${\displaystyle f}$ has at most one zero.
${\displaystyle f}$ is differentiable on ${\displaystyle (1,2)}$ as it is a composition of differentiable functions. In particular, ${\displaystyle f'(x)=-4{\tfrac {2}{x^{5}}}-1=-{\tfrac {8}{x^{5}}}
-1}$. We now assume that on ${\displaystyle [1,2]}$, the function ${\displaystyle f}$ has two distinct zeros ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$. Let us assume that ${\displaystyle x_
{1}<x_{2}}$. There is therefore ${\displaystyle f(x_{1})=0=f(x_{2})}$.
Now, ${\displaystyle f}$ is continuous on ${\displaystyle [x_{1},x_{2}]\subseteq [1,2]}$ and differentiable on ${\displaystyle (x_{1},x_{2})\subseteq (1,2)}$. According to Rolle's theorem, there
hence is a ${\displaystyle \xi \in (x_{1},x_{2})}$ with ${\displaystyle f'(\xi )=-{\tfrac {8}{\xi ^{5}}}-1=0}$. But since
${\displaystyle f'(x)=\underbrace {-{\tfrac {8}{x^{5}}}} _{\leq -{\frac {8}{2^{5}}}=-{\tfrac {1}{4}}}-1\leq -{\tfrac {5}{4}}<0}$
${\displaystyle f'}$ has no zeros. So ${\displaystyle f}$ has at most one zero.
It follows from both steps of proof that ${\displaystyle f}$ has exactly one zero.
Outlook: Rolle's theorem and the mean value theorem
As mentioned above, Rolle's theorem is a special case of the mean value theorem. This is one of the most important theorems from real Analysis, as many other useful results can be derived from it.
Conversely, we will show that the mean value theorem follows from Rolle's theorem. Both theorems are thus equivalent. | {"url":"https://de.wikibooks.org/wiki/Serlo:_EN:_Rolle%27s_theorem","timestamp":"2024-11-14T11:21:42Z","content_type":"text/html","content_length":"312197","record_id":"<urn:uuid:00abacfb-782a-4042-9dbb-d2a0a0f5841f>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00087.warc.gz"} |
Title : Succinct Data Structure for Graphs with d-Dimensional t-Representation
Speaker : Girish Balakrishnan (IITM)
Details : Mon, 4 Mar, 2024 11:00 AM @ SSB-334
Abstract: : Erdos and West (Discrete Mathematics'85) considered the class of n vertex intersection graphs which have
a d-dimensional t-representation, that is, each vertex of a graph in the class has an associated set
consisting of at most t d-dimensional axis-parallel boxes. In particular, for a graph G and for each d
≥ 1, they consider i_d(G) to be the minimum t for which G has such a representation. For fixed t and
d, they consider the class of n vertex labeled graphs for which i_d(G) ≤ t, and prove an upper bound
of (2nt+½)d log n - (n - ½)d log(4π t) on the logarithm of size of the class. In this work, for fixed t
and d we consider the class of n vertex unlabeled graphs which have a d-dimensional t-representation,
denoted by mathcal{G}_{t,d}. We address the problem of designing a succinct data structure for the class
mathcal{G}_{t,d} in an attempt to generalize the relatively recent results on succinct data structures
for interval graphs (Algorithmica'21). To this end, for each n such that td^2 is in o(n / log n), we
first prove a lower bound of (2dt-1)n log n - O(ndt log log n)-bits on the size of any data structure
for encoding an arbitrary graph that belongs to mathcal{G}_{t,d}. We then present a ((2dt-1)n log n +
dtlog t + o(ndt log n))-bit data structure for mathcal{G}_{t,d} that supports navigational queries
efficiently. Contrasting this data structure with our lower bound argument, we show that for each fixed
t and d, and for all n ≥ 0 when td^2 is in o(n/log n) our data structure for mathcal{G}_{t,d} is
succinct. As a byproduct, we also obtain succinct data structures for graphs of bounded boxicity
(denoted by d and t = 1) and graphs of bounded interval number (denoted by t and d=1) when td^2 is in o
(n/log n).
Web Conference Link : meet.google.com/dzj-nekm-tcc | {"url":"https://cse.iitm.ac.in/seminar_details.php?arg=NjA0","timestamp":"2024-11-07T04:18:21Z","content_type":"application/xhtml+xml","content_length":"14799","record_id":"<urn:uuid:caa590df-6daa-4eec-a8ff-c1853b46cdac>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00437.warc.gz"} |
Best AI Calculus Solver Tools (Free + Paid) 2024
What is an AI Calculus Solver
An AI calculus solver is a powerful tool that uses artificial intelligence to solve complex mathematical problems, particularly in calculus. These advanced systems can tackle a wide range of calculus
questions, from basic derivatives to advanced calculus concepts, providing step-by-step solutions that help students and professionals alike.
In addition to calculus, there are various other specialized tools available in the realm of mathematics. For instance, an AI Math Solver can assist with a broader spectrum of mathematical problems
beyond calculus, including algebra and trigonometry. Similarly, an AI Statistics Solver is designed to handle statistical analyses and computations, making it invaluable for data-driven fields.
Moreover, for those dealing with spatial relationships and properties, an AI Geometry Solver offers targeted assistance in solving geometric problems. Lastly, an AI Physics Solver can address complex
physics equations and concepts, integrating mathematical principles with physical laws. Together, these tools enhance the learning experience and provide comprehensive support across multiple
disciplines in science and mathematics.
How AI Calculus Solvers Work
Calculus solving AI combines sophisticated algorithms with machine learning to understand and solve mathematical problems. When you input an algebra equation or calculus question, the AI analyzes it,
breaks it down into smaller parts, and applies relevant mathematical rules to solve it. This process happens in seconds, making AI calculus solvers incredibly efficient.
Benefits of Using a Calculus AI Solver
1. Quick Results: AI solvers can solve complex problems much faster than humans.
2. Step-by-Step Solutions: They provide detailed explanations, helping users understand the problem-solving process.
3. Learning Aid: Students can use these tools to check their work and learn from their mistakes.
4. Versatility: Most AI math solvers can handle various types of calculus problems, from basic to advanced calculus. | {"url":"https://stackviv.ai/ai-categories/calculus-solver/","timestamp":"2024-11-12T19:42:52Z","content_type":"text/html","content_length":"81005","record_id":"<urn:uuid:e2f5afa9-8b06-464d-8f8f-3a86c5728297>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00531.warc.gz"} |
Why numbers manifest living planets – Sacred Number Sciences
Why numbers manifest living planets
above: The human essence class related to four other classes in J.G. Bennett’s Gurdjieff: Making a New World. Appendix II. page 290. This systematics presents the human essence class which eats the
germinal essence of Life, but is “eaten” by cosmic individuality, the purpose of the universe. The range of human potential is from living like an animal to living like an angel or demiurge, then
helping the cosmic process.
Please enjoy the text below which is ©2023 Richard Heath: all rights reserved.
The human essence class is a new type of participation within the universe where the creation can form its own creative Will, in harmony with the will that creates the universe. The higher
intelligences have a different relationship to the creation than human intelligence. It is based upon this Universal Will (to create the universe) which has manifested a world we can only experience
from outside of it. And the creative tip of creation* is the universal life principle that led to the human world where it is possible to participate in the intelligence behind the world, through a
transformation into an Individuality, creative according their own pattern while harmonious with the universal will.
*creative tip: The evolving part of organic life is humanity. Humanity also has its evolving part but we will speak of this later; in the meantime we will take humanity as a whole. If humanity
does not evolve it means that the evolution of organic life will stop and this in its turn will cause the growth of the ray of creation to stop. At the same time if humanity ceases to evolve it
becomes useless from the point of view of the aims for which it was created and as such it may be destroyed. In this way the cessation of evolution may mean the destruction of humanity.
In Search of the Miraculous. P.D. Ouspensky. 306.
Will is not something one does. Rather, it is a participation of one’s being with Will. This creates a transformational action of Will within a human that is receptive to it (rather than merely
assertive on their own account). We are born able, through our unique pattern, to participate in our own understanding of the meaning that is this world. In this, numbers are more than data: they
form structures of will which do not rely on complexity and are therefore directly intelligible for an intelligent lifeform, enabling what to do, by seeing more deeply what is in the present moment.
For example, number is the foundation of that universal invariance: the Present Moment of selfhood*.
The myth of a philosopher’s stone presents a challenge, to find the “stone” itself, which we shall see is probably the numerically favourable environment upon the earth. The stone has been rendered
invisible to modern humans by our functional science of infinite complexity, also called instrumental determinism. This has downgraded human expectations to being a walk-on part, an unintentional
result of evolution, by natural selection, of intelligent life. To think otherwise it is necessary to see what is not complex about the sky, which is a designed phenomenon related to Life on Earth.
Once-upon-a-time, the stone age understood the sky in this right way, the way it had been designed to be read by us, corresponding with the way intelligent life was intended to be, on a habitable
planet with a large moon.
1.1 Geocentric Numbers in the Sky
Our pre-digested meanings are those of modern science. Whilst accurate they cannot be trusted in the spiritual sense, if one is to continue looking at phenomena rather than at their preformed
conceptual wrapper. Numbers in themselves are these days largely ignored except by mathematicians who, loving puzzles, have yet largely failed to query the megalithsStructures built out of large
little-altered stones in the new stone age or neolithic between 5,000-2,500 (bronze age), in the pursuit of astronomical knowledge.** but, if or when anyone might say the megaliths had a technical
purpose, this has annoyed most archaeologists, who live by the spade and not by the ancient number sciences or astronomy.
**Fred Hoyle, Hawkins, Alexander ThomScottish engineer 1894-1985. Discovered, through surveying, that Britain's megalithic circles expressed astronomy using exact measures, geometrical forms and,
where possible, whole numbers., Merritt and others all found something new in Stonehenge but still failed to explore stone age numeracy as well as the numeracy of metrologyThe application of units of
length to problems of measurement, design, comparison or calculation.. Rather, they assumed measures unlike our own were used, yet the megaliths would continue to have no meaning “above ground”,
except as vaguely ritualistic venues in loose synchronization with a primitive calendar.
Numbers are not abstract once incarnated within Existence. In their manifestations as measurements, they have today become abstracted due to our notation and how we transform them using arithmetic,
using a positional notation based upon powers of two and five {10}, called the decimal system* (https://en.wikipedia.org/wiki/Decimal). The so-called ordinal numbers {1 2 3 4 5 6 7 8 9 10 …etc.} are
then no longer visually ordinal due to the form in which they are written, number-by-number, from right-to-left {ones, tens, hundreds, thousands …} * (the reversal of the left-to-right of western
languages). Positional notation awaited the invention of zero, standing for no powers of ten, as in 10 (one ten plus no units). But zero is not a number or, for that matter, a starting point in the
development of number and, with the declaring of zero, to occupy the inevitable spaces in base-10 notation, there came a loss of ordinality as being the distance from one.
Before the advance of decimal notation, groups within the ancient world had seen that everything came from one. By 3000BC, the Sumerian then Old Babylonian civilization, saw the number 60 perfect as
a positional base since 60 has so many harmonious numbers as its factors {3 4 5}, the numbers of the first Pythagorean triangle’s side lengths. Sixty was the god Anu, of the “middle path”, who formed
a trinity with Fifty {50}, Enlil* (who would flood humanity to destroy it) and Forty {40} who was Ea-Enki, the god of the waters. Anu presided over the Equatorial stars, Enlil over those of the North
and Ea-Enki over those of the South. In their positional notation, the Sumerians might leave a space instead of a zero, calling Sixty, “the Big One”, a sort of reciprocal meaning of 60 parts as with
360 degrees in a circle from its center. So the Sumerians were resisting the concept of zero as a number and instead left a space. And because 60 was seen as also being ONE, 60 was seen as the most
harmonious division of ONE using only the first three prime numbers {3 4 5}.
These days we are encouraged to think that everything comes from zero in the form of a big bang, and the zeros in our decimal notation have the unfortunate implication that nothing is a number,
“raining on the parade” of ordinal numbers, Nothing usurping One {1} as the start of the world of number. The Big Bang, vacuum energy, background temperature, and so on, see the physical world
springing from a quantum mechanical nothingness or from inconceivable prior situations where, perhaps two strings (within string theory) briefly touched each other. However, it is observations that
distinguish meaning.
In what follows we will nevertheless need to use decimal numbers in their position notation, to express ordinal numbers while remembering they have no positional order apart from their algorithmic
order as an infinite series in which each number is an increment, by one, from the previous number; a process starting with one and leading to the birth of two, the first number.
Whole (or integer) numbers are only seen clearly when defined by
• (a) their distance from One (their numerical value) and
• (b) their distance from one another (their difference).
In the Will that manifested the Universe, zero did not exist and numerical meaning was to be a function of distances between numbers!
Zero is part of one and the first true number is 2, of doubling; Two’s distance from one is one and in the definition of doubling and the octave, the distance from a smaller number doubled to a
number double it, is the distance of the smaller number from One. This “strange type of arithmetic” *(Ernest G McClain email) is seen in the behavior of a musical string as, in that kind of
resonator, half of the string merely provides the basis for the subsequent numerical division of its second half, to make musical notes – as in a guitar where the whole string provides low do and the
frets when pressed then define higher notes up to high do (half way) and beyond, through shortening the string.
This suggests that a tonal framework was given to the creation by Gurdjieff’s Universal Will, within which many inner and outer connections can then most easily arise within octaves, to
1. overcome the mere functionality of complexity,
2. enable Will to come into Being,
3. equip the venue of Life with musical harmony and
4. make the transformation of Life more likely.
Harmony is most explicit as musical harmony, in which vibrations arise through the ratios between wavelengths which are the very same distance functions of ordinal numbers, separated by a common unit
Take the number three, which is 3/2 larger than two. Like all ordinal numbers, succeeding and preceding numbers differ by plus or minus one respectively, and the most basic musical tuning emerges
from the very earliest six numbers to form Just intonationA musical tuning system improving the Pythagorean system of tuning by fifths (3/2), by introducing thirds (5/4 and 6/5) to obtain multiple
scales., whose scales within melodic music result as a sequence of three small intervals {9/8 10/9 16/15}, two tones and a semitone. Between one and those numbers {8 9 10 15 16} are the first six
numbers {1 2 3 4 5 6}* (note absence of seven between these sets), whose five ratios {1/2 2/3 3/4 4/5 5/6} provide any octave doubling with a superstructure for the melodic tone-semitone sequences;
their combined interdivision, directly realizes (in their wake) the tones and semitones of modal music.
We will see that the medium for such a music of the spheres was both the relationship of the sun and planets to the Moon and Earth, and this manifested quite literally in the lunar months and years,
when counted. But Gurdjieff’s octaves cannot be understood without disengaging modern numerical thinking, procedures and assumptions. It is always the whole being divided and not a line of numbers
being extended, though it is easier to look within wholes by expressing their boundary as a large number: Hence the large numbers of gods, cities, time and so on. For example, creating life on earth
requires a lot of stuff: perhaps the whole solar nebula has been necessary for that alone and billions of our years. Were you worth it? | {"url":"https://sacred.numbersciences.org/2024/02/17/why-numbers-manifest-living-planets/","timestamp":"2024-11-09T06:09:30Z","content_type":"text/html","content_length":"121184","record_id":"<urn:uuid:3a721750-a98c-4aa6-a08e-59654b3a3d6e>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00390.warc.gz"} |
Understanding Mathematical Functions: What Is A Function In Google She
Introduction to Mathematical Functions in Google Sheets
When it comes to organizing, analyzing, and manipulating data, mathematical functions play a crucial role in modern tools like Google Sheets. In this chapter, we will delve into the definition of
functions in Google Sheets, their importance in data handling, and how they enhance efficiency and accuracy in mathematical calculations.
A Definition and importance of functions in data manipulation and analysis
Functions in Google Sheets refer to predefined formulas that take input, perform a specific operation, and return a result. These functions are designed to simplify complex calculations and automate
repetitive tasks, making data manipulation more efficient and accurate.
Importance of functions in data manipulation and analysis:
• Reduce manual errors: Functions eliminate the need for manual calculations, reducing the chances of errors in data analysis.
• Save time: By automating calculations, functions save time and allow users to focus on interpreting results rather than performing calculations.
• Enhance accuracy: Functions ensure consistency in calculations, leading to more accurate and reliable results.
B Overview of Google Sheets as a powerful tool for mathematical calculations
Google Sheets is a cloud-based spreadsheet application that allows users to create, edit, and collaborate on spreadsheets online. It offers a wide range of tools and features, including mathematical
functions, that make it a powerful tool for data analysis and manipulation.
Key features of Google Sheets for mathematical calculations:
• Predefined functions: Google Sheets provides a variety of built-in functions for performing common mathematical operations.
• Custom functions: Users can create their own custom functions using Google Apps Script to extend the capabilities of Google Sheets.
• Data visualization: Google Sheets allows users to create charts and graphs to visualize data and identify trends.
C The role of functions in enhancing the efficiency and accuracy of data handling in Google Sheets
Functions play a crucial role in enhancing the efficiency and accuracy of data handling in Google Sheets by:
• Automating calculations: Functions automate repetitive calculations, saving time and reducing the chances of errors.
• Improving data analysis: By using functions to perform complex calculations, users can gain valuable insights from their data.
• Enhancing collaboration: Functions in Google Sheets allow multiple users to work on the same spreadsheet simultaneously, increasing collaboration and productivity.
Key Takeaways
• Definition of a function in Google Sheets
• How to use functions in Google Sheets
• Commonly used functions in Google Sheets
• Examples of functions in Google Sheets
• Tips for using functions effectively in Google Sheets
Understanding the Basics of Functions
Functions in Google Sheets are powerful tools that allow users to perform various calculations and operations on their data. Understanding the basics of functions is essential for utilizing Google
Sheets effectively.
The syntax of functions in Google Sheets
The syntax of functions in Google Sheets follows a specific format. Functions begin with an equal sign (=) followed by the function name and any arguments enclosed in parentheses. For example, the
SUM function in Google Sheets looks like this: =SUM(A1:A10).
Differentiating between functions and formulas
It is important to differentiate between functions and formulas in Google Sheets. Functions are predefined operations that perform specific calculations, while formulas are user-defined expressions
that can include functions, operators, and cell references. Understanding this distinction is crucial for creating accurate and efficient spreadsheets.
Basic examples of functions: SUM, AVERAGE, MIN, MAX
There are several basic functions in Google Sheets that are commonly used for data analysis. These include:
• SUM: The SUM function adds up a range of cells. For example, =SUM(A1:A10) would add up the values in cells A1 to A10.
• AVERAGE: The AVERAGE function calculates the average of a range of cells. For example, =AVERAGE(B1:B5) would calculate the average of the values in cells B1 to B5.
• MIN: The MIN function returns the smallest value in a range of cells. For example, =MIN(C1:C8) would return the smallest value in cells C1 to C8.
• MAX: The MAX function returns the largest value in a range of cells. For example, =MAX(D1:D6) would return the largest value in cells D1 to D6.
Types of Functions Available in Google Sheets
Google Sheets offers a wide range of functions to help users perform various mathematical calculations, data analysis, and data manipulation tasks. Understanding the different types of functions
available can greatly enhance your productivity and efficiency when working with spreadsheets. Let's take a closer look at some of the main types of functions in Google Sheets:
A. Mathematical and Trigonometric Functions
Mathematical and trigonometric functions in Google Sheets allow users to perform common mathematical operations and trigonometric calculations easily. These functions can be used to calculate values
such as sine, cosine, tangent, square root, logarithm, and more. Some of the commonly used mathematical and trigonometric functions in Google Sheets include:
• SIN: Calculates the sine of an angle.
• COS: Calculates the cosine of an angle.
• TAN: Calculates the tangent of an angle.
B. Statistical Functions for Data Analysis
Statistical functions in Google Sheets are essential for data analysis tasks, such as calculating averages, medians, modes, standard deviations, and more. These functions help users analyze and
interpret data effectively. Some of the statistical functions available in Google Sheets include:
• COUNT: Counts the number of cells in a range that contain numbers.
• MEDIAN: Calculates the median value in a range of numbers.
• MODE: Returns the most frequently occurring value in a dataset.
C. Lookup and Reference Functions to Work with Data Sets
Lookup and reference functions in Google Sheets are used to search for specific values within a dataset, retrieve information from different parts of a spreadsheet, and create dynamic formulas that
reference other cells. These functions are particularly useful for organizing and managing large sets of data. Some of the lookup and reference functions available in Google Sheets include:
• VLOOKUP: Searches for a value in the first column of a range and returns a value in the same row from another column.
• HLOOKUP: Searches for a value in the first row of a range and returns a value in the same column from another row.
• INDEX: Returns the value of a cell in a specified row and column of a range.
How to Insert and Use Functions in Google Sheets
Google Sheets offers a wide range of functions that can help you perform various calculations and data analysis tasks. Understanding how to insert and use functions in Google Sheets is essential for
maximizing the potential of this powerful tool. In this guide, we will walk you through the process step by step.
A Step-by-step guide to inserting a function
• Step 1: Open your Google Sheets document and select the cell where you want the result of the function to appear.
• Step 2: Type an equal sign (=) in the selected cell to indicate that you are entering a formula or function.
• Step 3: Begin typing the name of the function you want to use. Google Sheets will provide suggestions as you type, making it easier to find the function you need.
• Step 4: Once you have selected the function you want to use, enter the required arguments within parentheses. These arguments are the values or cell references that the function will operate on.
• Step 5: Press Enter to apply the function and display the result in the selected cell.
Understanding function arguments and how to input them
Function arguments are the inputs that a function requires to perform its calculation. These arguments can be values, cell references, ranges, or other functions. It is essential to understand how to
input function arguments correctly to ensure that the function works as intended.
• Values: Simply type the value directly into the function's parentheses. For example, =SUM(5, 10) will add 5 and 10 together.
• Cell references: To use a cell reference as an argument, simply enter the cell's address (e.g., A1) within the parentheses. For example, =A1+B1 will add the values in cells A1 and B1.
• Ranges: You can also specify a range of cells as an argument by entering the range (e.g., A1:A10) within the parentheses. For example, =SUM(A1:A10) will add the values in cells A1 to A10.
• Other functions: You can nest functions within each other by using one function's result as another function's argument. For example, =SUM(A1:A10) * 2 will multiply the sum of cells A1 to A10 by
Troubleshooting common errors when using functions
While using functions in Google Sheets, you may encounter errors that prevent the function from working correctly. Understanding common errors and how to troubleshoot them can help you resolve issues
quickly and efficiently.
• #NAME?: This error occurs when Google Sheets does not recognize the function you entered. Double-check the function name for typos or errors.
• #DIV/0!: This error occurs when you try to divide by zero. Check the values or cell references used in the function to ensure there are no zero values.
• #VALUE!: This error occurs when the function's arguments are of the wrong data type. Make sure the arguments are compatible with the function you are using.
• #REF!: This error occurs when a cell reference in the function is invalid. Check the cell references used in the function to ensure they are correct.
Advanced Functions and Their Applications
When it comes to working with mathematical functions in Google Sheets, understanding advanced functions can greatly enhance your data manipulation and analysis capabilities. In this chapter, we will
explore conditional functions, array functions, and real-world examples where these advanced functions can save time and improve analysis.
Exploring conditional functions (eg, IF, AND, OR)
Conditional functions in Google Sheets allow you to perform different calculations based on specified conditions. The IF function, for example, allows you to test a condition and return one value if
the condition is true, and another value if it is false. This can be useful for creating dynamic spreadsheets that adjust based on certain criteria.
The AND and OR functions are logical functions that allow you to test multiple conditions at once. The AND function returns true if all conditions are met, while the OR function returns true if at
least one condition is met. These functions can be combined with other functions to create complex logical tests in your spreadsheets.
Utilizing array functions for complex data manipulation (eg, ARRAYFORMULA)
Array functions in Google Sheets allow you to perform calculations on multiple cells at once, making it easier to manipulate large sets of data. The ARRAYFORMULA function, for example, allows you to
apply a formula to an entire range of cells, rather than having to copy and paste the formula into each individual cell.
By using array functions, you can streamline your data manipulation processes and save time when working with large datasets. These functions are especially useful for tasks such as calculating sums,
averages, or other aggregate functions across multiple rows or columns.
Real-world examples where advanced functions save time and enhance analysis
Advanced functions in Google Sheets can be incredibly valuable in real-world scenarios where time is of the essence and accurate analysis is crucial. For example, imagine you are managing a sales
team and need to calculate commissions based on sales targets and performance.
• Using conditional functions like IF, you can automatically calculate commissions based on predefined criteria.
• Utilizing array functions such as ARRAYFORMULA, you can apply the commission calculation to the entire sales team at once, saving you time and reducing the risk of errors.
• By leveraging these advanced functions, you can quickly analyze sales data, identify top performers, and make data-driven decisions to improve overall sales performance.
Overall, understanding and utilizing advanced functions in Google Sheets can significantly enhance your ability to manipulate data, perform complex calculations, and derive valuable insights from
your spreadsheets.
Integrating Functions for Comprehensive Data Analysis
Integrating functions in Google Sheets is a powerful way to perform comprehensive data analysis. By combining multiple functions and utilizing features like Pivot Tables, users can gain valuable
insights from their data. Let's explore how functions can be integrated for insightful analysis.
A. Combining multiple functions to solve complex problems
One of the key benefits of using functions in Google Sheets is the ability to combine multiple functions to solve complex problems. For example, you can use the IF function in combination with the
SUM function to calculate different values based on specific conditions. This allows for more dynamic and customized data analysis.
By nesting functions within each other, users can create sophisticated formulas that perform a series of calculations in a single cell. This not only saves time but also ensures accuracy in data
analysis. Functions like AVERAGE, MAX, and MIN can be combined to provide a comprehensive overview of data trends.
B. Using functions in tandem with Google Sheets features like Pivot Tables
Another way to integrate functions for comprehensive data analysis is by using them in tandem with Google Sheets features like Pivot Tables. Pivot Tables allow users to summarize and analyze large
datasets quickly and efficiently. By incorporating functions within Pivot Tables, users can perform complex calculations and generate meaningful insights.
Functions like SUMIF, COUNTIF, and AVERAGEIF can be used within Pivot Tables to filter and analyze data based on specific criteria. This enables users to drill down into their data and uncover
patterns that may not be immediately apparent. By combining functions with Pivot Tables, users can create dynamic reports that provide a comprehensive view of their data.
C. Practical scenarios demonstrating the integration of functions for insightful data analysis
To better understand how functions can be integrated for insightful data analysis, let's consider some practical scenarios:
• Scenario 1: Using the VLOOKUP function to retrieve data from another sheet and analyze it in conjunction with the SUM function to calculate total sales for each product.
• Scenario 2: Utilizing the IF function to categorize data into different groups and then using the AVERAGE function to calculate the average value for each group.
• Scenario 3: Combining the INDEX and MATCH functions to search for specific data points within a dataset and analyze them using the MAX function to identify the highest value.
By applying functions in these practical scenarios, users can gain valuable insights and make informed decisions based on their data analysis. Integrating functions for comprehensive data analysis in
Google Sheets is a powerful tool that can enhance the way users interpret and utilize their data.
Conclusion & Best Practices
A Recap of the key points covered on functions in Google Sheets
• Functions in Google Sheets: Functions in Google Sheets are predefined formulas that perform calculations or manipulate data in your spreadsheet.
• Common Functions: Some common functions in Google Sheets include SUM, AVERAGE, IF, VLOOKUP, and CONCATENATE.
• Arguments: Functions in Google Sheets take arguments, which are the input values that the function operates on.
• Output: Functions in Google Sheets return an output based on the input values and the operation defined by the function.
Best practices for using functions effectively, including testing and documentation for complex sheets
• Testing: Before using a function in a complex sheet, it is important to test the function with sample data to ensure it is working correctly.
• Documentation: Documenting the functions used in your sheet can help you and others understand the logic behind the calculations.
• Organize: Organize your functions in a logical manner to make it easier to troubleshoot and make changes in the future.
• Use Comments: Adding comments to your functions can provide additional context and explanations for others who may be reviewing the sheet.
Encouragement to experiment with functions and explore Google Sheets' documentation for continuous learning and improvement
Don't be afraid to experiment with different functions in Google Sheets. The best way to learn is by trying out new functions and seeing how they work in your specific use case.
Take advantage of Google Sheets' documentation to learn more about the available functions and how to use them effectively. The more you explore and practice, the more proficient you will become in
using functions in Google Sheets. | {"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-what-is-a-function-in-google-sheets","timestamp":"2024-11-09T04:25:12Z","content_type":"text/html","content_length":"230261","record_id":"<urn:uuid:a15a9812-6bfd-43f8-a4da-68c94e98d319>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00024.warc.gz"} |
Free Printable Math Worksheets 6Th Grade Order Operations Free Printable | Order of Operation Worksheets
Free Printable Math Worksheets 6Th Grade Order Operations Free Printable
Free Printable Math Worksheets 6Th Grade Order Operations Free Printable
Free Printable Math Worksheets 6Th Grade Order Operations Free Printable – You may have come across an Order Of Operations Worksheet, but just what is it? In this post, we’ll discuss what it is, why
it’s vital, as well as just how to get a Order Of Operations Worksheet 6th Grade With any luck, this info will be helpful for you. Your pupils are entitled to a fun, reliable way to review the most
important principles in mathematics. On top of that, worksheets are a wonderful means for pupils to exercise brand-new skills and testimonial old ones.
What is the Order Of Operations Worksheet?
An order of operations worksheet is a kind of math worksheet that requires students to perform math operations. Students who are still finding out how to do these jobs will discover this kind of
worksheet useful.
The main objective of an order of operations worksheet is to assist students learn the correct means to address math equations. They can examine it by referring to an explanation web page if a
trainee does not yet recognize the principle of order of operations. Additionally, an order of operations worksheet can be split right into numerous groups, based on its trouble.
Another crucial objective of an order of operations worksheet is to show trainees how to perform PEMDAS operations. These worksheets begin with basic problems related to the basic rules and also
accumulate to more complicated troubles including every one of the policies. These worksheets are a terrific means to introduce young students to the exhilaration of fixing algebraic formulas.
Why is Order of Operations Important?
One of the most important things you can find out in mathematics is the order of operations. The order of operations makes certain that the mathematics problems you address are consistent.
An order of operations worksheet is a terrific way to educate students the right means to address mathematics formulas. Prior to students start using this worksheet, they may require to review ideas
associated with the order of operations. To do this, they should evaluate the idea page for order of operations. This principle page will offer students a summary of the keynote.
An order of operations worksheet can aid trainees establish their skills in addition as well as subtraction. Prodigy’s worksheets are an ideal method to help pupils discover concerning the order of
Order Of Operations Worksheet 6th Grade
Mrs White s 6th Grade Math Blog ORDER OF OPERATIONS WHAT DO I DO FIRST
Order Of Operations Worksheet 6th Grade
Order Of Operations Worksheet 6th Grade supply a great source for young learners. These worksheets can be quickly tailored for specific requirements. They can be located in 3 degrees of difficulty.
The first level is easy, requiring students to practice making use of the DMAS approach on expressions containing four or even more integers or three operators. The second level needs students to
utilize the PEMDAS approach to simplify expressions using outer and also internal parentheses, brackets, as well as curly braces.
The Order Of Operations Worksheet 6th Grade can be downloaded free of charge as well as can be printed out. They can then be reviewed making use of addition, multiplication, division, and
subtraction. Students can also utilize these worksheets to examine order of operations and also the use of exponents.
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defmodule Ecto.Query.WindowAPI do
@moduledoc """
Lists all windows functions.
Windows functions must always be used as the first argument
of `over/2` where the second argument is the name of a window:
from e in Employee,
select: {e.depname, e.empno, e.salary, over(avg(e.salary), :department)},
windows: [department: [partition_by: e.depname]]
In the example above, we get the average salary per department.
`:department` is the window name, partitioned by `e.depname`
and `avg/1` is the window function.
However, note that defining a window is not necessary, as the
window definition can be given as the second argument to `over`:
from e in Employee,
select: {e.depname, e.empno, e.salary, over(avg(e.salary), partition_by: e.depname)}
Both queries are equivalent. However, if you are using the same
partitioning over and over again, defining a window will reduce
the query size. See `Ecto.Query.windows/3` for all possible window
expressions, such as `:partition_by` and `:order_by`.
@dialyzer :no_return
@doc """
Counts the entries in the table.
from p in Post, select: count()
def count, do: doc! []
@doc """
Counts the given entry.
from p in Post, select: count(p.id)
def count(value), do: doc! [value]
@doc """
Calculates the average for the given entry.
from p in Payment, select: avg(p.value)
def avg(value), do: doc! [value]
@doc """
Calculates the sum for the given entry.
from p in Payment, select: sum(p.value)
def sum(value), do: doc! [value]
@doc """
Calculates the minimum for the given entry.
from p in Payment, select: min(p.value)
def min(value), do: doc! [value]
@doc """
Calculates the maximum for the given entry.
from p in Payment, select: max(p.value)
def max(value), do: doc! [value]
@doc """
Defines a value based on the function and the window. See moduledoc for more information.
from e in Employee, select: over(avg(e.salary), partition_by: e.depname)
def over(window_function, window_name), do: doc! [window_function, window_name]
@doc """
Returns number of the current row within its partition, counting from 1.
from p in Post,
select: row_number() |> over(partition_by: p.category_id, order_by: p.date)
Note that this function must be invoked using window function syntax.
def row_number(), do: doc! []
@doc """
Returns rank of the current row with gaps; same as `row_number/0` of its first peer.
from p in Post,
select: rank() |> over(partition_by: p.category_id, order_by: p.date)
Note that this function must be invoked using window function syntax.
def rank(), do: doc! []
@doc """
Returns rank of the current row without gaps; this function counts peer groups.
from p in Post,
select: dense_rank() |> over(partition_by: p.category_id, order_by: p.date)
Note that this function must be invoked using window function syntax.
def dense_rank(), do: doc! []
@doc """
Returns relative rank of the current row: (rank - 1) / (total rows - 1).
from p in Post,
select: percent_rank() |> over(partition_by: p.category_id, order_by: p.date)
Note that this function must be invoked using window function syntax.
def percent_rank(), do: doc! []
@doc """
Returns relative rank of the current row:
(number of rows preceding or peer with current row) / (total rows).
from p in Post,
select: cume_dist() |> over(partition_by: p.category_id, order_by: p.date)
Note that this function must be invoked using window function syntax.
def cume_dist(), do: doc! []
@doc """
Returns integer ranging from 1 to the argument value, dividing the partition as equally as possible.
from p in Post,
select: ntile(10) |> over(partition_by: p.category_id, order_by: p.date)
Note that this function must be invoked using window function syntax.
def ntile(num_buckets), do: doc! [num_buckets]
@doc """
Returns value evaluated at the row that is the first row of the window frame.
from p in Post,
select: first_value(p.id) |> over(partition_by: p.category_id, order_by: p.date)
Note that this function must be invoked using window function syntax.
def first_value(value), do: doc! [value]
@doc """
Returns value evaluated at the row that is the last row of the window frame.
from p in Post,
select: last_value(p.id) |> over(partition_by: p.category_id, order_by: p.date)
Note that this function must be invoked using window function syntax.
def last_value(value), do: doc! [value]
@doc """
Applies the given expression as a FILTER clause against an
aggregate. This is currently only supported by Postgres.
from p in Post,
select: avg(p.value)
|> filter(p.value > 0 and p.value < 100)
|> over(partition_by: p.category_id, order_by: p.date)
def filter(value, filter), do: doc! [value, filter]
@doc """
Returns value evaluated at the row that is the nth row of the window
frame (counting from 1); `nil` if no such row.
from p in Post,
select: nth_value(p.id, 4) |> over(partition_by: p.category_id, order_by: p.date)
Note that this function must be invoked using window function syntax.
def nth_value(value, nth), do: doc! [value, nth]
@doc """
Returns value evaluated at the row that is offset rows before
the current row within the partition.
If there is no such row, instead return default (which must be of the
same type as value). Both offset and default are evaluated with respect
to the current row. If omitted, offset defaults to 1 and default to `nil`.
from e in Events,
windows: [w: [partition_by: e.name, order_by: e.tick]],
select: {
lag(e.action) |> over(:w), # previous_action
lead(e.action) |> over(:w) # next_action
Note that this function must be invoked using window function syntax.
def lag(value, offset \\ 1, default \\ nil), do: doc! [value, offset, default]
@doc """
Returns value evaluated at the row that is offset rows after
the current row within the partition.
If there is no such row, instead return default (which must be of the
same type as value). Both offset and default are evaluated with respect
to the current row. If omitted, offset defaults to 1 and default to `nil`.
from e in Events,
windows: [w: [partition_by: e.name, order_by: e.tick]],
select: {
lag(e.action) |> over(:w), # previous_action
lead(e.action) |> over(:w) # next_action
Note that this function must be invoked using window function syntax.
def lead(value, offset \\ 1, default \\ nil), do: doc! [value, offset, default]
defp doc!(_) do
raise "the functions in Ecto.Query.WindowAPI should not be invoked directly, " <>
"they serve for documentation purposes only" | {"url":"https://preview.hex.pm/preview/ecto/3.8.4/show/lib/ecto/query/window_api.ex","timestamp":"2024-11-13T07:26:29Z","content_type":"text/html","content_length":"55369","record_id":"<urn:uuid:017cce44-83b1-4b67-a284-b8534d7e83b9>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00331.warc.gz"} |
BWT (Burrows Wheeler Transform) Encoding Algorithm
Author: Deepak
Procedure for implementing the algorithm:
1. Select a block size to be used. Block size is directly related to effectiveness of encoding and inversely related to the time required. Hence a compromise has to be reached.
2. Convert the data byte stream to blocks of n bytes where n is the block size chosen.
3. The following example illustrates the procedure to be done next.
Let n=6 and the first string be “kerala”. By wrap around rotations of the string by one character each the following strings can be derived. i.e., first the string is rotated one character to the
right and the result taken and then one more character and so on until when one more rotation results in recovery of the original string.
Then these strings are sorted and the following order is obtained:
The string formed by taking the last letters of the sorted array of string forms the encoded data. i.e., here the encoded data is “lrkaae” . But obviously we cannot get back the original string from
this data alone. We have to have something called the primary_index. It is very easy to get. While rotating the original string, remember the string formed by rotating the original string once
i.e.,”eralak”. It’s position in the array of sorted strings gives the primary index. It occurs third in the sorted array of strings and hence the primary_index=2 (counting from 0).
4. Do this on each block of data until the data is exhausted. It is advisable to write the encoded string followed by the primary index on to the output file. Here we have dealt with data as string
but it is not possible in the actual implementation as the data stream will have in it non-alphabetic characters also. But the adapatation to be done is very trivial.
The strings should not be kept in memory unless we have a hell lot of memory. Just keep an array of indices each element of which points to a position in the block and sort them. So the first element
should point to the first data string and so on.
The data that we now have with us is the encoded version and the primary index. So we now have “lrkaae” and the primary index (2) with us. Now we have to prepare a vector of n elements and an array
to hold the sorted version of the encoded data. Sort the encoded data encoded_data and store it in sorted_data.
So now, sorted_data=”aaeklr” and encoded_data=”lrkaae”.
No we move to prepare the vector. We give the pseudo code below.
for(int i=0;i {
for(int j=0;j {
if(encoded_data[j]==sorted_data[i]&& encoded_data[j] is not flagged)
flag encoded_data[j];
So here the vector is {3,4,5,2,0,1}. Now the following simple code from Mark Nelson’s article on BWT in the DDJ does the job.
for(int i=0;i {
Thus we get back “kerala”, the original string.
Now the question is, is “lrkaae” more suitable for compression compared to the original string “kerala”. Here in the encoded string both a’s have come together. When the block size chosen is large it
causes more such occurrences. MTF (Move to front) encoder converts this data to one having a high frequency of characters having ASCII values less near 0. Encoding such data by an entropy encoder
results in very good performance. Huffman algorithm may be used as the entropy encoder.
Sincere thanks to Mark Nelson’s article on Data compression by the Burrows Wheeler Transform from which I came to know of this fantastic algorithm.
Now the performance table using BeWT and HuffPack compared with WinZip, the famed archiver:
│File │Full Size│BeWT+HuffPack │HuffPack│WinZip│
│a.htm │221514 │061743 │145632 │043472│
│t.pdf │056553 │051808 │055969 │045332│
│l.exe │040960 │013159 │020349 │009680│
│notepad.exe │053248 │023033 │034176 │017769│
The sizes are in bytes after compression. Although the sizes do not quite match those of the famed commercial archiver WinZip (WinZip Computing Inc.) , the improvements over using HuffPack alone is
great and it illustrates how well the method works. A better algorithm such as a combination of Run Length Encoding and Arithmetic encoding if used instead of Huffman should improve the compression
and exploit the full potential of BWT as illustrated by Mark Nelson in his article.
Tags: BWT, encoding
3 Responses to “BWT (Burrows Wheeler Transform) Encoding Algorithm”
1. Hi, really interesting article!
I’m not quite sure, but inside the code for bwt decompression:
for(int i=0;i
shouldn’t it be “sorted(encoded_data[index]);” instead?
Because when you imagine the decoder matrix, the sorted line is on the left and this is where the message reconstruction begins.
2. sry I meant “output(sorted_data[index]);” =)
3. Thomas, you’re right. Thank you for your correction.
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N1d – Ordering fractions
This is the students’ version of the page. Log in above for the teachers’ version.
Ordering fractions
Click the tabs for extension tasks…
You have four cards, numbered 5, 6, 7 and 8. You may pick one card to be the numerator and one card to be the denominator of a fraction.
1. What is the smallest fraction you can make?
2. What is the largest fraction you can make?
Hover for answers:
1. \(\dfrac{5}{8}\)
2. \(\dfrac{8}{5}\)
You have four cards, numbered 5, 6, 7 and 8. You may pick one card to be the numerator and one card to be the denominator of a fraction.
1. What is the second largest fraction you can make?
2. What is the third largest fraction you can make?
3. What is the largest proper fraction you can make? (A proper fraction must be strictly less than 1.)
Hover for answers:
1. \(\dfrac{7}{5}\)
2. \(\dfrac{8}{6}\)
3. \(\dfrac{7}{8}\)
You have four cards, numbered 5, 6, 7 and 8. You may pick one card to be the numerator and one card to be the denominator of a fraction.
• How many different fractions can you create?
Hover for answer:
Teacher resources
Teachers: log in to access the following:
Links to past exam and UKMT questions
Teachers: log in to access these.
Unlimited practice questions: Ordering fractions
In the real world
Teachers: log in to view this content. | {"url":"https://bossmaths.com/n1d/","timestamp":"2024-11-06T04:12:04Z","content_type":"text/html","content_length":"91704","record_id":"<urn:uuid:6aa51040-c9e6-4bf7-8982-e8555ca22cd7>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00544.warc.gz"} |
Catherine Greenhill
In Term 2, 2024 I taught MATH2601 Higher Linear Algebra. I am not lecturing in Term 1 or Term 3, 2024.
In Term 1, 2023 I taught my Honours course MATH5425 Graph Theory. For more detail on this course, see below. It will be offered again in Term 1, 2025.
MATH5425 Graph Theory
MATH5425 Graph Theory is a 6 UOC level V course which covers several topics in classical graph theory, as well as results proved using the probabilistic method. This course has run roughly every 2
years since 2006, and will be offered next in Term 1, 2025.
Graphs are fundamental objects in combinatorics, which can be used to model the relationships between the members of a network or system. They have many applications in areas such as computer
science, statistical physics and computational biology. Specifically, a graph is a set of vertices and a set of edges, where (generally) an edge is an unordered pair of distinct vertices. The course
covers various combinatorial aspects of graph theory and introduces some of the tools used to tackle graph theoretical questions. A particular focus will be on the use of probability to answer
questions in graph theory. This is known as the "Probabilistic Method", initiated by Erdős.
Topics include:
• matchings, coverings and packings,
• connectivity,
• graph colourings: vertex colourings and edge colourings,
• planar graphs,
• Ramsey theory,
• the probabilistic method,
• random graphs.
There are no formal prerequisites for this course, though students should be familiar with set theory, logic and proofs. There is a strong emphasis on proof in the course, and students are asked to
construct their own proofs in the assessment tasks.
The main textbook is R. Diestel, Graph Theory 5th edn. (Springer, 2017), which is also available online at diestel-graph-theory.com.
Some material is also drawn from
• B. Bollobás, Modern Graph Theory (Cambridge University Press, 1998),
• N. Alon and J. Spencer, The Probabilistic Method (Wiley 2000).
Here is a list of current and past Honours students, as well as some possible topics for an Honours project. | {"url":"https://web.maths.unsw.edu.au/~csg/teaching.html","timestamp":"2024-11-02T05:53:56Z","content_type":"application/xhtml+xml","content_length":"6253","record_id":"<urn:uuid:c7f359b2-de3c-4ae7-8759-a6ff557577d9>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00369.warc.gz"} |
Statistical cluster points and turnpike
File(s) under permanent embargo
Statistical cluster points and turnpike
Version 2 2024-06-05, 03:23
Version 1 2019-07-18, 15:09
journal contribution
posted on 2024-06-05, 03:23 authored by S Pehlivan, MA Mamedov
In this paper we study an asymptotic behaviour of optimal paths of a difference inclusion. The turnpike property in some wording [5,8, and so on] provided that there is a certain stationary point and
optimal paths converge to that point. In this case only a finite number terms of the path (sequence) remain on the outside of every neighbourhood of that point. In the present paper a statistical
cluster point introduced in [1] instead of the usual concept of limit point is considered and the turnpike theorem is proved. Here it is established that there exists a stationary point which is a
statistical cluster point for the all optimal paths. In this case not only a finite number but also infinite number terms of the path may remain on the outside of every small neighbourhood of the
stationary point, but the number of these terms in comparison with the number of terms in the neighbourhood is so small that we can say: the path "almost" remains in this neighbourhood. Note that the
main results are obtained under certain assumptions which are essentially weaker than the usual convexity assumption. These assumptions first were introduced for continuous systems in [6].
Abingdon, Eng.
Publisher DOI
Publication classification
C1.1 Refereed article in a scholarly journal
Copyright notice
2000, OPA (Overseas Publishers Association) N.V.
Taylor & Francis | {"url":"https://dro.deakin.edu.au/articles/journal_contribution/Statistical_cluster_points_and_turnpike/20765842","timestamp":"2024-11-06T17:31:48Z","content_type":"text/html","content_length":"149586","record_id":"<urn:uuid:23aab0bb-c8ed-4741-a545-76f59b217345>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00697.warc.gz"} |
How To Find Surface Area In Geometry - Geometry Spot
How To Find Surface Area In Geometry
Understanding surface area is very important, especially when dealing with three-dimensional geometric shapes. This article will guide you through the methods of finding the surface area of basic
geometric shapes.
Understanding surface area is very important, especially when dealing with three-dimensional geometric shapes. In many fields, such as chip design, heat sinks, packaging, and pharmaceuticals, surface
area is a critical parameter that directly impacts the output. Therefore, knowing how to calculate surface area is a key academic skill with practical applications. This article will guide you
through the methods of finding the surface area of basic geometric shapes.
To put it simply, the surface area of a three-dimensional object refers to the total area of the object exposed to the surrounding environment. Just imagine painting the exterior of a shape; the
amount of paint required to cover it would be directly proportional to its surface area. Unlike two-dimensional shapes that only have area, three-dimensional shapes have both volume and surface area.
Calculating the Surface Area of Common Shapes
3D shapes consist of multiple faces, which can be either flat, curved or a combination of both. Regardless of their type, all these faces contribute to the total surface area of a shape. When
calculating the surface area, it is useful to categorize the faces properly and calculate their areas individually, then add them together to obtain the total surface area of the shape.
Let’s look at some step-by-step examples of calculating surface area for common shapes.
Cuboid (Rectangular Prism):
A cuboid has six faces, each a rectangle. To find its surface area, calculate the area of each face and sum them up.
The formula is Surface Area = 2(lw + lh + wh) where l = length, w = width, h = height.
Since all faces of a cube are squares of equal size, its surface area is simply six times the area of one face. The formula is Surface Area = 6a^2 where a = side of the cube.
A cylinder has a curved surface and two circular bases. The area of the flat circular bases can be given as Area = 2πr^2 square units. Curved surfaces are more challenging to work with. It is
beneficial to map the net of the cylinder to tackle this. As the illustration shows, the net of the curved surface is a rectangle of height (h) and width equivalent to the circumference of its base.
Hence, curved Surface Area = 2πr*h square units. The total surface area is the sum of the area of these parts. The formula is: Surface Area = 2πr(h + r) where r = radius of the base, h = height.
The sphere, being entirely curved, has a surface area calculated by: Surface Area = 4πr^2 where r = radius of the sphere.
A cone has a circular base and a curved surface. Like the cylinder, visualizing the net of a cone can be helpful!
The curved surface, when unrolled, resembles a circular sector with a radius equal to the slant height of the cone, and an arc length that matches the circumference of the base. The surface area
includes the area of the base and the curved surface, given by: Surface Area = πr(r + l) where r = radius of the base, l = slant height.
Composite Shape surface area
As you advance in geometry, you’ll encounter more complex shapes like pyramids, prisms, and irregular polyhedra. The key to finding the surface area of these shapes is breaking them down into simpler
shapes (like triangles, rectangles, etc.), calculating the area of each, and then summing them up. When separating composite shapes into more basic 3D shapes be mindful about the new surfaces that
emerge upon separation. These additional surfaces should be excluded from your surface area calculations as they are not part of the exterior of the original shape.
Tips for Finding Surface Area
• Understand the Shape: Familiarize yourself with the shape you are dealing with. Know how many faces it has and what shapes these faces are.
• It’s important to note that recognizing congruent faces or symmetry in shapes can help save time in calculations. This is because it allows you to group them and multiply them instead of
calculating each face individually. By using this method, you can simplify the process and make the calculations more efficient.
• Measure Accurately: Ensure that you have the correct measurements for all the dimensions needed for your calculations. Incorrect measurements will lead to the incorrect surface area.
• Consider visualizing the net of an object. A net is essentially a 2D layout that unfolds to form a 3D shape. This technique is beneficial for curved surfaces and intricate 3D objects.
• Memorize Formulas: While understanding concepts is crucial, memorizing the formulas for surface area will save time and help in quick calculations.
• Unit Consistency: Always check that all your measurements are in the same unit. Convert if necessary before starting your calculation.
Understanding how to calculate surface area has numerous real-life applications. For example, in architecture and construction, knowing the surface area helps in estimating the material required for
building or painting a structure. In packaging and manufacturing, surface area calculations are vital for designing packaging materials. Even in fields like biology or meteorology, calculating
surface areas can be crucial for understanding processes like osmosis or heat exchange.
The ability to calculate surface area is not just a key academic skill but also a practical tool. Whether it’s for solving problems in your textbook, designing a project, or just satisfying your
curiosity about the world around you, knowing how to find the surface area of various shapes is invaluable. Keep practicing, and soon, these calculations will become second nature to you. Remember,
geometry is not just about numbers and formulas; it’s a way of understanding the space and shapes that make up our world. | {"url":"https://geometryspot.school/how-to-find-surface-area-in-geometry/","timestamp":"2024-11-04T09:09:40Z","content_type":"text/html","content_length":"156190","record_id":"<urn:uuid:29da39ac-eea8-43f4-b4e9-2bf81b03f60a>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00301.warc.gz"} |
GCSE 1 - Global Math Institute
8.1- Introduction • calculate the probability of a single event as either a fraction or a decimal • understand that the probability of an event occurring = 1 – the probability of the event not
occurring • understand relative frequency as an estimate of probability • calculate the probability of simple combined events using possibility diagrams and tree diagrams where appropriate
8.2- Probability experiments e.g. Use results of experiments with a spinner to estimate the probability of a given outcome e.g. use probability to estimate from a population
8.3- Theoretical probability e.g. P(blue) = 0.8, find P(not blue)
8.4- Mutually Exclusive Events Mutually exclusive events are events that cannot happen simultaneously, or such that the occurrence of one means that the other cannot subsequently occur. if you throw
a dice, only one face can be on top, so the events which described by the uppermost face are mutually exclusive. The probabilities of mutually exclusive events can be added to find the overall
probability of one of the events happening, so that if A and B are mutually exclusive events, then P(A∪B) =P(A)+P(b) In possibility diagrams outcomes will be represented by points on a grid and in
tree diagrams outcomes will be written at the end of branches and probabilities by the side of the branches
8.5- Summary and Review Probability – P(A ∪ B) and Mutually Exclusive Events P(A ∪ B) = P(A) + P(B) – P(A ∩ B) For mutually exclusive events, P(A ∩ B) = 0.
8.6- Assessment 8 Question 1 : What is the probability of a dice showing a 2 or a 5? Question 2 : The probabilities of three teams A, B and C winning a badminton competition are 1/3, 1/5 and 1/9
respectively. Calculate the probability that a) either A or B will win b) either A or B or C will win c) none of these teams will win d) neither A nor B will win Question 3 : Three questions now from
the higher level non-calculator paper. A fair spinner has five equal sections numbered 1, 2, 3, 4 and 5. A fair six-sided dice has five red faces and one green face. The spinner is spun. If the
spinner shows an even number, the dice is thrown. Calculate the probability of getting an even number and the colour green? | {"url":"https://www.globalmathinstitute.com/course/gcse-1/lessons/8-probability-lesson/","timestamp":"2024-11-04T20:20:27Z","content_type":"text/html","content_length":"1049490","record_id":"<urn:uuid:ff6bfdc3-7e1b-4a29-98de-defcfbb0303e>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00476.warc.gz"} |
It would be very good for me to understand this. At the moment I do not: Victor Chernozhukov et al.: [1608.00060] Double/Debiased Machine Learning for Treatment and Causal Parameters: "Most modern
supervised statistical/machine learning (ML) methods are explicitly designed to solve prediction problems very well...
...Achieving this goal does not imply that these methods automatically deliver good estimators of causal parameters. Examples of such parameters include individual regression coefficients,
average treatment effects, average lifts, and demand or supply elasticities. In fact, estimates of such causal parameters obtained via naively plugging ML estimators into estimating equations for
such parameters can behave very poorly due to the regularization bias. Fortunately, this regularization bias can be removed by solving auxiliary prediction problems via ML tools.
Specifically, we can form an orthogonal score for the target low-dimensional parameter by combining auxiliary and main ML predictions. The score is then used to build a de-biased estimator of the
target parameter which typically will converge at the fastest possible 1/root(n) rate and be approximately unbiased and normal, and from which valid confidence intervals for these parameters of
interest may be constructed. The resulting method thus could be called a "double ML" method because it relies on estimating primary and auxiliary predictive models. In order to avoid overfitting,
our construction also makes use of the K-fold sample splitting, which we call cross-fitting.
This allows us to use a very broad set of ML predictive methods in solving the auxiliary and main prediction problems, such as random forest, lasso, ridge, deep neural nets, boosted trees, as
well as various hybrids and aggregators of these methods... | {"url":"https://www.bradford-delong.com/2018/07/should-read-author-160800060-doubledebiased-machine-learning-for-treatment-and-causal-parametershttpsarxi.html","timestamp":"2024-11-02T23:38:58Z","content_type":"text/html","content_length":"33401","record_id":"<urn:uuid:cecc1c34-10c5-4649-ac34-052faff1820a>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00551.warc.gz"} |
NP Chart: Definition & Example
What is an NP Chart?
An NP chart allows a researcher to keep track of whether a measurement process is within bounds or ‘out of control’. It records the number of non conforming units or defective instances in the
measurement process.
The data it records is simple, binary data: nonconforming vs. conforming, fail vs. pass.
The NP chart is very similar to the p-chart. However, an NP chart plots the number of items while the p-chart plots proportions of items.
Reading an NP chart
The vertical (Y) axis of an NP chart will typically tell the number of defectives or nonconforming instances in each subgroup, and the horizontal axis carries the sub group designations. Sub-groups
are often time sequences (for instances, daily production in a factory). They should be equally sized, and ideally are large enough that they generally contain a few defective items.
After an initial period during which the process is known to be ‘within control’, control limits are calculated and represented on the graph as horizontal lines. They are calculated as:
• n = the number of items,
• p = proportion of defective items.
• The lower control line is bounded by 0.
These control lines allow us to see immediately when future measurements point to a process that has gone out of control and is producing too many defects.
We use NP charts to monitor whether or not our process is predictable and stable, or whether it meets standards. This makes them a key tool in statistical quality control, such as is routine in
1. Heckert & Filliben. Graphics Commands: NP Control Chart. Dataplot Reference Manual, Volume 1, Chapter 2. Retrieved from https://www.itl.nist.gov/div898/software/dataplot/refman1/ch2/np_cont.pdf
on May 13, 2018
2. NP Charts. NCSS Statistical Software, Chapter 257. Retrieved from https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/NCSS/NP_Charts.pdf on May 13, 2018.
3. McNeese, Bill. np Control Charts. SPC for Excel. Retrieved from https://www.spcforexcel.com/knowledge/attribute-control-charts/np-control-charts on May 13, 2018. | {"url":"https://www.statisticshowto.com/np-chart/","timestamp":"2024-11-04T06:04:24Z","content_type":"text/html","content_length":"68407","record_id":"<urn:uuid:25f53bdd-75ac-474b-a152-380e8153a3ff>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00437.warc.gz"} |
Analysis of the Results of Metadynamics Simulations by metadynminer and metadynminer3d
Molecular simulations and their pioneers Martin Karplus, Michael Levitt, and Arieh Warshel have been awarded the Nobel Prize in 2013 (Karplus 2013). Their methods, in particular the method of
molecular dynamics simulation, computationally simulate the motions of atoms in a molecular system. A simulation starts from a molecular system defined by positions (Cartesian coordinates) of the
individual atoms. The heart of the method is in a calculation of forces acting on individual atoms and their numerical integration in the spirit of Newtonian dynamics, i.e., the conversion of a force
vector to an acceleration vector, then velocity vector and, finally, to a new position of an atom. By repeating these steps, it is possible to reconstruct a record of atomic motions known as a
Molecular simulations have great potential in drug discovery. A molecule of drug influences (enhances or blocks) the function of some biomolecule in the patient’s body, typically a receptor, enzyme
or other protein. These molecules are called drug targets. The process of design for a new drug can be significantly accelerated with knowledge of the 3D structure (Cartesian coordinates of atoms) of
the target. With such knowledge, it is possible to find a “druggable” cavity in the target and a molecule that fits and favorably binds to this cavity to influence its function. Strong binding
implies that the drug influences the target even in low doses, hence does not cause side effects by interacting with unwanted targets.
Experimental determination of the 3D structures of proteins and other biomolecules is a very expensive and laborious process. Molecular simulations can, at least in principle, replace such expensive
and laborious experiments by computing. In principle, a molecular simulation starting from virtually any 3D shape of a molecule would end up in energetically the most favorable shape. This is
analogous with water flowing from mountains to valleys and not in the opposite way.
Unfortunately, this approach is extremely computationally expensive. The integration step of a simulation must be small enough to comprise the fastest motions in the molecular system. In practical
simulations, it is necessary to use femtosecond integration steps. This means that it is necessary to carry out thousands of steps to simulate picoseconds, millions of steps to simulate nanoseconds,
and so forth. In each step, it is necessary to evaluate a substantial number of interactions between atoms. As a result, it is possible to routinely simulate nano- to microseconds. Longer simulations
require special high-performance computing resources.
Protein folding, i.e., the transition from a quasi-random to the biologically relevant 3D structure, takes place in microseconds for very small proteins and in much longer time scales for
pharmaceutically interesting proteins. For this reason, prediction of a 3D structure by molecular simulations is limited to few small and fast folding proteins. For large proteins, it is currently
impossible or at least far from being routine.
Several methods have been developed to address this problem. Metadynamics (Laio and Parrinello 2002) uses artificial forces to force the system to explore states that have not been previously
explored in the simulation. At the beginning of the simulation, it is necessary to chose some parameters of the system referred to as collective variables. For example, numerically expressed
compactness of the protein can be used as a collective variable to accelerate its folding from a noncompact to a compact 3D structure. Metadynamics starts as a usual simulation. After a certain
number of steps (typically 500), the values of the collective variables are calculated and from this moment this state becomes slightly energetically disfavored due to the addition of an artificial
bias potential in the shape of a Gaussian hill. After another 500 steps, another hill is added to the bias potential and so forth. These Gaussian hills accumulate until they “flood” some energy
minimum and help the system to escape this minimum and explore various other states (Figure 1). In the analogy of water floating from mountains to valleys, metadynamics adds “sand” to fill valleys to
make water flow from valleys back to mountains. This makes the simulation significantly more efficient compared to a conventional simulation because the “water” does not get stuck anywhere.
Using the application of metadynamics, it is possible to significantly accelerate the process of folding. Hopefully, by the end of metadynamics we can see folded, unfolded, and many other states of
the protein. However, the interpretation of the trajectory is not straightforward. In standard molecular dynamics simulation (without metadynamics), the state which is the most populated is the most
populated in reality. This is not true anymore with metadynamics.
Packages metadynminer and metadynminer3d use the results of metadynamics simulations to calculate the free energy surface of the molecular system. The most favored states (states most populated in
reality) correspond to minima on the free energy surface. The state with the lowest free energy is the most populated state in the reality, i.e., the folded 3D structure of the protein.
As an example to illustrate metadynamics and our package, we use an ultrasimple molecule of “alanine dipeptide” (Figure 1). This molecule can be viewed as a “protein” with just one amino acid residue
(real proteins have hundreds or thousands of amino acid residues). As a collective variable it is possible to use an angle \(\phi\) defined by four atoms. Biasing of this collective variable
accelerates a slow rotation around the corresponding bond. Figure 1 shows the free energy surface of alanine dipeptide as the black thick line. It is not known before the simulation. The simulation
starts from the state B. After 500 simulation steps, the hill is added (the hill is depicted as the red line, the flooding potential (“sand”) at the top, the free energy surface with added flooding
potential at the bottom). The sum of 1, 10, 100, 200, 500, and 700 hills are depicted as red to blue lines.
At the end of simulation the free energy surface is relatively well flattened (blue line in Fig. 1 bottom). Therefore, the free energy surface can be estimated as a negative imprint of added “sand”:
\[$$G(s) = -kT \log(P(s)) = -V(s) = \sum_i w_i \exp(-(s-S_i)^2/{2 \sigma^2}), \tag{1}$$\]
where \(G\), \(V\), and \(P\) are free energy, metadynamics bias (flooding) potential, and probability, respectively, of a state with a collective variable \(s\), \(k\) is Boltzmann constant, \(T\)
is temperature in Kelvins, \(w_i\) is height, \(S_i\) is position and \(\sigma_i\) is width of each hill. The equation can be easily generalized for two or more collective variables. | {"url":"https://journal.r-project.org/articles/RJ-2022-057/","timestamp":"2024-11-03T11:17:45Z","content_type":"text/html","content_length":"1048952","record_id":"<urn:uuid:ab1fb19c-c2c2-4c3d-9575-19e9399ab250>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00199.warc.gz"} |
Using future value tables
You can then look up FV in the table and use this value as a factor in calculating the future value of an investment amount. Since PV = 1 the FV is the Future Value Future value tables are used to
carry out future value calculations without using a financial calculator. Examples and free PDF download are available. tion of the concept of present value in making capital investment decisions.
The concept itself is by no means new. Its use in the financial field dates back several.
APPENDIX A: FINANCIAL TABLES Table A1 Future Value Factors for One Dollar Com pounded at r. Percent for n. Periods. %,. (1. )n rn. F. VF r. =+ Period. 1%. Interest is an amount charged for the use of
money. Single Sum of $1 Present Value Table: How much $1 in the future is worth today, discounted at i% interest Appendix: Present Value Tables. Figure 17.1 Present Value of $1. Figure 17.2 Present
Value of Annuity Due (annuity in advance—beginning of period payments). Definition: A present value (PV) table allows you to convert a future sum, or a stream of money to be received at regular
intervals in the future, into its current The future value factor is calculated in the following way, where r is the You can also use the future value factor table to find the value of a future
value factor.
These values are often displayed in tables where the interest rate and time are specified. Find, Given, Formula. Future value
Present Value and Future Value Tables The future value factor is generally found on a table which is used to simplify calculations for amounts greater than one dollar (see example below). The future
value factor formula is based on the concept of time value of money. The concept of time value of money is that an amount today is worth more than if that same nominal amount is The present value
factor of 0.3405, is found using the tables by looking along the row for n = 14, until reaching the column for i = 8%, as shown in the preview below. Present Value Tables Download The PV tables are
available for download in PDF format by following the link below. To calculate future value with simple interest, you can use the mathematical formula FV = P times the sum of 1 + rt. In this formula,
FV is future value, and is the variable you’re solving for. P is the principal amount, r is the rate of interest per year, expressed as a decimal, and t is the number of years in the equation.
Present value and Future value tables Visit KnowledgEquity.com.au for practice questions, videos, case studies and support for your CPA studies
The future value calculator demonstrates power of the compound interest rate, or rate of return. For example, a $10,000.00 investment into an account with a 5% annual rate of return would grow to
$70,399.89 in 40 years. The 10% rate of return would increase your initial $10,000.00 to $452,592.56 in the same 40 years.
Using Present Value Tables: First question: Which table do I use? Rule: If it is a one-time payment, use the. “Present Calculates a table of the future value and interest of periodic payments. No.
year, future value, interest, effective rate Purpose of use. Future value of monthly Use this present value calculator to find today's net present value ( npv ) of a future lump sum payment
discounted to reflect the time value of money. The term "present value" plays an important part in your retirement planning. The table at the bottom of this article shows the respective present
values taking In present value, future value is given whereas in case of future value present value is already specified. Present Value vs Future Value Comparison Table. Let's The purpose of the
future value tables or FV tables is to carry out future value calculations without the use of a financial calculator. They provide the value at the end of period n of 1 received now at a discount
rate of i%. Use it as a factor to calculate $10,000 * 2.15443 = $21,544.30 which is the value of your investment, future value, after 15 years. Future value table example with monthly compounding:
You want to invest $10,000 at an annual interest rate of 5.25% that compounds monthly for 15 years.
The present value factor of 0.3405, is found using the tables by looking along the row for n = 14, until reaching the column for i = 8%, as shown in the preview below. Present Value Tables Download
The PV tables are available for download in PDF format by following the link below.
You can then look up FV in the table and use this value as a factor in calculating the future value of an investment amount. Since PV = 1 the FV is the Future Value Future value tables are used to
carry out future value calculations without using a financial calculator. Examples and free PDF download are available. tion of the concept of present value in making capital investment decisions.
The concept itself is by no means new. Its use in the financial field dates back several. 2 Aug 2019 A Present Value table is a tool that assists in the calculation of PV. A PV table includes
different coefficients depending on discount rate and APPENDIX A: FINANCIAL TABLES Table A1 Future Value Factors for One Dollar Com pounded at r. Percent for n. Periods. %,. (1. )n rn. F. VF r. =+
Period. 1%. Interest is an amount charged for the use of money. Single Sum of $1 Present Value Table: How much $1 in the future is worth today, discounted at i% interest
The future value factor is calculated in the following way, where r is the You can also use the future value factor table to find the value of a future value factor.
The future value factor is generally found on a table which is used to simplify calculations for amounts greater than one dollar (see example below). The future value factor formula is based on the
concept of time value of money. The concept of time value of money is that an amount today is worth more than if that same nominal amount is
Interest is an amount charged for the use of money. Single Sum of $1 Present Value Table: How much $1 in the future is worth today, discounted at i% interest Appendix: Present Value Tables. Figure
17.1 Present Value of $1. Figure 17.2 Present Value of Annuity Due (annuity in advance—beginning of period payments). Definition: A present value (PV) table allows you to convert a future sum, or a
stream of money to be received at regular intervals in the future, into its current The future value factor is calculated in the following way, where r is the You can also use the future value
factor table to find the value of a future value factor. But at the 5th year,discount factor at 0.784,you use present value table.So would like to know in exam how do I know which table should I be
Future Value Factor for an Ordinary Annuity. (Interest rate = r, Number of periods = n) n \ r. 1%. 2%. 3%. 4%. 5%. 6%. 7%. 8%. 9%. 10%. 11%. 12%. 13%. 14%. | {"url":"https://cryptooswpqx.netlify.app/meadors48957ryv/using-future-value-tables-cah.html","timestamp":"2024-11-13T07:43:37Z","content_type":"text/html","content_length":"35227","record_id":"<urn:uuid:c13290e3-39e0-4917-b1e8-e86895323116>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00128.warc.gz"} |
What is the Charge on Nitric acid (HNO3)? And Why?
The Charge of HNO3 (Nitric acid) is 0.
But the question is how can you say that the charge on HNO3 (Nitric acid) is 0?
Well you can say this by calculating its formal charge.
So let’s calculate the formal charge of HNO3 (Nitric acid).
If you are a visual learner like me, then here is a short two minute video for you.
Calculating the formal charge of HNO3 using lewis structure
In order to calculate the formal charge on HNO3 (Nitric acid), you should know the Lewis dot structure of HNO3 (Nitric acid).
Here is the lewis structure of HNO3.
Now using the above lewis structure of HNO3, you have to find the formal charge on each atom that is present in the HNO3 molecule.
For calculating the formal charge, you need to remember this formula;
Formal charge = Valence electrons – Nonbonding electrons – (Bonding electrons)/2
You can see the bonding and nonbonding electrons of HNO3 from the image given below.
So now let’s calculate the formal charge on each individual atom present in HNO3.
Formal charge on Hydrogen atom:
Valence electron = 1 (as it is in group 1 on periodic table) ^[1]
Nonbonding electrons = 0
Bonding electrons = 2
So according to the formula of formal charge, you will get;
Formal charge on Hydrogen = Valence electrons – Nonbonding electrons – (Bonding electrons)/2 = 1 – 0 – (2/2) = 0
So the formal charge on hydrogen atom is 0.
Formal charge on Nitrogen atom:
Valence electrons = 5 (as it is in group 15 on periodic table) ^[2]
Nonbonding electrons = 0
Bonding electrons = 8
So according to the formula of formal charge, you will get;
Formal charge on Nitrogen = Valence electrons – Nonbonding electrons – (Bonding electrons)/2 = 5 – 0 – (8/2) = 1+
So the formal charge on nitrogen atom is 1+
Formal charge on Oxygen atom (right side):
Valence electron = 6 (as it is in group 16 on periodic table) ^[3]
Nonbonding electrons = 6
Bonding electrons = 2
So according to the formula of formal charge, you will get;
Formal charge on Oxygen = Valence electrons – Nonbonding electrons – (Bonding electrons)/2 = 6 – 6 – (2/2) = 1-
So the formal charge on oxygen atom (which is on the right side) is 1-.
Formal charge on remaining Oxygen atoms:
Valence electron = 6 (as it is in group 16 on periodic table)
Nonbonding electrons = 4
Bonding electrons = 4
So according to the formula of formal charge, you will get;
Formal charge on Oxygen = Valence electrons – Nonbonding electrons – (Bonding electrons)/2 = 6 – 4 – (4/2) = 0
So the formal charge on oxygen atom is 0.
Now let’s put all these charges on the lewis dot structure of HNO3.
So there is overall 0 charge left on the entire molecule.
This indicates that the HNO3 (Nitric acid) has 0 charge.
I hope you have understood the above calculations of HNO3 (Nitric acid).
Check out some other related topics for your practice.
Related topics:
Charge on Acetic acid (CH3COOH)
Charge of O3 (Ozone)
Charge of Neon (Ne)
Charge on IO3 (Iodate ion)
Charge of Argon (Ar)
Jay is an educator and has helped more than 100,000 students in their studies by providing simple and easy explanations on different science-related topics. With a desire to make learning accessible
for everyone, he founded Knords Learning, an online learning platform that provides students with easily understandable explanations.
Read more about our Editorial process.
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Generic Class
◀ Friend Class and Friend Function▶ Traps and TipsAmazon T
he main purpose of a generic class is to provide genericity so that the same member function does not need to be rewritten to accept a different type of argument. Different programming languages
support genericity in different ways. In C++, a class template is used to provide genericity for a class. The word “template” in C++ in fact is linked to genericity.
The Standard Template Library we will be covering in
Chapter 7
is a case in point. Most of you should be well familiar with a template class; if you don’t, consult the Internet or a book for its syntax.
The following is a sample program using a template to represent a two-dimensional array of various types. In
I create one array of
and one array of
and manipulate them the same way to show you how the template class
is written to provide genericity.
#include <iostream>
using namespace std;
/* you can replace <class Type> with <typename Type> */
template <class Type>
class matrix{
Type **array;
int rows;
int cols;
matrix(int r, int c);
matrix(matrix & m); /* copy constructor */
Type* & operator[](int row) { return array[row]; }
int getRows() const { return rows; }
int getCols() const { return cols; }
void showArray();
template <class Type>
matrix<Type>::matrix(int r, int c) {
rows = r;
cols = c;
array = new Type*[r];
for(int i=0; i<r; i++)
array[i] = new Type[c];
template <class Type>
matrix<Type>::matrix(matrix & m) {
int i, j;
delete [] array;
rows = m.getRows();
cols = m.getCols();
array = new Type*[rows];
for(i=0; i<rows; i++)
array[i] = new Type[cols];
for(i=0; i<rows; i++)
for(j=0; j<cols; j++)
array[i][j] = m[i][j];
template <class Type>
void matrix<Type>::showArray() {
int i, j;
for(i=0; i<rows; i++) {
for(j=0; j<cols; j++)
cout << array[i][j] << '\t';
cout << endl;
int main(){
int i, j;
matrix<int> b(10,5); /* 10-by-5 array */
matrix<char> d(10,5);
for(i=0; i<10; i++)
for(j=0; j<5; j++)
b[i][j] = i+j; /* assigning values to every single cell */
cout << "Here are the contents of b:\n";
b.showArray(); /* display the array */
matrix<int> a(b); /* copy b to a */
cout << "\nHere are the contents of a:\n";
for(i=0; i<10; i++)
for(j=0; j<5; j++)
d[i][j] = 'a'+i+j; /* assigning values to every single cell */
cout << "Here are the contents of d:\n";
d.showArray(); /* display the array */
matrix<char> c(d); /* copy d to c */
cout << "\nHere are the contents of c:\n";
return 0;
Go ahead and add additional functions to this class template so that it can handle row and column additions and deletions.
Pay attention to the syntax of a class template and discern the differences between a class template and a normal class.
The primary use of a class template is, obviously, to accommodate a number of different data types. In most situations, however, a class is designed specifically to work with certain data types.
Therefore, don’t get intimidated by a class template; it is not as important as you may think.
◀ Friend Class and Friend Function▶ Traps and Tips | {"url":"https://cppprogramming.chtoen.com/generic-class.html","timestamp":"2024-11-01T22:51:09Z","content_type":"text/html","content_length":"34065","record_id":"<urn:uuid:d34cda9e-07ef-48d1-92ed-0e2680661074>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00506.warc.gz"} |
An improved limit on the charge of antihydrogen from stochastic acceleration
Antimatter continues to intrigue physicists because of its apparent absence in the observable Universe. Current theory requires that matter and antimatter appeared in equal quantities after the Big
Bang, but the Standard Model of particle physics offers no quantitative explanation for the apparent disappearance of half the Universe. It has recently become possible to study trapped atoms of
antihydrogen to search for possible, as yet unobserved, differences in the physical behaviour of matter and antimatter. Here we consider the charge neutrality of the antihydrogen atom. By applying
stochastic acceleration to trapped antihydrogen atoms, we determine an experimental bound on the antihydrogen charge, Qe, of |Q|< 0.71 parts per billion (one standard deviation), in which e is the
elementary charge. This bound is a factor of 20 less than that determined from the best previous measurement of the antihydrogen charge. The electrical charge of atoms and molecules of normal matter
is known to be no greater than about 10−21e for a diverse range of species including H2, He and SF6. Charge– parity–time symmetry and quantum anomaly cancellation demand that the charge of
antihydrogen be similarly small. Thus, our measurement constitutes an improved limit and a test of fundamental aspects of the Standard Model. If we assume charge superposition and use the best
measured value of the antiproton charge , then we can place a new limit on the positron charge anomaly (the relative difference between the positron and elementary charge) of about one part per
billion (one standard deviation), a 25-fold reduction compared to the current best measurement.
M. Ahmadi , M. Baquero-Ruiz, W. Bertsche, E. Butler, A. Capra, C. Carruth, C. L. Cesar, M. Charlton, A. E. Charman, S. Eriksson, L. T. Evans, N. Evetts, J. Fajans, T. Friesen, M. C. Fujiwara, D. R.
Gill, A. Gutierrez, J. S. Hangst, W. N. Hardy, M. E. Hayden, C. A. Isaac, A. Ishida, S. A. Jones, S. Jonsell, L. Kurchaninov, N. Madsen, D. Maxwell, J. T. K. McKenna, S. Menary, J. M. Michan, T.
Momose, J. J. Munich, P. Nolan, K. Olchanski, A. Olin, A. Povilus, P. Pusa, C. Ø. Rasmussen, F. Robicheaux, R. L. Sacramento, M. Sameed, E. Sarid, D. M. Silveira, C. So, T. D. Tharp, R. I. Thompson,
D. P. van der Werf, J. S. Wurtele & A. I. Zhmoginov
Nature 529, 373–376 (2016) | {"url":"https://alpha.web.cern.ch/publications/improved-limit-charge-antihydrogen-stochastic-acceleration","timestamp":"2024-11-15T03:00:32Z","content_type":"text/html","content_length":"37629","record_id":"<urn:uuid:5abd28a9-a0fc-4e0c-b176-77f799dc3675>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00807.warc.gz"} |
Introduction to Knowing Our Numbers Key Points,Notes,concept,Class - 6 (Maths) CHAPTER-1Introduction to Knowing Our Numbers Key Points,Notes,concept,Class - 6 (Maths) CHAPTER-1
Key Points of Knowing Our Numbers
Hi friends and my dear students! In this post, I have covered Knowing Our Numbers Key Points Class -6,Notes,concept. After Reading Key Points Knowing Our Numbers, Please do share it with your
friends. You can Learn Maths for All Classes here.
Knowing Our Numbers Key Points Class - 6
Knowing Our Numbers Key Points Class - 6
* Given two numbers, one with more digits is the greater number. If the number of digits in two given numbers is the same, that number is larger, which has a greater leftmost digit.
* Informing numbers from given digits, we should be careful to see if the conditions under which the numbers are to be formed are satisfied. Thus, to form the greatest four-digit number from 7, 8, 3,
5 without repeating a single digit, we need to use all four digits, the greatest number can have only 8 as the leftmost digit.
* The smallest four-digit number is 1000 (one thousand), It follows the largest three-digit number 999. Similarly, the smallest five-digit number is 10,000. It is ten thousand and follows the largest
four-digit number 9999. Further, the smallest six-digit number is 1,00,000. It is one Lakh and follows the largest five-digit number 99,999. This carries on for higher digit numbers in a similar
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* Use of commas helps in reading and writing large numbers. In the Indian system of numeration, we have commas after 3 digits starting from the right and thereafter every 2 digits. The commas after
3rd, 5th, and digits to separate thousand, lake, and crore respectively. In the International system of numeration, commas are placed after every 3 digits starting from the right. The Commas after 3^
rd and 6^th digits to separate thousand and million respectively.
* Estimation involves approximating a quantity to accuracy required. Thus, 3,107 may be approximated to 3,108 or to 3,000, i.e., to the nearest hundred or to the nearest thousand depending on our
* In a number of situations, we have to estimate the outcome of number operations. This is done by rounding off the numbers involved and getting a quick, rough answer.
* Use of numbers in Indo-Arabic system and International system.
* Estimation and Rounding off Numbers: We usually round off the numbers to the nearest 10's, 100's, 1000's, 10000's ... etc.
Introduction of large numbers:
Write the smallest and greatest of all two-digit, three-digit, four-digit, five-digit, six" digit, seven-digit, eight-digit numbers.
Sol. The smallest two-digit number is 10.
The greatest two-digit number is 99.
The smallest three-digit number is 100.
The greatest three-digit number is 999.
The smallest four-digit number is 1000.
The greatest four-digit number is 9999.
The smallest five-digit number is 10000.
The greatest five-digit number is 99999.
The smallest six-digit number is 100000.
The greatest six-digit number is 999999.
The smallest seven-digit number is 1000000.
The greatest seven-digit number is 9999999.
The smallest eight-digit number is 10000000.
The greatest eight digit number is 99999999.
Place value of large numbers: Use of 'comma' helps us in reading and writing of large numbers. Ex. 95,940 ; 1, 90, 407; 95,04,159; 1,82,09,370.
1 crore = 100 lakhs = 10,000 thousands
1 lakh = 100 thousands = 1000 hundreds
10 lakhs = 1 million
1 crore 10 million
10 crore 100 million
100 crore 1 billion
* Large numbers used in daily life situations:
The unit of length is meter.
The unit of weight is the kilogram.
The unit of volume is the liter. The unit of time is second.
* We use Centimetre of measuring the length of a pencil.
* We use meter length of a pen. We use kilometres for length of a saree.
* measuring the distance between We use millimeter for measuring the thickness of paper.
10 millimetres 1 Centimetre
100 centimetres 1 meter
1000 meters 1 kilometre
1 kilometre = 1000 x 100 x 10 millimetres 10,00,000 mm
1 kilogram = 1000 gms and 1 gram 1000 milligrams
1 kilolitre = 1000 litres
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