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Writing Scalar Objective Functions - MATLAB & Simulink - MathWorks Australia
Local Functions and Nested Functions
Anonymous Function Objectives
Provide Derivatives For Solvers
How to Include Gradients
Including Hessians
Benefits of Including Derivatives
Choose Input Hessian Approximation for interior-point fmincon
A scalar objective function file accepts one input, say x, and returns one real scalar output, say f. The input x can be a scalar, vector, or matrix. A function file can return more outputs (see Including Gradients and Hessians).
For example, suppose your objective is a function of three variables, x, y, and z:
Write this function as a file that accepts the vector xin = [x;y;z] and returns f:
Save it as a file named myObjective.m to a folder on your MATLAB® path.
Check that the function evaluates correctly:
For information on how to include extra parameters, see Passing Extra Parameters. For more complex examples of function files, see Minimization with Gradient and Hessian Sparsity Pattern or Minimization with Bound Constraints and Banded Preconditioner.
Functions can exist inside other files as local functions or nested functions. Using local functions or nested functions can lower the number of distinct files you save. Using nested functions also lets you access extra parameters, as shown in Nested Functions.
For example, suppose you want to minimize the myObjective.m objective function, described in Function Files, subject to the ellipseparabola.m constraint, described in Nonlinear Constraints. Instead of writing two files, myObjective.m and ellipseparabola.m, write one file that contains both functions as local functions:
Solve the constrained minimization starting from the point [1;1;1]:
Use anonymous functions to write simple objective functions. For more information about anonymous functions, see What Are Anonymous Functions?. Rosenbrock's function is simple enough to write as an anonymous function:
Check that anonrosen evaluates correctly at [-1 2]:
Minimizing anonrosen with fminunc yields the following results:
For fmincon and fminunc, you can include gradients in the objective function. Generally, solvers are more robust, and can be slightly faster when you include gradients. See Benefits of Including Derivatives. To also include second derivatives (Hessians), see Including Hessians.
The following table shows which algorithms can use gradients and Hessians.
fmincon active-set Optional No
interior-point Optional Optional (see Hessian for fmincon interior-point algorithm)
sqp Optional No
trust-region-reflective Required Optional (see Hessian for fminunc trust-region or fmincon trust-region-reflective algorithms)
fminunc quasi-newton Optional No
trust-region Required Optional (see Hessian for fminunc trust-region or fmincon trust-region-reflective algorithms)
Write code that returns:
The objective function (scalar) as the first output
The gradient (vector) as the second output
Set the SpecifyObjectiveGradient option to true using optimoptions. If appropriate, also set the SpecifyConstraintGradient option to true.
Optionally, check if your gradient function matches a finite-difference approximation. See Checking Validity of Gradients or Jacobians.
For most flexibility, write conditionalized code. Conditionalized means that the number of function outputs can vary, as shown in the following example. Conditionalized code does not error depending on the value of the SpecifyObjectiveGradient option. Unconditionalized code requires you to set options appropriately.
For example, consider Rosenbrock's function
f\left(x\right)=100{\left({x}_{2}-{x}_{1}^{2}\right)}^{2}+{\left(1-{x}_{1}\right)}^{2},
which is described and plotted in Solve a Constrained Nonlinear Problem, Solver-Based. The gradient of f(x) is
\nabla f\left(x\right)=\left[\begin{array}{c}-400\left({x}_{2}-{x}_{1}^{2}\right){x}_{1}-2\left(1-{x}_{1}\right)\\ 200\left({x}_{2}-{x}_{1}^{2}\right)\end{array}\right],
rosentwo is a conditionalized function that returns whatever the solver requires:
nargout checks the number of arguments that a calling function specifies. See Find Number of Function Arguments.
The fminunc solver, designed for unconstrained optimization, allows you to minimize Rosenbrock's function. Tell fminunc to use the gradient and Hessian by setting options:
Run fminunc starting at [-1;2]:
You can include second derivatives with the fmincon 'trust-region-reflective' and 'interior-point' algorithms, and with the fminunc 'trust-region' algorithm. There are several ways to include Hessian information, depending on the type of information and on the algorithm.
You must also include gradients (set SpecifyObjectiveGradient to true and, if applicable, SpecifyConstraintGradient to true) in order to include Hessians.
Hessian for fminunc trust-region or fmincon trust-region-reflective algorithms
Hessian for fmincon interior-point algorithm
Hessian for fminunc trust-region or fmincon trust-region-reflective algorithms. These algorithms either have no constraints, or have only bound or linear equality constraints. Therefore the Hessian is the matrix of second derivatives of the objective function.
Include the Hessian matrix as the third output of the objective function. For example, the Hessian H(x) of Rosenbrock’s function is (see How to Include Gradients)
H\left(x\right)=\left[\begin{array}{cc}1200{x}_{1}^{2}-400{x}_{2}+2& -400{x}_{1}\\ -400{x}_{1}& 200\end{array}\right].
Include this Hessian in the objective:
Set HessianFcn to 'objective'. For example,
Hessian for fmincon interior-point algorithm. The Hessian is the Hessian of the Lagrangian, where the Lagrangian L(x,λ) is
L\left(x,\lambda \right)=f\left(x\right)+\sum {\lambda }_{g,i}{g}_{i}\left(x\right)+\sum {\lambda }_{h,i}{h}_{i}\left(x\right).
g and h are vector functions representing all inequality and equality constraints respectively (meaning bound, linear, and nonlinear constraints), so the minimization problem is
\underset{x}{\mathrm{min}}f\left(x\right)\text{ subject to }g\left(x\right)\le 0,\text{ }h\left(x\right)=0.
For details, see Constrained Optimality Theory. The Hessian of the Lagrangian is
{\nabla }_{xx}^{2}L\left(x,\lambda \right)={\nabla }^{2}f\left(x\right)+\sum {\lambda }_{g,i}{\nabla }^{2}{g}_{i}\left(x\right)+\sum {\lambda }_{h,i}{\nabla }^{2}{h}_{i}\left(x\right).
To include a Hessian, write a function with the syntax
hessian is an n-by-n matrix, sparse or dense, where n is the number of variables. If hessian is large and has relatively few nonzero entries, save running time and memory by representing hessian as a sparse matrix. lambda is a structure with the Lagrange multiplier vectors associated with the nonlinear constraints:
fmincon computes the structure lambda and passes it to your Hessian function. hessianfcn must calculate the sums in Equation 1. Indicate that you are supplying a Hessian by setting these options:
For example, to include a Hessian for Rosenbrock’s function constrained to the unit disk
{x}_{1}^{2}+{x}_{2}^{2}\le 1
, notice that the constraint function
g\left(x\right)={x}_{1}^{2}+{x}_{2}^{2}-1\le 0
has gradient and second derivative matrix
\begin{array}{c}\nabla g\left(x\right)=\left[\begin{array}{c}2{x}_{1}\\ 2{x}_{2}\end{array}\right]\\ {H}_{g}\left(x\right)=\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right].\end{array}
Write the Hessian function as
Save hessianfcn on your MATLAB path. To complete the example, the constraint function including gradients is
Solve the problem including gradients and Hessian.
For other examples using an interior-point Hessian, see fmincon Interior-Point Algorithm with Analytic Hessian and Calculate Gradients and Hessians Using Symbolic Math Toolbox.
Hessian Multiply Function. Instead of a complete Hessian function, both the fmincon interior-point and trust-region-reflective algorithms allow you to supply a Hessian multiply function. This function gives the result of a Hessian-times-vector product, without computing the Hessian directly. This can save memory. The SubproblemAlgorithm option must be 'cg' for a Hessian multiply function to work; this is the trust-region-reflective default.
The syntaxes for the two algorithms differ.
For the interior-point algorithm, the syntax is
The result W should be the product H*v, where H is the Hessian of the Lagrangian at x (see Equation 1), lambda is the Lagrange multiplier (computed by fmincon), and v is a vector of size n-by-1. Set options as follows:
Supply the function HessMultFcn, which returns an n-by-1 vector, where n is the number of dimensions of x. The HessianMultiplyFcn option enables you to pass the result of multiplying the Hessian by a vector without calculating the Hessian.
The trust-region-reflective algorithm does not involve lambda:
The result W = H*v. fmincon passes H as the value returned in the third output of the objective function (see Hessian for fminunc trust-region or fmincon trust-region-reflective algorithms). fmincon also passes v, a vector or matrix with n rows. The number of columns in v can vary, so write HessMultFcn to accept an arbitrary number of columns. H does not have to be the Hessian; rather, it can be anything that enables you to calculate W = H*v.
Set options as follows:
For an example using a Hessian multiply function with the trust-region-reflective algorithm, see Minimization with Dense Structured Hessian, Linear Equalities.
If you do not provide gradients, solvers estimate gradients via finite differences. If you provide gradients, your solver need not perform this finite difference estimation, so can save time and be more accurate, although a finite-difference estimate can be faster for complicated derivatives. Furthermore, solvers use an approximate Hessian, which can be far from the true Hessian. Providing a Hessian can yield a solution in fewer iterations. For example, see the end of Calculate Gradients and Hessians Using Symbolic Math Toolbox.
For constrained problems, providing a gradient has another advantage. A solver can reach a point x such that x is feasible, but, for this x, finite differences around x always lead to an infeasible point. Suppose further that the objective function at an infeasible point returns a complex output, Inf, NaN, or error. In this case, a solver can fail or halt prematurely. Providing a gradient allows a solver to proceed. To obtain this benefit, you might also need to include the gradient of a nonlinear constraint function, and set the SpecifyConstraintGradient option to true. See Nonlinear Constraints.
The fmincon interior-point algorithm has many options for selecting an input Hessian approximation. For syntax details, see Hessian as an Input. Here are the options, along with estimates of their relative characteristics.
'bfgs' (default) High (for large problems) High
'lbfgs' Low to Moderate Moderate
'fin-diff-grads' Low Moderate
'HessianMultiplyFcn' Low (can depend on your code) Moderate
'HessianFcn' ? (depends on your code) High (depends on your code)
Use the default 'bfgs' Hessian unless you
Run out of memory — Try 'lbfgs' instead of 'bfgs'. If you can provide your own gradients, try 'fin-diff-grads', and set the SpecifyObjectiveGradient and SpecifyConstraintGradient options to true.
Want more efficiency — Provide your own gradients and Hessian. See Including Hessians, fmincon Interior-Point Algorithm with Analytic Hessian, and Calculate Gradients and Hessians Using Symbolic Math Toolbox.
The reason 'lbfgs' has only moderate efficiency is twofold. It has relatively expensive Sherman-Morrison updates. And the resulting iteration step can be somewhat inaccurate due to the 'lbfgs' limited memory.
The reason 'fin-diff-grads' and HessianMultiplyFcn have only moderate efficiency is that they use a conjugate gradient approach. They accurately estimate the Hessian of the objective function, but they do not generate the most accurate iteration step. For more information, see fmincon Interior Point Algorithm, and its discussion of the LDL approach and the conjugate gradient approach to solving Equation 38.
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Geometry Problem: \(\cos x = \cosh x?\) - Geoff Pilling | Brilliant
\cos x = \cosh x?
Billy Tangent naively thought that the hyperbolic cosine function and the standard cosine function were the same. To make sure, he tried one real value and, sure enough, he got the same result.
What value did he try?
If you think there are multiple values for which this would work, enter 99999 as your answer.
If you think there are no values for which this would work, enter 88888 as your answer.
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\checkmark
Shuffling is used to randomly assign validators to committees and choose block proposers.
Ethereum 2 uses a "swap-or-not" shuffle.
Swap-or-not is an oblivious shuffle: it can be applied to single list elements and subsets.
This makes it ideal for supporting light clients.
Shuffling is used to randomly assign validators to committees, both attestation committees and sync committees. It is also used to select the block proposer at each slot.
Although there are pitfalls to be aware of, shuffling is a well understood problem in computer science. The gold standard is probably the Fisher–Yates shuffle. So why aren't we using that for Eth2? In short: light clients.
Other shuffles rely on processing the entire list of elements to find the final ordering. We wish to spare light clients this burden. Ideally, they should deal with only the subsets of lists that they are interested in. Therefore, rather than Fisher–Yates, we are using a construction called a "swap-or-not" shuffle. The swap-or-not shuffle can tell you the destination index (or, conversely, the origin index) of a single list element, so is ideal when dealing with subsets of the whole validator set.
For example, formally committees are assigned by shuffling the full validator list and then taking contiguous slices of the resulting permutation. If I only need to know the members of committee
k
, then this is very inefficient. Instead, I can run the swap-or-not shuffle backwards for only the indices in slice
k
to find out which of the whole set of validators would be shuffled into
k
. This is much more efficient.
Swap-or-not Specification
The algorithm for shuffling in the specification deals with only a single index at a time.
An index position in the list to be shuffled, index, is provided, along with the total number of indices, index_count, and a seed value. The output is the index that the initial index gets shuffled to.
The hash functions used to calculate pivot and source are deterministic, and are used to generate pseudo-random output from the inputs: given the same input, they will generate the same output. So we can see that, for given values of index, index_count, and seed, the routine will always return the same output.
The shuffling proceeds in rounds. In each round, a pivot index is pseudo-randomly chosen somewhere in the list, based only on the seed value and the round number.
Next, an index flip is found, which is pivot - index, after accounting for wrap-around due to the modulo function. The important points are that, given pivot, every index maps to a unique flip, and that the calculation is symmetrical, so that flip maps to index.
With index_count = 100, pivot = 70, index = 45, we get flip = 25.
Finally in the round, a decision is made as to whether to keep the index as-is, or to update it to flip. This decision is pseudo-randomly made based on the values of seed, the round number, and the higher of index and flip.
Note that basing the swap-or-not decision on the higher of index and flip brings a symmetry to the algorithm. Whether we are considering the element at index or the element at flip, the decision as to whether to swap the elements or not will be the same. This is the key to seeing the that full algorithm delivers a shuffling (permutation) of the original set.
The algorithm proceeds with the next iteration based on the updated index.
It may not be immediately obvious, but since we are deterministically calculating flip based only on the round number, the shuffle can be run in reverse simply by running from SHUFFLE_ROUND_COUNT - 1 to 0. The same swap-or-not decisions will be made in reverse. As described above, this reverse shuffle is perfect for finding which validators ended up in a particular committee.
A full shuffle
To get an intuition for how this single-index shuffle can deliver a full shuffling of a list of indices, we can consider how the algorithm is typically implemented in clients when shuffling an entire list at once.
As an optimisation, the loop over the indices to be shuffled is brought inside the loop over rounds. This hugely reduces the amount of hashing required since the pivot is fixed for the round (it does not depend on the index) and the bits of source can be reused for 256 consecutive indices, since the hash has a 256-bit output.
For each round, we do the following.
1. Choose a pivot and find the first mirror index
First, we pick a pivot index
p
. This is pseudo-randomly chosen, based on the round number and some other seed data. The pivot is fixed for the rest of the round.
With this pivot, we then pick the mirror index
m_1
halfway between
p
0
m_1 = p / 2
. (We will simplify by ignoring off-by-one rounding issues for the purposes of this explanation.)
The pivot and the first mirror index.
2. Traverse first mirror to pivot, swapping or not
For each index between the mirror index
m_1
and the pivot index
p
, we decide whether we are going to swap the element or not.
Consider the element at index
i
. If we choose not to swap it, we just move on to consider the next index.
If we do decide to swap, then we exchange the list element at
i
with that at
i'
, its image in the mirror index. That is,
is swapped with
i' = m_1 - (i - m_1)
i
i'
are equidistant from
m_1
. In practice we don't exchange the elements at this point, we just update the indices
i \rightarrow i'
i' \rightarrow i
We make the same swap-or-not decision for each index between
m_1
p
Swapping or not from the first mirror up to the pivot.
The decision as to whether to swap or not is based on hashing together the random seed, the round number, and some position data. A single bit is extracted from this hash for each index, and the swap is made or not according to whether this bit is one or zero.
3. Calculate the second mirror index
After considering all the indices
i
m_1
p
, mirroring in
m_1
, we now find a second mirror index at
m_2
, which is the point equidistant between
p
and the end of the list:
m_2 = m_1 + n / 2
The second mirror index.
4. Traverse pivot to second mirror, swapping or not
Finally, we repeat the swap-or-not process, considering all the points
j
from the pivot,
p
to the second mirror
m_2
. If we choose not to swap, we just move on. If we choose to swap then we exchange the element at
j
with its image at
j'
in the mirror index
m_2
j' = m_2 + (m_2 - j)
Swapping or not from the pivot to the second mirror.
At the end of the round, we have considered all the indices between
m_1
m_2
, which, by construction, is half of the total indices. For each index considered, we have either left the element in place, or swapped the element at a distinct index in the other half. Thus, all of the indices have been considered exactly once for swapping.
The next round begins by incrementing (or decrementing for a reverse shuffle) the round number, which gives us a new pivot index, and off we go again.
The whole process running from one mirror to the other in a single round.
When deciding whether to swap or not for each index, the algorithm cleverly bases its decision on the higher of the candidate index or its image in the mirror. That is,
i
i'
(when below the pivot), and
j'
j
(when above the pivot). This means that we have flexibility when running through the indices of the list: we could do
0
m_1
p
m_2
as two separate loops, or do it with a single loop from
m_1
m_2
as I outlined above. The result will be the same: it doesn't matter if we are considering
i
or its image
i'
; the decision as to whether to swap or not has the same outcome.
In Ethereum 2.0 we do 90 rounds of the algorithm per shuffle, set by the constant SHUFFLE_ROUND_COUNT [TODO - link]. The original paper on which this technique is based suggests that
6\lg{N}
rounds is required "to start to see a good bound on CCA-security", where
N
is the list length. In his annotated spec Vitalik says "Expert cryptographer advice told us ~
4\log_2{N}
is sufficient for safety". The absolute maximum number of validators in Eth2, thus the maximum size of the list we would ever need to shuffle, is about
2^{22}
(4.2 million). On Vitalik's estimate that gives us 88 rounds required, on the paper's estimate, 92 rounds (assuming that
\lg
is the natural logarithm). So we are in the right ballpark, especially as we are very, very unlikely to end up with that many active validators.
It might be interesting to make the number of rounds adaptive based on list length. But we don't do that; it's probably an optimisation too far.
Fun fact: when Least Authority audited the beacon chain specification, they initially found bias in the shuffling used for selecting block proposers (see Issue F in their report). This turned out to be due to mistakenly using a configuration that had only 10 rounds of shuffling. When they increased it to the 90 we use for mainnet, the bias no longer appeared.
(Pseudo) randomness
The algorithm requires that we select a pivot point randomly in each round, and randomly choose whether to swap each element or not in each round.
In Eth2, we deterministically generate the "randomness" from a seed value, such that the same seed will always generate the same shuffling.
The pivot index is generated from eight bytes of a SHA256 hash of the seed concatenated with the round number, so it usually changes each round.
The decision bits used to determine whether or not to swap elements are bits drawn from SHA256 hashes of the seed, the round number, and the index of the element within the list.
This shuffling algorithm is much slower than Fisher–Yates. That algorithm requires
N
swaps. Our algorithm will require
90N/4
swaps on average to shuffle
N
We should also consider the generation of pseudo-randomness, which is the most expensive part of the algorithm. Fisher–Yates needs something like
N\log_2{N}
bits of randomness, and we need
90(\log_2{N} + N/2)
bits, which, for the range of
N
we need in Eth2, is many more bits (about twice as many when
N
is a million).
Why swap-or-not?
Why would we use such an inefficient implementation?
Shuffling single elements
The brilliance is that, if we are interested in only a few indices, we do not need to compute the shuffling of the whole list. In fact, we can apply the algorithm to a single index to find out which index it will be swapped with.
So, if we want to know where the element with index 217 gets shuffled to, we can run the algorithm with only that index; we do not need to shuffle the whole list. Moreover, if we want to know the converse, which element gets shuffled into index 217, we can just run the algorithm backwards for element 217 (backwards means running the round number from high to low rather than low to high).
In summary, we can compute the destination of element
in
O(1)
operations, and the source of element
i'
(the inverse operation) also in
O(1)
, not dependent on the length of the list. Shuffles like the Fisher–Yates shuffle do not have this property and cannot work with single indices, they always need to iterate the whole list. The technical term for a shuffle having this property is that it is oblivious (to all the other elements in the list).
Keeping light clients light
This property is important for light clients. Light clients are observers of the Eth2 beacon and shard chains that do not store the entire state, but do wish to be able to securely access data on the chains. As part of verifying that they have the correct data—that no-one has lied to them—it is necessary to compute the committees that attested to that data. This means shuffling, and we don't want light clients to have to hold and shuffle the entire list of validators. By using the swap-or-not shuffle, light clients need only to consider the small subset of validators that they are interested in, which is vastly more efficient overall.
The initial discussion about the search for a good shuffling algorithm is Issue 323 on the specs repo.
The winning algorithm was announced in Issue 563.
The original paper describing the swap-or-not shuffle is Hoang, Morris, and Rogaway, 2012, "An Enciphering Scheme Based on a Card Shuffle". See the "generalized domain" algorithm on page 3.
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Engora Data Blog: The datasaurus: always visualize your data
The datasaurus: always visualize your data
The summary is not the whole picture
If you just use summary statistics to describe your data, you can miss the bigger picture, sometimes literally so. In this blog post, I'm going to show you how relying on summaries alone can lead you catastrophically astray and I'm going to tell you how you can avoid making career-damaging mistakes.
The datasaurus is why you need to visualize your data. Source: Alberto Cairo. Open source.
Summary statistics are parameters like the mean, standard deviation, and correlation coefficient; they summarize the properties of the data and the relationship between variables. For example, if the correlation coefficient, r, is about 0.8 for two data sets x and y, we might think there's a relationship between them, but if it's about 0, we might think there isn't.
The use of summary statistics is widely taught, every textbook emphasizes them, and almost everyone uses them. But if you use summary statistics in isolation from other methods you might miss important relationships - you should always visualize your data as we'll see.
Take a look at the four plots below. They're obviously quite different, but they all have the same summary statistics!
Here are the summary statistics data:
Mean of x 9
Sample variance of x :
{\displaystyle \sigma ^{2}}
Mean of y 7.50
Sample variance of y :
{\displaystyle \sigma ^{2}}
Correlation between x and y 0.816
Linear regression line y = 3.00 + 0.500x
Coefficient of determination of the linear regression :
{\displaystyle R^{2}}
These plots were developed in 1973 by the statistician Francis Anscombe to make exactly this point: you can't rely on summary statistics, you need to visualize your data. The graphical relationship between the x and y variables are different in each case and imply different things. By plotting the data out, we can see what the relationships are, but summary statistics hide what's going on.
Let's zoom forward to 2016. The justly famous Alberto Cairo tweeted about Anscombe's quartet and illustrated the point with this cool set of summary statistics. He later expanded on his tweet in a short blog post.
x standard deviation 16.7651
y mean 47.8323
y standard deviation 26.9353
Pearson correlation -0.0645
What might you conclude from these summary statistics? I might say, the correlation coefficient is close to zero so there's not much of a relationship between the x and the y variables. I might conclude there's no interesting relationship between the x and y variables - but I would be wrong.
The summary might not mean anything to you, but the visualization surely will. This is the datasaurus data set, the x and the y variables draw out a dinosaur.
Two researchers at Autodesk Research took things a stage further. They started with Alberto Cairo's datasaurus and created a dozen other charts with exactly the same summary statistics as the datasaurus. Here they all are.
The summary statistics look like noise, but the charts reveal the underlying relationships between the x and y variables. Some of these relationships are obviously fun, like the star, but there are others that imply more meaningful relationships.
If all this sounds a bit abstract, let think about how this might manifest itself in business. Let's imagine you're an analyst working for a large company. You have data on sales by store size for Europe and you've been asked to analyze the data to gain insights. You're under time pressure, so you fire up a Python notebook and get some quick summary statistics. You get summary statistics that look like the ones I showed you above. So you conclude there's nothing interesting in the data; but you might be very wrong.
You should plot the data out and look at the chart. You might see something that looks like the slanting charts above, maybe something like this:
the individual diagonal lines might correspond to different European countries (different regulations, different planning rules, different competition, etc.). There could be a very significant relationship that you would have missed by relying on summary data.
(The Autodesk Research team haves posted their work as a paper you can read here.)
The lessons you should take away from all this are simple:
summary statistics hide a lot
there are many relationships between variables that will give summary statistics that look like noise
always visualize your data!
Labels: analytics, charts, data analytics, data science, statistics
The gamblers' fallacy
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Quadratic Equations: Level 4 Challenges Practice Problems Online | Brilliant
\alpha,\beta
x^2-\left( 3+2^{\sqrt{\log_23}}-3^{\sqrt{\log_32}} \right)x-2\left( 3^{\log_32}-2^{\log_23} \right) =0
\left( \alpha^2+\alpha\beta+\beta^2 \right)
Find the number of quadratic equations which are unchanged by squaring their roots.
a, b
c
satisfy the condition that the quadratic equation
ax^2 - bx +c =0
has 2 distinct real roots that are strictly between 0 and 1. What is the minimum value of
a
a
[-1000,1000]
{25}^x+(a+2).5^x-(a+3)<0
x
If the McDonald's logo were stored as a set of pixels, enlargement would quickly result in distorted or pixelated images, which are an eyesore. As such, companies often make vector images of their logos, in which the information is stored as mathematical formulae. Such vector images are easily scaled while maintaining sharp, crisp images.
As a first approximation, the logo is deconstructed and approximated as 2 parabolic curves of the form
y = -A(x-5)^2
y = - A (x+5)^2
. The McDonald's logo has a height to length ratio of 1.05. What is
A
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Andreas Dechant, Shin-ichi Sasa
We discuss how to use correlations between different physical observables to improve recently obtained thermodynamics bounds, notably the fluctuation-response inequality and the thermodynamic uncertainty relation (TUR). We show that increasing the number of measured observables will always produce a tighter bound. This tighter bound becomes particularly useful if one of the observables is a conserved quantity, whose expectation is invariant under a given perturbation of the system. For the case of the TUR, we show that this applies to any function of the state of the system. The resulting correlation TUR takes into account the correlations between a current and a noncurrent observable, thereby tightening the TUR. We demonstrate our finding on a model of the
{\mathrm{F}}_{1}
-ATPase molecular motor, a Markov jump model consisting of two rings and transport through a two-dimensional channel. We find that the correlation TUR is significantly tighter than the TUR and can be close to an equality even far from equilibrium.
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Autocorrelation and Partial Autocorrelation - MATLAB & Simulink - MathWorks Benelux
What Are Autocorrelation and Partial Autocorrelation?
Theoretical ACF and PACF
Sample ACF and PACF
Compute Sample ACF and PACF in MATLAB®
Generate Synthetic Time Series
Plot and Compute ACF
Plot and Compute PACF
Use Econometric Modeler
Autocorrelation is the linear dependence of a variable with itself at two points in time. For stationary processes, autocorrelation between any two observations depends only on the time lag h between them. Define Cov(yt, yt–h) = γh. Lag-h autocorrelation is given by
{\rho }_{h}=Corr\left({y}_{t},{y}_{t-h}\right)=\frac{{\gamma }_{h}}{{\gamma }_{0}}.
The denominator γ0 is the lag 0 covariance, that is, the unconditional variance of the process.
Correlation between two variables can result from a mutual linear dependence on other variables (confounding). Partial autocorrelation is the autocorrelation between yt and yt–h after the removal of any linear dependence on y1, y2, ..., yt–h+1. The partial lag-h autocorrelation is denoted
{\varphi }_{h,h}.
The autocorrelation function (ACF) for a time series yt, t = 1,...,N, is the sequence
{\rho }_{h},
h = 1, 2,...,N – 1. The partial autocorrelation function (PACF) is the sequence
{\varphi }_{h,h},
h = 1, 2,...,N – 1.
The theoretical ACF and PACF for the AR, MA, and ARMA conditional mean models are known, and are different for each model. These differences among models are important to keep in mind when you select models.
Conditional Mean Model
ACF Behavior
PACF Behavior
AR(p) Tails off gradually Cuts off after p lags
MA(q) Cuts off after q lags Tails off gradually
ARMA(p,q) Tails off gradually Tails off gradually
Sample autocorrelation and sample partial autocorrelation are statistics that estimate the theoretical autocorrelation and partial autocorrelation. Using these qualitative model selection tools, you can compare the sample ACF and PACF of your data against known theoretical autocorrelation functions [1].
For an observed series y1, y2,...,yT, denote the sample mean
\overline{y}.
The sample lag-h autocorrelation is given by
{\stackrel{^}{\rho }}_{h}=\frac{{\sum }_{t=h+1}^{T}\left({y}_{t}-\overline{y}\right)\left({y}_{t-h}-\overline{y}\right)}{{\sum }_{t=1}^{T}{\left({y}_{t}-\overline{y}\right)}^{2}}.
The standard error for testing the significance of a single lag-h autocorrelation,
{\stackrel{^}{\rho }}_{h}
, is approximately
S{E}_{\rho }=\sqrt{\left(1+2{\sum }_{i=1}^{h-1}{\stackrel{^}{\rho }}_{i}^{2}\right)/N}.
When you use autocorr to plot the sample autocorrelation function (also known as the correlogram), approximate 95% confidence intervals are drawn at
±2SE\rho
by default. Optional input arguments let you modify the calculation of the confidence bounds.
The sample lag-h partial autocorrelation is the estimated lag-h coefficient in an AR model containing h lags,
{\stackrel{^}{\varphi }}_{h,h}.
The standard error for testing the significance of a single lag-h partial autocorrelation is approximately
1/\sqrt{N}.
When you use parcorr to plot the sample partial autocorrelation function, approximate 95% confidence intervals are drawn at
±2/\sqrt{N}
This example shows how to compute and plot the sample ACF and PACF of a time series by using the Econometrics Toolbox™ functions autocorr and parcorr, and the Econometric Modeler app.
Simulate an MA(2) process
{\mathit{y}}_{\mathit{t}}
by filtering a series of 1000 standard Gaussian deviates
{\epsilon }_{\mathit{t}}\text{\hspace{0.17em}}
through the difference equation
{y}_{t}={\epsilon }_{t}-{\epsilon }_{t-1}+{\epsilon }_{t-2}.
e = randn(1000,1);
y = filter([1 -1 1],1,e);
Plot the sample ACF of
{\mathit{y}}_{\mathit{t}}
by passing the simulated time series to autocorr.
The sample autocorrelation of lags greater than 2 is insignificant.
Compute the sample ACF by calling autocorr again. Return the first output argument.
acf = autocorr(y)
acf = 21×1
acf(j) is the sample autocorrelation of
{\mathit{y}}_{\mathit{t}}
at lag j – 1.
Plot the sample PACF of
{\mathit{y}}_{\mathit{t}}
by passing the simulated time series to parcorr.
The sample PACF gradually decreases with increasing lag.
Compute the sample PACF by calling parcorr again. Return the first output argument.
pacf = parcorr(y)
pacf = 21×1
pacf(j) is the sample partial autocorrelation of
{\mathit{y}}_{\mathit{t}}
The sample ACF and PACF suggest that
{\mathit{y}}_{\mathit{t}}
is an MA(2) process.
Open the Econometric Modeler app by entering econometricModeler at the command prompt.
Load the simulated time series y.
On the Econometric Modeler tab, in the Import section, select Import > Import From Workspace.
In the Import Data dialog box, in the Import? column, select the check box for the y variable.
The variable y1 appears in the Data Browser, and its time series plot appears in the Time Series Plot(y1) figure window.
Plot the sample ACF by clicking ACF on the Plots tab.
Plot the sample PACF by clicking PACF on the Plots tab. Position the PACF plot below the ACF plot by dragging the PACF(y1) tab to the lower half of the document.
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Displaying equations on the web (Shallow Thoughts)
Displaying equations on the web
How do you show equations on a web page? Every now and then, I write an article that involves math, and I wrestle with that problem.
The obvious (but wrong) approach: MathML
It was nearly fifteen years ago that MathML was recommended as a standard for embedding equations inside an HTML page. I remember being excited about it back then. There were a few problems -- like the availability of fonts including symbols for integrals, summations and so forth -- but they seemed minor. That was 1998.
Now, in 2012, I found myself wanting to write an article involving an integral, so I looked into the state of MathML. I found that even now, all these years later, it wasn't widely supported.
In Firefox I could show some simple equations, like
{\int }_{\mathrm{x\; =\; 0}}^{\infty }\frac{dx}{x}
x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}
But when I tried them in Chromium, I learned that webkit-based browsers don't support MathML. At all. The exception is Safari: apparently Apple has added some MathML support into their browser but hasn't contributed that code back to webkit (yet?)
Besides that, MathML is ridiculously hard to use. Here's the code for that little integral:
<mn>x = 0</mn>
<mi>∞</mi>
Ugh! You can't even specify infinity without using an HTML numeric entity. And the code for the quadratic equation is even worse (use View Source if you want to see it).
Good ol' tables
Several years ago, I wrote about the Twelve Days of Christmas and how to calculate the total number of gifts represented in the song.
I needed summations, and I was rather proud of working out a way to use HTML tables to display all the sums and line up everything correctly. It wasn't exactly publication-quality graphics, but it was readable.
More recently, I worked out a way to do exponentials that way, and found a hint about how to do integrals:
∫ P (t) dt
P0 = ————
Looks a little better than the tiny MathML version. But the code isn't any easier to read:
<tr><td><td align="center"><small><i>now</i></small></td><td></td><td></td></tr>
<td rowspan="3" valign="middle"><font size="6" style="font-size:3em" class="bigsym">∫</font>
<td align="center"><i>P</i> (<i>t</i>)</td>
<td rowspan="3" valign="middle"> <i>dt</i></td></tr>
<tr><td>P<sub>0</sub> =<td align="center">————</td></tr>
<tr><td><td align="center">1 + <i>t</i></td></tr>
<tr><td><td valign="top"><small><i>0</i></small></td><td></td><td></td></tr>
The solution: MathJax
And then I discovered MathJAX. It was added recently to the Udacity forums, and I think it's also what MITx is using for their courses.
MathJax is fantastic. It's an open-source library that lets you specify equations in readable ways -- you can use MathML, but you can also use LaTEX or even ASCII math like `x = (-b +- sqrt(b^2-4ac))/(2a) .`
It uses Javascript: you put your equations in the text of the page with delimiters like $$ around them (you can control the delimiters), then run a function that scans the page content and replaces any equations it sees with pretty graphics. (Viewers using NoScript or similar extensions will need to allow mathjax.org to see the equations, unless you make a local copy of the mathjax.org libraries, which you probably should anyway if you're using a lot of equations.)
For displaying those graphics, MathJax might use MathML, HTML and CSS, or whatever, depending on the user's browser ... but you don't have to worry about that. (Alas, even in Firefox, MathML rendering isn't up to par so MathJax doesn't use it by default, though you can specify it as an option if you know your equations render well.)
Here's that integral again, using LaTeX format: $$ P_0 =\int_0^\infty \frac {P(t) dt}{1 + t} $$ and $$ x = {-b \pm \sqrt{b^2-4ac} \over 2a} $$
It's beautiful! And although I don't know LaTex at all -- been wanting an excuse to learn it -- I put together that integral with five minutes of web searching. (The quadratic code came from a MathJax demo page.) Here's what the code looks like:
$$ P_0 =\int_0^\infty \frac {P(t) dt}{1 + t} $$
MathJax is even smart enough to notice the code there is in a <pre> tag, so I didn't have to find a way to escape it.
I'm sold! The MathJax team has really put together a nice package, and I think we'll be seeing it on a lot more websites. If you want to try it, start here: Getting Started with MathJAX.
Tags: math, science, html, web, mathml
[ 16:45 Apr 03, 2012 More science | permalink to this entry | ]
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Finding the average distance from an ellipse to a point | Hadrian Hughes
Welcome to my blog 🎉 I'm a full stack software engineer, maths student and fan of functional programming. You can expect structured articles as well as occasional stream of consciousness posts.
Finding the average distance from an ellipse to a point
I've recently been reading a book called Infinite Powers by Steven Strogatz about the history that led to the development of calculus. This, combined with the subject of conics which I've been studying as part of my university work, led me to start thinking about the Astronomical Unit and how it's calculated.
My understanding of the Astronomical Unit was that it represents the average distance between the Earth and the Sun. This understanding leads to an interesting problem: the Earth orbits around the Sun in an ellipse, and there are infinitely many points on that ellipse at which you could measure the distance between the Earth and the Sun. How can we take an average of infinitely many things? This sounds like a problem well suited to calculus.
As it turns out, the Astronomical Unit was originally calculated by taking the average of the Earth's aphelion and perihelion (the furthest and closest points to the sun, respectively).
\begin{align*} \text{Astronomical Unit} = \frac{\text{Aphelion} + \text{Perihelion}}{2} \end{align*}
So it's quite a rough average. I therefore set myself the challenge of finding the true average - the average of the infinitely many distances from the sun to points along the Earth's orbit.
In this post, I'll outline the process I went through to find a general formula for finding the true average of the distances from an arbitrary point to all the possible points along an ellipse. I'll save the final step of deriving the equation of the Earth's elliptical orbit and applying the formula to it for another post.
I'll start this section by saying that I'm an undergraduate student just doing this for fun, so do take my conclusions with some skepticism.
I reasoned that if we can find a function
f(\theta)
for the distance from an arbitrary point
(p,q)
to the point on the ellipse which coincides with the straight line extending from the origin at angle
\theta
, we could then take
n
evenly spaced values of
\theta
and take the average of their respective
f(\theta)
\begin{align*} \text{Rough Average Distance} = \frac{\sum_{k=0}^{n - 1} f\left(\frac{2\pi \times k}{n}\right)}{n} \end{align*}
\theta
\frac{2\pi \times k}{n}
2\pi
n
divides the circle around the origin into the appropriate number of slices, and then we multiply the result by
k
to select the appropriate slice.
k
effectively becomes an index.
2\pi
divided into
n = 8
It's important for the values of
\theta
to be evenly spaced so as not to introduce a bias towards one or another extreme of the ellipse. The fact that the angles are evenly spaced also makes simplifying the equation much simpler later on, as we'll see.
This average would increase in precision as you take more and more distances (as
n
increases) - much like how a mean average better represents the population as the sample size increases. Continuing the statistics analogy, the population we're dealing with is infinitely large, meaning no matter how large (but finite) our sample size
n
, the result will still only be a rough average.
Average gets more accurate as
n\to\infty
The true average distance would be the limit as
n
\begin{align*} \text{True Average Distance} = \lim_{n\to\infty} \frac{\sum_{k=0}^{n - 1} f\left(\frac{2\pi \times k}{n}\right)}{n} \end{align*}
So the strategy can be summarised as follows:
Find a function for the distance from a given point on an ellipse in standard position to an arbitrary point
(p,q)
Find a function for the average of the distances between
evenly spaced points on the ellipse and the point
(p,q)
(using the function found in part 1).
Find the limit of the function found in part 2 as
n \to \infty
Step 1: A function for the distance from
(p,q)
to a point on the ellipse
To find a function for the distance from a point on the ellipse to the point
(p,q)
, we can start with the standard formula for the distance between two points
\begin{align*} \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \end{align*}
but since we're looking for a function, we'll need to square both sides. This will just mean taking the square root as a final step in calculating the average. So the function will take the form
\begin{align*} d(x) = (p - x)^2 + (q - y)^2 \end{align*}
x
y
are the coordinates of the point on the ellipse.
The equation of any ellipse in standard position takes the form
\begin{align*} \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \end{align*}
and
b
are constants. However, this can be parametrised into two separate formulas for
x
y
\begin{align*} x &= a\cos{\theta} \\ y &= b\sin{\theta} \end{align*}
Therefore, our function for the square of the distance will be
\begin{align*} d(\theta) = (p - a\cos{\theta})^2 + (q - b\sin{\theta})^2 \end{align*}
Step 2: A function for the average of
n
Now that we have a function
d(\theta)
for a single square distance from
(p,q)
to a point on the ellipse, the function
D(n)
for the average of
n
distances will take the form
\begin{align*} D(n) = \frac{\sum_{k = 0}^{n - 1} d\left(\frac{2\pi k}{n}\right)}{n} \end{align*}
n
, this is quite simple to calculate by hand. For example:
\begin{align*} D(2) &= \frac{d\left(\frac{2\pi \times 0}{2}\right) + d\left(\frac{2\pi \times 1}{2}\right)}{2} \\[0.5em] &= \frac{d(0) + d(\pi)}{2} \\[0.5em] &= \frac{(p - a\cos{0})^2 + (q - b\sin{0})^2 + (p - a\cos{\pi})^2 + (q - b\sin{\pi})^2}{2} \\[0.5em] &= \frac{(p - a)^2 + q^2 + (p + a)^2 + q^2}{2} \\[0.5em] &= \frac{(p - a)^2 + (p + a)^2 + 2q^2}{2} \end{align*}
However, the numerator becomes longer as
n
gets larger, and since we're trying to find the limit as
n \to \infty
this method will become unmanageable. Therefore, we need to find a way to express the same rule for the function without the need for series notation.
The first step in doing this is to use some of the rules for manipulating series to simplify as much as possible. For simplicity, let's focus on the numerator.
\begin{align*} \sum_{k=0}^{n-1} & d\left(\frac{2\pi k}{n}\right) \\ &= \sum_{k=0}^{n-1} \left(\left(p - a\cos{\left(\frac{2\pi k}{n}\right)}\right)^2 + \left(q - b\sin{\left(\frac{2\pi k}{n}\right)}\right)^2\right) \\[0.5em] &= \sum_{k=0}^{n-1} \left(p^2 - 2a\cos{\left(\frac{2\pi k}{n}\right)} + a^2\cos^2{\left(\frac{2\pi k}{n}\right)} + q^2 - 2b\sin{\left(\frac{2\pi k}{n}\right)} + b^2\sin^2{\left(\frac{2\pi k}{n}\right)}\right) \end{align*}
Next we can use the two trigonometric half angle identities
\begin{align*} \sin^2{\theta} &= \frac{1}{2}(1 - \cos{(2\theta})) \\ \cos^2{\theta} &= \frac{1}{2}(1 + \cos{(2\theta)}) \end{align*}
to remove the
\cos^2
\sin^2
\begin{align*} \sum_{k=0}^{n-1} \left(p^2 - 2a\cos{\left(\frac{2\pi k}{n}\right)} + a^2 \times \frac{1}{2}\left(1 + \cos{\left(\frac{4\pi k}{n}\right)}\right) + q^2 - 2b\sin{\left(\frac{2\pi k}{n}\right)} + b^2 \times \frac{1}{2}\left(1 - \cos{\left(\frac{4\pi k}{n}\right)}\right)\right) \end{align*}
The expression now looks very long and complex, but we can divide and conquer by using the following rule for manipulating a series
\begin{align*} \sum_{k=m}^{n} (x_k + y_k) = \sum_{k=m}^n x_k + \sum_{k=m}^n y_k \end{align*}
to replace the single long series with a simpler series for each subexpression, which we can then sum together.
\begin{align*} \sum_{k=0}^{n-1} p^2 - \sum_{k=0}^{n-1} 2a\cos{\left(\frac{2\pi k}{n}\right)} + \sum_{k=0}^{n-1} \frac{a^2}{2}\left(1 + \cos{\left(\frac{4\pi k}{n}\right)}\right) + \sum_{k=0}^{n-1} q^2 - \sum_{k=0}^{n-1} 2b\sin{\left(\frac{2\pi k}{n}\right)} + \sum_{k=0}^{n-1} \frac{b^2}{2}\left(1 - \cos{\left(\frac{4\pi k}{n}\right)}\right) \end{align*}
These resulting series can be simplified substantially if we remember that
a
b
p
q
are all constants, and therefore we can apply the following two further rules.
\begin{align*} \sum_{k=m}^n cx_k &= c\sum_{k=m}^n x_k \\[0.5em] \sum_{k=0}^n c &= cn\ \ \ \ \text{where $c$ is a constant} \end{align*}
p^2n - 2a\sum_{k=0}^{n-1} \cos{\left(\frac{2\pi k}{n}\right)} + \frac{a^2n}{2} + \sum_{k=0}^{n-1} \cos{\left(\frac{4\pi k}{n}\right)} + q^2n - 2b\sum_{k=0}^{n-1} \sin{\left(\frac{2\pi k}{n}\right)} + \frac{b^2n}{2} - \sum_{k=0}^{n-1} \cos{\left(\frac{4\pi k}{n}\right)}
Earlier I mentioned the importance of the values of
\theta
being evenly spaced. This step is where that becomes relevant. As it happens, the fact that the values are evenly spaced means that all our remaining sums (sums of sine and cosine) evaluate to zero.
\begin{align*} \sum_{k=0}^{n-1} \cos{\left(\frac{2\pi k}{n}\right)} &= 0 \\[0.5em] \sum_{k=0}^{n-1} \sin{\left(\frac{2\pi k}{n}\right)} &= 0 \\[0.5em] \sum_{k=0}^{n-1} \cos{\left(\frac{4\pi k}{n}\right)} &= 0 \\[0.5em] \sum_{k=0}^{n-1} \sin{\left(\frac{4\pi k}{n}\right)} &= 0 \end{align*}
I won't provide a proof for this here as this post is already becoming cumbersome with equations, but I found this article very helpful https://matthew-brett.github.io/teaching/sums_of_cosines.html. It also makes sense intuitively if we remember that the sine and cosine are just the
y
x
components of the point on the unit circle corresponding with the angle
\theta
, respectively. By virtue of the points on the unit circle being evenly spaced, each point's distance from the relevant axis is cancelled out by a point on the opposite side.
Sums of sines and cosines cancel to 0
So, we can remove the remaining sums from our expression which leaves us with the following much simpler one.
\begin{align*} p^2n + \frac{a^2n}{2} + q^2n + \frac{b^2n}{2} = n\left(p^2 + \frac{a^2}{2} + q^2 + \frac{b^2}{2}\right) \end{align*}
Bringing this back into context, this is the numerator in our original expression for the square of the average distance from the infinitely many points on the ellipse to the point
(p,q)
\begin{align*} D(n) &= \frac{\sum_{k = 0}^{n - 1} d\left(\frac{2\pi k}{n}\right)}{n} \\[0.5em] &= \frac{n\left(p^2 + \frac{a^2}{2} + q^2 + \frac{b^2}{2}\right)}{n} \\[0.5em] &= p^2 + \frac{a^2}{2} + q^2 + \frac{b^2}{2} \end{align*}
I was surprised at how concise this formula is. Moreover, Step 3 for finding the limit as
n\to\infty
has become irrelevant since
n
is entirely cancelled from the expression; as
n
approaches infinity it is always cancelled by the denominator.
Let's try the formula on an example point and ellipse.
\begin{align*} \text{Ellipse:} &\ \ \frac{x^2}{2^2} + \frac{y^2}{3^2} = 1 \\ \text{Point:} &\ \ (p,q) = (1,1) \\[0.5em] \end{align*}
\begin{align*} \text{Square distance:}\ \ 1^2 + \frac{2^2}{2} + 1^2 + \frac{3^2}{2} &= 1 + 2 + 1 + \frac{9}{2} \\ &= \frac{17}{2} \end{align*}
\begin{align*} \text{Average distance:}\ \ \sqrt{\frac{17}{2}} = \frac{\sqrt{34}}{2} = 2.91547594... \end{align*}
I won't prove the accuracy of the formula here, but this result matches the approximation we get from Maxima with a value of
n=100
(essentially calculating an approximate value by brute force).
In another post I'll return to the example of the Earth's orbit around the Sun, and attempt to find the average distance between them using this formula.
Copyright © 2021 Hadrian Hughes. All rights reserved.
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(Adding QNC0/🐱, the cutest complexity class.)
===== <span id="qnc0qpoly" style="color:red">QNC<sup>0</sup>/qpoly</span>: Quantum [[Complexity Zoo:N#nc0|NC<sup>0</sup>]] With Polynomial-Size Quantum Advice =====
Constant-depth quantum circuits with bounded fan-in, with polynomial size quantum advice. Contains [[#qnc0|QNC<sup>0</sup>]] and [[#qnc0cat|QNC<sup>0</sup>/🐱]]
===== <span id="qnc0qpoly" style="color:red">QNC<sup>0</sup>/🐱</span>: Quantum [[Complexity Zoo:N#nc0|NC<sup>0</sup>]] With Polynomial-Size Quantum Advice =====
Constant-depth quantum circuits without fanout gates, with an additional cat state as "advice": 🐱 = <math>\frac{|0^n\rangle+|1^n\rangle}{\sqrt 2}</math>.
Defined in [[zooref#wkst19|[WKST19]]].
Contains [[#qnc0|QNC<sup>0</sup>]]. Contained in [[#qnc0qpoly|QNC<sup>0</sup>/qpoly]] and [[ComplexityZoo:B#bqp|BQP]].
===== <span id="qncf0" style="color:red">QNC<sub>f</sub><sup>0</sup></span>: Quantum [[Complexity Zoo:N#nc0|NC<sup>0</sup>]] With Unbounded Fanout =====
{\displaystyle k\geq 4}
QNC0/qpoly: Quantum NC0 With Polynomial-Size Quantum Advice
Constant-depth quantum circuits with bounded fan-in, with polynomial size quantum advice. Contains QNC0 and QNC0/🐱
QNC0/🐱: Quantum NC0 With Polynomial-Size Quantum Advice
Constant-depth quantum circuits without fanout gates, with an additional cat state as "advice": 🐱 =
{\displaystyle {\frac {|0^{n}\rangle +|1^{n}\rangle }{\sqrt {2}}}}
Defined in [WKST19].
Contains QNC0. Contained in QNC0/qpoly and BQP.
{\displaystyle k\geq 2}
{\displaystyle k=1}
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EUDML | Event and apparent horizon finders for numerical relativity. EuDML | Event and apparent horizon finders for numerical relativity.
Event and apparent horizon finders for
3+1
numerical relativity.
Thornburg, Jonathan. "Event and apparent horizon finders for numerical relativity.." Living Reviews in Relativity [electronic only] 10 (2007): Article No. 2007-3. <http://eudml.org/doc/222790>.
@article{Thornburg2007,
author = {Thornburg, Jonathan},
title = {Event and apparent horizon finders for numerical relativity.},
AU - Thornburg, Jonathan
TI - Event and apparent horizon finders for numerical relativity.
Articles by Thornburg
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Generalized optimal subpattern assignment (GOSPA) metric - MATLAB - MathWorks España
trackGOSPAMetric
SwitchingPenalty
HasAssignmentInput
sGOSPA
missTarget
falseTrack
Evaluate Tracking Results Using GOSPA Metric
Generalized optimal subpattern assignment (GOSPA) metric
trackGOSPAMetric System object™ computes the generalized optimal subpattern assignment metric between a set of tracks and the known truths.
For more details, see GOSPA Metric and [1].
To compute the generalized subpattern alignment metric:
Create the trackGOSPAMetric object and set its properties.
GOSPAMetric = trackGOSPAMetric
GOSPAMetric = trackGOSPAMetric(Name,Value)
GOSPAMetric = trackGOSPAMetric creates a trackGOSPAMetric System object with default property values.
GOSPAMetric = trackGOSPAMetric(Name,Value) sets properties for the trackGOSPAMetric object using one or more name-value pairs. For example, GOSPAMetric = trackGOSPAMetric('CutoffDistance',5) creates a trackGOSPAMetric object with the cutoff distance equal to 5. Enclose property names in quotes.
CutoffDistance — Threshold for cutoff distance between track and truth
Threshold for cutoff distance between track and truth, specified as a real positive scalar. A truth is assigned to a track only if the distance between the track and the known truth is less than this distance.
Alpha parameter of GOSPA metric, specified as a positive scalar in the range [0, 2].
SwitchingPenalty — Penalty for assignment switching
0 | nonnegative real scalar
Distance — Distance type
'posnees' (default) | 'velnees' | 'posabserr' | 'velabserr' | 'custom'
Distance type, specified as 'posnees', 'velnees', 'posabserr', 'velabserr', or 'custom'. This property specifies the physical quantity used for distance calculations:
'posnees' – Normalized estimation error squared (NEES) of track position
'velnees' – NEES error of track velocity
'custom' – Custom distance error
If you specify the Distance property as 'custom', you must also specify the distance function in the DistanceFcn property.
DistanceFcn — Custom distance function
Custom distance function, specified as a function handle. The function must support this syntax:
where track is a structure or an object of track information, truth is a structure or an object of truth information, and d is the distance between the truth and the track. See objectTrack for an example on how to organize information for estimated tracks and truth tracks.
To enable this property, set the Distance property to 'custom'.
The motion models expect the 'State' field of the tracks input to have a column vector containing these values:
'constvel' — Constant velocity motion model of the form [x;vx;y;vy;z;vz], where x, y, and z are position coordinates and vx, vy, vz are velocity coordinates.
'constacc' — Constant acceleration motion model of the form [x;vx;ax;y;vy;ay;z;vz;az], where x, y, and z are position coordinates, vx, vy, vz are velocity coordinates, and ax, ay, az are acceleration coordinates.
'constturn' — Constant turn motion model of the form [x;vx;y;vy;theta;z;vz], where x, y, and z are position coordinates, vx, vy, vz are velocity coordinates, and theta is the yaw rate.
'singer' — Singer acceleration motion model of the form [x;vx;ax;y;vy;ay;z;vz;az], where x, y, and z are position coordinates, vx, vy, vz are velocity coordinates, and ax, ay, az are acceleration coordinates.
The 'StateCovariance' field of the tracks input must have position, velocity, and turn-rate covariances in the rows and columns corresponding to the position, velocity, and turn-rate of the 'State' field of the tracks input. 'StateCovariance' is required only if 'posnees' or 'velnees' is selected in the Distance property.
@defaultTrackIdentifier (default) | function handle
Track identifier function, specified as a function handle. The function extracts track ID from the tracks input. The function must support the following syntax:
tracks is an array of structures or objects containing the information of tracks.
trackids is a numeric array of the same size as tracks.
For an example of a track object, see objectTrack. If you use the default identifier function, defaultTrackIdentifier, you must include track ID in tracks as the value of the TrackID field or property.
Example: @myTrackIdetifier
@defaultTruthIdentifier (default) | function handle
Truth identifier function, specified as a function handle. The function extracts truth ID from truths input. The function must support the following syntax:
truths is an array of structures or objects containing the information of truths.
truthIDs is a numeric array of the same size as truths.
If you the use of the default identifier function, defaultTruthIdentifier, you must include the truth ID in truths as a value of the PlatformID field or property.
Example: @myTruthIdetifier
HasAssignmentInput — Enable assignment input
Enable assignment input, specified as true or false. This property enables providing the assignment input at each time step. The computed GOSPA metric uses the input assignment to compute the localization component.
sGOSPA = GOSPAMetric(tracks,truths)
[sGOSPA,GOSPA,switching] = OSPAMetric(tracks,truths)
[___] = GOSPAMetric(tracks,truths,assignment)
[sGOSPA,GOSPA,switching,localization,missTarget,falseTrack] = GOSPAMetric(___)
sGOSPA = GOSPAMetric(tracks,truths) returns the GOSPA metric between the set of tracks and truths, including the switching penalty. The value of the switching penalty included in the metric depends on the SwitchingPenalty property. By default, the metric uses the global nearest neighbor (GNN) assignments at the current and the previous step to decide if the tracks are switched.
[sGOSPA,GOSPA,switching] = OSPAMetric(tracks,truths) also returns the GOSPA component and the switching component.
[___] = GOSPAMetric(tracks,truths,assignment) allows you the specify the current assignments between tracks and truths used in the metric evaluation. You can return outputs as any of the previous syntaxes.
To use this syntax, set the HasAssignmentInput property to true.
[sGOSPA,GOSPA,switching,localization,missTarget,falseTrack] = GOSPAMetric(___) also returns the localization component, missed target component, and the false track component. You can use any of the input combinations in the previous syntaxes.
To use this syntax, set the value of the Alpha property to 2.
array of structures | array of objects
Track information, specified as an array of structures or objects for built-in distance functions. Each structure or object must contain State as a field or property. Additionally, if a NEES-based distance (posnees or velnees) is specified in the Distance property, each structure or object must also contain StateCovariance as a field or property. Moreover, if the default track identifier function is used in the TrackIdentifierFcn property, then each structure or object must also contain TrackID as a field or property. See objectTrack for an example of track object.
Truth information, specified as an array of structures or objects for built-in distance functions. Each structure or object must contain Position and Velocity as fields or properties. If the default truth identifier function is used in the TruthIdentifierFcn property, then each structure or object must also contain PlatformID as a field or property.
assignment — Known current assignment
Known current assignment, specified as an N-by-2 matrix of nonnegative integers. The first column elements are track IDs, and the second column elements are truth IDs. The IDs in the same row are tracks and truths assigned to each other. If a track (or a truth) is not assigned, specify 0 as the same row element for the truth (or the track).
Note that the assignment must be a unique assignment between tracks and truths. Redundant or false tracks should be treated as unassigned tracks by assigning them to the "0" TruthID.
sGOSPA — GOSPA metric including switching component
GOSPA metric including switching component, returned as a nonnegative real scalar.
GOSPA — GOSPA metric
GOSPA metric, returned as a nonnegative real scalar.
switching — Switching component
Switching component, returned as a nonnegative real scalar.
localization — Localization component
Localization component, returned as a nonnegative real scalar.
missTarget — Missed target component
Missed target component, returned as a nonnegative real scalar.
falseTrack — False track component
False track component, returned as a nonnegative real scalar.
Load prerecorded data.
load trackmetricex tracklog truthlog;
Create a trackGOSPAMetric object and set the SwitchingPenalty to 5.
tgm = trackGOSPAMetric('SwitchingPenalty',5);
Create output variables.
lgospa = zeros(numel(tracklog),1);
gospa = zeros(numel(tracklog),1);
switching = zeros(numel(tracklog),1);
localization = zeros(numel(tracklog),1);
missTarget = zeros(numel(tracklog),1);
falseTracks = zeros(numel(tracklog),1);
After extracting the tracks and ground truths, run the GOSPA metric.
for i = 1:numel(tracklog)
[lgospa(i),gospa(i),switching(i),localization(i),missTarget(i),falseTracks(i)] = tgm(tracks,truths);
plot([lgospa gospa switching localization missTarget falseTracks])
legend('Labeled GOSPA','GOSPA','Switching Component',...
'Localization Component','Missed Target Component','False Tracks Component')
X=\left[{x}_{1},{x}_{2},\dots ,{x}_{m}\right]
Y=\left[{y}_{1},{y}_{2},\dots ,{y}_{n}\right]
SGOSPA={\left(GOSP{A}^{p}+S{C}^{p}\right)}^{1/p}
GOSPA={\left[\sum _{i=1}^{m}{d}_{c}^{p}\left({x}_{i},{y}_{\pi \left(i\right)}\right)+\frac{{c}^{p}}{\alpha }\left(n-m\right)\right]}^{1/p}
{d}_{c}\left(x,y\right)=\mathrm{min}\left\{{d}_{b}\left(x,y\right),c\right\}
SC=SP×{n}_{s}^{1/p}
GOSPA={\left[lo{c}^{p}+mis{s}^{p}+fals{e}^{p}\right]}^{1/p}
loc={\left[\sum _{i=1}^{h}{d}_{b}^{p}\left({x}_{i},{y}_{\pi \left(i\right)}\right)\right]}^{1/p}
miss=\frac{c}{{2}^{1/p}}{\left({n}_{miss}\right)}^{1/p}
false=\frac{c}{{2}^{1/p}}{\left({n}_{false}\right)}^{1/p}
[1] Rahmathullash, A. S., A. F. García-Fernández, and L. Svensson. "Generalized Optimal Sub-Pattern Assignment Metric." 20th International Conference on Information Fusion (Fusion), pp. 1–8, 2017.
trackErrorMetrics | trackOSPAMetric | trackAssignmentMetrics
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group(deprecated)/centralizer - Maple Help
Home : Support : Online Help : group(deprecated)/centralizer
find the centralizer of a set of permutations
centralizer(pg, s)
group in which the centralizer is to be found
set of permutations or a single permutation
Important: The group package has been deprecated. Use the superseding command GroupTheory[Centralizer] instead.
This function finds the largest subgroup of the permutation group pg in which every element commutes with every element of s. s need not be contained in pg. The result is returned as an unevaluated permgroup call.
The command with(group,centralizer) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{group}\right):
\mathrm{centralizer}\left(\mathrm{permgroup}\left(7,{[[1,2]],[[1,2,3,4,5,6,7]]}\right),{[[3,6]]}\right)
\textcolor[rgb]{0,0,1}{\mathrm{permgroup}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}{[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]]}\right)
\mathrm{centralizer}\left(\mathrm{permgroup}\left(5,{[[1,2,3]],[[3,4,5]]}\right),[[1,2]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{permgroup}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}{[[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]]}\right)
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Sentence (mathematics) - zxc.wiki
Sentence (mathematics)
In mathematics, a sentence or theorem is a consistent logical statement that can be recognized as true by means of a proof , that is, can be derived from axioms , definitions and already known sentences.
A sentence is often referred to differently depending on its role, meaning or context. Within an article or a monograph (e.g. a dissertation or a textbook) one uses
Lemma (or auxiliary clause ) for a statement that is only used in the proof of other sentences in the same work and has no meaning regardless of it,
Proposition for a statement that is also mainly locally significant, such as a proposition that is used in more than one proof,
Theorem (or theorem ) for an essential knowledge presented in the work, and
Corollary (or sequential block ) for a trivial conclusion, the results from a set or definition without great effort.
The classification of a sentence in one of the above categories is subjective and has no consequences for the use of the sentence. Many authors dispense with the term proposition and use lemma or theorem for it. Also corollary is not always set distinguished. On the other hand, it is quite common and helpful for the reader if pure auxiliary clauses are recognizable as such.
Sentences that are generally known and are usually not cited with the original source bear the name of the subject matter about which they are making a statement, or the name of the author, or both. The terms fundamental clause or main clause (a field of mathematics) are also used in this context , and the distinction between proposition and lemma has often grown historically rather than determined by content and meaning. Many examples of such names can be found in the List of Mathematical Theorems .
1 examples of sentences
2.2 reverse theorem
2.2.2 Dependence on the division into prerequisites and statements
Below are a few simple sentences. The calculus to be used is given in brackets.
If every human is mortal and Socrates is human, then Socrates is mortal. ( Predicate logic ).
Every non-empty set has at least one element . ( Set theory )
The sum of the interior angles of a triangle is 180 degrees . ( Euclidean geometry )
For every real number there is a larger natural number . ( Archimedean order , analysis )
There is no such thing as a rational number whose square is 2. ( Number theory )
Let it be steady . Then it is also steady. (Analysis)
{\ displaystyle f, g: \ mathbb {R} \ to \ mathbb {R}}
{\ displaystyle f \ circ g: \ mathbb {R} \ to \ mathbb {R}}
Although a mathematical set of a statement can be made of any shape (eg "not V or A ."), A mathematical theorem mostly in the subjunctive formulated requirement and as a declarative sentence worded statement divided (example: "Be V . Then A. "), So that the impression of an implication arises.
Caution: Inconsiderate removal and use of individual parts of a sentence can lead to false conclusions, as these parts generally do not have to be valid.
{\ displaystyle n \ notin \ mathbb {N} \ quad \ vee \ quad n {\ mbox {is not prime}} \ quad \ vee \ quad n = 2 \ quad \ vee \ quad n {\ mbox {is odd} }}
“Let n be a prime number . For n the following applies: "
{\ displaystyle n = 2 \ quad \ vee \ quad n \ in 2 \ cdot \ mathbb {N} +1}
“ When it rains, the road gets wet. "(Not a sentence in the mathematical sense)
From the plane geometry: “ If a real square is a parallelogram , then opposite sides are the same length. ”(Here,“ real square ”means that degenerate and overturned squares are excluded from consideration).
If you swap the premise and statement of the sentence in a sentence , you get the corresponding reverse sentence. These are logical statements of the form “ requirement ⇐ statement ”. A distinction must then be made between the following cases:
If the reverse clause is not a clause - that is, it is false - then the premise of the clause is sufficient, but not necessary .
If the reverse sentence is a sentence - that is, it is applicable - then the premise of the sentence is necessary and sufficient. In this case one can formulate a further sentence in which the premise and statement of the sentence are equivalent (example: “ V holds, if and only if A holds ”).
“ If the street is wet, then it has rained. “This reverse sentence is wrong, because the water could have got onto the street differently. The prerequisite for the sentence “ it rained ” is therefore sufficient, but not necessary .
“ If opposite sides of a real square are the same length, then it's a parallelogram. “This reverse sentence is true. The premise of the sentence is necessary and sufficient . The sentence and the reverse sentence can be summarized: “ A real square is a parallelogram if and only if the opposite sides are the same length. "
Dependence on the division into prerequisite and statement
It is possible to have the same logical statement in various ways condition and statement divide, and the reversal rate depends on this division.
The logical statement can be written as a sentence in the following ways, for example:
{\ displaystyle \ lnot A \ lor \ lnot B \ lor C}
{\ displaystyle (A \ land B) \ Rightarrow C}
- reverse theorem:
{\ displaystyle C \ Rightarrow (A \ wedge B) \ quad \ equiv \ quad (A \ vee \ neg C) \ wedge (B \ vee \ neg C)}
{\ displaystyle A \ Rightarrow (\ lnot B \ lor C)}
{\ displaystyle (\ lnot B \ lor C) \ Rightarrow A \ quad \ equiv \ quad (A \ vee B) \ wedge (A \ vee \ neg C)}
As can be seen, it is generally not true that the two inverse theorems are equivalent.
Albrecht Beutelspacher : That is o. B. d. A. trivial! . Vieweg + Teubner Verlag, 9th edition (2009), ISBN 3-834-80771-0
This page is based on the copyrighted Wikipedia article "Satz_%28Mathematik%29" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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LPTX Financial Ratios - FinancialModelingPrep
\dfrac{Current Assets}{Current Liabilities}
14.29 A current ratio of 1.0 or greater is an indication that the company is well-positioned to cover its current or short-term liabilities.
\dfrac{Cash and Cash Equivalents + Short Term Investments + Account Receivables}{Current Liabilities}
14.06 The quick ratio is more conservative than the current ratio because it excludes inventory and other current assets, which generally are more difficult to turn into cash. A higher quick ratio means a more liquid current position.
\dfrac{Cash and Cash Equivalents}{Current Liabilities}
14.06 The cash ratio is almost like an indicator of a firm’s value under the worst-case scenario where the company is about to go out of business.
\dfrac{(Account Receivable (start) + Account Receivable (end))/2}{Revenue/365}
\dfrac{(Inventories (start) + Inventories (end))/2}{COGS/365}
\dfrac{DSO + DIO}{}
\dfrac{(Accounts Payable (start) + Accounts Payable (end))/2}{COGS/365}
\dfrac{DSO + DIO − DPO}{}
\dfrac{Gross Profit}{Revenue}
\dfrac{Operating Income}{Revenue}
\dfrac{Income Before Tax}{Revenue}
\dfrac{Net Income}{Revenue}
\dfrac{Provision For Income Taxes}{Income Before Tax}
\dfrac{Net Income}{Average Total Assets}
\dfrac{Net Income}{Average Total Equity}
\dfrac{EBIT}{Average Total Asset − Average Current Liabilities}
\dfrac{Net Income}{EBT}
\dfrac{EBT}{EBIT}
\dfrac{EBIT}{Revenue}
\dfrac{Total Liabilities}{Total Assets}
\dfrac{Total Debt}{Total Equity}
\dfrac{Long−Term Debt}{Long−Term Debt + Shareholders Equity}
\dfrac{Total Debt}{Total Debt + Shareholders Equity}
\dfrac{EBIT}{Interest Expense}
1,261.35 The lower a company’s interest coverage ratio is, the more its debt expenses burden the company.
\dfrac{Operating Cash Flows}{Total Debt}
\dfrac{Total Assets}{Total Equity}
\dfrac{Revenue}{NetPPE}
\dfrac{Revenue}{Total Average Assets}
\dfrac{Operating Cash Flow}{Revenue}
\dfrac{Free Cash Flow}{Operating Cash Flow}
\dfrac{Operating Cash Flow}{Total Debt}
\dfrac{Operating Cash Flow}{Short-Term Debt}
-75.40 The short-term debt coverage ratio compares the sum of a company's short-term borrowings and the current portion of its long-term debt to operating cash flow.
\dfrac{Operating Cash Flow}{Capital Expenditure}
\dfrac{Operating Cash Flow}{Dividend Paid + Capital Expenditure}
\dfrac{DPS (Dividend per Share)}{EPS (Net Income per Share Number}
\dfrac{Stock Price per Share}{Equity per Share}
\dfrac{Stock Price per Share}{Operating Cash Flow per Share}
\dfrac{Stock Price per Share}{EPS}
\dfrac{Price Earnings Ratio}{Expected Revenue Growth}
\dfrac{Stock Price per Share}{Revenue per Share}
\dfrac{Dividend per Share}{Stock Price per Share}
\dfrac{Entreprise Value}{EBITDA}
\dfrac{Stock Price per Share}{Intrinsic Value}
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Generate sine wave, using simulation time as time source - Simulink - MathWorks í•œêµ
y=amplitudeÃ\mathrm{sin}\left(frequencyÃtime+phase\right)+bias.
\begin{array}{l}\mathrm{sin}\left(t+\mathrm{Î}t\right)=\mathrm{sin}\left(t\right)\mathrm{cos}\left(\mathrm{Î}t\right)+\mathrm{sin}\left(\mathrm{Î}t\right)\mathrm{cos}\left(t\right)\\ \mathrm{cos}\left(t+\mathrm{Î}t\right)=\mathrm{cos}\left(t\right)\mathrm{cos}\left(\mathrm{Î}t\right)â\mathrm{sin}\left(t\right)\mathrm{sin}\left(\mathrm{Î}t\right)\end{array}
\left[\begin{array}{c}\mathrm{sin}\left(t+\mathrm{Î}t\right)\\ \mathrm{cos}\left(t+\mathrm{Î}t\right)\end{array}\right]=\left[\begin{array}{cc}\mathrm{cos}\left(\mathrm{Î}t\right)& \mathrm{sin}\left(\mathrm{Î}t\right)\\ â\mathrm{sin}\left(\mathrm{Î}t\right)& \mathrm{cos}\left(\mathrm{Î}t\right)\end{array}\right]\left[\begin{array}{c}\mathrm{sin}\left(t\right)\\ \mathrm{cos}\left(t\right)\end{array}\right]
Because Δt is constant, the following expression is a constant:
\left[\begin{array}{cc}\mathrm{cos}\left(\mathrm{Î}t\right)& \mathrm{sin}\left(\mathrm{Î}t\right)\\ â\mathrm{sin}\left(\mathrm{Î}t\right)& \mathrm{cos}\left(\mathrm{Î}t\right)\end{array}\right]
Therefore, the problem becomes one of a matrix multiplication of the value of
\mathrm{sin}\left(t\right)
by a constant matrix to obtain
\mathrm{sin}\left(t+\mathrm{Î}t\right)
y=A\mathrm{sin}\left(2\mathrm{Ï}\left(k+o\right)/p\right)+b
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Cube - New World Encyclopedia
Previous (Cuban Revolution)
Next (Cubic zirconia)
This article is about the geometric shape.
Symmetry Oh
A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and of three-sided trapezohedron. The cube is dual to the octahedron. It has cubical symmetry (also called octahedral symmetry).
For a cube centered at the origin, with edges parallel to the axes and with an edge length of two, the Cartesian coordinates of the vertices are
(±1,±1,±1)
while the interior consists of all points (x0, x1, x2) with -1
{\displaystyle <}
{\displaystyle <}
For a cube of edge length
{\displaystyle a}
{\displaystyle 6a^{2}}
{\displaystyle a^{3}}
radius of circumscribed sphere
{\displaystyle {\frac {{\sqrt {3}}a}{2}}}
radius of sphere tangent to edges
{\displaystyle {\frac {a}{\sqrt {2}}}}
radius of inscribed sphere
{\displaystyle {\frac {a}{2}}}
A cube construction has the largest volume among cuboids (rectangular boxes) with a given surface area (e.g., paper, cardboard, sheet metal, etc.). Also, a cube has the largest volume among cuboids with the same total linear size (length + width + height).
The familiar six-sided dice are cube shaped.
The analogue of a cube in four-dimensional Euclidean space has a special name—a tesseract or (rarely) hypercube.
The analog of the cube in n-dimensional Euclidean space is called a hypercube or n-dimensional cube or simply n-cube. It is also called a measure polytope.
In math theory you can also have lower dimensional cube. A 0th dimensional cube is simply a point. A first dimensional cube is a segment. A second dimensional cube is a square.
One such regular tetrahedron has a volume of one-third of that of the cube. The remaining space consists of four equal irregular polyhedra with a volume of one-sixth of that of the cube, each.
An extension is the three-dimensional k-ary Hamming graph, which for k = two is the cube graph. Graphs of this sort occur in the theory of parallel processing in computers.
↑ English cube from Old French, Latin cubus, Greek kubos, "a cube, a die, vertebra." In turn from PIE *keu(b)-, "to bend, turn".
Arnone, Wendy. Geometry for Dummies. Hoboken, NJ: For Dummies (Wiley), 2001. ISBN 0764553240
Cromwell, Peter R. Polyhedra. Cambridge, U.K.: Cambridge University Press, 1997. ISBN 0521664055
Hartshorne, Robin. Geometry: Euclid and Beyond. Undergraduate Texts in Mathematics. New York: Springer, 2002. ISBN 0387986502
Leff, Lawrence S. Geometry the Easy Way. Hauppauge, NY: Barron’s Educational Series, 1997. ISBN 0764101102
MacLean, Kenneth J.M. A Geometric Analysis of the Platonic Solids and Other Semi-Regular Polyhedra: With an Introduction to the Phi Ratio. Geometric Explorations Series. Ann Arbor, MI: Loving Healing Press, 2007. ISBN 978-1932690996
Pearce, Peter. Structure in Nature is a Strategy for Design. Cambridge: MIT Press, 1980. ISBN 0262660458
Smith, Joseph V. Geometrical and Structural Crystallography. Smith and Wyllie Intermediate Geology Series. New York: Wiley, 1982. ISBN 0471861685
Stillwell, John. The Four Pillars of Geometry. Undergraduate Texts in Mathematics. New York: Springer, 2005. ISBN 0387255303
Williams, Robert. The Geometrical Foundation of Natural Structure: A Source Book of Design. New York: Dover, 1979. ISBN 048623729X
Eric W. Weisstein. Cube. MathWorld
History of "Cube"
Retrieved from https://www.newworldencyclopedia.org/p/index.php?title=Cube&oldid=1068828
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View and Change Block Parameterization in Control System Tuner - MATLAB & Simulink - MathWorks Switzerland
View Block Parameterization
Fix Parameter Values or Limit Tuning Range
Custom Parameterization
Block Rate Conversion
Rate Conversion for Parameterized PID Blocks
Rate Conversion for Other Parameterized Blocks
Blocks with Fixed Rate Conversion Methods
Control System Tuner parameterizes every block that you designate for tuning.
When you tune a Simulink® model, Control System Tuner automatically assigns a default parameterization to tunable blocks in the model. The default parameterization depends on the type of block. For example, a PID Controller block configured for PI structure is parameterized by proportional gain and integral gain as follows:
u={K}_{p}+{K}_{i}\frac{1}{s}.
Kp and Ki are the tunable parameters whose values are optimized to satisfy your specified tuning goals.
When you tune a predefined control architecture or a MATLAB® (generalized state-space) model, you define the parameterization of each tunable block when you create it at the MATLAB command line. For example, you can use tunablePID to create a tunable PID block.
Control System Tuner lets you view and change the parameterization of any block to be tuned. Changing the parameterization can include changing the structure or current parameter values. You can also designate individual block parameters fixed (non-tunable) or limit their tuning range.
To access the parameterization of a block that you have designated as a tuned block, in the Data Browser, in the Tuned Blocks area, double-click the name of a block. The Tuned Block Editor dialog box opens, displaying the current block parameterization.
The fields of the Tuned Block Editor display the type of parameterization, such as PID, State-Space, or Gain. For more specific information about the fields, click .
To find a tuned block in the Simulink model, right-click the block name in the Data Browser and select Highlight.
You can change the current value of a parameter, fix its current value (make the parameter nontunable), or limit the parameter’s tuning range.
To change a current parameter value, type a new value in its text box. Alternatively, click to use a variable editor to change the current value. If you attempt to enter an invalid value, the parameter returns to its previous value.
Click to access and edit additional properties of each parameter.
Minimum — Minimum value that the parameter can take when the control system is tuned.
Maximum — Maximum value that the parameter can take when the control system is tuned.
Free — When the value is true, Control System Toolbox tunes the parameter. To fix the value of the parameter, set Free to false.
For array-valued parameters, you can set these properties independently for each entry in the array. For example, for a vector-valued gain of length 3, enter [1 10 100] to set the current value of the three gains to 1, 10, and 100 respectively. Alternatively, click to use a variable editor to specify such values.
For vector or matrix-valued parameters, you can use the Free parameter to constrain the structure of the parameter. For example, to restrict a matrix-valued parameter to be a diagonal matrix, set the current values of the off-diagonal elements to 0, and set the corresponding entries in Free to false.
When tuning a control system represented by a Simulink model or by a Predefined Feedback Architecture, you can specify a custom parameterization for any tuned block using a generalized state-space (genss) model. To do so, create and configure a genss model in the MATLAB workspace that has the desired parameterization, initial parameter values, and parameter properties. In the Change parameterization dialog box, select Custom. In the Parameterization area, the variable name of the genss model.
For example, suppose you want to specify a tunable low-pass filter, F = a/(s +a), where a is the tunable parameter. First, at the MATLAB command line, create a tunable genss model that represents the low-pass filter structure.
Generalized continuous-time state-space model with 1 outputs, 1 inputs,
1 states, and the following blocks:
Type "ss(F)" to see the current value, "get(F)" to see all properties, and
"F.Blocks" to interact with the blocks.
Then, in the Tuned Block Editor, enter F in the Parameterization area.
When you specify a custom parameterization for a Simulink block, you might not be able to write the tuned block value back to the Simulink model. When writing values to Simulink blocks, Control System Tuner skips blocks that cannot represent the tuned value in a straightforward and lossless manner. For example, if you reparameterize a PID Controller Simulink block as a third-order state-space model, Control System Tuner will not write the tuned value back to the block.
When Control System Tuner writes tuned parameters back to the Simulink model, each tuned block value is automatically converted from the sample time used for tuning, to the sample time of the Simulink block. When the two sample times differ, the Tuned Block Editor contains additional rate conversion options that specify how this resampling operation is performed for the corresponding block.
By default, Control System Tuner performs linearization and tuning in continuous time (sample time = 0). You can specify discrete-time linearization and tuning and change the sample time. To do so, on the Control System tab, click Linearization Options. Sample time for tuning reflects the sample time specified in the Linearization Options dialog box.
The remaining rate conversion options depend on the parameterized block.
For parameterization of continuous-time PID Controller and PID Controller (2-DOF) blocks, you can independently specify the rate-conversion methods as discretization formulas for the integrator and derivative filter. Each has the following options:
Trapezoidal (default) — Integrator or derivative filter discretized as (Ts/2)*(z+1)/(z-1), where Ts is the target sample time.
Forward Euler — Ts/(z-1).
Backward Euler — Ts*z/(z-1).
For more information about PID discretization formulas, see Discrete-Time Proportional-Integral-Derivative (PID) Controllers.
For discrete-time PID Controller and PID Controller (2-DOF) blocks, you set the integrator and derivative filter methods in the block dialog box. You cannot change them in the Tuned Block Editor.
For blocks other than PID Controller blocks, the following rate-conversion methods are available:
Zero-order hold — Zero-order hold on the inputs. For most dynamic blocks this is the default rate-conversion method.
Tustin — Bilinear (Tustin) approximation.
Tustin with prewarping — Tustin approximation with better matching between the original and rate-converted dynamics at the prewarp frequency. Enter the frequency in the Prewarping frequency field.
First-order hold — Linear interpolation of inputs.
Matched (SISO only) — Zero-pole matching equivalents.
For the following blocks, you cannot set the rate-conversion method in the Tuned Block Editor.
Discrete-time PID Controller and PID Controller (2-DOF) block. Set the integrator and derivative filter methods in the block dialog box.
Gain block, because it is static.
Transfer Fcn Real Zero block. This block can only be tuned at the sample time specified in the block.
Block that has been discretized using the Model Discretizer. Sample time for this block is specified in the Model Discretizer itself.
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Validators are slashed for breaking very specific protocol rules that could be part of an attack on the chain.
Slashed validators are exited from the beacon chain and receive three types of penalty.
Correlated penalties mean that punishment is light for isolated incidents, but severe when many validators are slashed in a short time period.
Block proposers receive rewards for reporting evidence of slashable offences.
Slashing occurs when validators break very specific protocol rules when submitting attestations or block proposals which could constitute attacks on the chain. Getting slashed means losing a potentially significant amount of stake and being ejected from the protocol. It is more "punishment" than "penalty". The good news is that stakers can take simple precautions to protect against ever being slashed.
Validators' stakes can be slashed for two distinct behaviours:
as attesters, for breaking the Casper commandments, the two rules on voting for source and target checkpoints; and
as proposers, for proposing two different blocks at the same height (equivocation).
The slashing of misbehaving attesters is what underpins Ethereum 2.0's economic finality guarantee by enforcing the Casper FFG protocol rules.
Proposer slashing, however, is not part of the Casper FFG protocol, and is not directly related to economic finality. It punishes a proposer that spams the block tree with multiple blocks that could partition the network, for example in a balancing attack.
As with penalties, the amounts removed from validators' beacon chain accounts due to slashing are effectively burned, reducing the overall net issuance of the beacon chain.
The cost of being slashed
When it comes to the punishment for being slashed it does not matter which rule was broken. All slashings are dealt with in the same way.
The initial penalty
Slashing is triggered by the evidence of the offence being included in a beacon chain block. Once the evidence is confirmed by the network, the offending validator (or validators) is slashed.
The offender immediately has one sixty-fourth (MIN_SLASHING_PENALTY_QUOTIENT_ALTAIR) of its effective balance deducted from its actual balance. This is a maximum of 0.5 ETH due to the cap on effective balance.
This initial penalty was introduced to make it somewhat costly for validators to self-slash for any reason.
Along with the initial penalty, the validator is queued for exit, and has its withdrawability epoch set to around 36 days (EPOCHS_PER_SLASHINGS_VECTOR, which is 8192 epochs) in the future.
During Phase 0, this initial penalty was
\frac{1}{128}
of the offender's effective balance. It is expected to be raised to its full value of
\frac{1}{32}
of the effective balance, a maximum of 1 ETH, as part of The Merge.
The correlation penalty
At the half way point of its withdrawability period (18 days after being slashed) the slashed validator is due to receive a second penalty.
This second penalty is based on the total amount of stake slashed during the 18 days before and after our validator was slashed. The idea is to scale the punishment so that a one-off event posing little threat to the chain is only lightly punished, while a mass slashing event that might be the result of an attempt to finalise conflicting blocks is punished to the maximum extent possible.
To be able to calculate this, the beacon chain maintains a record of the effective balances of all validators that were slashed during the most recent 8192 epochs (about 36 days).
The correlated penalty is calculated as follows.
Compute the sum of the effective balances (as they were when the validators were slashed) of all validators that were slashed in the previous 36 days. That is, for the 18 days preceding and the 18 days following our validator's slashing.
Multiply the slashed validator's effective balance by the result of #2 and then divide by the total_balance. This results in an amount between zero and the full effective balance of the slashed validator. That amount is subtracted from its actual balance as the penalty. Note that the effective balance could exceed the actual balance in odd corner cases, but decrease_balance() ensures the balance does not go negative.
The slashing multiplier in Altair is set to 2. With
S
being the sum of increments in the list of slashed validators over the last 36 days,
B
my effective balance, and
T
the total increments, the calculation looks as follows.
\text{Correlation penalty} = \min(B, \frac{2SB}{T})
Interestingly, due to the way the integer arithmetic is constructed in the implementation the result of this calculation will be zero if
2SB < T
. Effectively, the penalty is rounded down to the nearest whole amount of Ether. As a consequence, when there are few slashings there is no extra correlated slashing penalty at all, which is probably a good thing.
The intention is to raise the proportional slashing multiplier from 2 to 3 around The Merge, three being its "correct" cryptoeconomic value. To successfully finalise conflicting blocks at least one third of validators need to break attestation slashing rules. With the multiplier set to 3, that third would lose their entire stakes through this mechanism, which is optimal for security.
Validators that exit normally (by sending a voluntary exit message) are expected to participate only until their exit epoch, which is normally only a couple of epochs later.
A validator that is slashed continues to receive attestation penalties until its withdrawable epoch, which is set to 8192 epochs (36 days) after the slashing, and they are unable to receive any attestation rewards during this time. They are also subject for this entire period to any inactivity leak that might be in operation. It's not clear to me why there is this large hang-over from being slashed during which validators continue to receive penalties1; it seems like kicking a man when he's down, especially since slashed validators are locked in for twice as long as needed to calculate the correlation penalty.
So, in addition to the initial slashing penalty and the correlation penalty, there is a further penalty of up to
8192\frac{14 + 26}{64}32b = 106{,}987{,}520 \text{ Gwei} = 0.107 \text{ ETH}
, based on 300k validators, assuming that the chain is not in an inactivity leak. And (much) more if it is.
Slashed validators are eligible to be selected to propose blocks until they reach their exit epoch, but those blocks will be considered invalid, so there is no proposer reward available to them. This is in preference to immediately recomputing the duties assignments which would break the lookahead guarantees they have. (The proposer selection algorithm could easily be modified to skip slashed validators, but that is not how it is implemented currently.)
In an interesting edge case, however, slashed validators are eligible to be selected for sync committee duty until they reach their exit epoch and to receive the rewards for sync committee participation. Though the odds of this happening, absent a mass slashing event, are pretty tiny.
The value of reporting a slashing
In order for the beacon chain to verify slashings and take action against the offender, the evidence needs to be included in a beacon block. To incentivise validators to make the effort there is a specific reward for the proposer of a block that includes slashings.
The proposer reward
At the point of the the initial slashing report being included in a block, the proposer of the block receives a reward of validator.effective_balance / WHISTLEBLOWER_REWARD_QUOTIENT, which is
B / 512
B
is the effective balance of the validator being slashed.
A report of a proposer slashing violation can slash only one validator, but a report of an attestation slashing violation can simultaneously slash up to an entire committee, which might be hundreds of validators. This could be extremely lucrative for the proposer including the reports. A single block can contain up to 16 proposer slashing reports and up to 2 attester slashing reports.
Note that no new issuance is required to pay for this reward. The proposer reward is much less than the initial slashing applied to the validator, so the net issuance due to a slashing event is always negative.
In the code implementing the reward for reporting slashing evidence there is provision for a "whistleblower reward", with the whistleblower receiving
\frac{7}{8}
of the above reward and the proposer
\frac{1}{8}
The idea is to incentivise nodes that search for and discover evidence of slashable behaviour, which can be an intensive process.
However, this functionality is not currently used on the beacon chain, and the proposer receives both the whistleblower reward and the proposer reward, as above. The challenge is that it is too easy for a proposer just to steal a slashing report, so there's no point incentivising them separately. It's not an ideal situation, but so far there seem to be sufficient altruistic slashing detectors running on the beacon chain for slashings to be reported swiftly. There only needs to be one in practice.
This functionality may become useful in future upgrades.
The initial slashing penalty and proposer reward are applied in slash_validator() during block processing.
The correlation slashing penalty is applied in process_slashings() during epoch processing.
In the Serenity Design Rationale Vitalik gives some further background on why Ethereum 2.0 includes proposer slashing. It is specifically intended to discourage stakers from simultaneously running primary and backup nodes.
Vitalik says that this measure "is included to prevent self-slashing from being a way to escape inactivity leaks." But validators don't need to self-slash to avoid this; they could just make a normal voluntary exit. The exit mechanics are the same in each case.↩
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m (→QNC1: Quantum NC1: QNC1 is related to QNC, not QNC1.)
===== <span id="qnc1" style="color:red">QNC<sup>1</sup></span>: Quantum [[Complexity Zoo:N#nc1|NC<sup>1</sup>]] =====
Same as [[#qnc|QNC]]<sup>1</sup>, but for the exact rather than bounded-error case.
Same as [[#qnc|QNC]], but for the exact rather than bounded-error case.
In contrast to [[Complexity Zoo:N#nc1|NC<sup>1</sup>]], it is not clear how to simulate QNC<sup>1</sup> on a quantum computer in which one qubit is initialized to a pure state, and the remaining qubits are in the maximally mixed state [[zooref#asv00|[ASV00]]].
===== <span id="qp" style="color:red">QP</span>: Quasipolynomial-Time =====
Equals [[Complexity Zoo:D#dtime|DTIME]](2<sup>polylog(n)</sup>).
{\displaystyle k\geq 4}
{\displaystyle {\frac {|0^{n}\rangle +|1^{n}\rangle }{\sqrt {2}}}}
Same as QNC, but for the exact rather than bounded-error case.
{\displaystyle k\geq 2}
{\displaystyle k=1}
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Home : Support : Online Help : Mathematics : Linear Algebra : LinearAlgebra Package : SubOperations : Pivot
Pivot(A, i, j, L, ip, options)
(optional) an integer, a range with integer endpoints, or a list of integers and/or ranges with integer endpoints; selection of pivot rows
The Pivot(A, i, j) function pivots A about the non-zero entry A[i, j]. Multiples of the ith row are added to every other row in A, with the result that all of the entries in the jth column of A are zero except for the (i, j)th element.
The Pivot(A, i, j, L) function acts like Pivot(A, i, j) except that only rows indicated in L are modified. Rows not included in L are unaffected. If any rows in L are repeated or if i occurs in L, they are ignored. For more information regarding parameter L, see Matrix and Vector Entry Selection.
This function is part of the LinearAlgebra package, and so it can be used in the form Pivot(..) only after executing the command with(LinearAlgebra). However, it can always be accessed through the long form of the command by using LinearAlgebra[Pivot](..).
\mathrm{with}\left(\mathrm{LinearAlgebra}\right):
A≔〈〈1,5,9,3,7〉|〈2,6,0,4,8〉|〈3,7,1,5,9〉|〈4,8,2,6,0〉〉
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{6}& \textcolor[rgb]{0,0,1}{7}& \textcolor[rgb]{0,0,1}{8}\\ \textcolor[rgb]{0,0,1}{9}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{6}\\ \textcolor[rgb]{0,0,1}{7}& \textcolor[rgb]{0,0,1}{8}& \textcolor[rgb]{0,0,1}{9}& \textcolor[rgb]{0,0,1}{0}\end{array}]
A≔\mathrm{Pivot}\left(A,2,1\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{0}& \frac{\textcolor[rgb]{0,0,1}{4}}{\textcolor[rgb]{0,0,1}{5}}& \frac{\textcolor[rgb]{0,0,1}{8}}{\textcolor[rgb]{0,0,1}{5}}& \frac{\textcolor[rgb]{0,0,1}{12}}{\textcolor[rgb]{0,0,1}{5}}\\ \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{6}& \textcolor[rgb]{0,0,1}{7}& \textcolor[rgb]{0,0,1}{8}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{54}}{\textcolor[rgb]{0,0,1}{5}}& \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{58}}{\textcolor[rgb]{0,0,1}{5}}& \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{62}}{\textcolor[rgb]{0,0,1}{5}}\\ \textcolor[rgb]{0,0,1}{0}& \frac{\textcolor[rgb]{0,0,1}{2}}{\textcolor[rgb]{0,0,1}{5}}& \frac{\textcolor[rgb]{0,0,1}{4}}{\textcolor[rgb]{0,0,1}{5}}& \frac{\textcolor[rgb]{0,0,1}{6}}{\textcolor[rgb]{0,0,1}{5}}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{2}}{\textcolor[rgb]{0,0,1}{5}}& \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{4}}{\textcolor[rgb]{0,0,1}{5}}& \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{56}}{\textcolor[rgb]{0,0,1}{5}}\end{array}]
A≔\mathrm{Pivot}\left(A,1,2,[4..-1,3]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{0}& \frac{\textcolor[rgb]{0,0,1}{4}}{\textcolor[rgb]{0,0,1}{5}}& \frac{\textcolor[rgb]{0,0,1}{8}}{\textcolor[rgb]{0,0,1}{5}}& \frac{\textcolor[rgb]{0,0,1}{12}}{\textcolor[rgb]{0,0,1}{5}}\\ \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{6}& \textcolor[rgb]{0,0,1}{7}& \textcolor[rgb]{0,0,1}{8}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{10}& \textcolor[rgb]{0,0,1}{20}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-10}\end{array}]
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Learning patterns/Number of articles created or improved in Wikimedia projects - Meta
problemEdits, files, and bytes contributed and other metrics may not totally represent the significance of content contributions that involve creating or improving many pages.
solutionReport the number of articles (or other content pages) created or contributed to by members of your project.
This learning pattern is for the global metric on the number of articles created or improved in Wikimedia projects.
Creating new information resources is one of the most important ways to contribute to a Wikimedia project. Reporting on the number of articles created or improved over the course of a project provides a perspective on a project's contribution that is distinct from and complementary to measures like edits, bytes added, etc. It also provides you with another opportunity to demonstrate the impact of your project.
Articles Created + Articles Improved[edit]
Articles created and/or articles improved are two different numbers combined into one. Articles created can be found using Wikimetrics, while Articles improved can be found using an event page. Follow the steps below to find both.
To obtain the number of articles created, you just need to use Wikimetrics
3. In Wikimetrics, create a report to determine "Articles Created"
Using Wikimetrics, you want to use the Pages created metric.
Time Series by can be what you want
Namespaces should be: "0"
Note: This metric focuses ONLY on articles in the article namespace, not talk pages, etc.
In the Configure output section, to see results disaggregated by Wikimedia project, the Aggregate results should not be selected.
When you download or view the report, count how many pages were created!
Use an Event page
Getting the number of articles improved will vary based on the project. The simplest ways is to use an event page and ask participants to list the pages they worked on, and to indicate which pages they created. Just count the number of pages that were not new to get "Articles improved".
Or use Quarry
The following Quarry query will provide you with a list of pages that a list users edited or created: http://quarry.wmflabs.org/query/1053
Attention! The above query is limited to one project at a time and it lists all pages (i.e. not only articles in namespace 0 as requested by this metric). You will therefore need to iterate for every project and manually discriminate namespaces by looking up each page using its page_id to open it at https://your.wikiproject.org/w/index.php?curid=yourpageid
Add them up![edit]
{\displaystyle (PagesCreated)+(ArticlesImproved)=Total}
When computing total pages created or improved by your project, it usually makes sense to only count content pages, such as articles, Help pages, policy pages, and template pages. Creation of other types of pages, such as WikiProject pages or talk pages, should be tallied separately, or not reported at all. Unless the explicit goal of your project is to improve these kinds of pages, pages created in those spaces usually reflects coordination work, rather than content production.
Wikimetrics "Pages Created" metric
Retrieved from "https://meta.wikimedia.org/w/index.php?title=Learning_patterns/Number_of_articles_created_or_improved_in_Wikimedia_projects&oldid=16898636"
Wikimetrics learning patterns
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MaplePortal/ControlSystemDesign - Maple Help
Home : Support : Online Help : MaplePortal/ControlSystemDesign
Maple has tools for linear control system design in the DynamicSystems package. You can
Work with transfer functions, state space models, or differential equations
Linearize systems
Analyze the controllability, observability, phase and gain margin, and more
Generate control plots, including Bode, root-locus and Nyquist plots
Work symbolically or numerically
In this example, we will calculate the controllability matrix of a model of a DC motor, and generate a root-locus plot.
DC Motor System
\mathrm{restart}:\phantom{\rule[-0.0ex]{0.0em}{0.0ex}} \mathrm{with}\left(\mathrm{DynamicSystems}\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}
\mathrm{eq_sym}≔\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{L}\cdot \stackrel{.}{\mathrm{i}}\left(\mathrm{t}\right)+\mathrm{R}⋅\mathrm{i}\left(\mathrm{t}\right)=\mathrm{v}\left(\mathrm{t}\right)−\mathrm{K}⋅\stackrel{.}{\mathrm{\theta }}\left(\mathrm{t}\right),\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{J}\cdot \stackrel{..}{\mathrm{\theta }}\left(\mathrm{t}\right)+\mathrm{b}⋅\stackrel{.}{\mathrm{θ}}\left(\mathrm{t}\right)+\mathrm{Ks}⋅\mathrm{θ}\left(\mathrm{t}\right)=\mathrm{K}⋅\mathrm{i}\left(\mathrm{t}\right) :\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}
\mathrm{params}≔\left[\mathrm{J}=0.1, \mathrm{b}=0.1, \mathrm{K}=0.01, \mathrm{R}=1, \mathrm{L}=0.5, \mathrm{Ks} = 1\right]:
Controllability Matrix and Root-Locus Plot
\mathrm{eq_num}≔\mathrm{eval}\left(\mathrm{eq_sym},\mathrm{params}\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}
\mathrm{sys_num} ≔ \mathrm{StateSpace}\left(\left[\mathrm{eq_num}\right], \mathrm{inputvariable}=\left[\mathrm{v}\left(\mathrm{t}\right)\right], \mathrm{outputvariable}=\left[\mathrm{theta}\left(\mathrm{t}\right),\mathrm{i}\left(\mathrm{t}\right)\right]\right):
\mathrm{ControllabilityMatrix}\left(\mathrm{sys_num}\right)
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{-4}& \frac{\textcolor[rgb]{0,0,1}{1999}}{\textcolor[rgb]{0,0,1}{250}}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{5}}\\ \textcolor[rgb]{0,0,1}{0}& \frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{5}}& \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{5}}\end{array}]
\mathrm{RootLocusPlot}\left(\mathrm{sys_num}\right)
Symbolic Controllability Matrix
You can also work symbolically, and maintain the parameter relationships present in the original equation system. Here, for example, we generate a symbolic controllability matrix.
\mathrm{sys_sym} ≔ \mathrm{StateSpace}\left(\left[\mathrm{eq_sym}\right], \mathrm{inputvariable}=\left[\mathrm{v}\left(\mathrm{t}\right)\right], \mathrm{outputvariable}=\left[\mathrm{theta}\left(\mathrm{t}\right),\mathrm{i}\left(\mathrm{t}\right)\right]\right):
\mathrm{ControllabilityMatrix}\left(\mathrm{sys_sym}\right)
[\begin{array}{ccc}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{\mathrm{L}}}& \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{\mathrm{R}}}{{\textcolor[rgb]{0,0,1}{\mathrm{L}}}^{\textcolor[rgb]{0,0,1}{2}}}& \frac{{\textcolor[rgb]{0,0,1}{\mathrm{R}}}^{\textcolor[rgb]{0,0,1}{2}}}{{\textcolor[rgb]{0,0,1}{\mathrm{L}}}^{\textcolor[rgb]{0,0,1}{3}}}\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{\mathrm{K}}}^{\textcolor[rgb]{0,0,1}{2}}}{{\textcolor[rgb]{0,0,1}{\mathrm{L}}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{J}}}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \frac{\textcolor[rgb]{0,0,1}{\mathrm{K}}}{\textcolor[rgb]{0,0,1}{\mathrm{J}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{L}}}\\ \textcolor[rgb]{0,0,1}{0}& \frac{\textcolor[rgb]{0,0,1}{\mathrm{K}}}{\textcolor[rgb]{0,0,1}{\mathrm{J}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{L}}}& \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{\mathrm{K}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{R}}}{\textcolor[rgb]{0,0,1}{\mathrm{J}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{L}}}^{\textcolor[rgb]{0,0,1}{2}}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{\mathrm{b}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{K}}}{{\textcolor[rgb]{0,0,1}{\mathrm{J}}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{L}}}\end{array}]
LQR Controller for Inverted Pendulum
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Maximizing Profit for a Monopolist - Course Hero
Microeconomics/Monopoly Power/Maximizing Profit for a Monopolist
All profit-maximizing firms produce where their marginal cost (MC) (the cost of producing one more unit) is equal to their marginal revenue (MR) (the revenue received from selling an additional unit). This
\text{MR}=\text{MC}
rule is the same for monopolists as is it is for perfectly competitive firms. That is where the firm maximizes profit. How it sets its price, however, does differ in the two markets. It stems from the demand curve, the representation of the relationship between price and demand, that individual firms face in each market structure.
This graph shows the relationship of profit and quantity during a monopoly situation.
A monopolist has market power and can therefore affect the market price. Because the monopolist is the only supplier, its individual demand curve is also the market demand curve. As a result, it faces a downward-sloping demand curve that follows the law of demand: if the monopolist wants to sell more, it has to lower its price.
The law of demand states that as the price of a good decreases, the quantity demanded will increase, all other things being equal. Even though the monopolist has market power and can set its price, its pricing decision is still subject to the law of demand. As the only supplier in the market, the monopolist's demand curve is the downward-sloping market demand curve. So, in order to sell more, the monopolist must lower price assuming they are a single-price monopolist. To sell an extra unit the monopolist not only has to lower the price for that unit, it has to also lower the price on all units that it sells. This means that the contribution made to total revenue (MR) by the next unit sold is less than the contribution to total revenue (MR) made by the sale of the previous units. The result is that the marginal revenue curve for the monopolist is downward-sloping.
Thus, for the monopolist
\text{MR}\lt\text{P}
. The contribution to total revenue for selling an additional unit is less than the price charged for the unit (less by the amount it loses as a result of lowering the price on all units that it was selling already). Like a perfectly competitive firm, a monopolist determines the profit maximizing level of output where
\text{MC}=\text{MR}
. However, unlike a perfectly competitive firm, the monopolist does not face a given market price. With market power, the monopolist gets to set the price. Having set its production to an amount that maximizes profits, it then charges what consumers are willing to pay based on the market demand curve. For example, a monopolist sets its production where
\text{MC}=\text{MR}
, at 200,000 units. Given the monopolist is producing 200,000 units, it sets a price where the quantity demanded is also equal to 200,000 units (this is the price from the demand curve at the profit-maximizing level of output). The point where the quantity demanded meets the amount produced is at point B on the market demand curve, which corresponds to a price of $7. In order to calculate its profit, the monopolist must compare its price to its average total costs at the profit maximizing level of output or quantity. The equation for determining the amount of profit for a monopolist is:
\text{Profit}= (\text{P}-\text{ATC})\times\text{Q}
where P is price, ATC is average total cost, and Q is quantity. The monopolist will choose a level of output where
\text{MR}=\text{MC}
and will generate a profit as long as
\text{P}\gt\text{ATC}
at that level of output.
Both the perfectly competitive firm and monopoly choose output where marginal cost(MC) equals marginal revenue (MR). The perfectly competitive firm must set its price equal to the market price so its MR is equal to market price; thus
\text{P} = \text{MC}
, or 400,000. For the monopoly, it is where
\text{MR} = \text{MC}
or at 200,000. The perfectly competitive firm must sell at the market price of $4 while the monopoly can charge a higher price based on consumers' willingness to pay, or at $7.
The monopolist's profit can also be calculated from the graph using geometry. The area of a rectangle is its base multiplied by its height. The base of the rectangle is the line from the vertical axis to the profit maximizing quantity equal to 200,000. The height of the rectangle is the line
\text{P}-\text{ATC} = 7-4 = 3
. The base multiplied by the height is
200\text{,}000\times3=\$600\text{,}000
A monopolist will set production at its profit-maximizing quantity and then determine the market price. The monopolist will produce where marginal cost equals marginal revenue, then set the price where quantity demanded equals the monopolist's output.
<Reasons for a Monopoly>Social Effects of a Monopoly
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Shortest path between two single nodes - MATLAB shortestpath - MathWorks Benelux
Shortest Path Between Specified Nodes
Shortest Path in Weighted Graph
Shortest Path Ignoring Edge Weights
Shortest Path from Node Coordinates
Shortest path between two single nodes
P = shortestpath(G,s,t) computes the shortest path starting at source node s and ending at target node t. If the graph is weighted (that is, G.Edges contains a variable Weight), then those weights are used as the distances along the edges in the graph. Otherwise, all edge distances are taken to be 1.
P = shortestpath(G,s,t,'Method',algorithm) optionally specifies the algorithm to use in computing the shortest path. For example, if G is a weighted graph, then shortestpath(G,s,t,'Method','unweighted') ignores the edge weights in G and instead treats all edge weights as 1.
[P,d] = shortestpath(___) additionally returns the length of the shortest path, d, using any of the input arguments in previous syntaxes.
[P,d,edgepath] = shortestpath(___) additionally returns the edge indices edgepath of all edges on the shortest path from s to t.
Calculate the shortest path between nodes 7 and 8.
Find the shortest path between nodes 3 and 8, and specify two outputs to also return the length of the path.
Since the edges in the center of the graph have large weights, the shortest path between nodes 3 and 8 goes around the boundary of the graph where the edge weights are smallest. This path has a total length of 4.
Create and plot a graph with weighted edges, using custom node coordinates.
Find the shortest path between nodes 6 and 8 based on the graph edge weights. Highlight this path in green.
Specify Method as unweighted to ignore the edge weights, instead treating all edges as if they had a weight of 1. This method produces a different path between the nodes, one that previously had too large of a path length to be the shortest path. Highlight this path in red.
Find the shortest path between nodes in a graph using the distance between the nodes as the edge weights.
Create a graph with 10 nodes.
Create x- and y-coordinates for the graph nodes. Then plot the graph using the node coordinates by specifying the 'XData' and 'YData' name-value pairs.
Add edge weights to the graph by computing the Euclidean distances between the graph nodes. The distance is calculated from the node coordinates
\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{i}}\right)
\mathit{d}=\sqrt{{|\Delta \mathit{x}|}^{2}+{|\Delta \mathit{y}|}^{2}}=\sqrt{{|{\mathit{x}}_{\mathit{s}}-{\mathit{x}}_{\mathit{t}}|}^{2}+{|{\mathit{y}}_{\mathit{s}}-{\mathit{y}}_{\mathit{t}}|}^{2}}.
\Delta \mathit{x}
\Delta \mathit{y}
, first use findedges to obtain vectors sn and tn describing the source and target nodes of each edge in the graph. Then use sn and tn to index into the x- and y-coordinate vectors and calculate
\Delta \mathit{x}={\mathit{x}}_{\mathit{s}}-{\mathit{x}}_{\mathit{t}}
\Delta \mathit{y}={\mathit{y}}_{\mathit{s}}-{\mathit{y}}_{\mathit{t}}
. The hypot function computes the squareroot of the sum of squares, so specify
\Delta \mathit{x}
\Delta \mathit{y}
as the input arguments to calculate the length of each edge.
Add the distances to the graph as the edge weights and replot the graph with the edges labeled.
Calculate the shortest path between node 1 and node 10 and specify two outputs to also return the path length. For weighted graphs, shortestpath automatically uses the 'positive' method which considers the edge weights.
Use the highlight function to display the path in the plot.
s,t — Source and target node IDs (as separate arguments)
Example: shortestpath(G,2,5) computes the shortest path between node 2 and node 5.
Example: shortestpath(G,'node1','node2') computes the shortest path between the named nodes node1 and node2.
'auto' (default) | 'unweighted' | 'positive' | 'mixed' | 'acyclic'
'acyclic' (only for digraph)
Algorithm designed to improve performance for directed, acyclic graphs (DAGs) with weighted edges.
Use isdag to confirm if a directed graph is acyclic.
For most graphs, 'unweighted' is the fastest algorithm, followed by 'acyclic', 'positive', and 'mixed'.
Example: shortestpath(G,s,t,'Method','acyclic')
P — Shortest path between nodes
Shortest path between nodes, returned as a vector of node indices or an array of node names. P is empty, {}, if there is no path between the nodes.
If s and t contain numeric node indices, then P is a numeric vector of node indices.
If s and t contain node names, then P is a cell array or string array containing node names.
If there are multiple shortest paths between s and t, then P contains only one of the paths. The path that is returned can change depending on which algorithm Method specifies.
d — Shortest path distance
Shortest path distance, returned as a numeric scalar. d is the summation of the edge weights between consecutive nodes in P. If there is no path between the nodes, then d is Inf.
edgepath — Edges on shortest path
vector of edge indices
Edges on shortest path, returned as a vector of edge indices. For multigraphs, this output indicates which edge between two nodes is on the path. This output is compatible with the 'Edges' name-value pair of highlight, for example: highlight(p,'Edges',edgepath).
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Compute cos(X*pi) accurately - MATLAB cospi - MathWorks América Latina
Calculate Cosine of Multiples of π
Compute cos(X*pi) accurately
Y = cospi(X)
Y = cospi(X) computes cos(X*pi) without explicitly computing X*pi. This calculation is more accurate than cos(X*pi) because the floating-point value of pi is an approximation of π. In particular:
For odd integers, cospi(n/2) is exactly zero.
For integers, cospi(n) is +1 or -1.
Compare the accuracy of cospi(X) vs. cos(X*pi).
Calculate the cosine of X*pi using the normal cos function.
Y = cos(X*pi)
\pi
\mathrm{cos}\left(\frac{\pi }{2}\right)=0
Use cospi to calculate the same values. In this case, the results are exact.
Z = cospi(X)
cos | cosd | sinpi
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ExponentialWindow - Maple Help
Home : Support : Online Help : Science and Engineering : Signal Processing : Windowing Functions : ExponentialWindow
multiply an array of samples by an exponential windowing function
ExponentialWindow( A, alpha )
numeric value strictly between
0
1
The ExponentialWindow( A, alpha ) command multiplies the Array A by the exponential windowing function, with parameter
\mathrm{\alpha }
The exponential windowing function
w\left(k\right)
\mathrm{\alpha }
N
w\left(k\right)={\mathrm{\alpha }}^{\frac{k}{N}}
\mathrm{\alpha }
must lie in the open interval
0,1
The SignalProcessing[ExponentialWindow] command is thread-safe as of Maple 18.
\mathrm{with}\left(\mathrm{SignalProcessing}\right):
N≔1024:
a≔\mathrm{GenerateUniform}\left(N,-1,1\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893628825139122532}}
\mathrm{ExponentialWindow}\left(a,0.23\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893628824989257724}}
c≔\mathrm{Array}\left(1..N,'\mathrm{datatype}'='\mathrm{float}'[8],'\mathrm{order}'='\mathrm{C_order}'\right):
\mathrm{ExponentialWindow}\left(\mathrm{Array}\left(1..N,'\mathrm{fill}'=1,'\mathrm{datatype}'='\mathrm{float}'[8],'\mathrm{order}'='\mathrm{C_order}'\right),0.72,'\mathrm{container}'=c\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893628824989233148}}
u≔\mathrm{`~`}[\mathrm{log}]\left(\mathrm{FFT}\left(c\right)\right):
\mathbf{use}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{plots}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{display}\left(\mathrm{Array}\left(\left[\mathrm{listplot}\left(\mathrm{ℜ}\left(u\right)\right),\mathrm{listplot}\left(\mathrm{ℑ}\left(u\right)\right)\right]\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end use}
The SignalProcessing[ExponentialWindow] command was introduced in Maple 18.
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Black body - New World Encyclopedia
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Black-body radiation curves at different temperatures: 3000 K, 4000 K, and 5000 K. As the temperature decreases, the peak of the black-body radiation curve moves to lower intensities and longer wavelengths. The black-body radiation graph is also compared with the classical model of Rayleigh and Jeans.
In physics, a black body (in an ideal sense) is an object that absorbs all electromagnetic radiation that falls on it, without any of the radiation passing through it or being reflected by it. Because it does not reflect or transmit visible light, the object appears black when it is cold.
When heated, the black body becomes an ideal source of thermal radiation, which is called black-body radiation. If a perfect black body at a certain temperature is surrounded by other objects in equilibrium at the same temperature, it will on average emit exactly as much as it absorbs, at the same wavelengths and intensities of radiation that it had absorbed.
The temperature of the object is directly related to the wavelengths of the light it emits. At room temperature, black bodies emit infrared light, but as the temperature increases past a few hundred degrees Celsius, black bodies start to emit at visible wavelengths, from red through orange, yellow, and white before ending up at blue, beyond which the emission includes increasing amounts of ultraviolet radiation.
1.1 Black body simulators
4.5 Temperature of Earth
5 Doppler effect for a moving blackbody
The color (chromaticity) of black-body radiation depends on the temperature of the black body. The locus of such colors (shown here in CIE 1931 x,y space) is known as the Planckian locus.
Black bodies have been used to test the properties of thermal equilibrium because they emit radiation that is distributed thermally. In classical physics, each different Fourier mode in thermal equilibrium should have the same energy, leading to the theory of ultraviolet catastrophe that there would be an infinite amount of energy in any continuous field. Studies of black-body radiation led to the revolutionary field of quantum mechanics. In addition, black-body laws have been used to determine the black-body temperatures of planets.
If a small window is opened into an oven, any light that enters the window has a very low probability of leaving without being absorbed. Conversely, the hole acts as a nearly ideal black-body radiator. This makes peepholes into furnaces good sources of blackbody radiation, and some people call it cavity radiation for this reason.[1]
In the laboratory, black-body radiation is approximated by the radiation from a small hole entrance to a large cavity, a hohlraum. Any light entering the hole would have to reflect off the walls of the cavity multiple times before it escaped, in which process it is nearly certain to be absorbed. This occurs regardless of the wavelength of the radiation entering (as long as it is small compared to the hole). The hole, then, is a close approximation of a theoretical black body and, if the cavity is heated, the spectrum of the hole's radiation (i.e., the amount of light emitted from the hole at each wavelength) will be continuous, and will not depend on the material in the cavity (compare with emission spectrum). By a theorem proved by Gustav Kirchhoff, this curve depends only on the temperature of the cavity walls.[2] Kirchhoff introduced the term "black body" in 1860.
Calculating this curve was a major challenge in theoretical physics during the late nineteenth century. The problem was finally solved in 1901 by Max Planck as Planck's law of black-body radiation.[3] By making changes to Wien's Radiation Law (not to be confused with Wien's displacement law) consistent with thermodynamics and electromagnetism, he found a mathematical formula fitting the experimental data in a satisfactory way. To find a physical interpretation for this formula, Planck had then to assume that the energy of the oscillators in the cavity was quantized (i.e., integer multiples of some quantity). Einstein built on this idea and proposed the quantization of electromagnetic radiation itself in 1905 to explain the photoelectric effect.
These theoretical advances eventually resulted in the superseding of classical electromagnetism by quantum electrodynamics. Today, these quanta are called photons and the black-body cavity may be thought of as containing a gas of photons. In addition, it led to the development of quantum probability distributions, called Fermi-Dirac statistics and Bose-Einstein statistics, each applicable to a different class of particle, which are used in quantum mechanics instead of the classical distributions.
The temperature of a Pāhoehoe lava flow can be estimated by observing its color. The result agrees well with the measured temperatures of lava flows at about 1,000 to 1,200 °C.
The wavelength at which the radiation is strongest is given by Wien's displacement law, and the overall power emitted per unit area is given by the Stefan-Boltzmann law. So, as temperature increases, the glow color changes from red to yellow to white to blue. Even as the peak wavelength moves into the ultra-violet, enough radiation continues to be emitted in the blue wavelengths that the body will continue to appear blue. It will never become invisible — indeed, the radiation of visible light increases monotonically with temperature.[4]
WMAP image of the cosmic microwave background radiation anisotropy. It has the most precise thermal emission spectrum known and corresponds to a temperature of 2.725 kelvin (K) with an emission peak at 160.2 GHz.
Black body simulators
Although a black body is a theoretical object, (i.e. emissivity (e) = 1.0), common applications define a source of infrared radiation as a black body when the object approaches an emissivity of 1.0, (typically e = .99 or better). A source of infrared radiation less than .99 is referred to as a greybody.[5] Applications for black body simulators typically include the testing and calibration of infrared systems and infrared sensor equipment.
Much of a person's energy is radiated away in the form of infrared energy. Some materials are transparent to infrared light, while opaque to visible light (note the plastic bag). Other materials are transparent to visible light, while opaque or reflective to the infrared (note the man's glasses).
{\displaystyle P_{net}=P_{emit}-P_{absorb}.}
Applying the Stefan–Boltzmann law,
{\displaystyle P_{net}=A\sigma \epsilon \left(T^{4}-T_{0}^{4}\right)\,}
{\displaystyle P_{net}=100\ \mathrm {W} \,}
The total energy radiated in one day is about 9 MJ (Mega joules), or 2000 kcal (food calories). Basal metabolic rate for a 40-year-old male is about 35 kcal/(m²•h),[10] which is equivalent to 1700 kcal per day assuming the same 2 m² area. However, the mean metabolic rate of sedentary adults is about 50 percent to 70 percent greater than their basal rate.[11]
{\displaystyle \lambda _{peak}={\frac {2.898\times 10^{6}\ \mathrm {K} \cdot \mathrm {nm} }{305\ \mathrm {K} }}=9500\ \mathrm {nm} \,}
{\displaystyle I(\nu ,T)d\nu ={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{kT}}-1}}\,d\nu }
{\displaystyle I(\nu ,T)d\nu \,}
is the amount of energy per unit surface area per unit time per unit solid angle emitted in the frequency range between ν and ν+dν by a black body at temperature T;
{\displaystyle h\,}
is Planck's constant;
{\displaystyle c\,}
is the speed of light; and
{\displaystyle k\,}
is Boltzmann's constant.
The relationship between the temperature T of a black body, and wavelength
{\displaystyle \lambda _{max}}
at which the intensity of the radiation it produces is at a maximum is
{\displaystyle T\lambda _{\mathrm {max} }=2.898...\times 10^{6}\ \mathrm {nm\ K} .\,}
The total energy radiated per unit area per unit time
{\displaystyle j^{\star }}
(in watts per square meter) by a black body is related to its temperature T (in kelvins) and the Stefan–Boltzmann constant
{\displaystyle \sigma }
{\displaystyle j^{\star }=\sigma T^{4}.\,}
Here is an application of black-body laws to determine the black body temperature of a planet. The surface may be warmer due to the greenhouse effect.[13]
Earth's longwave thermal radiation intensity, from clouds, atmosphere and ground
Emitted radiation (for example [[Earth's_energy_budget#Outgoing_energy|Earth's infrared glow]])
For the inner planets, incident and emitted radiation have the most significant impact on temperature. This derivation is concerned mainly with that.
then we can derive a formula for the relationship between the Earth's temperature and the Sun's surface temperature.
{\displaystyle P_{Semt}=\left(\sigma T_{S}^{4}\right)\left(4\pi R_{S}^{2}\right)\qquad \qquad (1)}
{\displaystyle \sigma \,}
is the Stefan–Boltzmann constant,
{\displaystyle T_{S}\,}
is the surface temperature of the Sun, and
{\displaystyle R_{S}\,}
is the radius of the Sun.
{\displaystyle P_{Eabs}=P_{Semt}(1-\alpha )\left({\frac {\pi R_{E}^{2}}{4\pi D^{2}}}\right)\qquad \qquad (2)}
{\displaystyle R_{E}\,}
is the radius of the Earth and
{\displaystyle D\,}
is the distance between the Sun and the Earth.
{\displaystyle \alpha \ }
is the albedo of Earth.
Even though the earth only absorbs as a circular area
{\displaystyle \pi R^{2}}
, it emits equally in all directions as a sphere:
{\displaystyle P_{Eemt}=\left(\sigma T_{E}^{4}\right)\left(4\pi R_{E}^{2}\right)\qquad \qquad (3)}
{\displaystyle T_{E}}
is the black body temperature of the earth.
{\displaystyle P_{Eabs}=P_{Eemt}\,}
{\displaystyle \left(\sigma T_{S}^{4}\right)\left(4\pi R_{S}^{2}\right)(1-\alpha )\left({\frac {\pi R_{E}^{2}}{4\pi D^{2}}}\right)=\left(\sigma T_{E}^{4}\right)\left(4\pi R_{E}^{2}\right).\,}
{\displaystyle T_{S}{\sqrt {\frac {{\sqrt {1-\alpha }}R_{S}}{2D}}}=T_{E}}
{\displaystyle T_{S}\,}
is the surface temperature of the Sun,
{\displaystyle R_{S}\,}
is the radius of the Sun,
{\displaystyle D\,}
is the distance between the Sun and the Earth,
{\displaystyle \alpha }
is the albedo of the Earth, and
{\displaystyle T_{E}\,}
is the blackbody temperature of the Earth.
In other words, given the assumptions made, the temperature of Earth depends only on the surface temperature of the Sun, the radius of the Sun, the distance between Earth and the Sun and the albedo of Earth.
{\displaystyle T_{S}=5778\ \mathrm {K} ,}
{\displaystyle R_{S}=6.96\times 10^{8}\ \mathrm {m} ,}
{\displaystyle D=1.5\times 10^{11}\ \mathrm {m} ,}
{\displaystyle \alpha =0.3\ }
we'll find the effective temperature of the Earth to be
{\displaystyle T_{E}=255\ \mathrm {K} .}
This is the black body temperature as measured from space, while the surface temperature is higher due to the greenhouse effect
Doppler effect for a moving blackbody
The Doppler effect is the well known phenomenon describing how observed frequencies of light are "shifted" when a light source is moving relative to the observer. If f is the emitted frequency of a monochromatic light source, it will appear to have frequency f' if it is moving relative to the observer :
{\displaystyle f'=f{\frac {1}{\sqrt {1-v^{2}/c^{2}}}}(1-{\frac {v}{c}}\cos \theta )}
where v is the velocity of the source in the observer's rest frame, θ is the angle between the velocity vector and the observer-source direction, and c is the speed of light.[14] This is the fully relativistic formula, and can be simplified for the special cases of objects moving directly towards ( θ = π) or away ( θ = 0) from the observer, and for speeds much less than c.
To calculate the spectrum of a moving blackbody, then, it seems straightforward to simply apply this formula to each frequency of the blackbody spectrum. However, simply scaling each frequency like this is not enough. We also have to account for the finite size of the viewing aperture, because the solid angle receiving the light also undergoes a Lorentz transformation. (We can subsequently allow the aperture to be arbitrarily small, and the source arbitrarily far, but this cannot be ignored at the outset.) When this effect is included, it is found that a blackbody at temperature T that is receding with velocity v appears to have a spectrum identical to a stationary blackbody at temperature T', given by:[15]
{\displaystyle T'=T{\frac {1}{\sqrt {1-v^{2}/c^{2}}}}(1-{\frac {v}{c}}\cos \theta )}
For the case of a source moving directly towards or away from the observer, this reduces to
{\displaystyle T'=T{\sqrt {\frac {c-v}{c+v}}}}
This is an important effect in astronomy, where the velocities of stars and galaxies can reach significant fractions of c. An example is found in the cosmic microwave background radiation, which exhibits a dipole anisotropy from the Earth's motion relative to this blackbody radiation field.
↑ When used as a compound adjective, the term is typically hyphenated, as in "black-body radiation," or combined into one word, as in "blackbody radiation." The hyphenated and one-word forms should not generally be used as nouns.
↑ Kerson Huang. 1967. Statistical Mechanics. (New York, NY: John Wiley & Sons.)
↑ Max Planck, 1901. On the Law of Distribution of Energy in the Normal Spectrum. Annalen der Physik. 4:553. Retrieved December 15, 2008.
↑ L.D. Landau, and E.M. Lifshitz. 1996. Statistical Physics, 3rd Edition, Part 1. (Oxford, UK: Butterworth-Heinemann.)
↑ What is a Black Body and Infrared Radiation? Electro Optical Industries, Inc. Retrieved December 15, 2008.
↑ Emissivity Values for Common Materials. Infrared Services. Retrieved December 15, 2008.
↑ Emissivity of Common Materials. Omega Engineering. Retrieved December 15, 2008.
↑ Abanty Farzana, 2001. Temperature of a Healthy Human (Skin Temperature). The Physics Factbook. Retrieved December 15, 2008.
↑ B. Lee, Theoretical Prediction and Measurement of the Fabric Surface Apparent Temperature in a Simulated Man/Fabric/Environment System. dsto.defence.gov.au. Retrieved December 15, 2008.
↑ J. Harris, and F. Benedict. 1918. A Biometric Study of Human Basal Metabolism. Proc Natl Acad Sci USA 4(12):370–373.
↑ J. Levine, 2004. Nonexercise activity thermogenesis (NEAT): environment and biology. Am J Physiol Endocrinol Metab. 286:E675–E685. Retrieved December 15, 2008.
↑ Heat Transfer and the Human Body. DrPhysics.com. Retrieved December 15, 2008.
↑ George H.A. Cole, Michael M. Woolfson. 2002. Planetary Science: The Science of Planets Around Stars, 1st ed. (Institute of Physics Publishing. ISBN 075030815X), 36–37, 380–382.
↑ T.P. Gill, 1965. The Doppler Effect. (London, UK: Logos Press.)
↑ John M. McKinley, 1979. Relativistic transformations of light power. Am. J. Phys. 47(7).
Cole, George H.A., Michael M. Woolfson. Planetary Science: The Science of Planets Around Stars. Bristol, UK: Institute of Physics Publishing, 2002. ISBN 075030815X
Gill, T. P. The Doppler Effect. London, UK: Logos Press, 1965.
Harris, J. and F. Benedict. A Biometric Study of Human Basal Metabolism. Proc Natl Acad Sci USA 4(12) (1918): 370–373.
Huang, Kerson. Statistical Mechanics. New York, NY: John Wiley & Sons, 1967.
Kroemer, Herbert, and Charles Kittel. Thermal Physics, 2nd ed. W. H. Freeman Company, 1980. ISBN 0716710889
Landau, L. D., and E. M. Lifshitz. Statistical Physics, 3rd Edition, Part 1. Oxford, UK: Butterworth-Heinemann, 1996 (original 1958).
Tipler, Paul, and Ralph Llewellyn. Modern Physics, 4th ed. W. H. Freeman, 2002. ISBN 0716743450
Calculating Blackbody Radiation Interactive calculator with Doppler Effect. Includes most systems of units.
Cooling Mechanisms for Human Body - From Hyperphysics.
"Blackbody Spectrum" by Jeff Bryant, Wolfram Demonstrations Project.
Black body history
History of "Black body"
Retrieved from https://www.newworldencyclopedia.org/p/index.php?title=Black_body&oldid=1064472
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Calculus Problem on Related Rates - 2D Geometry: Popeye Needs His Spinach - Andrew Ellinor | Brilliant
Popeye Needs His Spinach
Popeye loves his spinach. He squeezes a can of his favorite spinach in such a way that it retains the shape of a cylinder and its volume remains constant, but the radius of the can decreases at
1 \text{ cm}
per second.
How fast is the height of the can changing at the moment the can has a radius of
4 \text{ cm}
10 \text{ cm}?
4 cm/s 5 cm/s 8 cm/s 10 cm/s
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Extra-base hit — Wikipedia Republished // WIKI 2
{\displaystyle XBH=2B+3B+HR}
In baseball, an extra-base hit (EB, EBH or XBH[1]), also known as a long hit, is any base hit on which the batter is able to advance past first base without the benefit of a fielder either committing an error or opting to make a throw to retire another base runner (see fielder's choice). Extra-base hits are often not listed separately in tables of baseball statistics, but are easily determined by calculating the sum total of a batter's doubles, triples, and home runs.[2] Extra-base hits are particularly valuable because they ensure that there will be no runners on base that will be forced to advance on the next fair ball.
Another related statistic of interest that can be calculated is "extra bases on long hits". A batter gets three of these for each home run, two for each triple, and one for each double. Thus, leading the league in "Most extra bases in long hits" is a significant accomplishment in power hitting.
The statistic Extra-Base Hits Allowed (for example by a pitcher or by the fielding team in general) is denoted by the abbreviation XBA.[1]
Watch all of Miguel Andujar's 76 extra-base hits in 2018
Every Yoán Moncada Extra-Base Hit in 2019
Miguel Andujar's extra-base hits in first 15 games
1 Major League Baseball leaders
1.2 Season
1.3 Single game
1.4 Consecutive games
Hank Aaron holds the record for most extra-base hits, at 1,477.
Further information: List of Major League Baseball career extra base hits leaders
The record for most career extra-base hits is 1,477, held by Hank Aaron.[2] Among players with at least 1,000 career hits, Mark McGwire is the only one to have had at least half of his hits go for extra bases.[3]
There have been 15 instances of a player recording 100 extra-base hits in a single season; Lou Gehrig, Chuck Klein and Todd Helton are the only players to have achieved this twice, with Helton the only one to do so in consecutive seasons.[4]
The top 5 are as follows: (totals are current through the end of the 2016 season)[5]
Babe Ruth (1921) – 119
Lou Gehrig (1927) – 117
Barry Bonds (2001) – 107
Chuck Klein (1930) – 107
Todd Helton (2001) – 105
The modern-era record for most extra-base hits by one batter, in one game, is five, held by 13 different players, including Lou Boudreau, Joe Adcock, Willie Stargell, Steve Garvey, Shawn Green, Kelly Shoppach, Josh Hamilton, Jackie Bradley, Jr., Kris Bryant, José Ramírez, Matt Carpenter, Alex Dickerson, and most recently Luis Urías.[6] Adcock, Green and Hamilton did so while hitting four home runs.[6] In the postseason, Albert Pujols, Hideki Matsui, Bob Robertson, Frank Isbell and Enrique Hernández have all recorded four extra-base hits in a game.[7]
Paul Waner (1927) and Chipper Jones (2006) jointly hold the longest hitting streak for extra bases. Both players recorded extra-base hits in 14 consecutive games.[8]
The Boston Red Sox recorded 17 extra-base hits in a 29–4 victory against the St. Louis Browns in 1950.[9] In the postseason, the team single game record for extra-base hits is 13, by the New York Yankees against the Red Sox in game 3 of the 2004 ALCS.[10] Two teams have 9 extra-base hits in a World Series game, namely the 1925 Pittsburgh Pirates (in game 7 vs the Washington Senators) and the 2007 Boston Red Sox (game 1, vs the Colorado Rockies).[10]
The 2003 Boston Red Sox had 649 extra-base hits, the most by one team in a single season.[11][12]
Slugging average
^ a b "Baseball Basics: Abbreviations". MLB.com. Retrieved April 20, 2014.
^ a b "Career Leaders & Records for Extra-Base Hits". Baseball-Reference.com. Retrieved April 20, 2014.
^ "Spanning Multiple Seasons or entire Careers, From 1871 to 2018, (requiring H>=1000 and XBH>=0.5*H), sorted by greatest Extra Base Hits". Baseball Reference. Retrieved July 20, 2018.
^ "Batting Season Finder, For single seasons, From 1901 to 2017, (requiring XBH>=100)". Baseball-Reference.com. Retrieved August 1, 2017.
^ "Single-Season Leaders & Records for Extra-Base Hits". Baseball-Reference.com. Retrieved June 11, 2015.
^ a b "Batting Game Finder: In years 1901 to 2020, (requiring XBH>=5), sorted by most recent date". Stathead. Retrieved September 3, 2020.
^ Schoenfield, David (October 11, 2011). "Pujols awesome; Brewer rotation in trouble". ESPN.com. Retrieved June 28, 2017.
^ "Paul Waner - BR Bullpen". Baseball-Reference.com. Retrieved April 25, 2018.
^ "Team Batting Game Finder: From 1913 to 2017, (requiring XBH>=15)". Baseball Reference. Retrieved August 6, 2017.
^ a b "Team Batting Game Finder: In the Postseason, From 1913 to 2017, (requiring XBH>=9)". Baseball Reference. Retrieved August 6, 2017.
^ "Red Sox announce 2004 Major League coaching staff". Boston Red Sox. January 9, 2004. Retrieved August 7, 2017.
^ "MLB Team Hitting Statistics". MLB.com. Retrieved August 7, 2017.
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The Red Dutchess: mental health
The importance of being Emotionally Intelligent
I could portray a 'perfect' life. It's the social norm as of late, but I choose not to. I choose to be authentic.
There are things I could change about myself physically; My insecurities and flaws. But I choose not to.
To be authentic, or at least endeavouring to be, means accepting yourself for the organic version of yourself in all its entirety without the sharp needle of ego, injecting you with notions that you too should be everything like those girls and guys parading a pseudo-idyllic life on social media. Their minds are as confined, changeable and flippant as the very boxes in which those euphoric scenes appear. It's like a game of human tetris; you only fit in if you meet the criteria: Abs, Plastic Surgery, some kind of diet/protein product in hand, airbrushed, probably unfulfilled and chronically unhappy. Now, don't get me wrong. I do enjoy seeing what people wear, makeup and beautiful art and photography. However, I do think there is a distinct difference between content that is led by the want to share experiences, work and beauty and content that is led by ego for status, an X amount of 'Likes' and validation.
Point is, anybody could feign this 'lifestyle'. Congratulations to those who choose to embrace their oftentimes not-so-easy-reality, you've unlocked the 'Emotional Intelligence' level. Game over for players who thought they could learn the 'cheats' and get ahead. Life isn't a game, there is no winner.
I believe it take a great degree of emotional intelligence to not conform to an unrealistic lifestyle as well as being responsible with what you show on social media, especially to our impressionable youth. It's because of our youth and their naivety that we should be cautious about what we turn into 'trends' and 'normative' human behaviour. So, since there is a huge amount of irresponsibility, I only feel it's my responsibility to highlight the importance of living as real as we humans ought to, and that's with humility, control of ego and with a good conscience. Not to mention that choosing to be vulnerable and to show your true self is in fact liberating.
The components of 'Emotional Intelligence' as given by Daniel Goleman is:
If we work on cultivating these aspects that are the definition of Emotional Intelligence then we can become better humans and create a better society. And it's possible because if you become aware of your inclination to 'think' a certain way then that's you becoming aware of your thinking styles and knowing which are helpful and unhelpful. For Example, I would have the tendency to 'Catastrophize', and so my 'meta-thinking', which basically means 'thinking about your thinking' allows me to work on developing another habit of thinking. I know this isn't easy, but it's possible. It's called neuroplasticity. It's where the brain physically "shapes itself according to repeated experiences"- Daniel Goldman
This is important because it can prevent you from getting caught up in unhelpful cyclical behaviours that you may always find yourself in, but never really know why.
The more you become aware of your own thinking and feeling then the more empathetic you become towards other people. The more empathetic you become, the more compassionate and calm you become. You see? Developing your emotional intelligence equips you to be a better human. When you're more compassionate you think twice about what influence you have over others, especially our young people. You choose to do what's good for others instead of you and your ego.
You'll find that people who have been through the wars with their mental health and tragic or negative life events are often the most empathetic people who care so much for others.
Being in the mental health field both as a blogger and a professional you would think you'd find the most wholesome people, but it's is actually the opposite. Just like any other career or area there is competition. And to practice what I preach and remain authentic I'll tell you the truth; It's very frustrating.
However, whenever people do compete with me I've already outdone them from that very moment. How? Well because I don't reciprocate, because for me there is no competition. I'm in mental health because I was born with a mental health disorder and my own suffering has impassioned me to help wherever I can. Not to mention over a decade of training and experience.
To compete is to be led by ego and to need constant recognition, to support and collaborate is to be led by compassion and empathy for the people around you.
You can have self-satisfaction without seeking validation on social media. You are enough, just as you have always been enough in your most imperfect, authentic state.
Having Emotional Intelligence helps to immunise you from being absorbed by this inane culture, a culture that only survive on ego and money. If you starve it, then we become a much healthier, compassionate and authentic culture.
Posted by The Red Dutchess at Tuesday, April 17, 2018 1 comment:
Labels: Beauty & Fashion Acting on Mental Health, Daniel Goleman, Emotional Intelligence, mental health, Replenish, Replenish: Acting on Mental Health, Social Media and Mental Health, The Red Dutchess
Today is Monday! 'Blue Monday' apparently?!
Like, who actually comes up with these 'trends'? Well, to answer my own question, and hopefully yours, Blue Monday was actually part of a marketing plan from a 'Travel Company' as an incentive to encourage people to book holidays. Relatively smart marketing yes, but a disregard for those of us who already struggle through tough, tarnished weeks upon weeks of mental torture.What about our hashtags and formulas?! Have you one for my O.C.D? Or my friend's P.T.S.D?
This pseudoscience was argued to have been discovered through using a "scientific formula"...
"The formula uses many factors, including: weather conditions, debt level (the difference between debt accumulated and our ability to pay), time since Christmas, time since failing our new year’s resolutions, low motivational levels and feeling of a need to take action. (By some fella Arnall in 2006).
Let me introduce you to Natasha, Counsellor, Mother, Mental Health Warrior and part of the Replenish Tribe.
Natasha speaks candidly about her struggle with her mental health and how she manages through all those days that nobody has a hashtag or marketing strategy for.
What Natasha has to say to you...
\frac{\left(C×R×ZZ\right)}{\left(\left(Tt+D\right)×St\right)}+\left(P×Pr\right)>
When I am having a 'bad day' (don't you just love that label by the way? How I wish it was confined to one day - I'd skip off to bed that evening, safe in the knowledge that tomorrow everything would be ok again), which sometimes turns into days & worst case scenario a week or longer, my confidence plummets.
When I DO find the courage to look in the mirror I mostly don't recognise the woman looking back at me. Where did my vibrant, assertive, funny, confident self go? The girl who loved concerts, nights out in the City, travel?
On the bad days those things mostly fill me with fear. How will I get out of a concert in a venue I'm not familiar with, if I need to? People on nights out who have consumed too much alcohol peak my anxiety as they can be unpredictable, argumentative, aggressive. Travel means airports, security, confined spaces.
I can only manage it with people I feel 'safe' with.
When I feel low & anxious I feel worthless, like I have nothing interesting to say to my friends, family, colleagues. Especially colleagues & customers, people who know me the least.
The paranoia is relentless, persistent and exhausting. The internal dialogue usually goes something like 'they think I'm boring/stupid, they wish I'd hurry up, they think I'm weird because I don't go on work nights out or drink, I'm the only one who makes mistakes' and on and on the list is endless.
When I'm having a 'bad day' I see no point in anything, there's no colour, everything is messy & dis-organised. I just want to stay in bed. I feel like I am merely going through the motions, functioning at the lowest level necessary, existing. NOT living.
That is what anxiety does to you. It robs you of your personality, robs you of your confidence and robs you of your identity.
My own experience caused me to feel as if my emotions and feelings had disappeared. I could not feel the highs of love that I used to feel, the intense happiness & excitement of seeing my favourite band, I couldn't grieve the loss of both my Aunts. Emotionless! This is exactly how I feel on a 'bad day'. Nothing anyone could do or say could make me happy. I feel numb and detached and there are times when I think I might never smile again.
My only thought can be HOPE. Recovering from the way I feel on those 'bad days'. I can tell those of you who feel like this that your emotions do come back in recovery. Your confidence and personality gradually return in little strips, building up in layers, until eventually you feel like the person you were before you became ill.
It takes commitment & tenacity. It takes speaking up, confiding in your 'tribe', being honest with yourself and with them. Totally honest! If you can't say it out loud, technology is your friend - put it in a text, just start the conversation.
Everyone's self-care is different. For me it's taking quiet time out, detaching, re-charging. My work is busy, both physically & mentally demanding, so quiet time is vital for me. I like to spend time with people who are close in my circle, people I feel safe with. Movies, pamper time, naps, meditation - these are all things I enjoy and I make time to incorporate them into my life. It's absolutely vital for me.
I read a lot about anxiety & obsessive thoughts.
Meeting Caroline has been an absolutely pivotal part in my recovery. Finally I felt like I could speak about how I was feeling, without fear. It was absolutely liberating. I drove home exhilarated after my first group session - I WAS NOT ALONE!
One thing I've read & utilise now on the daily is this :
'Never say yes when you mean no, and never say no when you means yes'
Simple but effective. Try it.
I was a people pleaser even to my detriment on most occasions, but now I realise I also need to please myself.
I often think to myself 'who am I kidding?'. I rarely stop thinking. I wonder about my internal dialogue. Would I speak to other people the way I speak to myself? Would I allow other people to speak to me the way I speak to myself? Absolutely not! Why then do I re-enforce the negative automatic thoughts? Example: 'You're useless/ugly/incapable/a laughing stock.....the list is endless'.
I've started challenging these thoughts when I have them and try to list facts to support the thoughts. The majority of the time they are unsupported.
I often feel tired of being tired. Obsessional thoughts are exhausting. Sometimes I just don't lend the energy to it. I concentrate on getting tasks done and nothing else. The more you learn to accept and let go, the more your body will respond to a new way of thinking.
I am the most impatient person, this I know. With everything in my life, not just wanting to be well. Recovery, I am told, will come in time. There is no time limit or magic cure. Everybody is different and some people will recover more quickly than others. Medication and therapy which works for one person might be totally ineffective for you - as I've discovered. Yes, it's frustrating - please trust me, just be patient and your body will take care of itself in its own time.
Remember this: you deserve to BE WELL. If you are struggling to be taken seriously by health professionals then be aware that you can take an advocate with you to help speak with you. I took Caroline with me to get the ball rolling. It started my journey towards reclaiming good mental health & has given me confidence to speak up to my GP since.
Go easy on yourself x
You do not have to be alone as you deal with your mental health. Replenish is developed by people with mental health issues who are compassionate about helping others who are similar to us.
www.Replenish.site
or find out where your local 'Replenish Tribe' is.
What the media doesn't tell us is how to manage those Blue Mondays after they've created a unnecessary hype.
However, it's a hype that does bring more global awareness to mental health.
See what we had to say in our YOUTUBE video/Podcast here:
Labels: Beauty & Fashion Acting on Mental Health, Blue Monday, Guest blogger, Irish Blogger, mental health, Mental Health awareness, Mental Health blogger, mental health ireland
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Symbols denoting vectors and matrices should be indicated in bold type. Scalar variable names should normally be expressed using italics.
Use decimal points (not commas); use a space for thousands (10 000 and above).
Follow internationally accepted rules and conventions. In particular use the international system of units (SI). If other quantities are mentioned, give their equivalent in SI.
Please insert tables as editable text and not as images. Tables should be placed next to the relevant text in the article. Number tables consecutively in accordance with their appearance in the text (table 1, table 2, etc.) and place any table notes below the table body. Be sparing in the use of tables and ensure that the data presented in them do not duplicate results described elsewhere in the article.
Young Modulus 12.74 MPa
Poisson coefficient 0.25
Figure 1. Scipedia logo.
Number the figures according to their sequence in the text (figure 1, figure 2, etc.). Ensure that each illustration has a caption. A caption should comprise a brief title. Keep text in the illustrations themselves to a minimum but explain all symbols and abbreviations used. Try to keep the resolution of the figures to a minimum of 300 dpi. If a finer resolution is required, the figure can be inserted as supplementary material
A bulleted list item
You may choose to number equations for easy referencing. In that case they must be numbered consecutively with Arabic numerals in parentheses on the right hand side of the page. Below is an example of formulae that should be referenced as eq. (1).
{\displaystyle {\nabla }^{2}\phi =0}
Citations in text will follow a citation-sequence system (i.e. sources are numbered by order of reference so that the first reference cited in the document is [1], the second [2], and so on) with the number of the reference in square brackets. Once a source has been cited, the same number is used in all subsequent references. If the numbers are not in a continuous sequence, use commas (with no spaces) between numbers. If you have more than two numbers in a continuous sequence, use the first and last number of the sequence joined by a hyphen (e.g. [1, 3] or [2-4]).
You should ensure that all references are cited in the text and that the reference list. References should preferably refer to documents published in Scipedia. Unpublished results should not be included in the reference list, but can be mentioned in the text. The reference data must be updated once publication is ready. Complete bibliographic information for all cited references must be given following the standards in the field (IEEE and ISO 690 standards are recommended). If possible, a hyperlink to the referenced publication should be given. See examples for Scipedia’s articles [1], other publication articles [2], books [3], book chapter [4], conference proceedings [5], and online documents [6], shown in references section below.
[1] Author, A. and Author, B. (Year) Title of the article. Title of the Publication. Article code. Available: http://www.scipedia.com/ucode.
[6] Institution or author. Title of the document. Year. [Online] (Date consulted: day, month and year). Available: http://www.scipedia.com/document.pdf. [Accessed day, month and year].
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2014 Strong Convergence for Hybrid Implicit S-Iteration Scheme of Nonexpansive and Strongly Pseudocontractive Mappings
Shin Min Kang, Arif Rafiq, Faisal Ali, Young Chel Kwun
K
be a nonempty closed convex subset of a real Banach space
E
S:K\to K
be nonexpansive, and let
T:K\to K
be Lipschitz strongly pseudocontractive mappings such that
p\in F\left(S\right)\cap F\left(T\right)=\left\{x\in K:Sx=Tx=x\right\}
∥x-Sy∥\le ∥Sx-Sy∥ \text{and} ∥x-Ty∥\le ∥Tx-Ty∥
x, y\in K
\left\{{\beta }_{n}\right\}
\left[0, 1\right]
satisfying (i)
{\sum }_{n=1}^{\infty }{\beta }_{n}=\infty
{\text{lim}}_{n\to \infty }{\beta }_{n}=0.
{x}_{0}\in K
\left\{{x}_{n}\right\}
be a sequence iteratively defined by
{x}_{n}=S{y}_{n}, {y}_{n}=\left(1-{\beta }_{n}\right){x}_{n-1}+{\beta }_{n}T{x}_{n}, n\ge 1.
\left\{{x}_{n}\right\}
converges strongly to a common fixed point
p
S
T
Shin Min Kang. Arif Rafiq. Faisal Ali. Young Chel Kwun. "Strong Convergence for Hybrid Implicit S-Iteration Scheme of Nonexpansive and Strongly Pseudocontractive Mappings." Abstr. Appl. Anal. 2014 (SI71) 1 - 5, 2014. https://doi.org/10.1155/2014/735673
Shin Min Kang, Arif Rafiq, Faisal Ali, Young Chel Kwun "Strong Convergence for Hybrid Implicit S-Iteration Scheme of Nonexpansive and Strongly Pseudocontractive Mappings," Abstract and Applied Analysis, Abstr. Appl. Anal. 2014(SI71), 1-5, (2014)
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Wavelet Scattering - MATLAB & Simulink - MathWorks 日本
Mallat, with Bruna and Andén, pioneered the creation of a mathematical framework for studying convolutional neural architectures [2][3][4][5]. Andén and Lostanlen developed efficient algorithms for wavelet scattering of 1-D signals [4] [6]. Oyallon developed efficient algorithms for 2-D scattering [7]. Andén, Lostanlen, and Oyallon are major contributors to the ScatNet [10] and Kymatio [11] software for computing scattering transforms.
\left\{{\mathrm{Ï}}_{j,k}\right\}
{\mathrm{Ï}}_{J}
f
{\mathrm{Ï}}_{j,k}
{\mathrm{Ï}}_{J}
fâ{\mathrm{Ï}}_{J}
Consider the tree view of the wavelet time scattering network. Suppose that there are M wavelets in the first filter bank, and N wavelets in the second filter bank. The number of wavelet filters in each filter bank do not have to be large before a naive implementation becomes unfeasible. Efficient implementations take advantage of the lowpass nature of the modulus function and critically downsample the scattering and scalogram coefficients. These strategies were pioneered by Andén, Mallat, Lostanlen, and Oyallon [4] [6] [7] in order to make scattering transforms computationally practical while maintaining their ability to produce low-variance data representations for learning. By default, waveletScattering and waveletScattering2 create networks that critically downsample the coefficients.
[4] Andén, J., and S. Mallat. "Deep Scattering Spectrum." IEEE Transactions on Signal Processing. Vol. 62, Number 16, 2014, pp. 4114–4128.
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Properties of Chemical Systems and Equilibrium - Course Hero
General Chemistry/Equilibrium Concepts/Properties of Chemical Systems and Equilibrium
Equilibrium constants can be calculated only if the system is closed and at equilibrium.
Values of equilibrium constants Kc and Kp provide valuable information about properties of chemical systems. However, some major factors must be noted while determining equilibrium constants.
Equilibrium can be achieved only in a closed system, which means no substance can enter or leave the system. If some substances are entering or exiting the system, an equilibrium cannot be reached.
The equilibrium constant Kc is equal to the reaction quotient Q at equilibrium. When Q is equal to Kc, no further changes in concentrations will happen.
When the reaction quotient is less than the equilibrium constant
(Q\lt K_{\rm{c}})
, the system is not at equilibrium, and the forward reaction is happening at a greater rate, producing more products.
When the reaction quotient is greater than the equilibrium constant
(Q>K_{\rm{c}})
, the system is not at equilibrium, and the reverse reaction is happening at a greater rate, producing more reactants.
Comparing the equilibrium constant Kp with the value of partial pressure term provides information about reactions when all of the reactants and products are gases. For a reaction at equilibrium
K_{\rm{p}}=\frac{\left({P_{\rm{C}}}^c\times{P_{\rm{D}}}^d\right)}{\left({P_{\rm{A}}}^a\times{P_{\rm{B}}}^b\right)}
The equilibrium constant Kp equals the partial pressure term only at equilibrium. At equilibrium, there will be no change in partial pressures of reactants or products.
When the partial pressure term is less than Kp, the system is not at equilibrium, and the forward reaction is happening at a greater rate, producing more products.
When the partial pressure term is greater than Kp, the system is not at equilibrium, and the reverse reaction is happening at a greater rate, producing more reactants.
The ideal gas law can be used to derive a relation between Kc and Kp. The ideal gas law is
PV = nRT
If all the reactants and the products are gases, the relation between them is given by the equation
{K_{\rm{p}}} = {K_{\rm{c}}}{\left( {RT} \right)^{\Delta {n_g}}}
Here, the term
\Delta n_g
is the difference between moles of products and moles of reactants.
<Law of Mass Action and Equilibrium Constant Kc and Kp>Changing Equilibria and Le Chatelier's Principle
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life expectancy - Brewer Inspections Services
Average service life or functional period in years, assuming regular maintenance.
the average life span of an individual
"Human lifespan" redirects here. For the lifespan of a person in stages, see [[:Maturation[disambiguation needed]|Maturation& § 91;disambiguation needed]]].
This article is about the measure of remaining life. For the Dean Koontz novel, see Life Expectancy (novel).
Human life expectancy at birth, measured by region, between 1950 and 2050
Life expectancy is a statistical measure of the average time an organism is expected to live, based on the year of its birth, its current age and other demographic factors including gender. The most commonly used measure of life expectancy is at birth (LEB), which can be defined in two ways. Cohort LEB is the mean length of life of an actual birth cohort (all individuals born a given year) and can be computed only for cohorts born many decades ago, so that all their members have died. Period LEB is the mean length of life of a hypothetical cohort assumed to be exposed, from birth through death, to the mortality rates observed at a given year.
National LEB figures reported by statistical national agencies and international organizations are indeed estimates of period LEB. In the Bronze Age and the Iron Age, LEB was 26 years; the 2010 world LEB was 67.2 years. For recent years, in Swaziland LEB is about 49, and in Japan, it is about 83. The combination of high infant mortality and deaths in young adulthood from accidents, epidemics, plagues, wars, and childbirth, particularly before modern medicine was widely available, significantly lowers LEB. But for those who survive early hazards, a life expectancy of 60 or 70 would not be uncommon. For example, a society with a LEB of 40 may have few people dying at precisely 40: most will die before 30 or after 55. In populations with high infant mortality rates, LEB is highly sensitive to the rate of death in the first few years of life. Because of this sensitivity to infant mortality, LEB can be subjected to gross misinterpretation, leading one to believe that a population with a low LEB will necessarily have a small proportion of older people. For example, in a hypothetical stationary population in which half the population dies before the age of five but everybody else dies at exactly 70 years old, LEB will be about 36, but about 25% of the population will be between the ages of 50 and 70. Another measure, such as life expectancy at age 5 (e5), can be used to exclude the effect of infant mortality to provide a simple measure of overall mortality rates other than in early childhood; in the hypothetical population above, life expectancy at 5 would be another 65. Aggregate population measures, such as the proportion of the population in various age groups, should also be used along individual-based measures like formal life expectancy when analyzing population structure and dynamics.
Mathematically, life expectancy is the mean number of years of life remaining at a given age, assuming age-specific mortality rates remain at their most recently measured levels. It is denoted by
{\displaystyle e_{x}}
, which means the mean number of subsequent years of life for someone now aged
{\displaystyle x}
, according to a particular mortality experience. Longevity, maximum lifespan, and life expectancy are not synonyms. Life expectancy is defined statistically as the mean number of years remaining for an individual or a group of people at a given age. Longevity refers to the characteristics of the relatively long life span of some members of a population. Maximum lifespan is the age at death for the longest-lived individual of a species. Moreover, because life expectancy is an average, a particular person may die many years before or many years after the "expected" survival. The term "maximum life span" has a quite different meaning and is more related to longevity.
Life expectancy is also used in plant or animal ecology;life tables (also known as actuarial tables). The term life expectancy may also be used in the context of manufactured objects, but the related term shelf life is used for consumer products, and the terms "mean time to breakdown" (MTTB) and "mean time between failures" (MTBF) are used in engineering.
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(Redirected from Block cipher modes)
{\displaystyle C_{i}=E_{K}(P_{i}\oplus C_{i-1}),}
{\displaystyle C_{0}=IV,}
{\displaystyle P_{i}=D_{K}(C_{i})\oplus C_{i-1},}
{\displaystyle C_{0}=IV.}
{\displaystyle C_{i}=E_{K}(P_{i}\oplus P_{i-1}\oplus C_{i-1}),P_{0}\oplus C_{0}=IV,}
{\displaystyle P_{i}=D_{K}(C_{i})\oplus P_{i-1}\oplus C_{i-1},P_{0}\oplus C_{0}=IV.}
{\displaystyle {\begin{aligned}C_{i}&={\begin{cases}{\text{IV}},&i=0\\E_{K}(C_{i-1})\oplus P_{i},&{\text{otherwise}}\end{cases}}\\P_{i}&=E_{K}(C_{i-1})\oplus C_{i},\end{aligned}}}
{\displaystyle I_{0}={\text{IV}}.}
{\displaystyle I_{i}={\big (}(I_{i-1}\ll s)+C_{i}{\big )}{\bmod {2}}^{b},}
{\displaystyle C_{i}=\operatorname {MSB} _{s}{\big (}E_{K}(I_{i-1}){\big )}\oplus P_{i},}
{\displaystyle P_{i}=\operatorname {MSB} _{s}{\big (}E_{K}(I_{i-1}){\big )}\oplus C_{i},}
{\displaystyle C_{j}=P_{j}\oplus O_{j},}
{\displaystyle P_{j}=C_{j}\oplus O_{j},}
{\displaystyle O_{j}=E_{K}(I_{j}),}
{\displaystyle I_{j}=O_{j-1},}
{\displaystyle I_{0}={\text{IV}}.}
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Inverse-producing extensions of Topological Algebras - Wikiversity
Inverse-producing extensions of Topological Algebras
Algebra extension
{\displaystyle B}
{\displaystyle A}
containing an inverse element
{\displaystyle b=z^{-1}\in B}
to a given
{\displaystyle z\in A}
The course covers a basic concept of considering mathematical properties in extensions of a given topological algebra. In doing so, we extends a algebra
{\displaystyle A}
to an extension
{\displaystyle B}
{\displaystyle A\subset B}
, checking a property of an element
{\displaystyle z\in A}
in the extension
{\displaystyle z\in B}
. In this course we treat multiplicative invertibility as a mathematical property and consider, among other things, topological properties that allow to create an algebra extension
{\displaystyle B}
in which an multiplicative inverse of a given element
{\displaystyle z\in A}
{\displaystyle \exists _{z^{-1}\in B}:\,z\cdot z^{-1}=z^{-1}\cdot z=e}
is satisfied and
{\displaystyle e\in A}
is the one-element of the multiplication. Essentially, this involves topological properties of the element
{\displaystyle z\in A}
that either allows invertibility in a particular extension
{\displaystyle B}
{\displaystyle A}
or never has an inverse element in any extensions
{\displaystyle B}
{\displaystyle A}
, i.e., is permanently singular. The basic sets with a multiplicative linkage here are topological algebras, where the linkages are.
(TA1) multiplication of a vector by a scalar as an outer linkage,
(TA2) addition of vectors in the vector space as an inner linkage, and
(TA3) multiplication of two vectors as inner linkage
are continuous in each case. Here, a vector space with properties (TA1) and (TA2) is called a topological vector space. If there is additionally a multiplication is additionally this multiplicative inner linkage continuous (TA3) then the vector space is called a topological algebra.
1 Origine of the Course
2 Wiki2Reveal Contents
2.1 Chapter 0: Preliminaries
2.2 Chapter 1: Introduction into topological algebras
2.3 Chapter 2: K-singular elements
2.4 Chapter 3: K-regular elements
2.5 (3.1) Normed and locally bounded algebras
2.6 (3.2) Multiplicative topological algebras
2.7 (3.3) Locally convex and locally pseudoconvex algebras
2.8 (3.4) Topological algebras
2.9 Chapter 4: Solvability of equations
Origine of the Course[edit | edit source]
This course was created in the german Wikiversity for a lecture with Wiki2Reveal slides, that can be annotated in the browser (no online storage of annotation). The course material will be translated as Open Educational Resources that can be maintained and updated in Wikiversity. The concept of inverse-producing algebra extensions is based on Richard Arens work on that topic for normed algebras[1]. Other classes of topological algebras were considered by Wieslaw Zelazko like locally convex and multiplicative locally convex algebras[2]. The course includes
Wiki2Reveal Contents[edit | edit source]
Chapter 0: Preliminaries[edit | edit source]
Chapter 1: Introduction into topological algebras[edit | edit source]
Chapter 2: K-singular elements[edit | edit source]
First, we shall discuss topological criteria which ensure that an element is permanently singular in any algebra extension of class
{\displaystyle {\mathcal {K}}}
. If the negation of the topological property causes the element to have an inverse element in an algebra extension of class
{\displaystyle {\mathcal {K}}}
, a topoligical invertibility criterion arises.
Chapter 3: K-regular elements[edit | edit source]
In this chapter, given topological criteria, we construct algebra extensions algebra extensions of class
{\displaystyle {\mathcal {K}}}
in which a given
{\displaystyle {\mathcal {K}}}
-regular element is invertible.
(3.1) Normed and locally bounded algebras[edit | edit source]
(3.2) Multiplicative topological algebras[edit | edit source]
(3.3) Locally convex and locally pseudoconvex algebras[edit | edit source]
(3.4) Topological algebras[edit | edit source]
Chapter 4: Solvability of equations[edit | edit source]
In this chapter, the invertibility
{\displaystyle z\cdot x=x\cdot z=e}
{\displaystyle x=z^{-1}}
is considered as a special case of the solvability of an equation
{\displaystyle z_{1}\cdot x=z_{2}}
{\displaystyle z_{1},z_{2}\in A}
. Here, an algebra expansion
{\displaystyle B}
{\displaystyle A}
is used to search for a solution
{\displaystyle x\in B}
{\displaystyle z_{1}\cdot x=z_{2}}
The lecture is provided in a PanDoc slide format (PanDocElectron-SLIDE) in Wikiversity, which can be transferred to annotatable slides using the Wiki2Reveal tool or using [PanDocElectron to load the Wikiversity source available online and convert it to presentation slides that can be used offline. You can also use Wiki2Reveal to create a RevealJS or DZSlides presentation directly from Wikiversity articles].
In the spirit of OER (Open Educational Resources), the lecture content should be made freely available. Initially, the slides created from the customizable wiki content were made available in a GitHib repository to facilitate download and use. However, maintaining and updating the content in a repository is not very efficient due to the fact that any update in Wikiversity requires also an update of the slides in the repository. Therefore, Wiki2Reveal was developed for the lecture slides, which allows to generate lecture slides directly from the Wikiversity content and to annotate the slides online as well. The Wikiversity slides are represented as sections and usually any annotation is performed either directly on the slide by pressing (C) for comment. Furthermore the lecturer is able to use a separate whiteboard for every slide by pressing (B) on the slide an return to the slide by pressing (B) again. Annotation mode for the slides can be switched by pressing (C) again. Keep the sections of Wiki2Reveal articles small so that the generated slides fit on the screen. More detailed text about the slides is usually created in separate articles. If the explanation pages explicitly refer to a slide, the explanation page gets a marker PanDocElectron-TEXT and SLIDE or TEXT version refer to each other reciprocally.
Help: Creating quizzes for lecture content
↑ Arens Richard, (1958), Inverse producing extensions of normed algebras, Trans. Amer. Math. Soc. 88, S. 536-548
↑ Zelazko Wieslaw, (1971), Selected topics in topological algebras, Aarhus University lecture notes, No. 31
Retrieved from "https://en.wikiversity.org/w/index.php?title=Inverse-producing_extensions_of_Topological_Algebras&oldid=2299155"
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VERU Financial Ratios - FinancialModelingPrep
\dfrac{Current Assets}{Current Liabilities}
\dfrac{Cash and Cash Equivalents + Short Term Investments + Account Receivables}{Current Liabilities}
\dfrac{Cash and Cash Equivalents}{Current Liabilities}
\dfrac{(Account Receivable (start) + Account Receivable (end))/2}{Revenue/365}
\dfrac{(Inventories (start) + Inventories (end))/2}{COGS/365}
\dfrac{DSO + DIO}{}
\dfrac{(Accounts Payable (start) + Accounts Payable (end))/2}{COGS/365}
\dfrac{DSO + DIO − DPO}{}
\dfrac{Gross Profit}{Revenue}
\dfrac{Operating Income}{Revenue}
\dfrac{Income Before Tax}{Revenue}
\dfrac{Net Income}{Revenue}
\dfrac{Provision For Income Taxes}{Income Before Tax}
\dfrac{Net Income}{Average Total Assets}
\dfrac{Net Income}{Average Total Equity}
\dfrac{EBIT}{Average Total Asset − Average Current Liabilities}
\dfrac{Net Income}{EBT}
\dfrac{EBT}{EBIT}
\dfrac{EBIT}{Revenue}
\dfrac{Total Liabilities}{Total Assets}
\dfrac{Total Debt}{Total Equity}
\dfrac{Long−Term Debt}{Long−Term Debt + Shareholders Equity}
\dfrac{Total Debt}{Total Debt + Shareholders Equity}
\dfrac{EBIT}{Interest Expense}
\dfrac{Operating Cash Flows}{Total Debt}
\dfrac{Total Assets}{Total Equity}
\dfrac{Revenue}{NetPPE}
\dfrac{Revenue}{Total Average Assets}
\dfrac{Operating Cash Flow}{Revenue}
\dfrac{Free Cash Flow}{Operating Cash Flow}
\dfrac{Operating Cash Flow}{Total Debt}
\dfrac{Operating Cash Flow}{Short-Term Debt}
\dfrac{Operating Cash Flow}{Capital Expenditure}
-30.81 The larger the operating cash flow coverage for these items, the greater the company's ability to meet its obligations, along with giving the company more cash flow to expand its business, withstand hard times, and not be burdened by debt servicing and the restrictions typically included in credit agreements.
\dfrac{Operating Cash Flow}{Dividend Paid + Capital Expenditure}
\dfrac{DPS (Dividend per Share)}{EPS (Net Income per Share Number}
\dfrac{Stock Price per Share}{Equity per Share}
\dfrac{Stock Price per Share}{Operating Cash Flow per Share}
-40.81 The price/cash flow ratio is used by investors to evaluate the investment attractiveness, from a value standpoint, of a company's stock.
\dfrac{Stock Price per Share}{EPS}
\dfrac{Price Earnings Ratio}{Expected Revenue Growth}
\dfrac{Stock Price per Share}{Revenue per Share}
\dfrac{Dividend per Share}{Stock Price per Share}
\dfrac{Entreprise Value}{EBITDA}
\dfrac{Stock Price per Share}{Intrinsic Value}
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Existence and Multiple Positive Solutions for Boundary Value Problem of Fractional Differential Equation with p -Laplacian Operator
2014 Existence and Multiple Positive Solutions for Boundary Value Problem of Fractional Differential Equation with
p
-Laplacian Operator
Min Jiang, Shouming Zhong
This paper investigates the existence, multiplicity, nonexistence, and uniqueness of positive solutions to a kind of two-point boundary value problem for nonlinear fractional differential equations with
p
-Laplacian operator. By using fixed point techniques combining with partially ordered structure of Banach space, we establish some criteria for existence and uniqueness of positive solution of fractional differential equations with
p
-Laplacian operator in terms of different value of parameter. In particular, the dependence of positive solution on the parameter was obtained. Finally, several illustrative examples are given to support the obtained new results. The study of illustrative examples shows that the obtained results are applicable.
Min Jiang. Shouming Zhong. "Existence and Multiple Positive Solutions for Boundary Value Problem of Fractional Differential Equation with
p
-Laplacian Operator." Abstr. Appl. Anal. 2014 (SI62) 1 - 18, 2014. https://doi.org/10.1155/2014/512426
Min Jiang, Shouming Zhong "Existence and Multiple Positive Solutions for Boundary Value Problem of Fractional Differential Equation with
p
-Laplacian Operator," Abstract and Applied Analysis, Abstr. Appl. Anal. 2014(SI62), 1-18, (2014)
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(Redirected from 16:9)
Aspect ratio with a width of 16 units and height of 9 units
"16x9" redirects here. For the TV series, see 16x9 (TV series).
16:9 (1.77:1) is a widescreen aspect ratio with a width of 16 units and height of 9.
Once seen as exotic,[1] since 2009, it has become the most common aspect ratio for televisions and computer monitors and is also the international standard format of digital television HDTV Full HD and SD TV. It has replaced the fullscreen 4:3 aspect ratio.
16:9 (1.77:1) (said as sixteen by nine or sixteen to nine) is the international standard format of HDTV, non-HD digital television and analog widescreen television PALplus. Japan's Hi-Vision originally started with a 15:9 (1.66:1) ratio but converted when the international standards group introduced a wider ratio of 16 to 9. Many digital video cameras have the capability to record in 16:9, and 16:9 is the only widescreen aspect ratio natively supported by the DVD standard. DVD producers can also choose to show even wider ratios such as 1.85:1 and 2.40:1 within the 16:9 DVD frame by hard matting or adding black bars within the image itself.
Derivation of the 16:9 aspect ratio
The main figure shows 4:3, 1.85:1, and 2.35:1 rectangles with the same area A, and 16:9 rectangles that covers (black) or is common to (grey) them. The calculation considers the extreme rectangles, where m and n are multipliers to maintain their respective aspect ratios and areas.
Dr. Kerns H. Powers, a member of the SMPTE Working Group on High-Definition Electronic Production, first proposed the 16:9 (1.77:1) aspect ratio in 1984,[2] when nobody was creating 16:9 videos. The popular choices in 1980 were 4:3 (based on TV standard's ratio at the time), 15:9 (the European "flat" 1.66:1 ratio), 1.85:1 (the American "flat" ratio) and 2.35:1 (the CinemaScope/Panavision) ratio for anamorphic widescreen.
Powers cut out rectangles with equal areas, shaped to match each of the popular aspect ratios. When overlapped with their center points aligned, he found that all of those aspect ratio rectangles fit within an outer rectangle with an aspect ratio of 1.77:1 and all of them also covered a smaller common inner rectangle with the same aspect ratio 1.77:1.[3] The value found by Powers is exactly the geometric mean of the extreme aspect ratios, 4:3 and 2.35:1,
{\displaystyle \textstyle {\sqrt {\frac {47}{15}}}}
≈ 1.77 which is coincidentally close to 16:9. Applying the same geometric mean technique to 16:9 and 4:3 yields an aspect ratio of around 1.5396:1, sometimes approximated as 14:9 (1.55:1), which is likewise used as a compromise between these ratios.[4]
While 16:9 (1.77:1) was initially selected as a compromise format, the subsequent popularity of HDTV broadcast has solidified 16:9 as perhaps the most common video aspect ratio in use.[5] Most 4:3 (1.33:1) and 2.40:1 video is now recorded using a "shoot and protect" technique[6] that keeps the main action within a 16:9 (1.77:1) inner rectangle to facilitate HD broadcast[citation needed]. Conversely it is quite common to use a technique known as center-cutting, to approach the challenge of presenting material shot (typically 16:9) to both an HD and legacy 4:3 audience simultaneously without having to compromise image size for either audience. Content creators frame critical content or graphics to fit within the 1.33:1 raster space. This has similarities to a filming technique called open matte.
After the original 16:9 Action Plan of the early 1990s, the European Union instituted the 16:9 Action Plan,[7] just to accelerate the development of the advanced television services in 16:9 aspect ratio, both in PALplus (compatible with regular PAL broadcasts) and also in HD-MAC (an early HD format). The Community fund for the 16:9 Action Plan amounted to €228,000,000.
Over a long period in the late 2000s and early 2010s, the computer industry switched from 4:3 to 16:9 as the most common aspect ratio for monitors and laptops. A 2008 report by DisplaySearch cited a number of reasons for this shift, including the ability for PC and monitor manufacturers to expand their product ranges by offering products with wider screens and higher resolutions, helping consumers to more easily adopt such products and "stimulating the growth of the notebook PC and LCD monitor market".[8] By using the same aspect ratio for both TVs and monitors, manufacturing can be streamlined and research costs reduced by not requiring two separate sets of equipment, and since a 16:9 is narrower than a 16:10 panel of the same length, more panels can be created per sheet of glass.[9][10][11]
In 2011, Bennie Budler, product manager of IT products at Samsung South Africa, confirmed that monitors capable of 1920 × 1200 resolutions are not being manufactured anymore. "It is all about reducing manufacturing costs. The new 16:9 aspect ratio panels are more cost-effective to manufacture locally than the previous 16:10 panels".[12]
In March 2011, the 16:9 resolution 1920 × 1080 became the most common used resolution among Steam's users. The previous most common resolution was 1680 × 1050 (16:10).[13]
16:9 is the only widescreen aspect ratio natively supported by the DVD format. An Anamorphic PAL+ DVD with a full frame, may contain 768 × 576p, but a video player software will stretch this to 1024 × 576p.
Producers can also choose to show even wider ratios such as 1.85:1 and 2.4:1 within the 16:9 DVD frame by hard matting or adding black bars within the image itself. Some films which were made in a 1.85:1 aspect ratio, such as the U.S.-Italian co-production Man of La Mancha and Kenneth Branagh's Much Ado About Nothing, fit quite comfortably onto a 1.77:1 HDTV screen and have been issued as an enhanced version on DVD without the black bars. Many digital video cameras have the capability to record in 16:9.
Common resolutions[edit]
320 180 QnHD
426 240 NTSC widescreen
In Europe, 16:9 is the standard broadcast format for most TV channels and all HD broadcasts. Some countries adopted the format for analogue television, first by using the PALplus standard (now obsolete) and then by simply using WSS on normal PAL broadcasts.
Albania All channels.
Andorra All channels.
Armenia All channels.
Austria All channels.
Azerbaijan All channels.
Belarus All channels.
Bosnia and Herzegovina All channels.
Cyprus All channels.
Czech Republic All channels.
Denmark All channels.
Estonia All channels.
Finland All channels.
France All channels.
Germany All channels.
Georgia All channels.
Greece All channels.
Hungary All channels.
Iceland All channels.
Ireland All channels.
Italy All channels.
Kazakhstan All channels.
Latvia All channels.
Lithuania All channels.
Luxembourg All channels.
Malta All channels.
Moldova All channels.
Monaco All channels.
Montenegro All channels.
Netherlands All channels.
North Macedonia All channels.
Norway All channels.
Poland All channels.
Portugal All channels.
Romania Always on 16:9: Antena channels (Antena 1, Antena Stars, Antena 3, Happy, ZU TV, Antena Internațional), RCS & RDS channels (including Digi24, U TV, Music Channel), Kiss TV, B1 TV, Telekom Sport, Look TV, Look Plus, WarnerMedia channels (Cartoon Network, Boomerang)
Often on 16:9: TVR channels (TVR 1, TVR 2, TVR 3, TVRi), PRO channels (Pro TV, Pro 2, Pro X, Pro Cinema, Pro Gold, Pro TV Internațional)
Always on 4:3 with 16:9 stretched: CNM channels (Național TV, Național 24 Plus, Favorit TV), TVR regional channels (TVR Cluj, TVR Craiova, TVR Iași, TVR Tîrgu-Mureș, TVR Timișoara), Prima TV.
Russia All channels.
San Marino All channels.
Serbia All channels.
Slovakia All channels.
Slovenia All channels.
Spain All channels.
Sweden All channels.
Switzerland All channels.
Turkey All channels.
Ukraine All channels.
United Kingdom All channels.
Australia All channels.
Fiji All channels.
New Zealand All channels.
Afghanistan All channels.
Cambodia All channels.
China CCTV channels 1–15, CCTV-5+, all CGTN channels. Older contents in 4:3 and news contents are stretched on SD variants of these channels as stretching on SD channels is common.
Hong Kong All channels.
India All HD channels. Most SD channels are still broadcasting in 4:3, either fullscreen or letterboxed.
Indonesia All channels except ANTV & tvOne.
Iran All channels.
Israel All channels.
Japan All channels.
Japan pioneered in its analogue HDTV system (MUSE) in 16:9 format, started in the 1980s. Currently all main channels have digital terrestrial television channels in 16:9. Many satellite broadcast channels are being broadcast in 16:9 as well.
Jordan All channels.
Kyrgyzstan All channels.
Lebanon All channels.
Malaysia All channels.
Mongolia MNB & MN2, TM Television, TV5, TV6, TV8, Channel 25, Эx Орон, SBN, ETV, MNC, Eagle News TV, Edutainment TV, Star TV, SPS, Sportbox and SHUUD TV.
Myanmar All channels.
Oman All channels.
Pakistan All HD channels. Most SD channels are still broadcasting in 4:3, either in fullscreen or letterboxed
Philippines 16:9 native:[a] PTV, ANC (both SD and HD),[b] Kapamilya Channel (both SD and HD),[b] CNN Philippines, One PH,[b] One News,[b] One Sports+, Hope Channel Philippines, 3ABN, Hope International, INCTV, Net 25, DZRH News Television, TeleRadyo, Colours, all TAP DMV channels (TAP TV, TAP Edge, TAP Movies, TAP Action Flix, TAP Sports, Premier Sports, Premier Tennis, and Premier Football), BuKo, NBA TV Philippines, PBA Rush, UAAP Varsity Channel, Golden Nation Network, Metro Channel
4:3 upscaled/stretched to 16:9:[c] ETC, 2nd Avenue, all BEAM's subchannels, Light Network, UNTV,[d] Ang Dating Daan TV, SMNI, TV5, One Sports, GMA 7, A2Z, GTV, IBC 13
Qatar All beIN Sports channels, Al Jazeera, Al Jazeera English, Al Jazeera Mubasher, Qatar TV HD, all Alkass channels.
Saudi Arabia All channels.
Singapore All channels, however 16:9 contents look squashed on older 4:3 sets. Also, all 4:3 contents including news clips are stretched as stretching is common.
South Korea All channels.
Sri Lanka All channels
Syria All channels.
Thailand All channels.
United Arab Emirates All channels.
Vietnam All of VTC's channels, VTV channels, HTV channels and K+'s channels (selected programmes), most of local channels.
^ Channels that are squeezed/letterboxed to 4:3 on analog terrestrial transmissions nor no letterbox on widescreen-produced programs
^ a b c d 16:9 versions available on pay-TV services only
^ channels that are originally broadcasting in 4:3 on analog terrestrial, but upscaled or stretched to 16:9 for digital terrestrial television, cable and satellite
^ Some programs are aired in true 16:9 formatting
Argentina All channels.
Barbados All channels.
Brazil Channels change between 16:9 and 4:3 pillarbox depending of what's airing.
Canada All channels.
Chile All channels. Expect Telecanal in 4.3 in ident 4:3 letterboxed in commercials
Colombia All channels.
Costa Rica All channels.
Dominican Republic All channels.
Ecuador All channels.
Jamaica All channels.
Mexico Free-to-air television: Las Estrellas, FOROtv, Canal 5, NU9VE, Televisa Regional, Azteca Uno, Azteca 7, a+, adn40, Imagen Televisión, Excélsior TV, Canal Once, Canal 22, Una Voz con Todos, Teveunam, Milenio Televisión, Multimedios Televisión, Teleritmo, and some local HD stations.
Pay television: U, Golden, Golden Edge, TL Novelas, Bandamax, De Película, De Película Clásico, Ritmoson Latino, TDN, TeleHit, Distrito Comedia, Tiin, Az Noticias, Az Clic!, Az Mundo, Az Corazón, Az Cinema, 52MX, TVC, TVC Deportes, Pánico, Cinema Platino, Cine Mexicano.
Panama All channels.
Paraguay Almost all channels on free-to-air television, especially HD feeds (ex.: RPC, NPY, Unicanal, channel 7 HD). SD feeds (usually found on pay television) are usually letterboxed and downscaled to 4:3 (ex.: SNT & Paravisión).
Peru All channels.
United States All HD channels. SD feeds (usually found on pay television) are usually letterboxed and downscaled to 4:3.
Uruguay All channels.
Venezuela All channels.
Angola All channels.
Botswana All channels.
Burkina Faso All channels.
Cameroon All channels.
Cape Verde All channels.
Comoros All channels.
Congo All channels.
Djibouti All channels.
Egypt All channels.
Equatorial Guinea All channels.
Eritrea All channels.
Ethiopia All channels.
Gabon All channels.
Ghana All channels.
Ivory Coast All channels.
Kenya All channels.
Lesotho All channels.
Liberia All channels.
Libya All channels.
Madagascar All channels.
Malawi All channels.
Mali All channels.
Morocco All channels except 2M.
Mozambique All channels.
Mauritius All channels.
Namibia All channels.
Nigeria All channels.
Rwanda All channels.
Senegal All channels.
Somalia All channels.
South Africa All channels.
Sudan All channels.
Togo All channels.
Tunisia All channels.
Uganda All channels.
Zimbabwe All channels.
Videos with display aspect ratio 16:9 on Commons
^ "A Brief Review on HDTV in Europe in the early 90's | LIVE-PRODUCTION.TV". www.live-production.tv.
^ "Understanding Aspect Ratios" (Technical bulletin). CinemaSource. The CinemaSource Press. 2001. Retrieved 2009-10-24.
^ EN 5956091, "Method of showing 16:9 pictures on 4:3 displays", issued 1999-09-21
^ "Why 16:9 aspect ratio was chosen for HD?". Guruprasad's Portal. 2014-06-13. Retrieved 2021-09-17.
^ Baker, I (1999-08-25). "Safe areas for widescreen transmission" (PDF). EBU. CH: BBC. Archived from the original (PDF) on 2010-10-11. Retrieved 2009-10-27.
^ "Television in the 16:9 screen format" (legislation summary). EU: Europa. Retrieved 2011-09-08.
^ "Product Planners and Marketers Must Act Before 16:9 Panels Replace Mainstream 16:10 Notebook PC and Monitor LCD Panels, New DisplaySearch Topical Report Advises". DisplaySearch. 2008-07-01. Retrieved 2011-09-08.
^ "Display Ratio Change (again)". 2009-04-14. Archived from the original on 2020-03-02. Retrieved 2020-01-22.
^ "16:10 vs 16:9 - the monitor aspect ratio conundrum". 2012-10-22. Retrieved 2020-01-22.
^ "Resurgence of 16:10 Aspect Ratio Laptop Computers to Occupy 2% Share of Non-Apple Market in 2020, Says TrendForce". 2019-04-11. Retrieved 2020-01-22.
^ "Widescreen monitors: Where did 1920×1200 go? « Hardware « MyBroadband Tech and IT News". Mybroadband.co.za. 2011-01-10. Retrieved 2011-09-08.
^ "Steam Hardware & Software Survey". Steam. Retrieved 2011-09-08.
"NEC Monitor Technology Guide". NEC. Archived from the original on 2006-05-21. Retrieved 2006-07-24.
Retrieved from "https://en.wikipedia.org/w/index.php?title=16:9_aspect_ratio&oldid=1088997948"
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Home : Support : Online Help : Science and Engineering : Signal Processing : Convolution and Correlation Computations : Convolution
compute the finite linear convolution of two arrays of samples
Convolution(A, B)
Arrays of real or complex numeric sample values
algorithm : symbol, algorithm to use for computation
The Convolution(A, B) command computes the convolution of the Arrays A and B of length
M
N
respectively, storing the result in a Array C of length
M+N-1
and having datatype float[8] or complex[8], which is then returned.
The convolution is defined by the formula
{C}_{k}=\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{i=1}^{k}{A}_{i}{B}_{k-i+1}
k
1
M+N-1
{A}_{j}=0
M<j
{B}_{j}=0
N<j
Before the code performing the computation runs, A and B are converted to datatype float[8] (if the values are all real-valued) or complex[8] (if all the values are complex-valued, but not all real-valued) if they do not have that datatype already. For this reason, it is most efficient if A and B have one of these datatypes beforehand.
If either A or B is an rtable that is not a 1-D Array, it is accepted by the command and converted to an Array. Should this not be possible, an error will be thrown.
M+N-1
having datatype float[8] or complex[8].
The algorithm=name option can be used to specify the algorithm used for computing the convolution. Supported algorithms:
auto - automatically choose the fastest algorithm based on input.
direct - use direct convolution formula for computation. This is the default.
fft - use an algorithm based on the Fast Fourier Transform (FFT). This is a much faster algorithm than the direct formula for large samples, but numerical roundoff can cause significant numerical artifacts, especially when the result has a large dynamic range.
The SignalProcessing[Convolution] command is thread-safe as of Maple 17.
\mathrm{with}\left(\mathrm{SignalProcessing}\right):
\mathrm{Convolution}\left(\mathrm{Array}\left([5,7]\right),\mathrm{Array}\left([-2,6,10]\right)\right)
[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{-10.}& \textcolor[rgb]{0,0,1}{16.}& \textcolor[rgb]{0,0,1}{92.}& \textcolor[rgb]{0,0,1}{70.}\end{array}]
a≔\mathrm{Array}\left([1,2,3],'\mathrm{datatype}'='\mathrm{float}'[8]\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{3.}\end{array}]
b≔\mathrm{Array}\left([1,-1,1,-1],'\mathrm{datatype}'='\mathrm{float}'[8]\right)
\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{-1.}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{-1.}\end{array}]
\mathrm{Convolution}\left(a,b,'\mathrm{algorithm}'='\mathrm{auto}'\right)
[\begin{array}{cccccc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{-2.}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{-3.}\end{array}]
c≔\mathrm{Array}\left(1..\mathrm{numelems}\left(a\right)+\mathrm{numelems}\left(b\right)-1,'\mathrm{datatype}'='\mathrm{float}'[8]\right):
\mathrm{Convolution}\left(a,b,'\mathrm{container}'=c,'\mathrm{algorithm}'='\mathrm{direct}'\right)
[\begin{array}{cccccc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{-2.}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{-3.}\end{array}]
c
[\begin{array}{cccccc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{2.}& \textcolor[rgb]{0,0,1}{-2.}& \textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{-3.}\end{array}]
A≔\mathrm{Vector}[\mathrm{row}]\left([2-I,0,5+3I,0,4I]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}\end{array}]
B≔\mathrm{Vector}[\mathrm{row}]\left([-7,3+10I,9-2I,1]\right)
\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{-7}& \textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{1}\end{array}]
\mathrm{C1}≔\mathrm{Convolution}\left(A,B,'\mathrm{algorithm}'='\mathrm{fft}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{C1}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccccccc}\textcolor[rgb]{0,0,1}{-14.}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{7.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{16.0000000000000}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{17.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{-19.}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{34.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{-13.}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{58.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{51.}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{11.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{-35.}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{15.0000000000000}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{8.}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{36.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4.00000000000000}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}\end{array}]
\mathrm{C2}≔\mathrm{`~`}[\mathrm{round}]\left(\mathrm{C1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{C2}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccccccc}\textcolor[rgb]{0,0,1}{-14.}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{7.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{16.}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{17.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{-19.}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{34.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{-13.}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{58.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{51.}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{11.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{-35.}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{15.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{8.}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{36.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}& \textcolor[rgb]{0,0,1}{0.}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4.}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}\end{array}]
The SignalProcessing[Convolution] command was introduced in Maple 17.
The SignalProcessing[Convolution] command was updated in Maple 2020.
The A, B parameter was updated in Maple 2020.
The algorithm option was introduced in Maple 2020.
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Line or vector perpendicular to a curve or a surface
{\displaystyle P}
{\displaystyle P}
{\displaystyle P.}
Normal to surfaces in 3D space[edit]
Calculating a surface normal[edit]
{\displaystyle ax+by+cz+d=0,}
{\displaystyle \mathbf {n} =(a,b,c)}
{\displaystyle \mathbf {r} (s,t)=\mathbf {r} _{0}+s\mathbf {p} +t\mathbf {q} ,}
{\displaystyle \mathbf {r} _{0}}
{\displaystyle \mathbf {p} ,\mathbf {q} }
{\displaystyle \mathbf {p} }
{\displaystyle \mathbf {q} ,}
{\displaystyle \mathbf {n} =\mathbf {p} \times \mathbf {q} .}
{\displaystyle S}
{\displaystyle \mathbb {R} ^{3}}
{\displaystyle \mathbf {r} (s,t)=(x(s,t),y(s,t),z(s,t)),}
{\displaystyle s}
{\displaystyle t}
{\displaystyle \mathbf {n} ={\frac {\partial \mathbf {r} }{\partial s}}\times {\frac {\partial \mathbf {r} }{\partial t}}.}
{\displaystyle S}
{\displaystyle (x,y,z)}
{\displaystyle F(x,y,z)=0,}
{\displaystyle (x,y,z)}
{\displaystyle \mathbf {n} =\nabla F(x,y,z).}
{\displaystyle S.}
{\displaystyle S}
{\displaystyle \mathbb {R} ^{3}}
{\displaystyle z=f(x,y),}
{\displaystyle \mathbf {r} (x,y)=(x,y,f(x,y)),}
{\displaystyle \mathbf {n} ={\frac {\partial \mathbf {r} }{\partial x}}\times {\frac {\partial \mathbf {r} }{\partial y}}=\left(1,0,{\tfrac {\partial f}{\partial x}}\right)\times \left(0,1,{\tfrac {\partial f}{\partial y}}\right)=\left(-{\tfrac {\partial f}{\partial x}},-{\tfrac {\partial f}{\partial y}},1\right);}
{\displaystyle F(x,y,z)=z-f(x,y)=0,}
{\displaystyle \mathbf {n} =\nabla F(x,y,z)=\left(-{\tfrac {\partial f}{\partial x}},-{\tfrac {\partial f}{\partial y}},1\right).}
Choice of normal[edit]
Transforming normals[edit]
{\displaystyle 3\times 3}
{\displaystyle \mathbf {M} ,}
{\displaystyle \mathbf {W} }
{\displaystyle \mathbf {n} }
{\displaystyle \mathbf {t} }
{\displaystyle \mathbf {n} ^{\prime }}
{\displaystyle \mathbf {Mt} ,}
{\displaystyle \mathbf {Wn} .}
{\displaystyle \mathbf {W} .}
{\displaystyle {\begin{alignedat}{5}W\mathbb {n} {\text{ is perpendicular to }}M\mathbb {t} \quad \,&{\text{ if and only if }}\quad 0=(W\mathbb {n} )\cdot (M\mathbb {t} )\\&{\text{ if and only if }}\quad 0=(W\mathbb {n} )^{\mathrm {T} }(M\mathbb {t} )\\&{\text{ if and only if }}\quad 0=\left(\mathbb {n} ^{\mathrm {T} }W^{\mathrm {T} }\right)(M\mathbb {t} )\\&{\text{ if and only if }}\quad 0=\mathbb {n} ^{\mathrm {T} }\left(W^{\mathrm {T} }M\right)\mathbb {t} \\\end{alignedat}}}
{\displaystyle \mathbf {W} }
{\displaystyle W^{\mathrm {T} }M=I,}
{\displaystyle W=(M^{-1})^{\mathrm {T} },}
{\displaystyle W\mathbb {n} }
{\displaystyle M\mathbb {t} ,}
{\displaystyle \mathbf {n} ^{\prime }}
{\displaystyle \mathbf {t} ^{\prime },}
Hypersurfaces in n-dimensional space[edit]
{\displaystyle (n-1)}
-dimensional hyperplane i{\displaystyle n}
{\displaystyle \mathbb {R} ^{n}}
{\displaystyle \mathbf {r} \left(t_{1},\ldots ,t_{n-1}\right)=\mathbf {p} _{0}+t_{1}\mathbf {p} _{1}+\cdots +t_{n-1}\mathbf {p} _{n-1},}
{\displaystyle \mathbf {p} _{0}}
{\displaystyle \mathbf {p} _{i}}
{\displaystyle i=1,\ldots ,n-1}
{\displaystyle \mathbf {n} }
{\displaystyle P={\begin{bmatrix}\mathbf {p} _{1}&\cdots &\mathbf {p} _{n-1}\end{bmatrix}},}
{\displaystyle P\mathbf {n} =\mathbf {0} .}
{\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=c,}
{\displaystyle \mathbb {n} =\left(a_{1},\ldots ,a_{n}\right)}
{\displaystyle (n-1)}
{\displaystyle \mathbb {R} ^{n}.}
{\displaystyle (x_{1},x_{2},\ldots ,x_{n})}
{\displaystyle F(x_{1},x_{2},\ldots ,x_{n})=0,}
{\displaystyle F}
{\displaystyle F}
{\displaystyle \mathbb {n} =\nabla F\left(x_{1},x_{2},\ldots ,x_{n}\right)=\left({\tfrac {\partial F}{\partial x_{1}}},{\tfrac {\partial F}{\partial x_{2}}},\ldots ,{\tfrac {\partial F}{\partial x_{n}}}\right)\,.}
{\displaystyle \{\mathbf {n} \}.}
Varieties defined by implicit equations in n-dimensional space[edit]
{\displaystyle n}
{\displaystyle \mathbb {R} ^{n}}
is the set of the common zeros of a finite set of differentiable functions i{\displaystyle n}
{\displaystyle f_{1}\left(x_{1},\ldots ,x_{n}\right),\ldots ,f_{k}\left(x_{1},\ldots ,x_{n}\right).}
{\displaystyle k\times n}
{\displaystyle i}
{\displaystyle f_{i}.}
{\displaystyle k.}
{\displaystyle P,}
{\displaystyle P}
{\displaystyle f_{i}.}
{\displaystyle k}
{\displaystyle P}
{\displaystyle P}
{\displaystyle P.}
{\displaystyle x\,y=0,\quad z=0.}
{\displaystyle x}
{\displaystyle y}
{\displaystyle (a,0,0),}
{\displaystyle a\neq 0,}
{\displaystyle (0,0,1)}
{\displaystyle (0,a,0).}
{\displaystyle x=a.}
{\displaystyle b\neq 0,}
{\displaystyle (0,b,0)}
{\displaystyle y=b.}
{\displaystyle (0,0,0)}
{\displaystyle (0,0,1)}
{\displaystyle (0,0,0).}
{\displaystyle z}
Normal in geometric optics[edit]
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Parabola - New World Encyclopedia
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In mathematics, the parabola (from the Greek word παραβολή) is a conic section generated by the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. A parabola can also be defined as locus of points in a plane which are equidistant from a given point (the focus) and a given line (the directrix).
A particular case arises when the plane is tangent to the conical surface. In this case, the intersection is a degenerate parabola consisting of a straight line.
1 Analytic geometry equations
6 What happens to a parabola when "b" varies?
7 Parabolas in the physical world
The parabola is an important concept in abstract mathematics, but it is also seen with considerable frequency in the physical world, and there are many practical applications for the construct in engineering, physics, and other domains.
A parabola (shaded green) is a conic section.
A graph showing the reflective property, the directrix (green), and the lines connecting the focus and directrix to the parabola (blue).
Analytic geometry equations
In Cartesian coordinates, a parabola with an axis parallel to the y axis with vertex (h, k), focus (h, k + p), and directrix y = k - p, with p being the distance from the vertex to the focus, has the equation with axis parallel to the y-axis
{\displaystyle (x-h)^{2}=4p(y-k)\,}
or, alternatively with axis parallel to the x-axis
{\displaystyle (y-k)={\frac {1}{4p}}(x-h)^{2}\,}
More generally, a parabola is a curve in the Cartesian plane defined by an irreducible equation of the form
{\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0\,}
{\displaystyle B^{2}=4AC\,}
, where all of the coefficients are real, where
{\displaystyle A\not =0\,}
{\displaystyle C\not =0\,}
, and where more than one solution, defining a pair of points (x, y) on the parabola, exists. That the equation is irreducible means it does not factor as a product of two not necessarily distinct linear equations.
Parabolas are conic sections.
A parabola may also be characterized as a conic section with an eccentricity of 1. As a consequence of this, all parabolas are similar. A parabola can also be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction. In this sense, a parabola may be considered an ellipse that has one focus at infinity. The parabola is an inverse transform of a cardioid.
(with vertex (h, k) and distance p between vertex and focus - note that if the vertex is below the focus, or equivalently above the directrix, p is positive, otherwise p is negative; similarly with horizontal axis of symmetry p is positive if vertex is to the left of the focus, or equivalently to the right of the directrix)
{\displaystyle (x-h)^{2}=4p(y-k)\,}
{\displaystyle y=a(x-h)^{2}+k\,}
{\displaystyle y=ax^{2}+bx+c\,}
{\displaystyle {\mbox{where }}a={\frac {1}{4p}};\ \ b={\frac {-h}{2p}};\ \ c={\frac {h^{2}}{4p}}+k;\ \ }
{\displaystyle h={\frac {-b}{2a}};\ \ k={\frac {4ac-b^{2}}{4a}}}
{\displaystyle x(t)=2pt+h;\ \ y(t)=pt^{2}+k\,}
{\displaystyle (y-k)^{2}=4p(x-h)\,}
{\displaystyle x=a(y-k)^{2}+h\,}
{\displaystyle x=ay^{2}+by+c\,}
{\displaystyle {\mbox{where }}a={\frac {1}{4p}};\ \ b={\frac {-k}{2p}};\ \ c={\frac {k^{2}}{4p}}+h;\ \ }
{\displaystyle h={\frac {4ac-b^{2}}{4a}};\ \ k={\frac {-b}{2a}}}
{\displaystyle x(t)=pt^{2}+h;\ \ y(t)=2pt+k\,}
In polar coordinates, a parabola with the focus at the origin and the directrix on the positive x-axis, is given by the equation
{\displaystyle r(1+\cos \theta )=l\,}
where l is the semilatus rectum: the distance from the focus to the parabola itself, measured along a line perpendicular to the axis. Note that this is twice the distance from the focus to the apex of the parabola or the perpendicular distance from the focus to the latus rectum.
A Gauss-mapped form:
{\displaystyle (\tan ^{2}\phi ,2\tan \phi )}
has normal
{\displaystyle (\cos \phi ,\sin \phi )}
Parabolic curve showing directrix (L) and focus (F). The distance from a given point Pn to the focus is always the same as the distance from Pn to a point Qn directly below, on the directrix.
{\displaystyle y=ax^{2},\qquad \qquad \qquad (1)}
then there is a point (0,f) — the focus — such that any point P on the parabola will be equidistant from both the focus and a line perpendicular to the axis of symmetry of the parabola (the linea directrix), in this case parallel to the x axis. Since the vertex is one of the possible points P, it follows that the linea directrix passes through the point (0,-f). So for any point P=(x,y), it will be equidistant from (0,f) and (x,-f). It is desired to find the value of f which has this property.
{\displaystyle \|FP\|={\sqrt {x^{2}+(y-f)^{2}}},}
{\displaystyle \|QP\|=y+f.}
{\displaystyle \|FP\|=\|QP\|}
{\displaystyle {\sqrt {x^{2}+(ax^{2}-f)^{2}}}=ax^{2}+f\qquad }
{\displaystyle x^{2}+(ax^{2}-f)^{2}=(ax^{2}+f)^{2}\qquad }
{\displaystyle =a^{2}x^{4}+f^{2}+2ax^{2}f\quad }
{\displaystyle x^{2}+a^{2}x^{4}+f^{2}-2ax^{2}f=a^{2}x^{4}+f^{2}+2ax^{2}f\quad }
{\displaystyle x^{2}-2ax^{2}f=2ax^{2}f,\quad }
{\displaystyle x^{2}=4ax^{2}f.\quad }
Cancel out the x² from both sides (x is generally not zero),
{\displaystyle 1=4af\quad }
{\displaystyle f={1 \over 4a}}
{\displaystyle x^{2}=4py\quad }
All this was for a parabola centered at the origin. For any generalized parabola, with its equation given in the standard form
{\displaystyle y=ax^{2}+bx+c}
{\displaystyle \left({\frac {-b}{2a}},{\frac {-b^{2}}{4a}}+c+{\frac {1}{4a}}\right)}
{\displaystyle y={\frac {-b^{2}}{4a}}+c-{\frac {1}{4a}}}
{\displaystyle {dy \over dx}=2ax={2y \over x}}
This line intersects the y-axis at the point (0,-y) = (0, - a x²), and the x-axis at the point (x/2,0). Let this point be called G. Point G is also the midpoint of points F and Q:
{\displaystyle F=(0,f),\quad }
{\displaystyle Q=(x,-f),\quad }
{\displaystyle {F+Q \over 2}={(0,f)+(x,-f) \over 2}={(x,0) \over 2}=({x \over 2},0).}
{\displaystyle \|FG\|\cong \|GQ\|,}
{\displaystyle \|PF\|\cong \|PQ\|,}
{\displaystyle \Delta FGP\cong \Delta QGP}
{\displaystyle \angle FPG\cong \angle GPQ}
Line QP can be extended beyond P to some point T, and line GP can be extended beyond P to some point R. Then
{\displaystyle \angle RPT}
{\displaystyle \angle GPQ}
are vertical, so they are equal (congruent). But
{\displaystyle \angle GPQ}
{\displaystyle \angle FPG}
{\displaystyle \angle RPT}
{\displaystyle \angle FPG}
Let a light beam travel down the vertical line TP and bounce off from P. The beam's angle of inclination from the mirror is
{\displaystyle \angle RPT}
, so when it bounces off, its angle of inclination must be equal to
{\displaystyle \angle RPT}
{\displaystyle \angle FPG}
has been shown to be equal to
{\displaystyle \angle RPT}
. Therefore the beam bounces off along the line FP: directly towards the focus.
What happens to a parabola when "b" varies?
Vertex of a parabola: Finding the y-coordinate
We know the x-coordinate at the vertex is
{\displaystyle x=-{\frac {b}{2a}}}
, so substitute it into the equation
{\displaystyle ax^{2}+bx+c}
{\displaystyle y=a\left(-{\frac {b}{2a}}\right)^{2}+b\left(-{\frac {b}{2a}}\right)+c\qquad {\textrm {Then~simplify\ldots }}}
{\displaystyle ={\frac {ab^{2}}{4a^{2}}}-{\frac {b^{2}}{2a}}+c}
{\displaystyle ={\frac {b^{2}}{4a}}-{\frac {2\cdot b^{2}}{2\cdot 2a}}+c\cdot {\frac {4a}{4a}}}
{\displaystyle ={\frac {-b^{2}+4ac}{4a}}}
{\displaystyle =-{\frac {b^{2}-4ac}{4a}}=-{\frac {D}{4a}}}
{\displaystyle \left(-{\frac {b}{2a}},-{\frac {D}{4a}}\right)}
Parabolas in the physical world
A bouncing ball captured with a stroboscopic flash at 25 images per second. Note that the ball becomes significantly non-spherical after each bounce, especially after the first. That, along with spin and air resistance, causes the curve swept out to deviate slightly from the expected perfect parabola
In nature, approximations of parabolas and paraboloids are found in many diverse situations. The most well-known instance of the parabola in the history of physics is the trajectory of a particle or body in motion under the influence of a uniform gravitational field without air resistance (for instance, a baseball flying through the air, neglecting air friction). The parabolic trajectory of projectiles was discovered experimentally by Galileo in the early seventeenth century, who performed experiments with balls rolling on inclined planes. The parabolic shape for projectiles was later proven mathematically by Isaac Newton. For objects extended in space, such as a diver jumping from a diving board, the object itself follows a complex motion as it rotates, but the center of mass of the object nevertheless forms a parabola. As in all cases in the physical world, the trajectory is always an approximation of a parabola. The presence of air resistance, for example, always distorts the shape, although at low speeds, the shape is a good approximation of a parabola. At higher speeds, such as in ballistics, the shape is highly distorted and does not resemble a parabola.
Parabolic shape formed by the surface of a Newtonian liquid under rotation
Another situation in which parabola may arise in nature is in two-body orbits, for example, of a small planetoid or other object under the influence of the gravitation of the sun. Such parabolic orbits are a special case that are rarely found in nature. Orbits that form a hyperbola or an ellipse are much more common. In fact, the parabolic orbit is the borderline case between those two types of orbit. An object following a parabolic orbit moves at the exact escape velocity of the object it is orbiting, while elliptical orbits are slower and hyperbolic orbits are faster.
Approximations of parabolas are also found in the shape of cables of suspension bridges. Freely hanging cables do not describe parabolas, but rather catenary curves. Under the influence of a uniform load (for example, the deck of bridge), however, the cable is deformed toward a parabola.
Paraboloids arise in several physical situations as well. The most well-known instance is the parabolic reflector, which is a mirror or similar reflective device that concentrates light or other forms of electromagnetic radiation to a common focal point. The principle of the parabolic reflector may have been discovered in the third century B.C.E. by the geometer Archimedes, who, according to a legend of debatable veracity,[1] constructed parabolic mirrors to defend Syracuse against the Roman fleet, by concentrating the sun's rays to set fire to the decks of the Roman ships. The principle was applied to telescopes in the seventeenth century. Today, paraboloid reflectors can be commonly observed throughout much of the world in microwave and satellite dish antennas.
↑ W.E. Knowles Middleton, 1961. Archimedes, Kircher, Buffon, and the Burning-Mirrors. Isis 52(4):533–543.
Eric W. Weisstein. Parabola. MathWorld.
Interactive parabola-drag focus, see axis of symmetry, directrix, standard and vertex forms.
Archimedes Triangle and Squaring of Parabola at cut-the-knot.
Two Tangents to Parabola at cut-the-knot.
Parabola As Envelope of Straight Lines at cut-the-knot.
Parabolic Mirror at cut-the-knot.
Circumcircle of Three Parabola Tangents at cut-the-knot.
Focal Properties of Parabola at cut-the-knot.
Generation of parabola via Apollonius' mesh at cut-the-knot.
Exploring Parabolas (JavaSketchpad).
Parabola history
History of "Parabola"
Retrieved from https://www.newworldencyclopedia.org/p/index.php?title=Parabola&oldid=1017186
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Some fixed point theorems for multifunctions with applications in game theory
Būi Cong Cuōng — 1985
IntroductionThe main result of this paper is concerned with the conditions which guarantee that a multifunction
f:C\to {2}^{X}
defined on an arbitrary subset C of a topological vector space X admits a point x of C such that x∈f(x).First, we give some definitions and propositions which are associated with semicontinuous multifunctions (Part 1).Next, in Part 2, we present a global convergence criterion on variable dimension algorithms for finding an approximate solution of the equation x∈f(x), and then we consider...
1 Cuōng, BC
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Predict state and state estimation error covariance at next time step using extended or unscented Kalman filter, or particle filter - MATLAB predict - MathWorks Nordic
\stackrel{^}{x}\left[k|k\right]
\stackrel{^}{x}\left[k+1|k\right]
\stackrel{^}{x}\left[k+1|k\right]
\stackrel{^}{x}\left[k+1|k\right]
\underset{}{\overset{ˆ}{x}}\left[k|k-1\right]
\underset{}{\overset{ˆ}{x}}\left[k|k-1\right]
\underset{}{\overset{ˆ}{x}}\left[k|k\right]
\underset{}{\overset{ˆ}{x}}\left[k+1|k\right]
\underset{}{\overset{ˆ}{x}}\left[k|k\right]
\underset{}{\overset{ˆ}{x}}\left[k|k-1\right]
\underset{}{\overset{ˆ}{x}}\left[k-1|k-1\right]
x\left[k\right]=\sqrt{x\left[k-1\right]+u\left[k-1\right]}+w\left[k-1\right]
y\left[k\right]=x\left[k\right]+2*u\left[k\right]+v\left[k{\right]}^{2}
\stackrel{^}{x}\left[k|k-1\right]
\stackrel{^}{x}\left[k|k\right]
\stackrel{^}{x}\left[k+1|k\right]
\stackrel{^}{x}\left[k-1|k-1\right]
\stackrel{^}{x}\left[k|k-1\right]
\stackrel{^}{x}\left[k|k\right]
\stackrel{^}{x}\left[k|k\right]
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From Jupyter Notebook to Scientific Paper - Curvenote
Jupyter Notebooks are a perfect tool for exploratory data analysis, data cleanup, and visualization. Jupyter is fantastic in creating rich, interactive visualizations, which are very different outputs than how most of science is communicated today — through PDFs, PPT Slides and WordDocs. This disconnect of how modern science is completed, using data analysis tools and high performance computing, and the way we communicate that work means it can be very difficult to share and to collaborate with stakeholders, especially if those stakeholders aren’t familiar with Jupyter!
At Curvenote, our mission is to help transition science out of PDFs — bringing science communication into the web: where data can be dynamic, linked to computation, and kept up to date. That vision is still a ways off, with most of science being communicated in PDFs (and still even behind paywalls!), and this disconnect is what we will address in this article: How to create a scientific paper using Jupyter Notebooks?
#Copy-and-Paste stops Reproducibility
Today, most scientific manuscripts are written in Word or LaTeX, with results and figures included via copy-and-paste screenshots into the documents. This copy-and-paste style writing means all interactivity and reproducibility that Jupyter Notebooks provide is lost, and the disconnect between computation, results, and communication ends up costing researchers valuable time. Even for the most organized researcher, this disconnect can cause issues knowing which images are the correct version or to which data and parameters they correspond.
Figure 1:Today communicating research through static screenshots of figures from Jupyter to papers and presentations. This copy-and-paste style writing creating a disconnect between communication and research.
#Linking your Notebooks through Curvenote
Curvenote is a scientific writing tool built to maintain an active connection between data, computation, visualization, and communication. Curvenote provides the version control and persistent links/ids at the level of a single Jupyter cell, enabling key data and visuals to be inserted directly into documents such as papers, reports, and presentations. The Curvenote editor also serves as a writing platform with collaborative features such as simultaneous editing, rich-text comments, notifications, and version control.
Figure 2:Curvenote connects research, writing, collaboration, and communication across a data science workflow by maintaining active links between Jupyter notebooks and Curvenote writing projects.
Curvenote presents a new streamlined approach to transition your Jupyter notebook into a paper, while maintaining the interactivity and direct connection between research and communication. Within this blog we will step through the creation of Steve Purves’ paper, La Palma Seismicity 2021 and explore the various technical writing and collaboration tools provided. The paper is based on recent analysis of openly available earthquake data (provided by Instituto Geográphico Nacional) from the island of La Palma in the Canary Islands, Spain, which is currently experiencing an ongoing volcanic eruption.
Curvenote is based around projects. Each project contains articles, notebooks, blocks, references, that can be organized, shared, referenced, and published. From your personal or team profile, you can create a new project based on a starter template - such as the Paper template which provides some example content to guide you.
You can invite your collaborators on your project, via their existing Curvenote username or an invitation email. Your collaborators can view, create, edit, comment, and save new versions of articles and notebooks. As the project owner you retain access to project administration.
#Jupyter Notebooks & Curvenote
Curvenote provides collaboration and version control features directly within the Jupyter environment via the Curvenote Jupyter browser extension, which is available for Chrome, Microsoft Edge, and Brave.
The Curvenote extension will recognize and connect to Jupyter, either your local machine or Jupyter Hubs, Amazon Sagemaker, Saturn Notebooks and any other environment that serves the Jupyter Notebook interface.
Once installed and logged in, the extension adds a toolbar and a new control panel to Jupyter, allowing you to save notebooks to the project you just created. Both your notebook and each individual cell is saved and given a persistent Curvenote identifier, called an OXA Link, along with a version number. OXA Links allow changes and new versions of a cell or notebook to be tracked and linked into other documents.
Figure 3:Save, version, collaborate, and comment on your Jupyter notebooks with the Curvenote Jupyter extension. Learn more about Curvenote for Jupyter.
The Jupyter extension also embeds this persistent identifier into the metadata fields of the .ipynb file, meaning that version information travels with the file should it be shared independently, or committed to a git repository. Multiple collaborators can each have their own local copies of a notebook, while making and receiving changes and comments that flow through Curvenote’s API in real time.
When any collaborator saves a new version of their notebook using Curvenote, all other collaborators immediately have access to those changes. The extension also adds features to Jupyter that anyone can take advantage of, even when working alone, such as inline diffs for cells and sharing notebooks via Curvenote profiles.
#Results and Figures from Jupyter
With your Jupyter notebooks saved, versioned, and linked through Curvenote, you can now easily include your figures from Jupyter within your manuscript. Rather than a typical copy-paste of a figure, you will copy the OXA Link for that figure in Jupyter then paste it into the necessary location within your paper.
Figure 4:Import Jupyter outputs into your Curvenote paper, any comments and changes are shown in both instances of the cell.
The OXA Link is active and any comments made to the figure on either Curvenote or Jupyter are visible and available in all locations, letting you have conversations with your collaborators, suggest changes, and explain updates. When you make updates to your figure in Jupyter and a new version of your notebooks is saved, you can automatically preview changes and update the figure within your manuscript. At any time you can also view and return to previous versions of the cell throughout the many drafts, reviews, and revisions your paper will go through.
#Writing Your Paper
When you are ready to start writing, the Paper template provides an outline of a standard research paper that you can rename, replace, and fill in with the necessary components for your project. The template also includes examples of the different writing features available within Curvenote and is a great way to get started.
Articles on Curvenote are made up of blocks - distinct sections of content which are versioned in the same way as Jupyter cells. They can also be individually saved, versioned and reused throughout your Curvenote project.
#Writing Features
Curvenote’s writing environment provides all of the features you would expect from a scientific text editor.
Most of these features can be accessed using an inline command menu (type / anywhere), then filtering through the drop down menu for the feature or component you want to add or reference. Some formatting like section headings can also be applied using Markdown accelerators, and selecting the subsequent formatted text provides a menu to edit the the heading size and numbering. Sections can be numbered and referenced throughout the text.
Figure 5:Add numbered section headings to your Curvenote paper and reference them throughout your text.
In addition to outputs from Jupyter, static images, videos, and GIFs can be uploaded to your paper. You can toggle on the figure numbering and add a caption. Figures can be referenced throughout the text and a preview will appear when you hover over the reference.
Figure 6:Upload, number, caption, and reference figures in your Curvenote paper.
Similarly you can create and format tables directly within Curvenote or copy in tables of data from other sources. You can toggle on the table numbering and add a caption. Use inline commands to add references to tables within you text.
Figure 7:Include tables with numbering and captions within your Curvenote paper.
Curvenote supports
\LaTeX
math, both inline and displayed as equation blocks. Equations can be numbered and referenced throughout the text. When you hover over an equation reference, the equation will appear.
Figure 8:Include
\LaTeX
math both inline and as equations that can be numbered and referenced throughout the text.
Citations are added to a Curvenote project via either a DOI or uploading a .bib file. After your citations have been uploaded, you can use the inline citation command to search through the available references. Citations can be formatted, combined, or separated by simply dragging the text.
Figure 9:Upload references, then use inline commands to search and add and format in-text citations.
#Communication and Publication
After several iterations of data analysis, drafting, and editing with your collaborators, it is time to share your work with the wider community. If you intend to publish online, you can share your paper and Jupyter notebooks publicly on Curvenote by adjusting the visibility settings of your project.
If you are publishing in a preprint service or traditional journal you can use Curvenote’s export feature to can export your paper in as a docx file for Microsoft Word, a collection of files for
\LaTeX
compilation or as a submission ready PDF file. Curvenote has a large collection of journal and preprint templates already available for services such as arXiv and journals, and are continually adding more. If you don’t see the template you need, request it via our open source template repository.
Throughout the export process, Curvenote will both suggest and check against the specific requirements of the template you are using to help you get to a submission ready document.
Figure 10:Export your paper using a variety of format and professional template options for submission to a preprint or academic journal. Learn more about export to these formats: Word, PDF, LaTeX.
Curvenote is designed to improve scientific communication, collaboration and reproducibility, and ultimately make it easier to publish interactive, reproducible scientific articles on the web.
Curvenote maintains an active link between your Jupyter Notebook outputs and written articles that can be worked on with stakeholders who are not familiar with Jupyter. These bi-directional links decrease the time and disconnect between researching and communicating your work.
Sign up for Curvenote for free and start versioning, collaborating, and writing in your project! We’re keen to hear your feedback, suggestions, and help you get the most out of Curvenote.
To learn more about writing a paper based on research from a Jupyter notebook, you can watch a webinar that Curvenote recently hosted. Steve Purves steps through the creation of his La Palma Seismicity 2021 paper, starting with earthquake data collected from the volcanic eruptions, creates visualizations in Jupyter, then demonstrates a how a paper can be brought together and exported to arXiv ready PDF using Curvenote.
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Similar Polygons: Level 3 Challenges Practice Problems Online | Brilliant
A square is inscribed in a circle and a circle is inscribed in that square. What will be the ratio of the area of the bigger circle to that of the smaller circle?
by Naman Sehgal
The sum of the areas of two similar polygons is 65 square units. If their perimeters are 12 units and 18 units, respectively, what is the area of the larger polygon?
by Ilham Saiful Fauzi
Two triangles have integral side lengths, with all sides being less than 50. They are similar but not congruent, and the smaller triangle has two side lengths identical with the larger triangle.
What is the sum of the side lengths of the larger triangle?
If a tan snowflake has area
T
, and a dark green snowflake has area
G
\frac{T}{G}
by Shabarish Ch
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GMS:UGrid Linear Interpolation - XMS Wiki
GMS:UGrid Linear Interpolation
The Interpolate – Linear dialog
The Linear interpolation scheme uses data points that are first triangulated to form a network of triangles. The network of triangles only covers the convex hull of the point data, making extrapolation beyond the convex hull not possible.
If the linear interpolation scheme is selected, the data points are first triangulated to form a network of triangles. The equation of the plane defined by the three vertices of a triangle is as follows:
where A, B, C, and D are computed from the coordinates of the three vertices (x1,y1,z1), (x2,y2,z2), and (x3,y3,z3):
{\displaystyle \ A=y_{1}(z_{2}-z_{3})+y_{2}(z_{3}-z_{1})+y_{3}(z_{1}-z_{2})}
{\displaystyle \ B=z_{1}(x_{2}-x_{3})+z_{2}(x_{3}-x_{1})+z_{3}(x_{1}-x_{2})}
{\displaystyle \ C=x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_{2})}
{\displaystyle \ D=-Ax_{1}-By_{1}-Dz_{1}}
The plane equation can also be written as:
{\displaystyle z=f(x,y)=-{\frac {A}{C}}x-{\frac {B}{C}}y-{\frac {D}{C}}}
which is the form of the plane equation used to compute the elevation at any point on the triangle.
Since the network of triangles only covers the convex hull of the point data, extrapolation beyond the convex hull is not possible with the linear interpolation scheme. Any points outside the convex hull of the point data are assigned the default extrapolation value entered at the bottom of the Interpolation Options dialog. The figure below shows a network of triangles created from point data.
Network of triangles
If the Linear interpolation method is selected in the Interpolate UGrid to UGrid dialog, options can be set in the Interpolate – Linear dialog. This dialog has the following options:
Truncate values – This section allows for limiting the interpolated values to lie between the minimum and maximum value.
Truncate to min/max of dataset – Limits the interpolated values to the minimum and maximum values in the original dataset.
Truncate to specified range – Allows setting a user specified minimum and maximum value range.
Min – Manually sets a minimum value.
Max – Manually sets a maximum value.
Clough-Tocher – When on, the Clough-Tocher interpolation technique will be used.
Advanced – This button will open the Interpolate – Advanced dialog where options for anisotropy and extrapolation can be adjusted.
Retrieved from "https://www.xmswiki.com/index.php?title=GMS:UGrid_Linear_Interpolation&oldid=137639"
UGrid Dialogs
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IMU simulation model - Simulink - MathWorks í•œêµ
Velocity random walk (m/s2/√Hz)
Acceleration random walk ((m/s2)(√Hz))
Bias from temperature ((m/s2)/℃)
Temperature scale factor (%/℃)
Angle random walk ((rad/s)/(√Hz))
Rate random walk ((rad/s)(√Hz))
Bias from temperature ((rad/s)/℃)
Maximum readings (μT)
Resolution ((μT)/LSB)
Constant offset bias (μT)
White noise PSD (μT/√Hz)
Bias Instability (μT)
Random walk ((μT)*√Hz)
Bias from temperature (μT/℃)
Orientation of the IMU sensor body frame with respect to the local navigation coordinate system, specified as an N-by-4 array of real scalars or a 3-by-3-by-N rotation matrix. Each row the of the N-by-4 array is assumed to be the four elements of a quaternion (Sensor Fusion and Tracking Toolbox). N is the number of samples in the current frame.
Mag — Magnetometer measurement of IMU in sensor body coordinate system (μT)
Magnetic field (NED) — Magnetic field vector expressed in NED navigation frame (μT)
MagneticField (ENU) — Magnetic field vector expressed in ENU navigation frame (μT)
Velocity random walk (m/s2/√Hz) — Velocity random walk (m/s2/√Hz)
Velocity random walk in (m/s2/√Hz), specified as a real scalar or 3-element row vector. This property corresponds to the power spectral density of sensor noise. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.
Acceleration random walk ((m/s2)(√Hz)) — Acceleration random walk ((m/s2)(√Hz))
Acceleration random walk of sensor in (m/s2)(√Hz), specified as a real scalar or 3-element row vector. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.
Bias from temperature ((m/s2)/℃) — Sensor bias from temperature ((m/s2)/℃)
Temperature scale factor (%/℃) — Scale factor error from temperature (%/℃)
Angle random walk ((rad/s)/(√Hz)) — Acceleration random walk ((rad/s)/(√Hz))
Acceleration random walk of sensor in (rad/s)/(√Hz), specified as a real scalar or 3-element row vector. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.
Rate random walk ((rad/s)(√Hz)) — Integrated white noise of sensor ((rad/s)(√Hz))
Bias from temperature ((rad/s)/℃) — Sensor bias from temperature ((rad/s)/℃)
Sensor bias from temperature in (rad/s)/℃, specified as a real scalar or 3-element row vector. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.
Maximum readings (μT) — Maximum sensor reading (μT)
Resolution ((μT)/LSB) — Resolution of sensor measurements ((μT)/LSB)
Resolution of sensor measurements in (μT)/LSB, specified as a real nonnegative scalar.
Constant offset bias (μT) — Constant sensor offset bias (μT)
White noise PSD (μT/√Hz) — Power spectral density of sensor noise (μT/√Hz)
Bias Instability (μT) — Instability of the bias offset (μT)
Random walk ((μT)*√Hz) — Integrated white noise of sensor ((μT)*√Hz)
Integrated white noise of sensor in (μT)*√Hz, specified as a real scalar or 3-element row vector. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.
Bias from temperature (μT/℃) — Sensor bias from temperature (μT/℃)
Sensor bias from temperature in μT/℃, specified as a real scalar or 3-element row vector. Any scalar input is converted into a real 3-element row vector where each element has the input scalar value.
totalAcc=âacceleration+g
a=\left(orientation\right){\left(totalAcc\right)}^{T}
b={\left(\left[\begin{array}{ccc}1& \frac{{\mathrm{α}}_{2}}{100}& \frac{{\mathrm{α}}_{3}}{100}\\ \frac{{\mathrm{α}}_{1}}{100}& 1& \frac{{\mathrm{α}}_{3}}{100}\\ \frac{{\mathrm{α}}_{1}}{100}& \frac{{\mathrm{α}}_{2}}{100}& 1\end{array}\right]\left({a}^{T}\right)\right)}^{T}+\text{ConstantBias}
where ConstantBias is a property of accelparams, and α1, α2, and α3 are given by the first, second, and third elements of the AxesMisalignment property of accelparams.
{\mathrm{β}}_{1}={h}_{1}*\left(w\right)\left(\text{BiasInstability}\right)
{H}_{1}\left(z\right)=\frac{1}{1â\frac{1}{2}{z}^{â1}}
{\mathrm{β}}_{2}=\left(w\right)\left(\sqrt{\frac{\text{SampleRate}}{2}}\right)\left(\text{NoiseDensity}\right)
{\mathrm{β}}_{3}={h}_{2}*\left(w\right)\left(\frac{\text{RandomWalk}}{\sqrt{\frac{\text{SampleRate}}{2}}}\right)
{H}_{2}\left(z\right)=\frac{1}{1â{z}^{â1}}
{\mathrm{Î}}_{e}=\left(\text{Temperature}â25\right)\left(\text{TemperatureBias}\right)
scaleFactorError=1+\left(\frac{\text{Temperature}â25}{100}\right)\left(\text{TemperatureScaleFactor}\right)
e=\left\{\begin{array}{c}\begin{array}{c}\text{MeasurementRange}\\ â\text{MeasurementRange}\end{array}\\ d\end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}\text{if}\\ \text{if}\\ \text{else}\end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}d>\text{MeasurementRange}\\ âd>\text{MeasurementRange}\\ \end{array}
accelReadings=\left(\text{Resolution}\right)\left(\mathrm{round}\left(\frac{e}{\text{Resolution}}\right)\right)
a=\left(orientation\right){\left(angularVelocity\right)}^{T}
b={\left(\left[\begin{array}{ccc}1& \frac{{\mathrm{α}}_{2}}{100}& \frac{{\mathrm{α}}_{3}}{100}\\ \frac{{\mathrm{α}}_{1}}{100}& 1& \frac{{\mathrm{α}}_{3}}{100}\\ \frac{{\mathrm{α}}_{1}}{100}& \frac{{\mathrm{α}}_{2}}{100}& 1\end{array}\right]\left({a}^{T}\right)\right)}^{T}+\text{ConstantBias}
where ConstantBias is a property of gyroparams, and α1, α2, and α3 are given by the first, second, and third elements of the AxesMisalignment property of gyroparams.
{\mathrm{β}}_{1}={h}_{1}*\left(w\right)\left(\text{BiasInstability}\right)
{H}_{1}\left(z\right)=\frac{1}{1â\frac{1}{2}{z}^{â1}}
{\mathrm{β}}_{2}=\left(w\right)\left(\sqrt{\frac{\text{SampleRate}}{2}}\right)\left(\text{NoiseDensity}\right)
{\mathrm{β}}_{3}={h}_{2}*\left(w\right)\left(\frac{\text{RandomWalk}}{\sqrt{\frac{\text{SampleRate}}{2}}}\right)
{H}_{2}\left(z\right)=\frac{1}{1â{z}^{â1}}
{\mathrm{Î}}_{e}=\left(\text{Temperature}â25\right)\left(\text{TemperatureBias}\right)
scaleFactorError=1+\left(\frac{\text{Temperature}â25}{100}\right)\left(\text{TemperatureScaleFactor}\right)
e=\left\{\begin{array}{c}\begin{array}{c}\text{MeasurementRange}\\ â\text{MeasurementRange}\end{array}\\ d\end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}\text{if}\\ \text{if}\\ \text{else}\end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}d>\text{MeasurementRange}\\ âd>\text{MeasurementRange}\\ \end{array}
gyroReadings=\left(\text{Resolution}\right)\left(\mathrm{round}\left(\frac{e}{\text{Resolution}}\right)\right)
a=\left(orientation\right){\left(totalAcc\right)}^{T}
b={\left(\left[\begin{array}{ccc}1& \frac{{\mathrm{α}}_{2}}{100}& \frac{{\mathrm{α}}_{3}}{100}\\ \frac{{\mathrm{α}}_{1}}{100}& 1& \frac{{\mathrm{α}}_{3}}{100}\\ \frac{{\mathrm{α}}_{1}}{100}& \frac{{\mathrm{α}}_{2}}{100}& 1\end{array}\right]\left({a}^{T}\right)\right)}^{T}+\text{ConstantBias}
where ConstantBias is a property of magparams, and α1, α2, and α3 are given by the first, second, and third elements of the AxesMisalignment property of magparams.
{\mathrm{β}}_{1}={h}_{1}*\left(w\right)\left(\text{BiasInstability}\right)
{H}_{1}\left(z\right)=\frac{1}{1â\frac{1}{2}{z}^{â1}}
{\mathrm{β}}_{2}=\left(w\right)\left(\sqrt{\frac{\text{SampleRate}}{2}}\right)\left(\text{NoiseDensity}\right)
{\mathrm{β}}_{3}={h}_{2}*\left(w\right)\left(\frac{\text{RandomWalk}}{\sqrt{\frac{\text{SampleRate}}{2}}}\right)
{H}_{2}\left(z\right)=\frac{1}{1â{z}^{â1}}
{\mathrm{Î}}_{e}=\left(\text{Temperature}â25\right)\left(\text{TemperatureBias}\right)
scaleFactorError=1+\left(\frac{\text{Temperature}â25}{100}\right)\left(\text{TemperatureScaleFactor}\right)
e=\left\{\begin{array}{c}\begin{array}{c}\text{MeasurementRange}\\ â\text{MeasurementRange}\end{array}\\ d\end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}\text{if}\\ \text{if}\\ \text{else}\end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}d>\text{MeasurementRange}\\ âd>\text{MeasurementRange}\\ \end{array}
magReadings=\left(\text{Resolution}\right)\left(\mathrm{round}\left(\frac{e}{\text{Resolution}}\right)\right)
accelparams (Sensor Fusion and Tracking Toolbox) | gyroparams | magparams
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Some extensions of the Marcinkiewicz interpolation theorem in terms of modular inequalities
April, 2003 Some extensions of the Marcinkiewicz interpolation theorem in terms of modular inequalities
María J. CARRO, Ludmila NIKOLOVA
Given a quasi-subaditive operator
T:{L}_{0}\left(\mu \right)
\to {L}_{0}\left(v\right)
, we want to study mapping properties of interpolation type for which the following modular inequality holds
{\int }_{N}P\left(|Tf\left(x\right)|\right)dv\left(x\right)\le {\int }_{M}Q\left(|f\left(x\right)|\right)d\mu \left(x\right)
P
Q
are modular functions. These results generalize the Marcinkiewicz interpolation theorem.
María J. CARRO. Ludmila NIKOLOVA. "Some extensions of the Marcinkiewicz interpolation theorem in terms of modular inequalities." J. Math. Soc. Japan 55 (2) 385 - 394, April, 2003. https://doi.org/10.2969/jmsj/1191419122
Keywords: Boundedness of operators , interpolation , Modular Inequalities
María J. CARRO, Ludmila NIKOLOVA "Some extensions of the Marcinkiewicz interpolation theorem in terms of modular inequalities," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 55(2), 385-394, (April, 2003)
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Units/Astronomy - Wikiversity
Units/Astronomy
A unit, especially in radiation astronomy, is often a quantity chosen as a standard in terms of which quantities may be expressed.
8 Theoretical astronomical units
Each unit so chosen is likely to have a sign, symbol, or notation to represent the quantity.
Notation: let m represent a metre.
Notation: let km represent a kilometre.
{\displaystyle R_{\oplus }}
{\displaystyle R_{J}}
{\displaystyle R_{\odot }}
Notation: an astronomical unit is usually represented by au, or AU.
"The International Astronomical Union (IAU) is responsible for maintaining and approving a special set of units in astronomy, formally defined in 1976."[1]
"One of the most important of these is the astronomical unit. It is a unit of length approximating the Sun-Earth distance (of about 150 million kilometres) which is of convenient use in astronomy. According to its definition adopted by the XXVIIIth General Asssembly of the IAU (IAU 2012 Resolution B2), the astronomical unit is a conventional unit of length equal to 149 597 870 700 m exactly. This definition is valid irrespective of the used time scale. The unique symbol for the astronomical unit is au."[1]
Def. "a conventional unit of length equal to 149 597 870 700 m exactly"[1] is called an astronomical unit (au).
"Beyond the Solar System the distances in astronomy are so great that using the au becomes too cumbersome."[1]
"The IAU also defines other astronomical units: the astronomical unit of time is 1 day (d) of 86,400 SI seconds (s) (SI is the International System of Units)."[1]
Masses[edit | edit source]
"The astronomical unit of mass is equal to the mass of the Sun, [
{\displaystyle M_{\odot }}
], 1.9891 × 1030 kg."[1]
{\displaystyle M_{J}=1.8986\times 10^{27}kg}
{\displaystyle M_{S}=5.6846\times 10^{24}kg}
{\displaystyle M_{U}=5.6846\times 10^{24}kg}
{\displaystyle M_{N}=1.0243\times 10^{26}kg}
Parsecs[edit | edit source]
This sketch illustrates the definition of the parsec unit. Credit: jd.
The IAU recognises several other distance units to be used on different scales. For studies of the structure of the Milky Way, our local galaxy, the parsec (pc) is the usual choice. This is equivalent to about 30.857 × 1012 km, or about 206,000 aus, and is itself defined in terms of the au – as the distance at which one Astronomical Unit subtends an angle of one arcsecond.
Def. a unit of length, used in astronomy, defined as the distance from the Earth [T in the second image down on the right] of an object that exhibits a parallax of 1 arcsecond is called a parsec, or a parallax second.
Light-years[edit | edit source]
"Alternatively the light-year (ly) is sometimes used in scientific papers as a distance unit, although its use is mostly confined to popular publications and similar media. The light-year is roughly equivalent to 0.3 parsecs, and is equal to the distance traveled by light in one Julian year in a vacuum, according to the IAU. To think of it in easily accessible terms, the light-year is 9,460,730,472,580.8 km or 63,241 au. While smaller than the parsec, it is still an incredibly large distance."[1]
Def. "9,460,730,472,580.8 km" is called the light-year (ly).[1]
1 Parsec = 3.08568025 × 1016 meters = 3.2616 light-years.
"Defining a unit is often more complex than first appears. For instance, to define a light-year it is necessary to understand exactly what a year is. When referring to a year in the precisely defined astronomical sense, it should be written with the indefinite article “a” as “a year”. Although there are several different kinds of year, the IAU regards a year as a Julian year of 365.25 days (31.5576 million seconds) unless otherwise specified. The IAU also recognises a Julian century of 36,525 days in the fundamental formulas for precession (more info). Other measurements of time such as sidereal, solar and universal time are not suitable for measuring precise intervals of time, since the rate of rotation of Earth, on which they ultimately depend, is variable with respect to the second."[1]
Theoretical astronomical units[edit | edit source]
Def. a standard measure of a quantity is called a unit.
Lengths[edit | edit source]
Main articles: Distances/Lengths and Lengths
This is a foldable, or pliant, wood, double meter, or double-meter stick. Credit: Isabelle Grosjean ZA.
This is a scanning electron micrograph of a gold nanowire 30 nm x 21.57 µm. Credit: Goldnanoparticles.
This is an integrated circuit die image. Credit: ZeptoBars.
HRTEM lattice images and electron diffraction patterns (top-right insets) are taken along the [0001] direction. Credit: Materialscientist.
Image size is 1,750 x 810 pm. Credit: Boris J. Albers, Todd C. Schwendemann, Mehmet Z. Baykara, Nicolas Pilet, Marcus Liebmann, Eric I. Altman and Udo D. Schwarz.
The symmetry of the protrusions in this image can only be explained by identifying them with the locations of the atoms in the lattice. Credit: Boris J. Albers, Todd C. Schwendemann, Mehmet Z. Baykara, Nicolas Pilet, Marcus Liebmann, Eric I. Altman and Udo D. Schwarz.
Def. a basic unit of length in the International System of Units (SI: Système International d'Unités) is called a metre, or meter.
Def. a unit of length; the thousandth part of one millimeter; the millionth part of a meter is called a micron.
"As one spacecraft lurches and drags through the Earth's uneven gravity field, the second follows 210 km behind, measuring changes in their separation to the nearest micron (a thousandth of a millimetre)."[2]
Def. an SI subunit of length equal to 10-9 metres is called a nanometre.
The second image down on the right is an integrated circuit die image of a STM32F103VGT6 ARM Cortex-M3 MCU (microcontroller) with 1 Mbyte Flash, 72 MHz CPU, motor control, USB and CAN. Die size is 5339x5188 µm. View is from a Scanning Electron Microscope looking at the 180 nanometre SRAM cells on the die.
The third image down on the right is a high-resolution, transmission electron micrograph (HRTEM) lattice image with an electron diffraction pattern (top-right inset) taken along the [0001] direction; i.e., looking down the [0001] axis. An image simulation is added in the bottom-left inset, and a model fragment of the crystal structure is on the right of the bottom-left inset. On the lower right of the composite image is a 1 nanometre (nm) marker.
Def. 10-12 of a metre is called a picometre.
The fourth lower image down on the right has an image of 1,750 x 810 pm.[3]
It is an atomic force microscope image of stacked graphene sheets in graphite.[3]
This compares with the image on the left which was taken at a height of ~97 pm above the surface.[3]
The image on the lowest right was at ~12 pm.[3]
For most heights above the surface the features display a three-fold symmetry as on the right.[3]
Public meter standard by Chalgrin is from 18th century. Credit: Airair.
The public meter standard by Chalgrin in the image on the right is from the 18th century (36, rue de Vaugirard, 6th arrondissement of Paris, near the Luxembourg Palace). The public could bring meter rulers and check if they were accurate. A bar that just fit between the end stops was exactly one meter.
any "trench, channel, or groove, as in wood or metal
Def. an SI unit of length equal to 103 metres is called a kilometre, or kilometer.
Times[edit | edit source]
Main article: Times
Main articles: Physics/Units and Physical units
Units have changed over time.
Energy phantoms/Lecture
↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 P. K. Seidelmann (1992). Measuring the Universe, The IAU and astronomical units. International Astronomical Union. http://www.iau.org/public/themes/measuring/. Retrieved 2015-08-09.
↑ Jonathan Amos (2009). Satellites weigh California water. BBC. http://news.bbc.co.uk/2/hi/science/nature/8414252.stm. Retrieved 2015-08-10.
↑ 3.0 3.1 3.2 3.3 3.4 Boris J. Albers, Todd C. Schwendemann, Mehmet Z. Baykara, Nicolas Pilet, Marcus Liebmann, Eric I. Altman and Udo D. Schwarz (May 2009). "Three-dimensional imaging of short-range chemical forces with picometre resolution". Nature Nanotechnology 4: 307-10. doi:10.1038/NNANO.2009.57. http://web.pdx.edu/~larosaa/RESEARCH_GROUP_Current_assignment/2009_3D%20imaging%20of%20short%20range%20chemical%20forces%20with%20picometer%20resolution.pdf. Retrieved 2015-08-10.
↑ 4.0 4.1 4.2 4.3 4.4 4.5 4.6 chain. San Francisco, California: Wikimedia Foundation, Inc. December 17, 2013. https://en.wiktionary.org/wiki/chain. Retrieved 2013-12-25.
Learn more about Astronomical units
Retrieved from "https://en.wikiversity.org/w/index.php?title=Units/Astronomy&oldid=2120109"
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A hierarchy of local symplectic filling obstructions for contact 3-manifolds
15 September 2013 A hierarchy of local symplectic filling obstructions for contact
3
We generalize the familiar notions of overtwistedness and Giroux torsion in
3
-dimensional contact manifolds, defining an infinite hierarchy of local filling obstructions called planar torsion, whose integer-valued order
k\ge 0
can be interpreted as measuring a gradation in “degrees of tightness” of contact manifolds. We show in particular that any contact manifold with planar torsion admits no contact-type embeddings into any closed symplectic
4
-manifold, and has vanishing contact invariant in embedded contact homology, and we give examples of contact manifolds that have planar
k
-torsion for any
k\ge 2
but no Giroux torsion. We also show that the complement of the binding of a supporting open book never has planar torsion. The unifying idea in the background is a decomposition of contact manifolds in terms of contact fiber sums of open books along their binding. As the technical basis of these results, we establish existence, uniqueness, and compactness theorems for certain classes of
J
-holomorphic curves in blown-up summed open books; these also imply algebraic obstructions to planarity and embeddings of partially planar domains.
Chris Wendl. "A hierarchy of local symplectic filling obstructions for contact
3
-manifolds." Duke Math. J. 162 (12) 2197 - 2283, 15 September 2013. https://doi.org/10.1215/00127094-2348333
Secondary: 32Q65 , 53D10 , 53D42
Chris Wendl "A hierarchy of local symplectic filling obstructions for contact
3
-manifolds," Duke Mathematical Journal, Duke Math. J. 162(12), 2197-2283, (15 September 2013)
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Chow group of 0-cycles with modulus and higher-dimensional class field theory
15 October 2016 Chow group of
0
-cycles with modulus and higher-dimensional class field theory
Moritz Kerz, Shuji Saito
One of the main results of this article is a proof of the rank-one case of an existence conjecture on lisse
{\overline{\mathbb{Q}}}_{\ell }
-sheaves on a smooth variety
U
over a finite field due to Deligne and Drinfeld. The problem is translated into the language of higher-dimensional class field theory over finite fields, which describes the abelian fundamental group of
U
by Chow groups of
0
-cycles with moduli. A key ingredient is the construction of a cycle-theoretic avatar of a refined Artin conductor in ramification theory originally studied by Kazuya Kato.
Moritz Kerz. Shuji Saito. "Chow group of
0
-cycles with modulus and higher-dimensional class field theory." Duke Math. J. 165 (15) 2811 - 2897, 15 October 2016. https://doi.org/10.1215/00127094-3644902
Received: 6 May 2014; Revised: 24 October 2015; Published: 15 October 2016
Keywords: Chow group modulus , higher-dimensional class field theory , ramification theory , refined Artin theory , Smooth I-adic
Moritz Kerz, Shuji Saito "Chow group of
0
-cycles with modulus and higher-dimensional class field theory," Duke Mathematical Journal, Duke Math. J. 165(15), 2811-2897, (15 October 2016)
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Partial trace - Wikipedia
Find sources: "Partial trace" – news · newspapers · books · scholar · JSTOR (July 2009) (Learn how and when to remove this template message)
Left hand side shows a full density matrix
{\displaystyle \rho _{AB}}
of a bipartite qubit system. The partial trace is performed over a subsystem of 2 by 2 dimension (single qubit density matrix). The right hand side shows the resulting 2 by 2 reduced density matrix
{\displaystyle \rho _{A}}
In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar valued function on operators, the partial trace is an operator-valued function. The partial trace has applications in quantum information and decoherence which is relevant for quantum measurement and thereby to the decoherent approaches to interpretations of quantum mechanics, including consistent histories and the relative state interpretation.
1.1 Invariant definition
1.2 Category theoretic notion
2 Partial trace for operators on Hilbert spaces
2.1 Computing the partial trace
3 Partial trace and invariant integration
4 Partial trace as a quantum operation
4.1 Comparison with classical case
{\displaystyle V}
{\displaystyle W}
are finite-dimensional vector spaces over a field, with dimensions
{\displaystyle m}
{\displaystyle n}
, respectively. For any space
{\displaystyle A}
{\displaystyle L(A)}
denote the space of linear operators on
{\displaystyle A}
. The partial trace over
{\displaystyle W}
is then written as
{\displaystyle \operatorname {Tr} _{W}:\operatorname {L} (V\otimes W)\to \operatorname {L} (V)}
It is defined as follows: For
{\displaystyle T\in \operatorname {L} (V\otimes W)}
{\displaystyle e_{1},\ldots ,e_{m}}
{\displaystyle f_{1},\ldots ,f_{n}}
, be bases for V and W respectively; then T has a matrix representation
{\displaystyle \{a_{k\ell ,ij}\}\quad 1\leq k,i\leq m,\quad 1\leq \ell ,j\leq n}
relative to the basis
{\displaystyle e_{k}\otimes f_{\ell }}
{\displaystyle V\otimes W}
Now for indices k, i in the range 1, ..., m, consider the sum
{\displaystyle b_{k,i}=\sum _{j=1}^{n}a_{kj,ij}.}
This gives a matrix bk, i. The associated linear operator on V is independent of the choice of bases and is by definition the partial trace.
Among physicists, this is often called "tracing out" or "tracing over" W to leave only an operator on V in the context where W and V are Hilbert spaces associated with quantum systems (see below).
Invariant definition[edit]
The partial trace operator can be defined invariantly (that is, without reference to a basis) as follows: it is the unique linear operator
{\displaystyle \operatorname {Tr} _{W}:\operatorname {L} (V\otimes W)\rightarrow \operatorname {L} (V)}
{\displaystyle \operatorname {Tr} _{W}(R\otimes S)=\operatorname {Tr} (S)\,R\quad \forall R\in \operatorname {L} (V)\quad \forall S\in \operatorname {L} (W).}
To see that the conditions above determine the partial trace uniquely, let
{\displaystyle v_{1},\ldots ,v_{m}}
{\displaystyle V}
{\displaystyle w_{1},\ldots ,w_{n}}
{\displaystyle W}
{\displaystyle E_{ij}\colon V\to V}
be the map that sends
{\displaystyle v_{i}}
{\displaystyle v_{j}}
(and all other basis elements to zero), and let
{\displaystyle F_{kl}\colon W\to W}
{\displaystyle w_{k}}
{\displaystyle w_{l}}
. Since the vectors
{\displaystyle v_{i}\otimes w_{k}}
{\displaystyle V\otimes W}
{\displaystyle E_{ij}\otimes F_{kl}}
{\displaystyle \operatorname {L} (V\otimes W)}
From this abstract definition, the following properties follow:
{\displaystyle \operatorname {Tr} _{W}(I_{V\otimes W})=\dim W\ I_{V}}
{\displaystyle \operatorname {Tr} _{W}(T(I_{V}\otimes S))=\operatorname {Tr} _{W}((I_{V}\otimes S)T)\quad \forall S\in \operatorname {L} (W)\quad \forall T\in \operatorname {L} (V\otimes W).}
Category theoretic notion[edit]
It is the partial trace of linear transformations that is the subject of Joyal, Street, and Verity's notion of Traced monoidal category. A traced monoidal category is a monoidal category
{\displaystyle (C,\otimes ,I)}
together with, for objects X, Y, U in the category, a function of Hom-sets,
{\displaystyle \operatorname {Tr} _{X,Y}^{U}\colon \operatorname {Hom} _{C}(X\otimes U,Y\otimes U)\to \operatorname {Hom} _{C}(X,Y)}
satisfying certain axioms.
Another case of this abstract notion of partial trace takes place in the category of finite sets and bijections between them, in which the monoidal product is disjoint union. One can show that for any finite sets, X,Y,U and bijection
{\displaystyle X+U\cong Y+U}
there exists a corresponding "partially traced" bijection
{\displaystyle X\cong Y}
Partial trace for operators on Hilbert spaces[edit]
The partial trace generalizes to operators on infinite dimensional Hilbert spaces. Suppose V, W are Hilbert spaces, and let
{\displaystyle \{f_{i}\}_{i\in I}}
be an orthonormal basis for W. Now there is an isometric isomorphism
{\displaystyle \bigoplus _{\ell \in I}(V\otimes \mathbb {C} f_{\ell })\rightarrow V\otimes W}
Under this decomposition, any operator
{\displaystyle T\in \operatorname {L} (V\otimes W)}
can be regarded as an infinite matrix of operators on V
{\displaystyle {\begin{bmatrix}T_{11}&T_{12}&\ldots &T_{1j}&\ldots \\T_{21}&T_{22}&\ldots &T_{2j}&\ldots \\\vdots &\vdots &&\vdots \\T_{k1}&T_{k2}&\ldots &T_{kj}&\ldots \\\vdots &\vdots &&\vdots \end{bmatrix}},}
{\displaystyle T_{k\ell }\in \operatorname {L} (V)}
First suppose T is a non-negative operator. In this case, all the diagonal entries of the above matrix are non-negative operators on V. If the sum
{\displaystyle \sum _{\ell }T_{\ell \ell }}
converges in the strong operator topology of L(V), it is independent of the chosen basis of W. The partial trace TrW(T) is defined to be this operator. The partial trace of a self-adjoint operator is defined if and only if the partial traces of the positive and negative parts are defined.
Computing the partial trace[edit]
Suppose W has an orthonormal basis, which we denote by ket vector notation as
{\displaystyle \{|\ell \rangle \}_{\ell }}
{\displaystyle \operatorname {Tr} _{W}\left(\sum _{k,\ell }T^{(k\ell )}\,\otimes \,|k\rangle \langle \ell |\right)=\sum _{j}T^{(jj)}.}
The superscripts in parentheses do not represent matrix components, but instead label the matrix itself.
Partial trace and invariant integration[edit]
In the case of finite dimensional Hilbert spaces, there is a useful way of looking at partial trace involving integration with respect to a suitably normalized Haar measure μ over the unitary group U(W) of W. Suitably normalized means that μ is taken to be a measure with total mass dim(W).
Theorem. Suppose V, W are finite dimensional Hilbert spaces. Then
{\displaystyle \int _{\operatorname {U} (W)}(I_{V}\otimes U^{*})T(I_{V}\otimes U)\ d\mu (U)}
commutes with all operators of the form
{\displaystyle I_{V}\otimes S}
and hence is uniquely of the form
{\displaystyle R\otimes I_{W}}
. The operator R is the partial trace of T.
Partial trace as a quantum operation[edit]
The partial trace can be viewed as a quantum operation. Consider a quantum mechanical system whose state space is the tensor product
{\displaystyle H_{A}\otimes H_{B}}
of Hilbert spaces. A mixed state is described by a density matrix ρ, that is a non-negative trace-class operator of trace 1 on the tensor product
{\displaystyle H_{A}\otimes H_{B}.}
The partial trace of ρ with respect to the system B, denoted by
{\displaystyle \rho ^{A}}
, is called the reduced state of ρ on system A. In symbols,
{\displaystyle \rho ^{A}=\operatorname {Tr} _{B}\rho .}
To show that this is indeed a sensible way to assign a state on the A subsystem to ρ, we offer the following justification. Let M be an observable on the subsystem A, then the corresponding observable on the composite system is
{\displaystyle M\otimes I}
. However one chooses to define a reduced state
{\displaystyle \rho ^{A}}
, there should be consistency of measurement statistics. The expectation value of M after the subsystem A is prepared in
{\displaystyle \rho ^{A}}
{\displaystyle M\otimes I}
when the composite system is prepared in ρ should be the same, i.e. the following equality should hold:
{\displaystyle \operatorname {Tr} (M\cdot \rho ^{A})=\operatorname {Tr} (M\otimes I\cdot \rho ).}
We see that this is satisfied if
{\displaystyle \rho ^{A}}
is as defined above via the partial trace. Furthermore, such operation is unique.
Let T(H) be the Banach space of trace-class operators on the Hilbert space H. It can be easily checked that the partial trace, viewed as a map
{\displaystyle \operatorname {Tr} _{B}:T(H_{A}\otimes H_{B})\rightarrow T(H_{A})}
is completely positive and trace-preserving.
The density matrix ρ is Hermitian, positive semi-definite, and has a trace of 1. It has a spectral decomposition:
{\displaystyle \rho =\sum _{m}p_{m}|\Psi _{m}\rangle \langle \Psi _{m}|;\ 0\leq p_{m}\leq 1,\ \sum _{m}p_{m}=1}
Its easy to see that the partial trace
{\displaystyle \rho ^{A}}
also satisfies these conditions. For example, for any pure state
{\displaystyle |\psi _{A}\rangle }
{\displaystyle H_{A}}
{\displaystyle \langle \psi _{A}|\rho ^{A}|\psi _{A}\rangle =\sum _{m}p_{m}\operatorname {Tr} _{B}[\langle \psi _{A}|\Psi _{m}\rangle \langle \Psi _{m}|\psi _{A}\rangle ]\geq 0}
{\displaystyle \operatorname {Tr} _{B}[\langle \psi _{A}|\Psi _{m}\rangle \langle \Psi _{m}|\psi _{A}\rangle ]}
represents the probability of finding the state
{\displaystyle |\psi _{A}\rangle }
when the composite system is in the state
{\displaystyle |\Psi _{m}\rangle }
. This proves the positive semi-definiteness of
{\displaystyle \rho ^{A}}
The partial trace map as given above induces a dual map
{\displaystyle \operatorname {Tr} _{B}^{*}}
between the C*-algebras of bounded operators on
{\displaystyle \;H_{A}}
{\displaystyle H_{A}\otimes H_{B}}
{\displaystyle \operatorname {Tr} _{B}^{*}(A)=A\otimes I.}
{\displaystyle \operatorname {Tr} _{B}^{*}}
maps observables to observables and is the Heisenberg picture representation of
{\displaystyle \operatorname {Tr} _{B}}
Comparison with classical case[edit]
Suppose instead of quantum mechanical systems, the two systems A and B are classical. The space of observables for each system are then abelian C*-algebras. These are of the form C(X) and C(Y) respectively for compact spaces X, Y. The state space of the composite system is simply
{\displaystyle C(X)\otimes C(Y)=C(X\times Y).}
A state on the composite system is a positive element ρ of the dual of C(X × Y), which by the Riesz-Markov theorem corresponds to a regular Borel measure on X × Y. The corresponding reduced state is obtained by projecting the measure ρ to X. Thus the partial trace is the quantum mechanical equivalent of this operation.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Partial_trace&oldid=1068657388"
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What is a Molecule? — lesson. Science State Board, Class 7.
A molecule is the smallest particle of an element or compound capable of an independent life and exhibiting all of the substance's properties.
In general, a molecule is a set of two or more atoms that are chemically bound or held together by attractive forces.
Atoms from the same or different elements may combine to form a molecule.
To understand it, consider a water molecule
{H}_{2}O
, the two elements, hydrogen and oxygen atoms combined, to form a water molecule.
Calcium oxide \(CaO\) contains two elements, calcium and oxygen atoms combined to form a calcium oxide molecule.
Oxygen \(O_2\) contains two elements of an oxygen atom.
Chlorine \(Cl_2\) contains two elements of a chlorine atom.
Molecules of elements:
The atoms that make up an element's molecules are all of the same nature.
Many elements' molecules are made up of just one electron, such as argon \(Ar\), helium \(He\), and so on.
Monoatomic Elements:
In their molecular form, certain elements are monatomic, meaning they are made up of just one atom (mono-atomic).
Helium \(He\) is an example of a monoatomic element.
Diatomic Elements:
If the molecule constitutes two atoms, we can classify it as a diatomic molecule. In their molecular shape, other elements have two or more atoms.
Each molecule of hydrogen \(H_2\), oxygen \(O_2\), and chlorine \(Cl_2\), for example, has two atoms.
Triatomic Elements:
If another atom is combined with
{O}_{2}
, it forms ozone
{O}_{3}
, which is a triatomic molecule.
Monoatomic and diatomic molecules are more stable than triatomic molecules.
Most non-metal elements are diatomic molecules.
Molecules of compounds:
Compound molecules contain atoms from two or more separate elements.
Water \(H_2O\), for example, has three atoms: Two hydrogen (H) atoms and one oxygen (O).
Methane \(CH_4\) with five atoms: One carbon (C) and four hydrogen (H) atoms.
Glucose \(C_6H_{12}O_6\) contains elements such as \(6\) carbon, \(12\) hydrogen and \(6\) oxygen, combined to form a glucose molecule.
Sodium chloride \(NaCl\) contains the one element of each sodium and chlorine combined to form a sodium chloride molecule.
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Medical Entrance Exam Question and Answers | Motion in Straight Line - Zigya
A ball is thrown vertically upwards. Which of the following plots represents the speed-time graph of the ball during its flight if the air resistance is not ignored?
During the upward motion, the speed of the body decreases and will be zero at the highest point (since the gravitational force acting downward), afterward, the body starts downward motion and its speed increases.
The radius of earth is 6400 km and g = 10 m/s2. In order that a body of 5 kg weight is zero at the equator, the angular speed is
\frac{1}{800}\mathrm{rad}/\mathrm{sec}
\frac{1}{1600}\mathrm{rad}/\mathrm{sec}
\frac{1}{400}\mathrm{rad}/\mathrm{sec}
\frac{1}{80}\mathrm{rad}/\mathrm{sec}
\frac{1}{800}\mathrm{rad}/\mathrm{sec}
Given:- radius of earth=6400km=6400
\times
103, g=10m/s2
g'=0, m=0(weight is Zero at equator)
g=R
\omega
\Rightarrow
\omega
\frac{g}{R}
\omega
\sqrt{\frac{\mathrm{g}}{\mathrm{R}}}
\sqrt{\frac{10}{6400\times {10}^{3}}}
\frac{1}{800}\mathrm{rad}/\mathrm{sec}
A particle starts from rest and has an acceleration of 2 m/s2 for 10 sec. After that, it travels for 30 sec with constant speed and then undergoes a retardation of 4 m/s2 and comes back to rest. The total distance covered by the particle is
Initial velocity ( u ) = 0
{\mathrm{\alpha }}_{1}
) = 2 m/s2
time during acceleration (t1) = 10 sec
Time during constant velocity (t2 ) = 30 sec
and retardation (
{\mathrm{\alpha }}_{2}
-
( negative sign due to retardation )
Distance covered by the particle during acceleration
s1 = ut1 +
\frac{1}{2}
{\mathrm{\alpha }}_{1}
{\mathrm{t}}_{1}^{\quad 2}
= ( 0 × 10 ) +
\frac{1}{2}
× 2 × (10)2
s1 = 100 m ......(i)
And velocity of the particle a the end of acceleration
v = u +
\frac{1}{2}\quad {\mathrm{\alpha }}_{1}{\mathrm{t}}_{1}^{\quad 2}
= 0 + ( 2 × 10 )
Therefore distance covered by the particle during constant velocity
s2 = v × t2
= 20 × 30⇒
s2 = 600 m .....(ii)
Relation for the distance covered by the particle during retardation ( s3 ) is
v2 = u2 + 2
{\mathrm{\alpha }}_{2}
⇒ ( 0 )2 = ( 20 )2 + 2 × (
-
4 ) × s3
⇒ 400 = 8 s3
⇒ s3 =
\frac{400}{8}
⇒ s3 = 50 m
Therefore total distance covered by the particle
= 100 +600 + 50
Speed in kilometre per hour in S.I unit is represented as
Kmhr-1
Kilometres per hour is a unit of measurement, which measures speed or velocity. The unit symbol is km/h or km h-1.
By definition, an object travelling at a speed of 1 km/h in a hour moves 1 kilometre.
S.I unit of velocity is
m sec-1
m hr-1
Velocity is a physical quantity, both magnitude and sirection are needed to define it.
The SI unit of distance is Meter (m)
The SI unit of time is second ( sec)
So the unit of velocity in SI unit is meter per sec i.e m/s.
Apparent frequency of sound of engine is changing in the ratio 5/3 under the condition that engine is first aprroaching and then receding away from the observer. If velocity of sound is 340 m/s, then velocity of engine is
When source is moving towards stationary observer, its apparent frequency
{\mathrm{\eta }}_{1}\quad =\quad \mathrm{\eta }\left(\frac{\mathrm{\nu }}{\mathrm{\nu }\quad -\quad {\mathrm{v}}_{\mathrm{s}}}\right)
where, ν
\to
velocity of sound
νs
\to
velocity of source
When source is moving away from observer, its apparent frequency
{\mathrm{\eta }}_{2}\quad =\quad \mathrm{\eta }\left(\frac{\mathrm{\nu }}{\mathrm{\nu }\quad +\quad {\mathrm{v}}_{\mathrm{s}}}\right)
\therefore
\frac{{\mathrm{\eta }}_{1}}{{\mathrm{\eta }}_{2}}\quad =\quad \left(\frac{\mathrm{\nu }}{\mathrm{\nu }\quad -\quad {\mathrm{\nu }}_{\mathrm{s}}}\right)\quad \times \quad \left(\frac{\mathrm{\nu }\quad +\quad {\mathrm{\nu }}_{\mathrm{s}}}{\mathrm{v}}\right)
\frac{5}{3}\quad =\quad \frac{\mathrm{\nu }\quad +\quad {\mathrm{\nu }}_{\mathrm{s}}}{\mathrm{\nu }\quad -\quad {\mathrm{\nu }}_{\mathrm{s}}}
{\mathrm{\nu }}_{\mathrm{s}}\quad =\quad \frac{\mathrm{\nu }}{4}\quad =\quad \frac{340}{4}
A body starting from rest moves along a straight line with a constant acceleration. The variation of speed (v) with distance ( s ) is represented by the graph
For a body moving with constant acceleration
\mathrm{\alpha }
\mathrm{\alpha }
Since the body starts from rest
∴ v =
\mathrm{\alpha }
which is a straight line passing through the origin. Hence the graph is (b)
An earthquake generates both transverse (S) and longitudinal (P) sound waves in the earth. The speed of S waves is about 4.5 km/s and that of P waves is about 8.0 km/s. A seismograph records P and S waves from an earthquake. The first P wave arrives 4.0 min before the first S wave. The epicentre of the earthquake is located at a distance about
Velocity of the S wave is v1 = 4.5 km/s
The velocity of the P waves is v2 = 8.0 km/s
Let the time taken by the S and P waves to reach the seismograph be t1 and t2
-
t2 = 4 min
= 4 × 60 sec
-
t2 = 240 sec .....(i)
Let the distance of the epicentre (km) be S. Then
S = v1t1 = v2t2
⇒ 4.5 × t1
-
8t2 = 0
⇒ t1 =
\frac{4.5}{8}
t1 ....(ii)
-
t2 = 240
⇒ t1
\left(1\quad -\quad \frac{4.5}{8}\right)
\frac{240\quad \times \quad 8}{3.5}
⇒ = 548.5 s
∴ S = v1t1
= 4.5 × 548.5
S = 2468.6
S ≈ 2500 km
A car covers 2/5 of a certain distance with speed v1 and rest 3/5 part with velocity v2. The average speed of the car is :
\frac{1}{2}\sqrt{{\mathrm{v}}_{1}{\mathrm{v}}_{2}}
\frac{5{\mathrm{v}}_{1}{\mathrm{v}}_{2}}{3{\mathrm{v}}_{1}+2{\mathrm{v}}_{2}}
\frac{2{\mathrm{v}}_{1}{\mathrm{v}}_{2}}{{\mathrm{v}}_{1}+{\mathrm{v}}_{2}}
\frac{{\mathrm{v}}_{1}+{\mathrm{v}}_{2}}{2}
\frac{5{\mathrm{v}}_{1}{\mathrm{v}}_{2}}{3{\mathrm{v}}_{1}+2{\mathrm{v}}_{2}}
Let total distance covered = s
t1=
\frac{\left(2/5\right)\mathrm{s}}{{\mathrm{v}}_{1}}
and t2 =
\frac{\left(3/5\right)s}{{v}_{2}}
\frac{\mathrm{total}\quad \mathrm{distance}\quad }{\mathrm{total}\quad \mathrm{time}\quad }
\frac{\mathrm{s}}{{\mathrm{t}}_{1}+\quad {\mathrm{t}}_{2}}
or v =
\frac{\mathrm{s}}{\frac{2\mathrm{s}}{5{\mathrm{v}}_{1}}+\frac{3\mathrm{s}}{5{\mathrm{v}}_{2}}}
Hence, v =
\frac{5{\mathrm{v}}_{1}{\mathrm{v}}_{2}}{2{\mathrm{v}}_{2}+3{\mathrm{v}}_{1}}
A wheel rotates with constant acceleration of 2.0 radian/sec2 .If the wheel starts from rest the number of revolution it makes in the first ten second, will be approximately
Angular acceleration= 2 rad/s2
Time interval t= 10sec
Angular displacement θ = ?
N\mathrm{ow}\quad \mathrm{by}\quad \mathrm{\theta }\quad =\mathrm{\omega t}+\frac{1}{2}{\mathrm{\alpha t}}^{2}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ....\left(\mathrm{i}\right)\phantom{\rule{0ex}{0ex}}\left[\mathrm{It}\quad \mathrm{is}\quad \mathrm{an}\quad \mathrm{equation}\quad \mathrm{of}\quad \mathrm{angular}\quad \mathrm{variables}\quad \mathrm{of}\quad \mathrm{motion}\quad \mathrm{corresponding}\quad \mathrm{to}\quad \mathrm{the}\quad \mathrm{linear}\quad \mathrm{variable}\quad \mathrm{based}\quad \mathrm{equations}\quad \mathrm{s}\right]\phantom{\rule{0ex}{0ex}}\left(\mathrm{s}=\mathrm{ut}+\frac{1}{2}{\mathrm{\alpha t}}^{2}\right)\phantom{\rule{0ex}{0ex}}\left(\mathrm{i}\right)\quad \Rightarrow \quad \mathrm{\theta }\quad =\quad 0+\frac{1}{2}{\mathrm{\alpha t}}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{\theta }\quad =\frac{{\mathrm{\alpha t}}^{2}}{2}=\frac{2\times 100}{2}=100\phantom{\rule{0ex}{0ex}}\mathrm{n}=\frac{\mathrm{\theta }}{2\mathrm{\pi }}=\frac{100}{2\times 3.14}=15.92\quad \approx 16
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Cross-correlation - MATLAB xcorr - MathWorks Deutschland
Cross-Correlation of Two Vectors
Autocorrelation of Vector
Cross-Correlation and Autocorrelation
r = xcorr(x,y) returns the cross-correlation of two discrete-time sequences. Cross-correlation measures the similarity between a vector x and shifted (lagged) copies of a vector y as a function of the lag. If x and y have different lengths, the function appends zeros to the end of the shorter vector so it has the same length as the other.
r = xcorr(x) returns the autocorrelation sequence of x. If x is a matrix, then r is a matrix whose columns contain the autocorrelation and cross-correlation sequences for all combinations of the columns of x.
r = xcorr(___,maxlag) limits the lag range from -maxlag to maxlag for either of the previous syntaxes.
r = xcorr(___,scaleopt) also specifies a normalization option for the cross-correlation or autocorrelation. Any option other than 'none' (the default) requires x and y to have the same length.
[r,lags] = xcorr(___) also returns the lags at which the correlations are computed.
Create a vector x and a vector y that is equal to x shifted by 5 elements to the right. Compute and plot the estimated cross-correlation of x and y. The largest spike occurs at the lag value when the elements of x and y match exactly (-5).
Compute and plot the estimated autocorrelation of a vector x. The largest spike occurs at zero lag, when x matches itself exactly.
Compute and plot the normalized cross-correlation of vectors x and y with unity peak, and specify a maximum lag of 10.
Input array, specified as a vector, matrix, or multidimensional array. If x is a multidimensional array, then xcorr operates column-wise across all dimensions and returns each autocorrelation and cross-correlation as the columns of a matrix.
Maximum lag, specified as an integer scalar. If you specify maxlag, the returned cross-correlation sequence ranges from -maxlag to maxlag. If you do not specify maxlag, the lag range equals 2N – 1, where N is the greater of the lengths of x and y.
'none' — Raw, unscaled cross-correlation. 'none' is the only valid option when x and y have different lengths.
'biased' — Biased estimate of the cross-correlation:
{\stackrel{^}{R}}_{xy,\text{biased}}\left(m\right)=\frac{1}{N}{\stackrel{^}{R}}_{xy}\left(m\right).
'unbiased' — Unbiased estimate of the cross-correlation:
{\stackrel{^}{R}}_{xy,\text{unbiased}}\left(m\right)=\frac{1}{N-|m|}{\stackrel{^}{R}}_{xy}\left(m\right).
'normalized' or 'coeff' — Normalizes the sequence so that the autocorrelations at zero lag equal 1:
{\stackrel{^}{R}}_{xy,\text{coeff}}\left(m\right)=\frac{1}{\sqrt{{\stackrel{^}{R}}_{xx}\left(0\right){\stackrel{^}{R}}_{yy}\left(0\right)}}{\stackrel{^}{R}}_{xy}\left(m\right).
r — Cross-correlation or autocorrelation
Cross-correlation or autocorrelation, returned as a vector or matrix.
If x is an M × N matrix, then xcorr(x) returns a (2M – 1) × N2 matrix with the autocorrelations and cross-correlations of the columns of x. If you specify maxlag, then r has size (2 × maxlag + 1) × N2.
\text{S}=\left(\begin{array}{ccc}{x}_{1}& {x}_{2}& {x}_{3}\end{array}\right)
, then the result of R = xcorr(S) is organized as
\text{R}=\left(\begin{array}{lllllllll}{R}_{{x}_{1}{x}_{1}}\hfill & {R}_{{x}_{1}{x}_{2}}\hfill & {R}_{{x}_{1}{x}_{3}}\hfill & {R}_{{x}_{2}{x}_{1}}\hfill & {R}_{{x}_{2}{x}_{2}}\hfill & {R}_{{x}_{2}{x}_{3}}\hfill & {R}_{{x}_{3}{x}_{1}}\hfill & {R}_{{x}_{3}{x}_{2}}\hfill & {R}_{{x}_{3}{x}_{3}}\hfill \end{array}\right).
The result of xcorr can be interpreted as an estimate of the correlation between two random sequences or as the deterministic correlation between two deterministic signals.
The true cross-correlation sequence of two jointly stationary random processes, xn and yn, is given by
{R}_{xy}\left(m\right)=E\left\{{x}_{n+m}{y}_{n}^{*}\right\}=E\left\{{x}_{n}{y}_{n-m}^{*}\right\},
where −∞ < n < ∞, the asterisk denotes complex conjugation, and E is the expected value operator. xcorr can only estimate the sequence because, in practice, only a finite segment of one realization of the infinite-length random process is available.
By default, xcorr computes raw correlations with no normalization:
{\stackrel{^}{R}}_{xy}\left(m\right)=\left\{\begin{array}{ll}\sum _{n=0}^{N-m-1}{x}_{n+m}{y}_{n}^{\ast },\hfill & m\ge 0,\hfill \\ {\stackrel{^}{R}}_{yx}^{*}\left(-m\right),\hfill & m<0.\hfill \end{array}
The output vector, c, has elements given by
\text{c}\left(\text{m}\right)={\stackrel{^}{R}}_{xy}\left(m-N\right),\text{ }\text{ }m=1,\text{\hspace{0.17em}}2,\dots ,2N-1.
In general, the correlation function requires normalization to produce an accurate estimate. You can control the normalization of the correlation by using the input argument scaleopt.
The syntax xcorr(x) is not supported.
Leading ones in size(x) (dimension lengths of 1 before the first dimension length not equal to 1) must be constant for every input x. If x is variable-size and is a row vector, it must be 1-by-:. It cannot be :-by-: with size(x,1) = 1 at run time.
For example, create a gpuArray object from a signal x and compute the normalized autocorrelation.
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Divergence of vector field - MATLAB divergence - MathWorks España
Find Divergence of Vector Field
Find Electric Charge Density from Electric Field
divergence(V,X) returns the divergence of vector field V with respect to the vector X in Cartesian coordinates. Vectors V and X must have the same length.
Find the divergence of the vector field V(x,y,z) = (x, 2y2, 3z3) with respect to vector X = (x,y,z).
Show that the divergence of the curl of the vector field is 0.
Find the divergence of the gradient of this scalar function. The result is the Laplacian of the scalar function.
Gauss’ Law in differential form states that the divergence of electric field is proportional to the electric charge density.
\underset{}{\overset{\to }{\nabla }}\cdot \underset{}{\overset{\to }{E}}\left(\underset{}{\overset{\to }{r}}\right)=\frac{\rho \left(\underset{}{\overset{\to }{r}}\right)}{{ϵ}_{0}}\phantom{\rule{0.2777777777777778em}{0ex}}.
Find the electric charge density for the electric field
\underset{}{\overset{\to }{E}}={x}^{2}\underset{}{\overset{ˆ}{i}}+{y}^{2}\underset{}{\overset{ˆ}{j}}
{\mathrm{ep}}_{0} \left(2 x+2 y\right)
Visualize the electric field and electric charge density for -2 < x < 2 and -2 < y < 2 with ep0 = 1. Create a grid of values of x and y using meshgrid. Find the values of electric field and charge density by substituting grid values using subs. Simultaneously substitute the grid values xPlot and yPlot into the charge density rho by using cells arrays as inputs to subs.
Plot the electric field using quiver. Overlay the charge density using contour. The contour lines indicate the values of the charge density.
symbolic expression | symbolic function | vector of symbolic expressions | vector of symbolic functions
Vector field to find divergence of, specified as a symbolic expression or function, or as a vector of symbolic expressions or functions. V must be the same length as X.
X — Variables with respect to which you find the divergence
Variables with respect to which you find the divergence, specified as a symbolic variable or a vector of symbolic variables. X must be the same length as V.
The divergence of the vector field V = (V1,...,Vn) with respect to the vector X = (X1,...,Xn) in Cartesian coordinates is the sum of partial derivatives of V with respect to X1,...,Xn.
div\left(\stackrel{\to }{V}\right)=\nabla \cdot \stackrel{\to }{V}=\sum _{i=1}^{n}\frac{\partial {V}_{i}}{\partial {x}_{i}}.
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Moving averages are a favorite tool of active traders. However, when markets consolidate, this indicator leads to numerous whipsaw trades, resulting in a frustrating series of small wins and losses. Analysts have spent decades trying to improve the simple moving average. In this article, we look at these efforts and find that their search has led to useful trading tools. (For background reading on simple moving averages, check out Simple Moving Averages Make Trends Stand Out.)
Pros and Cons of Moving Averages
The advantages and disadvantages of moving averages were summed up by Robert Edwards and John Magee in the first edition of Technical Analysis of Stock Trends, when they said "and, it was back in 1941 that we delightedly made the discovery (though many others had made it before) that by averaging the data for a stated number of days…one could derive a sort of automated trendline which would definitely interpret the changes of trend…It seemed almost too good to be true. As a matter of fact, it was too good to be true."
With the disadvantages outweighing the advantages, Edwards and Magee quickly abandoned their dream of trading from a beach bungalow. But 60 years after they wrote those words, others persist in trying to find a simple tool that would effortlessly deliver the riches of the markets.
To calculate a simple moving average, add the prices for the desired time period and divide by the number of periods selected. Finding a five-day moving average would require summing the five most recent closing prices and dividing by five.
If the most recent close is above the moving average, the stock would be considered to be in an uptrend.
Downtrends are defined by prices trading below the moving average. (For more, see our Moving Averages tutorial.)
This trend-defining property makes it possible for moving averages to generate trading signals. In its simplest application, traders buy when prices move above the moving average and sell when prices cross below that line. An approach such as this is guaranteed to put the trader on the right side of every significant trade. Unfortunately, while smoothing the data, moving averages will lag behind the market action and the trader will almost always give back a large part of their profits on even the biggest winning trades.
Analysts seem to like the idea of the moving average and have spent years trying to reduce the problems associated with this lag. One of these innovations is the exponential moving average (EMA). This approach assigns a relatively higher weighting to recent data, and as a result it stays closer to the price action than a simple moving average. The formula to calculate an exponential moving average is:
\begin{aligned}&\text{EMA}=(\text{Weight}\times\text{Close})+((1-\text{Weight})\times\text{EMAy)}\\&\textbf{where:}\\&\text{Weight}=\text{the smoothing constant selected by the analyst}\\&\text{EMAy}=\text{the exponential moving average from yesterday}\end{aligned}
EMA=(Weight×Close)+((1−Weight)×EMAy)where:Weight=the smoothing constant selected by the analyst
A common weighting value is 0.181, which is close to a 20-day simple moving average. Another is 0.10, which is approximately a 10-day moving average.
Although it reduces the lag, the exponential moving average fails to address another problem with moving averages, which is that their use for trading signals will lead to a large number of losing trades. In New Concepts in Technical Trading Systems, Welles Wilder estimates that markets only trend a quarter of the time. Up to 75% of trading action is confined to narrow ranges, when moving-average buy-and-sell signals will be repeatedly generated as prices rapidly move above and below the moving average. To address this problem, several analysts have suggested varying the weighting factor of the EMA calculation. (For more, see How are moving averages used in trading?)
Adapting Moving Averages to Market Action
One method of addressing the disadvantages of moving averages is to multiply the weighting factor by a volatility ratio. Doing this would mean that the moving average would be further from the current price in volatile markets. This would allow winners to run. As a trend comes to an end and prices consolidate, the moving average would move closer to the current market action and, in theory, allow the trader to keep most of the gains captured during the trend. In practice, the volatility ratio can be an indicator such as the Bollinger Band®width, which measures the distance between the well-known Bollinger Bands®. (For more on this indicator, see The Basics Of Bollinger Bands®.)
Perry Kaufman suggested replacing the "weight" variable in the EMA formula with a constant based on the efficiency ratio (ER) in his book, New Trading Systems and Methods. This indicator is designed to measure the strength of a trend, defined within a range from -1.0 to +1.0. It is calculated with a simple formula:
\begin{aligned}&\text{ER}\ =\ \frac{\text{total price change for period}}{\text{sum of absolute price changes for each bar}}\\&\textbf{where:}\\&\text{ER}\ =\ \text{efficiency ratio}\end{aligned}
ER = sum of absolute price changes for each bartotal price change for periodwhere:
Consider a stock that has a five-point range each day, and at the end of five days has gained a total of 15 points. This would result in an ER of 0.67 (15 points upward movement divided by the total 25-point range). Had this stock declined 15 points, the ER would be -0.67. (For more trading advice from Perry Kaufman, read Losing To Win, which outlines strategies for coping with trading losses.)
The principle of a trend's efficiency is based on how much directional movement (or trend) you get per unit of price movement over a defined time period. An ER of +1.0 indicates that the stock is in a perfect uptrend; -1.0 represents a perfect downtrend. In practical terms, the extremes are rarely reached.
To apply this indicator to find the adaptive moving average (AMA), traders will need to calculate the weight with the following, rather complex, formula:
\begin{aligned}&\text{C}\ =\ [(\text{ER}\ \times\ (\text{SCF}\ - \ \text{SCS}))\ +\ \text{SCS}]^{2}\\&\textbf{where:}\\&\text{SCF}\ =\ \text{the exponential constant for the fastest}\\&\qquad\quad\ \ \text{ EMA allowable (usually 2)}\\&\text{SCS}\ =\ \text{the exponential constant for the slowest}\\&\qquad\quad\ \ \text{ EMA allowable (often 30)}\\&\text{ER}\ =\ \text{the efficiency ratio that was noted above}\end{aligned}
C = [(ER × (SCF − SCS)) + SCS]2where:SCF = the exponential constant for the fastest EMA allowable (usually 2)SCS = the exponential constant for the slowest EMA allowable (often 30)
The value for C is then used in the EMA formula instead of the simpler weight variable. Although difficult to calculate by hand, the adaptive moving average is included as an option in almost all trading software packages. (For more on the EMA, read Exploring The Exponentially Weighted Moving Average.)
Examples of a simple moving average (red line), an exponential moving average (blue line) and the adaptive moving average (green line) are shown in Figure 1.
Figure 1: The AMA is in green and shows the greatest degree of flattening in the range-bound action seen on the right side of this chart. In most cases, the exponential moving average, shown as the blue line, is closest to the price action. The simple moving average is shown as the red line.
The three moving averages shown in the figure are all prone to whipsaw trades at various times. This drawback to moving averages has thus far been impossible to eliminate.
Robert Colby tested hundreds of technical-analysis tools in The Encyclopedia of Technical Market Indicators. He concluded, "Although the adaptive moving average is an interesting newer idea with considerable intellectual appeal, our preliminary tests fail to show any real practical advantage to this more complex trend smoothing method." This doesn't mean traders should ignore the idea. The AMA could be combined with other indicators to develop a profitable trading system. (For more on this topic, read Discovering Keltner Channels And The Chaikin Oscillator.)
The ER can be used as a stand-alone trend indicator to spot the most profitable trading opportunities. As one example, ratios above 0.30 indicate strong uptrends and represent potential buys. Alternatively, since volatility moves in cycles, the stocks with the lowest efficiency ratio might be watched as breakout opportunities.
For more, see Basics Of Weighted Moving Averages.
Robert D. Edwards and John Magee, Jr. "Technical Analysis of Stock Trends." Stock Trend Service, 1948.
Duke University. "Moving Average and Exponential Smoothing Models." Accessed June 8, 2020.
J. Welles Wilder Jr. "New Concepts in Technical Trading Systems." Trend Research, 1978. Accessed June 8, 2020.
Bollinger Bands. "Bollinger Bands." Accessed June 8, 2020.
Perry Kaufman. "Trading Systems and Methods." Wiley Trading, 2013.
Robert W. Colby. "The Encyclopedia of Technical Market Indicators." McGraw-Hill Education, 2002.
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Fractional_ideal Knowpia
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed integral ideals for clarity.
Definition and basic resultsEdit
{\displaystyle R}
{\displaystyle K=\operatorname {Frac} R}
A fractional ideal of
{\displaystyle R}
{\displaystyle R}
-submodule
{\displaystyle I}
{\displaystyle K}
such that there exists a non-zero
{\displaystyle r\in R}
{\displaystyle rI\subseteq R}
{\displaystyle r}
can be thought of as clearing out the denominators in
{\displaystyle I}
, hence the name fractional ideal.
The principal fractional ideals are those
{\displaystyle R}
-submodules of
{\displaystyle K}
generated by a single nonzero element of
{\displaystyle K}
. A fractional ideal
{\displaystyle I}
{\displaystyle R}
if, and only if, it is an ('integral') ideal of
{\displaystyle R}
A fractional ideal
{\displaystyle I}
is called invertible if there is another fractional ideal
{\displaystyle J}
{\displaystyle IJ=R}
{\displaystyle IJ=\{a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}:a_{i}\in I,b_{j}\in J,n\in \mathbb {Z} _{>0}\}}
is called the product of the two fractional ideals).
In this case, the fractional ideal
{\displaystyle J}
is uniquely determined and equal to the generalized ideal quotient
{\displaystyle (R:_{K}I)=\{x\in K:xI\subseteq R\}.}
The set of invertible fractional ideals form an abelian group with respect to the above product, where the identity is the unit ideal
{\displaystyle (1)=R}
itself. This group is called the group of fractional ideals of
{\displaystyle R}
. The principal fractional ideals form a subgroup. A (nonzero) fractional ideal is invertible if, and only if, it is projective as an
{\displaystyle R}
-module. Geometrically, this means an invertible fractional ideal can be interpreted as rank 1 vector bundles over the affine scheme
{\displaystyle {\text{Spec}}(R)}
Every finitely generated R-submodule of K is a fractional ideal and if
{\displaystyle R}
is noetherian these are all the fractional ideals of
{\displaystyle R}
Dedekind domainsEdit
In Dedekind domains, the situation is much simpler. In particular, every non-zero fractional ideal is invertible. In fact, this property characterizes Dedekind domains:
An integral domain is a Dedekind domain if, and only if, every non-zero fractional ideal is invertible.
The set of fractional ideals over a Dedekind domain
{\displaystyle R}
{\displaystyle {\text{Div}}(R)}
Its quotient group of fractional ideals by the subgroup of principal fractional ideals is an important invariant of a Dedekind domain called the ideal class group.
For the special case of number fields
{\displaystyle K}
{\displaystyle \mathbb {Q} (\zeta _{n})}
) there is an associated ring denoted
{\displaystyle {\mathcal {O}}_{K}}
called the ring of integers of
{\displaystyle K}
{\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {d}})}=\mathbb {Z} [{\sqrt {d}}]}
{\displaystyle d}
square free and equal to
{\displaystyle 2,3{\text{ }}({\text{mod }}4)}
. The key property of these rings
{\displaystyle {\mathcal {O}}_{K}}
is they are Dedekind domains. Hence the theory of fractional ideals can be described for the rings of integers of number fields. In fact, class field theory is the study of such groups of class rings.
Associated structuresEdit
For the ring of integers[1]pg 2
{\displaystyle {\mathcal {O}}_{K}}
of a number field, the group of fractional ideals forms a group denoted
{\displaystyle {\mathcal {I}}_{K}}
and the subgroup of principal fractional ideals is denoted
{\displaystyle {\mathcal {P}}_{K}}
. The ideal class group is the group of fractional ideals modulo the principal fractional ideals, so
{\displaystyle {\mathcal {C}}_{K}:={\mathcal {I}}_{K}/{\mathcal {P}}_{K}}
and its class number
{\displaystyle h_{K}}
is the order of the group
{\displaystyle h_{K}=|{\mathcal {C}}_{K}|}
. In some ways, the class number is a measure for how "far" the ring of integers
{\displaystyle {\mathcal {O}}_{K}}
is from being a unique factorization domain. This is because
{\displaystyle h_{K}=1}
{\displaystyle {\mathcal {O}}_{K}}
is a UFD.
Exact sequence for ideal class groupsEdit
{\displaystyle 0\to {\mathcal {O}}_{K}^{*}\to K^{*}\to {\mathcal {I}}_{K}\to {\mathcal {C}}_{K}\to 0}
associated to every number field.
Structure theorem for fractional idealsEdit
One of the important structure theorems for fractional ideals of a number field states that every fractional ideal
{\displaystyle I}
decomposes uniquely up to ordering as
{\displaystyle I=({\mathfrak {p}}_{1}\ldots {\mathfrak {p}}_{n})({\mathfrak {q}}_{1}\ldots {\mathfrak {q}}_{m})^{-1}}
for prime ideals
{\displaystyle {\mathfrak {p}}_{i},{\mathfrak {q}}_{j}\in {\text{Spec}}({\mathcal {O}}_{K})}
in the spectrum of
{\displaystyle {\mathcal {O}}_{K}}
{\displaystyle {\frac {2}{5}}{\mathcal {O}}_{\mathbb {Q} (i)}}
{\displaystyle (1+i)(1-i)((1+2i)(1-2i))^{-1}}
Also, because fractional ideals over a number field are all finitely generated we can clear denominators by multiplying by some
{\displaystyle \alpha }
to get an ideal
{\displaystyle J}
{\displaystyle I={\frac {1}{\alpha }}J}
Another useful structure theorem is that integral fractional ideals are generated by up to 2 elements. We call a fractional ideal which is a subset of
{\displaystyle {\mathcal {O}}_{K}}
{\displaystyle {\frac {5}{4}}\mathbb {Z} }
is a fractional ideal over
{\displaystyle \mathbb {Z} }
{\displaystyle K=\mathbb {Q} (i)}
the ideal
{\displaystyle (5)}
splits in
{\displaystyle {\mathcal {O}}_{\mathbb {Q} (i)}=\mathbb {Z} [i]}
{\displaystyle (2-i)(2+i)}
{\displaystyle \mathbb {Q} _{\zeta _{3}}}
we have the factorization
{\displaystyle (3)=(2\zeta _{3}+1)^{2}}
This is because if we multiply it out, we get
{\displaystyle {\begin{aligned}(2\zeta _{3}+1)^{2}&=4\zeta _{3}^{2}+4\zeta _{3}+1\\&=4(\zeta _{3}^{2}+\zeta _{3})+1\end{aligned}}}
{\displaystyle \zeta _{3}}
{\displaystyle \zeta _{3}^{2}+\zeta _{3}=-1}
, our factorization makes sense.
{\displaystyle \mathbb {Q} ({\sqrt {-23}})}
we can multiply the fractional ideals
{\displaystyle I=(2,(1/2){\sqrt {-23}}-(1/2))}
{\displaystyle J=(4,(1/2){\sqrt {-23}}+(3/2))}
to get the ideal
{\displaystyle IJ=(-(1/2){\sqrt {-23}}-(3/2))}
Divisorial idealEdit
{\displaystyle {\tilde {I}}}
denote the intersection of all principal fractional ideals containing a nonzero fractional ideal
{\displaystyle I}
{\displaystyle {\tilde {I}}=(R:(R:I)),}
where as above
{\displaystyle (R:I)=\{x\in K:xI\subseteq R\}.}
{\displaystyle {\tilde {I}}=I}
then I is called divisorial.[2]
In other words, a divisorial ideal is a nonzero intersection of some nonempty set of fractional principal ideals.
If I is divisorial and J is a nonzero fractional ideal, then (I : J) is divisorial.
Let R be a local Krull domain (e.g., a Noetherian integrally closed local domain).
Then R is a discrete valuation ring if and only if the maximal ideal of R is divisorial.[3]
An integral domain that satisfies the ascending chain conditions on divisorial ideals is called a Mori domain.[4]
Divisorial sheaf
Dedekind-Kummer theorem
^ Childress, Nancy (2009). Class field theory. New York: Springer. ISBN 978-0-387-72490-4. OCLC 310352143.
^ Bourbaki 1998, §VII.1
^ Bourbaki & Ch. VII, § 1, n. 7. Proposition 11. harvnb error: no target: CITEREFBourbakiCh._VII,_§_1,_n._7._Proposition_11. (help)
^ http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdffirstpage_1&handle=euclid.rmjm/1187453107
Stein, William, A Computational Introduction to Algebraic Number Theory (PDF)
Chapter 9 of Atiyah, Michael Francis; Macdonald, I.G. (1994), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8
Chapter VII.1 of Bourbaki, Nicolas (1998), Commutative algebra (2nd ed.), Springer Verlag, ISBN 3-540-64239-0
Chapter 11 of Matsumura, Hideyuki (1989), Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8 (2nd ed.), Cambridge University Press, ISBN 978-0-521-36764-6, MR 1011461
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DEFINE LIGHTYEAR AND POLESTAR - Science - Stars and the Solar System - 16911451 | Meritnation.com
DEFINE LIGHTYEAR AND POLESTAR.
Lightyear: It is the distance ravelled by light in 1 year. It is normally used to represent far-off distances of the celestial bodies.
1 Lightyear=ct=3×{10}^{8} ×365×24×60×60=9.461×{10}^{15}m
Polestar is the North star that lies closely with the rotation axis of the earth above its north pole. Pole star is the only star in the sky which appears stationary to an observer on Earth.
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Acid-Base Titrations - Course Hero
General Chemistry/Acid-Base Equilibria/Acid-Base Titrations
Acid-base indicators are weak acids or weak bases that can be used to detect changes in pH during a titration.
A titration is quantitative method that relies on measuring the volume of a solution of a known concentration necessary to neutralize a given volume of acid or base. In acid-base titration, strong acid or strong base is added to the unknown solution up to the point of neutralization. The concentration of acid or base in the unknown solution can then be calculated from the known amount of acid or base added.
An important aspect of acid-base titrations is being able to detect changes in pH as a volume of acid or base is being added to the solution of unknown concentration. An acid-base indicator is a weak acid or base that is different colors in the dissociated and nondissociated states. As the amount of dissociated acid or base increases, the solution changes color, making the change in pH visible.
Often an acid-base titration is used to determine the concentration of a solution of acid or base. This is done by adding acid or base of a known concentration (the titrant) to the unknown solution (the analyte). At the equivalence point all the acid or base molecules in the acidic or basic solution have been neutralized. At this point, the number of molecules added to the solution is stoichiometrically equal to the number originally present. In other words, if the ratio of acid to base in the balanced equation is 1:1, then the number of molecules added equals the number originally present. If the balanced equation calls for two molecules of acid to neutralize every molecule of base, the ratio of acid added to base neutralized will be 2:1.
Like the analyte, the indicator used in an acid-base titration also absorbs acid or base ions and will also reach an equivalence point. Ideally, this will occur at the pH at which the acid-base indicator changes color, called the end point.
Because the indicator is a weak acid or base, the end point can be expressed as an equilibrium:
K_{\rm a}=\frac{\lbrack\rm H^+\rbrack\lbrack\rm{Ind}^-\rbrack}{\lbrack\rm{HInd}\rbrack}
where [HInd] is the concentration of the indicator, a weak acid in this case, and [Ind–] is the concentration of its conjugate base. An acid-base indicator works because HInd and Ind– are different colors. The solution will be one color for most of the pH range below the end point and another color for most of the pH range above the end point. The color of the solution will change from the first to the second as the titration takes the pH through and past the end point.
The Henderson-Hasselbalch equation relates pH and pKa by the concentrations of the acid and its conjugate base. If
\frac{\lbrack\rm{Ind}^-\rbrack}{\lbrack\rm{HInd}\rbrack}
is greater than 1, then pH will be greater than pKa, and the solution will be more basic than before. If
\frac{\lbrack\rm{Ind}^-\rbrack}{\lbrack\rm{HInd}\rbrack}
is less than 1, then pH will be less than pKa, and the solution will be more acidic than before.
Many indicators will begin to change color when they get within one pH point of their end point and will continue to change until they are about one pH point past the end point.
Common Indicators and Their Colors
Each acid-base indicator has a specific pH and color range. Some indicators, such as thymol blue, undergo two color changes within the pH spectrum.
Titration curves show changes in pH as titrant is added to the analyte.
During titration, as more and more titrant is added to an analyte solution, the pH of the solution changes. A titration curve is a graph showing the change in pH of an analyte solution as titrant is added. Initially, the pH changes slowly, until the solution approaches its equivalence point. At that point the pH rises rapidly and then levels off.
Titration curves are similar, with slight differences, in different titration scenarios.
When a strong acid analyte, such as hydrochloric acid (HCl), is titrated against a strong base titrant, such as sodium hydroxide (NaOH), the titration curve will typically be steep.
Titration Curve for Strong Acid-Strong Base Titration
The titration of a strong acid analyte with a strong base titrant starts at low pH because only the strong acid is present. As base is added, the curve rises sharply when the OH- concentration increases. The final solution has a pH close to 14 because few H+ ions remain.
Since the hydrochloric acid (HCl) completely dissociates, the analyte solution initially contains a high concentration of hydrogen (H+) ions, giving it a low pH. The sodium hydroxide (NaOH) that is added also completely dissociates. As the titration begins, the hydroxide (OH–) ions entering solution will neutralize the H+ ions, and the pH slowly rises. Eventually, so many of the H+ ions are neutralized that the concentration of OH– ions begins to increase. At that point the pH rises sharply, and the slope of the curve becomes nearly vertical. The equivalence point occurs midway along this vertical portion of the curve, at pH 7. As more titrant is added, very few H+ ions remain, and the solution is dominated by OH– ions. Once this happens, the pH levels off at a high pH.
To calculate the pH after titrating an acid with a base, follow these steps:
1. Calculate the number of moles of titrant added. This is used to determine the amount of OH– ions that have been added and therefore the amount of H+ ions that have been neutralized.
2. Calculate the number of moles of H+ initially, and subtract the number of neutralized ions to find the number of moles remaining.
3. Calculate the total volume, and determine the concentration of protons (H+), which can be used to calculate pH.
Calculation of pH for a Known Amount of Titrant
Suppose 200.0 mL of 1.0 M HCl is titrated with 2.0 M NaOH. What is the pH when 80.0 mL of NaOH has been added?
Determine the moles of HCl present at the start of the titration.
(200.0\;\rm{mL}\;\rm{HCl}\;)\left(\frac{1\;\rm L}{1{,}000\;\rm{mL}}\right)\!\left(\frac{1.0\;\rm{mol}}{\rm L}\right)=0.20\;\rm{mol}\;\rm{HCl}
Determine the moles of NaOH that have been added.
(0.0800\;\rm L\;\rm{NaOH})\!\left(\frac{2.0\;\rm{mol}}{\rm L}\right)=0.16\;\rm{mol}\;\rm{NaOH}
Because HCl is a strong acid and NaOH is a strong base, assume all 0.16 mol NaOH reacts with 0.16 mol HCl, leaving 0.04 mol H3O+. The total volume is now 280.0 mL of solution containing 0.04 mol H3O+.
\left[{\rm H}_3\rm O^+\right]=\frac{0.04\;\rm{mol}}{0.2800\;\rm L}=\;0.14\;\rm M
\rm{pH}=-\log\left[{\rm H}_3\rm O^+\right]=-\log \;(0.14)=0.85
A similar approach can be used to calculate pH anywhere along the curve in any of the three titration scenarios.
When a weak base is titrated with a strong acid titrant, a slightly different curve forms. The initial pH is moderately high because the solution is basic but not strongly so. The presence of some undissociated base in the analyte solution will act as a buffer, able to accept the added protons. Therefore, although the pH falls, it does so more slowly than it would if the base were strong. Once nearing the equivalence point, the curve slopes steeply, and once again the equivalence point is in the middle of the steep region. The curve then levels out again at a low pH because the solution is acidic. Note that the solution at the equivalence point is not neutral but slightly acidic. This means the best indicator for this type of titration would be one that reaches its end point in slightly acidic conditions.
Titration Curve for Weak Base-Strong Acid Titration
The titration of a weak base analyte with strong acid titrant starts at a relatively high pH because only the weak base is present. As acid is added, the curve falls slowly at first and then falls steeply to a low pH.
When a weak acid is titrated with a strong base titrant, the initial pH is moderately low. The pH rises slowly. Near the equivalence point, the curve slopes steeply. It then levels out again at a high pH because the solution is basic. When a strong base is titrated with a weak acid titrant, the initial pH is very high, given the total dissociation of the base. Addition of the titrant lowers the pH slowly, until nearing the equivalence point. The curve then levels out at a moderately low pH because the solution is weakly acidic.
Titration Curve for Strong Base-Weak Acid Titration
The titration of a strong base analyte with a weak acid titrant starts at high pH because only the strong base is present. At first the decrease in pH is very slow because the base is completely dissociated. Once the base is neutralized, the curve falls steeply to a moderately low pH.
When a strong acid is titrated with a weak base, the shape of the curve is opposite the shape of a strong base titrated with a weak acid. In this case, the initial pH is very low, with total dissociation of the acid. Addition of the titrant slowly increases the pH, until near the equivalence point. The curve then levels out at a moderately high pH because the solution is weakly basic.
<Buffers>Suggested Reading
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Numerical Equivalence Testing - MATLAB & Simulink - MathWorks Benelux
Create Live Hardware Connection Object
GPU Acceleration or PIL Simulation with a Top Model
Create Mandelbrot Top Model
Configure the Model for GPU Acceleration
Run Normal and PIL Simulations
Test numerical equivalence between model components and production code that you generate from the components by using GPU acceleration and processor-in-the-loop (PIL) simulations.
With a GPU acceleration simulation, you test source code on your development computer. With a PIL simulation, you test the compiled object code that you intend to deploy on a target hardware by running the object code on real target hardware. To determine whether model components and generated code are numerically equivalent, compare GPU acceleration and PIL results to normal mode results.
Support communication between Simulink® and the target.
To produce a target connectivity configuration for hardware platforms such as NVIDIA® DRIVE and Jetson, install the MATLAB® Coder™ Support Package for NVIDIA Jetson® and NVIDIA DRIVE™ Platforms.
NVIDIA CUDA® Toolkit installed on the board.
The support package software uses an SSH connection over TCP/IP to execute commands while building and running the generated CUDA code on the DRIVE or Jetson platforms. Connect the target platform to the same network as the host computer or use an Ethernet crossover cable to connect the board directly to the host computer. For how to set up and configure your board, see NVIDIA documentation.
To communicate with the NVIDIA hardware, create a live hardware connection object by using the jetson (MATLAB Coder Support Package for NVIDIA Jetson and NVIDIA DRIVE Platforms) or drive (MATLAB Coder Support Package for NVIDIA Jetson and NVIDIA DRIVE Platforms) function. To create a live hardware connection object by using the function, provide the host name or IP address, user name, and password of the target board. For example, to create live object for Jetson hardware:
hwobj = jetson('192.168.1.15','ubuntu','ubuntu');
Fetching hardware details...
Fetching hardware details is now complete. Displaying details.
Board name : NVIDIA Jetson TX2
CUDA Version : 9.0
cuDNN Version : 7.0
TensorRT Version : 3.0
Available Webcams : UVC Camera (046d:0809)
Available GPUs : NVIDIA Tegra X2
hwobj = drive('92.168.1.16','nvidia','nvidia');
{z}_{k+1}={z}_{k}^{2}+{z}_{0},\text{ }k=0,\text{\hspace{0.17em}}1,\text{ }\text{\hspace{0.17em}}\dots
Test the generated model code by running a top-model PIL simulation. With this approach:
You can easily switch the top model between the normal, GPU acceleration, and PIL simulation modes.
model = 'mandelbrot_top';
Configure the input stimulus data. The following lines of code generate a 1000-by-1000 grid of real parts (x) and imaginary parts (y) between the limits specified by xlim and ylim.
[xG, yG] = meshgrid( x, y );
maxIterations = timeseries(500,0);
xGrid = timeseries(xG,0);
yGrid = timeseries(yG,0);
set_param(model, 'ExternalInput','maxIterations, xGrid, yGrid');
set_param(model,'GPUAcceleration','on');
count_normal = sim_output.yout{1}.Values.Data(:,:,1);
Run a top-model PIL simulation.
set_param(model,'SimulationMode','Processor-in-the-Loop (PIL)')
count_pil = sim_output.yout{1}.Values.Data(:,:,1);
### Target device has no native communication support.
Checking connectivity configuration registrations...
### Starting build procedure for: mandelbrot_top
### Generating code into build folder:
/mathworks/examples/sil_pil/mandelbrot_top_ert_rtw
### Generated code for 'mandelbrot_top' is up to date because no structural,
parameter or code replacement library changes were found.
### Using toolchain: NVCC for NVIDIA Embedded Processors
### '/mathworks/examples/sil_pil/mandelbrot_top_ert_rtw/mandelbrot_top.mk' is
### Building 'mandelbrot_top': make -f mandelbrot_top.mk buildobj
### Successful completion of build procedure for: mandelbrot_top
Model Action Rebuild Reason
mandelbrot_top Code compiled Compilation artifacts were out of date.
### Target device has no native communication support. Checking connectivity
configuration registrations...
### Connectivity configuration for component "mandelbrot_top": NVIDIA Jetson ###
PIL execution is using 30 Sec(s) for receive time-out.
### '/mathworks/examples/sil_pil/mandelbrot_top_ert_rtw/pil/mandelbrot_top.mk' is
### Building 'mandelbrot_top': make -f mandelbrot_top.mk all
### Starting application: 'mandelbrot_top_ert_rtw/pil/mandelbrot_top.elf'
### Launching application mandelbrot_top.elf...
Plot and compare the results of the normal and PIL simulations. Observe that the results match.
imagesc(x, y, count_normal);
title('Mandelbrot Set Normal Simulation');
imagesc(x, y, count_pil);
title('Mandelbrot Set PIL');
simResults = {'count_sil','count_normal','model'};
MAT-file logging is not supported for Processor-in-the-loop (PIL) simulation with GPU Coder™.
|
What is the least number of cuts that must be made in order to completely cut through this rope fence?
Note: The knots where multiple ropes meet are too thick to cut through. There is a beautiful way to solve this that is much more elegant than randomly attacking the fence with scissors.
Alice invited seven of her friends to a party. At the party, several pairs of people shook hands, although no one shook hands with themselves or shook hands with the same person more than once.
After the party, Alice asked each of her seven friends how many people they shook hands with during the party, and was surprised when they responded with seven distinct positive integers.
Given that her friends were truthful, how many hands did Alice shake?
The Queen plans to build a number of new cities in the wilderness, connected by a network of roads. She expects the annual tax revenues from each city to be numerically equal to the square of the population. Road maintenance will be expensive, though. The annual cost for each road is expected to be numerically equal to the product of the populations of the two cities that road connects. To keep the costs for road maintenance down, the Queen decrees that there should be only one way to get from any city to any other, and not more than three roads should connect to any given city (the attached design would be acceptable, for example).
Unfortunately, Parliament is responsible for deciding which roads will be built (subject to the two Queen's decrees) and how many people live in each city, but has no concern for attempting to stay financially solvent (meaning tax revenue
\geq
road maintenance). Regardless of what Parliament decides, what is the largest number of cities that the Queen can build without risking to lose money?
A graph is a collection of vertices (points) joined by edges (line segments). A pair of vertices are adjacent if they are connected by an edge.
Coloring a graph refers to assigning colors to its vertices. A graph is said to be properly colored if every pair of adjacent vertices receive different colors.
Consider the above graph with 4 vertices and 4 edges. This graph is denoted as
C_{4}
. If you are permitted to use 6 different colors, how many proper coloring does
C_{4}
have ?
1. Vertices are adjacent if they are joined by an edge (line).
2.The coloring that is in the figure is an example of proper coloring.
by Pramath Anamby
|
Knights and Formal Logic Practice Problems Online | Brilliant
Propositional logic was derived for the rigorous exploration of both philosophy and mathematics and has been used to study logic since the
3^\text{rd}
century BC. We want to teach you this language and show you how to apply it both when solving puzzles and, later in this course, to more serious applications in programming and AI.
First, go ahead and solve the puzzle below without using formal logic, then we'll show you how formal logic works and how you can use it to solve this puzzle and much more.
On the island of knights and knaves (where knights always tell the truth and knaves always lie), you meet two islanders named Penny and Quinru:
What are Penny and Quinru?
Penny is a knight, Quinru is a knight. Penny is a knight, Quinru is a knave. Penny is a knave, Quinru is a knight. Penny is a knave, Quinru is a knave.
Now let's dive into the world of formal propositional logic using the previous question as a gateway. Formal logic is a very powerful language for working with logic similarly to how algebra is a powerful language for working with numerical unknowns.
When translating a problem into formal logic, we give statements single-letter nicknames known as propositional variables. For example, we might refer to the statement "Penny is a knight." as P.
\text{ }
{\large\equiv}
is called a biconditional. You can think of it as meaning "if and only if." That is, P
\equiv
Q means "P is true if and only if Q is true."
Given P
\equiv
Q, and also knowing Q is false, what must be true about P?
P is true. P is false. P could be either true or false.
\neg
P is the negation of statement P. It's read as "NOT P."
Using "Penny is a knight" as P, what does
\neg
P indicate?
Penny is a knight. Penny is a knave. Quinru is a knight. Quinru is a knave.
Similarly to how
+
\times
are operators in algebra, AND, OR, and
\equiv
are operators in logic. Truth tables are simply a tabular organization of all of the cases: you fill them in left to right, and the left-most columns enumerate all possible true/false combinations of the propositional variables.
For example, if you have two statements, X, and Y, and each of them might be either true or false, then there are four possible cases to consider.
X Y X OR Y X
\equiv
\neg
Y X OR
\neg
X and Y can be any propositional statements
X AND Y is true only when both X and Y are true.
X OR Y is true when one or both of X and Y are true.
\neg
X is false when X is true and true when X is false.
\equiv
Y is true only when X and Y have the same truth values.
In which cases will "X OR
\neg
Y" be true?
When X is true and Y is true. When X is true and Y is false. When X is false and Y is true. When X is false and Y is false.
Suppose P is the statement "Penny is a knight" and Q is the statement "Quinru is a knight."
How would the statement "Both Quinru and Penny are knaves" be written in formal logic?
(Remember we're using
\neg
to stand for "NOT.")
\neg
\neg
\neg
\neg
\neg
Q P AND
\neg
Now, here's where it gets a bit tricky! On the island of knights and knaves, knights always speak the truth and knaves always lie. To translate this idea into formal logic, we say:
A statement spoken by an islander is true if and only if the person making the statement is a knight.
What formal logic expresses the scenario of Penny saying, "Quinru and I are knaves" in this puzzle?
P stands for the statement "Penny is a knight,"
Q stands for the statement "Quinru is a knight," and
\equiv
means "if and only if."
\equiv
\neg
\neg
\neg
\equiv
\neg
\neg
Q) P
\equiv
\neg
\neg
Now that we've written our statement in formal logic, let's solve the puzzle in a way that doesn't resort to exhaustively listing every case.
\equiv
\neg
\neg
We're going to attempt a proof by contradiction, where we assume something is true, then show it leads to an impossible scenario.
If we suppose that P is true,
\neg
\neg
Q (the right side of the biconditional) is also true.
\neg
\neg
Q is true, how many of the following statements must be true?
\neg
\neg
Neither are true Only one of them is true Both are true
In the prior problem, supposing P is true (that is, supposing P is a knight) led to these statements being true.
\neg
\neg
\neg
\neg
Notice the first and third lines: P and
\neg
P are true at the same time! That means Penny is both a knight and a knave, which can't happen! So our original assumption of P being true doesn't work: P can't be a knight and must be a knave instead.
Since Penny is a knave, we now know
\neg
P is true and P is false. Let's substitute in our original P
\equiv
\neg
\neg
Q statement:
\equiv
\neg
This line is enough to get: is Quinru a knight or a knave? (You already worked this out before, but try thinking it through just from the line above.)
Quinru is a knight Quinru is a knave
Formal propositional logic is in its essence a type of algebra that allows transformation of statements from one form to another. One important use is to systemize and simplify logic problems, and let us turn statements like
Penny says, "It's not true that: if I am a knave then Quinru is a knight."
into statements like
Penny says, "I am a knave and Quinru is a knave."
(Yes, these are equivalent! Which is easier to understand? Which would be easier to make using circuitry, or implement in computer code?)
It lets us handle situations where we might have tens or hundreds or even thousands of variables (perhaps referring to legal codes, or DNA strands), where a truth table would be impractical even by computer.
In this course, you'll use formal logic to solve some very challenging puzzles that are really only approachable using the logical tools you just learned, and we'll also go far beyond puzzles, applying formal logic both to create artificial intelligences and to define some of their limits.
One last puzzle!
The three treasure chests below each contain either gold coins or silver coins, but not both.
Given the facts below, are the coins in the third chest silver or gold?
1) The first and third chest do not contain the same type of coins.
2) The first chest contains gold coins if and only if the second chest contains silver coins and the third chest contains gold coins.
The third chest contains gold coins. The third chest contains silver coins. There isn't enough information to know.
|
Single-phase induction machine direct torque control - Simulink - MathWorks í•œêµ
{\mathrm{Ï}}_{a}=\left({v}_{a}â{i}_{a}{R}_{as}\right)\frac{{T}_{s}z}{zâ1}
{\mathrm{Ï}}_{b}=\left({v}_{b}â{i}_{b}{R}_{bs}\right)\frac{{T}_{s}z}{zâ1}
ψa and ψb are the main and auxiliary winding flux, respectively.
T=âp\left(a{\mathrm{Ï}}_{a}{i}_{b}â\frac{1}{a}{\mathrm{Ï}}_{b}{i}_{a}\right)
{\mathrm{Ï}}_{s}=\sqrt{{\mathrm{Ï}}_{a}{}^{2}+{\mathrm{Ï}}_{b}{}^{2}}
cψ, cT, S(θ)
cψ = 1 cT = 1 1, 0 1, 1 0, 1 0, 0
{\mathrm{Ï}}_{s}{}^{*}=\frac{2\mathrm{Ï}{f}_{n}{\mathrm{Ï}}_{n}}{p\mathrm{min}\left(|{\mathrm{Ï}}_{r}|,\frac{2\mathrm{Ï}{f}_{n}}{p}\right)}
ωr is the rotor angular mechanical speed in rad/s.
|
Axiom_of_power_set Knowpia
In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory.
The elements of the power set of the set {x, y, z} ordered with respect to inclusion.
{\displaystyle \forall x\,\exists y\,\forall z\,[z\in y\iff \forall w\,(w\in z\Rightarrow w\in x)]}
where y is the Power set of x,
{\displaystyle {\mathcal {P}}(x)}
Given any set x, there is a set
{\displaystyle {\mathcal {P}}(x)}
such that, given any set z, this set z is a member of
{\displaystyle {\mathcal {P}}(x)}
if and only if every element of z is also an element of x.
More succinctly: for every set
{\displaystyle x}
{\displaystyle {\mathcal {P}}(x)}
consisting precisely of the subsets of
{\displaystyle x}
Note the subset relation
{\displaystyle \subseteq }
is not used in the formal definition as subset is not a primitive relation in formal set theory; rather, subset is defined in terms of set membership,
{\displaystyle \in }
. By the axiom of extensionality, the set
{\displaystyle {\mathcal {P}}(x)}
The axiom of power set appears in most axiomatizations of set theory. It is generally considered uncontroversial, although constructive set theory prefers a weaker version to resolve concerns about predicativity.
The Power Set Axiom allows a simple definition of the Cartesian product of two sets
{\displaystyle X}
{\displaystyle Y}
{\displaystyle X\times Y=\{(x,y):x\in X\land y\in Y\}.}
{\displaystyle x,y\in X\cup Y}
{\displaystyle \{x\},\{x,y\}\in {\mathcal {P}}(X\cup Y)}
and, for example, considering a model using the Kuratowski ordered pair,
{\displaystyle (x,y)=\{\{x\},\{x,y\}\}\in {\mathcal {P}}({\mathcal {P}}(X\cup Y))}
and thus the Cartesian product is a set since
{\displaystyle X\times Y\subseteq {\mathcal {P}}({\mathcal {P}}(X\cup Y)).}
One may define the Cartesian product of any finite collection of sets recursively:
{\displaystyle X_{1}\times \cdots \times X_{n}=(X_{1}\times \cdots \times X_{n-1})\times X_{n}.}
Note that the existence of the Cartesian product can be proved without using the power set axiom, as in the case of the Kripke–Platek set theory.
This article incorporates material from Axiom of power set on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
|
Please explain this point - Physics - Dual nature of radiation and Matter - 16911021 | Meritnation.com
Please explain this point
the saturation current generated in photoelectric effect depends on intensity of light, not in the frequency of light. So if we change the frequency of incident light, without changing the intensity, then the saturation current remains the same.
now from Einstein's theory of photoelectric effect,
h\nu =\varphi +K.E. \left(electron\right)
now K.E. of the electron =
e{V}_{0}
Where V0 is the stopping potential
Now if frequency changes, the L.H.S. of the equation changes.
Now, workfunction of the material is the characteristic property of the material so it doesn't change with time.
Hence, the stopping potential changes with the frequency of incident light change.
|
Track error and NEES - MATLAB - MathWorks España
trackErrorMetrics
ErrorFunctionFormat
EstimationErrorLabels
EstimationErrorFcn
truthIDs
posRMSE
velRMSE
accRMSE
yawRateRMSE
posANEES
velANEES
accANEES
yawRateANEES
out1out2outN
Specific to trackErrorMetrics
Track error and NEES
The trackErrorMetrics System object™ provides quantitative comparisons between tracks and known truth trajectories.
Create the trackErrorMetrics object and set its properties.
errorMetrics = trackErrorMetrics(Name,Value)
errorMetrics = trackErrorMetrics creates a trackErrorMetrics System object with default property values.
errorMetrics = trackErrorMetrics(Name,Value) sets properties for the trackErrorMetrics object using one or more name-value pairs. For example, metrics = trackErrorMetrics('MotionModel','constvel') creates a trackErrorMetrics object with a constant velocity motion model. Enclose property names in quotes.
ErrorFunctionFormat — Error function format
Error function format specified as 'built-in' or 'custom'.
'built-in' – Enable the MotionModel property.
This property is a convenient interface when using tracks reported by any built-in multi-object tracker, and truths reported by the platformPoses object function of a trackingScenario object. The default estimation error function assumes tracks and truths are arrays of structures or arrays of objects.
'custom' – Enable custom properties: EstimationErrorLabels, EstimationErrorFcn, TruthIdentifierFcn, and TrackIdentifierFcns. These properties can be used to construct error functions for arbitrary tracks and truths input arrays.
EstimationErrorLabels — Labels for outputs of error estimation function
'posMSE' (default) | array of strings | cell array of character vectors
Labels for outputs of error estimation function, specified as an array of strings or cell array of character vectors. The number of labels must correspond to the number of outputs of the error estimation function. Specify the error estimation functions using the EstimationErrorFcn property.
Example: {'posMSE','velMSE'}
To enable this property, set the ErrorFunctionFormat property to 'custom'.
EstimationErrorFcn — Error estimation function
Error estimation function, specified as a function handle. The function determines estimation errors of truths to tracks.
The error estimation function can have multiple scalar outputs and must have the following syntax.
[out1,out2, ...,outN] = estimationerror(onetrack,onetruth)
The number of outputs must match the number of entries in the labels array specified in the EstimationErrorLabels property.
onetrack is an element of the tracks array passed in as input trackErrorMetric at object updates. onetruth is an element of the truths array passed in at object updates. The trackErrorMetrics object averages each output arithmetically when reporting across tracks or truths.
Example: @errorFunction
@trackIDFunction (default) | function handle
Track identifier function, specified as a function handle. Specifies the track identifiers for the tracks input at object update. The track identifiers are unique string or numeric values.
The track identifier function must have the following syntax:
trackID = trackIDentifier(tracks)
tracks is the same as the tracks array passed as input for trackErrorMetric at object update. trackID is the same size as tracks. The default identification function handle, @defaultTrackIdentifier, assumes tracks is an array of structures or objects with a 'TrackID' field name or property.
@truthIDFunction (default) | function handle
Truth identifier function, specified as a function handle. Specifies the truth identifiers for the truths input at object update. The truth identifiers are unique string or numeric values.
The truth identifier function must have the following syntax:
truthID = truthIDentifier(truths)
truths is the same as the truths array passed as input for trackErrorMetric updates. truthID must have the same size as truths. The default identification function handle, @defaultTruthIdentifier, assumes truths is an array or structures or objects with a 'PlatformID' field name or property.
To estimate errors, call the track error metrics object with arguments, as if it were a function (described here).
[posRMSE,velRMSE,posANEES,velANEES] = errorMetrics(tracks,trackIDs,truths,truthIDs)
[posRMSE,velRMSE,accRMSE,posANEES,velANEES,accANEES] = errorMetrics(tracks,trackIDs,truths,truthIDs)
[posRMSE,velRMSE,yawRateRMSE,posANEES,velANEES,yawRateANEES] = errorMetrics(tracks,trackIDs,truths,truthIDs)
[out1,out2, ... ,outN] = errorMetrics(tracks,trackIDs,truths,truthIDs)
[posRMSE,velRMSE,posANEES,velANEES] = errorMetrics(tracks,trackIDs,truths,truthIDs) returns the metrics
posRMSE – Position root mean squared error
velRMSE – Velocity root mean squared error
posANEES – Position average normalized-estimation error squared
velANEES – Velocity average normalized-estimation error squared
for constant velocity motion at the current time step. trackIDs is the set of track identifiers for all tracks. truthIDs is the set of truth identifiers. tracks are the set of tracks, and truths are the set of truths. trackIDs and truthIDs are each a vector whose corresponding elements match the track and truth identifiers found in tracks and truths, respectively.
The RMSE and ANEES values for different states are calculated by averaging the errors of all tracks at the current time step. For example, the position RMSE value, posRMSE, is defined as:
\text{posRMSE}=\sqrt{\frac{1}{M}\sum _{i=1}^{M}‖\Delta {p}_{i}{‖}^{2}}
where M is the total number of tracks with associated truth trajectories in the current time step, and
\Delta {p}_{i}={p}_{track,i}-{p}_{truth,i}
is the position difference between the position of track i, ptrack,i, and the position of the corresponding truth, ptruth,i, at the current time step. The RMSE values for other states (vel, pos, acc, and yawRate) are defined similarly.
The position ANEES value, posANEES, is defined as:
\text{posANEES}=\frac{1}{M}\sum _{i=1}^{M}\Delta {p}_{i}{}^{T}{C}_{p,i}^{-1}\Delta {p}_{i}
where Cp,i is the covariance matrix corresponding to the position of track i at the current time step. The ANEES values for other states (vel, pos, acc, and yawRate) are defined similarly.
To enable this syntax, set the ErrorFunctionFormat property to 'built-in' and the MotionModel property to 'constvel'.
[posRMSE,velRMSE,accRMSE,posANEES,velANEES,accANEES] = errorMetrics(tracks,trackIDs,truths,truthIDs) also returns the metrics
accRMS – Acceleration root mean squared error
accANEES – acceleration average normalized-estimation error squared
for constant acceleration motion at the current time step.
To enable this syntax, set the ErrorFunctionFormat property to 'built-in' and the MotionModel property to 'constacc'.
[posRMSE,velRMSE,yawRateRMSE,posANEES,velANEES,yawRateANEES] = errorMetrics(tracks,trackIDs,truths,truthIDs) also returns the metrics
yawRateRMSE – yaw rate root mean squared error
yawRateANEES – yaw rate average normalized-estimation error squared
for constant turn-rate motion at the current time step.
To enable this syntax, set the ErrorFunctionFormat property to 'built-in' and the MotionModel property to 'constturn'.
[out1,out2, ... ,outN] = errorMetrics(tracks,trackIDs,truths,truthIDs) returns the user-defined metrics out1, out2, ... , outN.
To enable this syntax, set the ErrorFunctionFormat property to 'custom'. The number of outputs corresponds to the number of elements listed in the EstimationErrorLabels property, and must match the number of outputs in the EstimationErrorFcn. The results of the estimation errors are averaged arithmetically over all track-to-truth assignments.
These usage syntaxes only calculate the RMSE and ANEES values of all tracks with associated truths at the current time step. To obtain the cumulative RMSE and ANEES values for each track and truth, use the cumulativeTrackMetrics and cumulativeTruthMetrics object functions, respectively. To obtain the current RMSE and ANEES values for each track and truth, use the currentTrackMetrics and currentTruthMetrics object functions, respectively.
Track information, specified as an array of structures or objects. For built-in trackers such as trackerGNN or trackerTOMHT, the objectTrack output contains 'State', 'StateCovariance', and 'TrackID' information.
trackIDs — Track identifiers
Track identifiers, specified as a real-valued vector. trackIDs elements match the tracks found in tracks.
Truth information, specified as an array of structures or objects. When using a trackingScenario, truth information can be obtained from the platformPoses object function.
truthIDs — Truth identifiers
Truth identifiers, specified as a real-valued vector. truthIDs elements match the truths found in truths.
Position root mean squared error for all tracks associated with truths, returned as a scalar.
To enable this argument, set the ErrorFunctionFormat property to 'built-in'.
Velocity root mean squared error for all tracks associated with truths, returned as a scalar.
accRMSE — Acceleration root mean squared error
Acceleration root mean squared error for all tracks associated with truths, returned as a scalar.
Yaw rate root mean squared error for all tracks associated with truths, returned as a scalar.
posANEES — Position average normalized estimation error squared
Position average normalized estimation error squared for all tracks associated with truths, returned as a scalar.
velANEES — Velocity average normalized estimation error squared
Velocity average normalized estimation error squared for all tracks associated with truths, returned as a scalar.
accANEES — Acceleration average normalized estimation error squared
Acceleration average normalized estimation error squared for all tracks associated with truths, returned as a scalar.
yawRateANEES — Yaw rate average normalized estimation error squared
Yaw rate average normalized estimation error squared for all tracks associated with truths, returned as a scalar.
out1, out2, outN — Custom error metric outputs
Custom error metric outputs, returned as scalars. These errors are the output of the error estimation function specified in the EstimationErrorFcn property.
To enable these arguments, set the ErrorFunctionFormat property to 'custom'.
cumulativeTrackMetrics Cumulative metrics for recent tracks
cumulativeTruthMetrics Cumulative metrics for recent truths
currentTrackMetrics Metrics for recent tracks
currentTruthMetrics Metrics for recent truths
fusionRadarSensor | trackerGNN | trackerJPDA | trackerTOMHT | trackerPHD | trackAssignmentMetrics | trackOSPAMetric
|
Moving channel propagation conditions - MATLAB lteMovingChannel - MathWorks í•œêµ
\mathrm{Î}\mathrm{Ï}=\frac{A}{2}\left(1+\mathrm{sin}\left(\mathrm{Î}\mathrm{Ï}\left(t+{t}_{0}\right)\right)\right)
{t}_{0}=InitTime+\frac{3\mathrm{Ï}}{2\left(\mathrm{Î}\mathrm{Ï}\right)}
If model.InitTime is 0, the delay of the first multipath component is 0. If t = 0,
\mathrm{Î}\mathrm{Ï}=0
. Relative delay between all multipath components is fixed.
\left[0,{2}^{31} –\text{ }1\text{ }–\text{ }K\left(K-1\right)/2\right],
Where K = P × model.NRxAnts, the product of the number of transmit and receive antennas. Seed values outside of this recommended range should be avoided as they may result in random sequences that repeat results produced using Seed values inside the recommended range.
 NTerms Optional
 ModelType Optional
 NormalizePathGains Optional
[1] 3GPP TS 36.104. “Evolved Universal Terrestrial Radio Access (E-UTRA); Base Station (BS) Radio Transmission and Reception.†3rd Generation Partnership Project; Technical Specification Group Radio Access Network. URL: https://www.3gpp.org.
[2] Dent, P., G. E. Bottomley, and T. Croft. “Jakes Fading Model Revisited.†Electronics Letters. Vol. 29, 1993, Number 13, pp. 1162–1163.
[3] Pätzold, Matthias, Cheng-Xiang Wang, and Bjørn Olav Hogstad. “Two New Sum-of-Sinusoids-Based Methods for the Efficient Generation of Multiple Uncorrelated Rayleigh Fading Waveforms.†IEEE Transactions on Wireless Communications. Vol. 8, 2009, Number 6, pp. 3122–3131.
|
Classical Mechanics Problem: Entangled or Not 2 - Matt DeCross | Brilliant
Entangled or Not 2
|\Psi\rangle = \frac{1}{\sqrt{5}}|\uparrow\uparrow\rangle +\frac{\sqrt{2}}{\sqrt{5}} |\downarrow\uparrow\rangle - \frac{\sqrt{2}}{\sqrt{5}} |\downarrow\downarrow\rangle
|\Psi\rangle
above entangled?
|\uparrow\downarrow\rangle = |\uparrow\rangle \otimes |\downarrow\rangle
is a common shorthand for tensor products of spin states.
|
On the prediction of human intelligence from neuroimaging: A systematic review of methods and reporting
tags: behavior; fMRI; resting-state; deep learning; intelligence; prediction; systematic review
B.H. Vieira, G.S.P. Pamplona, K. Fachinello, A.K. Silva, M.P. Foss, C.E.G. Salmon, Intelligence, 2022-05-12
Open Access 10.1016/j.intell.2022.101654
This paper stems from my Doctoral thesis, which was completed in November 2021. Human intelligence differences are a frequente object of study in neuroscience, particularly in neuroimaging studies. Along the years, the application of machine learning to the prediction of traits, e.g., behavior, phenotypes and demographics, became a widespread methodology to the study of neural bases of interindividual differences. From the earliest applications, the prediction of intelligence differences was already often included. Today, there is a large corpus of studies that corroborate that this prediction is, in fact, possible, with moderate levels of accuracy.
In this work, we recapitulate all the literature on the prediction of human intelligence differences from brain imaging. Specifically, we performed a preregistered systematic search with adaptations, retrieving 37 documents that fulfilled inclusion criteria. Using TRIPOD, we assessed the reporting quality across studies. Using PROBAST, we evaluated risk of bias and concerns regarding applicability in each set of results from each study. Results with low risk of bias and low concerns regarding applicability were selected for meta-analytic synthesis. Due to the number of studies, we performed meta-analysis only on papers using fMRI. We show that, on average, results on general intelligence are significantly higher than results on the prediction of fluid intelligence, which hints towards the moderating effect of measurement quality, which has been reported in other instances[1]. We follow the PRISMA reporting guidelines throughout the paper.
We describe several methodological aspects of the studies, starting from the earliest one, in 2008, such as datasets, sample sizes, machine learning models employed, evaluation of performance. This application of machine learning to cognition demonstrates exponential growth. For this reason, we believe that the time for this review is oportune, and we hope to foment better practices in the field, as well as foundations for future reviews and neurobiological theories of intelligence differences.
Reviews and meta-analyses have proved to be fundamental to establish neuroscientific theories on intelligence. The prediction of intelligence using in-vivo neuroimaging data and machine learning has become a widely accepted and replicated result. We present a systematic review of this growing area of research, based on studies that employ structural, functional, and/or diffusion MRI to predict intelligence in cognitively normal subjects using machine learning. We systematically assessed methodological and reporting quality using the PROBAST and TRIPOD in 37 studies. We observed that fMRI is the most employed modality, resting-state functional connectivity is the most studied predictor. A meta-analysis revealed a significant difference between the performance obtained in the prediction of general and fluid intelligence from fMRI data, confirming that the quality of measurement moderates this association. Studies predicting general intelligence from Human Connectome Project fMRI averaged
r = 0.42 \ (\text{CI}_\text{95\%} = [0.35, 0.50])
r=0.42 (CI95%=[0.35,0.50]) while studies predicting fluid intelligence averaged
r = 0.15 \ (\text{CI}_\text{95\%} = [0.13, 0.17])
r=0.15 (CI95%=[0.13,0.17]). We identified virtues and pitfalls in the methods for the assessment of intelligence and machine learning. The lack of treatment of confounder variables and small sample sizes were two common occurrences in the literature which increased risk of bias. Reporting quality was fair across studies, although reporting of results and discussion could be vastly improved. We conclude that the current literature on the prediction of intelligence from neuroimaging data is reaching maturity. Performance has been reliably demonstrated, although extending findings to new populations is imperative. Current results could be used by future works to foment new theories on the biological basis of intelligence differences.
[1] Gignac, G. E., & Bates, T. C. (2017). Brain volume and intelligence: The moderating role of intelligence measurement quality. Intelligence, 64(May), 18–29. https://doi.org/10.1016/j.intell.2017.06.004.
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LinearSystemPlotTutor - Maple Help
Home : Support : Online Help : Education : Student Packages : Linear Algebra : Interactive : LinearSystemPlotTutor
Student[LinearAlgebra][LinearSystemPlotTutor] - plot a system of 2-D or 3-D linear equations
LinearSystemPlotTutor(M)
(optional) Matrix; augmented matrix corresponding to linear system
The LinearSystemPlotTutor(M) command graphically displays the linear system corresponding to the Matrix M.
The Matrix can have at most 4 rows and 4 columns. If M is not provided, a default Matrix is used. In either case, you can edit the entries of the Matrix interactively from within the Maplet application.
When LinearSystemPlotTutor is running, interaction with the worksheet is not possible.
\mathrm{with}\left(\mathrm{Student}[\mathrm{LinearAlgebra}]\right):
M≔〈〈2,-2,-18,-52〉|〈0,-3,-18,-51〉|〈-4,-6,-20,-52〉|〈-10,-11,-24,-55〉〉
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-4}& \textcolor[rgb]{0,0,1}{-10}\\ \textcolor[rgb]{0,0,1}{-2}& \textcolor[rgb]{0,0,1}{-3}& \textcolor[rgb]{0,0,1}{-6}& \textcolor[rgb]{0,0,1}{-11}\\ \textcolor[rgb]{0,0,1}{-18}& \textcolor[rgb]{0,0,1}{-18}& \textcolor[rgb]{0,0,1}{-20}& \textcolor[rgb]{0,0,1}{-24}\\ \textcolor[rgb]{0,0,1}{-52}& \textcolor[rgb]{0,0,1}{-51}& \textcolor[rgb]{0,0,1}{-52}& \textcolor[rgb]{0,0,1}{-55}\end{array}]
\mathrm{LinearSystemPlotTutor}\left(M\right)
Student[LinearAlgebra], Student[LinearAlgebra][EigenPlotTutor], Student[LinearAlgebra][LinearSystemPlot]
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Einstein–de Haas effect - Wikipedia
The Einstein–de Haas effect is a physical phenomenon in which a change in the magnetic moment of a free body causes this body to rotate. The effect is a consequence of the conservation of angular momentum. It is strong enough to be observable in ferromagnetic materials. The experimental observation and accurate measurement of the effect demonstrated that the phenomenon of magnetization is caused by the alignment (polarization) of the angular momenta of the electrons in the material along the axis of magnetization. These measurements also allow the separation of the two contributions to the magnetization: that which is associated with the spin and with the orbital motion of the electrons. The effect also demonstrated the close relation between the notions of angular momentum in classical and in quantum physics.
The effect was predicted[1] by O. W. Richardson in 1908. It is named after Albert Einstein and Wander Johannes de Haas, who published two papers[2][3] in 1915 claiming the first experimental observation of the effect.
4 Literature about the effect and its discovery
5 Later measurements and applications
The orbital motion of an electron (or any charged particle) around a certain axis produces a magnetic dipole with the magnetic moment of
{\displaystyle {\boldsymbol {\mu }}=e/2m\cdot \mathbf {j} ,}
wher{\displaystyle e}
{\displaystyle m}
are the charge and the mass of the particle, while
{\displaystyle \mathbf {j} }
is the angular momentum of the motion ( SI units are used). In contrast, the intrinsic magnetic moment of the electron is related to its intrinsic angular momentum (spin) as
{\displaystyle {\boldsymbol {\mu }}\approx {}2\cdot {}e/2m\cdot \mathbf {j} }
(see Landé g-factor and anomalous magnetic dipole moment).
If a number of electrons in a unit volume of the material have a total orbital angular momentum of
{\displaystyle \mathbf {J_{o}} }
with respect to a certain axis, their magnetic moments would produce the magnetization of
{\displaystyle \mathbf {M_{o}} =e/2m\cdot \mathbf {J_{o}} }
. For the spin contribution the relation would be
{\displaystyle \mathbf {M_{s}} \approx e/m\cdot \mathbf {J_{s}} }
. A change in magnetization,
{\displaystyle \Delta \mathbf {M} ,}
implies a proportional change in the angular momentum,
{\displaystyle \Delta \mathbf {J} \propto {}\Delta \mathbf {M} ,}
of the electrons involved. Provided that there is no external torque along the magnetization axis applied to the body in the process, the rest of the body (practically all its mass) should acquire an angular momentum
{\displaystyle -\Delta \mathbf {J} }
due to the law of conservation of angular momentum.
The experiments involve a cylinder of a ferromagnetic material suspended with the aid of a thin string inside a cylindrical coil which is used to provide an axial magnetic field that magnetizes the cylinder along its axis. A change in the electric current in the coil changes the magnetic field the coil produces, which changes the magnetization of the ferromagnetic cylinder and, due to the effect described, its angular momentum. A change in the angular momentum causes a change in the rotational speed of the cylinder, monitored using optical devices. The external field
{\displaystyle \mathbf {B} }
interacting with a magnetic dipole
{\displaystyle {\boldsymbol {\mu }}}
cannot produce any torque (
{\displaystyle {\boldsymbol {\tau }}={\boldsymbol {\mu }}\times \mathbf {B} }
) along the field direction. In these experiments the magnetization happens along the direction of the field produced by the magnetizing coil, therefore, in absence of other external fields, the angular momentum along this axis must be conserved.
In spite of the simplicity of such a layout, the experiments are not easy. The magnetization can be measured accurately with the help of a pickup coil around the cylinder, but the associated change in the angular momentum is small. Furthermore, the ambient magnetic fields, such as the Earth field, can provide a 107 - 108 times larger[4] mechanical impact on the magnetized cylinder. The later accurate experiments were done in a specially constructed demagnetized environment with active compensation of the ambient fields. The measurement methods typically use the properties of the torsion pendulum, providing periodic current to the magnetization coil at frequencies close to the pendulum's resonance.[2][4] The experiments measure directly the ratio:
{\displaystyle \lambda =\Delta \mathbf {J} /\Delta \mathbf {M} }
and derive the dimensionless gyromagnetic factor
{\displaystyle g'}
of the material from the definition:
{\displaystyle g'\equiv {}{\frac {2m}{e}}{\frac {1}{\lambda }}}
{\displaystyle \gamma \equiv {\frac {1}{\lambda }}\equiv {\frac {e}{2m}}g'}
is called gyromagnetic ratio.
The expected effect and a possible experimental approach was first described by Owen Willans Richardson in a paper[1] published in 1908. The electron spin was discovered in 1925, therefore only the orbital motion of electrons was considered before that. Richardson derived the expected relation of
{\displaystyle \mathbf {M} =e/2m\cdot \mathbf {J} }
. The paper mentioned the ongoing attempts to observe the effect at Princeton.
In that historical context the idea of the orbital motion of electrons in atoms contradicted classical physics. This contradiction was addressed in the Bohr model in 1913, and later was removed with the development of quantum mechanics.
S.J. Barnett, motivated by the Richardson's paper realized that the opposite effect should also happen - a change in rotation should cause a magnetization (the Barnett effect). He published[5] the idea in 1909, after which he pursued the experimental studies of the effect.
Einstein and de Haas published two papers[2][3] in April 1915 containing a description of the expected effect and the experimental results. In the paper "Experimental proof of the existence of Ampere's molecular currents"[3] they described in details the experimental apparatus and the measurements performed. Their result for the ratio of the angular momentum of the sample to its magnetic moment (the authors called it
{\displaystyle \lambda }
) was very close (within 3%) to the expected value of
{\displaystyle 2m/e}
. It was realized later that their result with the quoted uncertainty of 10% was not consistent with the correct value which is close to
{\displaystyle m/e}
. Apparently, the authors underestimated the experimental uncertainties.
S.J. Barnett reported the results of his measurements at several scientific conferences in 1914. In October 1915 he published the first observation of the Barnett effect in a paper[6] titled "Magnetization by Rotation". His result for
{\displaystyle \lambda }
was close to the right value of
{\displaystyle m/e}
, which was unexpected at that time.
In 1918 J.Q. Stewart published[7] the results of his measurements confirming the Barnett's result. In his paper he was calling the phenomenon 'The Richardson effect'.
The following experiments demonstrated that the gyromagnetic ratio for iron is indeed close to
{\displaystyle e/m}
{\displaystyle e/2m}
. This phenomenon, dubbed "gyromagnetic anomaly" was finally explained after the discovery of the spin and introduction of the Dirac equation in 1928.
Literature about the effect and its discovery[edit]
Detailed accounts of the historical context and the explanations of the effect can be found in literature[8][9] Commenting on the papers by Einstein, Calaprice in The Einstein Almanac writes:[10]
52. "Experimental Proof of Ampère's Molecular Currents" (Experimenteller Nachweis der Ampereschen Molekularströme) (with Wander J. de Hass). Deutsche Physikalische Gesellschaft, Verhandlungen 17 (1915): 152-170.
Considering Ampère's hypothesis that magnetism is caused by the microscopic circular motions of electric charges, the authors proposed a design to test Lorentz's theory that the rotating particles are electrons. The aim of the experiment was to measure the torque generated by a reversal of the magnetisation of an iron cylinder.
Calaprice further writes:
53. "Experimental Proof of the Existence of Ampère's Molecular Currents" (with Wander J. de Haas) (in English). Koninklijke Akademie van Wetenschappen te Amsterdam, Proceedings 18 (1915-16).
Einstein wrote three papers with Wander J. de Haas on experimental work they did together on Ampère's molecular currents, known as the Einstein–De Haas effect. He immediately wrote a correction to paper 52 (above) when Dutch physicist H. A. Lorentz pointed out an error. In addition to the two papers above [that is 52 and 53] Einstein and de Haas cowrote a "Comment" on paper 53 later in the year for the same journal. This topic was only indirectly related to Einstein's interest in physics, but, as he wrote to his friend Michele Besso, "In my old age I am developing a passion for experimentation."
The second paper by Einstein and de Haas[3] was communicated to the "Proceedings of the Royal Netherlands Academy of Arts and Sciences" by Hendrik Lorentz who was the father-in-law of de Haas. According to Frenkel[8] Einstein wrote in a report to the German Physical Society: "In the past three months I have performed experiments jointly with de Haas–Lorentz in the Imperial Physicotechnical Institute that have firmly established the existence of Ampère molecular currents." Probably, he attributed the hyphenated name to de Haas, not meaning both de Haas and H. A. Lorentz.
Later measurements and applications[edit]
The effect was used to measure the properties of various ferromagnetic elements and alloys.[4] The key to more accurate measurements was better magnetic shielding, while the methods were essentially similar to those of the first experiments. The experiments measure the value of the g-factor
{\displaystyle g'={\frac {2m}{e}}{\frac {M}{J}}}
(here we use the projections of the pseudovectors
{\displaystyle \mathbf {M} }
{\displaystyle \mathbf {J} }
onto the magnetization axis and omit the
{\displaystyle \Delta }
sign). The magnetization and the angular momentum consist of the contributions from the spin and the orbital angular momentum:
{\displaystyle M=M_{s}+M_{o}}
{\displaystyle J=J_{s}+J_{o}}
Using the known relations
{\displaystyle M_{o}={\frac {e}{2m}}J_{o}}
{\displaystyle M_{s}=g\cdot {}{\frac {e}{2m}}J_{s}}
{\displaystyle g\approx {}2.002}
is the g-factor for the anomalous magnetic moment of the electron, one can derive the relative spin contribution to magnetization as:
{\displaystyle {\frac {M_{s}}{M}}={\frac {(g'-1)g}{(g-1)g'}}}
For pure iron the measured value is
{\displaystyle g'=1.919\pm {}0.002}
,[11] and
{\displaystyle {\frac {M_{s}}{M}}\approx {}0.96}
. Therefore, in pure iron 96% of the magnetization is provided by the polarization of the electrons' spins, while the remaining 4% is provided by the polarization of their orbital angular momenta.
^ a b Richardson, O. W. (1908). "A Mechanical Effect Accompanying Magnetization". Physical Review. Series I. 26 (3): 248–253. Bibcode:1908PhRvI..26..248R. doi:10.1103/PhysRevSeriesI.26.248.
^ a b c Einstein, A.; de Haas, W. J. (1915). "Experimenteller Nachweis der Ampereschen Molekularströme" [Experimental Proof of Ampère's Molecular Currents]. Deutsche Physikalische Gesellschaft, Verhandlungen (in German). 17: 152–170.
^ a b c d Einstein, A.; de Haas, W. J. (1915). "Experimental proof of the existence of Ampère's molecular currents" (PDF). Koninklijke Akademie van Wetenschappen te Amsterdam, Proceedings. 18: 696–711. Bibcode:1915KNAB...18..696E.
^ a b c Scott, G. G. (1962). "Review of Gyromagnetic Ratio Experiments". Reviews of Modern Physics. 34 (1): 102–109. Bibcode:1962RvMP...34..102S. doi:10.1103/RevModPhys.34.102.
^ Barnett, S. J. (1908). "On Magnetization by Angular Acceleration". Science. 30 (769): 413. Bibcode:1909Sci....30..413B. doi:10.1126/science.30.769.413. PMID 17800024.
^ Barnett, S. J. (1915). "Magnetization by Rotation". Physical Review. 6 (4): 239–270. Bibcode:1915PhRv....6..239B. doi:10.1103/PhysRev.6.239.
^ Stewart, J. Q. (1918). "The Moment of Momentum Accompanying Magnetic Moment in Iron and Nickel". Physical Review. 11 (2): 100–270. Bibcode:1918PhRv...11..100S. doi:10.1103/PhysRev.11.100.
^ a b Frenkel, Viktor Ya. (1979). "On the history of the Einstein–de Haas effect". Soviet Physics Uspekhi. 22 (7): 580–587. doi:10.1070/PU1979v022n07ABEH005587.
^ David R Topper (2007). Quirky sides of scientists: true tales of ingenuity and error from physics and astronomy. Springer. p. 11. ISBN 978-0-387-71018-1.
^ Alice Calaprice, The Einstein Almanac (Johns Hopkins University Press, Baltimore, 2005), p. 45. ISBN 0-8018-8021-1
^ Reck, R. A.; Fry, D. L. (1969). "Orbital and Spin Magnetization in Fe-Co, Fe-Ni, and Ni-Co". Physical Review. 184 (2): 492–495. Bibcode:1969PhRv..184..492R. doi:10.1103/PhysRev.184.492.
"Einsteins's only experiment" [1] (links to a directory of the Home Page of Physikalisch-Technische Bundesanstalt (PTB), Germany [2]). Here is a replica to be seen of the original apparatus on which the Einstein–de Haas experiment was carried out.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Einstein–de_Haas_effect&oldid=1082235596"
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Youth Scouts - MetaSoccer
Each Youth Scout has specific skills. The characteristics that make up a Youth Scout are as follows:
The age of the Youth Scout. New Youth Scouts joining your team will start between the ages of 25 and 30.
Youth Scouts will retire at age 65 and may not serve in any additional capacity.
Is the club to which a Youth Scout belongs.
Mentorships done
Mentoring is the ability of the scouts to find and form new Youth Scouts that will join your club. This attribute represents the number of mentorships that a Youth Scout has done to date. You can see more information in the Mentoring section:
Expertise has a direct consequence on the position of the players a scout will encounter during a Scouting activity. It is divided into three main categories, Role, Specific Role and Player Special Ability.
A Youth Scout can have four different roles:
Players found by Scouts will have the same role, ensuring that all those players have optimized and higher attributes from the scouts’ knowledge.
There are 17 specific roles that correspond to the players’ primary positions on the field, namely:
Right Wing-Back (RWB)
Central Defensive Midfielder (CDM)
Some scouts may have a specific role, in addition to the primary role. If the scout has any specific role, there is a 100% probability that at least one player from Scouting will have the same specific role.
Player Special ability
They can also be experts in finding players with a Special Ability, in this case, enhancing the search for players who have that special ability. There are 17 available special Abilities, detailed in the Players section.
Scout Special abilities
Some scouts can also be experts in some domains that greatly improve their performance as a scout.
These Scout Special Skills are:
Motivational leader: Accelerate the improving speed of any scout working with him. Impact: +10% pace in improving for all scouts working with him. (maxim +50% if more than one leader)
This ability will be inactive until a new learning feature will be in place.
Extra motivated: Impact: +50% in his improving when learning.
Mentoring star: Accelerate the mentoring process. Impact: +50% pace in mentoring process.
Master of relations: With the contact network, accelerates the scouting process. Impact: -50% time in the scouting process.
Youth specialist: He can detect younger players. Impact: 50% probability of finding players from 14 to 15 age.
Scouting expert: 20% extra probability of getting special abilities in found players.
Developed players: find players who are closer to the maximum potential (60%-70% of their potential).
The knowledge of a scout has a direct impact on the characteristics of the players he will be able to find in the Scouting.
It is divided into five different areas, each with its own value, that is also connected to the role of the scout, namely:
Defending: With a value between 1 and 99
Attacking: With a value between 1 and 99
Goalkeeping: With a value between 1 and 99
Physical: With a value between 1 and 99
Mental: With a value between 1 and 99
For every scout role, some knowledge skills are more valuable than others:
The overall knowledge is obtained through a formula that takes into account his four main knowledge areas:
OverallKnowledge Skill = 0.7 * K_1+0.2*K_2+0.05*K_3+0.05*K_4
K_1, K_2, K_3
K_4
are first, second, third, and fourth knowledge.
There are 6 different ranges of overall knowledge:
The range in which a scout is gives a general overview of how good he is at finding players, and has a direct impact on the Scouting, changing its price, number of players found and their potential abilities. Scouts with different overalls will have different results on their scouting. Depending on the Scout Overall, a scout receives a different number of players. Also, the player found will have a potential related to the Overall of the Scout. If the Scout has an Overall of 65, players found will have a potential close to this value.
Knowledge will improve after each Scouting. You can view more information about Scouting by clicking below.
Youth scouts will improve their knowledge skills after Scouting, according to the importance of the knowledge to their role.
The learning process will be implemented in the future and continuously improved to make it easier.
Game Assets - Previous
Next - Game Assets
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\mathrm{c1}=A\left(a\right)
\mathrm{c2}=B\left(b\right)
be two nonconcentric circles and let I and E divide SensedMagnitude(AB) internally and externally in the ratio a/b. Then I and E are called the internal and the external centers of similitude of the two circles c1 and c2
\mathrm{with}\left(\mathrm{geometry}\right):
\mathrm{_EnvHorizontalName}≔'x':
\mathrm{_EnvVerticalName}≔'y':
\mathrm{circle}\left(\mathrm{c1},{x}^{2}+{y}^{2}=1\right),\mathrm{circle}\left(\mathrm{c2},[\mathrm{point}\left(A,3,3\right),4]\right):
\mathrm{circle}\left(\mathrm{c3},{\left(x-2\right)}^{2}+{y}^{2}=1\right):
\mathrm{similitude}\left(\mathrm{obj1},\mathrm{c1},\mathrm{c2}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{in_similitude_of_c1_c2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{ex_similitude_of_c1_c2}}]
\mathrm{map}\left(\mathrm{coordinates},\mathrm{obj1}\right)
[[\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{5}}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{-1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-1}]]
\mathrm{similitude}\left(\mathrm{obj2},\mathrm{c1},\mathrm{c3},[M,N]\right)
[\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{N}]
\mathrm{detail}\left(\mathrm{obj2}\right)
[\begin{array}{ll}\textcolor[rgb]{0,0,1}{\text{name of the object}}& \textcolor[rgb]{0,0,1}{M}\\ \textcolor[rgb]{0,0,1}{\text{form of the object}}& \textcolor[rgb]{0,0,1}{\mathrm{point2d}}\\ \textcolor[rgb]{0,0,1}{\text{coordinates of the point}}& [\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]\end{array}\textcolor[rgb]{0,0,1}{,}\begin{array}{ll}\textcolor[rgb]{0,0,1}{\text{name of the object}}& \textcolor[rgb]{0,0,1}{N}\\ \textcolor[rgb]{0,0,1}{\text{form of the object}}& \textcolor[rgb]{0,0,1}{\mathrm{point2d}}\\ \textcolor[rgb]{0,0,1}{\text{coordinates of the point}}& [\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]\end{array}]
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Chemical Kinetics - Vocabulary - Course Hero
General Chemistry/Chemical Kinetics/Vocabulary
intermediate configuration of atoms during a chemical reaction with high potential energy
equation that relates temperature to the rate constant; given as
k=Ae^{-\frac{E_a}{RT}}
elementary reaction that occurs between two reactants
field that studies the rates of chemical reactions
theory that describes how chemical reactions occur through molecular collisions and why reaction rates vary between reactions
reaction that takes place in a single step
reaction in which the rate is dependent on the concentration of one of the reactants, raised to the power of one; given as a rate law with the form
r=k\lbrack\rm{A}\rbrack
catalyst that is in a different phase than that of the reactants
catalyst that is in the same phase as that of the reactants and is present in the reaction medium
rate of a chemical reaction at a particular moment
rate law that defines concentration as a function of time
species that is produced in one step and consumed in another step of a chemical reaction
number of molecules that participate in the rate-determining step of a chemical reaction
constant relating reaction rate to concentrations of reactants
mathematical expression that relates rate with a rate constant and concentrations of the reactants; given as the general form
r=k\lbrack{\rm{A}}\rbrack^m\lbrack{\rm{B}}\rbrack^n
slowest reaction in a multistep reaction that determines the overall rate
reaction half-life
time it takes for a reactant to drop to half its starting concentration during a chemical reaction
exact step or steps required to convert reactants into products
sum of the powers that the reactant concentrations are raised to in the rate expression
speed at which a reaction occurs
reaction in which the sum of the powers that the reactant concentrations are raised to in the rate expression is equal to two; given as a rate law with the form
r=k\lbrack{\rm A}\rbrack\lbrack{\rm B}\rbrack
r=k\lbrack{\rm A}\rbrack^2
elementary reaction that occurs between three reactants
elementary reaction with one reactant
reaction in which rate is not dependent on reactant concentrations; given as a rate law with the form
r=k
<Overview>Reaction Rates
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Home : Support : Online Help : Education : Student Packages : Statistics : Mean
Mean(A, numeric_option, output_option)
Mean(M, numeric_option, output_option)
Mean(X, numeric_option, output_option, inert_option)
The Mean function computes and/or plots the arithmetic mean of the specified random variable or data set. This is the same as the expected value of the random variable. The same command can be obtained as ExpectedValue.
The first parameter can be a data sample (e.g., a Vector), a Matrix data set, a random variable, or an algebraic expression involving random variables (see Student[Statistics][RandomVariable]).
If the option output is not included or is specified to be output=value, then the function will return the value of the mean. If output=plot is specified, then the function will return a plot of the input data set and its mean. If output=both is specified, then both the value and the plot of the mean will be returned.
By default, the mean is computed according to the rules mentioned above. To always compute the mean numerically, specify the numeric or numeric = true option.
\mathrm{with}\left(\mathrm{Student}[\mathrm{Statistics}]\right):
Compute the mean of data containing floating point values. This leads to a floating point result.
\mathrm{Mean}\left([2,4,4.0]\right)
\textcolor[rgb]{0,0,1}{3.333333333}
\mathrm{Mean}\left(\mathrm{Vector}[\mathrm{column}]\left([\mathrm{sqrt}\left(14.0\right),\mathrm{\pi },33]\right)\right)
\textcolor[rgb]{0,0,1}{13.29441668}
Compute the mean of data not containing any floating point values. This leads to an exact result.
\mathrm{Mean}\left([2,4,4]\right)
\frac{\textcolor[rgb]{0,0,1}{10}}{\textcolor[rgb]{0,0,1}{3}}
\mathrm{Mean}\left([100,20,\mathrm{\pi }]\right)
\textcolor[rgb]{0,0,1}{40}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}}{\textcolor[rgb]{0,0,1}{3}}
\mathrm{Mean}\left(\mathrm{Vector}[\mathrm{row}]\left([\mathrm{sqrt}\left(2\right),3,\mathrm{\pi }]\right)\right)
\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\frac{\sqrt{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}}{\textcolor[rgb]{0,0,1}{3}}
p
q
\mathrm{Mean}\left(\mathrm{BetaRandomVariable}\left(p,q\right)\right)
\frac{\textcolor[rgb]{0,0,1}{p}}{\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{q}}
\mathrm{Mean}\left(\mathrm{BetaRandomVariable}\left(3,5\right)\right)
\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{8}}
\mathrm{Mean}\left(\mathrm{BetaRandomVariable}\left(3,5\right),\mathrm{numeric}\right)
\textcolor[rgb]{0,0,1}{0.3750000000}
\mathrm{Mean}\left(\mathrm{BetaRandomVariable}\left(3,5\right),\mathrm{inert}\right)
{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }}_{\textcolor[rgb]{0,0,1}{0}}^{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{105}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_t}}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{}{\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{_t}}\right)}^{\textcolor[rgb]{0,0,1}{4}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{ⅆ}\textcolor[rgb]{0,0,1}{\mathrm{_t}}
\mathrm{evalf}\left(\mathrm{Mean}\left(\mathrm{BetaRandomVariable}\left(3,5\right),\mathrm{inert}\right)\right)
\textcolor[rgb]{0,0,1}{0.3750000000}
\mathrm{Mean}\left(\mathrm{BetaRandomVariable}\left(3,5\right),\mathrm{output}=\mathrm{plot}\right)
Compute the mean of
x
y
z
\mathrm{Mean}\left([x,y,z]\right)
\frac{\textcolor[rgb]{0,0,1}{x}}{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{y}}{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{z}}{\textcolor[rgb]{0,0,1}{3}}
Consider the following Matrix data sample with entries that have floating point values.
M≔\mathrm{Matrix}\left([[2.0,7.5,10,18],[3,5\mathrm{ln}\left(2\right),1,\mathrm{\pi }],[4,2,7,4]]\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{2.0}& \textcolor[rgb]{0,0,1}{7.5}& \textcolor[rgb]{0,0,1}{10}& \textcolor[rgb]{0,0,1}{18}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\right)& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{\mathrm{\pi }}\\ \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{7}& \textcolor[rgb]{0,0,1}{4}\end{array}]
We compute the mean of each column according to the computation rules. (If a column has floating point values, then a floating point value will be given for that column. Otherwise, it will result in an exact expression.)
\mathrm{Mean}\left(M\right)
[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{3.000000000}& \textcolor[rgb]{0,0,1}{4.321911968}& \textcolor[rgb]{0,0,1}{6}& \frac{\textcolor[rgb]{0,0,1}{22}}{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}}{\textcolor[rgb]{0,0,1}{3}}\end{array}]
Using the command ExpectedValue will give the same result.
\mathrm{ExpectedValue}\left(M\right)
[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{3.000000000}& \textcolor[rgb]{0,0,1}{4.321911968}& \textcolor[rgb]{0,0,1}{6}& \frac{\textcolor[rgb]{0,0,1}{22}}{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}}{\textcolor[rgb]{0,0,1}{3}}\end{array}]
If the numeric option is included, then a floating point value will be given independently of the presence of floating point numbers in the input.
\mathrm{Mean}\left([1,2,3,4],\mathrm{numeric}\right)
\textcolor[rgb]{0,0,1}{2.50000000000000}
If the output=both option is included, then both the value of the mean and its plot will be returned.
\mathrm{mean},\mathrm{graph}≔\mathrm{Mean}\left([1,2,3,4],\mathrm{numeric},\mathrm{output}=\mathrm{both}\right)
\textcolor[rgb]{0,0,1}{\mathrm{mean}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{graph}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{2.50000000000000}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{PLOT}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{...}}\right)
\mathrm{mean}
\textcolor[rgb]{0,0,1}{2.50000000000000}
\mathrm{graph}
The Student[Statistics][Mean] and Student[Statistics][ExpectedValue] commands were introduced in Maple 18.
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find the area of the trapezoid by composing into - OnlineMathe - das mathe-forum
Startseite » Forum » find the area of the trapezoid by composing into
dear tutors and supporters:
I am attaching the problemm.
I draw a line and I can see I have a rectangle and a triangle now.
is this a good way to find the area of this trapezoid?
I am going to post some work here and see if it is good
I am gonna find the area of the triangle first.
the hypotenuse and a side
hypothenuse=
10
a\phantom{\rule{0.12em}{0ex}}=7
b\phantom{\rule{0.12em}{0ex}}=
{c\phantom{\rule{0.12em}{0ex}}}^{2}={a\phantom{\rule{0.12em}{0ex}}}^{2}+{b\phantom{\rule{0.12em}{0ex}}}^{2}
{10}^{2}={7}^{2}+{b\phantom{\rule{0.12em}{0ex}}}^{2}
100=49+{b\phantom{\rule{0.12em}{0ex}}}^{2}
100-49=49-49+{b\phantom{\rule{0.12em}{0ex}}}^{2}
51={b\phantom{\rule{0.12em}{0ex}}}^{2}
√51=√b
b\phantom{\rule{0.12em}{0ex}}=7.14
this is the area of the triangle
now I will find the area of the rectangle and add the two together. is this right?
Decomposing the trapezoid into a rectangle and a triangle, as you did, is certainly a good way to determine the total area.
Unfortunately, the creator of the problem has made a serious mistake, because he has given too many dimensions. The side of your triangle, which you called
b\phantom{\rule{0.12em}{0ex}},
6c\phantom{\rule{0.12em}{0ex}}m\phantom{\rule{0.12em}{0ex}}
(difference of the lengths
18c\phantom{\rule{0.12em}{0ex}}m\phantom{\rule{0.12em}{0ex}}
12c\phantom{\rule{0.12em}{0ex}}m\phantom{\rule{0.12em}{0ex}}\right)
. From this we can calculate the length of the hypotenuse with
\sqrt{85}c\phantom{\rule{0.12em}{0ex}}m\phantom{\rule{0.12em}{0ex}}\approx 9.22c\phantom{\rule{0.12em}{0ex}}m\phantom{\rule{0.12em}{0ex}}
(and not
10
cm).
So depending on which of the four dimensions given you omit in your calculation you will arrive at a different total area.
I guess its best to omit the
10
cm which leads to a total area of
105c\phantom{\rule{0.12em}{0ex}}{m\phantom{\rule{0.12em}{0ex}}}^{2}
The picture has a mistake
if the lower side is 12cm and the upper side 18cm the difference, your side
b\phantom{\rule{0.12em}{0ex}}
should be 6cm but your calculation with Pythagoras for
b\phantom{\rule{0.12em}{0ex}}
is also right .
b\phantom{\rule{0.12em}{0ex}}
the area is not
b\phantom{\rule{0.12em}{0ex}}
like you suggest but
\frac{a\phantom{\rule{0.12em}{0ex}}\cdot b\phantom{\rule{0.12em}{0ex}}}{2}
so if the 18cm are right it would be 6*7cm^2 and then the rectangle with Pythagoras you find 7*7,14cm^2
so either the 18cm or the
10
cm are wrong,
calculating with the formula for trapezoids you get the area with b=6cm
but I thought the hypothenuse side
c\phantom{\rule{0.12em}{0ex}}
=10cm. why is it not the hypothenuse? I'm in doubt here.
The side you call
c\phantom{\rule{0.12em}{0ex}}
IS the hypotenuse, but if you rely on the given measures
18c\phantom{\rule{0.12em}{0ex}}m\phantom{\rule{0.12em}{0ex}},7c\phantom{\rule{0.12em}{0ex}}m\phantom{\rule{0.12em}{0ex}}
12c\phantom{\rule{0.12em}{0ex}}m\phantom{\rule{0.12em}{0ex}},
its length can't be
10c\phantom{\rule{0.12em}{0ex}}m\phantom{\rule{0.12em}{0ex}}
So EITHER of the four given length
7c\phantom{\rule{0.12em}{0ex}}m\phantom{\rule{0.12em}{0ex}},10c\phantom{\rule{0.12em}{0ex}}m\phantom{\rule{0.12em}{0ex}},12c\phantom{\rule{0.12em}{0ex}}m\phantom{\rule{0.12em}{0ex}}
18c\phantom{\rule{0.12em}{0ex}}m\phantom{\rule{0.12em}{0ex}}
must be omitted and it seems that its best to omit the
10
Verständigungs-Stütze:
I get it now. I solved for
c\phantom{\rule{0.12em}{0ex}}
and got
9.22
cm like you.
okay, now
i\phantom{\rule{0.12em}{0ex}}
c\phantom{\rule{0.12em}{0ex}}=9.22
a\phantom{\rule{0.12em}{0ex}}=7
b\phantom{\rule{0.12em}{0ex}}=6
I am adding all this up to find the area
9.22+7+6
=22.22
cm^2
A\phantom{\rule{0.12em}{0ex}}=L\phantom{\rule{0.12em}{0ex}}\cdot W\phantom{\rule{0.12em}{0ex}}
=7\cdot 12
=84
22.22+84
(cm^2)
Area of the whole trapezoid
=106.22
thanks ledum as well. If I Did not reply directly to you was becasue your comment was similar to Roman
>
=22.22
Forget about the side
c\phantom{\rule{0.12em}{0ex}}!
You don't need it. Look at the sketch I attached at the end of my last post.
Since you have given both cathets, you can simply calculate the area with
\frac{6c\phantom{\rule{0.12em}{0ex}}m\phantom{\rule{0.12em}{0ex}}\cdot 7c\phantom{\rule{0.12em}{0ex}}m\phantom{\rule{0.12em}{0ex}}}{2}=21c\phantom{\rule{0.12em}{0ex}}{m\phantom{\rule{0.12em}{0ex}}}^{2}
BTW, another way to get the total area of the trapezoid would be to duplicate the whole trapezoid and compose a rectangle from the two pieces with the side lenghts 7 cm and
30c\phantom{\rule{0.12em}{0ex}}m\phantom{\rule{0.12em}{0ex}}\left(=12c\phantom{\rule{0.12em}{0ex}}m\phantom{\rule{0.12em}{0ex}}+18c\phantom{\rule{0.12em}{0ex}}m\phantom{\rule{0.12em}{0ex}}\right)
You sure can calculate the area of this rectangular and divide it by 2 ;-)
oh okay, I see it.
A\phantom{\rule{0.12em}{0ex}}=\frac{b\phantom{\rule{0.12em}{0ex}}\cdot h\phantom{\rule{0.12em}{0ex}}}{2}
A\phantom{\rule{0.12em}{0ex}}=21
now to find the area of the rectangle
A\phantom{\rule{0.12em}{0ex}}=l\phantom{\rule{0.12em}{0ex}}\cdot w\phantom{\rule{0.12em}{0ex}}
A\phantom{\rule{0.12em}{0ex}}=12\cdot 7
A\phantom{\rule{0.12em}{0ex}}=44
so, The trapezoid's are is
=21
+44
=65
thanks, Nuele, too.
thanks, N8eule!.
>
Now your area of the rectangle is wrong. But you already knew the correct result of 7 times
12
17:22
Combining the correct results should given you the
105c\phantom{\rule{0.12em}{0ex}}{m\phantom{\rule{0.12em}{0ex}}}^{2}
already mentioned above.
BTW, if you type a space between "c" and "m" when typing "7 cm^2" in text mode, you get the desired
7c\phantom{\rule{0.12em}{0ex}}{m\phantom{\rule{0.12em}{0ex}}}^{2}
I like that way best!. thanks.
Roman you said the area of the rectangle is wrong. can you explain why? I did not get that.
i\phantom{\rule{0.12em}{0ex}}
see it. it is the wrong calculation
7\cdot 12=84c\phantom{\rule{0.12em}{0ex}}{m\phantom{\rule{0.12em}{0ex}}}^{2}
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Difference between revisions of "Complexity Garden" - Complexity Zoo
Difference between revisions of "Complexity Garden"
Hbarnum (talk | contribs)
m (→Permanent: What is a 0-1 matrix's permanent: minor grammar fix)
(→Table of Contents: Added isomorphism problems and algebraic problems as categories)
[[#ksat|k-SAT]] -
[[#unique-ksat|Unique <math>k</math>-SAT]]
''Isomorphism problems:''
[[#graph_automorphism|Graph Automorphism]] -
[[#graph_isomorphism|Graph Isomorphism]] -
[[#ra|Ring Automorphism]] -
[[#ri|Ring Isomorphism]]
''Algebraic problems:''
[[#ri|Ring Isomorphism]] -
[[#approximate_shortest_lattice_vector|Approximate Shortest Lattice Vector]] -
[[#group_nonmembership|Group Nonmembership]] -
[[#integer_factorization|Integer Factorization]] -
[[#integer_factor|Integer Factor ≤ k]] -
[[#integer_multiplication|Integer Multiplication]] -
[[#matrix_multiplication|Matrix Multiplication]] -
[[#permanent|Permanent]] -
[[#square_root|Square Root mod n]]
''Sets and partitions:''
''Uncategorized problems:''
[[#boolean_matrix_multiplication|Boolean Matrix Multiplication]] -
[[#3sum|3SUM]] -
[[#boolean_convolution|Boolean Convolution]] -
[[#boolean_sorting|Boolean Sorting]] -
[[#discrete_logarithm|Discrete Logarithm]] -
[[#equality|Equality]] -
[[#k-local-ham|k-Local Hamiltonians]] -
[[#k-round_sorting|k-Round Sorting]] -
[[#linear_programming|Linear Programming]] -
[[#majority|Majority]] -
[[#parity|Parity]] -
[[#shortest_implicant|Short Implicant]] -
[[#stochastic_games|Stochastic Games]] -
[[#tautology|Tautology]] -
[[#square_root|Square Root mod n]] -
[[#threshold(k)|Threshold(k)]]
Satisfiability problems: #SAT - #WSAT - Constraint Satisfaction - QBF - Quantum k-SAT - SAT - k-SAT - Unique
{\displaystyle k}
-SAT
Given a Boolean formula, count the number of satisfying assignments of Hamming weight
{\displaystyle k}
. This is the canonical problem for #WT.
3SUM: Do there exist members of a list satisfying
{\displaystyle a+b+c=0}
Given a list
{\displaystyle X=\{x_{i}\}_{i=1}^{n}}
of integers, do there exist elements
{\displaystyle a,b,c\in X}
{\displaystyle a+b+c=0}
? This problem is important enough to computational geometry that there is defined a class of problems to which it is reducible: 3SUM-hard.
Memberships: ∈ NP ∩ coNP,
{\displaystyle \notin }
Compute the convolution of two n-bit integers in binary notation
{\displaystyle \sum _{i+j=k}x_{i}\cdot y_{j}}
, where the multiplicands are respectively the i-1 and j-1 bits of the two integer inputs, and k is the k-1 bit of the output. It is a Boolean function with n input bits and 2n-1 output bits.
It has monotone circuit complexity
{\displaystyle \Omega (n^{3/2})}
(Weiss) and nonmonotone circuit complexity
{\displaystyle n\log n2^{O(\log ^{*}n)}}
(Furer [Fur07]).
{\displaystyle \log ^{*}(n)}
means iteratively taking
{\displaystyle \log }
of n until the result is less than 2. Nonmonotone circuits use the discrete Fourier transform (Wegener [Weg87]).
It has monotone circuit complexity of exactly
{\displaystyle 2n^{3}-n^{2}}
(Pratt [Pra74]), but its nonmonotone circuit complexity is the same as matrix multiplication, presently
{\displaystyle O(n^{2.376})}
(Wegener [Weg87]).
{\displaystyle \Theta (n\log n)}
(Lamagna and Savage [LS74]) and nonmonotone circuit complexity
{\displaystyle \Theta (n)}
(Muller and Preparata [MP75]). The nonmonotone circuits use binary rather than unary addition to count the 0 inputs.
Clique-Like is a Boolean function with
{\displaystyle n^{2}}
inputs (an adjacency matrix) and one output.
It has monotone circuit complexity of
{\displaystyle \Omega \left(e^{cn^{1/6-o(1)}}\right)}
(Tardos [Tar88]). By contrast it has polynomial nonmonotone circuit complexity, because it reduces to Linear Programming.
Given two sets of relations
{\displaystyle I,T}
{\displaystyle I}
is the instance (not to be confused with an instance of CSP) and
{\displaystyle T}
is the template, over the same vocabulary, where the vocabulary defines the names and arities of allowed relations, is there a mapping
{\displaystyle h}
such that for all relations
{\displaystyle R(x_{1},x_{2},\dots ,x_{k})\in I}
{\displaystyle R\left(h(x_{1},x_{2},\dots ,x_{k})\right)\in T}
The discrete logarithm problem is to solve for x in the equation ax = b in some number-theoretic abelian group, typically either the group of units of a finite field or an elliptic curve over a finite field. Like the related Integer Factorization, the fastest classical algorithm is the number field sieve, with heuristic time complexity
{\displaystyle 2^{O(n^{1/3}(\log n)^{2/3})}}
. Also like Integer Factorization, Shor's algorithm solves Discrete Logarithm in quantum polynomial time. In fact, Shor's algorithm solves Discrete Logarithm even in a black-box abelian group, provided that group elements have unique names.
If Alice has a string
{\displaystyle x\in \left\{0,1\right\}^{n}}
and Bob has a string
{\displaystyle y\in \left\{0,1\right\}^{n}}
, define EQUALITY(x, y) = 1 if and only if x = y.
Luks showed that Graph Isomorphism for bounded-valence graphs is in P non-uniformly (the exponent of algorithm's running time depends on the bound). Combining Luks' algorithm with a trick due to Zemlyachenko yields a time complexity upper bound of
{\displaystyle 2^{O({\sqrt {v\log v}})}}
for graphs with v vertices. However, some practical Graph Isomorphism algorithms, such as NAUTY, seem to run much faster than this rigorous upper bound.
Defined by [Wat00]. Let
{\displaystyle G}
be a group, whose elements are represented by polynomial-size strings. We're given a "black box" that correctly multiplies and inverts elements of
{\displaystyle G}
. Then given elements
{\displaystyle g\in G}
{\displaystyle h_{1},\dots ,h_{k}\in G}
, we can asked to find if
{\displaystyle g\notin \left\langle h_{1},\dots ,h_{k}\right\rangle }
(the subgroup generated by
{\displaystyle h_{1},\dots ,h_{k}\in G}
{\displaystyle G}
, exists one Hamiltonian circuit in
{\displaystyle G}
Algorithms: The fastest known algorithm for integer factorization is the number field sieve. It has heuristic randomized time complexity
{\displaystyle 2^{O(n^{1/3}(\log n)^{2/3})}}
for inputs with n digits.
Algorithms: The fastest algorithm is either the number field sieve or Lenstra's elliptic curve method, depending on the relative size of n and k. Lenstra's algorithm has heuristic randomized time complexity
{\displaystyle \mathrm {poly} (n)2^{O({\sqrt {k(\log k))}})}}
if n and k have n and k digits, respectively.
Given an n-qubit Hilbert space, as well as a collection H1,...,Hm of Hamiltonians (i.e. Hermitian positive semidefinite matrices), each of which acts on at most k qubits of the space. Also given real numbers a,b such that
{\displaystyle b-a\in \Theta (1/{\mathsf {poly}}(n))}
. Decide whether the smallest eigenvalue of
{\displaystyle H=\sum _{i=1}^{m}H_{i}}
is less than a or greater than b, promised that one of these is the case.
{\displaystyle \in }
P. Majority
{\displaystyle \notin }
REG. Therefore, REG
{\displaystyle \subsetneq }
Matrix Multiplication: Multiply two
{\displaystyle n\times n}
Multiply two dense
{\displaystyle n\times n}
{\displaystyle F}
. Matrix Multiplication has O-speedup among Strassen-type bilinear algorithms [CW82]. Determining the minimal number of multiplications needed to compute a bilinear form (of which Matrix Multiplication is one) is NP-complete ([Has90]). This suggests that Matrix Multiplication is P-nonuniform over bilinear algorithms if NP
{\displaystyle \neq }
coNP.
If the group-theoretic algorithms of Cohn et al can perform Matrix Multiplication in
{\displaystyle O(n^{2})}
, then Matrix Multiplication has O-speedup among algorithms of the type they consider [CKSU05].
{\displaystyle \in }
P. Parity
{\displaystyle \notin }
FO ([Ajt83] and [FSS84]). Therefore, FO
{\displaystyle \subsetneq }
{\displaystyle n^{\Omega (\log n)}}
(Razborov [Raz85b]) but polynomial nonmonotone circuit complexity.
The permanent of an n-by-n 0-1 matrix (ai,j) is defined as
{\displaystyle \sum _{\sigma }\prod _{i=1}^{n}a_{i,\sigma (i)}}
where σ ranges over all permutations of the numbers 1, 2, ..., n. The value of the permanent is equivalent to #Perfect Matching
Note that [Pap94] calls this problem QSAT to emphasize its relationship to SAT. In particular, any instance of QBF can be written as
{\displaystyle \exists x_{1}\forall x_{2}\cdots Q_{n}x_{n}\phi (x_{1},x_{2},\dots ,x_{n})}
{\displaystyle \phi }
is a Boolean formula as in SAT, and where
{\displaystyle Q_{n}}
is either a universal or existential qualifier, depending on whether
{\displaystyle n}
is even or odd. This characterization lends itself well as a candidate for reductions from two-player games.
Given a universe
{\displaystyle {\mathcal {U}}}
{\displaystyle {\mathcal {S}}}
{\displaystyle {\mathcal {U}}}
, a cover is a subfamily
{\displaystyle {\mathcal {C}}\subseteq {\mathcal {S}}}
of sets whose union is
{\displaystyle {\mathcal {U}}}
. In the set covering decision problem, the input is a pair
{\displaystyle ({\mathcal {U}},{\mathcal {S}})}
{\displaystyle k}
; the question is whether there is a set covering of size
{\displaystyle k}
or less. In the set covering optimization problem, the input is a pair
{\displaystyle ({\mathcal {U}},{\mathcal {S}})}
, and the task is to find a set covering that uses the fewest sets.
Algorithms: A simple greedy algorithm yields an
{\displaystyle H(l)}
-approximation (and hence,
{\displaystyle (\ln(l)+1)}
-approximation)where
{\displaystyle l}
is the size of largest subset in
{\displaystyle {\mathcal {S}}}
{\displaystyle H(l)}
{\displaystyle l}
th Harmonic number.
Set covering cannot be approximated in polynomial time to within a factor of
{\displaystyle {\bigl (}1-o(1){\bigr )}\cdot \ln {n}}
, unless NP has quasi-polynomial time algorithms [Fie98]. In addition, Set covering cannot be approximated in polynomial time to within a factor of
{\displaystyle c\cdot \ln {n}}
{\displaystyle c}
is a constant, unless P
{\displaystyle =}
NP. The largest value of
{\displaystyle c}
is proved in [AMS06].
{\displaystyle k}
-SAT: Satisfiability with clauses of length
{\displaystyle k}
{\displaystyle k}
, an instance of
{\displaystyle k}
-SAT is an instance of SAT in conjunctive normal form (CNF) where all clauses are the logical OR of
{\displaystyle k}
Boolean variables.
{\displaystyle \notin }
Properties: p-speedup? (If so then NP
{\displaystyle \neq }
coNP)
Boolean Sorting is equivalent to the n Threshold(k) functions in reverse order. Majority is equivalent to Threshold(
{\displaystyle \lceil n/2\rceil }
The Unique
{\displaystyle k}
-SAT problem is a promise problem variant of
{\displaystyle k}
-SAT, where the promise is that each instance has either no satisfying assignment, or has exactly one satisfying assignment.
Intuitively, adding this promise restricts us to only the hardest instances, as the more satisfying assignments there are, the easier it should be to find one. This intuition is justified by [CIK+03], where it is also shown that if Unique
{\displaystyle k}
-SAT can be solved in deterministic time
{\displaystyle O(2^{\epsilon n})}
{\displaystyle \epsilon >0}
, then so can
{\displaystyle k}
-SAT for all
{\displaystyle k\geq 3}
|
The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form
\mathrm{div}\phantom{\rule{0.166667em}{0ex}}a\left(\nabla u\right)+F\left[u\right]\left(x\right)=0,
over the functions
u\in {W}^{1,1}\left(\Omega \right)
that assume given boundary values
\phi
\partial \Omega .
a:{ℝ}^{n}\to {ℝ}^{n}
satisfies an ellipticity condition and for a fixed
x,F\left[u\right]\left(x\right)
denotes a non-linear functional of
u.
In considering the same problem, Hartman and Stampacchia [Acta Math. 115 (1966) 271-310] have obtained existence results in the space of uniformly Lipschitz continuous functions when
\phi
satisfies the classical bounded slope condition. In a variational context, Clarke [Ann. Sc. Norm. Super. Pisa Cl. Sci. 4 (2005) 511-530] has introduced a new type of hypothesis on the boundary condition
\phi :
the lower (or upper) bounded slope condition. This condition, which is less restrictive than the previous one, is satisfied if
\phi
is the restriction to
\partial \Omega
of a convex function. We show that if
and
F
satisfy hypotheses similar to those of Hartman and Stampacchia, the lower bounded slope condition implies the existence of solutions in the space of locally Lipschitz continuous functions on
\Omega .
Mots clés : non-linear elliptic PDE's, Lipschitz continuous solutions, lower bounded slope condition
author = {Bousquet, Pierre},
title = {Local {Lipschitz} continuity of solutions of non-linear elliptic differential-functional equations},
AU - Bousquet, Pierre
TI - Local Lipschitz continuity of solutions of non-linear elliptic differential-functional equations
Bousquet, Pierre. Local Lipschitz continuity of solutions of non-linear elliptic differential-functional equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 4, pp. 707-716. doi : 10.1051/cocv:2007035. http://www.numdam.org/articles/10.1051/cocv:2007035/
[1] P. Bousquet, The lower bounded slope condition. J. Convex Anal. 14 (2007) 119-136. | Zbl 1132.49031
[2] P. Bousquet and F. Clarke, Local Lipschitz continuity of solutions to a problem in the calculus of variations. J. Differ. Eq. (to appear). | MR 2371797 | Zbl 1141.49033
[3] F. Clarke, Continuity of solutions to a basic problem in the calculus of variations. Ann. Sc. Norm. Super. Pisa Cl. Sci. 4 (2005) 511-530. | Numdam | Zbl 1127.49001
[4] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Classics in Mathematics, Springer-Verlag, Berlin (2001). Reprint of the 1998 edition. | MR 1814364 | Zbl 1042.35002
[5] P. Hartman, On the bounded slope condition. Pacific J. Math. 18 (1966) 495-511. | Zbl 0149.32001
[6] P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations. Acta Math. 115 (1966) 271-310. | Zbl 0142.38102
[7] G.M. Lieberman, The quasilinear Dirichlet problem with decreased regularity at the boundary. Comm. Partial Differential Equations 6 (1981) 437-497. | Zbl 0458.35039
[8] G.M. Lieberman, The Dirichlet problem for quasilinear elliptic equations with Hölder continuous boundary values. Arch. Rational Mech. Anal. 79 (1982) 305-323. | Zbl 0497.35010
[9] G.M. Lieberman, The Dirichlet problem for quasilinear elliptic equations with continuously differentiable boundary data. Comm. Partial Differential Equations 11 (1986) 167-229. | Zbl 0589.35036
[10] M. Miranda, Un teorema di esistenza e unicità per il problema dell’area minima in
variabili. Ann. Scuola Norm. Sup. Pisa 19 (1965) 233-249. | Numdam | Zbl 0137.08201
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Tomáš Mrkvička — 2017
Obálkové metody představují populární nástroj pro testování hypotéz o vhodnosti statistického modelu. Tyto testy graficky porovnávají funkci
T:I\to ℝ
vypočtenou ze statistických dat s jejím protějškem získaným simulacemi. Chyba prvního druhu
\alpha
, tj. pravděpodobnost zamítnutí platné hypotézy, je obvykle kontrolována pouze pro fixní hodnotu
r\in I
, zatímco funkce
T
je definována na intervalu hodnot
I
. V tomto článku představíme nový globální obálkový test, který umožňuje kontrolovat chybu prvního druhu současně...
On testing of general random closed set model hypothesis
A new method of testing the random closed set model hypothesis (for example: the Boolean model hypothesis) for a stationary random closed set
\Xi \subseteq {ℝ}^{d}
with values in the extended convex ring is introduced. The method is based on the summary statistics – normalized intrinsic volumes densities of the
\epsilon
-parallel sets to
\Xi
. The estimated summary statistics are compared with theirs envelopes produced from simulations of the model given by the tested hypothesis. The p-level of the test is then computed via approximation...
Estimation of intersection intensity in a Poisson process of segments
The minimum variance unbiased estimator of the intensity of intersections is found for stationary Poisson process of segments with parameterized distribution of primary grain with known and unknown parameters. The minimum variance unbiased estimators are compared with commonly used estimators.
Estimation variances for parameterized marked Poisson processes and for parameterized Poisson segment processes
A complete and sufficient statistic is found for stationary marked Poisson processes with a parametric distribution of marks. Then this statistic is used to derive the uniformly best unbiased estimator for the length density of a Poisson or Cox segment process with a parametric primary grain distribution. It is the number of segments with reference point within the sampling window divided by the window volume and multiplied by the uniformly best unbiased estimator of the mean segment length.
On estimation of intrinsic volume densities of stationary random closed sets via parallel sets in the plane
Tomáš Mrkvička; Jan Rataj — 2009
A method of estimation of intrinsic volume densities for stationary random closed sets in
{ℝ}^{d}
based on estimating volumes of tiny collars has been introduced in T. Mrkvička and J. Rataj, On estimation of intrinsic volume densities of stationary random closed sets, Stoch. Proc. Appl. 118 (2008), 2, 213-231. In this note, a stronger asymptotic consistency is proved in dimension 2. The implementation of the method is discussed in detail. An important step is the determination of dilation radii in the...
On the Bayesian estimation for the stationary Neyman-Scott point processes
Jiří Kopecký; Tomáš Mrkvička — 2016
The pure and modified Bayesian methods are applied to the estimation of parameters of the Neyman-Scott point process. Their performance is compared to the fast, simulation-free methods via extensive simulation study. Our modified Bayesian method is found to be on average 2.8 times more accurate than the fast methods in the relative mean square errors of the point estimates, where the average is taken over all studied cases. The pure Bayesian method is found to be approximately as good as the fast...
Spatial prediction of the mark of a location-dependent marked point process: How the use of a parametric model may improve prediction
Tomáš Mrkvička; François Goreaud; Joël Chadoeuf — 2011
We discuss the prediction of a spatial variable of a multivariate mark composed of both dependent and explanatory variables. The marks are location-dependent and they are attached to a point process. We assume that the marks are assigned independently, conditionally on an unknown underlying parametric field. We compare (i) the classical non-parametric Nadaraya-Watson kernel estimator based on the dependent variable (ii) estimators obtained under an assumption of local parametric model where explanatory...
7 Mrkvička, T
1 Chadoeuf, J
1 Goreaud, F
1 Kopecký, J
1 Rataj, J
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Trig Identities - Angle Addition - Maple Help
Home : Support : Online Help : Math Apps : Trigonometry : Trig Identities - Angle Addition
Trig Identities: Angle Addition
There are several trig identities used in mathematics, among which are the angle addition and subtraction formulas. These formulas are summarized as follows:
Angle Addition or Angle Subtraction
\mathrm{Equivalent} \mathrm{Identity}
\mathrm{sin}\left(A+B\right)
\mathrm{sin}\left(A\right)\mathrm{cos}\left(B\right)+\mathrm{cos}\left(A\right)\mathrm{sin}\left(B\right)
\mathrm{sin}\left(A-B\right)
\mathrm{sin}\left(A\right)\mathrm{cos}\left(B\right)-\mathrm{cos}\left(A\right)\mathrm{sin}\left(B\right)
\mathrm{cos}\left(A+B\right)
\mathrm{cos}\left(A\right)\mathrm{cos}\left(B\right)-\mathrm{sin}\left(A\right)\mathrm{sin}\left(B\right)
\mathrm{cos}\left(A-B\right)
\mathrm{cos}\left(A\right)\mathrm{cos}\left(B\right)+\mathrm{sin}\left(A\right)\mathrm{sin}\left(B\right)
\mathrm{tan}\left(A+B\right)
\frac{\mathrm{tan}\left(A\right)+\mathrm{tan}\left(B\right)}{1-\mathrm{tan}\left(A\right)\mathrm{tan}\left(B\right)}
\mathrm{tan}\left(A-B\right)
\frac{\mathrm{tan}\left(A\right)-\mathrm{tan}\left(B\right)}{1+ \mathrm{tan}\left(A\right)\mathrm{tan}\left(B\right)}
The following questions focus on angle addition. Both the identities for
\mathrm{sin}\left(A+B\right)
\mathrm{cos}\left(A+B\right)
can be derived using geometry as shown below.
Select the appropriate radio button and click "Next" to see, step by step, how the identity is derived.
1. Using the identities in the above table, determine the value of sin(75°).
2. Prove that
\mathrm{tan}\left(A+B\right)=\frac{\mathrm{tan}\left(A\right)+\mathrm{tan}\left(B\right)}{1-\mathrm{tan}\left(A\right)\mathrm{tan}\left(B\right)}.
You can use the identity
\mathrm{sin}\left(A+B\right)=\mathrm{sin}\left(A\right)\mathrm{cos}\left(B\right)+\mathrm{cos}\left(A\right)\mathrm{sin}\left(B\right)
to determine the value of sin(75°):
First, note that 45° + 30° = 75°. This is important, as these angles form one of the special triangles:
From this, you can substitute the two angles into the identity for
\mathrm{sin}\left(A+B\right)
\mathrm{sin}\left(A+B\right)=\mathrm{sin}\left(A\right)\mathrm{cos}\left(B\right)+\mathrm{cos}\left(A\right)\mathrm{sin}\left(B\right)
\mathrm{sin}\left(30°+45°\right)=\mathrm{sin}\left(30°\right)\mathrm{cos}\left(45°\right)+\mathrm{cos}\left(30°\right)\mathrm{sin}\left(45°\right)
\mathrm{sin}\left(75°\right)=\left(\frac{1}{2}\right)\left(\frac{1}{\sqrt{2}}\right)+\left(\frac{\sqrt{3}}{2}\right)\left(\frac{1}{\sqrt{2}}\right)
\mathrm{sin}\left(75°\right)=\frac{\sqrt{2}}{4}+\frac{\sqrt{6}}{4}
\mathrm{sin}\left(75°\right)=\frac{\sqrt{2}+\sqrt{6}}{4}
First, note that:
\mathrm{tan}\left(A+B\right)=\frac{\mathrm{sin}\left(A+B\right)}{\mathrm{cos}\left(A+B\right)}
Keeping this in mind, you can substitute the trig identities for
\mathrm{sin}\left(A+B\right)
\mathrm{cos}\left(A+B\right)
\frac{\mathrm{sin}\left(A\right)\mathrm{cos}\left(B\right)+\mathrm{cos}\left(A\right)\mathrm{sin}\left(B\right)}{\mathrm{cos}\left(A\right)\mathrm{cos}\left(B\right)-\mathrm{sin}\left(A\right)\mathrm{sin}\left(B\right)}
Now, divide every term by
\mathrm{cos}\left(A\right)\mathrm{cos}\left(B\right)
\frac{\frac{\mathrm{sin}\left(A\right)\mathrm{cos}\left(B\right)}{\mathrm{cos}\left(A\right)\mathrm{cos}\left(B\right)}+\frac{\mathrm{cos}\left(A\right)\mathrm{sin}\left(B\right)}{\mathrm{cos}\left(A\right)\mathrm{cos}\left(B\right)}}{\frac{\mathrm{cos}\left(A\right)\mathrm{cos}\left(B\right)}{\mathrm{cos}\left(A\right)\mathrm{cos}\left(B\right)}-\frac{\mathrm{sin}\left(A\right)\mathrm{sin}\left(B\right)}{\mathrm{cos}\left(A\right)\mathrm{cos}\left(B\right)}}
Simplify this expression as follows:
\frac{\frac{\mathrm{sin}\left(A\right)}{\mathrm{cos}\left(A\right)}+\frac{\mathrm{sin}\left(B\right)}{\mathrm{cos}\left(B\right)}}{1-\frac{\mathrm{sin}\left(A\right)\mathrm{sin}\left(B\right)}{\mathrm{cos}\left(A\right)\mathrm{cos}\left(B\right)}}
Finally, you can simplify this expression further by recalling that
\mathrm{tan}\left(Q\right)=\frac{\mathrm{sin}\left(Q\right)}{\mathrm{cos}\left(Q\right)}
\frac{\mathrm{tan}\left(A\right)+\mathrm{tan}\left(B\right)}{1-\mathrm{tan}\left(A\right)\mathrm{tan}\left(B\right)}
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Skygazing/Solar eclipse lab on a sunny day - Wikiversity
Skygazing/Solar eclipse lab on a sunny day
< Skygazing
2 Why this is a scale model
5 How far from the model "Earth" should we hold the model "Moon"?
6 Calculating the scale
7 Scaling the USA
8 Other photos from the experience
9 Understanding the size of the penumbra
11 Test the hypothesis:How big is the umbra in this nasa gif?
12 W:On the Sizes and Distances (Aristarchus)
12.1 Aristarchus simplified
12.2 The distance to the Moon in Earth diameters
12.3 Two more images re:Aristarchus
12.4 References and footnotes
Today we did a lab that creates a scale model of a solar eclipse. In his left hand is an iPhone that can take the shadow's photograph.
Is this shadow of the 1-inch ball the umbra or the penumbra?[1] The vertical shadow is the support that held the Styrofoam ball in place.[2]
A 1-inch diameter "Moon" must be held 110 inches above the ground for a proper scale model[3]
The digital image of the penumbra was imported into Inkscape [4] and converted into grey-scale objects. A map of USA was found on commons and placed next to it at the proper scale. This evidence suggests that the penumbra associated with a solar eclipse is larger than the United States.
To measure the size of the Moon's penumbra during a solar eclipse using a scale model.
Why this is a scale model[edit | edit source]
A scale model is a drawing that preserves all the angles. Here we represent the Moon as a small Styrofoam ball. The Earth is most easily represented as a map of a region where a solar eclipse is expected. Here we use the United States.
For the Sun, we use the actual object, which of course is not to the same scale. But for our purposes, any object that produces light with the proper angular distribution can be used. In other words, the actual Sun provides the required angular spread of light rays (i.e., light spread over a range of about half a degree). The actual size of the Sun is irrelevant (otherwise one could use a solar eclipse to calculate the size and distance to the Sun).
10 ft rod to hold up the model "Moon" (we clamped a two-meter stick to a one-meter stick)
One inch diameter ball to server as the Moon (rod and ball dimensions may vary proportionally)
Tape measure or other way to measure distance on the ground (we used 2 metersticks)
Camera (we used an Apple iPhone)
White screen. We used about a dozen sheets of paper. It would also be worthwhile to print onto this paper a properly scaled image of a region on Earth, or perhaps the Earth itself.
At least two people (three works better)
The Sun on a sunny day
The instructor and students should be mentally prepared to do calculations and proportional reasoning on three different scales:
The solar system (especially Earth/Moon system)
A scale where the Moon is reduced to a small ball
A scale on which photos of the shadow are analyzed
The Moon was a one inch diameter styrofoam ball attached to a stick that was 110 inches long (chosen to match the Earth-Moon distance).
We forgot to measure the altitude of the Sun. Fortunately the images on commons contain a time stamp and we know where we were (Celina, Ohio). Somewhere a website will give us that angle.
Photograph the shadow alongside a ruler for scale.
Record the location of the shadow so that you may later calculate the distance from the "Moon" at the end of the stick and the "Earth", represented by the white paper on the ground.
How far from the model "Earth" should we hold the model "Moon"?[edit | edit source]
The Moon is about 60 earth-radii from Earth.
The Moon has a diameter about 1/4 that of the Earth.
Our Styrofoam "Moon" had a diameter of one inch.
Keeping track of scaled objects can be confusing. One trick is to use capital letters for the real thing and lower case for the scale:
{\displaystyle d}
equals scaled distance from Earth to Moon
{\displaystyle D}
is the actual distance.
{\displaystyle r_{E}}
equals scaled radius of Earth
{\displaystyle R_{E}}
is the actual radius.
{\displaystyle r_{M}}
equals scaled radius of the Moon
{\displaystyle R_{M}}
Focusing on only astronomical data, we have:
{\displaystyle D\approx 60R_{E}{\text{ and }}R_{E}\approx 4R_{M}\rightarrow D\approx (100)(2R_{M})}
{\displaystyle 2r_{M}=1{\text{ inch}}}
we conclude that the Styrofoam ball should be held at a distance of
{\displaystyle d=120{\text{ inches}}}
away from the Earth. Our "model" Earth was just a piece of paper, but we can add a scale to it using the fact that a one inch diameter ball represents the 2159 mile diameter Moon.
Calculating the scale[edit | edit source]
{\displaystyle 1\ {\widehat {\text{in}}}=2159{\text{ miles }}}
where we have used the "hat" notation to define the "scaled" inch. This "scaled" inch equals 2159 miles, but is also one (actual) inch on the scale model.
{\displaystyle 1\ {\widehat {\text{in}}}=2159{\text{ miles }}\left({\frac {5280\ {\text{ft }}}{1{\text{ mile }}}}\right)\left({\frac {12\ {\text{in }}}{1{\text{ft}}}}\right)=1.37\times 10^{8}{\text{ inches}}}
In other words, our scale is 1.37 x 108 to 1.
Scaling the USA[edit | edit source]
http://www.distance.to/New-York/San-Francisco says that the distance between NYC and SF is 2566 miles = 1430 km. Convert this to centimeters and divide by the scaling factor to get:
{\displaystyle {\frac {1.43\times 10^{8}{\text{ cm}}}{1.37\times 10^{8}}}=1.04\approx 1{\widehat {\text{cm}}}={\text{ scaled length of the US}}}
This says that the penumbra is larger than the USA? This seems to big, but see below. This experiment should be repeated and the results reported here.
Other photos from the experience[edit | edit source]
showing how we did the lab
0 cm[5]
Understanding the size of the penumbra[edit | edit source]
This diagram explains why the penumbra of a solar eclipse is approximately twice the size of the Moon. Since the Sun and Moon have nearly the same angular diameter, the umbra is reduced to nearly a single point in length. Therefore the three angles marked in red are nearly equal, all with a base equal to the Moon's diameter.
If you go here http://eclipse.gsfc.nasa.gov/SEgoogle/SEgoogle2001/SE2017Aug21Tgoogle.html, you will see that the penumbra is very large and extends far outside the USA. Perhaps our results are correct.
We measured the length of the shadow on the photograph marked 0cm. It was projected on a classroom screen at a scale where 4.9 cm* equals 1 cm. In other words, the classroom projected image of the photo expanded the meter stick so that 1 cm on the stick measured 4.9cm* on the screen.[6]. The shadow's length on the screen was 21 plus/minus 1 cm*. We concluded that the umbra was
1.69 ± 0.08 inches[7]
This much smaller than the 2 inch umbra we expected from a 1 inch moon. Perhaps the brightest part of the penumbra was indistinguishable from the illuminated portion of the paper.[8]
Test the hypothesis:How big is the umbra in this nasa gif?[edit | edit source]
2017 August 21, a total eclipse of the Sun occurs in United States of America.
File:Total solar eclipse NASA TSE2017-1.gif has been annotated to show the outline of the USA, as well as the 50% and 0% partial eclipse contours.
It looks to me that the height of the umbra is approximately equal to the radius of the Earth. According to https://nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html, the ratio of diameters for the Earth to the Moon is:
{\displaystyle {\frac {R_{M}}{R_{E}}}=0.2725}
{\displaystyle d_{U}}
be the diameter of the umbra, which we estimate to be equal to the radius of Earth and also twice the diameter of the Moon, but since the diameter of the Moon is twice its radius, we have:
{\displaystyle d_{U}\approx R_{E}\approx 4R_{M}=(4)(0.2725)\cdot R_{E}=1.09R_{E}}
I think the greyscale stuff was wrong (see above)
W:On the Sizes and Distances (Aristarchus)[edit | edit source]
The article on Wikipedia is excellent:
copied from w:Special:Permalink/773165157
Aristarchus began with the premise that, during a half moon, the moon forms a right triangle with the Sun and Earth. By observing the angle between the Sun and Moon, φ, the ratio of the distances to the Sun and Moon could be deduced using a form of trigonometry.
From the diagram and trigonometry, we can calculate that
{\displaystyle {\frac {S}{L}}={\frac {1}{\cos \varphi }}=\sec \varphi .}
The diagram is greatly exaggerated, because in reality, S = 390 L, and φ is extremely close to 90°. Aristarchus determined φ to be a thirtieth of a quadrant (in modern terms, 3°) less than a right angle: in current terminology, 87°. Trigonometric functions had not yet been invented, but using geometrical analysis in the style of Euclid, Aristarchus determined that
{\displaystyle 18<{\frac {S}{L}}<20.}
In other words, the distance to the Sun was somewhere between 18 and 20 times greater than the distance to the Moon. This value (or values close to it) was accepted by astronomers for the next two thousand years, until the invention of the telescope permitted a more precise estimate of solar parallax.
Aristarchus also reasoned that as the angular size of the Sun and the Moon were the same, but the distance to the Sun was between 18 and 20 times further than the Moon, the Sun must therefore be 18-20 times larger.
Aristarchus then used another construction based on a lunar eclipse:
By similarity of the triangles,
{\displaystyle {\frac {D}{L}}={\frac {t}{t-d}}\quad }
{\displaystyle \quad {\frac {D}{S}}={\frac {t}{s-t}}.}
Dividing these two equations and using the observation that the apparent sizes of the Sun and Moon are the same,
{\displaystyle {\frac {L}{S}}={\frac {\ell }{s}}}
{\displaystyle {\frac {\ell }{s}}={\frac {t-d}{s-t}}\ \ \Rightarrow \ \ {\frac {s-t}{s}}={\frac {t-d}{\ell }}\ \ \Rightarrow \ \ 1-{\frac {t}{s}}={\frac {t}{\ell }}-{\frac {d}{\ell }}\ \ \Rightarrow \ \ {\frac {t}{\ell }}+{\frac {t}{s}}=1+{\frac {d}{\ell }}.}
The rightmost equation can either be solved for ℓ/t
{\displaystyle {\frac {t}{\ell }}(1+{\frac {\ell }{s}})=1+{\frac {d}{\ell }}\ \ \Rightarrow \ \ {\frac {\ell }{t}}={\frac {1+{\frac {\ell }{s}}}{1+{\frac {d}{\ell }}}}.}
or s/t
{\displaystyle {\frac {t}{s}}(1+{\frac {s}{\ell }})=1+{\frac {d}{\ell }}\ \ \Rightarrow \ \ {\frac {s}{t}}={\frac {1+{\frac {s}{\ell }}}{1+{\frac {d}{\ell }}}}.}
The appearance of these equations can be simplified using n = d/ℓ and x = s/ℓ.
{\displaystyle {\frac {\ell }{t}}={\frac {1+x}{x(1+n)}}}
{\displaystyle {\frac {s}{t}}={\frac {1+x}{1+n}}}
The above equations give the radii of the Moon and Sun entirely in terms of observable quantities.
The following formulae give the distances to the Sun and Moon in terrestrial units:
{\displaystyle {\frac {L}{t}}=\left({\frac {\ell }{t}}\right)\left({\frac {180}{\pi \theta }}\right)}
{\displaystyle {\frac {S}{t}}=\left({\frac {s}{t}}\right)\left({\frac {180}{\pi \theta }}\right)}
where θ is the apparent radius of the Moon and Sun measured in degrees.
It is unlikely that Aristarchus used these exact formulae, yet these formulae are likely a good approximation to those of Aristarchus.
Aristarchus simplified[edit | edit source]
I have a theory that great minds not only do much, they omit much. Suppose Aristarchus was more interested in knowing approximate facts and less interested in exact theories of geometry. Let's replace what he either knew, or should have known by the following facts:
The angular size of the Moon and Sun are half a degree.
The Earth's shadow (umbra) at the Moon's distance is twice the diameter of the Moon.
The angle between the Sun and the Moon when visible Moon is 50% illuminated is effectively 90 degrees.
The third of these statements would have permitted Aristarchus to set the Sun at "infinity" (i.e. very far away), and used parallel lines to bypass a lot of complicated geometry (by "complicated" I mean too sophisticated for the average mind of a 21st century educated person.) Here is how Aristarchus might have concluded that the Earth is three times larger than the Moon (in diameter).
How Aristarchus might have calculated the size of the Moon if he knew the Sun is extremely far away. To the left we see a full moon, and to the right we see two adjacent Moons that represent a statement attributed to Aristarchus to the effect that the Earth's umbra is twice as large as the Moon at one lunar distance from the Sun. At two lunar distances the base of the triangle is one lunar diameter, and the apex of the Earth's umbra is situated three lunar distances from Earth (seen at extreme right of figure). All triangles are similar if the Sun and Moon have the same angular size. Hence, Aristarchus might have calculated that Earth is 3 times as large as the Moon (the actual ratio is closer to 3.67).The bottom figure was created by compressing the top one in the vertical direction by a factor of 3. These figures are not too scale in that they assume that the angular diameter of the Sun and Moon are 6.7 degrees (instead of 0.5 degrees.) Hence the Moon is represented as being over 10 times closer to Earth than it really is.
The distance to the Moon in Earth diameters[edit | edit source]
Since one radian is 57.3 degrees, the distance to the Moon in Earth diameters can be written as:
{\displaystyle \theta ={\frac {2\cdot {\text{Moon's radius}}}{\text{Earth/Moon distance}}}={\frac {2R_{\text{Moon}}}{\text{Earth/Moon distance}}}={\frac {1}{2}}{\text{deg}}={\frac {1{\text{ deg}}}{2}}{\frac {\pi {\text{ rad}}}{180{\text{ deg}}}}={\frac {\pi }{360}}}
Since the Moon is three times smaller than Earth, we have:
{\displaystyle R_{Moon}={\frac {1}{3}}R_{Earth}}
Solving, we obtain the distance to the Moon in Earth radii:
{\displaystyle {\text{Earth/Moon distance}}={\frac {2}{3}}{\frac {360}{pi}}R_{Earth}=76R_{Earth}}
The accepted distance is closer to 60 Earth radii. Taking the ratio to extimate the percent error that Aristarchus would have made:
{\displaystyle {\frac {76}{60}}\approx 1.3}
which corresponds to only a 30% error that Aristarchus might have made if only he understood the vast distance to the Sun.
Two more images re:Aristarchus[edit | edit source]
↑ It is almost entirely penumbra because the ball was held so that the angular size of the ball nearly equaled the angular size of the Sun. If the two angular sizes are equal, the umbra is essentially reduced to the area of a single point.
↑ We use a ballpoint pen that was rammed into the Styrofoam ball to hold it in place.
↑ Although our calculation stipulated a 120 inch rod, a more careful calculation indicated that 111 inches was closer to the condition that the angular size of the Styrofoam ball equals the angular size of the Moon.
↑ People interested in editing for Wikipedia should know about Inkscape because it is open source and makes the editable svg files preferred by Wikipedia and its "sisters".
↑ See also File:Solar eclipse lab on a sunny day 7 cropped.jpg
↑ Here the asterisk indicates that the cm* was as measured on a screen projected in the classroom. This forms a third scale. In this context, cm refers to the actual photograph.
↑ More properly, this shadow was 1.69 hat-inches, since the hat is used on measurement made on the scale model.
↑ Another factor is that the student was not instructed to try for the largest possible shadow. After class, the instructor could discern a faint shadow 30 cm* in length on the screen that projected the image.
Retrieved from "https://en.wikiversity.org/w/index.php?title=Skygazing/Solar_eclipse_lab_on_a_sunny_day&oldid=1689340"
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CHTR Financial Ratios - FinancialModelingPrep
\dfrac{Current Assets}{Current Liabilities}
\dfrac{Cash and Cash Equivalents + Short Term Investments + Account Receivables}{Current Liabilities}
\dfrac{Cash and Cash Equivalents}{Current Liabilities}
\dfrac{(Account Receivable (start) + Account Receivable (end))/2}{Revenue/365}
\dfrac{(Inventories (start) + Inventories (end))/2}{COGS/365}
\dfrac{DSO + DIO}{}
\dfrac{(Accounts Payable (start) + Accounts Payable (end))/2}{COGS/365}
8.29 DPO tells you how many days the company takes to pay its suppliers.
\dfrac{DSO + DIO − DPO}{}
9.35 The cash conversion cycle (CCC = DSO + DIO – DPO) measures the number of days a company's cash is tied up in the production and sales process of its operations and the benefit it derives from payment terms from its creditors. The shorter this cycle, the more liquid the company's working capital position is. The CCC is also known as the "cash" or "operating" cycle.
\dfrac{Gross Profit}{Revenue}
\dfrac{Operating Income}{Revenue}
\dfrac{Income Before Tax}{Revenue}
\dfrac{Net Income}{Revenue}
\dfrac{Provision For Income Taxes}{Income Before Tax}
\dfrac{Net Income}{Average Total Assets}
\dfrac{Net Income}{Average Total Equity}
\dfrac{EBIT}{Average Total Asset − Average Current Liabilities}
\dfrac{Net Income}{EBT}
\dfrac{EBT}{EBIT}
\dfrac{EBIT}{Revenue}
\dfrac{Total Liabilities}{Total Assets}
\dfrac{Total Debt}{Total Equity}
\dfrac{Long−Term Debt}{Long−Term Debt + Shareholders Equity}
\dfrac{Total Debt}{Total Debt + Shareholders Equity}
\dfrac{EBIT}{Interest Expense}
\dfrac{Operating Cash Flows}{Total Debt}
\dfrac{Total Assets}{Total Equity}
\dfrac{Revenue}{NetPPE}
\dfrac{Revenue}{Total Average Assets}
\dfrac{Operating Cash Flow}{Revenue}
\dfrac{Free Cash Flow}{Operating Cash Flow}
\dfrac{Operating Cash Flow}{Total Debt}
\dfrac{Operating Cash Flow}{Short-Term Debt}
\dfrac{Operating Cash Flow}{Capital Expenditure}
\dfrac{Operating Cash Flow}{Dividend Paid + Capital Expenditure}
\dfrac{DPS (Dividend per Share)}{EPS (Net Income per Share Number}
\dfrac{Stock Price per Share}{Equity per Share}
\dfrac{Stock Price per Share}{Operating Cash Flow per Share}
\dfrac{Stock Price per Share}{EPS}
\dfrac{Price Earnings Ratio}{Expected Revenue Growth}
\dfrac{Stock Price per Share}{Revenue per Share}
\dfrac{Dividend per Share}{Stock Price per Share}
\dfrac{Entreprise Value}{EBITDA}
\dfrac{Stock Price per Share}{Intrinsic Value}
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Complex Solutions of Quadratic Equations - Course Hero
College Algebra/Complex Numbers/Complex Solutions of Quadratic Equations
Solving Equations without a Linear Term
Quadratic equations with no linear term can be solved by isolating
x^2
and taking the square root of both sides of the equation. If
x^2
is equal to a negative number, then the solutions of the equation are pure imaginary numbers.
A quadratic equation is an equation, where
a
b
c
a\neq0
ax^2 + bx + c = 0
Solving an equation of the form
ax^2 = -c
combines the processes of solving a quadratic equation and simplifying imaginary numbers.
Solving a Quadratic Equation of the Form
ax^2 = -c
5x^2=-80
Divide both sides of the equation by the coefficient of
x^2
\begin{aligned}\frac {5x^2}{5}&=\frac{-80}{5}\\x^2&=-16\end{aligned}
\begin{aligned}\sqrt{(x^2)}&=\sqrt{-16}\\x&=\pm \sqrt{-16}\end{aligned}
Simplify the negative square root.
\begin{aligned}x&=\pm \sqrt{-16}\\&=\pm \sqrt{-1\cdot 16}\\&=\pm (\sqrt{-1}\cdot \sqrt{16})\\&=\pm (i\cdot4)\\&=\pm 4i\end{aligned}
x=4i
x=-4i
Quadratic equations with no real roots can be solved by completing the square.
The roots of a quadratic equation are its solutions. A real root is a solution of an equation that is also a real number. Some quadratic equations do not have any real roots.
One way to tell that an equation of the form
ax^2+bx+c=0
has no real roots is by looking at the graph of the related quadratic function:
f(x)=ax^2+bx+c
If the related function has no real zeros, or values of
x
f(x)
equal to zero, then the quadratic equation has no real roots.
The equation of a related function without any real zeros does not have any real roots. For example, the function
f(x)=x^2-6x+10
has no real zeros, or values of
x
f(x)
equal to zero, because the graph does not intersect the
x
-axis. Therefore, the related quadratic equation
x^2-6x+10=0
When a quadratic equation has no real roots, it has a pair of complex roots that are complex conjugates. One strategy for solving quadratic equations with complex roots is completing the square.
Completing the square is the process of adding a constant term to a quadratic expression to form a perfect square trinomial. A perfect square trinomial is a trinomial, or polynomial with three terms, that can be written as the square of a binomial. For example,
x^2+8x+16
is a perfect square trinomial because its factored form is
(x+4)^2
To complete the square for a quadratic expression of the form
x^2+bx
, add the value of:
\left(\frac{b}{2}\right)^{2}
Complete the Square to Solve a Quadratic Equation with Complex Solutions
x^2-6x+10=0
Write the equation in the form
x^2+bx=c
x^2-6x=-10
Find the value of this formula:
\begin{aligned}\left (\frac{b}{2} \right )^{2}&=\left ( \frac{-6}{2} \right )^{2}\\&=(-3)^{2}\\&=9\end{aligned}
Add 9 to
x^2+bx
to complete the square. Add 9 to the other side of the equation to keep the equation balanced.
\begin{aligned}x^2-6x+9&=-10+9\\x^2-6x+9&=-1\end{aligned}
Factor the perfect square trinomial, or a polynomial with three terms that can be written as the square of a binomial.
\left(x-3\right)^2=-1
\begin{aligned}x-3&=\pm \sqrt{-1}\\x-3&=\pm i\end{aligned}
\begin{aligned}x-3&=i\\x&=3+i\end{aligned}\;\;\;\;\; \text{or}\;\;\;\;\;\begin{aligned}x-3&=-i\\x&=3-i\end{aligned}
x=3+i\;\;\;\;\; \text{or}\;\;\;\;\;x=3-i
The solutions can be checked by substituting them into the original equation. For
x=3+i
\begin{aligned}x^2-6x+10&=0\\(3+i)^2-6(3+i)+10&=0\\(3+i)(3+i)-6(3+i)+10&=0\\9+3i+3i+i^2-18-6i+10&=0\\1+(-1)&=0\\0&=0\end{aligned}
x=3-i
\begin{aligned}x^2-6x+10&=0\\(3-i)^2-6(3-i)+10&=0\\(3-i)(3-i)-6(3-i)+10&=0\\9-3i-3i+i^2-18+6i+10&=0\\1+(-1)&=0\\0&=0\end{aligned}
Solving by Using the Quadratic Formula
Quadratic equations with no real roots can be solved by using the quadratic formula.
Another way to find solutions to quadratic equations with no real roots is by using the quadratic formula. The quadratic formula is a formula used to solve a quadratic equation of the form
ax^2+bx+c=0
x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}
The quadratic formula can be used to find the solutions of any quadratic equation. Once the equation is in the standard form of a quadratic equation,
ax^2+bx+c=0
, substitute the values of
a
b
c
into the formula and simplify. The discriminant is the part of the quadratic formula that is under the radical sign:
b^2-4ac
The value of the discriminant indicates the type and number of roots of the quadratic equation.
If the discriminant is positive, then the quadratic equation has two real roots.
If the discriminant is zero, then the quadratic equation has one real root.
If the discriminant is negative, then the quadratic equation has two complex roots, which come in conjugate pairs.
Using the Quadratic Formula with Complex Solutions
4x^2-16x+41=0
ax^2+bx+c=0
The given equation is in standard form, where
a=4
b=-16
c=41
a
b
c
into the discriminant from the quadratic formula.
\begin{aligned}b^2-4ac&=(-16)^2-4(4)(41)\\&=-400\end{aligned}
The discriminant is negative. So the equation has two complex solutions.
a
b
, and the discriminant into the quadratic formula.
\begin{aligned}x&=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\x&=\frac{-(-16)\pm\sqrt{-400}}{2(4)}\end{aligned}
\begin{aligned}x&=\frac{16\pm\sqrt{-400}}{8}\\&=\frac{16\pm 20i}{8}\\&=2\pm\frac{5}{2}i\end{aligned}
x=2+\frac{5}{2}i\;\;\;\;\;\text{or}\;\;\;\;\;x=2-\frac{5}{2}i
<Operations with Complex Numbers
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On isolated, respectively consecutive large values of arithmetic functions
A. Sárközy — 1994
On sums and products of residues modulo p
On Irregularities of Distribution in Shifts and Dilations of Integer Sequences. I.
A. Sárközy; C.L. Stewart — 1986
On divisors of sums of integers. II.
On Differences and sums of Integers, II
A. Sarkozy; P. Erdos — 1977
Some solved and unsolved problems in combinatorial number theory, ii
P. Erdős; A. Sárközy — 1993
In an earlier paper [9], the authors discussed some solved and unsolved problems in combinatorial number theory. First we will give an update of some of these problems. In the remaining part of this paper we will discuss some further problems of the two authors.
On prime factors of integers of the form (ab+1)(bc+1)(ca+1)
K. Győry; A. Sárközy — 1997
1. Introduction. For any integer n > 1 let P(n) denote the greatest prime factor of n. Győry, Sárközy and Stewart [5] conjectured that if a, b and c are pairwise distinct positive integers then (1) P((ab+1)(bc+1)(ca+1)) tends to infinity as max(a,b,c) → ∞. In this paper we confirm this conjecture in the special case when at least one of the numbers a, b, c, a/b, b/c, c/a has bounded prime factors. We prove our result in a quantitative form by showing that if is a finite set of triples (a,b,c)...
Problems and Results on Additive Properties of General Sequemnces, V.
P. Erdös; A. Sárközy; V.T. Sós — 1986
On partitions without small parts
J.-L. Nicolas; A. Sárközy — 2000
r\left(n,m\right)
denote the number of partitions of
n
into parts, each of which is at least
m
. By applying the saddle point method to the generating series, an asymptotic estimate is given for
r\left(n,m\right)
, which holds for
n\to \infty
1\le m\le {c}_{1}\frac{n}{{\left(log\phantom{\rule{0.166667em}{0ex}}n\right)}^{{c}_{2}}}
On the number of pairs of partitions of n without common subsums
P. Erdős; J. Nicolas; A. Sárközy — 1992
On pseudorandom binary lattices
P. Hubert; C. Mauduit; A. Sárközy — 2006
Sommes de sous-ensembles
P. Erdös; J.-L. Nicolas; A. Sárkozy — 1991
On the number of prime factors of integers of the form ab + 1
K. Győry; A. Sárközy; C. L. Stewart — 1996
1 Δελτίο της Ελληνικής Μαθηματικής Εταιρίας
11 Sárközy, A
3 Stewart, CL
2 Erdős, P
2 Nicolas, JL
1 Erdos, P
1 Hubert, P
1 Mauduit, C
1 Nicolas, J
1 Sarkozy, A
1 Sárkozy, A
1 Sós, VT
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Boolean model of information retrieval
(Redirected from Standard Boolean model)
The (standard) Boolean model of information retrieval (BIR)[1] is a classical information retrieval (IR) model and, at the same time, the first and most-adopted one. It is used by many IR systems to this day.[citation needed] The BIR is based on Boolean logic and classical set theory in that both the documents to be searched and the user's query are conceived as sets of terms (a bag-of-words model). Retrieval is based on whether or not the documents contain the query terms.
5.1 Hash sets
5.2 Signature file
5.3 Inverted file
An index term is a word or expression, which may be stemmed, describing or characterizing a document, such as a keyword given for a journal article. Let
{\displaystyle T=\{t_{1},t_{2},\ \ldots ,\ t_{m}\}}
be the set of all such index terms.
A document is any subset of
{\displaystyle T}
{\displaystyle D=\{D_{1},\ \ldots \ ,D_{n}\}}
be the set of all documents.
A query is a Boolean expression
{\textstyle Q}
in normal form:
{\displaystyle Q=(W_{1}\ \lor \ W_{2}\ \lor \ \cdots )\land \ \cdots \ \land \ (W_{i}\ \lor \ W_{i+1}\ \lor \ \cdots )}
{\textstyle W_{i}}
{\displaystyle D_{j}}
{\displaystyle t_{i}\in D_{j}}
. (Equivalently,
{\textstyle Q}
could be expressed in disjunctive normal form.)
We seek to find the set of documents that satisfy
{\textstyle Q}
. This operation is called retrieval and consists of the following two steps:
{\textstyle W_{j}}
{\textstyle Q}
, find the set
{\textstyle S_{j}}
of documents that satisfy
{\textstyle W_{j}}
{\displaystyle S_{j}=\{D_{i}\mid W_{j}\}}
2. Then the set of documents that satisfy Q is given by:
{\displaystyle (S_{1}\cup S_{2}\cup \cdots )\cap \cdots \cap (S_{i}\cup S_{i+1}\cup \cdots )}
Let the set of original (real) documents be, for example
{\displaystyle O=\{O_{1},\ O_{2},\ O_{3}\}}
{\textstyle O_{1}}
= "Bayes' principle: The principle that, in estimating a parameter, one should initially assume that each possible value has equal probability (a uniform prior distribution)."
{\textstyle O_{2}}
= "Bayesian decision theory: A mathematical theory of decision-making which presumes utility and probability functions, and according to which the act to be chosen is the Bayes act, i.e. the one with highest subjective expected utility. If one had unlimited time and calculating power with which to make every decision, this procedure would be the best way to make any decision."
{\textstyle O_{3}}
= "Bayesian epistemology: A philosophical theory which holds that the epistemic status of a proposition (i.e. how well proven or well established it is) is best measured by a probability and that the proper way to revise this probability is given by Bayesian conditionalisation or similar procedures. A Bayesian epistemologist would use probability to define, and explore the relationship between, concepts such as epistemic status, support or explanatory power."
Let the set
{\textstyle T}
of terms be:
{\displaystyle T=\{t_{1}={\text{Bayes' principle}},t_{2}={\text{probability}},t_{3}={\text{decision-making}},t_{4}={\text{Bayesian epistemology}}\}}
Then, the set
{\textstyle D}
of documents is as follows:
{\displaystyle D=\{D_{1},\ D_{2},\ D_{3}\}}
{\displaystyle {\begin{aligned}D_{1}&=\{{\text{probability}},\ {\text{Bayes' principle}}\}\\D_{2}&=\{{\text{probability}},\ {\text{decision-making}}\}\\D_{3}&=\{{\text{probability}},\ {\text{Bayesian epistemology}}\}\end{aligned}}}
Let the query
{\textstyle Q}
{\displaystyle Q={\text{probability}}\land {\text{decision-making}}}
Then to retrieve the relevant documents:
Firstly, the following sets
{\textstyle S_{1}}
{\textstyle S_{2}}
of documents
{\textstyle D_{i}}
are obtained (retrieved):
{\displaystyle {\begin{aligned}S_{1}&=\{D_{1},\ D_{2},\ D_{3}\}\\S_{2}&=\{D_{2}\}\end{aligned}}}
Finally, the following documents
{\textstyle D_{i}}
are retrieved in response to
{\textstyle Q}
{\displaystyle Q:\{D_{1},\ D_{2},\ D_{3}\}\ \cap \ \{D_{2}\}\ =\ \{D_{2}\}}
This means that the original document
{\textstyle O_{2}}
(corresponding to
{\textstyle D_{2}}
) is the answer to
{\textstyle Q}
Obviously, if there is more than one document with the same representation, every such document is retrieved. Such documents are indistinguishable in the BIR (in other words, equivalent).
Clean formalism
Intuitive concept
If the resulting document set is either too small or too big, it is directly clear which operators will produce respectively a bigger or smaller set.
It gives (expert) users a sense of control over the system. It is immediately clear why a document has been retrieved given a query.
Exact matching may retrieve too few or too many documents
Hard to translate a query into a Boolean expression
All terms are equally weighted
More like data retrieval than information retrieval
Retrieval based on binary decision criteria with no notion of partial matching
No ranking of the documents is provided (absence of a grading scale)
Information need has to be translated into a Boolean expression, which most users find awkward
The Boolean queries formulated by the users are most often too simplistic
The model frequently returns either too few or too many documents in response to a user query
From a pure formal mathematical point of view, the BIR is straightforward. From a practical point of view, however, several further problems should be solved that relate to algorithms and data structures, such as, for example, the choice of terms (manual or automatic selection or both), stemming, hash tables, inverted file structure, and so on.[2]
Hash sets[edit]
Main article: feature hashing
Another possibility is to use hash sets. Each document is represented by a hash table which contains every single term of that document. Since hash table size increases and decreases in real time with the addition and removal of terms, each document will occupy much less space in memory. However, it will have a slowdown in performance because the operations are more complex than with bit vectors. On the worst-case performance can degrade from O(n) to O(n2). On the average case, the performance slowdown will not be that much worse than bit vectors and the space usage is much more efficient.
Signature file[edit]
Each document can be summarized by Bloom filter representing the set of words in that document, stored in a fixed-length bitstring, called a signature. The signature file contains one such superimposed code bitstring for every document in the collection. Each query can also be summarized by a Bloom filter representing the set of words in the query, stored in a bitstring of the same fixed length. The query bitstring is tested against each signature.[3][4][5]
The signature file approached is used in BitFunnel.
Inverted file[edit]
Main article: inverted index
An inverted index file contains two parts: a vocabulary containing all the terms used in the collection, and for each distinct term an inverted index that lists every document that mentions that term.[3][4]
^ Lancaster, F.W.; Fayen, E.G. (1973), Information Retrieval On-Line, Melville Publishing Co., Los Angeles, California
^ Wartik, Steven (1992). "Boolean operations". Information Retrieval Data Structures & Algorithms. Prentice-Hall, Inc. ISBN 0-13-463837-9. Archived from the original on 2013-09-28.
^ a b Justin Zobel; Alistair Moffat; and Kotagiri Ramamohanarao. "Inverted Files Versus Signature Files for Text Indexing".
^ a b Bob Goodwin; et al. "BitFunnel: Revisiting Signatures for Search". 2017.
^ Richard Startin. "Bit-Sliced Signatures and Bloom Filters".
Lashkari, A.H.; Mahdavi, F.; Ghomi, V. (2009), "A Boolean Model in Information Retrieval for Search Engines", 2009 International Conference on Information Management and Engineering, pp. 385–389, doi:10.1109/ICIME.2009.101, ISBN 978-0-7695-3595-1
Retrieved from "https://en.wikipedia.org/w/index.php?title=Boolean_model_of_information_retrieval&oldid=1083758872"
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Version 40 (177×177) Content: "Version 40 QR Code can contain up to 1852 chars..." (and followed by four paragraphs of ASCII text describing QR Codes).
{\displaystyle \mathbb {F} _{256}}
{\displaystyle b_{7}b_{6}b_{5}b_{4}b_{3}b_{2}b_{1}b_{0}}
{\displaystyle \textstyle \sum _{i=0}^{7}b_{i}2^{i}}
{\displaystyle \textstyle \sum _{i=0}^{7}b_{i}\alpha ^{i}}
{\displaystyle \alpha \in \mathbb {F} _{256}}
{\displaystyle \alpha ^{8}+\alpha ^{4}+\alpha ^{3}+\alpha ^{2}+1=0}
{\displaystyle \mathbb {F} _{256}}
{\textstyle \prod _{i=0}^{n-1}(x-\alpha ^{i})}
{\displaystyle n}
{\displaystyle \mathbb {F} _{256}}
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Least Squares - First Approach - Blog | Julio Marquez
Least Squares - First Approach
The problem of Least Squares is a very interesting and recursive one in Geophysics, especially in interpolation, inversion and filtering techniques.
The problem of Least Squares is a very interesting and recursive one in Geophysics, especially in interpolation, inversion and filtering techniques. My first approach to students into the subject is through the simple fitting of a three-point curve in
R^2
I start by assuming a model to fit the data, a linear model to begin with. A linear model in
R^2
is of the form:
y=ax+b
Once the model has been established is time to set the conditions for Least Squares. The purpose is to minimize a quadratic function (
L^2
) that contains the squared sum of differences or errors between the data and the prediction of the model assumed for that data.
L^2(a,b)=(ax_1+b-y_1)^2+(ax_2+b-y_2)^2+(ax_3+b-y_3)^2
As can be seen
L^2
is a function of "a" and "b" which are the fitting parameters of the model.
What comes next? Well, minimization of the function! As you may recall from math, one has to take the partial derivatives of the variables involved and set them to zero. As shown below:
\frac{\partial L^2}{\partial b}=0
Those equations lead to a system of equations that most of the time in Geophysics is either determined or overdetermined. The system of equations looks like this:
Ap=t
Where "p" is a vector with the parameters to fit. Once you solve that system of equations, you will have found the fitting parameters. Pretty simple, isn't it?.
If we take a closer look at the structure of the system of equations of the Least Squares problem, one can see a structure. What is the purpose of "A" and "t"?
At this moment I say: "A" relates the data to the chosen model, one can say that it shows how your data is seen through the model, and "t" is related to the desired output of the fit, in this case, the best output in the sense of minimizing the quadratic error. At this point students tend to blindly believe me, so I take another approach to the Least Squares problem to clarify the tricky points of my dubious interpretation.
In order to make my point, I start the exercise from another formulation, a formulation that resembles somehow the system of equations from the very beginning. What I want is this:
ax_1+b=y_1
ax_2+b=y_2
ax_3+b=y_3
This system of equations can be written as:
Ap=t
\left(\begin{array}{cc}x_1&1\\ x_2&1\\ x_3&1\end{array}\right)\left(\begin{array}{c}a\\ b\end{array}\right)=\left(\begin{array}{c}y_1\\ y_2\\ y_3\end{array}\right)
That's an overdetermined problem, three equations and two variables. This system is solved as follows:
(A^T.A).p=A^T.t
p= (A^T.A)^{-1}.A^T.t
Following this formulation one reaches the same results but without partial derivation.
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\mathrm{sin}\left(1\right);\mathrm{sin}\left(2\right);\mathrm{sin}\left(3\right)
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Determinant Representations of Polynomial Sequences of Riordan Type
Sheng-liang Yang, Sai-nan Zheng, "Determinant Representations of Polynomial Sequences of Riordan Type", Journal of Discrete Mathematics, vol. 2013, Article ID 734836, 6 pages, 2013. https://doi.org/10.1155/2013/734836
Sheng-liang Yang1 and Sai-nan Zheng1
1Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China
Academic Editor: Gi Sang Cheon
In this paper, using the production matrix of a Riordan array, we obtain a recurrence relation for polynomial sequence associated with the Riordan array, and we also show that the general term for the sequence can be expressed as the characteristic polynomial of the principal submatrix of the production matrix. As applications, a unified determinant expression for the four kinds of Chebyshev polynomials is given.
The concept of a Riordan array is very useful in combinatorics. The infinite triangles of Pascal, Catalan, Motzkin, and Schröder are important and meaningful examples of Riordan array, and many others have been proposed and developed (see, e.g., [1–7]). In the recent literature, Riordan arrays have attracted the attention of various authors from many points of view and many examples and generalizations can be found (see, e.g., [8–12]).
A Riordan array denoted by is an infinite lower triangular matrix such that its column () has generating function , where and are formal power series with , , and . That is, the general term of matrix is ; here denotes the coefficient of in power series . Given a Riordan array and column vector , the product of and gives a column vector whose generating function is , where . If we identify a vector with its ordinary generating function, the composition rule can be rewritten as This property is called the fundamental theorem for Riordan arrays and this leads to the matrix multiplication for Riordan arrays: The set of all Riordan arrays forms a group under the previos operation of a matrix multiplication. The identity element of the group is . The inverse element of is where is compositional inverse of .
A Riordan array can be characterized by two sequences and such that, for If and are the generating functions for the - and -sequences, respectively, then it follows that [9, 13] If the inverse of is , then the - and -sequences of are
For an invertible lower triangular matrix , its production matrix (also called its Stieltjes matrix; see [11, 14]) is the matrix , where is the matrix with its first row removed. The production matrix can be characterized by the matrix equality , where ( is the usual Kronecker delta).
Lemma 1 (see [14]). Assume that is an infinite lower triangular matrix with . Then is a Riordan array if and only if its production matrix is of the form where is the A-sequence and is the Z-sequence of the Riordan array .
Definition 2. Let be a sequence of polynomials where is of degree and . We say that is a polynomial sequence of Riordan type if the coefficient matrix is an element of the Riordan group; that is, there exists a Riordan array such that . In this case, we say that is the polynomial sequence associated with the Riordan array .
Letting , , then in matrix form we have Hence, by using (1), we have the following lemma.
Lemma 3. Let be the polynomial sequence associated with a Riordan array , and let be its generating function. Then
In [15], Luzón introduced a new notation to represent the Riordan arrays and gave a recurrence relation for the family of polynomials associated to Riordan arrays. In recent works [16, 17], a new definition by means of a determinant form for Appell polynomials is given. Sequences of Appell polynomials are special of the Sheffer sequences [18]. In [19], the author obtains a determinant representation for the Sheffer sequence. The aim of this work is to propose a similar approach for polynomial sequences of Riordan type, which are special of the generalized Sheffer sequences [12, 18]. A determinant representation for polynomial sequences of Riordan type is obtained by using production matrix of Riordan array. In fact, we will show that the general formula for the polynomial sequences of Riordan type can be expressed as the characteristic polynomial of the principal submatrix of the production matrix. As applications, determinant expressions for some classical polynomial sequences such as Fibonacci, Pell, and Chebyshev are derived, and a unified determinant expression for the four kinds of Chebyshev polynomials [20, 21] is established.
In this section we are going to develop our main theorem.
Theorem 4. Let be a Riordan array with the Z-sequence and the A-sequence . Let be the polynomial sequence associated with . Then satisfies the recurrence relation: with initial condition , and . In general, for all , is given by the following Hessenberg determinant:
Proof. Let and . Then from definition and (3), we have and . Hence and . Letting , then and , where . Letting be the production matrix of , then , and . Thus , and . In matrix form, we have
Using the block matrix method, we get
Since The previous matrix equation can be rewritten as Therefore, , and for , we have or equivalently
By applying the Cramer’s rule, we can work out the unknown operating with the first equations in (15):
After transferring the last column to the first position, an operation which introduces the factor , the theorem follows.
Corollary 5. Let be a Riordan array with production matrix . Let be the polynomial sequence associated with . Then , and for all , where , is the principal submatrix of order of the production matrix and is the identity matrix of order .
A useful application of Theorem 4 is to find the determinant expression of a well-known sequence. We illustrate the ideal in the following examples. In the final paragraph, we will give a unified determinant expression for the four kinds of Chebyshev polynomials.
Example 6. Considering the Riordan array , we have . The generating functions of the - and -sequences of are
Let be the polynomial sequence associated with . Then satisfies the recurrence relation: with initial condition , and . In general, is also given by the following Hessenberg determinant: If , , then becomes the Fibonacci polynomials: If , , then gives the Pell polynomials: In case , , becomes the Chebyshev polynomials of the second kind:
Example 7. Considering the Riordan array , we have . Then the generating functions of the - and -sequences of are
Let be the polynomial sequence associated with . Then satisfies the recurrence relation: with initial condition , and , .
In general, is also given by the following Hessenberg determinant:
If , , then become the Chebyshev polynomials of the first kind :
In case , , , give the Fermat polynomials (see [15]):
If , , , then becomes the Chebyshev polynomials of the third kind :
If , , , then gives the Chebyshev polynomials of the fourth kind :
Finally, considering the Riordan array , we have . Then the generating functions of the - and -sequences of are
Let be the polynomial sequence associated with . Then satisfies the recurrence relation:
with initial condition , and , . For , we have
Therefore we can give, now, the following.
Definition 9. The Chebyshev polynomial of degree , denoted by , is defined by where is represented by a Hessenberg determinant of order .
Note that , , , and . Hence, Definition 9 can be considered as a unified form for the four kinds of Chebyshev polynomials.
The authors wish to thank the editor and referee for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grant no. 11261032) and the Natural Science Foundation of Gansu Province (Grant no. 1010RJZA049).
G.-S. Cheon, H. Kim, and L. W. Shapiro, “Combinatorics of Riordan arrays with identical
A
Z
sequences,” Discrete Mathematics, vol. 312, no. 12-13, pp. 2040–2049, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
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Chua's circuit - Wikipedia
Chua's circuit. The component NR is a nonlinear negative resistance called a Chua's diode. It is usually made of a circuit containing an amplifier with positive feedback.
The current–voltage characteristic of the Chua diode
Chua's circuit (also known as a Chua circuit) is a simple electronic circuit that exhibits classic chaotic behavior. This means roughly that it is a "nonperiodic oscillator"; it produces an oscillating waveform that, unlike an ordinary electronic oscillator, never "repeats". It was invented in 1983 by Leon O. Chua, who was a visitor at Waseda University in Japan at that time.[1] The ease of construction of the circuit has made it a ubiquitous real-world example of a chaotic system, leading some to declare it "a paradigm for chaos".[2]
1 Chaotic criteria
3 Self-excited and hidden Chua attractors
Chaotic criteria[edit]
One version of Chua's circuit, where the nonlinear Chua's diode is synthesized by an op amp negative impedance converter (OPA1) and a diode–resistor network (D1, D2, R2)
An autonomous circuit made from standard components (resistors, capacitors, inductors) must satisfy three criteria before it can display chaotic behaviour.[3] It must contain:
one or more nonlinear elements,
one or more locally active resistors,
three or more energy-storage elements.
Chua's circuit is the simplest electronic circuit meeting these criteria.[3] As shown in the top figure, the energy storage elements are two capacitors (labeled C1 and C2) and an inductor (labeled L; L1 in lower figure).[4] A "locally active resistor" is a device that has negative resistance and is active (it can amplify), providing the power to generate the oscillating current. The locally active resistor and nonlinearity are combined in the device NR, which is called "Chua's diode". This device is not sold commercially but is implemented in various ways by active circuits. The circuit diagram shows one common implementation. The nonlinear resistor is implemented by two linear resistors and two diodes. At the far right is a negative impedance converter made from three linear resistors and an operational amplifier, which implements the locally active resistance (negative resistance).
Computer simulation of Chua's circuit after 100 seconds, showing chaotic "double scroll" attractor pattern
Chua's attractor for different values of the α parameter
Analyzing the circuit using Kirchhoff's circuit laws, the dynamics of Chua's circuit can be accurately modeled by means of a system of three nonlinear ordinary differential equations in the variables x(t), y(t), and z(t), which represent the voltages across the capacitors C1 and C2 and the electric current in the inductor L1 respectively. These equations are:
{\displaystyle {\frac {dx}{dt}}=\alpha [y-x-f(x)],}
{\displaystyle RC_{2}{\frac {dy}{dt}}=x-y+Rz,}
{\displaystyle {\frac {dz}{dt}}=-\beta y.}
The function f(x) describes the electrical response of the nonlinear resistor, and its shape depends on the particular configuration of its components. The parameters α and β are determined by the particular values of the circuit components.
A computer-assisted proof of chaotic behavior (more precisely, of positive topological entropy) in Chua's circuit was published in 1997.[5] A self-excited chaotic attractor, known as "the double scroll" because of its shape in the (x, y, z) space, was first observed in a circuit containing a nonlinear element such that f(x) was a 3-segment piecewise-linear function.[6]
The easy experimental implementation of the circuit, combined with the existence of a simple and accurate theoretical model, makes Chua's circuit a useful system to study many fundamental and applied issues of chaos theory. Because of this, it has been object of much study and appears widely referenced in the literature.
Further, Chua' s circuit can be easily realized by using a multilayer CNN (cellular nonlinear network). CNNs were invented by Leon Chua in 1988.
The Chua diode can also be replaced by a memristor; an experimental setup that implemented Chua's chaotic circuit with a memristor was demonstrated by Muthuswamy in 2009; the memristor was actually implemented with active components in this experiment.[7]
Self-excited and hidden Chua attractors[edit]
Two hidden chaotic attractors and one hidden periodic attractor coexist with two trivial attractors in Chua circuit (from the IJBC cover[8]).
The classical implementation of Chua circuit is switched on at the zero initial data, thus a conjecture was that the chaotic behavior is possible only in the case of unstable zero equilibrium. In this case a chaotic attractor in mathematical model can be obtained numerically, with relative ease, by standard computational procedure where after transient process a trajectory, started from a point of unstable manifold in a small neighborhood of unstable zero equilibrium, reaches and computes a self-excited attractor. To date, a large number of various types of self-excited chaotic attractors in Chua's system have been discovered.[9] However, in 2009, N. Kuznetsov discovered hidden Chua's attractors coexisting with stable zero equilibrium,[10][11] and since then various scenarios of the birth of hidden attractors have been described.[8]
Experimental confirmation[edit]
First experimental confirmation of chaos from Chua's circuit was reported in 1985 at the Electronics Research Lab at U.C. Berkeley.[12]
^ Matsumoto, Takashi (December 1984). "A Chaotic Attractor from Chua's Circuit" (PDF). IEEE Transactions on Circuits and Systems. IEEE. CAS-31 (12): 1055–1058. doi:10.1109/TCS.1984.1085459. Retrieved 2008-05-01.
^ Madan, Rabinder N. (1993). Chua's circuit: a paradigm for chaos. River Edge, N.J.: World Scientific Publishing Company. Bibcode:1993ccpc.book.....M. ISBN 981-02-1366-2.
^ a b Kennedy, Michael Peter (October 1993). "Three steps to chaos – Part 1: Evolution" (PDF). IEEE Transactions on Circuits and Systems. Institute of Electrical and Electronics Engineers. 40 (10): 640. doi:10.1109/81.246140. Retrieved February 6, 2014.
^ Kennedy, Michael Peter (October 1993). "Three steps to chaos – Part 2: A Chua's circuit primer" (PDF). IEEE Transactions on Circuits and Systems. Institute of Electrical and Electronics Engineers. 40 (10): 658. doi:10.1109/81.246141. Retrieved February 6, 2014.
^ Z. Galias, "Positive topological entropy of Chua's circuit: a computer-assisted proof", Int. J. Bifurcations and Chaos, 7 (1997), pp. 331–349.
^ Chua, Leon O.; Matsumoto, T.; Komuro, M. (August 1985). "The Double Scroll". IEEE Transactions on Circuits and Systems. IEEE. CAS-32 (8): 798–818. doi:10.1109/TCS.1985.1085791.
^ Bharathwaj Muthuswamy, "Implementing memristor based chaotic circuits", International Journal of Bifurcation and Chaos, Vol. 20, No. 5 (2010) 1335–1350, World Scientific Publishing Company, doi:10.1142/S0218127410026514.
^ a b Stankevich N. V.; Kuznetsov N. V.; Leonov G. A.; Chua L. (2017). "Scenario of the birth of hidden attractors in the Chua circuit". International Journal of Bifurcation and Chaos. 27 (12): 1730038–188. arXiv:1710.02677. Bibcode:2017IJBC...2730038S. doi:10.1142/S0218127417300385. S2CID 45604334.
^ Bilotta, E.; Pantano, P. (2008). Gallery of Chua Attractors. World Scientific. ISBN 978-981-279-062-0.
^ Leonov G. A.; Vagaitsev V. I.; Kuznetsov N. V. (2011). "Localization of hidden Chua's attractors" (PDF). Physics Letters A. 375 (23): 2230–2233. Bibcode:2011PhLA..375.2230L. doi:10.1016/j.physleta.2011.04.037.
^ Leonov G. A.; Kuznetsov N. V. (2013). "Hidden attractors in dynamical systems. From hidden oscillations in Hilbert–Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits". International Journal of Bifurcation and Chaos. 23 (1): 1330002–219. Bibcode:2013IJBC...2330002L. doi:10.1142/S0218127413300024.
^ Zhong, G.-Q.; Ayrom, F. (January 1985). "Experimental confirmation of chaos from Chua's circuit". International Journal of Circuit Theory and Applications. 13 (1): 93–98. doi:10.1002/cta.4490130109.
Chaos synchronization in Chua's circuit, Leon O Chua, Berkeley: Electronics Research Laboratory, College of Engineering, University of California, [1992], OCLC: 44107698
Chua's Circuit Implementations: Yesterday, Today and Tomorrow, L. Fortuna, M. Frasca, M. G. Xibilia, World Scientific Series on Nonlinear Science, Series A - Vol. 65, 2009, ISBN 978-981-283-924-4
Recai Kilic (2010). A Practical Guide for Studying Chua's Circuits. World Scientific. Bibcode:2010pgsc.book.....K. ISBN 978-981-4291-14-9.
Chua's Circuit: Diagram and discussion
NOEL laboratory. Leon O. Chua's laboratory at the University of California, Berkeley
Chua and Memristors
Hidden attractor in Chua's system
https://eecs.berkeley.edu/~chua/papers/Arena95.pdf
Interactive Chua's circuit 3D simulation
Chua's circuit 3D numerical interactive experiment, experiences.math.cnrs.fr
Retrieved from "https://en.wikipedia.org/w/index.php?title=Chua%27s_circuit&oldid=1077187850"
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Medical Entrance Exam Question and Answers | Thermodynamics - Zigya
The density of a substance at 0°C is 10 g/cc and at 100° C, its density is 9.7 g/cc. The coefficient of linear expansion of the substance is
Initial temperature ( T1 ) = 0o C
Initial density ( ρ1 ) = 10 g/cc
Final temperature ( T2 ) = 100oC
Final density ( ρ2 ) = 9.7 g/cc
{\mathrm{\alpha }}_{\mathrm{v}}
\frac{∆\mathrm{\rho }}{{\mathrm{\rho }}_{1}\quad \times \quad ∆\mathrm{T}}
\frac{{\mathrm{\rho }}_{1}\quad -\quad {\mathrm{\rho }}_{2}}{{\mathrm{\rho }}_{1}\quad \left(\quad {\mathrm{T}}_{2}\quad -\quad {\mathrm{T}}_{1}\quad \right)}
\frac{10\quad -\quad 9.7}{10\quad \left(100\quad -\quad 0\right)}
\frac{0.3}{1000}
\mathrm{\alpha }
v = 3 × 10-4
Therefore linear coefficient of linear expansion
\mathrm{\alpha }
\frac{\mathrm{\gamma }}{3}
\frac{3\quad \times \quad {10}^{-4}}{3}
\mathrm{\alpha }
1 = 10-4
An engine is working. It takes 100 calories of heat from source and leaves 80 calories of heat to sink. If the temperature of source is 127°C, then temperature of sink is
Heat taken from source Q1 = 100 cal
Heat left to sink Q2 = 80 cal
∴ efficiency of the engine
-\quad \frac{{\mathrm{Q}}_{2}}{{\mathrm{Q}}_{1}}
-\quad \frac{80}{100}
Temperature of the sink
-\quad \frac{{\mathrm{T}}_{2}}{{\mathrm{T}}_{1}}
0.2 = 1
-\quad \frac{{\mathrm{T}}_{2}}{400}
\frac{{\mathrm{T}}_{2}}{400}
-
T2 = 0.8 × 400
⇒ T2 = 47°C
Assertion: Mass and volume are extensive properties.
Reason: Mass/volume is also an extensive parameter.
Extensive properties are dependent upon the amount of the substance. e.g. mass, volume, etc.
Intensive properties are independent of the amount of the substance e.g. temperature, density.
Air is expanded from 50 litres to 150 litres at atmosphere the external work done is (1 atmospheric pressure =1
\times
{10}^{5}\mathrm{N}/{\mathrm{m}}^{2}
2\times {10}^{-8}\mathrm{J}
2\times {10}^{4}\mathrm{J}
2\times {10}^{4}\mathrm{J}
\mathrm{Work}\quad \mathrm{done}\quad \mathrm{is}\quad \mathrm{W}=\quad \mathrm{P}∆\mathrm{V}\phantom{\rule{0ex}{0ex}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =2\times {10}^{5}\times (150-50)\times {10}^{-3}\phantom{\rule{0ex}{0ex}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =2\times {10}^{4}\mathrm{J}
Anatomy and Bismuth are usually used in thermocouple, because
a constant e..m.f is produced
higher thermo e.m.f is produced
a negative e.m.f is produced
lower thermo e.m.f is produced
When two wires of dissimilar metals like Antimony and Bismuth are joined to form a closed circuit with two junctions and if the junctions are at different temperatures emf is generated and current flows through the circuit.
The heat produced in a long wire is characterized by resistance, current and time through which the current passes. If the errors in measuring these quantities are respectively 1 %, 2% and 1%, then the total error in calculating the energy produced is
The heat produced in a wire due to current flown is given by
\frac{∆\mathrm{H}}{\mathrm{H}}\quad \quad =\quad \quad \frac{2∆\mathrm{I}}{\mathrm{I}}\quad +\quad \quad \frac{∆\mathrm{R}}{\mathrm{R}}\quad +\quad \frac{∆\mathrm{t}}{\mathrm{t}}
= 2 × 0.02 + 0.01 + 0.01
\frac{∆\mathrm{H}}{\mathrm{H}}
Helium at 270C has a volume 8 litres. it is suddenly compressed to a volume of 1 litre. The temperature of the gas will be
\left(\mathrm{V}=\quad \frac{5}{3}\right)
{{\mathrm{TV}}_{1}}^{\mathrm{\gamma }-1}={{\mathrm{TV}}_{2}}^{\mathrm{\gamma }-2}\phantom{\rule{0ex}{0ex}}{\mathrm{T}}_{2}=\left(\frac{{\mathrm{V}}_{1}}{{\mathrm{V}}_{2}}\right){\mathrm{T}}_{1}\phantom{\rule{0ex}{0ex}}{\mathrm{T}}_{2}={\left(\frac{{\mathrm{V}}_{1}}{{\mathrm{V}}_{2}}\right)}^{\frac{5}{3}-2}\times 300\phantom{\rule{0ex}{0ex}}{\mathrm{T}}_{2}={\left(\frac{8}{1}\right)}^{\frac{2}{3}}\times 300=4\times 300=1200\mathrm{k}\phantom{\rule{0ex}{0ex}}{\mathrm{T}}_{2}=1200-273={927}^{0}\mathrm{C}
∆
\therefore
∆
Assertion: During an adiabatic process, heat energy is not exchanged between the system and its surroundings.
Reason: The temperature of a gas increases when it undergoes an adiabatic expansion.
In the adiabatic process, no heat enters or leaves the system. The system is completely insulated from its surroundings. From the first law of thermodynamics, change in internal energy,
∆
E = q-W = 0-W = -W
Since the work is done at the expense of internal energy, therefore internal energy of the system decreases and hence temperature of the gas falls.
The latent heat of vapourisation of water is 2240. If the work done in the process of vapourisation of 1 g is 168 J, then increase in internal energy is
Latent heat of vapourisation of water ( L ) = 2240 J
Mass of water (m) = 1 g
Work done (dW ) = 168 J
From first law of thermodynamics, heat supplied in vapourisation
(dQ) = mL
= dU + dW
1 × 2240 = dU + 168
( where dU = increase in internal energy )
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When the beacon chain is not finalising it enters a special "inactivity leak" mode.
Attesters receive no rewards. Non-participating validators receive increasingly large penalties based on their track records.
This is designed to eventually restore finality in the event of a permanent failure of large numbers of validators.
If the beacon chain hasn't finalised a checkpoint for longer than MIN_EPOCHS_TO_INACTIVITY_PENALTY (4) epochs, then it enters "inactivity leak" mode.
The inactivity leak is a kind of emergency state in which rewards and penalties are modified as follows.
Attesters receive no attestation rewards while attestation penalties are unchanged.
Any validators deemed inactive have their inactivity scores raised, leading to an additional inactivity penalty that potentially grows quadratically with time. This is the inactivity leak, sometimes known as the quadratic leak.
Proposer and sync committee rewards are unchanged.
The idea for the inactivity leak (aka the quadratic leak) was proposed in the original Casper FFG paper. The problem it addresses is that of how to recover finality (liveness, in some sense) in the event that over one-third of validators goes offline. Finality requires a majority vote from validators representing 2/3 of the total stake.
It works as follows. When loss of finality is detected the inactivity leak gradually reduces the stakes of validators who are not making attestations until, eventually, the participating validators control 2/3 of the remaining stake. They can then begin to finalise checkpoints once again.
In any case, it provides a powerful incentive for stakers to fix any issues they have and to get back online.
The reason why no validators receive attestation rewards during an inactivity leak is once again due to the possibility of discouragement attacks. An attacker might deliberately drive the beacon chain into an inactivity leak, perhaps by a combination of censorship and denial of service attack on other validators. This would cause the non-participants to suffer the leak, while the attacker continues to attest normally. We need to increase the cost to the attacker in this scenario, which we do by not rewarding attestations at all during an inactivity leak.
As with penalties, the amounts subtracted from validators' beacon chain accounts due to the inactivity leak are effectively burned, reducing the overall net issuance of the beacon chain.
Let's study the effect of the leak on a single validator's balance, assuming that during the period of the inactivity leak (non-finalisation) the validator is completely offline.
At each epoch, the offline validator will be penalised an amount proportional to
tB / \alpha
t
is the number of epochs since the chain last finalised,
B
\alpha
is the INACTIVITY_PENALTY_QUOTIENT.
B
t
t(t+1)B / 2\alpha
B
\frac{dB}{dt}=-\frac{tB}{\alpha}
, which can be solved to give the exponential
B(t)=B_0e^{-t^2/2\alpha}
\alpha
1 / \sqrt{e}
, or around 60.7% of its initial value. With the value of INACTIVITY_PENALTY_QUOTIENT at 3 * 2**24, this equates to around seven thousand epochs, or 31.5 days.
For Phase 0 of the beacon chain, the value of INACTIVITY_PENALTY_QUOTIENT was increased by a factor of four from
2^{24}
2^{26}
, so that validators would be penalised less severely if there were non-finalisation due to implementation problems in the early days. As it happens, there were no instances of non-finalisation during the whole eleven months of Phase 0 of the beacon chain.
The value was decreased by one quarter in the Altair upgrade from
2^{26}
3 \times 2^{24}
as a step towards eventually setting it to its final value. Decreasing the inactivity penalty quotient speeds up recovery of finalisation in the event of an inactivity leak.
During Phase 0, the inactivity penalty was an increasing global amount applied to all validators that did not participate in an epoch, regardless of their individual track records of participation. So a validator that was able to participate for a significant fraction of the time could still be quite severely penalised due to the growth of the inactivity penalty. Vitalik gives a simplified example: "if fully [off]line validators get leaked and lose 40% of their balance, someone who has been trying hard to stay online and succeeds at 90% of their duties would still lose 4% of their balance. Arguably this is unfair." We found during the Medalla testnet incident that keeping a validator online when all around you is chaos is not easy. We don't want to punish stakers who are honestly doing their best.
To improve this, the Altair upgrade introduced individual validator inactivity scores that are stored in the state. The scores are updated each epoch as follows.
increase the score by INACTIVITY_SCORE_BIAS (four) otherwise.
When not in an inactivity leak,
decrease every validator's score by INACTIVITY_SCORE_RECOVERY_RATE (sixteen).
Graphically, the flow-chart looks like this.
Note that there is a floor of zero on the score.
When not in an inactivity leak validators' inactivity scores are reduced by INACTIVITY_SCORE_RECOVERY_RATE + 1 per epoch when they make a timely target vote, and by INACTIVITY_SCORE_RECOVERY_RATE - INACTIVITY_SCORE_BIAS when they don't. So, even for non-performing validators, scores decrease outside a leak.
p
0
1
\lambda
N
\max (0, N((1-p)\lambda - p))
\lambda = 4
\max (0, N(4 - 5p))
This is nice because, if many validators are able to participate intermittently, it indicates that whatever event has befallen the chain is potentially recoverable, unlike a permanent network partition, or a super-majority network fork, for example. The inactivity leak is intended to bring finality to irrecoverable situations, so prolonging the time to finality if it's recoverable is likely a good thing.
The following graph illustrates some scenarios. We have an inactivity leak that starts at zero, and ends after 100 epochs, after which finality is recovered and we are no longer in the leak. There are five validators. Working up from the lowest line, they are:
Always online: correctly registering a timely target vote in every epoch. The inactivity score remains at zero.
90% online: the inactivity score remains bounded near zero. From the analysis above, it is expected that anything better than 80% online will bound the score near zero.
70% online: the inactivity score grows slowly over time.
Generally online, but offline between epochs 50 and 75: the inactivity score is zero during the initial online period; grows linearly and fairly rapidly while offline during the leak; declines slowly when back online during the leak; and declines rapidly once the leak is over.
Always offline: the inactivity score increases rapidly during the leak, and declines even more rapidly once the leak is over.
The inactivity scores of five different validator personas in an inactivity leak that starts at zero and ends at epoch 100 (labelled "End" and shown with a dashed line). The dotted lines labelled "A" and "B" mark the start and end of the offline period for the fourth validator.
The inactivity penalty is applied to all validators at every epoch based on their individual inactivity scores, irrespective of whether a leak is in progress or not. When there is no leak, the scores return to zero (rapidly for active validators, less rapidly for inactive ones), so most of the time this is a no-op.
The penalty for validator
is calculated as
\begin{split} s_iB_i / (\tt{INACTIVITY\_SCORE\_BIAS} \times \tt{INACTIVITY\_PENALTY\_QUOTIENT\_ALTAIR}) \\ = \frac{s_iB_i}{4 \times 50{,}331{,}648} \end{split}
s_i
is the validator's inactivity score, and
B_i
is the validator's effective balance.
This penalty is applied at each epoch, so (for constant
B_i
) the total penalty is proportional to the area under the curve of the inactivity score, above. With the same five validator persona's we can quantify the penalties in the following graph.
Always online: no penalty due to the leak.
90% online: negligible penalty due to the leak.
70% online: the total penalty grows quadratically but slowly during the leak, and rapidly stops after the leak ends.
Generally online, but offline between epochs 50 and 75: a growing penalty during the leak, that rapidly stops when the leak ends.
Always offline: we can clearly see the quadratic nature of the penalty in the initial parabolic shape of the curve. After the end of the leak it takes around 35 epochs for the penalties to return to zero.
The balance retained by each of the five validator personas after the inactivity leak penalty has been applied. The scenario is identical to the chart above.
We can see that the new scoring system means that some validators will continue to be penalised due to the leak even after finalisation starts again. This is intentional. When the leak causes the beacon chain to finalise, at that point we have just two-thirds of the stake online. If we immediately stop the leak (as we used to), then the amount of stake online would remain close to two-thirds and the chain would be vulnerable to flipping in and out of finality as small numbers of validators come and go. We saw this behaviour on some of the testnets prior to launch. Continuing the leak after finalisation serves to increase the balances of participating validators to greater than two-thirds, providing a buffer that should mitigate such behaviour.
Inactivity scores are updated during epoch processing in process_inactivity_updates().
Inactivity penalties are calculated in def_get_inactivity_penalty_deltas().
For the original description of the mechanics of the inactivity leak, see the Casper paper, section 4.2.
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Logistic Distribution - MATLAB & Simulink - MathWorks í•œêµ
â\infty <\mathrm{μ}<\infty
\mathrm{Ï}â¥0
f\left(x|\mathrm{μ},\mathrm{Ï}\right)=\frac{\mathrm{exp}\left\{\frac{xâ\mathrm{μ}}{\mathrm{Ï}}\right\}}{\mathrm{Ï}{\left(1+\mathrm{exp}\left\{\frac{xâ\mathrm{μ}}{\mathrm{Ï}}\right\}\right)}^{2}}\text{â};\text{â}â\infty <x<\infty \text{â}.
The loglogistic distribution is closely related to the logistic distribution. If x is distributed loglogistically with parameters μ and σ, then log(x) is distributed logistically with parameters μ and σ.
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Euler simulation of stochastic differential equations (SDEs) for SDE, BM, GBM, CEV, CIR, HWV, Heston, SDEDDO, SDELD, or SDEMRD models - MATLAB simByEuler - MathWorks í•œêµ
d{X}_{t}=S\left(t\right)\left[L\left(t\right)-{X}_{t}\right]dt+D\left(t,{X}_{t}^{\frac{1}{2}}\right)V\left(t\right)dW
D
d{X}_{t}=0.2\left(0.1-{X}_{t}\right)dt+0.05{X}_{t}^{\frac{1}{2}}dW
Paths = 250×1
Times = 250×1
d{X}_{t}=S\left(t\right)\left[L\left(t\right)-{X}_{t}\right]dt+D\left(t,{X}_{t}^{\frac{1}{2}}\right)V\left(t\right)dW
D
d{X}_{t}=0.2\left(0.1-{X}_{t}\right)dt+0.05{X}_{t}^{\frac{1}{2}}dW
d{X}_{1t}=B\left(t\right){X}_{1t}dt+\sqrt{{X}_{2t}}{X}_{1t}d{W}_{1t}
d{X}_{2t}=S\left(t\right)\left[L\left(t\right)-{X}_{2t}\right]dt+V\left(t\right)\sqrt{{X}_{2t}}d{W}_{2t}
d{X}_{t}=0.25{X}_{t}dt+0.3{X}_{t}d{W}_{t}
The simByEuler function partitions each time increment dt into NSteps subintervals of length dt/NSteps, and refines the simulation by evaluating the simulated state vector at NSteps − 1 intermediate points. Although simByEuler does not report the output state vector at these intermediate points, the refinement improves accuracy by allowing the simulation to more closely approximate the underlying continuous-time process.
Direct specification of the dependent random noise process used to generate the Brownian motion vector (Wiener process) that drives the simulation. This argument is specified as the comma-separated pair consisting of 'Z' and a function or as an (NPeriods ⨉ NSteps)-by-NBrowns-by-NTrials three-dimensional array of dependent random variates.
{X}_{t}=P\left(t,{X}_{t}\right)
Dependent random variates used to generate the Brownian motion vector (Wiener processes) that drive the simulation, returned as a (NPeriods ⨉ NSteps)-by-NBrowns-by-NTrials three-dimensional time series array.
d{X}_{t}=F\left(t,{X}_{t}\right)dt+G\left(t,{X}_{t}\right)d{W}_{t}
[1] Deelstra, G. and F. Delbaen. “Convergence of Discretized Stochastic (Interest Rate) Processes with Stochastic Drift Term.†Applied Stochastic Models and Data Analysis., 1998, vol. 14, no. 1, pp. 77–84.
[2] Higham, Desmond, and Xuerong Mao. “Convergence of Monte Carlo Simulations Involving the Mean-Reverting Square Root Process.†The Journal of Computational Finance, vol. 8, no. 3, 2005, pp. 35–61.
[3] Lord, Roger, et al. “A Comparison of Biased Simulation Schemes for Stochastic Volatility Models.†Quantitative Finance, vol. 10, no. 2, Feb. 2010, pp. 177–94
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SurfacePlot - Maple Help
Home : Support : Online Help : Statistics and Data Analysis : Statistics Package : Visualization : SurfacePlot
SurfacePlot(X, Y, Z, plotoptions)
SurfacePlot['interactive'](X, Y, Z)
The SurfacePlot command generates a surface plot for the specified data.
The third parameter Z is the third data sample - given as e.g. a Vector. Note that X, Y and Z must have the same number of elements.
The plotoptions argument can contain one or more plot options. All plot options will be passed to the plots[display] command. See plot/options for details.
\mathrm{with}\left(\mathrm{Statistics}\right):
X≔〈1,2,3,4,5,6,7,8,9,9〉:
Y≔〈1,4,2,3,3,1,5,1,5,5〉:
Z≔〈5,2,-3,-4,5,2,-1,2,1,1〉:
\mathrm{SurfacePlot}\left(X,Y,Z,\mathrm{title}="Surface Plot"\right)
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Post 8: Sampling the variance of person ability with a Gibbs step | R-bloggers
Post 8: Sampling the variance of person ability with a Gibbs step
The Bayesian 2PL IRT model we defined in Post 1 set a hyper-prior on the variance of the person ability parameters. This post implements the sampler for this parameter as a Gibbs step. We will check that Gibbs step is working by running it on the fake data from Post 2, visualizing the results, and checking that the true value is recovered.
Implementing the variance sampler
pi_{pi}^{u_{pi}} (1 – pi_{pi})^{1-u_{pi}}
f(sigma^2_theta|text{rest}) & propto
f_text{Inverse-Gamma}left(sigma^2_theta left|
alpha_theta + frac{P}{2},
beta_theta + frac{1}{2} sum_{p=1}^P theta^2_p right.
right) ,
text{rest}
f_star(dagger|dots)
dagger
star
pi_{pi}
ln{frac{pi_{pi}}{1+pi_{pi}}} = a_i ( theta_p – b_i) label{eq:pipi} quad.
Note that the complete conditional for
sigma^2_theta
only depends on the data (
boldsymbol U
) via the values of
boldsymbol theta
. This is common for parameters that are at higher levels of model hierarchy.
Since the complete conditional for
sigma^2_theta
is distributed as an inverse-gamma, we can sample from the complete conditional directly without the need for the Metropolis-Hastings algorithm. This is referred to as a Gibbs step.
We implement the sampler in R by noting that an Inverse-Gamma is defined to be the reciprocal of a Gamma random variable, and that R implements a draw from a Gamma with the rgamma function:
sample.s2 <- function(U.data, old) {
## Grab the appropriate values
alpha.th <- old$hyperpar$alpha.th
beta.th <- old$hyperpar$beta.th
P.persons <- length(old$th)
cur <- old
cur$s2 <- 1/rgamma(1, shape = alpha.th + P.persons/2,
rate = beta.th + sum((old$th)^2)/2 )
Note that we do not need to calculate an acceptance rate because it is always 1.
Testing that the sampler works
Since the complete conditional for the variance depends only on the person ability parameters, we simplify our test of the variance sampler code by setting the item parameter samplers to the dummy sampler from Post 3.
## Reset the item sampler to dummy samplers
sample.a <- dummy.cc.sampler
sample.b <- dummy.cc.sampler
run.D <- run.chain.2pl(
## Keep item parameters at their true values
## Generate starting values uniformly from 0 to 5
s2.init = runif( 1, min=0, max=5),
MH.th=1, MH.a=NA, MH.b=NA,
run.D.mcmc <- mcmc( t(run.D) )
plot( run.D.mcmc[, get.2pl.params(1,1,1,1)],
density=FALSE, smooth=TRUE )
It looks like the sampler burns in almost immediately. We discard the first 100 iterations to be conservative
run.D.mcmc.conv <- window( run.D.mcmc, start=100)
and then make a parameter recovery plot for the variance:
Where we first define a new helper function check.sigma as
check.sigma <- function( mcmc.conv , xylim) {
traceplot(mcmc.conv[, 'sig2.theta'],
smooth=TRUE, ylim=xylim,
main='Trace plot sig2.theta')
abline(h=sig2.theta , col='blue')
densplot( mcmc.conv[, 'sig2.theta'],
main='Density plot sig2.theta',
xlab='sig2.theta', xlim=xylim,)
abline(v=sig2.theta , col='blue')
abline(v=var(theta.abl), col='purple')
legend( "topright",
c('true value','var(theta.abl)'),
col=c('blue', 'purple'),
lty='solid')
Then we call check.sigma for this example to make the graph:
check.sigma( run.D.mcmc.conv, c(1.2,2))
The MCMC estimate is off from the true value of
sigma^2_theta
, which is plotted in blue. This is not a concern for two reasons:
The true value is contained within the credible interval of the estimate.
The MCMC estimator for
sigma^2_theta
is (roughly) an estimate of the sample variance of the
theta_p
values (see Post 1 for details). Note that our MCMC estimate of the posterior mode is quite close to the sample variance of the generated
theta_p
values, which is plotted in purple. Indeed, if we generate different data, the posterior mode will find the sample variance of the person ability parameters each time (not shown).
As an additional check that the Gibbs step is working, we can make sure that the person ability parameters are still recovered well:
## Calculate the EAP estimates
all.eap <- apply( run.D.mcmc.conv, MARGIN=2, mean )
The person ability parameters are, indeed, still recovered well.
The sampler for the variance of the person parameters is working well.
In the next post we combine all of the samplers and tune them.
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