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Fun with Natural Language Processing’s “Secret Sauce“
Computers don’t understand the nuances of language. It's because they only understand numbers and, as you can imagine, it’s impossible for us to enumerate every single nuance in understanding human language (let alone as numbers). But, we’ve seen a lot of progress in recent years of computers understanding language. So how do these work? More specifically, how are these system representing words as numbers?
In this post, we’re going to talk about one of the coolest advances in machine learning and natural language processing: word embeddings. I hope that this post serves as a launch pad to get started with embeddings, including code from how the models are trained to using pre-trained embeddings. In addition, we'll be looking at interesting use cases of word embeddings and recent research in the area of representation.
So how do we represent one of the most basic units of natural language, words? If this is the first time you’ve thought about this, you might be tempted to say the appropriate data structure is obviously a string! However, this comes with some design considerations:
Your compiler doesn’t understand the contents of strings - it just recognizes the ASCII values that correspond to each symbol. This representation has room for improvement. Recognizing the word cat as being [3, 1, 20] is not very useful for capturing the meaning of words.
Strings can have a variable length. If we wanted to give a machine learning model a string, it might truncate (or artificially elongate) values for consistency’s sake and we would lose (or create useless) information.
For a long time, one of best approaches we could do was representing a word as an vector (or array) with all 0s that had 1 in the index corresponding to a specific word. This is called one-hot encoding. For example, in the vocabulary {"I", "am", "a", "cat"}, the word I would correspond to [1, 0, 0, 0], the word am as [0, 1, 0, 0], and so forth. We’ll see that word representation has shifted recent from one-hot encoding to something better.
So what can we do about our representation problem? Surprise, surprise: we can turn to linguistics for the answer. Distributional semantics, in particular, holds the key:
"You shall know a word by the company it keeps." - Firth (1957)
In essence, it makes sense to be defining words in relation to other words. Intuitively, this is easy to see; if I asked you to describe the word “avocado”, you would probably define it in terms of other words like “fruit” and “green” because the meanings of the word are close. This idea of word “closeness” gives rise to the notion of placing our words in some kind of space where we can measure their distance.
To be specific, this means these words exist in some vector spaces such as
\mathbb{R}^2
. This means every word can be thought as a vector, like with one-hot encoding! This may seem scary, but this is just because we want to mathematically determine the “closeness” between two words. We could define a one dimensional scalar value for each word (e.g. {"cat": 1}), but the nuances of language extend far beyond a single dimension. That’s why we place them in higher order dimensions - typically in the tens or hundreds. Our brains can’t really imagine dimensions higher than three, so for the sake of visualization we’ll work in two dimensions for now: the classic
x-y
We’ll discuss how to obtain word vectors (hereby referred to as word embeddings) in just a little bit. For now, let’s imagine that in our vocabulary, we have the word “cat”. Furthermore, let’s imagine that we also have access to a word embedding fairy that gives us the perfect vector for each word.
As such, she gives us the vector [1, 4] to represent the word cat.
Why do we want to represent a three-letter word as a vector with potentially a vector of hundreds of values? In short, we want to create the embeddings such that the vectors capture the meaning of a given word. This can intuitively be visualized as the vectors for similar words being group together. For example, if the vector for "cat" is [1, 4], the vector for "kitten" would be something like [2, 4] whereas the vector for "dog" would be close by, for example [1, 5].
Since words are now vectors, we are also able to perform linear algebra operations on the given language. Although it may feel weird to subtract dog from cat, it turns out performing such operations tends to be useful for a variety of tasks. For example, the cosine distance (which encodes similarity) between two vectors is a powerful function that is easily applied to tasks involving natural language. For word vectors
u
v
, we can define cosine similarity as:
\operatorname{cos}(u, v) = \frac{u \cdot v}{|u| \cdot |v|} = \frac{\sum_{i = 1}^{n} u_i v_i}{\sqrt{\sum_{i = 1}^{n} u^2_i} \cdot \sqrt{\sum_{i = 1}^{n} v^2_i}}
As a result, something interesting we can do is train our word embeddings to create analogies. For example, a classic example in the field is using word embeddings to see that "king" - "man" = "queen" - "woman". We can even generalize this to fill-in-the-blanks for sentences like “Bill Gates is to Microsoft as ____ is to Apple” by predicting "Steve Jobs". This prediction is relatively straightforward when you have good embeddings and can be computed as:
d = \operatorname*{arg\, max}_{v \in V} ~ \operatorname{cos} (v, ~ a - b + c)
s
is our cosine similarity function from earlier. In the above example, we have that a = "Bill Gates", b = "Microsoft", c = "Apple". Finally, this gives us d = "Steve Jobs".
Why Are Word Embeddings "Secret Sauce"?
Analogy generation was a fun result that was discovered by researchers, but this isn't where the true potential of word emebddings lie. Although these ad-hoc analyses are interesting to think about, the real use of word embeddings is to serve as a semantically-aware representation of words for other downstream tasks. For example, providing word embeddings to a neural network that powers a chatbot will let it generate sentences that make more sense than if we represented words using a string-to-index mapping.
The use of pre-trained word embeddings galvanized progress in natural language processing research since representation is often at the root of most machine learning problems. It's hard to think of mathematical grounding for this kind of phenomenon, but intuitively it's clear that better representation of language means that our neural networks can better understand semantics and therefore model language.
It seems like perfect word embeddings will end up being too specific and are therefore good to be exist. These concerns are true thanks to the curse of dimensionality, but we can actually still approximate very powerful word embeddings that capture semantic meaning by training neural networks using stochastic gradient descent.
But, why neural networks? One previous method of word embedding generation was to perform dimensionality reduction on word co-occurrence matrices (which doesn’t involve deep learning). This procedure captures the intuition behind distributional semantics, but doesn’t have the powerful non-linearity capabilities of neural networks. However, it’s still useful to think about since certain methods of generation word embeddings draw upon this as reference.
But to be clear, I would actually like you to forget about pre-neural network methods, for now. It turns out that NLP can be implemented “from scratch“, i.e. purely through statistical and neural means. (You can read more about this from Collobert et al.).
How Do We Create Word Embeddings?
So you now know we can generate word embeddings using a neural network. But exactly how do we do that? A commonly used implementation to generate word embeddings is called word2vec, which is what we will use as reference in this guide. This model was conceptualized at Google around 5 years ago and has gone on to push the state of the art in natural language processing.
The word2vec model generates word embeddings through one of two related models. Both models are be trained using different objectives and as such, we can build two simple neural networks that performs the following tasks:
Continuous Bag Of Word: predicts a given missing word in a sentence/phrase based on context (faster but less specific)
Skip Gram: given a word, predicts the words that will co-appear near it (slower but works better for infrequent words)
If you notice, they are in essence the inverse of the other. This is good for our intuition of how word2vec works to generate word embeddings as both are really good examples of the distributional hypothesis from earlier!
A simple implementation of the above objectives would be logistic regression, which is nothing more than a fancy perceptron. You might be wondering: how do we get the word vectors from this process? Turns out what we're actually doing is making the network perform a fake that we training the network off of - we actually won’t use the model that’s trained. Instead, the goodies are encoded in the parameters of the neural network layers: the weights and biases of each neuron. The network’s internal representation of different words encodes the embeddings that we are looking for.
Word embeddings are an idea from as early as 2003. In the nearly two decades, there has been a huge surge in its use across the machine learning landscape. From powering virtually every NLP system at Google to inspiring a slew of new models, here are some recent developments regarding word embeddings:
Spotify, Airbnb, etc.
BERT, etc.
Let’s use pre-trained word embeddings from Google (trained by reading through Google News). Using trusted pre-trained models will allow us to quickly play with word vectors as well as prototype with deep learning faster since such models already been worked well in practice.
We first want to run pip install pymagnitude to install the embedding format. Then we can download the pre-trained word2vec embeddings using some wget magic:
wget http://magnitude.plasticity.ai/word2vec/light/GoogleNews-vectors-negative300.magnitude
Finally, we can import the package and start writing queries:
vectors = Magnitude("GoogleNews-vectors-negative300.magnitude")
print(vectors.distance("cat", "dog"))
There's a lot of great documentation for how you can query the vectories and gain interesting insights available at the GitHub repository for Magnitude here.
Training Word Embeddings
Training word embeddings with a given dataset is easy using gensim, a Python package that abstracts the implementation of the word2vec neural network. This is one of the most commonly used Python package for generating word embeddings.
Extra: Visualizing Word Embeddings with t-SNE
It would be cool to visualize the word vectors. Sadly, we humans are mostly incapable of visualizing in the 300th dimension.
Instead, we can use a process called dimensionality reduction which will allow us to turn our 300 dimensions into regular 2D vectors that we can visualize. We will be using an algorithm called t-SNE (t-Distributed Stochastic Neighbouring Entities) to perform our dimensionality reduction from 300 dimensions to 2 dimensions.
You might be wondering how we can find a correspondance of vectors in
\mathbb{R}^{300}
to vectors in
\mathbb{R}^2
. Why don't we just work with these 2D vectors in the first place? The truth is, these new embeddings (in 2D) actually do lose information. This seems obvious since we are going from a high dimensional space to a lower dimensional space. But more concretely, if we imagine an embedding space as being a distribution of points, we can measure the Kullback-Leiber Divergence of the original vectors and the transformed vectors:
KL(p || q) = \sum_x {~ p(x) \cdot \operatorname{log} ~ \frac{p(x)}{q(x)}}
This is a measure of how "different" two probability distributions are. As a result, minimizing the KL Divergence between the two distributions using gradient descent "learns" us new a representation of the original embeddings such that they preserve information. This is a very powerful result! It helps us build intuition on high-dimensional embeddings for otherwise blackbox systems.
This is a multi-core implementation of t-SNE available here. In my usage, Sci-Kit Learn's implementation of the algorithm is much slower, especially when running on more powerful machines (e.g. Google Colab).
embeddings = TSNE(n_jobs=4).fit_transform(vectors)
vis_x = embeddings[:, 0]
vis_y = embeddings[:, 1]
plt.scatter(vis_x, vis_y, c=digits.target, cmap=plt.cm.get_cmap("jet", 10), marker='.')
Extra: Gensim Compatibility with Gensim
A lot of natural language processing might use the gensim package, which has a different API as the faster pymagnitude package we’ve been using. In order to interface with the pymagnitude model, we can write a wrapper class to use the same API as the gensim model:
class Word2Vec:
self.layer1_size = self.vectors.dim
return self.vectors.query(word)
return word in self.vectors
return self.vectors.dim
Using this, we can wrap our magnitude model as follows:
vectors = Magnitude('vectors.magnitude')
w2v = Word2Vec(vectors)
And we can access the vector exactly the same as we would with gensim as follows:
cat_vector = w2v['cat']
This should resolve a lot of compatibility issues if you choose to leverage faster Magnitude embeddings with an existing Gensim codebase.
Epilogue: why am I writing about this?
I read about word embeddings sometime during my freshman year of university. I’m not really sure where I learned about it, but I found the idea really enchanting.
Word embeddings are a good introduction to neural networks as well as computational linguistics, and it’s what eventually introduced me to academia - which has made me a better developer and computer scientist. I decided to pay homage by writing this tutorial. I know lots of good resources about word embeddings exist already, but I wanted to help excite others to the wonders of NLP!
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Estimate Models Using armax - MATLAB & Simulink - MathWorks Switzerland
Load a sample data set z8 with three inputs and one output, measured at 1 -second intervals and containing 500 data samples.
Use armax to both construct the idpoly model object, and estimate the parameters:
A\left(q\right)y\left(t\right)=\sum _{i=1}^{nu}{B}_{i}\left(q\right){u}_{i}\left(t-n{k}_{i}\right)+C\left(q\right)e\left(t\right)
Typically, you try different model orders and compare results, ultimately choosing the simplest model that best describes the system dynamics. The following command specifies the estimation data set, z8 , and the orders of the A , B , and C polynomials as na , nb , and nc, respectively. nk of [0 0 0] specifies that there is no input delay for all three input channels.
opt.SearchOptions.Tolerance = 1e-5;
m_armax = armax(z8, [na nb nc nk], opt);
Focus, Tolerance, and MaxIter are estimation options that configure the estimation objective function and the attributes of the search algorithm. The Focus option specifies whether the model is optimized for simulation or prediction applications. The Tolerance and MaxIter search options specify when to stop estimation. For more information about these properties, see the armaxOptions reference page.
armax is a version of polyest with simplified syntax for the ARMAX model structure. The armax method both constructs the idpoly model object and estimates its parameters.
View information about the resulting model object.
m_armax
m_armax =
A(z) = 1 - 1.284 z^-1 + 0.3048 z^-2 + 0.2648 z^-3 - 0.05708 z^-4
B1(z) = -0.07547 + 1.087 z^-1 + 0.7166 z^-2
B2(z) = 1.019 + 0.1142 z^-1
B3(z) = -0.06739 + 0.06828 z^-1 + 0.5509 z^-2
C(z) = 1 - 0.06096 z^-1 - 0.1296 z^-2 + 0.02489 z^-3 - 0.04699 z^-4
Polynomial orders: na=4 nb=[3 2 3] nc=4 nk=[0 0 0]
m_armax is an idpoly model object. The coefficients represent estimated parameters of this polynomial model. You can use present(m_armax) to show additional information about the model, including parameter uncertainties.
View all property values for this model.
get(m_armax)
A: [1 -1.2836 0.3048 0.2648 -0.0571]
B: {[-0.0755 1.0870 0.7166] [1.0188 0.1142] [-0.0674 ... ]}
C: [1 -0.0610 -0.1296 0.0249 -0.0470]
F: {[1] [1] [1]}
Report: [1x1 idresults.polyest]
The Report model property contains detailed information on the estimation results. To view the properties and values inside Report, use dot notation. For example:
m_armax.Report
Status: 'Estimated using ARMAX with simulation focus'
Method: 'ARMAX'
This action displays the contents of estimation report such as model quality measures (Fit), search termination criterion (Termination), and a record of estimation data (DataUsed) and options (OptionsUsed).
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Shear rate - Wikipedia
Rate of change in the shear deformation of a material with respect to time
In physics, shear rate is the rate at which a progressive shearing deformation is applied to some material.
Simple shear[edit]
The shear rate for a fluid flowing between two parallel plates, one moving at a constant speed and the other one stationary (Couette flow), is defined by
{\displaystyle {\dot {\gamma }}={\frac {v}{h}},}
{\displaystyle {\dot {\gamma }}}
is the shear rate, measured in reciprocal seconds;
v is the velocity of the moving plate, measured in meters per second;
h is the distance between the two parallel plates, measured in meters.
{\displaystyle {\dot {\gamma }}_{ij}={\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}.}
For the simple shear case, it is just a gradient of velocity in a flowing material. The SI unit of measurement for shear rate is s−1, expressed as "reciprocal seconds" or "inverse seconds".[1]
The shear rate at the inner wall of a Newtonian fluid flowing within a pipe[2] is
{\displaystyle {\dot {\gamma }}={\frac {8v}{d}},}
{\displaystyle {\dot {\gamma }}}
v is the linear fluid velocity;
The linear fluid velocity v is related to the volumetric flow rate Q by
{\displaystyle v={\frac {Q}{A}},}
where A is the cross-sectional area of the pipe, which for an inside pipe radius of r is given by
{\displaystyle A=\pi r^{2},}
{\displaystyle v={\frac {Q}{\pi r^{2}}}.}
Substituting the above into the earlier equation for the shear rate of a Newtonian fluid flowing within a pipe, and noting (in the denominator) that d = 2r:
{\displaystyle {\dot {\gamma }}={\frac {8v}{d}}={\frac {8\left({\frac {Q}{\pi r^{2}}}\right)}{2r}},}
which simplifies to the following equivalent form for wall shear rate in terms of volumetric flow rate Q and inner pipe radius r:
{\displaystyle {\dot {\gamma }}={\frac {4Q}{\pi r^{3}}}.}
For a Newtonian fluid wall, shear stress (τw) can be related to shear rate by
{\displaystyle \tau _{w}={\dot {\gamma }}_{x}\mu }
where μ is the dynamic viscosity of the fluid. For non-Newtonian fluids, there are different constitutive laws depending on the fluid, which relates the stress tensor to the shear rate tensor.
^ "Brookfield Engineering - Glossary section on Viscosity Terms". Archived from the original on 2007-06-09. Retrieved 2007-06-10.
^ Darby, Ron (2001). Chemical Engineering Fluid Mechanics (2nd ed.). CRC Press. p. 64. ISBN 9780824704445.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Shear_rate&oldid=1087918970"
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Pseudo-localisation of singular integrals in $L^p$ | EMS Press
L^p
As a step in developing a non-commutative Calderón-Zygmund theory, J. Parcet (J. Funct. Anal. {\bf 256} (2009), no. 2, 509-593) established a new pseudo-localisation principle for classical singular integrals, showing that
Tf
has small
L^2
norm outside a set which only depends on
f\in L^2
but not on the arbitrary normalised Calderón-Zygmund operator
T
. Parcet also asked if a similar result holds true in
L^p
p\in(1,\infty)
. This is answered in the affirmative in the present paper. The proof, which is based on martingale techniques, even somewhat improves on the original
L^2
result.
Tuomas Hytönen, Pseudo-localisation of singular integrals in
L^p
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Proactive Market Making Algorithm | DODO Docs
PMM: A Universal Liquidity Framework#
To keep our market-making algorithm running smoothly and efficiently, we need to boil the vast sea of market information down to its most crucial core metric. So, what is a market’s “most important metric”? The answer is liquidity. Liquidity can be graphically represented by a market depth chart, as shown below.
A depth chart consists of two roughly triangular (though not necessarily symmetrical) shapes, representing bids (buy orders) on the left and asks (sell orders) on the right, along the price x-axis and the depth y-axis. The two triangles can be mathematically described by two parameters, mid price and slope, or how “steep” the triangle is.
Let us closely examine the depth triangle on the right hand side first. This is the ask side, where ask (sell) prices are quoted. We can see that the more base tokens are sold, the higher the price. This linear relationship can be captured by the following formula:
P = i + ik(\frac{B_0-B}{B_0})
where i is Parameter 1, the mid price, and k is Parameter 2, the slope. B is the number of base tokens currently in the inventory and B_0 is the initial number of base tokens in the inventory. (B_0-B)/B_0 is the portion of base tokens that have been removed from the ask side due to transactions, relative to the initial base token balance. This formula stipulates that as the number of base tokens that have been traded increases, the base token price rises linearly.
Is this an accurate representation of market reality? Not exactly, as this linear model has two limitations:
In practice, most liquidity is concentrated near (immediately above or below) the mid price, because that is the most capital-efficient strategy for market makers. The linear model does not reflect this uneven distribution and is thus an oversimplification
The linear model returns a liquidity of zero after the price exceeds or goes below a certain threshold. However, in reality, no matter how favourable the quoted price is (e.g. for ETH/USDC, a bid order at $100 and an ask order at $1,500), there is liquidity present at that price. This model fails to take such scenarios into account
Therefore, we need to make this pricing curve/depth chart nonlinear to align it with market patterns, but we also don’t want to introduce additional parameters. How should we go about doing that?
We want to make the depth chart nonlinear to depict the fact that depth is more concentrated in the vicinity of the mid price.
Mathematically, the most obvious and straightforward solution is to change the addition in the aforementioned linear formula to multiplication, like this:
P = i(\frac{B_0}{B})
In this formula, P increases as B decreases, and it also doesn’t have an upper or lower bound (technically it has a lower limit of 0, but a subzero price doesn’t make sense anyway). But what about the slope? The solution is to refactor the B_0/B term and add a new parameter k that we can use to control the magnitude of the change in price due to B.
P = i(1-k + k\frac{B_0}{B})
When B_0/B >= 1, P is directly proportional to B_0/B in the previous formula, but in this new formula, k dictates the extent of which P is affected by B_0/B. More specifically, k is in the range [0, 1] and governs the slope of the pricing curve.
When k = 0, the formula becomes P = i, so the price does not change regardless of other parameters.
When k = 1, the formula reverts back to (2).
When k is in (0, 1), as k increases, so does the price elasticity, meaning that the price becomes more sensitive to changes in base token quantity (i.e. B). Conversely, as k decreases, the price elasticity also decreases.
This model seems sufficiently complete to cover all scenarios, but there is another issue. In a transaction, the total amount of tokens that needs to be paid is the area under the pricing curve, so we will have to take the integral of the curve, but the curve formula above makes this calculation cumbersome as B_0/B introduces a logarithmic term during derivation. To make computation easier, we square the B_0/B term to eliminate all instances of log:
P = i(1-k + k(\frac{B_0}{B}^2))
Incredibly, when k = 1, this curve is identical to the AMM bonding curve. This reaffirms our belief that this algorithm has captured the essence of market activities and patterns.
Similarly, without loss of generality, we apply the same derivation procedure for the bid side depth chart, substituting base tokens with quote tokens (denoted by Q) and using division instead of multiplication. We get:
P=i/(1-k+(\frac{Q_0}{Q})^2k)
Combining both formulae, we get the proactive market maker (PMM) pricing formula, described in mathematical terms below.
P_{margin}=iR
R
determined by the following formula:
if B<B_0, then R=1-k+(\frac{B_0}{B})^2k
if Q<Q_0, then R=1/(1-k+(\frac{Q_0}{Q})^2k)
else R=1
The PMM algorithm is a “high-fidelity” abstraction of the orderbook-based market, defined and regulated by a handful of simple parameters, but it is also highly flexible and optimized for on-chain operations.
We will now enumerate several promising use cases for PMM that can be achieved by fine-tuning parameters and instituting different withdrawal/deposit rules.
Use case1#
For mainstream assets, such as BTC and ETH, external markets have much higher volumes and are thus a price source for other platforms from which to retrieve market prices. PMM is capable of proactively adjusting these fetched mid prices to minimize impermanent loss (IL) and achieve higher capital efficiency than AMM platforms. This mechanic also means unlocking single-token liquidity provision — market makers are not forced to deposit tokens Uniswap-style.
The configurations required for this use case are:
Mid price i is set to the price retrieved from external sources.
Parameter k is set to below 1.
Everyone is given the single-token liquidity provision option.
We call this use case DODO Classic Pool, as this was first pioneered in DODO v1.0 in August 2020.
This use case mainly applies to long-tail asset markets (i.e. predominantly newly issued assets with little sell-side liquidity on AMM platforms). PMM can help these assets with the initial liquidity they desperately require for their long-term growth and sustainability. With PMM, asset issuers do not need large amounts of capital on standby to pair up with their assets when initializing liquidity pools. For instance, if a team wants to issue their token X on PMM, they have the option to initialize liquidity with 100% X and 0% stables or ETH. This drastically reduces the barrier-to-entry for smaller projects.
In this use case, PMM gives the pricing power to takers entirely — makers have no control over the price discovery mechanic whatsoever.
Mid price i is set to the initial offering price designated by the asset issuers.
Parameter k can be set to any arbitrary number in [0, 1].
The first liquidity deposit can be made in arbitrary proportions, and it does not change the price.
All subsequent liquidity deposits and withdrawals must be made in proportion to the current pool ratio (i.e. similar to Uniswap liquidity pools).
We call this use case the DODO Vending Machine.
Fully customizable and free market making
This use case is intended for experienced and ambitious market makers (both institutions and individuals), who want the highest degree of freedom and customizability possible to execute their own market making strategies. In this use case, all liquidity in the liquidity pools belongs to the market makers themselves and they also have full control over all the pool parameters. Market makers can dynamically adjust the asset price by changing these parameters based on their assessment of market sentiment, valuation, and other factors. Moreover, market makers can deposit to and withdraw from these liquidity pools in arbitrary ratios, without affecting the asset price.
For a more concrete example, a ETH/USDT market maker in this use case can choose to market-make near ETH=700USDT with a very small k in order to provide highly competitive liquidity and earn considerable transaction/swap fees from trading activity. When the market maker foresees or predicts an increase in ETH price, they can then react accordingly by removing some ETH from their pool to reduce their market risk exposure. This maneuver does not affect the liquidity on the USDT side, however, so trading activity can continue as usual.
This use case also applies to issuers of new assets, who can choose to only deposit the tokens they are issuing, without any capital (e.g. ETH, USDT, or other stablecoins). They can set the initial offering price and a small k to ensure low price elasticity, so that the token price does not fluctuate too dramatically due to the influx of trading activity. This design also means that when token issuers need capital for development and operations, they can simply withdraw capital from the liquidity pool without affecting the sell-side liquidity.
The only configuration required for this use case is that:
Deposits/Withdrawals are set so that only market makers (owners/creators of the pools) are allowed to perform such operations.
Single-token liquidity provision/removal is allowed.
We call this use case the DODO Private Pool.
Crowdpooling is a portmanteau of “crowdsourcing” and “liquidity pools”, and this use case is an innovation compared to current asset issuance mechanics. For a newly issued token,
The platform does not allow token trading immediately upon launch; all sale participants receive tokens (regardless of the amount they purchased or the timing of the purchase) at the same unit price.
After the conclusion of the sale, token trading is enabled, and the remaining unsold tokens and liquidity collected from the last phase are used to construct a DODO Vending Machine.
We call this use case CrowdPooling.
Reversion to Traditional AMM
This use case ties into our aforementioned claim that PMM is essentially a generalization of AMMs. When:
k is set to 1.
Deposits/Withdrawals are made in proportion to the currently pool ratio.
PMM behaves exactly the same as AMMs.
capable of supporting stablecoin trading scenarios
k = 0.001 (when k = 0, the exchange rate becomes exactly 1 to 1)
PMM is virtually identical to Curve in terms of performance and capital efficiency, with the added benefits of flexibility, since k can be tweaked to ensure 1-to-1 ratio or make it closer to AMMs where price fluctuation is more pronounced.
« DODO V2 docs
PMM Core Concepts »
PMM: A Universal Liquidity Framework
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A ball is thrown horizontally from the top of a tower with a velocity of 40 ms-1 Take g - Physics - Motion In A Straight Line - 11663691 | Meritnation.com
For horizontal distance, gravity plays no role.
Horizontal Dis\mathrm{tan}ce=vt\phantom{\rule{0ex}{0ex}}So, in 1,2,3,4,5 sec the dis\mathrm{tan}ce will be\phantom{\rule{0ex}{0ex}}40m, 80m,120m,160m,200m\phantom{\rule{0ex}{0ex}}For vertical dis\mathrm{tan}ce\phantom{\rule{0ex}{0ex}}S=ut+0.5g{t}^{2}\phantom{\rule{0ex}{0ex}}For 1sec, s=45m\phantom{\rule{0ex}{0ex}}for 2 sec\phantom{\rule{0ex}{0ex}}S=80+20=100m\phantom{\rule{0ex}{0ex}}For 3 sec\phantom{\rule{0ex}{0ex}}S=120+45=165m\phantom{\rule{0ex}{0ex}}For 4 sec\phantom{\rule{0ex}{0ex}}S=160+80=240m\phantom{\rule{0ex}{0ex}}For 5sec\phantom{\rule{0ex}{0ex}}S=200+125=325m\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}
The path of the ball will be parabolic
(b) When the ball hits the ground, its final velocity is zero
H=\frac{{u}^{2}}{2g}\phantom{\rule{0ex}{0ex}}H=\frac{{40}^{2}}{20}=80m
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Conserved energies for the cubic nonlinear Schrödinger equation in one dimension
15 November 2018 Conserved energies for the cubic nonlinear Schrödinger equation in one dimension
Herbert Koch, Daniel Tataru
Duke Math. J. 167(17): 3207-3313 (15 November 2018). DOI: 10.1215/00127094-2018-0033
We consider the cubic nonlinear Schrödinger (NLS) equation as well as the modified Korteweg–de Vries (mKdV) equation in one space dimension. We prove that for each
s>-\frac{1}{2}
there exists a conserved energy which is equivalent to the
{H}^{s}
-norm of the solution. For the Korteweg–de Vries (KdV) equation, there is a similar conserved energy for every
s\ge -1
The current version of this article supersedes the original advance publication version posted on 26 October 2018. Corrections have been made in the following locations: equations (1.2), (2.2), (2.4), and (2.11); the displays in the proof of Lemma 4.3; the last two displays in the proof of Proposition B.2; the second display in the proof of Theorem B.18; and the third paragraph in Appendix C.
Herbert Koch. Daniel Tataru. "Conserved energies for the cubic nonlinear Schrödinger equation in one dimension." Duke Math. J. 167 (17) 3207 - 3313, 15 November 2018. https://doi.org/10.1215/00127094-2018-0033
Received: 12 August 2017; Revised: 23 May 2018; Published: 15 November 2018
Secondary: 35Q53 , 37K10
Keywords: fractional Sobolev bounds , Korteweg-de Vries , new conserved energies , nonlinear Schrödinger , transmission coefficient
Herbert Koch, Daniel Tataru "Conserved energies for the cubic nonlinear Schrödinger equation in one dimension," Duke Mathematical Journal, Duke Math. J. 167(17), 3207-3313, (15 November 2018)
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Regularity, local behavior and partial uniqueness for self-similar profiles of Smoluchowski’s coagulation equation | EMS Press
We consider Smoluchowski's equation with a homogeneous kernel of the form
a(x,y) = x^\alpha y ^\beta + x^\beta y^\alpha
-1 < \alpha \leq \beta < 1
\lambda := \alpha + \beta \in (-1,1)
. We first show that self-similar solutions of this equation are infinitely differentiable and prove sharp results on the behavior of self-similar profiles at
y = 0
\alpha < 0
. We also give some partial uniqueness results for self-similar profiles: in the case
\alpha = 0
we prove that two profiles with the same mass and moment of order
\lambda
are necessarily equal, while in the case
\alpha < 0
we prove that two profiles with the same moments of order
\alpha
\beta
, and which are asymptotic at
y = 0
, are equal. Our methods include a new representation of the coagulation operator, and estimates of its regularity using derivatives of fractional order.
José A. Cañizo, Stéphane Mischler, Regularity, local behavior and partial uniqueness for self-similar profiles of Smoluchowski’s coagulation equation. Rev. Mat. Iberoam. 27 (2011), no. 3, pp. 803–839
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§ Centroid of a tree
A centroid is a node which upon removal creates subtrees of size at most ceil(n/2).
§ Existence of centroid for rooted tree (algorithm to compute centroid)
If tree has exactly one node, we are done, the centroid is the root.
Suppose for induction a centroid exists for trees of size
n-1
. We will now prove the existence of a centroid for tree of size
n
Otherwise, if the root has all children whose subtree sizes are at most ceil(n/2), the root is the centroid and we are done.
Otherwise, the root has one child with subtree size strictly greater than ceil(n/2). There can't be two such children, because their combined size would be 2*ceil(n/2) >= n. This is nonsensical, as the size of the subtrees plus the root node would mean the tree has 2*ceil(n/2) + 1 >= n+1 nodes, a contradiction.
We recurse into the subtree. The size of the subtree of the child is at least one less than the size of the root, thus we are decreasing on the size of the tree.
By recursion, we must terminate this process and find a centroid.
§ Centroid decomposition
Once we find the centroid of a tree, we see that all of its subtrees has size less than ceil(n/2).
We can now recurse, and find sizes of centroids of these subtrees.
These subtrees are disjoint, so we will take at most O(n) to compute sizes and whatnot.
We can do this log(n) many steps since we're halving the size of the subtree each time.
In total, this implies that we can recursively find centroids to arrive at a "centroid decomposition" of a tree.
Note that the centroid decomposition of the tree constructs a new tree, which is different from the original tree, sorta how the dominator tree is a different tree from the original tree.
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CALCULLA - Least Common Multiple (LCM) calculator
Least Common Multiple (LCM) calculator - finds LCM for up to 10 given numbers and shows process of dividing by primes with the school-like vertical notation.
The multiples of number 36 are:
36 (1 × 36),
108 (3 × 36),
31644 (879 × 36),
The multiples of number 31752 are:
31752 (1 × 31752),
127008 (4 × 31752),
The least common multiple is 31752.
Least common multiple (in short: LCM) is the smallest positive integer that is a multiple of two or more numbers. This means also, it can be divided by these numbers without a reminder.
ⓘ Example: Numbers 2 and 3 have LCM of 6 because 6 divides completely by both two and three.
ⓘ Example: Numbers 4 and 10 have LCM of 20 because 20 divides completely by both 4 and 10.
As you can see in the first example, LCM was simply the product (multiplication) of given numbers. However, in the second example is a much smaller number than multiplication.
Least common multiple is sometimes called lowest common multiple or smallest common multiple
The least common multiple of the numbers a and b is usually denoted by LCM(a, b) or lcm(a, b).
\text{LCM}(a, b) = \text{least common multiple of numbers} \left\{a, b\right\}
The least common multiple can also be determined for more numbers e.g. LCM(4, 6, 3) is 12 because it is the lowest number, which is divisible by all those three numbers.
\text{LCM}(a, b, c, ...) = \text{least common multiple of numbers} \left\{a, b, c, ... \right\}
The least common multiple is used for operations on fractions, for example, to calculate the common denominator needed when we add or subtract the fractions.
ⓘ Example: We want to add 1/3 to 1/4. The least common multiple of the denominators 3 and 4 is 12 because this number is divided by both of them:
\text{LCM}(3, 4) = 12
To add fractions, we convert them to a common denominator being the least common multiple of both denominators of the input fractions:
\dfrac{1}{3} + \dfrac{1}{4} = \dfrac{1 \times 4}{3 \times 4} + \dfrac{1 \times 3}{4 \times 3} = \dfrac{4}{12} + \dfrac{3}{12} = \dfrac{7}{12}
ⓘ Hint: If you want to learn more about adding or subtracting fractions check our other calculator: Fractions: add and subtract step by step
A property similar to LCM is the greatest common divisor (in short: GCD), which is the largest natural number by which all of the given numbers divide.
ⓘ Hint: If you want to learn more about GCD check our other calculator: GCD.
\text{GCD}(a, b) = \dfrac{a \times b} {\text{LCM}(a, b)}
least_common_multiple · lcm · lowest_common_multiple · smallest_common_multiple
https://calculla.com/least_common_multiple?inText=36%2C%2031752
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Delay spread - Wikipedia
In telecommunications, the delay spread is a measure of the multipath richness of a communications channel. In general, it can be interpreted as the difference between the time of arrival of the earliest significant multipath component (typically the line-of-sight component) and the time of arrival of the last multipath components. The delay spread is mostly used in the characterization of wireless channels, but it also applies to any other multipath channel (e.g. multipath in optical fibers).
Delay spread can be quantified through different metrics, although the most common one is the root mean square (rms) delay spread. Denoting the power delay profile of the channel by
{\displaystyle A_{c}(\tau )}
, the mean delay of the channel is
{\displaystyle {\overline {\tau }}={\frac {\int _{0}^{\infty }\tau A_{c}(\tau )d\tau }{\int _{0}^{\infty }A_{c}(\tau )d\tau }}}
and the rms delay spread is given by [1]
{\displaystyle \tau _{\text{rms}}={\sqrt {\frac {\int _{0}^{\infty }(\tau -{\overline {\tau }})^{2}A_{c}(\tau )d\tau }{\int _{0}^{\infty }A_{c}(\tau )d\tau }}}}
The formula above is also known as the root of the second central moment of the normalised delay power density spectrum
The importance of delay spread is how it affects the Inter Symbol Interference (ISI). If the symbol duration is long enough compared to the delay spread (typically 10 times as big would be good enough), one can expect an equivalent ISI-free channel. The correspondence with the frequency domain is the notion of coherence bandwidth (CB), which is the bandwidth over which the channel can be assumed flat(i.e. channel that passes all spectral components with approximately equal gain and linear phase.). Coherence bandwidth is related to the inverse of the delay spread. The shorter the delay spread, the larger is the coherence bandwidth. Delay spread has a significant impact on Intersymbol interference.
^ Goldsmith, Andrea (2005), Wireless Communications, Cambridge University Press, p. 86, ISBN 978-0-521-83716-3
Saunders, Antennas and propagation for Wireless communication systems, 2nd ed, pp246–250, 2007
Retrieved from "https://en.wikipedia.org/w/index.php?title=Delay_spread&oldid=904873204"
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Solve this: - Maths - Statistics - 11175637 | Meritnation.com
Please refer the following question for answer to similar query
https://www.meritnation.com/ask-answer/question/find-the-mean-and-variance-for-the-following-frequency-dist/statistics/6301318
For s\mathrm{tan}dard deviation find the value of \sigma \phantom{\rule{0ex}{0ex}}\sigma =\sqrt{ {\sigma }^{2}}=\sqrt{variance}\phantom{\rule{0ex}{0ex}}=\sqrt{132}=11.489
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Study of Multimode Combustion System With Gasoline Direct Injection | J. Eng. Gas Turbines Power | ASME Digital Collection
Jian-Xin Wang,
Shi-Jin Shuai,
Shi-Jin Shuai
Yan-Jun Wang,
Guo-Hong Tian,
Guo-Hong Tian
Xin-Liang An
J. Eng. Gas Turbines Power. Oct 2007, 129(4): 1079-1087 (9 pages)
Wang, Z., Wang, J., Shuai, S., Wang, Y., Tian, G., and An, X. (October 2, 2006). "Study of Multimode Combustion System With Gasoline Direct Injection." ASME. J. Eng. Gas Turbines Power. October 2007; 129(4): 1079–1087. https://doi.org/10.1115/1.2718221
In this paper, a multimode combustion system was developed in a gasoline direct injection engine. A two-stage fuel-injection strategy, including flexible injection timings and flexible fuel quantity, is adopted as a main means to form desired mixture in the cylinder. The combustion system can realize five combustion modes. The homogeneous charge spark ignition (HCSI) mode was used at high load to achieve high-power output density; stratified charge spark ignition (SCSI) was adopted at intermediate load to get optimum fuel economy; stratified charge compression ignition (SCCI) was introduced at transient operation between SI and CI mode. Homogeneous charge compression ignition (HCCI) was utilized at part load to obtain ultralow emissions. Reformed charge compression ignition (RCCI) was imposed at low load to extend the HCCI operation range. In SI mode, the stratified concentration is formed by introducing a second fuel injection in the compression stroke. This kind of stratified mixture has a faster heat release than the homogeneous mixture and is primarily optimized to reduce the fuel consumption. In CI mode, the cam phase configurations are switched from positive valve overlap to negative valve overlap (NVO). The test results reveal that the CI combustion is featured with a high gradient pressure after ignition and has advantages in high thermal efficiency and low
NOx
emissions over SI combustion at part load.
internal combustion engines, ignition, valves
Combustion, Emissions, Fuels, Gasoline, Homogeneous charge compression ignition engines, Ignition, Engines, Cylinders, Stress, Compression, Valves, Combustion systems, Temperature
Homogenous Charge Compression Ignition (HCCI) Engine: Key Research and Development Issues
Investigation of Hydrocarbon Emissions From a Direct Injection-Gasoline Premixed Charge Compression Ignited Engine
HCCI Combustion: The Sources of Emissions at Low Loads and the Effects of GDI Fuel Injection
,” 8th Diesel Engine Emissions Reduction Workshop, August 25–29.
Study on Homogeneous Charge Compression Ignition Gasoline Engine
,” 5th International Symposium on Diagnostics and Modeling of Combustion in Internal Combustion Engines (COMODIA 2001) July, Nagoya.
Fuerhapter
CSI-Controlled Auto Ignition—The Best Solution for the Fuel Consumption Versus Emission Trade-Off?
” SAE Paper No. 2003-01-0754.
The new AVL CSI Engine–HCCI Operation on a Multicylinder Gasoline Engine
Direct Gasoline Injection in the Negative Valve Overlap of a Homogeneous Charge Compresson Ignition Engine
Expansion of HCCI Operating Region by the Combination of Direct Fuel Injection, Negative Valve Overlap and Internal Fuel Reformation
Managing SI∕HCCI Dual-Mode Engine Operation
Numerical Simulation of Mixture Formation and Combustion of Gasoline Engines With Multi-Stage Direct Injection Compression Ignition (DICI)
,” ASME Paper No. GTP05-1167.
A Comparison of Imaging of Compression Ignition Phenomena of Iso-Octane, Indolene, and Gasoline Fuels in a Single-Cylinder Research Engine
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A Transversely Isotropic Viscoelastic Constitutive Equation for Brainstem Undergoing Finite Deformation | J. Biomech Eng. | ASME Digital Collection
Xinguo Ning,
Xinguo Ning
Yoram Lanir,
e-mail: margulie@seas.upenn.edu
Ning, X., Zhu, Q., Lanir, Y., and Margulies, S. S. (June 29, 2006). "A Transversely Isotropic Viscoelastic Constitutive Equation for Brainstem Undergoing Finite Deformation." ASME. J Biomech Eng. December 2006; 128(6): 925–933. https://doi.org/10.1115/1.2354208
The objective of this study was to define the constitutive response of brainstem undergoing finite shear deformation. Brainstem was characterized as a transversely isotropic viscoelastic material and the material model was formulated for numerical implementation. Model parameters were fit to shear data obtained in porcine brainstem specimens undergoing finite shear deformation in three directions: parallel, perpendicular, and cross sectional to axonal fiber orientation and determined using a combined approach of finite element analysis (FEA) and a genetic algorithm (GA) optimizing method. The average initial shear modulus of brainstem matrix of 4-week old pigs was
12.7Pa
, and therefore the brainstem offers little resistance to large shear deformations in the parallel or perpendicular directions, due to the dominant contribution of the matrix in these directions. The fiber reinforcement stiffness was
121.2Pa
, indicating that brainstem is anisotropic and that axonal fibers have an important role in the cross-sectional direction. The first two leading relative shear relaxation moduli were 0.8973 and 0.0741, respectively, with corresponding characteristic times of 0.0047 and
1.4538s
, respectively, implying rapid relaxation of shear stresses. The developed material model and parameter estimation technique are likely to find broad applications in neural and orthopaedic tissues.
biomechanics, brain, viscoelasticity, shear deformation, finite element analysis, genetic algorithms, parameter estimation, neurophysiology, brain tissues, brainstem, shear, large deformation, viscoelastic, hyperelastic, anisotropic, genetic algorithm, finite element analysis
Biological tissues, Deformation, Fibers, Finite element analysis, Genetic algorithms, Shear (Mechanics), Brain, Die cutting, Shearing (Deformation), Shear deformation, Parameter estimation, Relaxation (Physics), Anisotropy
Finite Element Analysis of Traumatic Subdural Hematoma
Proc. of 31st Stapp Car Crash Conference
Finite Element Modeling of the Brain to Establish Diffuse Axonal Injury Criteria
,” Ph.D. dissertation, Ohio State University.
Proc. of Proceedings of 42nd Stapp Car Crash Conference
Proc. of 39th Stapp Car Crash Conference
Secondary Brainstem Involvement in Blunt Head Injury
Advance in Neurotraumatology
, Princeton.
Nonlinear Viscoelastic Effects in Oscillatory Shear Deformation of Brain Tissue
Design and Numerical Implementation of a 3-D Non-Linear Viscoelastic Constitutive Model for Brain Tissue During Impact
Regional, Directional, and Age Dependent Properties of the Brain Undergoing Large Deformation
Theory of the Mechanics of Fiber-Reinforced Composites
, Karlsson, & Sorensen, Inc., 2003, ABAQUS Theory Manual Version 6.3.
Handbook of Human Tolerance
Immediate Coma Following Inertia Brain Injury Dependent on Axonal Damage in the Brainstem
Large Strain Behavior of Brain Tissue in Shear: Some Experimental Data and Differential Constitutive Model
A Characterization of the Anisotropic Mechanical Properties of The Brainstem
,” Ph.D. dissertation, University of Pennsylvania.
The Effects of Directionality of Blunt Impacts on Mechanical Response of the Brain
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Line bundles and the Thom construction in noncommutative geometry | EMS Press
The idea of a line bundle in classical geometry is transferred to noncommutative geometry by the idea of a Morita context. From this we construct
\mathbb{Z}
\mathbb{N}
-graded algebras, the
\mathbb{Z}
-graded algebra being a Hopf–Galois extension. A non-degenerate Hermitian metric gives a star structure on this algebra, and an additional star operation on the line bundle gives a star operation on the
\mathbb{N}
-graded algebra. In this case, we carry out the associated circle bundle and Thom constructions. Starting with a C*-algebra as base, and with some positivity assumptions, the associated circle and Thom algebras are also C*-algebras. We conclude by examining covariant derivatives and Chern classes on line bundles after the method of Kobayashi and Nomizu.
Edwin Beggs, Tomasz Brzeziński, Line bundles and the Thom construction in noncommutative geometry. J. Noncommut. Geom. 8 (2014), no. 1, pp. 61–105
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On the Mixed Problem for Quasilinear Partial Differential-Functional Equations of the First Order | EMS Press
On the Mixed Problem for Quasilinear Partial Differential-Functional Equations of the First Order
We consider the mixed problem for the quasilinear partial differential-functional equation of the first order
D_z(x,y) = \sum^n_{i=1}f_i(x, y, z_{(x,y)} D_{y,i}z(x, y) + G(x, y, z_{(x,y)})
z(x,y) = \phi(x,y) \ \ \ ((x,y) \in [-r,a] x [-b,b + h] \backslash [0,a] \times [-b,b])
z_{(x,y)} : [-r,0] \times [0,h] \to \mathbb R
is a function defined by
z_{(x,y)}(t,s) = z(x + t,y + s)
(t,s) \in [-r,0] \times [0,h]
. Using the method of characteristics and the fixed-point method we prove, under suitable assumptions, a theorem on the local existence and uniqueness of solutions of the problem.
Tomasz Człapiński, On the Mixed Problem for Quasilinear Partial Differential-Functional Equations of the First Order. Z. Anal. Anwend. 16 (1997), no. 2, pp. 463–478
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Quasi-Periodic Solutions in Nonlinear Asymmetric Oscillations | EMS Press
Quasi-Periodic Solutions in Nonlinear Asymmetric Oscillations
The existence of Aubry--Mather sets and infinitely many subharmonic solutions to the following
p
-Laplacian like nonlinear equation
(p-1)^{-1}(\phi_p(x'))'+[\al\phi_p(x^+)-\beta\phi_p(x^-)]+g(x) = h(t)
\phi_p(u)=|u|^{p-2}u, \,p>1
, \,
\al, \beta
are \vspace{-0.05cm} positive constants satisfying \linebreak
\al^{-\frac{1}{p}}+\beta^{-\frac{1}{p}}=\frac2n
n\in \N, \,h
is piece-wise two times differentiable and
2\pi_p
-periodic,
g\in C^1(R)
x^{\pm}=\max \{\pm x, 0\}, \,\pi_p=\frac{2\pi}{p\sin(\pi/p)}.
Xiaojing Yang, KUEIMING LO, Quasi-Periodic Solutions in Nonlinear Asymmetric Oscillations. Z. Anal. Anwend. 26 (2007), no. 2, pp. 207–220
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{\displaystyle x_{RMS}={\sqrt {{\tfrac {1}{n}}\sum _{i=1}^{n}x_{i}^{2}}}}
{\displaystyle f_{RMS}={\sqrt {{\tfrac {1}{T_{2}-T_{1}\int _{T_{1}}^{T_{2}}[f(t)]^{2}}}dt}}}
{\displaystyle f_{RMS}={\sqrt {{\tfrac {1}{T\int _{0}^{T}[f(t)]^{2}}}dt}}}
{\displaystyle y(t)=Asin(2\pi ft+\varphi )=sin(\omega t+\varphi )}
{\displaystyle Y_{RMS}={\sqrt {{\tfrac {1}{T}}\int _{0}^{T}[Asin(\omega t)]^{2}dt}}}
{\displaystyle =A{\sqrt {{\tfrac {1}{T}}\int _{0}^{T}{\tfrac {1-cos(2\omega t)}{2}}dt}}}
{\displaystyle =A{\sqrt {{\tfrac {1}{T}}[{\tfrac {T}{2}}-{\tfrac {sin(2\omega t)}{4\omega }}]_{0}^{T}}}}
{\displaystyle ={\tfrac {A}{\sqrt {2}}}}
Thus, the RMS amplitude is 0.707 times the maximum amplitude.
The amplitude of a periodic variable is a measure of its change over a single period. Although the amplitude allows the relative sizes of sine waves to be compared, it does not give a good idea of what a sine wave can deliver in absolute terms. For instance, a sine wave can have both positive and negative amplitude values (Figure 1). when calculating the arithmetic mean of a sine wave, the negative values would offset the positive values and the result would be zero, this approach is not informative about the average wave. Thus, it’s often useful to specify the magnitude of a sine wave in a way that facilitates direct comparison with a non-oscillatory source of energy. One benefit of this is that it enables to describe how big a non-oscillatory source would be needed to deliver the same energy as the sine wave delivers in a particular length of time.
An analysis used for the overall amplitude of a signal is called RMS amplitude. Conceptually, it describes the average signal amplitude. However, it is different than simply measuring the arithmetic mean of a signal, it is derived by calculating the average power of a sine wave. This is where the RMS level can be useful. It is based on the magnitude of a signal as a measure of signal strength, regardless of whether the amplitude is positive or negative. The magnitude is calculated by squaring each sample value, therefore, they are all positive, then the signal average is calculated, eventually followed by the square root operation.
RMS mainly used in the context of sine waves. It can be considered as an alternative way of specifying how big a sine wave is, but with the advantage of allowing direct comparison with a non-oscillating source of energy.
For seismic, RMS is a most commonly used post-stack amplitude attribute, it computes the square root of the sum of squared amplitude values divided by the number of samples within the specified window. The windowed amplitudes are basically used as a simple and quick means to identify interesting zones of hydrocarbons for resource estimates in the reconnaissance stage.
The window selection is critical as different windows will provide varying amplitude patterns having diverse geological implications and requires a careful choice of window for the purpose. And squaring offers the opportunity for the high amplitudes to stand out best zones of hydrocarbons, while since amplitudes are squared before averaging, it also increases the noise, thus, RMS is highly sensitive to noise.
Essentially the RMS amplitudes for all samples in a selected window are considered for estimating amplitudes to be displayed in a plan view. In clastic, RMS is often helpful in delineating thin hydrocarbon sands for which appropriate slice must be chosen for use. The relative advantages and limitations of each slicing technique must be weighed based on the specific geologic issue on hand. For example, delineation by RMS windowed amplitude may show more amplitude standouts leading to overestimate of the hydrocarbon rock volume. RMS amplitude may work well for a single reservoir but not for multiple reservoirs occurring at different levels within the specified window especially if it is chosen arbitrarily and wide. The horizon or stratal amplitude slices, on the other hand suffer lesser contamination and are preferred for delineating single reservoirs provided the horizon phase is correctly identified and tracked for correlation.
[1] A Dictionary of Physics (Sixth Edition.). Oxford University Press. 2009. [2] Encyclopedic Dictionary of Applied Geophysics (Fourth Edition). SEG.2002 [3] Weisstein, Eric W. "Root-Mean-Square". MathWorld.
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§ monic and epic arrows
This is trivial, I'm surprised it took me this long to internalize this fact. When we convert a poset
(X, \leq)
into a category, we stipulate that
x \rightarrow y \iff x \leq y
. If we now consider the category
Set
of sets and functions between sets, and arrow
A \xrightarrow{f} B
A
B
. If
is monic, then we know that
|A| = |Im(f)| \leq |B|
. That is, a monic arrow behaves a lot like a poset arrow! Similarly, an epic arrow behaves a lot like the arrow in the inverse poset. I wonder if quite a lot of category theoretic diagrams are clarified by thinking of monic and epic directly in terms of controlling sizes.
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On the Solutions of a Quadratic Integral and an Integral-Differential Equation | EMS Press
An integral equation and a related integral-differential equation of first order over
\mathbb R_+
with a quadratic integral term representing the so-called autocorrelation of the unknown function is dealt with. For both equations the general solution is constructed and estimated in the
L^2
-norm. Further, the asymptotic behaviour and the stability of the solution are investigated.
Lothar von Wolfersdorf, On the Solutions of a Quadratic Integral and an Integral-Differential Equation. Z. Anal. Anwend. 21 (2002), no. 2, pp. 381–398
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IsMetabelian - Maple Help
Home : Support : Online Help : Mathematics : Group Theory : IsMetabelian
attempt to determine whether a group is metabelian
IsMetabelian( G )
G
is metabelian if it is an extension of an Abelian group by another Abelian group. Equivalently,
G
is metabelian if its derived subgroup is Abelian.
The IsMetabelian( G ) command attempts to determine whether the group G is metabelian. It returns true if G is metabelian and returns false otherwise.
\mathrm{with}\left(\mathrm{GroupTheory}\right):
\mathrm{IsMetabelian}\left(\mathrm{Symm}\left(4\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{IsMetabelian}\left(\mathrm{Alt}\left(4\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
The GroupTheory[IsMetabelian] command was introduced in Maple 2019.
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§ Try and think of natural transformations as intertwinings
I'm comfrotable with elementary representation theory, but I feel far less at home manipulating natural transformations. I should try and simply think of them as the intertwinig operators in representation theory, since they do have the same diagram. Then the functors become two representations of the same category (group), and the natural transformation is an intertwining operator. If one does this, then Yoneda sort of begins to look like Schur's lemma. Schur's lemma tells us that intertwinings between irreducible representations are either zero or a scaling of the identity matrix. That is, they are one-dimensional, and the space of all intertwinings is morally isomorphic to the field
\mathbb C
. If we specialize to character theory of cyclic groups
Z/nZ
, let's pick one representation to be the "standard representation"
\sigma: x \mapsto e^{i 2 \pi x/n}
. Then, given some other representation
\rho: Z/nZ \rightarrow \mathbb C^\times
, the intertwining between
\sigma
\rho
\rho
1
\rho(1) = k \sigma(1)
k \in \mathbb R
, then the intertwining is scaling by
k
. Otherwise, the intertwining is zero. This is quite a lot like Yoneda, where the natural transformation is fixed by wherever the functor sends the identity element.
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(Redirected from Boundary value analysis)
Software testing technique that tests boundary values
Boundary-value analysis is a software testing technique in which tests are designed to include representatives of boundary values in a range. The idea comes from the boundary. Given that we have a set of test vectors to test the system, a topology can be defined on that set. Those inputs which belong to the same equivalence class as defined by the equivalence partitioning theory would constitute the basis. Given that the basis sets are neighbors, there would exist a boundary between them. The test vectors on either side of the boundary are called boundary values. In practice this would require that the test vectors can be ordered, and that the individual parameters follows some kind of order (either partial order or total order).
Formally the boundary values can be defined as below:
Let the set of the test vectors be X1,..., Xn.
Let's assume that there is an ordering relation defined over them, as ≤.
Let C1, C2 be two equivalent classes.
Assume that test vector X1 ∈ C1 and X2 ∈ C2.
{\displaystyle X_{1}\leq X_{2}}
{\displaystyle X_{2}\leq X_{1}}
then the classes
{\displaystyle C_{1},C_{2}}
are in the same neighborhood and the values
{\displaystyle X_{1},X_{2}}
are boundary values.
In plainer English, values on the minimum and maximum edges of an equivalence partition are tested. The values could be input or output ranges of a software component, can also be the internal implementation. Since these boundaries are common locations for errors that result in software faults they are frequently exercised in test cases.
The expected input and output values to the software component should be extracted from the component specification. The values are then grouped into sets with identifiable boundaries. Each set, or partition, contains values that are expected to be processed by the component in the same way. Partitioning of test data ranges is explained in the equivalence partitioning test case design technique. It is important to consider both valid and invalid partitions when designing test cases.
The demonstration can be done using a function written in Java.
if (a >= 0 && b >= 0 && c < 0)
System.err.println("Overflow!");
if (a < 0 && b < 0 && c >= 0)
System.err.println("Underflow!");
On the basis of the code, the input vectors of [a,b] are partitioned. The blocks we need to cover are the overflow statement and the underflow statement and neither of these 2. That gives rise to 3 equivalent classes, from the code review itself.
Demonstrating Boundary Values (Orange)
we note that there is a fixed size of integer hence:-
MIN_VALUE ≤ x + y ≤ MAX_VALUE
We note that the input parameter a and b both are integers, hence total order exists on them. When we compute the equalities:-
x + y = MAX_VALUE
MIN_VALUE = x + y
we get back the values which are on the boundary, inclusive, that is these pairs of (a,b) are valid combinations, and no underflow or overflow would happen for them.
x + y = MAX_VALUE + 1
gives pairs of (a,b) which are invalid combinations, Overflow would occur for them. In the same way:-
x + y = MIN_VALUE - 1
gives pairs of (a,b) which are invalid combinations, Underflow would occur for them.
Boundary values (drawn only for the overflow case) are being shown as the orange line in the right hand side figure.
For another example, if the input values were months of the year, expressed as integers, the input parameter 'month' might have the following partitions:
... -2 -1 0 1 .............. 12 13 14 15 .....
--------------|-------------------|-------------------
invalid partition 1 valid partition invalid partition 2
The boundary between two partitions is the place where the behavior of the application changes and is not a real number itself. The boundary value is the minimum (or maximum) value that is at the boundary. The number 0 is the maximum number in the first partition, the number 1 is the minimum value in the second partition, both are boundary values. Test cases should be created to generate inputs or outputs that will fall on and to either side of each boundary, which results in two cases per boundary. The test cases on each side of a boundary should be in the smallest increment possible for the component under test, for an integer this is 1, but if the input was a decimal with 2 places then it would be .01. In the example above there are boundary values at 0,1 and 12,13 and each should be tested.
Boundary value analysis does not require invalid partitions. Take an example where a heater is turned on if the temperature is 10 degrees or colder. There are two partitions (temperature≤10, temperature>10) and two boundary values to be tested (temperature=10, temperature=11).
Where a boundary value falls within the invalid partition the test case is designed to ensure the software component handles the value in a controlled manner. Boundary value analysis can be used throughout the testing cycle and is equally applicable at all testing phases.
The Testing Standards Working Party website.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Boundary-value_analysis&oldid=1045395085"
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Derivative (mathematics) - Simple English Wikipedia, the free encyclopedia
In mathematics (particularly in differential calculus), the derivative is a way to show instantaneous rate of change: that is, the amount by which a function is changing at one given point. For functions that act on the real numbers, it is the slope of the tangent line at a point on a graph. The derivative is often written as
{\displaystyle {\tfrac {dy}{dx}}}
("dy over dx" or "dy upon dx", meaning the difference in y divided by the difference in x). The d is not a variable, and therefore cannot be cancelled out. Another common notation is
{\displaystyle f'(x)}
—the derivative of function
{\displaystyle f}
{\displaystyle x}
, usually read as "
{\displaystyle f}
{\displaystyle x}
".[1][2][3]
1 Definition of a derivative
2 Derivatives of functions
3 Properties of derivatives
4 Uses of derivatives
Definition of a derivative[change | change source]
The derivative of y with respect to x is defined as the change in y over the change in x, as the distance between
{\displaystyle x_{0}}
{\displaystyle x_{1}}
becomes infinitely small (infinitesimal). In mathematical terms,[2][3]
{\displaystyle f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}}
Derivatives of functions[change | change source]
Linear functions[change | change source]
Derivatives of linear functions (functions of the form
{\displaystyle mx+c}
with no quadratic or higher terms) are constant. That is, the derivative in one spot on the graph will remain the same on another.
When the dependent variable
{\displaystyle y}
directly takes
{\displaystyle x}
's value (
{\displaystyle y=x}
), the slope of the line is 1 in all places, so
{\displaystyle {\tfrac {d}{dx}}(x)=1}
regardless of where the position is.
{\displaystyle y}
modifies
{\displaystyle x}
's number by adding or subtracting a constant value, the slope is still 1, because the change in
{\displaystyle x}
{\displaystyle y}
do not change if the graph is shifted up or down. That is, the slope is still 1 throughout the entire graph and its derivative is also 1.
Power functions[change | change source]
Power functions (in the form of
{\displaystyle x^{a}}
) behave differently from linear functions, because their exponent and slope vary.
Power functions, in general, follow the rule that
{\displaystyle {\tfrac {d}{dx}}x^{a}=ax^{a-1}}
.[2] That is, if we give a the number 6, then
{\displaystyle {\tfrac {d}{dx}}x^{6}=6x^{5}}
Another example, which is less obvious, is the function
{\displaystyle f(x)={\tfrac {1}{x}}}
. This is essentially the same, because 1/x can be simplified to use exponents:
{\displaystyle f(x)={\frac {1}{x}}=x^{-1}}
{\displaystyle f'(x)=-1(x^{-2})}
{\displaystyle f'(x)=-{\frac {1}{x^{2}}}}
{\displaystyle f(x)={\sqrt[{3}]{x^{2}}}=x^{\frac {2}{3}}}
{\displaystyle f'(x)={\frac {2}{3}}(x^{-{\frac {1}{3}}})}
Exponential functions[change | change source]
An exponential function is of the form
{\displaystyle ab^{f\left(x\right)}}
{\displaystyle a}nd
{\displaystyle b}
{\displaystyle f(x)}
{\displaystyle x}
. The difference between an exponential and a polynomial is that in a polynomial
{\displaystyle x}
is raised to some power, whereas in an exponential
{\displaystyle x}
is in the power.
Example 1[change | change source]
{\displaystyle {\frac {d}{dx}}\left(ab^{f\left(x\right)}\right)=ab^{f(x)}\cdot f'\left(x\right)\cdot \ln(b)}
{\displaystyle {\frac {d}{dx}}\left(3\cdot 2^{3{x^{2}}}\right)}
{\displaystyle a=3}
{\displaystyle b=2}
{\displaystyle f\left(x\right)=3x^{2}}
{\displaystyle f'\left(x\right)=6x}
{\displaystyle {\frac {d}{dx}}\left(3\cdot 2^{3x^{2}}\right)=3\cdot 2^{3x^{2}}\cdot 6x\cdot \ln \left(2\right)=\ln \left(2\right)\cdot 18x\cdot 2^{3x^{2}}}
Logarithmic functions[change | change source]
The derivative of logarithms is the reciprocal:[2]
{\displaystyle {\frac {d}{dx}}\ln(x)={\frac {1}{x}}}
{\displaystyle {\frac {d}{dx}}\ln \left({\frac {5}{x}}\right)}
. This can be reduced to (by the properties of logarithms):
{\displaystyle {\frac {d}{dx}}(\ln(5))-{\frac {d}{dx}}(\ln(x))}
The logarithm of 5 is a constant, so its derivative is 0. The derivative of
{\displaystyle \ln(x)}
{\displaystyle {\tfrac {1}{x}}}
{\displaystyle 0-{\frac {d}{dx}}\ln(x)=-{\frac {1}{x}}}
For derivatives of logarithms not in base e, such as
{\displaystyle {\tfrac {d}{dx}}(\log _{10}(x))}
, this can be reduced to:
{\displaystyle {\frac {d}{dx}}\log _{10}(x)={\frac {d}{dx}}{\frac {\ln {x}}{\ln {10}}}={\frac {1}{\ln {10}}}{\frac {d}{dx}}\ln {x}={\frac {1}{x\ln(10)}}}
Trigonometric functions[change | change source]
The cosine function is the derivative of the sine function, while the derivative of cosine is negative sine (provided that x is measured in radians):[2]
{\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x)}
{\displaystyle {\frac {d}{dx}}\cos(x)=-\sin(x)}
{\displaystyle {\frac {d}{dx}}\sec(x)=\sec(x)\tan(x)}
Properties of derivatives[change | change source]
Derivatives can be broken up into smaller parts where they are manageable (as they have only one of the above function characteristics). For example,
{\displaystyle {\tfrac {d}{dx}}(3x^{6}+x^{2}-6)}
can be broken up as:
{\displaystyle {\frac {d}{dx}}(3x^{6})+{\frac {d}{dx}}(x^{2})-{\frac {d}{dx}}(6)}
{\displaystyle =6\cdot 3x^{5}+2x-0}
{\displaystyle =18x^{5}+2x\,}
Uses of derivatives[change | change source]
↑ 2.0 2.1 2.2 2.3 2.4 Weisstein, Eric W. "Derivative". mathworld.wolfram.com. Retrieved 2020-09-15.
↑ 3.0 3.1 "The meaning of the derivative - An approach to calculus". themathpage.com. Retrieved 2020-09-15.
Online derivative calculator which shows the intermediate steps of calculation
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Derivative_(mathematics)&oldid=8225883"
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§ John Conway: The symmetries of things
Original way to classify wallpaper groups: think of geometric transforms that fix the pattern. Thurston's orbifold solution: think of quotients of
\mathbb R^2
by groups --- this gives you an orbifold (orbit manifold). Take a chair, surround it around by a sphere. The symmetries of a physical object fixes the center of gravity. So we pick the center of the sphere to be the center of gravity. The "celestial sphere" (the sphere around the chair) is a nice manifold (We only have the surface of the sphere). The vertical line that divides the chair also divides the sphere into two parts.
The points of the orbifold are orbits of the group.
So now the orbifold gives us a hemisphere in this case.
The topology of the orbifold determines the group.
This is astonishing, because the group is a metrical object: elements of the group preserve the inner product of the space.
And yet, geometrical groups are determined by the topology of their orbifolds!
Thurston's metrization conjecture: certain topological problems reduce to geometrical ones.
Conway came up with his notation for wallpaper groups/orbifolds. There are only four types of features.
The hemisphere orbifold is *. (group of order 2). * denotes the effect on the orbifold. * really means: what is left out of a sphere when I cut out a hemispherical hole. * is the name for a disk, because a hemisphere is a disk topologically. It has metrical information as well, but we're not going to speak about it, because all we need is the topological information.
One-fourth of a sphere (symmetry group of rectangular table) is denoted by * 2 2. The * for the hemisphere, and 2, 2for the angles of pi/2.
If the table is a sphere, then we have diagonal symmetry as well. In this case, the orbifold has angle pi/4. So the table is * 4 4.
If we take a cube, then we have an even more complicated orbifold. The "fundamental region" of the cube has 2, 3, and 4 mirrors going through them. So in the orbifold, we get triangles of angles pi/2, pi/3, pi/4. This would be * 4 3 2.
Draw a swastika. This has no reflection symmetry. This has a gyration : a point about which the figure can be rotated, but the point is NOT on a line of reflection. We can tear the paper and make it into a cone. This gives us a cone point . The angle around the cone point is 2pi/4. This is the orbifold of the original square with a swastika on it.
An orbifol can be made to carry some amount of metrical information. The cone point only has 90 degrees, so it is in some sense, "a quarter of a point".
Draw a cube with swastikas marked on each face. This has no reflection symmetry. Once again, we have a gyration, and again, only the gyration/singularities matter. This group is again 4, 3, 2 , but in blue . In this notation, red is reflection, blue is "true motion" (?).
Let us try to work out the euler characteristic of the rectangular table orbifold by using
V - E + F
. The orbifold as one face. The wrong thing to say is that the orbifold has two edges and two vertices. It is untrue because the edge of the orbifold is only half an edge --- let's say that lines have thickness. In this case, we will have
V = 2/4
E = 2/2
F = 1
. The euler characteristic works out to be a half. This is appropriate, because the orbifold is a type of divided manifold.
If we work this out for a cube, we get
2/48
. This is because the sphere gets divided into 48 pieces, and the sphere has an euler characteristic of 2!
Alternatively, we can think that we started out with 2 dollars, and we are then buying the various features of our orbifold. * costs 1$, a blue number after a star, for example: 2 costs 1/2 a dollar. 3 costs 2/3 of a dollar, 4 costs 3/4 of a dollar. In general, n costs 1 - 1/n. The red numbers are children, so they cost half an much: n consts 1/2(1 - 1/n) = (n-1)/2n.
Now, see that we started with positive euler characteristic (2), and we divide it by some n (the order of the group). So we end up with a positive euler characteric. By a sort of limiting argument, the euler characteristic of the wallpaper groups, which are infinite, is zero. However, see that we must get to the zero by starting with two dollars and buying things off the menu! If we try and figure out what all the possible ways are to start with 2 dollars and buy things till we are left with exactly 0 dollars, we get that there are 17 possible ways of buying things on the menu! Thus, this the reason for there being 17 wallpaper groups.
To buy more than two dollars, you are buying symmetries from the hyperbolic plane!
Because we can completely enumerate 2-manifolds, we can completely enumerate 2-orbifolds, which are essentially the same thing as symmetry groups. The real power is in the 3D case. We don't have a full classification of 3-manifolds. But we maybe able to go the other way. This is the metrization theorem.
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A position vector A of magnitude 10 units makes an angle of 53 with a position vector B of magnitude - Physics - Units And Measurements - 12301285 | Meritnation.com
A position vector A of magnitude 10 units makes an angle of 53 with a position vector B of magnitude 6 unit. The magnitude of position vector AB and the angle with vector A is:
A. 8 units, (180 - 37 degree)
B. 10 units, 37 degree
C. 10 Units, 53 degree
D. 8 Units, 53 degree
\left|AB\right|=\left|A-B\right|=\sqrt{{\left|A\right|}^{2}+{\left|B\right|}^{2}-2\left|A\right|\left|B\right|\mathrm{cos}\left(53\right)}=\sqrt{{10}^{2}+{6}^{2}-2×10×6×\frac{3}{5}}=\sqrt{64}=8 unit\phantom{\rule{0ex}{0ex}}Resul\mathrm{tan}t of AB and B will give A \phantom{\rule{0ex}{0ex}}by magnitude we cann see that these vectores are making a right angle tringle \phantom{\rule{0ex}{0ex}}So if one angle of right angle triangle is 53 then other will 37 \left(except right angle\right)\phantom{\rule{0ex}{0ex}}Angle between And AB will be \left(180-37\right) \phantom{\rule{0ex}{0ex}}
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Q given below A part of circuit shown in figure Find potential difference between point A and B - Physics - Current Electricity - 12892375 | Meritnation.com
Q given below
Q given below A part of circuit shown in figure. Find potential difference between point A and B. IOV IA 20
u\mathrm{sin}g KCL curent towards point B will be 2A and hence potential at B will be 4 volt\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Hence potential difference between A and B will be \left(-10-\left(-4\right)\right)=-6V\phantom{\rule{0ex}{0ex}}\left(potential at A is -10 V\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Regards
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§ Clean way to write burnside lemma
Burnside lemma says that
|Orb(G)| \equiv 1/|G| \sum_{g \in G} fix(g)
. We prove this as follows:
\begin{aligned} &\sum_{g \in G} fix(g) \\ &= \sum_{g \in G} |\{x : g(x) = x \}| \\ &= |\{(g, x) : g(x) = x \}| \\ &= \sum_{x \in X}|\{x : g(x) = x \}| \\ &= \sum_{x \in X} Stab(x) \end{aligned}
From orbit stabilizer, we know that
|Orb(x)||Stab(x)| = |G|
|Orb(x)
is the total cardinality of the orbit, each element in the orbit contributes
1/|Orb(x)|
towards cardinality of the full orbit.
Thus, the sum over an orbit
\sum_{x \in Orb(x)} 1/|Orb(x)|
will be 1.
Suppose a group action has two orbits,
O_1
O_2
. I can write the sum
\sum_{x \in g} 1/|Orb(x)|
\sum_{x \in O_1} 1/|O_1| + \sum_{x \in O_2} 1/|O_2|
, which is equal to 2.
I can equally write the sum as
\sum_{o \in Orbits} \sum_{x \in o} 1/|o|
. But this sum is equal to
\sum_{o \in Orbits} \sum_{x \in o} 1/|Orb(x)|
This sum sums over the entire group, so it can be written as
\sum_{x \in G} 1/|Orb(x)|
In general, the sum over the entire group
\sum_{x \in g} 1/|Orb(x)|
will be the number of orbits, since the same argument holds for each orbit .
\begin{aligned} &= \sum_{x \in X} Stab(x) \\ &= \sum_{x \in X} |G|/|Orb(x)| \\ &= |G| \sum_{o \in orbits} \sum_{x \in o} 1/|o| \\ &= |G| \texttt{num.orbits} \\ \end{aligned}
So we have derived:
\begin{aligned} &\sum_{g \in G} fix(g) = |G| \texttt{num.orbits} \\ &1/|G| (\sum_{g \in G} fix(g)) = \texttt{num.orbits} \\ \end{aligned}
If we have a transformation that fixes many things, ie,
fix(g)
is large, then this
g
is not helping "fuse" orbits of
x
together, so the number of orbits will increase.
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Tags: machine learning all
There is accompanying code for reproducing some of the results in this blog post. Check out the repository here!
Decoding Strategies for Text Generation
Finding the humanity in approximating a NP-Hard problem
Recently, machine learning models have seen incredible progress towards computers being able to generate text that sounds human. This is an area of research that involves both furthering our understanding of machine intelligence as well as language use in society. I think it’s an interesting problem because it once against prompts a long-standing question in machine intelligence: what does it mean to be human?
So how do we make computers sound human? We'll explore this in two ways. The first is understanding why this problem is so computationally challenging and the second is to outline the different approaches we can take using a special kind of machine learning model known as a language model. In its simplest sense, a language model assigns a probability to a sequence of words. As you can imagine, language is infinite and so we can’t possibly know the probability of the phrase “cat in the hat”. However, our machine learning model (by definition) is going to approximate this likelihood for us. And, as it turns out, this is a powerful technique that works incredibly well for practical purposes of generating human-sounding text.
In this blog post, I want to motivate why this task of generating text is computationally difficult, and even if we had the "optimal solution", why it may not fit our idea of what good text sounds like. Afterward, we'll walk through a few modern approaches to remedy this. As we'll see, these approaches work really well.
We can use the chain rule in probability to break down a sequence (or sentence) of words into step-by-step calculations. Let's say we are looking for the probability of the phrase "cat in the hat":
P(\text{cat in the hat})
We can break this value down into the product of the following terms (where
\text{<s>}
denotes the starting token):
P(\text{<s>}) \newline P(\text{cat} ~|~ \text{<s>}) \newline P(\text{in} ~|~ \text{<s> cat}) \newline P(\text{the} ~|~ \text{<s> cat in}) \newline P(\text{hat} ~|~ \text{<s> cat in the})
But how do we get these probabilities to start? That's what the language model is for. In essence, a language model’s purpose is to give us
P(w ~|~ c)
w
is a particular target word (i.e. the next word) and
c
is the context that precedes the target word. Using a trained model, we can use
P(w ~|~ c)
to create a distribution of the likelihood for the next word.
Now, we turn our focus to using these probabilities to create text. Let's determine what the first word of our generation would be. Similar to the previous example, where we have a
\text{<s>}
token to signify the beginning of a sequence, we can ask the model what the value of
P(w ~|~ \text{<s>})
for a variety of different values of
w
. But how do we select what
w
should be when we're given the probabilities for every possible word? Is it simply the highest value?
More broadly, our goal is to select the words that maximize:
\prod _{i = 0} ^{n} P(w_i ~ | ~ c_0 ~ ... ~ c_{i - 1})
But how do we do this using the model's outputted probabilities? As we will see, this is the million dollar question - literally.
Showing NP-Hardness
Before moving on to approaching a solution, it’s worth gaining a little appreciation for how difficult this problem truly is. To solve it in its entirety, you would make a million dollars! Literally!
The challenges we have with using our machine model to generate text is yet another manifestation of the NP-Complete class of problems (in it’s decision form). If you are unfamiliar, these problems are known to be the toughest problems in computer science. What’s more interesting, is that problems existing in this class can all be reduced to one another, implying that they are all in essence the same problem.
Wait it's all 3-SAT? (Always has been.)
Let’s show that our issue of finding the most likely sequence is just as hard as the other famous NP-hard problems, like the Traveling Salesman Problem and Knapsack Problem.
Generating text is akin to the problem of finding the highest probability sequence that starts with
\text{<s>}
. The easiest reduction to see is if we construct a directed graph, starting with
\text{<s>}
, and layers consisting of each word in our vocabulary. Each edge
(u, v)
P(v ~|~ u)
u
v
are words. Note that this graph goes on forever, and that we have a
\text{</s>}
token for ending a sequence. Thus, finding out the most likely sequence of words is equivalent to finding the longest path in this graph.
(an illustration of how we "search" for the most probable sequence)
As we know, the longest path problem is famously NP-Hard, which means that trying to maximize
\prod _{i = 0} ^{n} P(w_i ~ | ~ c_0 ~ ... ~ c_{i - 1})
for an entire sequence is thereby also NP-Hard since they are equivalent problems. As a result, solving our little text decoding problem in polynomial time could net you one million dollars! People have tried to do this for a long time with little luck, so let's look into approximating this problem instead.
There are a lot of ways to approximate the task of generating natural language. I really like the paper The Curious Case of Neural Text DeGeneration, which introduces Top-P Sampling and reflects on previous approaches.
Our first and most intuitive approximation is known as Greedy Decoding, where we take the most probable word over a vocabulary
V
for a context
c
as the next word.
w_i = \operatorname*{arg\, max}_{w \in V} ~ P(w_i ~ | ~ c_0 ~ ... ~ c_{i - 1})
Repeatedly taking the most performing this operation will allow us to create a sentence, one word at a time. Unfortunately, this approach doesn't produce very convincing results, and tends to exploit weird patterns, even repeating itself due to cyclic dependencies (i.e. going back and forth between words that predict each other like "I went to the place that the place that the place that the place ..."). This approach also creates deterministic answers for a given start token, which definitely is not how humans approach language generation.
As with most naive approaches, Greedy Decoding doesn't always produce the best outcomes. This is especially true in certain domains such as translating between different languages, where tokens at the beginning of the sentence may dramatically alter the likehood of its following tokens. This creates the problem of a high-probability token "hiding behind" a low-probability token that preceeds it in the order of the sentence. As a result, Greedy Decoding will forgo the low-probabilty token in favour of another one, regardless of that token's subsequently generated tokens.
One approach that mitigates this problem is Beam Search, which is another greedy algorithm that approximates the search process by maintaining multiple possible candidates for a path (which each represent a sentence). This "beam" of results is ultimately a search heuristic that allows us to deterministically approximate the most likely sequence of words.
Beam Search works really well in translation settings, since there is often not too much creativity involved with translating sentences between languages, and it works extremely well for the task. However, for tasks such as dialogue and story generation, Beam Search is rarely used, since it can create boring text.
(from Holtzman et al.)
As it turns out, there are more direct (and elegant) ways to approach this search tasks non-deterministically.
The last way that we can generate text is to let uncertainy do it's thing and randomly sample directly from the distribution of
P(w ~|~ c)
. This might also feels like a naive way to do text generation, but in reality this allows for a good middle-ground between creativity and greediness. In expectation, the strategy tends to produce statistically likely sequences. For machine learning in general, relying on expectation tends to do well for us!
However, one issue with Random Sampling (and decoding strategies in general) is the lack of control we have over how the signal is used to select candidate words. Fear not, because we have ways of tuning the way our model uses the token probabilities to generate text. One main way to impose behaviour on decoding strategies by altering the output distribution of
P(w ~|~ c)
The first way we can alter the probability distribution of the next word is to truncate the vocabulary to a certain length of the
k
most likely tokens, and redistributing the truncated probability mass to these selected tokens. This has the effect of constraining the generation process to select words that the model itself has deemed to be more sensible in the context. This tends to produce more creative and human-sounding text!
However there's a tradeoff in the hyperparameter
k
k
is too large, then we basically have no benefit over vanilla random sampling. Conversely, when
k
is too small, we have a limited vocabulary and the generated text will make even less sense.
A nice implementation tidbit about Top-K Sampling is that it's equal to vanilla random sampling when
k = |V|
, so the results of using a specific value of
k
can be quickly evaluated against a distributional baseline.
Top-P Sampling
Top-P Sampling (or Nucleus Sampling) was introduced by Holtzman et al. in an aptly named paper: The Curious Case of Text Degeneration. In this sampling method, instead of predefining the
k
most likely tokens, we instead only consider the words which have a cumulative probability mass larger than the hyperparameter
p
Intuitively, this is a natural extension of Top-K Sampling, since we're trying to mitigate the effects of the "unreliable" tail end of the distribution over the next word. The main difference is that we rely on a probabalistic approach (which can be more robust) instead of a function of the vocaulary size.
Yet another hyperparameter we can use to alter the probability distribution of the next word (specifically when generating text with neural networks) is known as temperature
T
. Intuitively, this controls how "confident" the model must be before making a prediction about the next word.
P(w_i ~ | ~ c_0 ~ ... ~ c_{i - 1}) = \frac{e^{(z_i / T)}}{\sum_j e^{(z_j / T)}}
When temperature is high (e.g.
T = 1
), the model can rely on its original predictions, without worrying about how confident it has to be. This has the effect of the network being more creative and diverse, at the risk of making more language errors.
However, lowering the value of
T
closer to zero makes the output distribution more "soft", meaning that the neural network has to be more confident in its predictions. As a result, the network's most confident prediction will trump the probability of the others, resulting in less diverse, but more coherent text.
So it seems like there are a lot of decoding strategies for generating text. For deciding which to use, it's best to think about what aspect of human language you are trying to capture. If it's an accuracy-maximizing translation task, then Beam Search is the way to go (determinism in selection can be helpful). If you want the expressiveness and character of a chatbot, then random sampling with a distributional change would make the most sense.
Another interesting note is that I've largely tried to abstract how we get the distribution
P(w ~|~ c)
. In modern times, we use neural networks to do this (the current state of the art at the time of writing this post are Transformer networks), but the logic from this blog posts applies even if you do a simpler Markov Chain to yield the distribution of the next word.
Trying to use statistical and mathematical tools to decipher what makes text sound human is an interesting avenue of research and lately I've been exploring it in earnest. Check out this project here to see our latest efforts!
Theory: (Ippolito et al.)
Code: (HuggingFace)
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Flow past a Sphere
Received: March 5, 2019 ; Accepted: March 24, 2019 ; Published: March 27, 2019
A new theoretical framework is applied to the steady fluid flow past a solid smooth sphere. Bernoulli’s law along a streamline is combined with the cross-stream force balance: centrifugal force on the curved flow equals a pressure gradient. When compared with the standard potential theory for flow past a sphere in a text book, the prospect of a major discrepancy is found. Whereas the decay rate of the velocity perturbation away from the sphere goes as the inverse cube of the distance in the text book, the decay rate computed here is in all likelihood very different, and it depends on an unknown constant function, the radius of curvature of the streamlines versus distance from the sphere. When that function is supplied either from another theory or from detailed observations (probably streak photographs), then the new approach can be solved completely. In any case, accurate measurements of flow rates at different positions with respect to the solid are badly needed.
Leonardo da Vinci wrote down in a notebook his observation of the flow past a rock in a stream: the flow is fastest at the sides of a covered rock rather than above it [ 1 ]. Unfortunately he did not write about the rate at which the fastest flow decayed outward from the rock toward the normal mean stream velocity. How far apart should two rocks in a stream be in order that their respective perturbed velocities do not interfere with each other? To this day, I have not seen observations relevant to answering this question. It is not that the subject is so unimportant either, because it is tied up with understanding how birds and planes fly.
A theory exists, but to my knowledge no observations have ever been compared with it. Potential flow is the name of the theory, which has been applied to flow past a circular cylinder in many text books as well as flow past a sphere in one text that I have [ 2 ]. It would not seem to be very difficult to obtain experimental evidence, such as carefully made streak photographs. After all, Leonardo used bird seed and other small floating objects to obtain enlightening visual images.
As presented in the text books, the potential flow theory is incomplete because the pressure field surrounding the sold body is never shown, only the velocity field. Not just that limitation, but within itself the theory looks to be inconsistent or contradictory. For example, the die off rate for the velocity perturbation extending radially out from a circular cylinder varies as the inverse square of the distance, whereas for the sphere the rate varies as the inverse cube of the distance. That makes no sense physically.
There is another theoretical framework available, which has some advantages over the classical method, and it has already been applied to the steady flow past a circular cylinder [ 3 ]. This new approach can also just as easily be applied to flow past a sphere. One of the advantages is that the die off rate for velocity perturbations is the same for both the sphere and the cylinder, which is a more reasonable outcome. Another advantage is that the pressure perturbation is readily calculated and presented.
Consider a steady uniform flow encountering a fixed smooth solid sphere and limit the mean speed so that no eddies form at the back face of the sphere. Gravity and friction are not included in the model. In a plane parallel to the mean flow that slices through the middle of the sphere the streamlines separate in going around the circular cross-section of the sphere. Let the z-axis point up at the top of the circle.
Bernoulli’s law applies to all streamlines going over the top (and bottom) of the solid circle.
p=const-\frac{1}{2}\rho {V}^{2}
where p is the pressure, V is the speed of flow in the vicinity of the solid body and
\rho
is the constant fluid density. Far away from the sphere V trans forms into the uniform flow speed U. For simplicity the constant in (1) is taken the same for all streamlines.
Fluid following a curving path above the sphere’s circular cross-section experiences an upward centrifugal force. When the flow is steady there must be an equal but opposite force, and in this situation it can only be a pressure gradient. Therefore the cross-stream force balance is
\frac{\text{d}p}{\text{d}z}=\frac{\rho {V}^{2}}{R}
In (2), R(z) is the variable radius of curvature of the streamlines, which at the top of the sphericalcross-section becomes the constant radius of the circle,
{R}_{o}
Equations (1) and (2) are two equations in the two unknowns, velocity and pressure. Each equation by itself is nonlinear. But between the two equations one of the variables can be eliminated, and it does not matter which, because the result is a linear equation in each case. For example, the velocity equation is
\frac{\text{d}V}{\text{d}z}=-\frac{V}{R}
which is linear although the unknown constant R(z) varies with z. Equations (1) and (2) are exactly the same for the circular cross-section of the sphere as they are for the circular cylinder [ 2 ]. As a consequence, by skipping steps the solution can be written down immediately
V={V}_{0}\left({e}^{f\left(z\right)}-1\right)+U
{V}_{0}
is a constant that can be determined by conserving mass between two vertical cross-sections, one on top of the sphere and the other far away from it, and
f\left(z\right)=\underset{0}{\overset{z}{\int }}\frac{\text{d}Z}{R\left(Z\right)}
Equation (4) is by no means a complete solution of the problem for the velocity perturbation, but it contains strong hints that the potential flow method has flaws that may eventually render its usefulness questionable. If Equation (5) could be integrated, either algebraically or numerically, that would be a good start. Detailed measurements of the radius of curvature function R(z) are needed to evaluate (5) numerically. While waiting for that to happen a couple of algebraic representations of R(z) have been tried out to see what the die off rate looks like for the velocity perturbation. So far nothing approximating the inverse square or inverse cube law of the distance has emerged.
As mentioned above the pressure equation is quickly obtained from (1) and (2) by eliminating the velocity to give
\frac{\text{d}p}{\text{d}z}=-\frac{2p}{R}
which is similar to (3) with p replacing V except for the factor of 2 in the numerator of the RHS. That factor of 2 is critical in the sense that it means the pressure perturbation is more tightly bound to the solid body (cylinder or sphere) than the velocity perturbation is no matter what the radius of curvature function turns out to be.
Finally comes a thought experiment, within the same general theoretical framework, but strictly speaking outside the bounds of Equation (2) in the above model. Drop the sphere and greatly expand the (horizontal) scale such that the centrifugal force on the RHS of Equation (2) is replaced by the Coriolis force. Then when the pressure is substituted out between (1) and (2), a linear equation in the horizontal velocity shear results
\frac{\text{d}V}{\text{d}s}=-f
where f is the Coriolis parameter, which locally is a constant (sign/hemisphere left unspecified), and measures horizontal distance normal to the mean flow direction. This combination of Bernoulli’s law with the geostrophic relation has not occurred in print before as far as I am aware. One reason for such an omission may be that until recently Bernoulli’s law has never been associated with large-scale phenomena, such as certain weather systems over the North Pacific [ 4 ].
Based on an apparent internal inconsistency in the theory and a complete lack of any comparisons between theory and measurements, it is predicted that the applications of potential flows past circular cylinders and spheres will not lead to increased understanding of these phenomena. Although incomplete at this time, a new theoretical approach is under development as outlined above that may be more helpful in this regard in the future.
1. MacCurdy, E., Ed. (1948) The Notebooks of Leonardo Da Vinci. Volume 2, Jonathan Cape, London, 90.
2. Faber, T.E. (1995) Fluid Dynamics for Physicists. Cambridge University Press, Cambridge, 133. https://doi.org/10.1017/CBO9780511806735
3. Kenyon, K.E. (2013) Flow past a Cylinder. Journal of Scientific Theory and Methods, 2013, 211-222.
4. Kenyon, K.E. (2018) Bernoulli Weather or Not? Natural Science, 10, 178-181.
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Cabtaxi number - Wikipedia
In mathematics, the n-th cabtaxi number, typically denoted Cabtaxi(n), is defined as the smallest positive integer that can be written as the sum of two positive or negative or 0 cubes in n ways. Such numbers exist for all n, which follows from the analogous result for taxicab numbers.
1 Known cabtaxi numbers
Known cabtaxi numbers[edit]
Only 10 cabtaxi numbers are known (sequence A047696 in the OEIS):
{\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (1)&=&1&=&1^{3}\pm 0^{3}\end{matrix}}}
{\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (2)&=&91&=&3^{3}+4^{3}\\&&&=&6^{3}-5^{3}\end{matrix}}}
{\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (3)&=&728&=&6^{3}+8^{3}\\&&&=&9^{3}-1^{3}\\&&&=&12^{3}-10^{3}\end{matrix}}}
{\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (4)&=&2741256&=&108^{3}+114^{3}\\&&&=&140^{3}-14^{3}\\&&&=&168^{3}-126^{3}\\&&&=&207^{3}-183^{3}\end{matrix}}}
{\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (5)&=&6017193&=&166^{3}+113^{3}\\&&&=&180^{3}+57^{3}\\&&&=&185^{3}-68^{3}\\&&&=&209^{3}-146^{3}\\&&&=&246^{3}-207^{3}\end{matrix}}}
{\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (6)&=&1412774811&=&963^{3}+804^{3}\\&&&=&1134^{3}-357^{3}\\&&&=&1155^{3}-504^{3}\\&&&=&1246^{3}-805^{3}\\&&&=&2115^{3}-2004^{3}\\&&&=&4746^{3}-4725^{3}\end{matrix}}}
{\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (7)&=&11302198488&=&1926^{3}+1608^{3}\\&&&=&1939^{3}+1589^{3}\\&&&=&2268^{3}-714^{3}\\&&&=&2310^{3}-1008^{3}\\&&&=&2492^{3}-1610^{3}\\&&&=&4230^{3}-4008^{3}\\&&&=&9492^{3}-9450^{3}\end{matrix}}}
{\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (8)&=&137513849003496&=&22944^{3}+50058^{3}\\&&&=&36547^{3}+44597^{3}\\&&&=&36984^{3}+44298^{3}\\&&&=&52164^{3}-16422^{3}\\&&&=&53130^{3}-23184^{3}\\&&&=&57316^{3}-37030^{3}\\&&&=&97290^{3}-92184^{3}\\&&&=&218316^{3}-217350^{3}\end{matrix}}}
{\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (9)&=&424910390480793000&=&645210^{3}+538680^{3}\\&&&=&649565^{3}+532315^{3}\\&&&=&752409^{3}-101409^{3}\\&&&=&759780^{3}-239190^{3}\\&&&=&773850^{3}-337680^{3}\\&&&=&834820^{3}-539350^{3}\\&&&=&1417050^{3}-1342680^{3}\\&&&=&3179820^{3}-3165750^{3}\\&&&=&5960010^{3}-5956020^{3}\end{matrix}}}
{\displaystyle {\begin{matrix}\mathrm {Cabtaxi} (10)&=&933528127886302221000&=&77480130^{3}-77428260^{3}\\&&&=&41337660^{3}-41154750^{3}\\&&&=&18421650^{3}-17454840^{3}\\&&&=&10852660^{3}-7011550^{3}\\&&&=&10060050^{3}-4389840^{3}\\&&&=&9877140^{3}-3109470^{3}\\&&&=&9781317^{3}-1318317^{3}\\&&&=&9773330^{3}-84560^{3}\\&&&=&8444345^{3}+6920095^{3}\\&&&=&8387730^{3}+7002840^{3}\end{matrix}}}
Cabtaxi(5), Cabtaxi(6) and Cabtaxi(7) were found by Randall L. Rathbun; Cabtaxi(8) was found by Daniel J. Bernstein. Cabtaxi(9) was found by Duncan Moore, using Bernstein's method. Cabtaxi(10) was first reported as an upper bound by Christian Boyer in 2006 and verified as Cabtaxi(10) by Uwe Hollerbach and reported on the NMBRTHRY mailing list on May 16, 2008.
Announcement of Cabtaxi(9)
Announcement of Cabtaxi(10)[permanent dead link]
Cabtaxi at Euler
Retrieved from "https://en.wikipedia.org/w/index.php?title=Cabtaxi_number&oldid=994407024"
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§ Leapfrog Integration
We have a system we wish to simulate using hamilton's equations:
\begin{aligned} \frac{\partial q}{\partial t} = \frac{\partial H}{\partial p}|_{(p_0, q_0)} \\ \frac{\partial p}{\partial t} = -\frac{\partial H}{\partial q}|_{(p_0, q_0)} \\ \end{aligned}
We want to simulate a system using these differential equations. We will begin with some initial position and momentum
(q_0, p_0)
\frac{\partial q}{\partial t} \rvert_{(q_0, p_0)}
\frac{\partial p}{\partial t} \rvert_{(q_0, p_0)}
, and use these to find
(q_{next}, p_{next})
. An integrator is a general algorithm that produces the next position and momentum using current information:
(q_{next}, p_{next}) = I \left(q_0, p_0, \frac{\partial q}{\partial t}\rvert_{(q_0, p_0)}, \frac{\partial p}{\partial t}\rvert_{(q_0, p_0)} \right)
The design of
I
is crucial: different choices of
I
will have different trade-offs for accuracy and performance. Another interesting property we might want is for
I
to be a symplectic integrator --- that is, it preserves the total energy of the system. For example, here's a plot of the orbits of planets using two integrators, one that's symplectic (leapfrog) and one that isn't (Euler) Notice that since leapfrog attempts to keep energy conserved, the orbits stay as orbits! On the other hand, the euler integrator quickly spirals out, since we lose energy during the integration. Note that this is not an issue of numerical precision : The euler integrator is ineherently such that over long timescales, it will lose energy. On the other hand, the leapfrog integrator will always remain stable , even with very large timesteps and low precision. I present the equations of the leapfrog integrator, a proof sketch that it is symplectic, and the code listing that was used to generate the above plot. Often, code makes most ideas very clear!
§ The integrator
§ Code listing
§ Incantations
# Run HMC with a particular choice of potential
# dq/dt = dH/dp|_{p0, q0}
# dp/dt = -dH/dq|_{p0, q0}
def leapfrog(dhdp, dhdq, q0, p0, dt):
p0 += -dhdq(q0, p0) * 0.5 * dt
# full step position
# q += dt * p
q0 += dhdp(q0, p0) * dt
# half step position
return (q0, p0)
For reference, we also implement an euler integrator, that uses the derivative to compute the position and momentum of the next timestep independently.
return (qnew, pnew)
Finally, we implement planet(integrator, n, dt) which simulates gravitational potential and usual kinetic energy, using the integrator given by integrator for n steps, with each timestep taking dt.
# minimise potential V(q): q, K(p, q) p^2
q = np.array([0.0, 1.0])
p = np.array([-1.0, 0.0])
print("q: %10s | p: %10s | H: %6.4f" % (q, p, H(q, p)))
We plot the simulations using matplotlib and save them.
print("planet simulation with leapfrog")
planet_leapfrog = planet(leapfrog, NITERS, TIMESTEP)
print(planet_leapfrog)
print("planet simulation with euler")
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Boundedness of the twisted paraproduct | EMS Press
\mathrm{L}^p
estimates for a two-dimensional bilinear operator of paraproduct type. This result answers a question posed by Demeter and Thiele.
Vjekoslav Kovač, Boundedness of the twisted paraproduct. Rev. Mat. Iberoam. 28 (2012), no. 4, pp. 1143–1164
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The District of Columbia has a very high population density. There are
646,449
people in
68.3
Calculate the unit rate in terms of people per square mile for the District of Columbia. Make sure the precision of your answer is reasonable.
The problem indicates the rate is to be in terms of people per square mile.
This indicates a division problem.
\frac{\text{People}}{\text{Square Mile}}=\frac{646449}{68.3}=9464.84
people per square mile.
But is this amount of precision reasonable?
We should round our answer to the nearest whole person:
9465
Alaska is
570,374
square miles in area. If Alaska had the same population density as the District of Columbia, how many people would live in Alaska?
Each square mile would have a population density of
9465
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Guardrails & Impermanent Loss Mitigation - Tokemak: The Utility for Sustainable Liquidity
1. Guardrails & LP Safeguards
Mechanics Goals
The goals can be generally defined as follows:
1. Make the LPs whole (QTY): Hold sufficient quantities of the deployed assets in the system reserve
2. Maintain or increase the PCA (system reserve): Pull operational surpluses and system revenue into the PCA
1.1 The Asset Stack & Mitigation Waterfall
For all examples in this document a mitigation for a relative change in the exchange rate of up to 100% between the two paired assets will be demonstrated.
If a certain threshold (as defined below) of net-withdrawals by the LPs of a particular reactor is reached, mitigation mechanics become necessary in order to make the LPs whole while simultaneously maintaining or increasing the PCA:
Make LPs whole
By drawing the asset in deficit from assets in reserve
Assets in Surplus / System Revenue
Draw system-wide asset surpluses and system revenue into the PCA. This will cover a large portion of the negative asset flow. The amount of surpluses that can be pulled into the reserve also depends on the net-withdrawal requests.
TOKE staked
TOKE that was staked to direct the asset (whether Pair Reactor or Token Reactor staked TOKE) are used to cure and pulled into the PCA. There is a difference in how staked TOKE are being used depending on which side of the pool the deficit occurred (asset vs base asset).
Protocol Controlled Assets
In a last step, should the above steps not be sufficient to make users whole, the system will resort to:
Using ETH and/or stable coins from the reserve to make the LPs whole.
Should not enough ETH or stable coins be available, highly liquid reserve assets are sold for ETH or stables on external venues.
This last step would be performed without regard to a net-negative effect on the reserve. As previously mentioned, this should not occur under circumstances within the range defined by the guardrails. It will be initiated by the DAO multi-sig and executed using best practices in order to mitigate front running and mitigate other manipulations. This is highly unlikely and we don't anticipate this stage will ever be reached.
1.2 Deployment Guardrails
Deployment guardrails are restrictions imposed on the protocol limiting the maximum amount of assets deployed per individual Token Reactor (asset pairs) and deployment cycle. The guardrails allow for managing market risk by ensuring that the mitigation mechanics are effective within predetermined market conditions.
Defining and Setting Guardrail Parameters
In order to define and set appropriate parameters for the deployment guardrails the range of market conditions (exchange rates) under which the changes in quantity of the deployed assets are to be mitigated has to be defined and potential parameters have to be identified.
More refined metrics (such as volatility) will be implemented soon for asset pairs, initially the goal is to set parameters conservatively such that relative changes in exchange rates of up to 100% can be covered by the reserve. Below is an example of a fictitious asset paired with ETH and deployed to a 50:50 AMM pool under such conditions:
Table 1. Status of the pooled assets at the beginning
After a 100% price increase of ABC the status of the pool is as follows:
Table 2. Status of the pooled assets upon withdrawal
Examining the changes in QTY and associated notional values produces the following results:
Table 3. Changes in quantity and resulting loss in notional value
The following can be observed in a scenario of a 100% (doubling or halving of value of one of the assets) relative change in price:
A QTY-deficit of 29.3% in ABC will have to be covered by assets from the ABC reserve to make the LPs whole.
A net withdrawal of over 70.7% of ABC would be required before mitigation mechanics would become necessary.
In conclusion it can be inferred that the protocol will be able to make LPs whole after a relative change in price of 100% as long as no more than 3x (as 29.3% of the initial qty will have to be covered) the reserve QTY held in the PCA of an asset are deployed.
It should also be noted that even stricter, gradually lowering guardrails will be implemented in the beginning stages of the protocol, which will provide additional mitigation against other potential issues.
Guardrails to be Implemented
In accordance with the above definitions the guardrails will be controlled via parameters that can be altered by changing variables, which as the protocol matures will be adapted to match asset pair-specific risks, market conditions and other factors such as overall PCA value and DeFi-landscape (e.g. new protocols that can be leveraged).
For XXX / YYY pool, deploy only the liquidity that obeys the following rules (use most restrictive):
Deploy only V-multiple of the quantity of XXX available in the reserve Deploy only Z-multiple of the quantity of YYY available in reserve
Deploy only M% of XXX contributed to the reactor by LPs Deploy only N% of YYY contributed to the reactor by LPs (this guardrail which is not based on any above calculate percentages will be loosened over time, as its primary function is to ensure safety during the first deployments of liquidity.)
1.3 TOKE Backstop
In order to further safeguard the reactor and its LPs, TOKE staked to reactor can be used to cure the deficit if the price moves outside of the aforementioned assumptions.
1.4 Asset Stack and Reactor Health
The below graphic illustrates the asset stack at the beginning of the cycle, from which the reactor health (RH) and the health of Genesis Pools can be calculated using the quantities:
RH = \frac {Asset_{Reserve}+Asset_{LP}}{tAsset_{Deployed}}
According to the 3x reserve multiplier the RH is 1.33 at beginning of cycles and can increase and decrease depending on the market price of the asset.
Additionally, the total collateral (TC) of the reactor can be calculated using the notional values, which includes the TOKE staked to the reactor:
TC = \frac {Asset_{Reserve}+Asset_{LP} + TOKE}{tAsset_{Deployed}}
Illustration of the asset stack at the start of a cycle
1.5 Surplus / Deficit Balancing
At the end of every cycle the reactors will rebalance. In the case that there are more LP ABC than tABC, the surplus ABC is flowed into the reserve. In the case there are less LP ABC than tABC, ABC from the reserve is flowed out into the LP.
Using these mechanics, each cycle Tokemak is flowing out the same amount of LP ABC as there are tABC deposited into the system (while adhering to the above guardrails).
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§ LCS DP: The speedup is from filtration
I feel like I finally see where the power of dynamic programming lies.
Consider the longest common subsequence problem over arrays
A
B
n
m
Naively, we have
2^n \times 2^m
pairs of subsequences and we need to process each of them.
How does the LCS DP solution manage to solve this in
O(nm)
Key idea 1: create "filtration" of the problem
F_{i, j} \subseteq 2^n\times2^m
(i, j)
, consider the "filter"
F_{i, j}
containing all pairs of subsequences
(s \in 2^n, t \in 2^m)
where maxix(s) \leq i and maxix(t) \leq j.
These filters of the filtration nest into one another, so
F_{i, j} \subseteq F_{i', j'}
i \leq i'
j \leq j'
Key idea 2: The value of max LCS(filter) is (a) monotonic, and (b) can be computed efficiently from the values of lower filtration. So we have a monotone map from the space of filters to the solution space, and this monotone map is efficiently computable, given the values of filters below this in the filtration.
This gives us a recurrence, where we start from the bottom filter and proceed to build upward.
See that this really has nothing to do with recursion. It has to do with problem decomposition . We decompose the space
2^n \times 2^m
cleverly via filtration
F_{i, j}
such that max LCS(F[i, j]) was efficiently computable.
To find a DP, think of the entire state space, then think of filtrations, such that the solution function becomes a monotone map, and the solution function is effeciently computable given the values of filters below it.
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Asymptotic Behaviour of Relaxed Dirichiet Problems Involving a Dirichlet-Poincar Form | EMS Press
Asymptotic Behaviour of Relaxed Dirichiet Problems Involving a Dirichlet-Poincar Form
N. Tchou
We study the convergence of the solutions of a sequence of relaxed Dirichlet prob lems relative to Dirichlet forms to the solution of the Γ-limit problem. In particular we prove the strong convergence in
D^P_0[a,Ω](1≤p≤2)
and the existence of "correctors" for the strong convergence in
D0[a,Ω]
. The above two results are generalizations to our framework of previous results proved in [10] in the usual uniformly elliptic setting.
Marco Biroli, N. Tchou, Asymptotic Behaviour of Relaxed Dirichiet Problems Involving a Dirichlet-Poincar Form. Z. Anal. Anwend. 16 (1997), no. 2, pp. 281–309
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How to Find the Magnitude of a Vector: 7 Steps (with Pictures)
1 Finding the Magnitude of a Vector at the Origin
2 Finding the Magnitude of a Vector Away from the Origin
A vector is a geometrical object that has both a magnitude and direction.[1] X Research source The magnitude is the length of the vector, while the direction is the way it's pointing. Calculating the magnitude of a vector is simple with a few easy steps. Other important vector operations include adding and subtracting vectors, finding the angle between two vectors, and finding the cross product.
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Determine the components of the vector. Every vector can be numerically represented in the Cartesian coordinate system with a horizontal (x-axis) and vertical (y-axis) component.[2] X Research source It is written as an ordered pair
{\displaystyle v=<x,y>}
For example, the vector above has a horizontal component of 3 and a vertical component of -5, therefore the ordered pair is <3, -5>.
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Draw a vector triangle. When you draw the horizontal and vertical components, you end up with a right triangle. The magnitude of the vector is equal to the hypotenuse of the triangle so you can use the Pythagorean theorem to calculate it.[3] X Research source
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Rearrange the Pythagorean theorem to calculate the magnitude. The Pythagorean theorem is A2 + B2 = C2. "A" and "B" are the horizontal and vertical components of the triangle while "C" is the hypotenuse. Since the vector is the hypotenuse you want to solve for "C".
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Solve for the magnitude. Using the equation above, you can plug in the numbers of the ordered pair of the vector to solve for the magnitude.[4] X Research source
For example, v = √((32+(-5)2))
v =√(9 + 25) = √34 = 5.831
Don't worry if your answer is not a whole number. Vector magnitudes can be decimals.
Finding the Magnitude of a Vector Away from the Origin
Determine the components of both points of the vector. Every vector can be numerically represented in the Cartesian coordinate system with a horizontal (x-axis) and vertical (y-axis) component.[5] X Research source It is written as an ordered pair
{\displaystyle v=<x,y>}
. If you are given a vector that is placed away from the origin of the Cartesian coordinate system, you must define the components of both points of the vector.
Point A has a horizontal component of 5 and a vertical component of 1, so the ordered pair is <5, 1>.
Point B has a horizontal component of 1 and a vertical component of 2, so the ordered pair is <1, 2>.
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Use a modified formula to solve for the magnitude. Because you now have two points you are dealing with, you must subtract the x and y components of each point before you solve using the equation v = √((x2-x1)2 +(y2-y1)2).
Point A is ordered pair 1 <x1, y1> and point B is ordered pair 2 <x2, y2>
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Solve for the magnitude. Plug in the numbers of your ordered pairs and calculate the magnitude. Using our above example the calculation looks like this:[6] X Research source
v = √(16+1) = √(17) = 4.12
The coordinates of head and tail of a vector are (2, 1, 0), (-4, 2, -3). What is the magnitude of the vector?
You can use the same formula: |→a| = √((x2 – x1)^2 + (y2 – y1)^2), but add on (z2 – z1)^2 at the end for the third set of coordinates!
Use this formula: tan(y component / x component). If the vector is in quadrant 3 or 4, add a half rotation.
What is the magnitude of the resultant vector of vector A-12.66 and vector B-11.93?
To get the magnitude you need to square both the vectors' magnitude and then take the underroot.
How do I find the magnitude of a vertical and horizontal component if a vector is shown in a scale diagram?
If you're given an angle, use that angle and the vector's magnitude to calculate. Vx= (vector's mag)*cos(angle), Vy= (vector's mag)*sin(angle).
How can I find the magnitude of vectors if there is no coordinates, an angle, and then a force?
If you are given a value for work, you can divide that value by the magnitude of the force multiplied by the cosine of the angle, since W=|F||d|cos⍺ (|d|=W/(|F|cos⍺)).
How do I find the vector when only its modulus is given?
One modulus can apply to more than one vector, so any coordinates that fit the formula |a|= square root of (x^2 = y^2) should work, where |a|= the magnitude/modulus.
See Find the Angle Between Two Vectors.
Add or Subtract Vectors
Find Perpendicular Vectors in 2 Dimensions
Find the Equations of the Asymptotes of a Hyperbola
↑ https://www.physicsclassroom.com/class/vectors/Lesson-1/Vector-Addition
↑ https://sciencing.com/calculate-magnitude-force-physics-6209165.html
↑ https://www.khanacademy.org/math/precalculus/vectors-precalc/component-form-of-vectors/a/vector-magnitude-and-direction-review
To find the magnitude of a vector, first determine its horizontal and vertical components on their respective number lines around the origin. Next, draw the horizontal and vertical components to plot the point where they intersect. Then draw a line from the origin to that point, creating a vector triangle, which is a right triangle. Finally, use the Pythagorean theorem to calculate the triangle’s hypotenuse, which is the same as the vector’s magnitude. For more information on finding the magnitude of a vector, including using a modified formula when the vector is away from the origin, scroll down!
"Had to lookup information for my IGCSE's. This website was my saving grace. Seriously, thank you so much!"
"Visualization with graphs was helpful, good step-by-step explanation of algebraic formula."
Pat Myhiney
"This article was much more clear and easier to understand than the class recommended book."
"I learned how to calculate the magnitude using theorems."
"It's way of representation is good."
|
Fast subsonic combustion as a free-interface problem | EMS Press
Leonid S. Kagan
The paper is concerned with the recently identified fast, yet subsonic, combustion waves occurring in obstacle-laden (e.g. porous) systems and driven not by thermal diffusivity but rather by the drag-induced diffusion of pressure. In the framework of a quasi-one-dimensional formulation where the impact of obstacles is accounted for through a frictional drag term, an asymptotic expression for the wave propagation velocity
D
is derived. The propagation velocity is controlled by the temperature
(T_+)
at the entrance to the reaction zone rather than at its exit
(T_b)
as occurs in deflagrative combustion. The evaluated
D(T_+)
dependence allows description of the subsonic detonation in terms of a free-interface problem. The latter is found to be dynamically akin to the problem of gasless combustion known for its rich pattern-forming dynamics.
Gregory I. Sivashinsky, Peter V. Gordon, Leonid S. Kagan, Fast subsonic combustion as a free-interface problem. Interfaces Free Bound. 5 (2003), no. 1, pp. 47–62
|
Thermal Conductivity Measurements of Nylon 11-Carbon Nanofiber Nanocomposites | J. Heat Transfer | ASME Digital Collection
Arden L. Moore,
Justin M. Jensen,
Moore, A. L., Cummings, A. T., Jensen, J. M., Shi, L., and Koo, J. H. (June 24, 2009). "Thermal Conductivity Measurements of Nylon 11-Carbon Nanofiber Nanocomposites." ASME. J. Heat Transfer. September 2009; 131(9): 091602. https://doi.org/10.1115/1.3139110
Carbon nanofibers (CNFs) were incorporated into nylon 11 to form nylon 11-carbon nanofiber nanocomposites via twin screw extrusion. Injection molding has been employed to fabricate specimens that possess enhanced mechanical strength and fire retardancy. The thermal conductivity of these polymer nanocomposites was measured using a guarded hot plate method. The measurement results show that the room temperature thermal conductivity increases with the CNF loading from
0.24±0.01 W/m K
for pure Nylon 11 to
0.30±0.02 W/m K
7.5 wt %
CNF loading. The effective medium theory has been used to determine the interface thermal resistance between the CNFs and the matrix to be in the range of
2.5–5.0×10−6 m2 K/W
from the measured thermal conductivity of the nanocomposite.
carbon fibres, extrusion, filled polymers, flame retardants, injection moulding, mechanical strength, nanocomposites, thermal conductivity, thermal resistance, thermal conductivity, nanocomposites, carbon nanofibers, nylon, measurement, thermal interface resistance
Carbon, Nanocomposites, Nanofibers, Nylon fabrics, Thermal conductivity, Temperature, Injection molding, Thermal conductivity measurement
Thermal and Flammability Properties of Polypropylene/Carbon Nanotube Nanocomposites
Flammability and Mechanical Properties of Nylon 11 Nanocomposites
Proceedings of the SAMPE 2005 ISSE, SAMPE
, Covina, CA, pp.
Synthesis and Properties of Syndiotactic Poly(propylene)/Carbon Nanofiber and Nanotube Composites Prepared by In Situ Polymerization With Metallocene/MAO Catalysts
Carbon Nanofibers for Composite Applications
Sugimotot
Mechanical and Thermal Properties of Vapor-Grown Carbon Nanofiber and Polycarbonate Composite Sheets
C177-04 Standard Test Method for Steady-State Heat Flux Measurements and Thermal Transmission Properties by Means of the Guarded-Hot-Plate Apparatus
Available online at http://www.apsci.com/ngm-pyro1.htmlhttp://www.apsci.com/ngm-pyro1.html.
Available online at http://www.goodfellow.com/E/Polyamide_-_Nylon_11.HTMLhttp://www.goodfellow.com/E/Polyamide_-_Nylon_11.HTML.
Thermal Contact Resistance Evaluation in Polymer-Based Carbon Fiber Composites
Oxisik
Keblinksi
Effect of Chemical Functionalization on Thermal Transport of Carbon Nanotube Composites
Single Wall Carbon Nanotube/Polyethylene Nanocomposites: Thermal and Electrical Conductivity
Effective Thermal Conductivity of Functionally Graded Particulate Nanocomposites With Interfacial Thermal Resistance
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LAN Manager - Wikipedia
(Redirected from LM hash)
Microsoft, 3Com
2.2a / 1994; 28 years ago (1994)
MS-Net, Xenix-NET, 3+Share
LAN Manager is a discontinued network operating system (NOS) available from multiple vendors and developed by Microsoft in cooperation with 3Com Corporation. It was designed to succeed 3Com's 3+Share network server software which ran atop a heavily modified version of MS-DOS.
2 Password hashing algorithm
3 Security weaknesses
5 Reasons for continued use of LM hash
The LAN Manager OS/2 operating system was co-developed by IBM and Microsoft. It originally used the Server Message Block (SMB) protocol atop either the NetBIOS Frames (NBF) protocol or a specialized version of the Xerox Network Systems (XNS) protocol. These legacy protocols had been inherited from previous products such as MS-Net for MS-DOS, Xenix-NET for MS-Xenix, and the afore-mentioned 3+Share. A version of LAN Manager for Unix-based systems called LAN Manager/X was also available.
In 1990, Microsoft announced LAN Manager 2.0 with a host of improvements, including support for TCP/IP as a transport protocol. The last version of LAN Manager, 2.2, which included an MS-OS/2 1.31 base operating system, remained Microsoft's strategic server system until the release of Windows NT Advanced Server in 1993.
1987 – MS LAN Manager 1.0 (Basic/Enhanced)
1989 – MS LAN Manager 1.1
1992 – MS LAN Manager 2.1a
Password hashing algorithm[edit]
The LM hash is computed as follows:[1][2]
The user's password is restricted to a maximum of fourteen characters.[Notes 1]
The user’s password is converted to uppercase.
The user's password is encoded in the System OEM code page.[3]
This password is NULL-padded to 14 bytes.[4]
The “fixed-length” password is split into two 7-byte halves.
These values are used to create two DES keys, one from each 7-byte half, by converting the seven bytes into a bit stream with the most significant bit first, and inserting a parity bit after every seven bits (so 1010100 becomes 10101000). This generates the 64 bits needed for a DES key. (A DES key ostensibly consists of 64 bits; however, only 56 of these are actually used by the algorithm. The parity bits added in this step are later discarded.)
Each of the two keys is used to DES-encrypt the constant ASCII string “KGS!@#$%”,[Notes 2] resulting in two 8-byte ciphertext values. The DES CipherMode should be set to ECB, and PaddingMode should be set to NONE.
These two ciphertext values are concatenated to form a 16-byte value, which is the LM hash.
Security weaknesses[edit]
LAN Manager authentication uses a particularly weak method of hashing a user's password known as the LM hash algorithm, stemming from the mid 1980s when viruses transmitted by floppy disks were the major concern.[5] Although it is based on DES, a well-studied block cipher, the LM hash has several weaknesses in its design.[6] This makes such hashes crackable in a matter of seconds using rainbow tables, or in a few minutes using brute force. Starting with Windows NT, it was replaced by NTLM, which is still vulnerable to rainbow tables, and brute force attacks unless long, unpredictable passwords are used, see password cracking. NTLM is used for logon with local accounts except on domain controllers since Windows Vista and later versions no longer maintain the LM hash by default.[5] Kerberos is used in Active Directory Environments.
Password length is limited to a maximum of 14 characters chosen from the 95 ASCII printable characters.
Passwords are not case sensitive. All passwords are converted into uppercase before generating the hash value. Hence LM hash treats PassWord, password, PaSsWoRd, PASSword and other similar combinations same as PASSWORD. This practice effectively reduces the LM hash key space to 69 characters.
A 14-character password is broken into 7+7 characters and the hash is calculated for each half separately. This way of calculating the hash makes it dramatically easier to crack, as the attacker only needs to brute-force 7 characters twice instead of the full 14 characters. This makes the effective strength of a 14-character password equal to only
{\displaystyle 2\times 69^{7}\approx 2^{44}}
, or twice that of a 7-character password, which is 3.7 trillion times less complex than the
{\displaystyle 69^{14}\approx 2^{86}}
theoretical strength of a 14-character single-case password. As of 2020, a computer equipped with a high-end graphics processor (GPUs) can compute 40 billion LM-hashes per second.[8] At that rate, all 7-character passwords from the 95-character set can be tested and broken in half an hour; all 7-character alphanumeric passwords can be tested and broken in 2 seconds.
If the password is 7 characters or less, then the second half of hash will always produce same constant value (0xAAD3B435B51404EE). Therefore, a password is less than or equal to 7 characters long can be identified visibly without using tools (though with high speed GPU attacks, this matters less).
The hash value is sent to network servers without salting, making it susceptible to man-in-the-middle attacks such as replay the hash. Without salt, time–memory tradeoff pre-computed dictionary attacks, such as a rainbow table, are feasible. In 2003, Ophcrack, an implementation of the rainbow table technique, was published. It specifically targets the weaknesses of LM encryption, and includes pre-computed data sufficient to crack virtually all alphanumeric LM hashes in a few seconds. Many cracking tools, such as RainbowCrack, Hashcat, L0phtCrack and Cain, now incorporate similar attacks and make cracking of LM hashes fast and trivial.
To address the security weaknesses inherent in LM encryption and authentication schemes, Microsoft introduced the NTLMv1 protocol in 1993 with Windows NT 3.1. For hashing, NTLM uses Unicode support, replacing LMhash=DESeach(DOSCHARSET(UPPERCASE(password)), "KGS!@#$%") by NThash=MD4(UTF-16-LE(password)), which does not require any padding or truncating that would simplify the key. On the negative side, the same DES algorithm was used with only 56-bit encryption for the subsequent authentication steps, and there is still no salting. Furthermore, Windows machines were for many years configured by default to send and accept responses derived from both the LM hash and the NTLM hash, so the use of the NTLM hash provided no additional security while the weaker hash was still present. It also took time for artificial restrictions on password length in management tools such as User Manager to be lifted.
While LAN Manager is considered obsolete and current Windows operating systems use the stronger NTLMv2 or Kerberos authentication methods, Windows systems before Windows Vista/Windows Server 2008 enabled the LAN Manager hash by default for backward compatibility with legacy LAN Manager and Windows ME or earlier clients, or legacy NetBIOS-enabled applications. It has for many years been considered good security practice to disable the compromised LM and NTLMv1 authentication protocols where they aren't needed.[9] Starting with Windows Vista and Windows Server 2008, Microsoft disabled the LM hash by default; the feature can be enabled for local accounts via a security policy setting, and for Active Directory accounts by applying the same setting via domain Group Policy. The same method can be used to turn the feature off in Windows 2000, Windows XP and NT.[9] Users can also prevent a LM hash from being generated for their own password by using a password at least fifteen characters in length.[4] -- NTLM hashes have in turn become vulnerable in recent years to various attacks that effectively make them as weak today as LanMan hashes were back in 1998.[citation needed]
Reasons for continued use of LM hash[edit]
Many legacy third party SMB implementations have taken considerable time to add support for the stronger protocols that Microsoft has created to replace LM hashing because the open source communities supporting these libraries first had to reverse engineer the newer protocols—Samba took 5 years to add NTLMv2 support, while JCIFS took 10 years.
Availability of NTLM protocols to replace LM authentication
Windows NT 3.1 RTM (1993) Not supported
Windows NT 3.51 RTM (1995) Not supported
Windows NT 4 RTM (1996) Service Pack 4[10] (25 October 1998)
Windows 95 Not supported Directory services client (released with Windows 2000 Server, 17 February 2000)
Windows 98 RTM Directory services client (released with Windows 2000 Server, 17 February 2000)
Windows 2000 RTM (17 February 2000) RTM (17 February 2000)
Windows Me RTM (14 September 2000) Directory services client (released with Windows 2000 Server, 17 February 2000)
Samba ? Version 3.0[11] (24 September 2003)
JCIFS Not supported Version 1.3.0 (25 October 2008)[12]
IBM AIX (SMBFS) 5.3 (2004)[13] Not supported as of v7.1[14]
Poor patching regimes subsequent to software releases supporting the feature becoming available have contributed to some organisations continuing to use LM Hashing in their environments, even though the protocol is easily disabled in Active Directory itself.
Lastly, prior to the release of Windows Vista, many unattended build processes still used a DOS boot disk (instead of Windows PE) to start the installation of Windows using WINNT.EXE, something that requires LM hashing to be enabled for the legacy LAN Manager networking stack to work.
^ If the password is more than fourteen characters long, the LM hash cannot be computed.
^ The string “KGS!@#$%” could possibly mean Key of Glen and Steve and then the combination of Shift + 12345. Glen Zorn and Steve Cobb are the authors of RFC 2433 (Microsoft PPP CHAP Extensions).
^ "Chapter 3 - Operating System Installation: The LMHash". Microsoft Technet. Retrieved 2015-05-12.
^ Glass, Eric (2006). "The NTLM Authentication Protocol and Security Support Provider: The LM Response". Retrieved 2015-05-12.
^ "List of Localized MS Operating Systems". Microsoft Developer Network. Retrieved 2015-05-12.
^ a b "Cluster service account password must be set to 15 or more characters if the NoLMHash policy is enabled". Microsoft. 2006-10-30. Retrieved 2015-05-12.
^ a b Jesper Johansson. "The Most Misunderstood Windows Security Setting of All Time". TechNet Magazine. Microsoft. Retrieved 2 November 2015. Although Windows Vista has not been released yet, it is worthwhile to point out some changes in this operating system related to these protocols. The most important change is that the LM protocol can no longer be used for inbound authentication—where Windows Vista is acting as the authentication server.
^ Johansson, Jasper M. (2004-06-29). "Windows Passwords: Everything You Need To Know". Microsoft. Retrieved 2015-05-12.
^ Rahul Kokcha
^ Benchmark Hashcat v6.1.1 on RTX 2070S (SUPER), Mode 3000 LM, accessed November 29, 2020
^ a b "How to prevent Windows from storing a LAN manager hash of your password in Active Directory and local SAM databases". Microsoft Knowledge Base. 2007-12-03. Retrieved 2015-05-12.
^ "Windows NT 4.0 Service Pack 4 Readme.txt File (40-bit)". Microsoft. 1998-10-25. Retrieved 2015-05-12.
^ "The Samba Team announces the first official release of Samba 3.0". SAMBA. 2003-09-24. Retrieved 2015-05-12.
^ "The Java CIFS Client Library". Retrieved 2015-05-12.
^ "AIX 5.3 Networks and communication management: Server Message Block file system". IBM. 2010-03-15. p. 441. Retrieved 2015-05-12.
^ "AIX 7.1 Networks and communication management: Server Message Block file system". IBM. 2011-12-05. Retrieved 2015-05-12.
Wikibooks has a book on the topic of: Reverse Engineering/Cracking Windows XP Passwords
"Microsoft LAN Manager". Archived from the original on 2017-02-12.
Oechslin, Philippe (2003). "Making a Faster Cryptanalytic Time-Memory Trade-Off" (PDF). Advances in Cryptology, CRYPTO 2003.
"Ophcrack, a well known password cracker".
"Cain and Abel, password recovery tool for Microsoft Operating Systems".
mudge (1997-07-24). "A L0phtCrack Technical Rant". Archived from the original on 2011-12-11. Alt URL
Novell S-Net
Retrieved from "https://en.wikipedia.org/w/index.php?title=LAN_Manager&oldid=1087303205#Password_hashing_algorithm"
|
Annotate Video Using Detections in Vehicle Coordinates - MATLAB & Simulink - MathWorks í•œêµ
Display a Frame with Video Annotations
Display a Clip with Video Annotations
Create the Mono Camera for On-Video Display
Using the Mono Camera Object to Update the Display
Configure and use a monoCamera object to display information provided in vehicle coordinates on a video display.
Displaying data recorded in vehicle coordinates on a recorded video is an integral part of ground truth labeling and analyzing tracking results. Using a two-dimensional bird's-eye view can help you understand the overall environment, but it is sometimes hard to correlate the video with the bird's-eye view display. In particular, this problem becomes worse when using a third-party sensor where you cannot access the raw video captured by the sensor, and you need to use a video captured by a separate camera.
Automated Driving Toolbox™ provides the monoCamera object that facilitates the conversion between vehicle coordinates and image coordinates. This example reads data recorded by a video sensor installed on a test vehicle. Then it displays the data on a video captured by a separate video camera installed on the same car. The data and video were recorded at the following rates:
Reported lane information: 20 times per second
Reported vision objects: 10 times per second
The selected frame corresponds to 5.9 seconds into the video clip, when there are several objects to show on the video.
% Set up video reader and player
videoFile = '01_city_c2s_fcw_10s.mp4';
videoPlayer = vision.DeployableVideoPlayer;
% Jump to the desired frame
videoReader.CurrentTime = time;
frameWithoutAnnotations = readFrame(videoReader);
imshow(frameWithoutAnnotations);
title('Original Video Frame')
Get the corresponding recorded data.
recordingFile = '01_city_c2s_fcw_10s_sensor.mat';
[visionObjects, laneReports, timeStep, numSteps] = readDetectionsFile(recordingFile);
currentStep = round(time / timeStep) + 1;
videoDetections = processDetections(visionObjects(currentStep));
laneBoundaries = processLanes(laneReports(currentStep));
% Set up the monoCamera object for on-video display
sensor = setupMonoCamera(videoReader);
frameWithAnnotations = updateDisplay(frameWithoutAnnotations, sensor, videoDetections, laneBoundaries);
imshow(frameWithAnnotations);
title('Annotated Video Frame')
To display the video clip with annotations, simply repeat the annotation frame-by-frame. The video shows that the car pitches slightly up and down, which changes the pitch angle. No attempt has been made to compensate for this pitch motion. As a result, the conversion from vehicle coordinates to image coordinates is a little inaccurate on some of the frames.
% Reset the time back to zero
currentStep = 0; % Reset the recorded data timestep
videoReader.CurrentTime = 0; % Reset the video reader time
while currentStep < numSteps && hasFrame(videoReader)
% Get the current time
% Prepare the detections to the tracker
videoDetections = processDetections(visionObjects(currentStep), videoDetections);
% Process lanes
% Update video frame with annotations from the reported objects
% Pause for 50 milliseconds for a more realistic display rate. If you
% process data and form tracks in this loop, you do not need this
% pause.
pause(0.05 - toc);
% Display annotated frame
The setupMonoCamera function returns a monoCamera sensor object, which is used for converting positions in vehicle coordinates to image coordinates.
Start by defining the camera intrinsic parameters. The parameters in this function were estimated based on the camera model. To obtain the parameters for your camera, use the Camera Calibrator app.
Because the data in this example has little distortion, this function ignores the lens distortion coefficients. The parameters are next stored in a cameraIntrinsics object.
Next, define the camera extrinsics. Camera extrinsics relate to the way the camera is mounted on the car. The mounting includes the following properties:
Height: Mounting height above the ground, in meters.
Pitch: Pitch of the camera, in degrees, where positive is angled below the horizon and toward the ground. In most cases, the camera is pitched slightly below the horizon.
Roll: Roll of the camera about its axis. For example, if the video is flipped upside down, use roll = 180.
Yaw: Angle of the camera sideways, where positive is in the direction of the positive y-axis (to the left). For example, a forward-facing camera has a yaw angle of 0 degrees, and a backward-facing camera has a yaw angle of 180 degrees.
function sensor = setupMonoCamera(vidReader)
% Define the camera intrinsics from the video information
focalLength = [1260 1100]; % [fx, fy] % pixels
principalPoint = [360 245]; % [cx, cy] % pixels
imageSize = [vidReader.height, vidReader.width]; % [numRows, numColumns] % pixels
% Define the camera mounting (camera extrinsics)
mountingHeight = 1.45; % height in meters from the ground
mountingPitch = 1.25; % pitch of the camera in degrees
mountingRoll = 0.15; % roll of the camera in degrees
mountingYaw = 0; % yaw of the camera in degrees
sensor = monoCamera(intrinsics, mountingHeight, ...
'Pitch', mountingPitch, ...
'Roll', mountingRoll, ...
'Yaw', mountingYaw);
The updateDisplay function displays all the object annotations on top of the video frame.
The display update includes the following steps:
Using the monoCamera sensor to convert reported detections into bounding boxes and annotating the frame.
Using the insertLaneBoundary method of the parabolicLaneBoundary object to insert the lane annotations.
function frame = updateDisplay(frame, sensor, videoDetections, laneBoundaries)
% Allocate memory for bounding boxes
bboxes = zeros(numel(videoDetections), 4);
% Create the bounding boxes
for i = 1:numel(videoDetections)
% Use monoCamera sensor to convert the position in vehicle coordinates
% to the position in image coordinates.
% 1. The width of the object is reported and is used to calculate the
% size of the bounding box around the object (half width on each
% side). The height of the object is not reported. Instead, the
% function uses a height/width ratio of 0.85 for cars and 3 for
% pedestrians.
% 2. The reported location is at the center of the object at ground
% level, i.e., the bottom of the bounding box.
xyLocation1 = vehicleToImage(sensor, videoDetections(i).positions' + [0,videoDetections(i).widths/2]);
xyLocation2 = vehicleToImage(sensor, videoDetections(i).positions' - [0,videoDetections(i).widths/2]);
dx = xyLocation2(1) - xyLocation1(1);
% Define the height/width ratio based on object class
if strcmp(videoDetections(i).labels, 'Car')
dy = dx * 0.85;
elseif strcmp(videoDetections(i).labels, 'Pedestrian')
dy = dx * 3;
% Estimate the bounding box around the vehicle. Subtract the height of
% the bounding box to define the top-left corner.
bboxes(i,:) =[(xyLocation1 - [0, dy]), dx, dy];
labels = {videoDetections(:).labels}';
% Add bounding boxes to the frame
frame = insertObjectAnnotation(frame, 'rectangle', bboxes, labels,...
'Color', 'yellow', 'FontSize', 10, 'TextBoxOpacity', .8, 'LineWidth', 2);
% Display the lane boundary on the video frame
xRangeVehicle = [1, 100];
xPtsInVehicle = linspace(xRangeVehicle(1), xRangeVehicle(2), 100)';
frame = insertLaneBoundary(frame, laneBoundaries(1), sensor, xPtsInVehicle, ...
'Color', 'red');
'Color', 'green');
This example showed how to create a monoCamera sensor object and use it to display objects described in vehicle coordinates on a video captured by a separate camera. Try using recorded data and a video camera of your own. Try calibrating your camera to create a monoCamera that allows for transformation from vehicle to image coordinates, and vice versa.
readDetectionsFile - Reads the recorded sensor data file. The recorded data is in a single structure that is divided into four struct arrays. This example uses only the following two arrays:
laneReports, a struct array that reports the boundaries of the lane. It has these fields: left and right. Each element of the array corresponds to a different timestep. Both left and right are structures with these fields: isValid, confidence, boundaryType, offset, headingAngle, and curvature.
visionObjects, a struct array that reports the detected vision objects. It has the fields numObjects (integer) and object (struct). Each element of the array corresponds to a different timestep. object is a struct array, where each element is a separate object with these fields: id, classification, position (x;y;z), velocity(vx;vy;vz), size(dx;dy;dz). Note: z=vy=vz=dx=dz=0
function [visionObjects, laneReports, timeStep, numSteps] = readDetectionsFile(filename)
A = load(strcat(filename));
timeStep = 0.05; % Lane data is provided every 50 milliseconds
processDetections - Reads the recorded vision detections. This example extracts only the following properties:
Position: A two-dimensional [x, y] array in vehicle coordinates
Width: The width of the object as reported by the video sensor (Note: The sensor does not report any other dimension of the object size.)
Labels: The reported classification of the object
function videoDetections = processDetections(visionData, videoDetections)
% The video sensor reports a classification value as an integer
% according to the following enumeration (starting from 0)
ClassificationValues = {'Unknown', 'Unknown Small', 'Unknown Big', ...
'Pedestrian', 'Bike', 'Car', 'Truck', 'Barrier'};
% The total number of objects reported by the sensor in this frame
numVideoObjects = visionData.numObjects;
% The video objects are reported only 10 times per second, but the video
% has a frame rate of 20 frames per second. To prevent the annotations from
% flickering on and off, this function returns the values from the previous
% timestep if there are no video objects.
if numVideoObjects == 0
if nargin == 1 % Returning a result even if there is no previous value
videoDetections = struct('positions', {}, 'labels', {}, 'widths', {});
% Prepare a container for the relevant properties of video detections
videoDetections = struct('positions', [], 'labels', [], 'widths', []);
for i = 1:numVideoObjects
videoDetections(i).widths = visionData.object(i).size(2);
videoDetections(i).positions = visionData.object(i).position(1:2);
videoDetections(i).labels = ClassificationValues{visionData.object(i).classification + 1};
processLanes - Reads reported lane information and converts it into parabolicLaneBoundary objects.
Lane boundaries are updated based on the laneReports from the recordings. The sensor reports the lanes as parameters of a parabolic model: %
y=a{x}^{2}+bx+c
function laneBoundaries = processLanes(laneReports)
% Return processed lane boundaries
% Boundary type information
types = {'Unmarked', 'Solid', 'Dashed', 'Unmarked', 'BottsDots', ...
'Unmarked', 'Unmarked', 'DoubleSolid'};
% Read the recorded lane reports for this frame
% Create parabolicLaneBoundary objects for left and right lane boundaries
leftParams = cast([leftLane.curvature, leftLane.headingAngle, leftLane.offset], 'double');
leftBoundaries.BoundaryType = types{leftLane.boundaryType};
rightParams = cast([rightLane.curvature, rightLane.headingAngle, rightLane.offset], 'double');
rightBoundaries.BoundaryType = types{rightLane.boundaryType};
insertObjectAnnotation | insertLaneBoundary
VideoReader | monoCamera | vision.DeployableVideoPlayer
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Multiphase Quasi-RMS Current Sensor - MapleSim Help
Home : Support : Online Help : MapleSim : MapleSim Component Library : Electrical : Multiphase Systems : Sensors : Multiphase Quasi-RMS Current Sensor
Current Quasi-RMS Sensor
The Current Quasi-RMS Sensor component measures the continuous quasi-RMS value of multiphase current.
If the current waveform deviates from a sine curve, the output of the sensor will not be exactly the average RMS value.
{I}_{\mathrm{rms}}=\sqrt{\frac{1}{m}\sum _{k=1}^{m}{i}_{k}^{2}}
i={i}_{p}=-{i}_{n}
v={v}_{p}-{v}_{n}=0
{\mathrm{plug}}_{p}
m
{\mathrm{plug}}_{n}
m
I
Real output; quasi-RMS of current in
A
m
3
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Review the Math Notes box in 1.1.2 to help you describe the graph.
If you used a function to model this data, it would make sense to limit the maximum and minimum values for jump heights. Why? What are reasonable limits (maximum and minimum) for the jump heights?
Since distances are measured with positive numbers, it makes sense to set
0
as a reasonable minimum.
The maximum has more room for interpretation, perhaps
15
feet seems a reasonable maximum.
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Milling Processes With Active Damping: Modeling and Stability | J. Comput. Nonlinear Dynam. | ASME Digital Collection
David Lehotzky,
David Lehotzky
1Corresponding author. e-mail: d.lehotzky@northeastern.edu
Iker Mancisidor,
Iker Mancisidor
Ideko, Elgoibar, Basque Country 20870,
e-mail: imancisidor@ideko.es
e-mail: jmunoa@ideko.es
Department of Applied Mechanics, Budapest University of Technology and Economics
e-mail: dombovari@mm.bme.hu
Jokin Munoa Head of Department
Zoltan Dombovari Associate Professor
J. Comput. Nonlinear Dynam. Feb 2022, 17(2): 021005 (18 pages)
Lehotzky, D., Mancisidor, I., Munoa, J., and Dombovari, Z. (December 3, 2021). "Milling Processes With Active Damping: Modeling and Stability." ASME. J. Comput. Nonlinear Dynam. February 2022; 17(2): 021005. https://doi.org/10.1115/1.4052723
Active dampers are on the verge of appearing in commercial machines as devices that assist the avoidance of machine tool chatter. The adjustment of control parameters in these devices is mostly guided by models that do not consider the dynamics within the control loop of active damper. Therefore, these models neglect the dynamics of actuation, measurement, and filtering, which can result in inaccurate stability predictions that hinder the efficient tuning of active dampers. To formulate a more realistic model for milling processes assisted by active damping, this paper derives a novel mathematical model that takes into account the internal dynamics of the actuator, measuring device, and discrete filtering. This study shows that accurate stability prediction requires the incorporation of actuator and filter dynamics into the model, especially at high spindle speeds and large feedback gains.
Accelerometers, Actuators, Feedback, Filters, Milling, Modeling, Stability, Approximation, Filtration, Dynamics (Mechanics), Cutting, Active damping
Spezifische Schnittkräfte bei der Metallbearbeitung
Werkstattstech. Maschinenbau
Chatter Suppression in Turning Operations With a Tuned Vibration Absorber
Magnetorheological Fluid-Controlled Boring Bar for Chatter Suppression
Application of Spindle Speed Variation for Chatter Suppression in Turning
Design of Self-Tuneable Mass Damper for Modular Fixturing Systems
Optimal Control of Chatter in Turning
Active Dampers for Machine Tools
Köpken
Chatter Suppression in Ram Type Travelling Column Milling Machines Using a Biaxial Inertial Actuator
Stability of Turning Processes Subjected to Digital PD Control
.10.3311/pp.me.2012-1.06
Stabilizability Diagram for Turning Processes Subjected to Digital PD Control
Coupled Model for Simulating Active Inertial Actuators in Milling Processes
Vibration Control Strategies for Proof-Mass Actuators
Bilbao-Guillerna
Low Frequency Chatter Suppression Using an Inertial Actuator
Ninth CIRP International Conference on High Speed Machining
.https://www.researchgate.net/publication/281923756_Low_Frequency_Chatter_Suppression_using_an_Inertial_Actuator
,” 11th International Conference on High Speed Machining (
Prague, Czech Republic, Sept. 11–12, pp. 1–8
.https://hal.archives-ouvertes.fr/hal-01134192v1
System Modeling of Microaccelerometer Using Piezoelectric Thin Films
Robust Controller Design for Turning Operations Based on Measured Frequency Response Functions
Identification of Cutting Force Characteristics Based on Chatter Experiments
Numerical Methods for the Stability and Stabilizability Analysis of Delayed Dynamical Systems
Analysis and Computation of the
H 2
Norm of Delay Differential Algebraic Equations
Improved Prediction of Stability Lobes With Extended Multi Frequency Solution
Bisection Method in Higher Dimensions and the Efficiency Number
Spectral Element Method for Stability Analysis of Milling Processes With Discontinuous Time-Periodicity
A Comprehensive Dynamic End Milling Simulation Model
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A density version of the Carlson–Simpson theorem | EMS Press
We prove a density version of the Carlson–Simpson Theorem. Specifically we show the following.
k ≥ 2
A
k
\mathrm {lim \ sup}_{n\to\infty} |A \cap [k]^n|/k^n >0
c
k
(w_n)
k
{c}\cup \big\{c^{\smallfrown}w_0(a_0)^{\smallfrown}...^{\smallfrown}w_n(a_n):n\in\mathbb N \text{ and } a_0,...,a_n\in [k]\big\}
A
While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.
Pandelis Dodos, Vassilis Kanellopoulos, Konstantinos Tyros, A density version of the Carlson–Simpson theorem. J. Eur. Math. Soc. 16 (2014), no. 10, pp. 2097–2164
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Spectral and stochastic properties of the $f$-Laplacian, solutions of PDEs at infinity and geometric applications | EMS Press
Spectral and stochastic properties of the
f
-Laplacian, solutions of PDEs at infinity and geometric applications
The aim of this paper is to suggest a new perspective to study qualitative properties of solutions of semilinear elliptic partial differential equations defined outside a compact set. The relevant tools in this setting come from spectral theory and from a combination of stochastic properties of the differential operators in question. Possible links between spectral and stochastic properties are analyzed in detail.
G. Pacelli Bessa, Stefano Pigola, Alberto G. Setti, Spectral and stochastic properties of the
f
-Laplacian, solutions of PDEs at infinity and geometric applications. Rev. Mat. Iberoam. 29 (2013), no. 2, pp. 579–610
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in rational no what is greater -8/5 or -7/4 - Maths - Rational Numbers - 7912515 | Meritnation.com
in rational no. what is greater -8/5 or -7/4
First we have to find the L.C.M of denominators of both fractions
\frac{-8}{5} \mathrm{and} \frac{-7}{4}
And L.C.M of 5 and 4 = 20
Now making the denominator of both fractions common as;
\frac{-8}{5} = \frac{-8}{5}×\frac{4}{4} = \frac{-32}{20}\phantom{\rule{0ex}{0ex}} \mathrm{and} \frac{-7}{4}= \frac{-7}{4}×\frac{5}{5} = \frac{-35}{20}
Now since the denominator is same, so the fraction having greater numerator is greater.
-32>-35
\frac{-32}{20}>\frac{-35}{20} \mathrm{i}.\mathrm{e}. \frac{-8}{5}>\frac{-7}{4}
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Heat transfer by convection - MATLAB
Simscape / Foundation Library / Thermal / Thermal Elements
The Convective Heat Transfer block represents a heat transfer by convection between two bodies by means of fluid motion. The transfer is governed by the Newton law of cooling:
Q=k\cdot A\cdot \left({T}_{A}-{T}_{B}\right),
Q is heat flow.
k is convection heat transfer coefficient.
A is surface area.
D is distance between layers (that is, thickness of material).
TA and TB are temperatures of the two bodies.
Connections A and B are thermal conserving ports associated with the points between which the heat transfer by convection takes place. The block positive direction is from port A to port B. This means that the heat flow is positive if it flows from A to B.
A — Body A
Thermal conserving port associated with body A.
B — Body B
Thermal conserving port associated with body B.
Area — Area of heat transfer
0.0001 m^2 (default) | positive scalar
Surface area of heat transfer.
Heat transfer coefficient — Convection heat transfer coefficient
20 W/m^2/K (default) | positive scalar
Convection heat transfer coefficient.
Conductive Heat Transfer | Radiative Heat Transfer
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CALCULLA - Concentration of saturated solution calculator
Concentration of saturated solution calculator
Calculations finds out concentration (percentage or molar) of saturated solution based on substance solubility or vice versa.
Choose a scenario that best fits your needs I know solubility (ms) and want to calculate percentage concentration (Cp) I know percentage concentration (Cp) and want to calculate solubility (ms) I know solubility (ms), molar mass (M) and volume (V) and want to calculate molar concentration (Cm) I know molar concentration (Cm), molar mass (M) and volume (V) and want to calculate solubility (ms) I know solubility (ms), molar concentration (Cm) and volume (V) and want to calculate molar mass (M) I know solubility (ms), molar concentration (Cm) and molar mass (M) and want to calculate volume (V)
Percentage concentration (Cp) percent [%]permil [‰]basis point (permyriad) [‱] basis point (permyriad) [‱]percentage in point [pip]tick size (US bonds) [tick]tick size plus (US bonds) [tick+] parts per thousand [ppth]parts per million [ppm]parts per hundred million [pphm]parts per billion [ppb]parts per trillion [ppt]parts per quadrillion [ppq] yottauno [YU]zettauno [ZU]exauno [EU]petauno [PU]terauno [TU]gigauno [GU]megauno [MU]kilouno [kU]hektouno [hU]uno [U]deciuno [dU]centiuno [cU]miliuno [mU]microuno [µU]nanouno [nU]pikouno [pU]femtouno [fU]attouno [aU]zeptouno [zU]yoctouno [yU]
Solubility (ms)
(mass of substance in saturated soultion) kiloton (gigagram) [kton, Gg]ton (megagram) [ton, Mg]quintal [q]kilonewton [kN]kilogram [kg]hectogram [hg]decagram [dag]gram [g]carat [ct]centigram [cg]milligram [mg]microgram [µg]nanogram [ng]atomic mass unit [u] long ton [ton]short ton [sh tn]long hundredweight [cwt]short hundredweight [sh cwt]stone [st]pound [lb av]ounce [oz av]dram [dr av]grain [gr] pound [lb t]ounce [oz t]dram [dr t]pennyweight [dwt, pwt]scruple [s ap]grain [gr]carat [kt]mitedoite
Molar concentration (Cm)
Solubility (ms) Show source
10\ \left[g\right]
Percentage concentration (Cp)
Result: Percentage concentration (Cp)#
Cp=\frac{\mathrm{ms}}{\mathrm{ms}+100} \cdot 100
\frac{100}{11}
9.090909090909090909090909090909090909090909090909090909090909091\ \left[\mathrlap{\it{/}}{^0}_{\,0}\right]
\frac{10}{10+100} \cdot 100
Multipled fractions To multiply two fractions we need to multiply numberators and denominators from first and second fractions:
\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}
\frac{10 \cdot 100}{10+100}
Simplify arithmetic -
\frac{1000}{10+100}
\frac{\cancel{1000}}{\cancel{110}}
\frac{\cancel{a}}{\cancel{a}} = 1
\frac{100}{11}
\frac{100}{11}
Divided fraction -
9.090909090909090909090909090909090909090909090909090909090909091
9.090909090909090909090909090909090909090909090909090909090909091\ \left[\mathrlap{\it{/}}{^0}_{\,0}\right]
Saturated solution contains maximum dissolvable amount of substance.
Attempting to introduce more substance into the solution will end with the remaining undissolved substance (sludge).
The maximum amount of substance that can be dissolved is solubility. The solubility is often given in grams per 100 grams of solvent.
Solubility is substance specific. In addition, it depends on the kind of solvent and external conditions (temperature, pressure).
If we have the solubility of the substance (in grams per 100 grams of solvent), we can calculate the percentage concentration of saturated solution:
C_p = \dfrac{m_s}{m_s + 100} \times 100
C_p
- percentage concentration of saturated solution (%),
m_s
- solubility of substance (grams per 100 grams of solvent).
To calculate the molar concentration of saturated solution we additionally need to know the volume of the solution and the molar mass of the substance:
C_m = \dfrac{m_s}{M \times V}
C_m
- molar concentration of saturated solution (
g/dm^3
m_s
- solubility of substance (grams per 100 grams of solvent),
M
- molar mass of substance (
g/mol
V
- volume of saturated solution (
dm^3
concentration_of_saturated_solution · solubility_to_concentration_calculator
stezenie_roztworu_nasyconego · zamiana_rozpuszczalnosci_na_stezenie
CALCULATOR · CHEMISTRY · A -> Z (all)
https://calculla.com/concentration_of_saturated_solution?ioMassOfSubstance=$symbolic(10)&ioConcentrationPercentageUnitId=percent&ioConcentrationMolarUnitId=mol_per_dm3&ioMassOfSubstanceUnitId=g&ioMolarMassUnitId=g_per_mol&ioVolumeUnitId=dm3&ioIntentSchemeId=want_ioConcentrationPercentage
chem.libretexts.org: saturated solutions and solubility
youtube.com: saturated, unsaturated, & supersaturated solutions - concentration vs solubility
sciencing.com: how to convert from moles per liter to percentage
rechneronline.de: alternative saturated solution concentration
portal.ddsb.ca: concentration and solubility
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Dynamic range limiter - Simulink - MathWorks Switzerland
The Limiter block performs dynamic range limiting independently across each input channel. Dynamic range limiting suppresses the volume of loud sounds that cross a given threshold. The block uses specified attack and release times to achieve a smooth applied gain curve.
1-D vector | matrix
The Limiter block outputs a signal with the same data type as the input signal. The size of the output depends on the size of the input:
Knee width (dB) — Transition area in the limiter characteristic
y=x-\frac{{\left(x-T+\frac{W}{2}\right)}^{2}}{\left(2×W\right)}
\left(2×|x-T|\right)\le W
View static characteristic — Open static characteristic plot of dynamic range limiter
The plot is updated automatically when parameters of the Limiter block change.
0 (default) | scalar in the range 0 to 4 inclusive
Attack time is the time the limiter gain takes to rise from 10% to 90% of its final value when the input goes above the threshold. The Attack time (s) parameter smooths the applied gain curve.
Release time is the time the limiter gain takes to drop from 90% to 10% of its final value when the input goes below the threshold. The Release time (s) parameter smooths the applied gain curve.
Auto –– Make-up gain is applied at the output of the Limiter block such that a steady-state 0 dB input has a 0 dB output.
Make-up gain compensates for gain lost during limiting. It is applied at the output of the Limiter block.
When you select this parameter, an additional input port SC is added to the block. The SC port enables dynamic range limiting of the input signal x using a separate sidechain signal.
Suppress Volume of Loud Sounds
Suppress the volume of loud sounds and visualize the applied dynamic range control gain.
The Limiter block processes a signal frame by frame and element by element.
{x}_{\text{dB}}\left[n\right]=20×{\mathrm{log}}_{10}|x\left[n\right]|
xdB[n] passes through the gain computer. The gain computer uses the static characteristic properties of the dynamic range limiter to brickwall gain that is above the threshold.
{x}_{\text{sc}}\left({x}_{\text{dB}}\right)=\left\{\begin{array}{cc}{x}_{\text{dB}}& {x}_{\text{dB}}<\left(T-\frac{W}{2}\right)\\ {x}_{\text{dB}}-\frac{{\left({x}_{\text{dB}}-T+\frac{W}{2}\right)}^{2}}{2W}& \begin{array}{c}\\ \\ \end{array}\left(T-\frac{W}{2}\right)\le {x}_{\text{dB}}\le \left(T+\frac{W}{2}\right)\\ T& {x}_{\text{dB}}>\left(T+\frac{W}{2}\right)\end{array}\text{ },
{x}_{\text{sc}}\left({x}_{\text{dB}}\right)=\left\{\begin{array}{cc}{x}_{\text{dB}}& {x}_{\text{dB}}<T\\ T& {x}_{\text{dB}}\ge T\end{array}
{g}_{\text{c}}\left[n\right]={x}_{\text{sc}}\left[n\right]-{x}_{\text{dB}}\left[n\right].
{g}_{\text{s}}\left[n\right]=\left\{\begin{array}{cc}{\alpha }_{\text{A}}{g}_{\text{s}}\left[n-1\right]+\left(1-{\alpha }_{\text{A}}\right){g}_{c}\left[n\right],& {g}_{\text{c}}\left[n\right]\le {g}_{\text{s}}\left[n-1\right]\\ {\alpha }_{\text{R}}{g}_{\text{s}}\left[n-1\right]+\left(1-{\alpha }_{\text{R}}\right){g}_{c}\left[n\right],& {g}_{\text{c}}\left[n\right]>{g}_{\text{s}}\left[n-1\right]\end{array}
{\alpha }_{\text{A}}=\mathrm{exp}\left(\frac{-\mathrm{log}\left(9\right)}{Fs×{T}_{\text{A}}}\right)\text{\hspace{0.17em}}.
{\alpha }_{\text{R}}=\mathrm{exp}\left(\frac{-\mathrm{log}\left(9\right)}{Fs×{T}_{\text{R}}}\right)\text{\hspace{0.17em}}.
TA is the attack time period, specified by the Attack time (s) parameter. TR is the release time period, specified by the Release time (s) parameter. Fs is the input sampling rate, specified by the Inherit sample rate from input or Input sample rate (Hz) parameter.
If the Make-up gain (dB) parameter is set to Auto, the make-up gain is calculated as the negative of the computed gain for a 0 dB input:
M=-{x}_{\text{sc}}\left({x}_{\text{dB}}=0\right)
Given a steady-state input of 0 dB, this configuration achieves a steady-state output of 0 dB. The make-up gain is determined by the Threshold (dB) and Knee width (dB) parameters. It does not depend on the input signal.
{g}_{\text{m}}\left[n\right]={g}_{\text{s}}\left[n\right]+M
{g}_{\text{lin}}\left[n\right]={10}^{\left(\frac{{g}_{\text{m}}\left[n\right]}{20}\right)}
y\left[n\right]=x\left[n\right]×{g}_{\text{lin}}\left[n\right].
Compressor | Expander | Noise Gate | limiter
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Jefimenko's equations - Wikipedia
In electromagnetism, Jefimenko's equations (named after Oleg D. Jefimenko) give the electric field and magnetic field due to a distribution of electric charges and electric current in space, that takes into account the propagation delay (retarded time) of the fields due to the finite speed of light and relativistic effects. Therefore they can be used for moving charges and currents. They are the particular solutions to Maxwell's equations for any arbitrary distribution of charges and currents.[1]
1.2 Origin from retarded potentials
2 Heaviside–Feynman formula
Electric and magnetic fieldsEdit
Position vectors r and r′ used in the calculation
Jefimenko's equations give the electric field E and magnetic field B produced by an arbitrary charge or current distribution, of charge density ρ and current density J:[2]
{\displaystyle \mathbf {E} (\mathbf {r} ,t)={\frac {1}{4\pi \varepsilon _{0}}}\int \left[{\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{3}}}\rho (\mathbf {r} ',t_{r})+{\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{2}}}{\frac {1}{c}}{\frac {\partial \rho (\mathbf {r} ',t_{r})}{\partial t}}-{\frac {1}{|\mathbf {r} -\mathbf {r} '|}}{\frac {1}{c^{2}}}{\frac {\partial \mathbf {J} (\mathbf {r} ',t_{r})}{\partial t}}\right]dV',}
{\displaystyle \mathbf {B} (\mathbf {r} ,t)=-{\frac {\mu _{0}}{4\pi }}\int \left[{\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{3}}}\times \mathbf {J} (\mathbf {r} ',t_{r})+{\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{2}}}\times {\frac {1}{c}}{\frac {\partial \mathbf {J} (\mathbf {r} ',t_{r})}{\partial t}}\right]dV',}
where r′ is a point in the charge distribution, r is a point in space, and
{\displaystyle t_{r}=t-{\frac {|\mathbf {r} -\mathbf {r} '|}{c}}}
is the retarded time. There are similar expressions for D and H.[3]
These equations are the time-dependent generalization of Coulomb's law and the Biot–Savart law to electrodynamics, which were originally true only for electrostatic and magnetostatic fields, and steady currents.
Origin from retarded potentialsEdit
Jefimenko's equations can be found[2] from the retarded potentials φ and A:
{\displaystyle {\begin{aligned}&\varphi (\mathbf {r} ,t)={\dfrac {1}{4\pi \varepsilon _{0}}}\int {\dfrac {\rho (\mathbf {r} ',t_{r})}{|\mathbf {r} -\mathbf {r} '|}}dV',\\&\mathbf {A} (\mathbf {r} ,t)={\dfrac {\mu _{0}}{4\pi }}\int {\dfrac {\mathbf {J} (\mathbf {r} ',t_{r})}{|\mathbf {r} -\mathbf {r} '|}}dV',\end{aligned}}}
which are the solutions to Maxwell's equations in the potential formulation, then substituting in the definitions of the electromagnetic potentials themselves:
{\displaystyle \mathbf {E} =-\nabla \varphi -{\dfrac {\partial \mathbf {A} }{\partial t}}\,,\quad \mathbf {B} =\nabla \times \mathbf {A} }
{\displaystyle c^{2}={\frac {1}{\varepsilon _{0}\mu _{0}}}}
replaces the potentials φ and A by the fields E and B.
Heaviside–Feynman formulaEdit
Explanation of the variables relevant for the Heaviside–Feynman formula.
The Heaviside–Feynman formula, also known as the Jefimenko–Feynman formula, can be seen as the point-like electric charge version of Jefimenko's equations. Actually, it can be (non trivially) deduced from them using Dirac functions, or using the Liénard-Wiechert potentials.[4] It is mostly known from The Feynman Lectures on Physics, where it was used to introduce and describe the origin of electromagnetic radiation.[5] The formula provides a natural generalization of the Coulomb's law for cases where the source charge is moving:
{\displaystyle \mathbf {E} ={\frac {-q}{4\pi \epsilon _{0}}}\left[{\frac {\mathbf {e} _{r'}}{r'^{2}}}+{\frac {r'}{c}}{\frac {d}{dt}}\left({\frac {\mathbf {e} _{r'}}{r'^{2}}}\right)+{\frac {1}{c^{2}}}{\frac {d^{2}}{dt^{2}}}\mathbf {e} _{r'}\right]}
{\displaystyle \mathbf {B} =-\mathbf {e} _{r'}\times {\frac {\mathbf {E} }{c}}}
{\displaystyle \mathbf {E} }
{\displaystyle \mathbf {B} }
are the electric and magnetic fields respectively,
{\displaystyle q}
is the electric charge,
{\displaystyle \epsilon _{0}}
is the vacuum permittivity and
{\displaystyle c}
is the speed of light. The vector
{\displaystyle \mathbf {e} _{r'}}
is a unit vector pointing from the observer to the charge and
{\displaystyle r'}
is the distance between observer and charge. Since the electromagnetic field propagates at the speed of light, both these quantities are evaluated at the retarded time
{\displaystyle t-r'/c}
Illustration of the retarded charge position for a particle moving in one spatial dimension: the observer sees the particle where it was, not where it is.
The first term in the formula for
{\displaystyle \mathbf {E} }
represents the Coulomb's law for the static electric field. The second term is the time derivative of the first Coulombic term multiplied by
{\displaystyle {\frac {r'}{c}}}
which is the propagation time of the electric field. Heuristically, this can be regarded as nature "attempting" to forecast what the present field would be by linear extrapolation to the present time.[5] The last term, proportional to the second derivative of the unit direction vector
{\displaystyle e_{r'}}
, is sensitive to charge motion perpendicular to the line of sight. It can be shown that the electric field generated by this term is proportional to
{\displaystyle a_{t}/r'}
{\displaystyle a_{t}}
is the transverse acceleration in the retarded time. As it decreases only as
{\displaystyle 1/r'}
with distance compared to the standard
{\displaystyle 1/r'^{2}}
Coulumbic behavior, this term is responsible for the long-range electromagnetic radiation caused by the accelerating charge.
The Heaviside–Feynman formula can be derived from Maxwell's equations using the technique of the retarded potential. It allows, for example, the derivation of the Larmor formula for overall radiation power of the accelerating charge.
There is a widespread interpretation of Maxwell's equations indicating that spatially varying electric and magnetic fields can cause each other to change in time, thus giving rise to a propagating electromagnetic wave[6] (electromagnetism). However, Jefimenko's equations show an alternative point of view.[7] Jefimenko says, "...neither Maxwell's equations nor their solutions indicate an existence of causal links between electric and magnetic fields. Therefore, we must conclude that an electromagnetic field is a dual entity always having an electric and a magnetic component simultaneously created by their common sources: time-variable electric charges and currents."[8]
As pointed out by McDonald,[9] Jefimenko's equations seem to appear first in 1962 in the second edition of Panofsky and Phillips's classic textbook.[10] David Griffiths, however, clarifies that "the earliest explicit statement of which I am aware was by Oleg Jefimenko, in 1966" and characterizes equations in Panofsky and Phillips's textbook as only "closely related expressions".[2] According to Andrew Zangwill, the equations analogous to Jefimenko's but in the Fourier frequency domain were first derived by George Adolphus Schott in his treatise Electromagnetic Radiation (University Press, Cambridge, 1912).[11]
Essential features of these equations are easily observed which is that the right hand sides involve "retarded" time which reflects the "causality" of the expressions. In other words, the left side of each equation is actually "caused" by the right side, unlike the normal differential expressions for Maxwell's equations where both sides take place simultaneously. In the typical expressions for Maxwell's equations there is no doubt that both sides are equal to each other, but as Jefimenko notes, "... since each of these equations connects quantities simultaneous in time, none of these equations can represent a causal relation."[12]
^ Oleg D. Jefimenko, Electricity and Magnetism: An Introduction to the Theory of Electric and Magnetic Fields, Appleton-Century-Crofts (New-York - 1966). 2nd ed.: Electret Scientific (Star City - 1989), ISBN 978-0-917406-08-9. See also: David J. Griffiths, Mark A. Heald, Time-dependent generalizations of the Biot–Savart and Coulomb laws, American Journal of Physics 59 (2) (1991), 111-117.
^ a b c Introduction to Electrodynamics (3rd Edition), D. J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3.
^ Oleg D. Jefimenko, Solutions of Maxwell's equations for electric and magnetic fields in arbitrary media, American Journal of Physics 60 (10) (1992), 899–902.
^ Feynman, R. P., R .B. Leighton, and M. Sands, 1965, The Feynman Lectures on Physics, Vol. II, 21.5, Addison-Wesley, Reading, Massachusetts
^ a b Feynman, R. P., R .B. Leighton, and M. Sands, 1965, The Feynman Lectures on Physics, Vol. I, Addison-Wesley, Reading, Massachusetts
^ Kinsler, P. (2011). "How to be causal: time, spacetime, and spectra". Eur. J. Phys. 32 (6): 1687. arXiv:1106.1792. Bibcode:2011EJPh...32.1687K. doi:10.1088/0143-0807/32/6/022. S2CID 56034806.
^ Oleg D. Jefimenko, Causality Electromagnetic Induction and Gravitation, 2nd ed.: Electret Scientific (Star City - 2000) Chapter 1, Sec. 1-4, page 16 ISBN 0-917406-23-0.
^ Kirk T. McDonald, The relation between expressions for time-dependent electromagnetic fields given by Jefimenko and by Panofsky and Phillips, American Journal of Physics 65 (11) (1997), 1074-1076.
^ Wolfgang K. H. Panofsky, Melba Phillips, Classical Electricity And Magnetism, Addison-Wesley (2nd. ed - 1962), Section 14.3. The electric field is written in a slightly different - but completely equivalent - form. Reprint: Dover Publications (2005), ISBN 978-0-486-43924-2.
^ Andrew Zangwill, Modern Electrodynamics, Cambridge University Press, 1st edition (2013), pp. 726—727, 765
^ Oleg D. Jefimenko, Causality Electromagnetic Induction and Gravitation, 2nd ed.: Electret Scientific (Star City - 2000) Chapter 1, Sec. 1-1, page 6 ISBN 0-917406-23-0.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Jefimenko%27s_equations&oldid=1089567374"
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Counting Surfaces Singular Along a Line in P3
2022 Counting Surfaces Singular Along a Line in
{\mathbb{P}}^{3}
Shachar Carmeli, Lev Radzivilovsky
We enumerate the surfaces of degree d in
{\mathbb{P}}^{3}
having a singular line of order k and passing through δ generic points (where δ is the dimension of moduli space of such surfaces).
Shachar Carmeli. Lev Radzivilovsky. "Counting Surfaces Singular Along a Line in
{\mathbb{P}}^{3}
." Michigan Math. J. Advance Publication 1 - 22, 2022. https://doi.org/10.1307/mmj/20205956
Received: 27 July 2020; Revised: 8 January 2021; Published: 2022
Shachar Carmeli, Lev Radzivilovsky "Counting Surfaces Singular Along a Line in
{\mathbb{P}}^{3}
," Michigan Mathematical Journal, Michigan Math. J. Advance Publication, 1-22, (2022)
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An obstacle problem with gradient term and asymptotically linear reaction | EMS Press
An obstacle problem with gradient term and asymptotically linear reaction
Sidi Mohamed Bouguima
We will consider the following obstacle problem
\int_{\Omega}\nabla u\nabla T_{k}(v-u)dx +\int_{\Omega }h(u)\left\vert \nabla u\right\vert ^{q}T_{k}(v-u)dx\geq \int_{\Omega }\left(g(x,u)+f\right) T_{k}(v-u)dx,
with the condition that
u\geq\psi
a.e in
\Omega .
Under suitable condition relating
h
q
, we show the existence of a solution for all
f \in L^1(\Omega)
The main feature is, assuming that
g(x,s)
is asymptotically linear as
|s|\to \pm\infty
and independently of the values of
\lim\limits_{s\to \pm\infty }\dfrac{g(x,s)}{s},
to obtain a solution for all
\lambda>0
f\in L^1(\Omega)
. In this sense we could say that the first order term break down any resonant effect.
Boumediene Abdellaoui, Sidi Mohamed Bouguima, Ireneo Peral, An obstacle problem with gradient term and asymptotically linear reaction. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 22 (2011), no. 1, pp. 29–50
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Things to hopefully avoid this year
2020 definitely sucked, so naturally we all have incredible expectations for 2021.
Most blog posts I've seen on goal-setting for 2021 are a list of normative statements for what to do within the following year. From running
X
kilometers to learning a new programming language, there's nothing wrong in highlighting your aspirations.
However, whenever I write a list like that on my blog, I find myself either being too ambitious or inadvertently value signalling. So this year, I'm going to try something different: writing a list of things to not do.
Let's call these anti-goals.
1. Have a never-ending list of books to read.
Bonus points if you buy more books before finishing the ones you're currently reading. The actual reflections and musings don't matter as much as the length of your Goodreads list and the amount of overlap with Twitter's current "thought leader".
2. Prioritize schoolwork over a well-rounded senior year.
It's your last semester so why not finish strong with a good GPA? It's a pandemic anyway, so this very well could be your only productive use of time (beyond launching a SubStack or starting a new company).
3. Earn to give instead of giving.
Instead of getting your hands dirty, use the time you save to earn more money! Donate a fraction of that instead (say
Y
% of it) and pat yourself on the back for being responsible during a pandemic.
4. Doom-scroll into Twitter oblivion every day.
How else will you simultaneously know every single thing that occurs in the world? As a citizen of the 21st century, it is your responsibility to do so, and have a tweetable opinion on everything as well.
This post felt a bit cheeky to write, but all of my anti-goals are things I've been guilty of over the past few years. Sometimes (especially during a pandemic), it's nice to not expect too much from yourself and to let momentum take its course. Your best offense can just be a good defense.
Cheers to a marginally better year!
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Nonexistence results of positive solutions for semilinear elliptic equations in Rn
July, 2000 Nonexistence results of positive solutions for semilinear elliptic equations in
{R}^{n}
We consider the global properties of nonnegative solutions of the semilinear elliptic equations in the entire space. By employing Pohozaev identity in the entire space and the results concerning the asymptotic behavior of nonnegative solutions, we establish some theorems of Liouville type.
Yūki NAITO. "Nonexistence results of positive solutions for semilinear elliptic equations in
{R}^{n}
." J. Math. Soc. Japan 52 (3) 637 - 644, July, 2000. https://doi.org/10.2969/jmsj/05230637
Keywords: entire solutions , nonexistence results , Pohozaev identity , semilinear elliptic equations
Yūki NAITO "Nonexistence results of positive solutions for semilinear elliptic equations in
{R}^{n}
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Boltu vs Balti | Toph
Boltu vs Balti
By Fake_Death · Limits 500ms, 512 MB
Boltu and Balti are two friends. Boltu is a great programmer whereas Balti is a newbie programmer. But Balti thinks that he is also a great programmer, but Boltu can't accept Balti as a great programmer. Now Boltu challenges Balti to solve a task to prove himself as a great programmer. Balti accepts that challenge.
The task is Boltu will give him any 2 numbers N and M , he (Balti) has to find the sum of all numbers in the range N and M (inclusive).
Balti knows that the equation of finding sum from 1 to n is
n(n+1)/2.
Balti is getting stuck to find it from N to M. Now he wants your help. Help him to win this challenge by solving this problem.
Each line of input contains two integers N and M (0 ≤ N,M ≤ 109). The numbers are separated by exactly one space. The end of file will indicate the end of input.
You have to find the sum of numbers between N and M and print it as follows:
Sum of X to Y is: Z;
Here X is the smaller number, Y is the larger number and Z is the sum of all numbers between N and M (inclusive).
Sum of 2 to 6 is -> 20;
In the first sample test case the sum of numbers from 2 to 6 is
2+3+4+5+6 = 20
2+3+4+5+6=20.
In the second sample test case the sum of numbers from 3 to 7 is
3+4+5+6+7 = 25
Rizon_sunnyEarliest, Jul '17
Swarna_1603033Fastest, 0.0s
suhasiniLightest, 0 B
The equation to finding the summation of 1 to n is given in the description: n*(n+1) / 2 Let's assu...
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<em>D</em>-bundles and integrable hierarchies | EMS Press
\mathcal D
-bundles---locally projective
\mathcal D
-modules---on algebraic curves, and apply them to the study of integrable hierarchies, specifically the multicomponent Kadomtsev-Petviashvili (KP) and spin Calogero-Moser (CM) hierarchies. We show that KP hierarchies have a geometric description as flows on moduli spaces of
\mathcal D
\mathcal D
-bundles is captured by the full Sato Grassmannian. The rational, trigonometric, and elliptic solutions of KP are therefore captured by
\mathcal D
-bundles on cubic curves
E
, that is, irreducible (smooth, nodal, or cuspidal) curves of arithmetic genus
1
. We develop a Fourier-Mukai transform describing
\mathcal D
-modules on cubic curves
E
in terms of (complexes of) sheaves on a twisted cotangent bundle
{E^{\natural}}
E
. We then apply this transform to classify
\mathcal D
-bundles on cubic curves, identifying their moduli spaces with phase spaces of general CM particle systems (realized through the geometry of spectral curves in
{E^{\natural}}
). Moreover, it is immediate from the geometric construction that the flows of the KP and CM hierarchies are thereby identified and that the poles of the KP solutions are identified with the positions of the CM particles. This provides a geometric explanation of a much-explored, puzzling phenomenon of the theory of integrable systems: the poles of meromorphic solutions to KP soliton equations move according to CM particle systems.
David Ben-Zvi, Thomas Nevins, <em>D</em>-bundles and integrable hierarchies. J. Eur. Math. Soc. 13 (2011), no. 6, pp. 1505–1567
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Square_number Knowpia
{\displaystyle {\sqrt {9}}=3,}
{\displaystyle \textstyle {\frac {4}{9}}=\left({\frac {2}{3}}\right)^{2}}
{\displaystyle \lfloor {\sqrt {m}}\rfloor }
{\displaystyle \lfloor x\rfloor }
The squares (sequence A000290 in the OEIS) smaller than 602 = 3600 are:
{\displaystyle n^{2}=\sum _{k=1}^{n}(2k-1).}
For example, 52 = 25 = 1 + 3 + 5 + 7 + 9.
The sum of the first n odd integers is n2. 1 + 3 + 5 + ... + (2n − 1) = n2. Animated 3D visualization on a tetrahedron.
2 × 52 − 42 + 2 = 2 × 25 − 16 + 2 = 50 − 16 + 2 = 36 = 62.
One number less than a square (m − 1) is always the product of
{\displaystyle {\sqrt {m}}-1}
{\displaystyle {\sqrt {m}}+1}
(for example, 8 × 6 equals 48, while 72 equals 49). Thus, 3 is the only prime number one less than a square.
if the last digit of a number is 0, its square ends in 0 (in fact, the last two digits must be 00);
if the last digit of a number is 1 or 9, its square ends in 1;
if the last digit of a number is 4 or 6, its square ends in 6; and
if the last digit of a number is 5, its square ends in 5 (in fact, the last two digits must be 25).
if a number is divisible both by 2 and by 3 (that is, divisible by 6), its square ends in 0;
if a number is divisible neither by 2 nor by 3, its square ends in 1;
if a number is divisible by 2, but not by 3, its square ends in 4; and
if a number is not divisible by 2, but by 3, its square ends in 9.
Squarity testing can be used as alternative way in factorization of large numbers. Instead of testing for divisibility, test for squarity: for given m and some number k, if k2 − m is the square of an integer n then k − n divides m. (This is an application of the factorization of a difference of two squares.) For example, 1002 − 9991 is the square of 3, so consequently 100 − 3 divides 9991. This test is deterministic for odd divisors in the range from k − n to k + n where k covers some range of natural numbers
{\displaystyle k\geq {\sqrt {m}}.}
The sum of the n first square numbers is
{\displaystyle \sum _{n=0}^{N}n^{2}=0^{2}+1^{2}+2^{2}+3^{2}+4^{2}+\cdots +N^{2}={\frac {N(N+1)(2N+1)}{6}}.}
The first values of these sums, the square pyramidal numbers, are: (sequence A000330 in the OEIS)
Proof without words for the sum of odd numbers theorem
If the number is of the form m5 where m represents the preceding digits, its square is n25 where n = m(m + 1) and represents digits before 25. For example, the square of 65 can be calculated by n = 6 × (6 + 1) = 42 which makes the square equal to 4225.
If the number is of the form m0 where m represents the preceding digits, its square is n00 where n = m2. For example, the square of 70 is 4900.
If the number ends in 5, its square will end in 5; similarly for ending in 25, 625, 0625, 90625, ... 8212890625, etc. If the number ends in 6, its square will end in 6, similarly for ending in 76, 376, 9376, 09376, ... 1787109376. For example, the square of 55376 is 3066501376, both ending in 376. (The numbers 5, 6, 25, 76, etc. are called automorphic numbers. They are sequence A003226 in the OEIS.[3])
Brahmagupta–Fibonacci identity – Expression of a product of sums of squares as a sum of squares
Cubic number – Number raised to the third power
Euler's four-square identity – Product of sums of four squares expressed as a sum of four squares
Fermat's theorem on sums of two squares – Condition under which an odd prime is a sum of two squares
Some identities involving several squares
Integer square root – Greatest integer less than or equal to square root
Methods of computing square roots – Algorithms for calculating square roots
Power of two – Two raised to an integer power
Pythagorean triple – Three sides of an integer right triangle
Quadratic residue – Integer that is a perfect square modulo some integer
Quadratic function – Polynomial function of degree two
Square triangular number – Integer that is both a perfect square and a triangular number
^ Some authors also call squares of rational numbers perfect squares.
^ Olenick, Richard P.; Apostol, Tom M.; Goodstein, David L. (2008-01-14). The Mechanical Universe: Introduction to Mechanics and Heat. Cambridge University Press. p. 18. ISBN 978-0-521-71592-8.
^ Sloane, N. J. A. (ed.). "Sequence A003226 (Automorphic numbers: n^2 ends with n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
Kiran Parulekar. Amazing Properties of Squares and Their Calculations. Kiran Anil Parulekar, 2012 https://books.google.com/books?id=njEtt7rfexEC&source=gbs_navlinks_s
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Pneumatic cylinder (34100 views - Mechanical Engineering)
Pneumatic cylinder(s) (sometimes known as air cylinders) are mechanical devices which use the power of compressed gas to produce a force in a reciprocating linear motion. Like hydraulic cylinders, something forces a piston to move in the desired direction. The piston is a disc or cylinder, and the piston rod transfers the force it develops to the object to be moved. Engineers sometimes prefer to use pneumatics because they are quieter, cleaner, and do not require large amounts of space for fluid storage. Because the operating fluid is a gas, leakage from a pneumatic cylinder will not drip out and contaminate the surroundings, making pneumatics more desirable where cleanliness is a requirement. For example, in the mechanical puppets of the Disney Tiki Room, pneumatics are used to prevent fluid from dripping onto people below the puppets.
3D CAD Models - pneumatic cylinder
Licensed under Creative Commons Attribution-Share Alike 3.0 (I, RainerB.).
Pneumatic cylinder(s) (sometimes known as air cylinders) are mechanical devices which use the power of compressed gas to produce a force in a reciprocating linear motion.[1]:85
Like hydraulic cylinders, something forces a piston to move in the desired direction. The piston is a disc or cylinder, and the piston rod transfers the force it develops to the object to be moved.[1] :85 Engineers sometimes prefer to use pneumatics because they are quieter, cleaner, and do not require large amounts of space for fluid storage.
Because the operating fluid is a gas, leakage from a pneumatic cylinder will not drip out and contaminate the surroundings, making pneumatics more desirable where cleanliness is a requirement. For example, in the mechanical puppets of the Disney Tiki Room, pneumatics are used to prevent fluid from dripping onto people below the puppets.
1.2 Compressibility of gasses
1.3 Fail safe mechanisms
2.1 Single-acting cylinders
2.2 Double-acting cylinders
2.3 Multi-stage, telescoping cylinder
2.4.1 Rodless cylinders
4 Pressure, radius, area and force relationships
4.1 Rod stresses
4.2 Instroke and outstroke
4.2.1 Outstroke
4.2.2 Instroke
Once actuated, compressed air enters into the tube at one end of the piston and, hence, imparts force on the piston. Consequently, the piston becomes displaced.
Compressibility of gasses
One major issue engineers come across working with pneumatic cylinders has to do with the compressibility of a gas. Many studies have been completed on how the precision of a pneumatic cylinder can be affected as the load acting on the cylinder tries to further compress the gas used. Under a vertical load, a case where the cylinder takes on the full load, the precision of the cylinder is affected the most. A study at the National Cheng Kung University in Taiwan, concluded that the accuracy is about ± 30 nm, which is still within a satisfactory range but shows that the compressibility of air has an effect on the system.[2]
Pneumatic systems are often found in settings where even rare and brief system failure is unacceptable. In such situations locks can sometimes serve as a safety mechanism in case of loss of air supply (or its pressure falling) and, thus remedy or abate any damage arising in such a situation. Leakage of air from the input or output reduces the pressure and so the desired output.
Although pneumatic cylinders will vary in appearance, size and function, they generally fall into one of the specific categories shown below. However, there are also numerous other types of pneumatic cylinder available, many of which are designed to fulfill specific and specialized functions.
Single-acting cylinders (SAC) use the pressure imparted by compressed air to create a driving force in one direction (usually out), and a spring to return to the "home" position. More often than not, this type of cylinder has limited extension due to the space the compressed spring takes up. Another downside to SACs is that part of the force produced by the cylinder is lost as it tries to push against the spring
Double-acting cylinders (DAC) use the force of air to move in both extend and retract strokes. They have two ports to allow air in, one for outstroke and one for instroke. Stroke length for this design is not limited, however, the piston rod is more vulnerable to buckling and bending. Additional calculations should be performed as well.[1] :89
Multi-stage, telescoping cylinder
Telescoping cylinders, also known as telescopic cylinders can be either single or double-acting. The telescoping cylinder incorporates a piston rod nested within a series of hollow stages of increasing diameter. Upon actuation, the piston rod and each succeeding stage "telescopes" out as a segmented piston. The main benefit of this design is the allowance for a notably longer stroke than would be achieved with a single-stage cylinder of the same collapsed (retracted) length. One cited drawback to telescoping cylinders is the increased potential for piston flexion due to the segmented piston design. Consequently, telescoping cylinders are primarily utilized in applications where the piston bears minimal side loading.[3]
Although SACs and DACs are the most common types of pneumatic cylinder, the following types are not particularly rare:[1]:89
Through rod air cylinders: piston rod extends through both sides of the cylinder, allowing for equal forces and speeds on either side.
Cushion end air cylinders: cylinders with regulated air exhaust to avoid impacts between the piston rod and the cylinder end cover.
Rotary air cylinders: actuators that use air to impart a rotary motion.
Rodless air cylinders: These have no piston rod. They are actuators that use a mechanical or magnetic coupling to impart force, typically to a table or other body that moves along the length of the cylinder body, but does not extend beyond it.
Tandem air cylinder: two cylinders assembled in series
Impact air cylinder: high velocity cylinders with specially designed end covers that withstand the impact of extending or retracting piston rods.
Some rodless types have a slot in the wall of the cylinder that is closed off for much of its length by two flexible metal sealing bands. The inner one prevents air from escaping, while the outer one protects the slot and inner band. The piston is actually a pair of them, part of a comparatively long assembly. They seal to the bore and inner band at both ends of the assembly. Between the individual pistons, however, are camming surfaces that "peel off" the bands as the whole sliding assembly moves toward the sealed volume, and "replace" them as the assembly moves away from the other end. Between the camming surfaces is part of the moving assembly that protrudes through the slot to move the load. Of course, this means that the region where the sealing bands are not in contact is at atmospheric pressure.[4]
Another type has cables (or a single cable) extending from both (or one) end[s] of the cylinder. The cables are jacketed in plastic (nylon, in those referred to), which provides a smooth surface that permits sealing the cables where they pass through the ends of the cylinder. Of course, a single cable has to be kept in tension.[5]
Still others have magnets inside the cylinder, part of the piston assembly, that pull along magnets outside the cylinder wall. The latter are carried by the actuator that moves the load. The cylinder wall is thin, to ensure that the inner and outer magnets are near each other. Multiple modern high-flux magnet groups transmit force without disengaging or excessive resilience.
Depending on the job specification, there are multiple forms of body constructions available:[1]:91
Tie rod cylinders: The most common cylinder constructions that can be used in many types of loads. Has been proven to be the safest form.
Flanged-type cylinders: Fixed flanges are added to the ends of cylinder, however, this form of construction is more common in hydraulic cylinder construction.
One-piece welded cylinders: Ends are welded or crimped to the tube, this form is inexpensive but makes the cylinder non-serviceable.
Threaded end cylinders: Ends are screwed onto the tube body. The reduction of material can weaken the tube and may introduce thread concentricity problems to the system.
Upon job specification, the material may be chosen. Material range from nickel-plated brass to aluminum, and even steel and stainless steel. Depending on the level of loads, humidity, temperature, and stroke lengths specified, the appropriate material may be selected.[6]
Depending on the location of the application and machinability, there exist different kinds of mounts for attaching pneumatic cylinders:[1]:95
Type of Mount Ends
Clevis Bracket-single or double
Torque or eye Trunnion
Clevis etc.
Air cylinders are available in a variety of sizes and can typically range from a small 2.5 mm (1⁄10 in) air cylinder, which might be used for picking up a small transistor or other electronic component, to 400 mm (16 in) diameter air cylinders which would impart enough force to lift a car. Some pneumatic cylinders reach 1,000 mm (39 in) in diameter, and are used in place of hydraulic cylinders for special circumstances where leaking hydraulic oil could impose an extreme hazard.
Pressure, radius, area and force relationships
Rod stresses
Due to the forces acting on the cylinder, the piston rod is the most stressed component and has to be designed to withstand high amounts of bending, tensile and compressive forces. Depending on how long the piston rod is, stresses can be calculated differently. If the rods length is less than 10 times the diameter, then it may be treated as a rigid body which has compressive or tensile forces acting on it. In which case the relationship is:
{\displaystyle F=A\sigma }
{\displaystyle F}
is the compressive or tensile force
{\displaystyle A}
is the cross-sectional area of the piston rod
{\displaystyle \sigma }
is the stress
However, if the length of the rod exceeds the 10 times the value of the diameter, then the rod needs to be treated as a column and buckling needs to be calculated as well.[1] :92
Instroke and outstroke
The relationship between the force, radius, and pressure can derived from simple distributed load equation:[7]
{\displaystyle F_{r}=PA_{e}}
{\displaystyle F_{r}}
is the resultant force
{\displaystyle P}
is the pressure or distributed load on the surface
{\displaystyle A_{e}}
is the effective cross sectional area the load is acting on
Using the distributed load equation provided the
{\displaystyle A_{e}}
can be replaced with area of the piston surface where the pressure is acting on.
{\displaystyle F_{r}=P(\pi r^{2})}
{\displaystyle F_{r}}
represents the resultant force
{\displaystyle r}epresents the radius of the piston
{\displaystyle \pi }
is pi, approximately equal to 3.14159.
On instroke, the same relationship between force exerted, pressure and effective cross sectional area applies as discussed above for outstroke. However, since the cross sectional area is less than the piston area the relationship between force, pressure and radius is different. The calculation isn't more complicated though, since the effective cross sectional area is merely that of the piston surface minus the cross sectional area of the piston rod.
For instroke, therefore, the relationship between force exerted, pressure, radius of the piston, and radius of the piston rod, is as follows:
{\displaystyle F_{r}=P(\pi r_{1}^{2}-\pi r_{2}^{2})=P\pi (r_{1}^{2}-r_{2}^{2})}
{\displaystyle F_{r}}
{\displaystyle r_{1}}
represents the radius of the piston
{\displaystyle r_{2}}
represents the radius of the piston rod
{\displaystyle \pi }
PneumaticsTelescopic cylinderSMC Corporation가스용기실린더 블록Control valve기계공학Hydraulic cylinderPneumatic motor크랭크실
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Statistical Analysis in SerDes Systems - MATLAB & Simulink - MathWorks Italia
Init Subsystem Workflow
SerDes System Using Init Subsystem
Access Init Code
Reshape Impulse Response and Instantiate Tx System object
Tx Impulse Response Processing
Reshape Impulse Response and Instantiate Rx System object
Rx Impulse Response Processing
Custom User Code Area
PAM4 Thresholds
Advance Init Options
External Init
Disable Default Impulse Response Processing
Metrics Used in Statistical Analysis
A SerDes system simulation involves a transmitter (Tx) and a receiver (Rx) connected by a passive analog channel. There are two distinct phases to a SerDes system simulation: statistical analysis and time-domain analysis. Statistical analysis (also known as analytical, linear time-invariant, or Init analysis) is based on impulse responses enabling fast analysis and adaptation of equalization algorithms. Time-domain analysis (also known as empirical, bit-by-bit or GetWave analysis) is a waveform-based implementation of equalization algorithms that can optionally include nonlinear effects.
The reference flow of statistical analysis differs from time-domain analysis. During a statistical analysis simulation, an impulse response is generated. The impulse response represents the combined response of the transmitter’s analog output, the channel, and the receiver’s analog front end. The impulse response of the channel is modified by the transmitter model's statistical functions. The modified impulse response from the transmitter output is then further modified by the receiver model's statistical functions. The simulation is then completed using the final modified impulse response which represents the behavior of both AMI models combined with the analog channel.
During a time-domain simulation, a digital stimulus waveform is passed to the transmitter model's time-domain function. This modified time-domain waveform is then convolved with the analog channel impulse response used in the statistical simulation. The output of this convolution is then passed to the receiver model's time-domain function. The modified output of the receiver becomes the simulation waveform at the receiver latch.
In SerDes Toolbox™, the Init subsystem within both the Tx and Rx blocks uses an Initialize Function Simulink® block. The Initialize Function block contains a MATLAB® function to handle the statistical analysis of an impulse response vector. The impulse response vector is generated by the Analog Channel block.
The MATLAB code within the Init subsystems mimics the architecture of Simulink time-domain simulation by initializing and setting up the library blocks from the SerDes Toolbox that implement equalization algorithms. Each subsystem then processes the impulse response vector through one or more System objects representing the corresponding blocks.
Additionally, an Init subsystem can adapt or optimize the equalization algorithms and then apply the modified algorithms to the impulse response. The output of an Init subsystem is an adapted impulse response. If the Init subsystem adapts the equalization algorithms, it can also output the modified equalization settings as AMI parameters. These modified equalization parameters can also be passed to the time-domain analysis as an optimal setting or to provide a starting point for faster time-domain adaptation.
In a Simulink model of a SerDes system, there are two Init subsystems, one on the transmitter side (Tx block) and one on the receiver side (Rx block). During statistical analysis, the impulse response of the analog channel is first equalized by the Init subsystem inside the Tx block based on the System object™ properties. The modified impulse response is then fed as an input to the Rx block. The Init system inside the Rx block further equalizes the impulse response and produces the final output.
The System objects corresponding to the Tx and Rx blocks modify the impulse response in the same order as they were received. If there are multiple self-adapting System objects in a Tx or Rx block, each System object finds the best setting for the impulse response and modifies it before sending it to the next System object.
The final equalized impulse response is used to derive the pulse response, statistical eye, and the waveforms.
To understand how an Init subsystem handles statistical analysis in a SerDes system, create a SerDes system using the SerDes Designer app. The SerDes system contains an FFE block on the Tx side and CTLE and DFECDR blocks on the Rx side. Use the default settings for each block.
Export the SerDes system to a Simulink model. In Simulink, double-click the Tx block to open the Init block. Then double-click the Init block to open the Block Parameters dialog box. Click the Show Init button to open the code pertaining to the Init function of the transmitter.
The Init function first reshapes the impulse response vector of the analog channel into a 2-D matrix. The first column in the 2-D matrix represents the analog channel impulse response (victim). The subsequent columns (if any are present) represent the crosstalk (aggressors).
%% Impulse response formatting
% Size ImpulseOut by setting it equal to ImpulseIn
ImpulseOut = ImpulseIn;
% Reshape ImpulseIn vector into a 2D matrix using RowSize and Aggressors called LocalImpulse
LocalImpulse = zeros(RowSize,Aggressors+1);
AggressorPosition = 1;
for RowPosition = 1:RowSize:RowSize*(Aggressors+1)
LocalImpulse(:,AggressorPosition) = ImpulseIn(RowPosition:RowSize-1+RowPosition)';
AggressorPosition = AggressorPosition+1;
Then the Init function initializes the System objects that represent the blocks on the Tx side and sets up the simulation and AMI parameters and the block properties. In this SerDes system, there is only one block on the Tx side, FFE.
%% Instantiate and setup system objects
% Create instance of serdes.FFE for FFE
FFEInit = serdes.FFE('WaveType', 'Impulse');
% Setup simulation parameters
FFEInit.SymbolTime = SymbolTime;
FFEInit.SampleInterval = SampleInterval;
% Setup FFE In and InOut AMI parameters
FFEInit.Mode = FFEParameter.Mode;
FFEInit.TapWeights = FFEParameter.TapWeights;
% Setup FFE block properties
FFEInit.Normalize = true;
The channel impulse response is then processed by the System object on the Tx side.
%% Impulse response processing via system objects
% Return impulse response for serdes.FFE instance
LocalImpulse = FFEInit(LocalImpulse);
The modified impulse response in 2-D matrix form is reshaped back into an impulse response vector and sent to the Rx side for further equalization.
%% Impulse response reformating
% Reshape LocalImpulse matrix into a vector using RowSize and Aggressors
ImpulseOut(1:RowSize*(Aggressors+1)) = LocalImpulse;
Similarly, if you look at the Rx Init code, you can see that the Rx Init function first reshapes the output of the Tx Init function into a 2-D matrix.
Then the Init function initializes the System objects that represent the blocks on the Rx side and sets up the simulation and AMI parameters and the block properties. In this case, there are two blocks on the Rx side, CTLE and DFECDR.
% Create instance of serdes.CTLE for CTLE
CTLEInit = serdes.CTLE('WaveType', 'Impulse');
CTLEInit.SymbolTime = SymbolTime;
CTLEInit.SampleInterval = SampleInterval;
% Setup CTLE In and InOut AMI parameters
CTLEInit.Mode = CTLEParameter.Mode;
CTLEInit.ConfigSelect = CTLEParameter.ConfigSelect;
% Setup CTLE block properties
CTLEInit.Specification = 'DC Gain and Peaking Gain';
CTLEInit.DCGain = [0 -1 -2 -3 -4 -5 -6 -7 -8];
CTLEInit.ACGain = 0;
CTLEInit.PeakingGain = [0 1 2 3 4 5 6 7 8];
CTLEInit.PeakingFrequency = 5000000000;
CTLEInit.GPZ = [0 -23771428571 -10492857142 -13092857142;-1 -17603571428 -7914982142 -13344642857;...
-2 -17935714285 -6845464285 -13596428571;-3 -15321428571 -5574642857 -13848214285;...
-8 -16714285714 -3227142857 -15107142857];
% Create instance of serdes.DFECDR for DFECDR
DFECDRInit = serdes.DFECDR('WaveType', 'Impulse');
DFECDRInit.SymbolTime = SymbolTime;
DFECDRInit.SampleInterval = SampleInterval;
DFECDRInit.Modulation = Modulation;
% Setup DFECDR In and InOut AMI parameters
DFECDRInit.ReferenceOffset = DFECDRParameter.ReferenceOffset;
DFECDRInit.PhaseOffset = DFECDRParameter.PhaseOffset;
DFECDRInit.Mode = DFECDRParameter.Mode;
DFECDRInit.TapWeights = DFECDRParameter.TapWeights;
% Setup DFECDR block properties
DFECDRInit.EqualizationGain = 9.6e-05;
DFECDRInit.EqualizationStep = 1e-06;
DFECDRInit.MinimumTap = -1;
DFECDRInit.MaximumTap = 1;
DFECDRInit.Count = 16;
DFECDRInit.ClockStep = 0.0078;
DFECDRInit.Sensitivity = 0;
The impulse response that was previously modified by the System objects on the Tx side is then further modified by the System objects on the Rx side.
% Return impulse response and any Out or InOut AMI parameters for serdes.CTLE instance
[LocalImpulse, CTLEConfigSelect] = CTLEInit(LocalImpulse);
% Return impulse response and any Out or InOut AMI parameters for serdes.DFECDR instance
[LocalImpulse, DFECDRTapWeights, DFECDRPhase, ~, ~] = DFECDRInit(LocalImpulse);
The final equalized impulse response in 2-D matrix form is reshaped back into an impulse response vector.
Each Init function also contains a section, Custom user code area, where you can customize your own code.
%% BEGIN: Custom user code area (retained when 'Refresh Init' button is pressed)
% END: Custom user code area (retained when 'Refresh Init' button is pressed)
For more information on how you can use the Custom user code area, see Customizing Datapath Building Blocks and Implement Custom CTLE in SerDes Toolbox PassThrough Block.
The code generation of Init function (Refresh Init) can support one or multiple System objects when using the custom PassThrough block. If multiple system objects are present, they must be in series. The first input port must have a waveform as the input. If any waveform output is present, it must be the first output port.
If you are using a SerDes Toolbox datapath library block, PAM4 thresholds in the Init function are maintained for you automatically. If you are using a custom configuration using a PassThrough, the code generation of the Init function finds the Data Store Write blocks that reference the PAM4 threshold signals (PAM4_UpperThreshold, PAM4_CenterThreshold, PAM4_LowerThreshold) and determines connectivities. The connectivities that are supported are:
Direct connection to System object
Connection to System object through bus selector
Connection to System object through Gain block
Direct connection to Constant block
If the Init code generation cannot find a supported topology, it applies the default PAM4 thresholds.
You can export the Init code to an external MATLAB function, customize it, and then use the customized Init function for rapid analysis. To export the Init code, select the External Init option in the block parameters dialog box of either the Tx or Rx Init block, then click the Refresh Init button. This copies the contents of each of the Init MATLAB function blocks to txInit.m and rxInit.m files and links these functions back to the Simulink model. It also creates a runExternalInit.m file that runs these external Init files in MATLAB.
Once you have customized the Init code and you want to reintegrate the Init function back into the Simulink model, you can disable the External Init option and click the Refresh Init button again. This copies the contents of the Init function into the default Init files and deletes the external Init files.
You can comment out the default impulse processing section of the Init code. This option comments out the code preforming the impulse response processing as shown in the Tx Impulse Response Processing and Rx Impulse Response Processing sections. You can then customize the impulse response processing required for your system design in the Custom User Code Area.
\text{Linearity}=\frac{\text{Minimum amplitude of the different eye levels}}{\text{Maximum amplitude of the different eye levels}}
\text{COM}=20{\mathrm{log}}_{10}\left(\frac{\text{Mean eye height}}{\text{Mean eye height - Inner eye height}}\right)
\text{VEC}=\frac{\text{Mean eye height}}{\text{Inner eye height}}
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Credit card functionality is under development. In September 2016, the Reserve Bank of India (RBI) launched the eponymously named Bharat QR, a common QR code jointly developed by all the four major card payment companies – National Payments Corporation of India that runs RuPay cards along with MasterCard, Visa and American Express. It will also have the capability of accepting payments on the unified payments interface (UPI) platform.[28][29]
The amount of data that can be stored in the QR code symbol depends on the data type (mode, or input character set), version (1, ..., 40, indicating the overall dimensions of the symbol, i.e. 4 × version number + 17 dots on each side), and error correction level. The maximum storage capacities occur for version 40 and error correction level L (low), denoted by 40-L:[11][70]
2,953 8 ISO/IEC 8859-1
{\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}
When discussing the Reed–Solomon code phase there is some risk for confusion, in that the QR ISO/IEC standard uses the term codeword for the elements of
{\displaystyle \mathbb {F} _{256}}
IQR Code is an alternative to existing QR codes developed by Denso Wave. IQR codes can be created in square or rectangular formations; this is intended for situations where a rectangular barcode would otherwise be more appropriate, such as cylindrical objects. IQR codes can fit the same amount of information in 30% less space. There are 61 versions of square IQR codes, and 15 versions of rectangular codes. For squares, the minimum size is 9 × 9 modules; rectangles have a minimum of 19 × 5 modules. IQR codes add error correction level S, which allows for 50% error correction.[79] IQR Codes have not yet been given an ISO/IEC specification, and only proprietary Denso Wave products can create or read IQR codes.[80]
The barcode is not subject to licensing and was submitted to ISO/IEC standardization as ISO/IEC 23634 expected to be approved at the beginning of 2021[91] and finalized in 2022.[90] The software is open-source and published under the LGPL v2.1 license.[92] The specification is freely available.[89]
The use of QR code technology is freely licensed as long as users follow the standards for QR Code documented with JIS or ISO/IEC. Non-standardized codes may require special licensing.[93]
The text QR Code itself is a registered trademark and wordmark of Denso Wave Incorporated.[95] In UK, the trademark is registered as E921775, the term QR Code, with a filing date of 3 September 1998.[96] The UK version of the trademark is based on the Kabushiki Kaisha Denso (DENSO CORPORATION) trademark, filed as Trademark 000921775, the term QR Code, on 3 September 1998 and registered on 16 December 1999 with the European Union OHIM (Office for Harmonization in the Internal Market).[97] The U.S. Trademark for the term QR Code is Trademark 2435991 and was filed on 29 September 1998 with an amended registration date of 13 March 2001, assigned to Denso Corporation.[98]
^ "ISO/IEC DIS 23634 Information technology — Automatic identification and data capture techniques — JAB Code polychrome bar code symbology specification". ISO/IEC. Retrieved 17 February 2021.
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MedianDeviation - Maple Help
Home : Support : Online Help : Statistics and Data Analysis : Statistics Package : Quantities : MedianDeviation
compute the median absolute deviation from the median
MedianDeviation(A, ds_options)
MedianDeviation(X, rv_options)
(optional) equation(s) of the form option=value where option is one of ignore, or weights; specify options for computing the median absolute deviation of a data set
(optional) equation of the form numeric=value; specifies options for computing the median absolute deviation of a random variable
The MedianDeviation function computes the median absolute deviation from the median of the specified random variable or data set.
ignore=truefalse -- This option controls how missing data is handled by the MedianDeviation command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the MedianDeviation command will return undefined. If ignore=true all missing items in A will be ignored. The default value is false.
1
numeric=truefalse -- By default, the median absolute deviation is computed symbolically. To compute the median absolute deviation numerically, specify the numeric or numeric = true option.
\mathrm{with}\left(\mathrm{Statistics}\right):
Compute the median absolute deviation from the median of the Normal distribution with mean 3 and standard deviation 1.
\mathrm{MedianDeviation}\left(\mathrm{Normal}\left(3,1\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{RootOf}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{erf}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{_Z}}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{2}}
\mathrm{MedianDeviation}\left(\mathrm{Normal}\left(3,1\right),\mathrm{numeric}\right)
\textcolor[rgb]{0,0,1}{0.674489750196106}
Generate a random sample of size 1000000 drawn from the above distribution and compute the sample median absolute deviation.
A≔\mathrm{Sample}\left(\mathrm{Normal}\left(3,1\right),{10}^{6}\right):
\mathrm{MedianDeviation}\left(A\right)
\textcolor[rgb]{0,0,1}{0.674833282683731}
Compute the standard error of the median absolute deviation for the normal distribution with parameters 5 and 2.
X≔\mathrm{RandomVariable}\left(\mathrm{Normal}\left(5,2\right)\right):
B≔\mathrm{Sample}\left(X,{10}^{6}\right):
\mathrm{MedianDeviation}\left(X\right)
\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{RootOf}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{erf}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{_Z}}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{2}}
M≔\mathrm{Median}\left(X\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{5}
\mathrm{Median}\left(\mathrm{abs}\left(X-M\right)\right)
\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{RootOf}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{erf}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{_Z}}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{2}}
\mathrm{MedianDeviation}\left(X,\mathrm{numeric}\right)
\textcolor[rgb]{0,0,1}{1.34897950039221}
\mathrm{MedianDeviation}\left(B\right)
\textcolor[rgb]{0,0,1}{1.34820016524082}
Compute the median absolute deviation of a weighted data set.
V≔〈\mathrm{seq}\left(i,i=57..77\right),\mathrm{undefined}〉:
W≔〈2,4,14,41,83,169,394,669,990,1223,1329,1230,1063,646,392,202,79,32,16,5,2,5〉:
\mathrm{MedianDeviation}\left(V,\mathrm{weights}=W\right)
\textcolor[rgb]{0,0,1}{2.}
\mathrm{MedianDeviation}\left(V,\mathrm{weights}=W,\mathrm{ignore}=\mathrm{true}\right)
\textcolor[rgb]{0,0,1}{2.}
M≔\mathrm{Matrix}\left([[3,1130,114694],[4,1527,127368],[3,907,88464],[2,878,96484],[4,995,128007]]\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1130}& \textcolor[rgb]{0,0,1}{114694}\\ \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{1527}& \textcolor[rgb]{0,0,1}{127368}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{907}& \textcolor[rgb]{0,0,1}{88464}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{878}& \textcolor[rgb]{0,0,1}{96484}\\ \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{995}& \textcolor[rgb]{0,0,1}{128007}\end{array}]
We compute the median absolute deviation of each of the columns.
\mathrm{MedianDeviation}\left(M\right)
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1.}& \textcolor[rgb]{0,0,1}{117.}& \textcolor[rgb]{0,0,1}{13313.}\end{array}]
The A parameter was introduced in Maple 16.
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Linear motion (8910 views - Mechanical Engineering)
Linear motion (also called rectilinear motion) is a one dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion with constant velocity or zero acceleration; non uniform linear motion with variable velocity or non-zero acceleration. The motion of a particle (a point-like object) along a line can be described by its position x {\displaystyle x} , which varies with t {\displaystyle t} (time). An example of linear motion is an athlete running 100m along a straight track. Linear motion is the most basic of all motion. According to Newton's first law of motion, objects that do not experience any net force will continue to move in a straight line with a constant velocity until they are subjected to a net force. Under everyday circumstances, external forces such as gravity and friction can cause an object to change the direction of its motion, so that its motion cannot be described as linear. One may compare linear motion to general motion. In general motion, a particle's position and velocity are described by vectors, which have a magnitude and direction. In linear motion, the directions of all the vectors describing the system are equal and constant which means the objects move along the same axis and do not change direction. The analysis of such systems may therefore be simplified by neglecting the direction components of the vectors involved and dealing only with the magnitude. Neglecting the rotation and other motions of the Earth, an example of linear motion is the ball thrown straight up and falling back straight down.
3D CAD Models - linear motion
For the class of linkages, see straight line mechanism.
{\displaystyle {\vec {F}}=m{\vec {a}}}
Linear motion (also called rectilinear motion[1]) is a one dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion with constant velocity or zero acceleration; non uniform linear motion with variable velocity or non-zero acceleration. The motion of a particle (a point-like object) along a line can be described by its position
{\displaystyle x}
, which varies with
{\displaystyle t}
(time). An example of linear motion is an athlete running 100m along a straight track.[2]
Linear motion is the most basic of all motion. According to Newton's first law of motion, objects that do not experience any net force will continue to move in a straight line with a constant velocity until they are subjected to a net force. Under everyday circumstances, external forces such as gravity and friction can cause an object to change the direction of its motion, so that its motion cannot be described as linear.[3]
One may compare linear motion to general motion. In general motion, a particle's position and velocity are described by vectors, which have a magnitude and direction. In linear motion, the directions of all the vectors describing the system are equal and constant which means the objects move along the same axis and do not change direction. The analysis of such systems may therefore be simplified by neglecting the direction components of the vectors involved and dealing only with the magnitude.[2]
Neglecting the rotation and other motions of the Earth, an example of linear motion is the ball thrown straight up and falling back straight down.
6 Equations of kinematics
7 Analogy between linear and rotational motion
Main article: Displacement (vector)
The motion in which all the particles of a body move through the same distance in the same time is called translatory motion. There are two types of translatory motions: rectilinear motion; curvilinear motion. Since linear motion is a motion in a single dimension, the distance traveled by an object in particular direction is the same as displacement.[4] The SI unit of displacement is the metre.[5][6] If
{\displaystyle \,x_{1}}
is the initial position of an object and
{\displaystyle \,x_{2}}
is the final position, then mathematically the displacement is given by:
{\displaystyle \Delta x=x_{2}-x_{1}}
The equivalent of displacement in rotational motion is the angular displacement
{\displaystyle \theta }
measured in radian. The displacement of an object cannot be greater than the distance because it is also a distance but the shortest one. Consider a person travelling to work daily. Overall displacement when he returns home is zero, since the person ends up back where he started, but the distance travelled is clearly not zero.
Main article: velocity
Velocity = displacement / time. Velocity is defined as the rate of change of displacement with respect to time.[7] The SI unit of velocity is
{\displaystyle ms^{-1}}
or metre per second.[6]
The average velocity is the ratio of total displacement
{\displaystyle \Delta x}
taken over time interval
{\displaystyle \Delta t}
. Mathematically, it is given by:[8][9]
{\displaystyle \mathbf {v_{av}} ={\frac {\Delta x}{\Delta t}}={\frac {x_{2}-x_{1}}{t_{2}-t_{1}}}}
{\displaystyle t_{1}}
is the time at which the object was at position
{\displaystyle x_{1}}
{\displaystyle t_{2}}
{\displaystyle x_{2}}
The instantaneous velocity can be found by differentiating the displacement with respect to time.
{\displaystyle \mathbf {v} =\lim _{\Delta t\to 0}{\Delta x \over \Delta t}}
{\displaystyle ={\frac {dx}{dt}}}
Speed is the absolute value of velocity i.e. speed is always positive. The unit of speed is metre per second.[10] If
{\displaystyle v}
is the speed then,
{\displaystyle v=\left|\mathbf {v} \right|=\left|{\frac {dx}{dt}}\right|}
The magnitude of the instantaneous velocity is the instantaneous speed.
Acceleration is defined as the rate of change of velocity with respect to time. Acceleration is the second derivative of displacement i.e. acceleration can be found by differentiating position with respect to time twice or differentiating velocity with respect to time once.[11] The SI unit of acceleration is
{\displaystyle ms^{-2}}
or metre per second squared.[6]
{\displaystyle \mathbf {a_{av}} }
is the average acceleration and
{\displaystyle \Delta \mathbf {v} =\mathbf {v_{2}} -\mathbf {v_{1}} }
is the average velocity over the time interval
{\displaystyle \Delta t}
then mathematically,
{\displaystyle \mathbf {a_{av}} ={\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {\mathbf {v_{2}} -\mathbf {v_{1}} }{t_{2}-t_{1}}}}
The instantaneous acceleration is the limit of the ratio
{\displaystyle \Delta \mathbf {v} }
{\displaystyle \Delta t}
{\displaystyle \Delta t}
approaches zero i.e.,
{\displaystyle \mathbf {a} =\lim _{\Delta t\to 0}{\Delta \mathbf {v} \over \Delta t}}
{\displaystyle ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}x}{dt^{2}}}}
Main article: jerk (physics)
The rate of change of acceleration, the third derivative of displacement is known as jerk.[12] The SI unit of jerk is
{\displaystyle ms^{-3}}
. In the UK jerk is also known as jolt.
Main article: jounce
The rate of change of jerk, the fourth derivative of displacement is known as jounce.[12] The SI unit of jounce is
{\displaystyle ms^{-4}}
which can be pronounced as metres per quartic second.
Main article: Equations of motion
In case of constant acceleration, the four physical quantities acceleration, velocity, time and displacement can be related by using the Equations of motion[13][14][15]
{\displaystyle \mathbf {V_{f}} =\mathbf {V_{i}} +\mathbf {a} \mathbf {t} \;\!}
{\displaystyle \mathbf {d} =\mathbf {V_{i}} \mathbf {t} +{\begin{matrix}{\frac {1}{2}}\end{matrix}}\mathbf {a} \mathbf {t} ^{2}}
{\displaystyle {\mathbf {V_{f}} }^{2}={\mathbf {V_{i}} }^{2}+2{\mathbf {a} }\mathbf {d} }
{\displaystyle \mathbf {d} ={\tfrac {1}{2}}\left(\mathbf {V_{f}} +\mathbf {V_{i}} \right)\mathbf {t} }
{\displaystyle \mathbf {V_{i}} }
is the initial velocity
{\displaystyle \mathbf {V_{f}} }
is the final velocity
{\displaystyle \mathbf {a} }
is the acceleration
{\displaystyle \mathbf {d} }
is the displacement
{\displaystyle \mathbf {t} }
These relationships can be demonstrated graphically. The gradient of a line on a displacement time graph represents the velocity. The gradient of the velocity time graph gives the acceleration while the area under the velocity time graph gives the displacement. The area under an acceleration time graph gives the change in velocity.
Analogy between linear and rotational motion
See also: List of equations in classical mechanics § Equations of motion (constant acceleration)
{\displaystyle \mathbf {s} }
{\displaystyle \mathbf {r} }
{\displaystyle \mathbf {a} _{\mathbf {t} }}
{\displaystyle \mathbf {a} _{\mathbf {c} }=v^{2}/r=\omega ^{2}r}
, is perpendicular to the motion. The component of the force parallel to the motion, or equivalently, perpendicular to the line connecting the point of application to the axis is
{\displaystyle \mathbf {F} _{\perp }}
{\displaystyle \mathbf {j} \ =1\ \mathbf {to} \ N}
particles and/or points of application.
Analogy between Linear Motion and Rotational motion[16]
{\displaystyle \mathbf {x} }
{\displaystyle \theta }
{\displaystyle \theta =\mathbf {s} /\mathbf {r} }
{\displaystyle \mathbf {v} }
{\displaystyle \omega }
{\displaystyle \omega =\mathbf {v} /\mathbf {r} }
{\displaystyle \mathbf {a} }
{\displaystyle \alpha }
{\displaystyle \alpha =\mathbf {a_{\mathbf {t} }} /\mathbf {r} }
{\displaystyle \mathbf {m} }
{\displaystyle \mathbf {I} }
{\displaystyle \mathbf {I} =\sum \mathbf {m_{j}} \mathbf {r_{j}} ^{2}}
{\displaystyle \mathbf {F} =\mathbf {m} \mathbf {a} }
{\displaystyle \tau =\mathbf {I} \alpha }
{\displaystyle \tau =\sum \mathbf {r_{j}} \mathbf {F} _{\perp }\mathbf {_{j}} }
{\displaystyle \mathbf {p} =\mathbf {m} \mathbf {v} }
{\displaystyle \mathbf {L} =\mathbf {I} \omega }
{\displaystyle \mathbf {L} =\sum \mathbf {r_{j}} \mathbf {p} \mathbf {_{j}} }
{\displaystyle {\frac {1}{2}}\mathbf {m} \mathbf {v} ^{2}}
{\displaystyle {\frac {1}{2}}\mathbf {I} \omega ^{2}}
{\displaystyle {\frac {1}{2}}\sum \mathbf {m_{j}} \mathbf {v} ^{2}={\frac {1}{2}}\sum \mathbf {m_{j}} \mathbf {r_{j}} ^{2}\omega ^{2}}
The following table shows the analogy in derived SI units:
Motion graphs and derivatives
Linear actuatorControl knob무한궤도기계공학Sliding (motion)운동 에너지자전4절 링크Tribology
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Boundary Layer Correctors for the Solution of Laplace Equation in a Domain with Oscillating Boundary | EMS Press
Boundary Layer Correctors for the Solution of Laplace Equation in a Domain with Oscillating Boundary
O. Bodart
Université Blaise Pascal, Aubière Cedex, France
We study the asymptotic behaviour of the solution of Laplace equation in a domain with very rapidly oscillating boundary. The motivation comes from the study of a longitudinal flow in an infinite horizontal domain bounded at the bottom by a plane wall and at the top by a rugose wall. The rugose wall is a plane covered with periodic asperities which size depends on a small parameter
\epsilon > 0
. The assumption of sharp asperities is made, that is the height of the asperities does not vanish as
\epsilon \to 0
. We prove that, up to an exponentially decreasing error, the solution of Laplace equation can be approximated, outside a layer of width 2
\epsilon
, by a non-oscillating explicit function.
Y. Amirat, O. Bodart, Boundary Layer Correctors for the Solution of Laplace Equation in a Domain with Oscillating Boundary. Z. Anal. Anwend. 20 (2001), no. 4, pp. 929–940
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Simplify or convert Galois field element formatting - MATLAB gftuple - MathWorks France
gftuple
Simplify or convert Galois field element formatting
tp = gftuple(a,m)
tp = gftuple(a,prim_poly)
tp = gftuple(a,m,p)
tp = gftuple(a,prim_poly,p)
tp = gftuple(a,prim_poly,p,prim_ck)
[tp,expform] = gftuple(...)
This function performs computations in GF(pm), where p is prime. To perform equivalent computations in GF(2m), apply the .^ operator and the log function to Galois arrays. For more information, see Example: Exponentiation and Example: Elementwise Logarithm.
gftuple serves to simplify the polynomial or exponential format of Galois field elements, or to convert from one format to another. For an explanation of the formats that gftuple uses, see Representing Elements of Galois Fields.
In this discussion, the format of an element of GF(pm) is called “simplest” if all exponents of the primitive element are
Between 0 and m-1 for the polynomial format
Either -Inf, or between 0 and pm-2, for the exponential format
For all syntaxes, a is a matrix, each row of which represents an element of a Galois field. The format of a determines how MATLAB interprets it:
If a is a column of integers, MATLAB interprets each row as an exponential format of an element. Negative integers are equivalent to -Inf in that they all represent the zero element of the field.
If a has more than one column, MATLAB interprets each row as a polynomial format of an element. (Each entry of a must be an integer between 0 and p-1.)
The exponential or polynomial formats mentioned above are all relative to a primitive element specified by the second input argument. The second argument is described below.
tp = gftuple(a,m) returns the simplest polynomial format of the elements that a represents, where the kth row of tp corresponds to the kth row of a. The formats are relative to a root of the default primitive polynomial for GF(2^m), where m is a positive integer.
tp = gftuple(a,prim_poly) is the same as the syntax above, except that prim_poly is a polynomial character vector or a row vector that lists the coefficients of a degree m primitive polynomial for GF(2^m) in order of ascending exponents.
tp = gftuple(a,m,p) is the same as tp = gftuple(a,m) except that 2 is replaced by a prime number p.
tp = gftuple(a,prim_poly,p) is the same as tp = gftuple(a,prim_poly) except that 2 is replaced by a prime number p.
tp = gftuple(a,prim_poly,p,prim_ck) is the same as tp = gftuple(a,prim_poly,p) except that gftuple checks whether prim_poly represents a polynomial that is indeed primitive. If not, then gftuple generates an error and tp is not returned. The input argument prim_ck can be any number or character vector; only its existence matters.
[tp,expform] = gftuple(...) returns the additional matrix expform. The kth row of expform is the simplest exponential format of the element that the kth row of a represents. All other features are as described in earlier parts of this “Description” section, depending on the input arguments.
List of All Elements of a Galois Field (end of section)
As another example, the gftuple command below generates a list of elements of GF(p^m), arranged relative to a root of the default primitive polynomial. Some functions in this toolbox use such a list as an input argument.
Finally, the two commands below illustrate the influence of the shape of the input matrix. In the first command, a column vector is treated as a sequence of elements expressed in exponential format. In the second command, a row vector is treated as a single element expressed in polynomial format.
tp1 = gftuple([0; 1],3,3)
tp2 = gftuple([0, 0, 0, 1],3,3)
The outputs reflect that, according to the default primitive polynomial for GF(33), the relations below are true.
\begin{array}{l}{\alpha }^{0}=1+0\alpha +0{\alpha }^{2}\\ {\alpha }^{1}=0+1\alpha +0{\alpha }^{2}\\ 0+0\alpha +0{\alpha }^{2}+{\alpha }^{3}=2+\alpha +0{\alpha }^{2}\end{array}
gftuple uses recursive callbacks to determine the exponential format.
gfadd | gfmul | gfconv | gfdiv | gfdeconv | gfprimdf
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사용자:Hwangjy9/작업장3 - 위키백과, 우리 모두의 백과사전
사용자:Hwangjy9/작업장3
< 사용자:Hwangjy9
이 문서는 다른 언어판 위키백과의 문서(en:divergent series)를 번역 중이며, 한국어로 좀 더 다듬어져야 합니다.
수학에서, 발산급수는 수렴하지 않는 무한급수를 말한다.
급수가 수렴한다면, 급수의 각 항들은 0으로 접근한다. 그러므로 각 항이 0으로 접근하지 않는 모든 급수는 발산한다. 하지만, 수렴하는 것에 대한 조건이 더 강력한데, 각 항이 0으로 수렴하는 모든 급수가 수렴하는 것은 아니다. 가장 간단한 반례는 조화급수이다.
{\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots =\sum _{n=1}^{\infty }{\frac {1}{n}}.}
조화급수가 발산한다는 것은 중세 수학자 Nicole Oresme에 의해 증명되었다.
In specialized mathematical contexts, values can be usefully assigned to certain series whose sequence of partial sums diverges. A summability method or summation method is a partial function from the set of sequences of partial sums of series to values. For example, Cesàro summation assigns Grandi's divergent series
{\displaystyle 1-1+1-1+\cdots }
1 발산급수를 더하는 방법에 대한 정리
3 공리적 방법
4 Nörlund means
5 아벨리안 평균
5.1 아벨 합
발산급수를 더하는 방법에 대한 정리편집
A summability method M is regular if it agrees with the actual limit on all convergent series. Such a result is called an abelian theorem for M, from the prototypical Abel's theorem. More interesting and in general more subtle are partial converse results, called tauberian theorems, from a prototype proved by Alfred Tauber. Here partial converse means that if M sums the series Σ, and some side-condition holds, then Σ was convergent in the first place; without any side condition such a result would say that M only summed convergent series (making it useless as a summation method for divergent series).
The operator giving the sum of a convergent series is linear, and it follows from the Hahn–Banach theorem that it may be extended to a summation method summing any series with bounded partial sums. This fact is not very useful in practice since there are many such extensions, inconsistent with each other, and also since proving such operators exist requires invoking the axiom of choice or its equivalents, such as Zorn's lemma. They are therefore nonconstructive.
Summation of divergent series is also related to extrapolation methods and sequence transformations as numerical techniques. Examples for such techniques are Padé approximants, Levin-type sequence transformations, and order-dependent mappings related to renormalization techniques for large-order perturbation theory in quantum mechanics.
Properties of summation methods편집
Summation methods usually concentrate on the sequence of partial sums of the series. While this sequence does not converge, we may often find that when we take an average of larger and larger initial terms of the sequence, the average converges, and we can use this average instead of a limit to evaluate the sum of the series. So in evaluating a = a0 + a1 + a2 + ..., we work with the sequence s, where s0 = a0 and sn+1 = sn + an+1. In the convergent case, the sequence s approaches the limit a. A summation method can be seen as a function from a set of sequences of partial sums to values. If A is any summation method assigning values to a set of sequences, we may mechanically translate this to a series-summation method AΣ that assigns the same values to the corresponding series. There are certain properties it is desirable for these methods to possess if they are to arrive at values corresponding to limits and sums, respectively.
Linearity. A is linear if it is a linear functional on the sequences where it is defined, so that A(k r + s) = k A(r) + A(s) for sequences r, s and a real or complex scalar k. Since the terms an = sn+1 − sn of the series a are linear functionals on the sequence s and vice versa, this is equivalent to AΣ being a linear functional on the terms of the series.
Stability. If s is a sequence starting from s0 and s′ is the sequence obtained by omitting the first value and subtracting it from the rest, so that s′n = sn+1 − s0, then A(s) is defined if and only if A(s′) is defined, and A(s) = s0 + A(s′). Equivalently, whenever a′n = an+1 for all n, then AΣ(a) = a0 + AΣ(a′).
The third condition is less important, and some significant methods, such as Borel summation, do not possess it.
Finite Re-indexability. If s and s′ are two sequences such that there exists a bijection
{\displaystyle f:\mathbb {N} \rightarrow \mathbb {N} }
such that si = s′f(i) for all i, and if there exists some
{\displaystyle N\in \mathbb {N} }
such that si = s′i for all i > N, then A(s) = A(s′). (In other words, s′ is the same sequence as s, with only finitely many terms re-indexed.) Note that this is a weaker condition than Stability, because any summation method that exhibits Stability also exhibits Finite Re-indexability, but the converse is not true.
A desirable property for two distinct summation methods A and B to share is consistency: A and B are consistent if for every sequence s to which both assign a value, A(s) = B(s). If two methods are consistent, and one sums more series than the other, the one summing more series is stronger.
공리적 방법편집
Taking regularity, linearity and stability as axioms, it is possible to sum many divergent series by elementary algebraic manipulations. For instance, whenever r ≠ 1, the geometric series
{\displaystyle {\begin{aligned}G(r,c)&=\sum _{k=0}^{\infty }cr^{k}&&\\&=c+\sum _{k=0}^{\infty }cr^{k+1}&&{\mbox{ (stability) }}\\&=c+r\sum _{k=0}^{\infty }cr^{k}&&{\mbox{ (linearity) }}\\&=c+r\,G(r,c),&&{\mbox{ whence }}\\G(r,c)&={\frac {c}{1-r}},&&\\\end{aligned}}}
can be evaluated regardless of convergence. More rigorously, any summation method that possesses these properties and which assigns a finite value to the geometric series must assign this value. However, when r is a real number larger than 1, the partial sums increase without bound, and averaging methods assign a limit of ∞.
Nörlund means편집
{\displaystyle {\frac {p_{n}}{p_{0}+p_{1}+\cdots +p_{n}}}\rightarrow 0.}
{\displaystyle t_{m}={\frac {p_{m}s_{0}+p_{m-1}s_{1}+\cdots +p_{0}s_{m}}{p_{0}+p_{1}+\cdots +p_{m}}}}
then the limit of tn as n goes to infinity is an average called the Nörlund mean Np(s).
The Nörlund mean is regular, linear, and stable. Moreover, any two Nörlund means are consistent. The most significant of the Nörlund means are the Cesàro sums. Here, if we define the sequence pk by
{\displaystyle p_{n}^{k}={n+k-1 \choose k-1}}
then the Cesàro sum Ck is defined by Ck(s) = N(pk)(s). Cesàro sums are Nörlund means if k ≥ 0, and hence are regular, linear, stable, and consistent. C0 is ordinary summation, and C1 is ordinary Cesàro summation. Cesàro sums have the property that if h > k, then Ch is stronger than Ck.
아벨리안 평균편집
Suppose λ = {λ0, λ1, λ2, ...} is a strictly increasing sequence tending towards infinity, and that λ0 ≥ 0. Suppose
{\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}\exp(-\lambda _{n}x)}
converges for all real numbers x>0. Then the Abelian mean Aλ is defined as
{\displaystyle A_{\lambda }(s)=\lim _{x\rightarrow 0^{+}}f(x).}
A series of this type is known as a generalized 디리클레 급수; in applications to physics, this is known as the method of heat-kernel regularization.
Abelian means are regular, linear, and stable, but not always consistent between different choices of λ. However, some special cases are very important summation methods.
아벨 합편집
아벨 정리 문서를 참고하십시오.
{\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}\exp(-nx)=\sum _{n=0}^{\infty }a_{n}z^{n},}
where z = exp(−x). Then the limit of ƒ(x) as x approaches 0 through positive reals is the limit of the power series for ƒ(z) as z approaches 1 from below through positive reals, and the Abel sum A(s) is defined as
{\displaystyle A(s)=\lim _{z\rightarrow 1^{-}}\sum _{n=0}^{\infty }a_{n}z^{n}}
Lindelöf summation편집
If λn = n ln(n), then (indexing from one) we have
{\displaystyle f(x)=a_{1}+a_{2}2^{-2x}+a_{3}3^{-3x}+\cdots .}
Then L(s), the Lindelöf sum (Volkov 2001), is the limit of ƒ(x) as x goes to zero. The Lindelöf sum is a powerful method when applied to power series among other applications, summing power series in the Mittag-Leffler star.
오일러 합
Brezinski, C.; Zaglia, M. Redivo (1991), 《Extrapolation Methods. Theory and Practice》, North-Holland .
Volkov, I.I. (2001). “Lindelöf summation method”. 《Encyclopedia of Mathematics》 (영어). Springer-Verlag. ISBN 978-1-55608-010-4. .
Zakharov, A.A. (2001). “Abel summation method”. 《Encyclopedia of Mathematics》 (영어). Springer-Verlag. ISBN 978-1-55608-010-4. .
틀:Series (mathematics)
원본 주소 "https://ko.wikipedia.org/w/index.php?title=사용자:Hwangjy9/작업장3&oldid=11098891"
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Super Duality and Homology of Unitarizable Modules of Lie Algebras | EMS Press
National Changhua University of Eduation, Taiwan
\mathfrak u
-homology formulas for unitarizable modules at negative levels over classical Lie algebras of infinite rank of types
\mathfrak gl (n)
\mathfrak sp (2n)
\mathfrak so (2n)
are obtained. As a consequence, we recover the Enright's formulas for three Hermitian symmetric pairs of classical types
(SU(p,q),SU(p)\times SU(q))
(Sp(2n),U(n))
(SO^\ast(2n),U(n))
Po-Yi Huang, Ngau Lam, Tze-Ming To, Super Duality and Homology of Unitarizable Modules of Lie Algebras. Publ. Res. Inst. Math. Sci. 48 (2012), no. 1, pp. 45–63
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Data Points Given in Two Vectors - Maple Help
Home : Support : Online Help : Tasks : Plots : Data Points Given in Two Vectors
Plot a Set of Data Points Given by Two Vectors
Plot a set of data points defined by two vectors.
Enter the first vector of data points.
\mathrm{V1}≔〈\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{1.1}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{,}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{1.9}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{,}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{2.2}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{,}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{2.4}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{,}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{2.5}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{,}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{3.2}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{,}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{4.0}〉
\textcolor[rgb]{0,0,1}{\mathrm{V1}}\textcolor[rgb]{0,0,1}{:=}\left[\begin{array}{c}\textcolor[rgb]{0,0,1}{1.1}\\ \textcolor[rgb]{0,0,1}{1.9}\\ \textcolor[rgb]{0,0,1}{2.2}\\ \textcolor[rgb]{0,0,1}{2.4}\\ \textcolor[rgb]{0,0,1}{2.5}\\ \textcolor[rgb]{0,0,1}{3.2}\\ \textcolor[rgb]{0,0,1}{4.0}\end{array}\right]
Enter the second vector of data points.
\mathrm{V2}≔〈\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{0}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{,}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{0.1}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{,}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{0.2}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{,}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{-}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{0.3}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{,}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{-}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{0.2}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{,}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{0.5}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{,}\textcolor[rgb]{0.372549019607843,0,0.850980392156863}{0.7}〉
\textcolor[rgb]{0,0,1}{\mathrm{V2}}\textcolor[rgb]{0,0,1}{:=}\left[\begin{array}{c}\textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0.1}\\ \textcolor[rgb]{0,0,1}{0.2}\\ \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{0.3}\\ \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{0.2}\\ \textcolor[rgb]{0,0,1}{0.5}\\ \textcolor[rgb]{0,0,1}{0.7}\end{array}\right]
\mathrm{plot}([\mathrm{seq}(\left[\mathrm{V1}{\left[i\right],\mathrm{V2}\left[i\right]}_{}\right],i=1..\mathrm{LinearAlgebra}\left[\mathrm{Dimension}\right]\left(\mathrm{V1}\right))])
<>, LinearAlgebra[Dimension], plot, seq
plot3d, type/procedure, Vector
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Find and compare two standard scores and scores in two normally distributed examinations | ePractice - HKDSE 試題導向練習平台
Question Sample Titled 'Find and compare two standard scores and scores in two normally distributed examinations'
The table below shows the means and the standard deviations of the scores of a large group of students in a Mathematics examination and a Biology examination:
{58}
{14}
{68}
{15}
The standard score of Ethan in the Mathematics examination is
-{1.5}
Find the score of Ethan in the Mathematics examination.
Assume that the scores in each of the above examinations are normally distributed. Ethan gets
{56}
marks in the Biology examination. He claims that relative to other students, he performs better in the Biology examination than in the Mathematics examination. Is the claim correct? Explain your answer.
{x}
marks be the score of Ethan in the Mathematics examination.
\dfrac{{{x}-{58}}}{{14}}
=-{1.5}
{x}
={58}+{\left(-{1.5}\right)}{\left({14}\right)}
{x}
={37}
Thus, the score of Ethan in the Mathematics examination is
{37}
marks.
The standard score of Ethan in the Biology examination
=\dfrac{{{56}-{68}}}{{15}}
=-{0.8}
\lt-{1.5}
Relative to other students, Ethan performs better in the Biology examination than in the Mathematics examination.
Thus, the claim is correct.
Caution: Comparing the two absolute difference of Ethan's marks to the means should be awarded zero marks.
Note that, relative to other students, if Ethan performs equally in both examination, the standard score of both examination should be the same.
{y}
be his score in Biology examination so that Ethan performs equally in both examination.
\dfrac{{{y}-{68}}}{{15}}
=-{1.5}
{y}
={45.5}
He gets
{56}
marks in Biology examination, which is higher than
{45.5}
Therefore, relative to other students, Ethan performs better in the Biology examination than in the Mathematics examination.
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§ Geodesic equation, Extrinsic
The geodesic on a sphere must be a great circle. If it's not, so say we pick a circle at some fixed azimuth, then all the velocities point towards the center at this azimuth, not at the center of the sphere! But towards the center of the sphere is the real normal plane. So we get a deviation from the normal.
§ How do we know if a path is straight?
Velocity remains constant on a straight line.
So it has zero acceleration.
If we think of a curved spiral climbing a hill (or a spiral staircase), the acceleration vector will point upward (to allow us to climb the hill) and will the curved inward into the spiral (to allow us to turn as we spiral).
On the other hand, if we think of walking straight along an undulating plane, the acceleration with be positive/negative depending on whether the terrian goes upward or downward, but we won't have any left/right motion in the plane .
If the acceleration is always along the normal vectors, then we have a geodesic.
§ Geodesic curve
Curve with zero tangential acceleration when we walk along the curve with constant speed.
(u, v)
plane, and map it to
R(u, v) \equiv (R_x, R_y, R_z)
. Denote the curve as
c: I \to \mathbb R^3
c
always lies on
R
. Said differently, we have
c: I \to UV
, which we then map to
\mathbb R^3
R
R(u, v) = (\cos(u), \sin(u)\cos(v), \sin(u)\sin(v))
c(\lambda) = (\lambda, \lambda)
. Which is to say,
c(\lambda) = (\cos(\lambda), \sin(\lambda)\cos(\lambda), \sin(\lambda)\sin(\lambda))
e_u \equiv \partial_u R, e_v \equiv \partial_v R \in \mathbb R^3
are the basis of the tangent plane at
R_{u, v}
\partial_\lambda c
gives us the tangent vector along
c
on the surface.
\begin{aligned} &\frac{dc}{d \lambda} = \frac{du}{d\lambda}\frac{dR}{du} + \frac{dv}{d\lambda}\frac{dR}{dv} &\frac{d}{d\lambda}(\frac{dc}{d \lambda})\\ &=\frac{d}{d\lambda}(\frac{du}{d\lambda}\frac{dR}{du} + \frac{dv}{d\lambda}\frac{dR}{dv}) \\ &=\frac{d}{d\lambda}(\frac{du}{d\lambda}\frac{dR}{du}) + \frac{d}{d\lambda}(\frac{dv}{d\lambda}\frac{dR}{dv}) \\ &= \frac{d^2 u}{d\lambda^2}\frac{dR}{du} + (\frac{du}{d\lambda} \frac{d}{d\lambda} \frac{dR}{du}) \frac{d^2 v}{d\lambda^2}\frac{dR}{dv} + (\frac{dv}{d\lambda} \frac{d}{d\lambda} \frac{dR}{dv}) \end{aligned}
How to calculate
\frac{d}{d\lambda} \frac{dR}{ddu}
? Use chain rule, again!
\frac{d}{d\lambda} = \frac{du}{d \lambda}\frac{\partial}{\partial u} + \frac{dv}{d \lambda}\frac{\partial}{\partial v}
§ Geodesic curve with notational abuse
R(u, v)
the surface, and by
R(\lambda)
the equation of the curve. So for example,
R(u, v) = (\cos(u), \sin(u)\cos(v), \sin(u)\sin(v))
R(\lambda) = R(\lambda, \lambda) = (\cos(\lambda), \sin(\lambda)\cos(\lambda), \sin(\lambda)\sin(\lambda))
EigenChris videos
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Advances in Research, Development, and Testing of Single Cells at Forschungszentrum Jülich | J. Electrochem. En. Conv. Stor | ASME Digital Collection
Special Section On The 2Nd European Fuel Cell Technology And Applications Conference
V. A. C. Haanappel,
V. A. C. Haanappel
, D-52425 Jülich, Germany
e-mail: v.haanappel@fz-juelich.de
, Hegifeldstrasse 30, CH-8404 Winterthur, Switzerland
J. Mertens,
J. M. Serra,
J. M. Serra
, Avenida Los Naranjos S/N, E-46022 Valencia, Spain
F. Tietz,
S. Uhlenbruck,
I. C. Vinke,
I. C. Vinke
L. G. J. de Haart
J. Fuel Cell Sci. Technol. May 2009, 6(2): 021302 (10 pages)
Haanappel, V. A. C., Jordan, N., Mai, A., Mertens, J., Serra, J. M., Tietz, F., Uhlenbruck, S., Vinke, I. C., Smith, M. J., and de Haart, L. G. J. (February 26, 2009). "Advances in Research, Development, and Testing of Single Cells at Forschungszentrum Jülich." ASME. J. Fuel Cell Sci. Technol. May 2009; 6(2): 021302. https://doi.org/10.1115/1.3080547
This paper presents an overview of the main advances in solid oxide fuel cells (SOFCs) research and development (R&D), measurement standardization, and quality assurance in SOFC testing at the Forschungszentrum Jülich. These activities have resulted in both a significant improvement of the electrochemical performance and a better understanding of the electrochemical behavior of SOFCs. Research and development of SOFCs was mainly focused on two types of anode-supported cells, namely, those employing either
La0.65Sr0.3MnO3
(LSM) or
La0.58Sr0.4Co0.2Fe0.8O3−δ
(LSCF) cathode materials. In both cases the optimization of processing and microstructural parameters resulted in satisfactory power output and long-term stability at reduced operation temperatures. Standardization and quality assurance in SOFC testing was also addressed with the goal of producing consistent and reliable tests and measurement results. At present, under optimized experimental conditions, SOFCs with LSM or LSCF cathodes can deliver a power output of about
1.0 W/cm2
1.9 W/cm2
800°C
(700 mV), respectively.
measurement standards, quality assurance, solid oxide fuel cells, testing, SOFC, anode-supported, cathode, microstructure, electrochemical performance, standardisation, quality assurance
Anodes, Solid oxide fuel cells, Testing, Temperature
Component Manufacturing and Stack Integration of Anode-Supported Planar SOFC System
Proceedings of the Second European Solid Oxide Fuel Cell Forum
, Oslo, Norway, May 6–10,
European SOFC Forum
, Oberrohrdorf, Switzerland, pp.
Electrochemical Performance of an Anode Supported Planar SOFC System
Operation of Anode-Supported Thin Electrolyte Film Solid Oxide Fuel Cells at 800°C
Mallener
Cathode Processing by Wet Powder Spraying
Tropartz
Optimisation of Processing and Microstructural Parameters of LSM Cathodes to Improve the Electrochemical Performance of Anode-Supported SOFCs
Performance Improvement of (La,Sr)MnO3
and (La,Sr)(Co,Fe)O3-Type Anode-Supported SOFCs,”
Ferrite-Based Perovskites as Cathode Materials for Anode-Supported Solid Oxide Fuel Cells. Part I: Variation of Composition
Ferrite-Based Perovskites as Cathode Materials for Anode-Supported Solid Oxide Fuel Cells. Part II: Influence of the CGO Interlayer
Characterization of Anode-Supported SOFCs with PSCF Cathode
Manufacturing of High Performance Solid Oxide Fuel Cells (SOFCs) With Atmospheric Plasma Spraying (APS)
Characterisation of Ni-Cermets SOFCs With Varying Anode Densities
Laberty-Robert
Composition and Porosity Graded La2−xNiO4+δ
(x≥0) Interlayers for SOFC: Control of the Microstructure via a Sol-Gel Process,”
Oxygen Diffusion and Transport Properties in Non-Stoichiometric Ln2−xNiO4+δ
Vashook
Preparation, Thermal Expansion, Chemical Compatibility, Electrical Conductivity and Polarization of A2−αAα′MO4
(A=Pr, Sm; A′=Sr; M=Mn, Ni; α=0.3, 0.6) as a New Cathode for SOFC,”
Electrochemical Behaviour of Porous Nd2−xNiO4+δ
Proceedings of the Sixth European Solid Oxide Fuel Cells Forum
, Lucerne, Switzerland, June 28–July 2,
Quality Assurance and Solid Oxide Fuel Cell Testing at Forschungszentrum Jülich
A Review of Standardising SOFC Measurement and Quality Assurance at FZJ
Electrode Activation of Anode-Supported SOFCs With LSM- or LSCF-Type Cathodes
Quality Assurance and Solid Oxide Fuel Cell Testing at Forschungszentrum Juelich
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On Ill-Posedness Measures and Space Change in Sobolev Scales | EMS Press
On Ill-Posedness Measures and Space Change in Sobolev Scales
The degree of ill-posedness of a linear inverse problem is an important knowledge base to select appropriate regularization methods for the stable approximate solution of such a problem. In this paper, we consider ill-posedness measures for a linear ill-posed operator equation
Ax = y
, where the compact linear operator
A : X \to Y
maps between infinite dimensional Hilbert spaces. Using the decay rate of singular values of
A
tending to zero we define an interval of ill-posedness and motivate its meaning by considering lower and upper bounds for the rates of the condition numbers occurring in the numerical solution process of the discretized problem. An equivalent interval information is obtained when compactness measures as
\epsilon
-entropy or
\epsilon
-capacity are exploited alternatively. For the specific case
X := L^2(0, 1)
, the space change problem of shifting the space
X
along a Sobolev scale is treated. In detail, we study the change of the interval of ill-posedness if the solutions are restricted to the Sobolev space
W^1_2 [0, 1)
. The results of these considerations are a warning to characterize the ill-posedness of a problem superficial. Moreover, the interdependences between ill-posedess measures, embedding operators, Hilbert and Sobolev scales are discussed.
Bernd Hofmann, Ulrich Tautenhahn, On Ill-Posedness Measures and Space Change in Sobolev Scales. Z. Anal. Anwend. 16 (1997), no. 4, pp. 979–1000
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§ The number of pairs (a,b) such that ab≤x is O(xlogx)
Fix a given a. ab ≤ x implies that b ≤ x/a, or there are only x/a possible values for b. If we now consider all possible values for a from 1 upto x, we get:
\begin{aligned} |{ (a, b) : ab <= x }| = \sum_{a=1}^x |{ b: b <= x/a }| \leq \sum_{a=1}^x |x/a| \leq x \sum_{a=1}^x (1/a) \leq x \log x \end{aligned}
To show that the harmonic numbers are upper bounded by
\log
, can integrate:
\sum_{i=1}^n 1/i \leq \int_0^n 1/i = \log n
§ Relationship to Euler Mascheroni constant
This is the limit
\gamma \equiv \lim_{n \to \infty} H_n - \log n
. That this is a constant tells us that these functions grow at the same rate. To see that this si indeed a constant, consider the two functions:
f(n) \equiv H_n - \log n
which starts at
f(1) = 1
and strictly decreases.
g(n) \equiv H_n - \log(n+1)
start lower at
g(1) 1 - \log 2
and strictly increases. [why? ]
\lim_n f(n) - g(n) = 0
. So these sandwhich something in between, which is the constant
\gamma
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Exponential Admissibility and H ∞ Control of Switched Singular Time-Delay Systems: An Average Dwell Time Approach
2012 Exponential Admissibility and
{H}_{\infty }
Control of Switched Singular Time-Delay Systems: An Average Dwell Time Approach
Jinxing Lin, Zhifeng Gao
This paper deals with the problems of exponential admissibility and
{H}_{\infty }
control for a class of continuous-time switched singular systems with time-varying delay. The
{H}_{\infty }
controllers to be designed include both the state feedback (SF) and the static output feedback (SOF). First, by using the average dwell time scheme, the piecewise Lyapunov function, and the free-weighting matrix technique, an exponential admissibility criterion, which is not only delay-range-dependent but also decay-rate-dependent, is derived in terms of linear matrix inequalities (LMIs). A weighted
{H}_{\infty }
performance criterion is also provided. Then, based on these, the solvability conditions for the desired SF and SOF controllers are established by employing the LMI technique, respectively. Finally, two numerical examples are given to illustrate the effectiveness of the proposed approach.
Jinxing Lin. Zhifeng Gao. "Exponential Admissibility and
{H}_{\infty }
Control of Switched Singular Time-Delay Systems: An Average Dwell Time Approach." J. Appl. Math. 2012 1 - 28, 2012. https://doi.org/10.1155/2012/482792
Jinxing Lin, Zhifeng Gao "Exponential Admissibility and
{H}_{\infty }
Control of Switched Singular Time-Delay Systems: An Average Dwell Time Approach," Journal of Applied Mathematics, J. Appl. Math. 2012(none), 1-28, (2012)
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Props1SI - Maple Help
Home : Support : Online Help : Science and Engineering : ThermophysicalData : CoolProp : Props1SI
Props1SI(output, fluid, opts)
The Props1SI function interrogates the CoolProp library for thermophysical data.
The output parameter can be any of the numerical thermophysical properties in the Quantities and Maple-specific aliases columns of the following table, whenever that property makes sense for the given fluid. In almost all circumstances, you can use either one of the names used by the CoolProp library, or an alias defined by the Maple package.
If you supply the useunits = true option (which can be shortened to just useunits), then the result will always have the appropriate unit. If you supply useunits = false (the default), the result will never have a unit.
\mathrm{with}\left(\mathrm{ThermophysicalData}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{Atmosphere}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Chemicals}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{CoolProp}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{PHTChart}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Property}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{PsychrometricChart}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{TemperatureEntropyChart}}]
\mathrm{with}\left(\mathrm{CoolProp}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{HAPropsSI}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{PhaseSI}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Property}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Props1SI}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{PropsSI}}]
Determine the triple point of water.
\mathrm{Props1SI}\left(\mathrm{T_triple},\mathrm{Water}\right)
\textcolor[rgb]{0,0,1}{273.160000000000025}
\mathrm{Props1SI}\left(\mathrm{T_triple},\mathrm{Water},\mathrm{useunits}\right)
\textcolor[rgb]{0,0,1}{273.1600000}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{K}⟧
The ThermophysicalData[CoolProp][Props1SI] command was introduced in Maple 2016.
|
EUDML | A general lemma for fixed-point theorems involving more than two maps in -metric spaces with applications. EuDML | A general lemma for fixed-point theorems involving more than two maps in -metric spaces with applications.
A general lemma for fixed-point theorems involving more than two maps in
D
-metric spaces with applications.
Dhage, B. C.; Arya, Smrati; Ume, Jeong Sheok
Dhage, B. C., Arya, Smrati, and Ume, Jeong Sheok. "A general lemma for fixed-point theorems involving more than two maps in -metric spaces with applications.." International Journal of Mathematics and Mathematical Sciences 2003.11 (2003): 661-672. <http://eudml.org/doc/50738>.
author = {Dhage, B. C., Arya, Smrati, Ume, Jeong Sheok},
keywords = {common fixed point theorem; -metric; contractive condition; -metric},
title = {A general lemma for fixed-point theorems involving more than two maps in -metric spaces with applications.},
AU - Arya, Smrati
TI - A general lemma for fixed-point theorems involving more than two maps in -metric spaces with applications.
KW - common fixed point theorem; -metric; contractive condition; -metric
common fixed point theorem,
D
-metric, contractive condition,
D
|
Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity | EMS Press
We deal with a class on nonlinear Schr\"odinger equations \eqref{eq:1} with potentials
V(x)\sim |x|^{-\a}
0<\a<2
K(x)\sim |x|^{-\b}
\b>0
. Working in weighted Sobolev spaces, the existence of ground states
v_{\e}
W^{1,2}(\Rn)
is proved under the assumption that
p
satisfies \eqref{eq
}. Furthermore, it is shown that
v_{\e}
are {\em spikes} concentrating at a minimum of
{\cal A}=V^{\theta}K^{-2/(p-1)}
\theta= (p+1)/(p-1)-1/2
Antonio Ambrosetti, Andrea Malchiodi, Veronica Felli, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity. J. Eur. Math. Soc. 7 (2005), no. 1, pp. 117–144
|
Rosalinda examined the angles at right and wrote the equation below.
(2x + 1°) + (x – 10°) = 90°
Does her equation make sense? If so, explain why her equation must be true. If it is not correct, determine what is incorrect and rewrite the equation.
Is a right angle (as illustrated above), equal to 90°?
Yes. This means Rosalinda's equation makes sense, because the two angles make up a 90° angle.
If you have not already done so, solve her equation, clearly showing all your steps. What are the measures of the two angles?
Refer to the Math Notes box in Lesson 1.1.4 if you need help remembering how to solve a linear equation.
Verify that your answer is correct.
After solving for x, substitute your answer into the original equation to verify it is correct.
|
On global solutions to a defocusing semi-linear wave equation | EMS Press
We prove that the 3D cubic defocusing semi-linear wave equation is globally well-posed for data in the Sobolev space
\dot{H}^{s}
s>3/4
. This result was obtained in [Kenig-Ponce-Vega, 2000] following Bourgain's method ([Bourgain, 1998]). We present here a different and somewhat simpler argument, inspired by previous work on the Navier-Stokes equations ([Calderon, 1990], [Gallagher-Planchon, 2002]).
Isabelle Gallagher, Fabrice Planchon, On global solutions to a defocusing semi-linear wave equation. Rev. Mat. Iberoam. 19 (2003), no. 1, pp. 161–177
|
Line bundles with partially vanishing cohomology | EMS Press
Define a line bundle
L
on a projective variety to be
q
-ample, for a natural number
q
, if tensoring with high powers of
L
kills coherent sheaf cohomology above dimension
q
. Thus 0-ampleness is the usual notion of ampleness. We show that
q
-ampleness of a line bundle on a projective variety in characteristic zero is equivalent to the vanishing of an explicit finite list of cohomology groups. It follows that
q
-ampleness is a Zariski open condition, which is not clear from the definition.
Burt Totaro, Line bundles with partially vanishing cohomology. J. Eur. Math. Soc. 15 (2013), no. 3, pp. 731–754
|
Tags: machine learning research notes all
Some Interesting Research Papers
Notes on a few papers from my coursework
This page is for short outlines of papers I find interesting. These notes may vary in techincal detail as well as in core concepts - feel free to reach out to me with errata or questions.
In this paper, roboticist Rodney Brooks advocates for a research direction of artificial intelligence that is orthogonal to the traditional paradigms of the 1980s (and before). In particular, he argues against centralized representation for AI systems, and instead offers that for a given Creature, the environment itself should act as a representation of the world through its feedback. This is an extension of the greater debate against Symbolic AI, and Brooks introduces the Subsumption Architecture. A variety of robots were developed and deployed at MIT, that leveraged the actionist approach to AI system design.
Ji He et al. (2016)
This paper introduces a new deep reinforcement architecture for handling natural language spaces: the Deep Reinforcement Relevance Network (DRRN). The DRRN is shown to perform well in text-based game settings, with superior performance over standard Q-Learning architectures.
Q-Learning for Text Games
In text-based games, we can view the state as being a function of the description given to the player, and the action as one of the possible lists of actions for the current state. Formally, the environment is updated
s_{t + 1} = s'
in accordance with the distribution
P(s' | s, a)
. Then, the agent receives a reward
r_t
for the transition. The authors use a stochastic policy
\pi(a_t | s_t)
for time
t
. The Q-function
Q^\pi (s, a)
is defined as the expected return starting from state
s
, taking action
a
, and remaining optimal for the remainder of the horizon:
Q^\pi(s_t, a_t) = \mathop{\mathbb{E}}~[\thinspace \sum_{k = 0}^\infty \gamma^k r_{t+k} | s_t = s, a_t = a \thinspace]
One of the primary motivators of the Deep Reinforcement Relevance Network is semantic representation through word embeddings. As such, the DRRN uses two deep neural networks to approximate embeddings for both actions/states in the given textual environment. Then, a general interaction function can be defined between these finite vectors (e.g. dot product, inner product) to approximate the Q-function for the state-action pair.
The optimal policy and Q-function is determined using the canonical Q-learning update algorithm, where
\eta
is the learning rate:
Q(s_t, a_t) \leftarrow Q(s_t, a_t) + \eta_t \cdot (r_t + \gamma \cdot \operatorname{max}_a Q(s_{t+1}, a) - Q(s_t, a_t))
The authors also use a softmax selection strategy as the exploration policy during the learning state, where
A_t
is the set of actions at time
t
\pi(a_t = a_t^i | s_t) = \frac{\operatorname{exp} (\alpha \cdot Q(s_t, a_t^i))}{\sum _{j=1} ^{|A_t|} \operatorname{exp} (\alpha \cdot Q(s_t, a_t^j))}
\alpha
is used to facilitate the degree of exploration early in the model's training. As the model approximates the Q-function better, the
\alpha
factor will assign higher probability to the optimal action. This allows the model to practice exploitation of optimal policies later on in inference.
Natural Language Target Spaces
For an action space
A
, and state space
S
, vanilla Q-learning recursion requires maintaining a table of size
|A| \cdot |S|
, which can be intractable for large action/state spaces. In addition, depending on the type of text-game, the possible action set
A_t
t
can be unknown, so it's unrealistic to design an architecture around the cardinality of the action space.
From the paper: "it is not practical to have a DQN architecture of a size that is explicitly dependent on the large number of natural language options"
To mitigate this problem, the DRNN is used to compute Q-values with a single forward pass for each state/action pair. Then, softmax selection can be applied with the exploration/exploitation factor
\alpha
The authors manually anotate endings for two different text games, with the reward being proportional to the sentiment of the output. Small negative rewards are given for each non-ending state, to promote the agent to finish the game as quickly as possible.
A Max-Action DQN and a Per-Action DQN are used as baselines to test against. The DRNNs used have 1 or 2 hidden layers with a dimensionality of 20, 50 or 100.
Nathanael Chambers and Dan Jurafsky (2008)
In Fall 2019, I worked on an updated implementation of Unsupervised Learning of Narrative Event Chains by Chambers and Jurafsky (2008) as part of an independent study project at the University of Pennsylvania, advised by Chris Callison-Burch. The overall goal of the project is to learn discrete representations of narrative knowledge through Narrative Events and orderings known as Narrative Chains.
Hand-written scripts were used in NLP in the 1980s as a structured representation of a body of text. In this paper, such scripts are learned for narrative text, referred to as narrative chains. These chains not only provide a representation of the source text, but also encode subject/verb semantics and temporal orderings of events as well. From the paper: "Since we are focusing on a single actor in this study, a narrative event is thus a tuple of the event and the typed dependency of the protagonist". Let's formalize:
The contributions of this paper are three-fold: 1) learning unsupervised relations between entities, 2) temporal ordering of narrative events, 3) pruning of the narrative chains into discrete sets.
The Narrative Chain Model
The authors define two key terminology: narrative chains and narrative events. Narrative Events are defined as tuples of an event and its participants, represented as typed dependencies. This paper only considers single actors as protagonists and as such narrative events are a tuple of the event and the typed dependency of the protagonist: (event, dependency). Narrative Chains are therefore defined as a partially ordered set of narrative events that share a common protagonist/actor. Formally this is defined as
\{e_1, e_2, ..., e_n \}
n
is the length of the chain and relationship
B(e_i, e_j)
is true if and only if event
i
occurs strictly before event
j
Learning Narrative Relations
Given a list of observed verb/dependency frequencies, we can compute the pointwise mutual information between these occurances as:
PMI[e(w, d), e(v, g)] = \operatorname{log} \frac{P[e(w, d), e(v, g)]}{P[e(w, d)] \cdot P[e(v, g)]}
e(w, d)
is the verb/dependency pair between
w
and
Evaluation is performed using the Narrative Cloze Evaluation Task for narrative coherence. A narrative chain is provided to the task and an event is removed in order for the model to perform a prediction to be evaluated on. The aim of the task is to perform a fill-in-the-blanks task, which upon successful completion indicates the presence of coherent narrative knowledge by the model. Given of tuple list of (chain, event) where chain is missing the true prediction event, the evaluation module returns the average model position. The model position is defined as the true event's position in the model's ranked candidate outputs (lower is better).
My implementation of (Chambers and Jurafsky, 2008) uses updated libaries, classes and functions. Written in Python, using the Stanford CoreNLP library (updated dependency parsing from transition model to neural-based Universal Dependencies) as well as the SpaCy pipeline for neural network models (with extensions from HuggingFace). I also extended the project to use NLP's secret sauce: word embeddings. An interpolated model between pointwise mutual information and cosine similarity shows strong results with low amounts of training data.
The following libraries are used throughout the study:
Stanford CoreNLP Python Implementation (stanfordnlp)
SpaCy Dependency Parser (spacy)
HuggingFace Neural Coreference Resolution (neuralcoref)
Magnitude Embedding Library (pymagnitude)
Word2Vec Google News Skip-Gram Model
Examples of identified narrative events in the format (subject, verb, dependency, dependency_type, probability):
you kiss girl dobj 0.00023724792408066428
that enables users dobj 0.00023724792408066428
God bestows benefaction dobj 0.00023724792408066428
Astronomers observed planets dobj 0.00023724792408066428
Examples of generated narrative chains (using a Greedy Decoding strategy):
(Embedding-Similary Based)
seed event: play I dsubj -> I play
score nsubj -> I score
win nsubj -> I win
beat nubj -> I beat
(Pointwise Mutual Information Approximation Based)
seed event: go I nsubj -> I go
get nsubj -> I get
do nsubj -> I do
want nsubj -> I want
verb space too large -> lemmatizing verbs before parsing
events are similar to themselves -> removing seen verbs in chain from prediction candidates
coreference resolution fails occasionally -> increase chunk size
parsing is slow -> single grammatical pass and resolve entities ad-hoc
coreference count computation is slow -> refactor to matrix implementation
This was a really interesting approach to modelling narrative semantics. I'm currently taking an advanced seminar course in text generation and interactive fiction, and I hope to draw inspiration from this project to state of the art models/games such as GPT-2 and AI Dungeon 2!
|
Some Distributional Products of Mikusiński Type in the Colombeau Algebra $\mathcal G(R^m)$ | EMS Press
Some Distributional Products of Mikusiński Type in the Colombeau Algebra
\mathcal G(R^m)
Particular products of Schwartz distributions on the Euclidean space
\mathbb R^m
are derived when the latter have coinciding point singularities and the products are ’balanced’ so that their sum to give an ordinary distribution. These products follow the pattern of a known distributional product published by Jan Mikusiński in 1966. The results are obtained in the Colombeau algebra
\mathcal G (\mathbb R^m)
of generalized functions.
\mathcal G (\mathbb R^m)
is a relevant algebraic construction, with the distribution space linearly embedded, which by the notion of ’association’ allows the results to be evaluated on the level of distributions.
B. Damyanov, Some Distributional Products of Mikusiński Type in the Colombeau Algebra
\mathcal G(R^m)
|
Diffusion Phenomenon for Linear Dissipative Wave Equations | EMS Press
In this paper we prove the diffusion phenomenon for the linear wave equation. To derive the diffusion phenomenon, a new method is used. In fact, for initial data in some weighted spaces, prove that for
2 \leq p\leq \infty
,\,
\left\Vert u-v\right\Vert _{L^{p }(\mathbb{R}^{N})}
decays with the rate
%%t^{-\left(\!\frac{N}{2}(1-\frac{1}{p})\!\right)-1-\frac{\gamma}{2}}, t^{-\frac{N}{2}(1-\frac{1}{p})-1-\frac{\gamma}{2}}, \,\gamma \in \lbrack 0,1]
faster than that of either
% u
v
u
is the solution of the linear wave equation with initial data
\left( u_{0},u_{1}\right) \in \left( H^{1}(\mathbb{R}^{N})\cap L^{1,\gamma }(\mathbb{R}^{N})\right) \times \left( L^{2}(\mathbb{R}^{N})\cap L^{1,\gamma }(\mathbb{R}^{N})\right)
\gamma \in \left[ 0,1\right]
v
is the solution of the related heat equation with initial data
% v_{0}=u_{0}+u_{1}
. This result improves the result in H. Yang and A. Milani [Bull. Sci. Math. 124 (2000), 415–433] in the sense that, under the above restriction on the initial data, the decay rate given in that paper can be improved by
t^{-\frac{\gamma}{2}}
Belkacem Said-Houari, Diffusion Phenomenon for Linear Dissipative Wave Equations. Z. Anal. Anwend. 31 (2012), no. 3, pp. 267–282
|
11 For the reaction C3H8+5O2→3CO2+4H2O, if the standard enthalpy change is -2x103 KJ mol-1and bond enthalpy of C-C, C-H, - Chemistry - Thermodynamics - 12253683 | Meritnation.com
11. For the reaction {C}_{3}{H}_{8}+5{O}_{2}\to 3C{O}_{2}+4{H}_{2}O, if the s\mathrm{tan}dard enthalpy change is -2x{10}^{3} KJ mo{l}^{-1}\phantom{\rule{0ex}{0ex}}and bond enthalpy of C-C, C-H, C=O and O-H are 347, 414, 741 and 464 KJmo{l}^{-1} respectively, calculate the bond enthalpy of O=O.
Kindly refer to the similar link -https://www.meritnation.com/ask-answer/question/estimate-change-in-enthalpy-for-the-following-reaction-c/chemistry/9043255
in the above link change in enthalpy is calculated but you have to calculate bond energy of oxygen only with simple mathematics.
Possy answered this
For Q8 use formula delta G=delta H -T Delta S
Akshita Purohit answered this
bde of c3h8= 347*2+414*8= 4006
bde of 3co2= 741*2*3= 4446
bde of 4h2o= 464*2*4= 3712
enthalpy change= bond energy of reactant - bond energy of product
-2*10^3= (4006+ 5O2) - 8158
5o2 = 2125
bond energy of o2= 425
i hope that i have solved it correctly and can help you too❤
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On Kakeya–Nikodym averages, $L^p$-norms and lower bounds for nodal sets of eigenfunctions in higher dimensions | EMS Press
L^p
We extend a result of the second author [27, Theorem 1.1] to dimensions
d \geq 3
which relates the size of
L^p
-norms of eigenfunctions for
2 < p < 2(d+1) / d-1
to the amount of
L^2
-mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee [22] and a variable coefficient variant of an "
\epsilon
removal lemma" of Tao and Vargas [35]. We also use Hörmander's [20]
L^2
oscillatory integral theorem and the Cartan–Hadamard theorem to show that, under the assumption of nonpositive curvature, the
L^2
-norm of eigenfunctions
e_{\lambda}
over unit-length tubes of width
\lambda^{-1/2}
goes to zero. Using our main estimate, we deduce that, in this case, the
L^p
-norms of eigenfunctions for the above range of exponents is relatively small. As a result, we can slightly improve the known lower bounds for nodal sets in dimensions
d \ge 3
of Colding and Minicozzi [10] in the special case of (variable) nonpositive curvature.
Matthew D. Blair, Christopher D. Sogge, On Kakeya–Nikodym averages,
L^p
-norms and lower bounds for nodal sets of eigenfunctions in higher dimensions. J. Eur. Math. Soc. 17 (2015), no. 10, pp. 2513–2543
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Schismatic_temperament Knowpia
A schismatic temperament is a musical tuning system that results from tempering the schisma of 32805:32768 (1.9537 cents) to a unison. It is also called the schismic temperament, Helmholtz temperament, or quasi-Pythagorean temperament.
Tonnetz for Pythagorean tuning (above) and schismatic temperament (below)
In Pythagorean tuning all notes are tuned as a number of perfect fifths (701.96 cents
play (help·info)). The major third above C, E, is considered four fifths above C. This causes the Pythagorean major third, E+ (407.82 cents
play (help·info)), to differ from the just major third, E♮ (386.31 cents
play (help·info)): the Pythagorean third is sharper than the just third by 21.51 cents (a syntonic comma
play (help·info)).
C — G — D — A+ — E+
Ellis's "skhismic temperament"[1] instead uses the note eight fifths below C, F♭-- (384.36 cents
play (help·info)), the Pythagorean diminished fourth or schismatic major third. Though spelled "incorrectly" for a major third, this note is only 1.95 cents (a schisma) flat of E♮, and thus more in tune than the Pythagorean major third. As Ellis puts it, "the Fifths should be perfect and the Skhisma should be disregarded [accepted/ignored]."
{\displaystyle \approx }
F♭--
F♭-- — C♭-- — G♭-- — D♭-- — A♭- — E♭- — B♭- — F — C
In his eighth-schisma "Helmholtzian temperament"[1] the note eight fifths below C is also used as the major third above C. However, in the "skhismic temperament" pure perfect fifths are used to construct an approximate major third, while in the "Helmholtzian temperament" approximate perfect fifths are used to construct a pure major third. To raise the Pythagorean diminished fourth 1.95 cents to a just major third each fifth must be narrowed, or tempered, by 1.95/8 = 0.24 cents. Thus the fifth becomes 701.71 cents instead of 701.96 cents. As Ellis puts it, "the major Thirds are taken perfect, and the Skhisma is disregarded [tempered out]."
E♮
{\displaystyle \approx }
E♮ —
{\displaystyle \approx }
C♭-- —
{\displaystyle \approx }
G♭-- —
{\displaystyle \approx }
D♭-- —
{\displaystyle \approx }
A♭- —
{\displaystyle \approx }
E♭- —
{\displaystyle \approx }
B♭- —
{\displaystyle \approx }
Pythagorean (help·info) vs.
Skhismic (help·info).
Comparison with other tuningsEdit
Whereas schismatic temperaments achieve a ratio with a number of perfect fifths, each tempered by a fraction of the schisma; meantone temperaments achieve a ratio with perfect fifths, each tempered by a fraction of the syntonic comma (81:80, 21.51 cents). As meantone temperaments are often described by what fraction of the syntonic comma is used to alter the perfect fifths, schismatic temperaments are often described by what fraction of the schisma is used to alter the perfect fifths (thus quarter-comma meantone temperament, eighth-schisma temperament, etc.).
In both eighth-schisma tuning and quarter-comma meantone the octave and major third are just, but eighth-schisma has much much more accurate perfect fifths and minor thirds (less than a quarter of a cent off from just intonation). However, quarter-comma meantone has a large advantage in that the major third and minor third are spelled as such, whereas in schismatic tunings, they're represented by the diminished fourth and augmented second (if spelled according to their construction in the tuning). This places them well outside the span of a single diatonic scale, and requires both a larger number of pitches and more microtonal pitch-shifting when attempting common-practice Western music.
Various equal temperaments lead to schismatic tunings which can be described in the same terms. Dividing the octave by 53 provides an approximately 1/29-schisma temperament; by 65 a 1/5-schisma temperament, by 118 a 2/15-schisma temperament, and by 171 a 1/10-schisma temperament. The last named, 171, produces very accurate septimal intervals, but they are hard to reach, as to get to a 7/4 requires 39 fifths. The −1/11-schisma temperament of 94, with sharp rather than flat fifths, gets to a less accurate but more available 7:4 by means of 14 fourths. Eduardo Sabat-Garibaldi also had an approximation of 7:4 by means of 14 fourths in mind when he derived his 1/9-schisma tuning.
History of schismatic temperamentsEdit
Historically significant is the eighth-schisma tuning of Hermann von Helmholtz and Norwegian composer Eivind Groven. Helmholtz had a special Physharmonica (a harmonium by Schiedmayer) with 24 tones to the octave.[citation needed] Groven built an organ internally equipped with 36 tones to the octave which had the ability to adjust its tuning automatically during performances; the performer plays a familiar 12-key (per octave) keyboard and in most cases the mechanism will choose from among the three tunings for each key so that the chords played sound virtually in just intonation.[citation needed] A 1/9-schisma tuning has also been proposed by Eduardo Sabat-Garibaldi, who together with his students uses a 53-tone to the octave guitar with this tuning.[citation needed]
Mark Lindley and Ronald Turner-Smith argue that schismatic tuning was briefly in use during the late medieval period.[2][need quotation to verify] This was not temperament but merely 12-tone Pythagorean tuning. Justly tuned fifths and fourths generate a reasonable schismatic tuning and therefore schismatic is in some respects an easier way to introduce approach justly tuned thirds into a Pythagorean harmonic fabric than meantone. However, the result suffers from the same difficulties as just intonation – for example, the wolf B-G♭ here arises all too easily when availing oneself of the concordant schismatic substitutions just outlined – so it is not surprising that meantone temperament became the dominant tuning system by the early Renaissance. Helmholtz's and Groven's systems get around some, but not all, of these difficulties by including multiple tunings for each key on the keyboard, so that a particular note can be tuned as G♭ in some contexts and F♯ in others, for example.
^ a b Helmholtz, Hermann; Ellis, Alexander J. (1885), On the Sensations of Tone (Second English ed.), Dover Publications, p. 435 . On the Sensations of Tone at the Internet Archive
^ Lindley, Mark; Turner-Smith, Ronald (1993), "Chapter 17. Quasi-Pythagorean Temperaments", Mathematical Models of Musical Scales: A New Approach, Orpheus-Schriftenreihe zu Grundfragen der Musik, vol. 66, Bonn-Bad Godesberg: Verlag fuer systematische Musikwissenschaft, GmbH, pp. 55–57
"Schismic Temperaments", Intonation Information.
"Schismatic family", on Xenharmonic Wiki.
"Schismic", Tonalsoft(R) - Encyclopedia of microtonal music theory.
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Variational Inference | Zhiya Zuo
optimization variational-inference bayesian
It took me more than two weeks to finally to get the essence of variational inference. The painful but fulfilling process brought me to appreciate the really difficult (at least for me) but beautiful math behind it.
A couple of useful tutorials I found:
D. M. Blei, A. Kucukelbir, and J. D. McAuliffe, “Variational Inference: A Review for Statisticians,” J. Am. Stat. Assoc., vol. 112, no. 518, pp. 859–877, 2017.
D. G. Tzikas, A. C. Likas and N. P. Galatsanos, “The variational approximation for Bayesian inference,” in IEEE Signal Processing Magazine, vol. 25, no. 6, pp. 131-146, November 2008. doi: 10.1109/MSP.2008.929620
https://am207.github.io/2017/wiki/VI.html
Machine Learning: Variational Inference by Jordan Boyd-Graber
Mean Field Variational Family
Coordinate Ascent VI (CAVI)
Derivation of optimal var. dist.:
Applying VI on GMM
Choose $q$
Full joint probability
Entropy of variational distributions
Full ELBO
$m_j$
$s_j^2$
As with expectation maximization, I start by describing a problem to motivate variational inference. Please refer to Prof. Blei’s review for more details above.
Let’s start by considering a problem where we have data points sampled from mixtures of Gaussian distributions. Specifically, there are $K$ univariate Gaussian distributions with means $\mathbf{\mu} = { \mu_1, …, \mu_K }$ and unit variance ($\mathbf{\sigma}=\mathbf{1}$) (for simplicity):
Please refer to my EM post on details of this sample data
In a Bayesian setting, we can assume that all the means come from the same prior distribution, which is also a Gaussian $\mathcal{N}(0, \sigma^2)$, with variance $\sigma^2$ being a hyperparameter. Specifically, we can setup a very simple generative model:
For each data point $x^{(i)}$, where $i=1,…,n$
Sample a cluster assigment (or membership to which Gaussian mixture component it belongs) $c^{(i)}$ uniformally: $c^{(i)} \sim Uniform(K)$
Sample its value from the correpsonding component: $x^{(i)} \sim \mathcal{N}(\mu_{c_i}, 1)$
This gives us a straightforward view of how the joint probability can be written out:
\begin{align} p(\mathbf{c}, \mathbf{\mu}, \mathbf{x}) & = p(\mathbf{\mu})p(\mathbf{c})p(\mathbf{x} \vert \mathbf{c}, \mathbf{\mu}) \\ & = p(\mathbf{\mu}) \prod_{i} p(c^{(i)})p(x^{(i)} \vert c^{(i)}, \mathbf{\mu}) \end{align}
Summing/integrating out the latent variables, we can obtain the marginal likelihood (i.e., evidence):
\begin{align} p(\mathbf{x}) & = \int_{\mathbf{\mu}} p(\mathbf{\mu}) \prod_{i} \sum_{c^{(i)}} p(c^{(i)})p(x^{(i)} \vert c^{(i)}, \mathbf{\mu}) d \mathbf{\mu}\\ & [\text{We can switch the order of how we integrate/sum out the latent variables}] \\ & = \sum_{\mathbf{c}} p(\mathbf{c}) \int_{\mathbf{\mu}} p(\mathbf{\mu}) \prod_{i} p(x^{(i)} \vert c^{(i)}, \mathbf{\mu}) d \mathbf{\mu} \end{align}
Note that while it is possible to compute individual termins within the integral (Gaussian prior and Gaussian likelihood), the overall complexity will go up to $\mathcal{O}(K^n)$ (which is all possible configurations). Therefore, we need to consider approximate inference due to the intractability.
Actually, the motivation of VI is very similar to EM, which is to come up with an approximation of point estimates of the latent variables. Instead of point estimates, VI tries to find variational distributions that serve as good proxies for the exact solution.
Suppose we have $\mathbf{z}={ z^{(1)}, …, z^{(n)}}$ as observed data and $\mathbf{z}={ z^{(1)}, …, z^{(n)}}$ as latent variables. The inference problem is to find the posterior probability of the latent variables given observations $p(\mathbf{z} \vert \mathbf{x})$:
Often times, the denominator evidence is intractable. Therefore, we need approximations to find a relatively good solution in a reasonable amount of time. VI is exactly what we need!
In my EM post, we can prove that the log evidence $ln~p(\mathbf{x})$ can actually be decomposed as follows (note that we will use integral this time):
\begin{align} ln~p(\mathbf{x}) & = \int_{\mathbf{z}} q(\mathbf{z}) d\mathbf{z}~~ln~p(\mathbf{x}) \\ & [\text{Recall that } \int_{\mathbf{z}} q(\mathbf{z}) d\mathbf{z} = 1] \\ & = \int_{\mathbf{z}} q(\mathbf{z}) ln~ \frac{p(\mathbf{x}, \mathbf{z})}{p(\mathbf{z} \vert \mathbf{x})} d\mathbf{z}\\ & = \int_{\mathbf{z}} q(\mathbf{z}) ln~ \frac{p(\mathbf{x}, \mathbf{z})~q(\mathbf{z})}{p(\mathbf{z} \vert \mathbf{x}) ~q(\mathbf{z})} d\mathbf{z}\\ & = \int_{\mathbf{z}} q(\mathbf{z}) ln~ \frac{p(\mathbf{x}, \mathbf{z})}{q(\mathbf{z})} d\mathbf{z} + \int_{\mathbf{z}} q(\mathbf{z}) ln~ \frac{q(\mathbf{z})}{p(\mathbf{z} \vert \mathbf{x})} d\mathbf{z}\\ & = \mathcal{L}(\mathbf{x}) + KL(q\vert \vert p) \end{align}
where $\mathcal{L}(\mathbf{x})$ is defined as ELBO: KL divergence is bounded and nonnegative.
If we further decompose ELBO, we have:
\begin{align} \mathcal{L}(\mathbf{x}) & = \int_{\mathbf{z}} q(\mathbf{z}) ln~ \frac{p(\mathbf{x}, \mathbf{z})}{q(\mathbf{z})} d\mathbf{z} \\ & = \int_{\mathbf{z}} q(\mathbf{z}) ln~p(\mathbf{x} \vert \mathbf{z}) - q(\mathbf{z})ln~\frac{q(\mathbf{z})}{p(\mathbf{z})} d\mathbf{z}\\ & = E_q\big[ ln~p(\mathbf{z} \vert \mathbf{x}) \big] - KL(q(\mathbf{z})||p(\mathbf{z}))\\ & = \int_{\mathbf{z}} q(\mathbf{z}) ln~p(\mathbf{x}, \mathbf{z}) - q(\mathbf{z})ln~q(\mathbf{z}) d\mathbf{z}\\ & = E_q\big[ ln~p(\mathbf{x}, \mathbf{z}) \big] + \mathcal{H}(q) ~~\text{(Entropy of } q\text{)}\\ \end{align}
The last equation above shows that ELBO trades off between the two terms:
The first term prefers $q(\mathbf{z})$ to be high when complete likelihood $p(\mathbf{x}, \mathbf{z})$ is high
The second term encourages $q(\mathbf{z})$ to be diffuse across the space
Finally, we note that, in EM, we are able to compute $p(\mathbf{z}\vert \mathbf{x})$ so that we can easily maximize ELBO. However, VI is the way to do when we cannot.
By far, we haven’t say anything about what $q$’s should be. In this note, we only look at a classical type, called mean field variational family. Specifically, it assumes that latent variables are mutually independent. This means that we can easily factorize the variational distributions into groups:
By doing this, we are unable to capture the interdependence between the latent variables. A nice visualization from Blei et al. (2017):
By factorizing the variational distributions into invidual products, we can easily apply coordinate ascent optimization on each factor. A common procedure to conduct CAVI is:
Choose variational distributions $q$;
Compute ELBO;
Optimize individual $q_j$’s by taking the gradient for each latent variable;
Repeat until ELBO converges.
In fact, we can derive the optimal solutions without too much efforts:
\begin{align} ELBO & = E_q[log~p(x,z)] - E_q[log~q(z)] \\ & [\text{Here we use the fact that } q(z) \text{ can be factorized}]\\ & = E_q[log~p(x, z_j, z_{-j})] - \sum_{q_l}E_{q_l}[q_l(z_l)] \\ & [\text{Iterative expectation: } E[A] = E[E[A|B]]]\\ & = E_j\Big[E_{-j}\big[ log~p(x, z_j, z_{-j}) \vert z_{j} \big] \Big] - E_{q_j}[q_j] + const \\ \end{align}
Now, according to the definition of expectation, we have:
\begin{align} E_{-j}\big[ log~p(x, z_j, z_{-j}) \vert z_{j} \big] &= \int_{-j} log~p(x, z_j, z_{-j})~q(z_{-j}|z_j) dq_{-j} \\ & = \int_{-j} log~p(x, z_j, z_{-j})~q(z_{-j}) dq_{-j} \\ & = E_{-j}\big[ log~p(x, z_j, z_{-j}) \big] \end{align}
We assume independence between latent variables’ variational distributions $q(z)$
\begin{align} ELBO & = E_{j}\Big[E_{-j}\big[ log~p(x, \mathbf{z}) \big]\Big] - E_{j}[q_j] + const \\ \end{align}
We can see that the first two terms can be combined into a negative KL divergence between those within the $E_j\big[ \cdot \big]$. Therefore, we can write down the optimal solution as:
While the derivation through iterative expectation seems to be simpler, I personally still prefer taking partial derivatives to parameters of variational distributions, as in the following example, which seems to be more natural to me. After all, we will be using ELBO to check convergence anyway.
Let’s get back to our original problem with the univariate Gaussian mixtures with unit variance. The full parameterization is as follows:
\begin{align} \mu_j & \sim \mathcal{N}(0, \sigma^2)~\text{for } j = 1, ..., K \\ c_i & \sim \mathcal{U}(K)~\text{for } i = 1, ..., N \\ x_i & \sim \mathcal{N}(c_i^T \mu, 1)~\text{for } i = 1, ..., N \end{align}
Note that $c_i$ is a vector of one’s and zero’s such that $c_{ij} = 1; c_{il} = 0 \text{ for } j\neq l$ (a.k.a, one-hot vector).
By mean field VI, we can introduce variational distributions for the two latent variables $\mathbf{c}$ and $\mathbf{\mu}$:
According to what we have above, we will choose the following variational distributions for $c$ and $\mu$
\begin{align} \mu_j; m_j, s_j^2 & \sim \mathcal{N}(m_j, s_j^2) \\ c_i; \phi_i & \sim Multi(\phi_i) \end{align}
Therefore, $\phi_i$ is a vector of probabilities such that $p(c_i=j) = \phi_{ij}$
The most important thing is to write down ELBO, the evidence lower bound, which is needed for (i) parameter updates; (ii) convergence check. However, I’ve seen that convergence check could be done by the relative change of parameter estimates here. If parameters do not change much, VI will stop by thinking that it has converged.
Recall that $ELBO = E_q[log~p(x,z)] - E_q[log~q(z)] $. Let me split this task into two.
The hidden/latent variables in this problem are $c$ and $\mu$.
\begin{align} log~p(x, c, \mu) & = log~p(\mu)p(c)p(x~\vert~c, \mu) \\ & = \sum_j log~p(\mu_j) + \sum_i \big[ log~p(c_i) + log~p(x_i~\vert~c_i, \mu) \big] \\ \end{align}
$p(c_i) = \dfrac{1}{K}$ is a constant drop it. We then expand $p(\mu_j)$:
\begin{align} log~p(\mu_j) & = log~\Big\{ \dfrac{1}{\sqrt{2\pi \sigma^2}} exp\big[ -\dfrac{\mu_j^2}{2\sigma^2} \big] \Big\} \\ & [log~\dfrac{1}{\sqrt{2\pi \sigma^2}} \text{ is a constant} ]\\ & \propto -\dfrac{\mu_j^2}{2\sigma^2} \end{align}
For $log~p(x_i~\vert~c_i, \mu)$, it is a bit tricky. Recall that $c_i$ is a one-hot vector, where only one of the element is 1. We can make use of this property and rewrite:
Combine all the above, we can write the log full joint probability as:
Thanks to the mean field assumption, we can factorize the joint of variational easily:
Let’s expand these two terms seperately.
\begin{align} log~p(\mu_j; m_j, s_j^2) & = log~\Big\{ \dfrac{1}{\sqrt{2\pi s_j^2}} exp \big[ -\dfrac{(\mu_j-m_j)^2}{s_j^2} \big] \Big\} \\ & = -\dfrac{1}{2}log~(2\pi s_j^2) -\dfrac{(\mu_j-m_j)^2}{s_j^2} \end{align}
log~q(c, \mu) \propto \sum_i \sum_j log~\phi_{ij} + \sum_j -\dfrac{1}{2}log~(2\pi s_j^2) -\dfrac{(\mu_j-m_j)^2}{s_j^2}
Merge the results back, we have the ELBO written as:
\begin{align} ELBO \propto & \sum_j -E_q\Big[\dfrac{\mu_j}{2\sigma^2}\Big] + \sum_i\sum_j E_q\Big[c_{ij}\Big]E_q\Big[-\dfrac{(x_i-\mu_j)^2}{2}\Big] \\ &- \sum_i \sum_j E_q\Big[log~\phi_{ij}\Big] + \sum_j \dfrac{1}{2}log~(s_j^2) \end{align}
This is a contrained optimization because $\sum_j \phi_{ij} = 1~\forall i$. However, we do not need to add the Lagrange multiplier and the result can still be normalized (we are using a lot of $\propto$ here!)
\begin{align} \dfrac{\partial}{\partial \phi_{ij}}~ELBO & \propto \dfrac{\partial}{\partial \phi_{ij}}\Big\{E_q\Big[-\dfrac{(x_i-\mu_j)^2}{2}\Big] \phi_{ij} - E_q\Big[log~\phi_{ij}\Big] \Big\}\\ & = E_q\Big[-\dfrac{(x_i-\mu_j)^2}{2}\Big] - log~\phi_{ij} - 1 = 0 \\ & E[\mu_j] = m_j \text{; } E[\mu_j^2] = V[\mu] + E^2[\mu] = s_j^2 + \mu_j^2 \\ log~\phi_{ij} & \propto E_q\Big[-\dfrac{(x_i-\mu_j)^2}{2}\Big] \\ \phi_{ij}^* & \propto exp\{ -\tfrac{1}{2}(m_j^2+s_j^2) + x_i m_j \} \end{align}
\begin{align} \dfrac{\partial}{\partial m_{j}}~ELBO & \propto \dfrac{\partial}{\partial m_{j}}~\Big\{ -E\big[\dfrac{\mu_j^2}{2\sigma^2}\big] - \sum_i \phi_{ij} E[\dfrac{(x_i-\mu_j)^2}{2}] \Big\} \\ & \propto \dfrac{\partial}{\partial m_{j}}~\Big\{ -\dfrac{1}{2\sigma^2} m_j^2 - \sum_i \phi_{ij} \big[ -\dfrac{1}{2}m_j^2 + x_i m_j \big] \Big\} \\ & = -\dfrac{1}{\sigma^2}m_j - \sum_i\phi_{ij} m_j + \sum_i \phi_{ij} x_i = 0 \\ m_j^* &= \dfrac{\sum_i\phi_{ij}x_i}{\tfrac{1}{\sigma^2} + \sum_i\phi_{ij}} \end{align}
Note that we are considering $s_j^2$ as a whole.
\begin{align} \dfrac{\partial}{\partial s_j^2}~ELBO & \propto \dfrac{\partial}{\partial s_j^2}~ \big\{ -E \big[\dfrac{\mu_j^2}{2\sigma^2}\big] - \sum_i \phi_{ij} E[\dfrac{(x_i-\mu_j)^2}{2}] +\dfrac{1}{2}log~s_j^2 \big\} \\ & \propto \dfrac{\partial}{\partial s_j^2}~ \Big\{ -\dfrac{1}{2\sigma^2}s_j^2 - \sum_i\phi_{ij}(\dfrac{1}{2}s_j^2) +\dfrac{1}{2}log~s_j^2 \Big\} \\ & = -\dfrac{1}{2\sigma^2} - \sum_i\dfrac{\phi_{ij}}{2} + \dfrac{1}{2s_j^2} = 0 \\ \dfrac{1}{s_j^2} & = \dfrac{1}{\sigma^2} + \sum_i\phi_{ij} \\ (s_j^2)^{*} & = \dfrac{1}{\frac{1}{\sigma^2} + \sum_i\phi_{ij}} \end{align}
Now that we have the ELBO and paramter update formulas, we can setup our own VI algorithm for this simple Guassian Mixture!
class UGMM(object):
'''Univariate GMM with CAVI'''
def __init__(self, X, K=2, sigma=1):
self.N = self.X.shape[0]
self.sigma2 = sigma**2
self.phi = np.random.dirichlet([np.random.random()*np.random.randint(1, 10)]*self.K, self.N)
self.m = np.random.randint(int(self.X.min()), high=int(self.X.max()), size=self.K).astype(float)
self.m += self.X.max()*np.random.random(self.K)
self.s2 = np.ones(self.K) * np.random.random(self.K)
print('Init mean')
print('Init s2')
print(self.s2)
def get_elbo(self):
t1 = np.log(self.s2) - self.m/self.sigma2
t1 = t1.sum()
t2 = -0.5*np.add.outer(self.X**2, self.s2+self.m**2)
t2 += np.outer(self.X, self.m)
t2 -= np.log(self.phi)
t2 *= self.phi
def fit(self, max_iter=100, tol=1e-10):
self.elbo_values = [self.get_elbo()]
self.m_history = [self.m]
self.s2_history = [self.s2]
for iter_ in range(1, max_iter+1):
self._cavi()
self.m_history.append(self.m)
self.s2_history.append(self.s2)
self.elbo_values.append(self.get_elbo())
if iter_ % 5 == 0:
print(iter_, self.m_history[iter_])
if np.abs(self.elbo_values[-2] - self.elbo_values[-1]) <= tol:
print('ELBO converged with ll %.3f at iteration %d'%(self.elbo_values[-1],
iter_))
if iter_ == max_iter:
print('ELBO ended with ll %.3f'%(self.elbo_values[-1]))
def _cavi(self):
self._update_phi()
self._update_mu()
def _update_phi(self):
t1 = np.outer(self.X, self.m)
t2 = -(0.5*self.m**2 + 0.5*self.s2)
exponent = t1 + t2[np.newaxis, :]
self.phi = np.exp(exponent)
self.phi = self.phi / self.phi.sum(1)[:, np.newaxis]
def _update_mu(self):
self.m = (self.phi*self.X[:, np.newaxis]).sum(0) * (1/self.sigma2 + self.phi.sum(0))**(-1)
assert self.m.size == self.K
#print(self.m)
self.s2 = (1/self.sigma2 + self.phi.sum(0))**(-1)
assert self.s2.size == self.K
mu_arr = np.random.choice(np.arange(-10, 10, 2),
num_components) +\
np.random.random(num_components)
mu_arr
X = np.random.normal(loc=mu_arr[0], scale=1, size=SAMPLE)
for i, mu in enumerate(mu_arr[1:]):
X = np.append(X, np.random.normal(loc=mu, scale=1, size=SAMPLE))
sns.distplot(X[:SAMPLE], ax=ax, rug=True)
sns.distplot(X[SAMPLE:SAMPLE*2], ax=ax, rug=True)
sns.distplot(X[SAMPLE*2:], ax=ax, rug=True)
<matplotlib.axes._subplots.AxesSubplot at 0x10f5784e0>
ugmm = UGMM(X, 3)
ugmm.fit()
Init mean
Init s2
5 [ 8.78575069 -5.69598804 6.32040619]
10 [ 8.77126102 -5.69598804 6.30384436]
30 [ 8.77082381 -5.69598804 6.3034367 ]
ELBO converged with ll -1001.987 at iteration 35
ugmm.phi.argmax(1)
sorted(mu_arr)
[-5.704263600460798, 6.298034563379406, 8.791535506275245]
sorted(ugmm.m)
sns.distplot(X[:SAMPLE], ax=ax, hist=True, norm_hist=True)
sns.distplot(np.random.normal(ugmm.m[0], 1, SAMPLE), color='k', hist=False, kde=True)
sns.distplot(X[SAMPLE:SAMPLE*2], ax=ax, hist=True, norm_hist=True)
sns.distplot(X[SAMPLE*2:], ax=ax, hist=True, norm_hist=True)
« MLE vs. MAP Plot Lorenz Curve in Python »
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Zhiya Zuo © 2019
|
The Golden Mean Ratio Compass
When you open the Golden Mean Ratio Compass, the spacing between the yellow and the red arm and the (smaller) spacing between the red arm and the blue arm are in the golden ratio, which is often represented by the Greek letter phi or
Expressed algebraically, for quantities
{\displaystyle a}nd
{\displaystyle b}
{\displaystyle a>b>0,}
The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation.
It has also appeared in architecture and in art.
You can analyse images and photographs using the compass.
There are a number of books written about the Golden Ratio, one of which is
'The Golden Ratio: The Story of PHI, the World's Most Astonishing Number' by Mario Livio
Painted in Bauhaus colours, our Golden Mean Ratio Compass is made of wood, and comes from Japan.
|
Analyze and Plot RF Components - MATLAB & Simulink - MathWorks í•œêµ
Analyze Networks in Frequency Domain
Visualize Component and Network Data
Budget Plot
Mixer Spur Plot
Polar Plots and Smith Charts®
Compute and Plot Time-Domain Specifications
Compute Network Transfer Function
Fit Model Object to Circuit Object Data
Compute and Plot Time-Domain Response
RF Toolbox™ lets you analyze RF components and networks in the frequency domain. You use the analyze function to analyze a circuit object over a specified set of frequencies.
For example, to analyze a coaxial transmission line from 1 GHz to 2.9 GHz in increments of 10 MHz:
ckt = rfckt.coaxial;
f = [1.0e9:1e7:2.9e9];
analyze(ckt,f);
For all circuits objects except those that contain data from a file, you must perform a frequency-domain analysis with the analyze method before visualizing component and network data. For circuits that contain data from a file, the toolbox performs a frequency-domain analysis when you use the read method to import the data.
When you analyze a circuit object, the toolbox computes the circuit network parameters, noise figure values, and output third-order intercept point (OIP3) values at the specified frequencies and stores the result of the analysis in the object's AnalyzedResult property.
For more information, see the analyze function page.
The RF Toolbox lets you validate the behavior of circuit objects that represent RF components and networks by plotting the following data:
Large- and small-signal S-parameters
This table summarizes the available plots and charts, along with the functions you can use to create each one and a description of its contents.
Plot Contents
Parameters as a function of frequency or, where applicable, operating condition. The available parameters include:
Parameters as a function of frequency for each component in a cascade, where the curve for a given component represents the cumulative contribution of each RF component up to and including the parameter value of that component.
Mixer spur power as a function of frequency for an rfckt.mixer object or an rfckt.cascade object that contains a mixer.
Polar plot: Magnitude and phase of S-parameters as a function of frequency.
Smith plot: Real and imaginary parts of S-parameters as a function of frequency, used for analyzing the reflections caused by impedance mismatch.
For each plot you create, you choose a parameter to plot and, optionally, a format in which to plot that parameter. The plot format defines how the RF Toolbox displays the data on the plot. The available formats vary with the data you select to plot. The data you can plot depends on the type of plot you create.
You can use the listparam function to list the parameters of a specified circuit object that are available for plotting. You can use the listformat function to list the available formats for a specified circuit object parameter.
The following topics describe the available plots:
You can plot any parameters that are relevant to your object on a rectangular plot. You can plot parameters as a function of frequency for any object. When you import object data from a .p2d or .s2d file, you can also plot parameters as a function of any operating condition from the file that has numeric values, such as bias. In addition, when you import object data from a .p2d file, you can plot large-signal S-parameters as a function of input power or as a function of frequency. These parameters are denoted LS11, LS12, LS21, and LS22.
This table summarizes the methods that are available in the toolbox for creating rectangular plots and describes the uses of each one. For more information on a particular type of plot, follow the link in the table to the documentation for that method.
plot Plot of one or more object parameters
plotyy Plot of one or more object parameters with y-axes on both the left and right sides
semilogx Plot of one or more object parameters using a log scale for the X-axis
semilogy Plot of one or more object parameters using a log scale for the Y-axis
loglog Plot of one or more object parameters using a log-log scale
You use the link budget or budget plot to understand the individual contribution of each component to a plotted parameter value in a cascaded network with multiple components. The budget plot shows one or more curves of parameter values as a function of frequency, ordered by the circuit index of the cascaded network.
Consider the following cascaded network:
casc = rfckt.cascade('Ckts',...
{rfckt.amplifier,rfckt.lcbandpasspi,rfckt.txline})
This figure shows how the circuit index is assigned to each component in the cascade, based on its sequential position in the network.
You create a 3-D budget plot for this cascade using the plot method with the second argument set to 'budget', as shown in the following command:
analyze(casc,linspace(1e9,3e9,100));
plot(casc,'budget','s21')
Note that you have to analyze your circuit before plotting the budget plot and by default the budget plot is a 2-D plot. If you specify the array of frequencies in the analyze function you can visualize the budget results in 3-D. A curve on the budget plot for each circuit index represents the contributions to the parameter value of the RF components up to that index. This figure shows the budget plot.
If you specify two or more parameters, the RF Toolbox puts the parameters in a single plot. You can only specify a single format for all the parameters.
You use the mixer spur plot to understand how mixer nonlinearities affect output power at the desired mixer output frequency and at the intermodulation products that occur at the following frequencies:
{f}_{out}=Nâ{f}_{in}+Mâ{f}_{LO}
{f}_{in}
{f}_{LO}
is the local oscillator frequency.
N and M are integers.
The RF toolbox calculates the output power from the mixer intermodulation table (IMT). These tables are described in detail in the Visualize Mixer Spurs example.
The mixer spur plot shows power as a function of frequency for an rfckt.mixer object or an rfckt.cascade object that contains a mixer. By default, the plot is three-dimensional and shows a stem plot of power as a function of frequency, ordered by the circuit index of the object. You can create a two-dimensional stem plot of power as a function of frequency for a single circuit index by specifying the index in the mixer spur plot command.
rfdata.network('Type', 'S', 'Freq', 2.1e9, ...
'Data', [0,0;10,0]), 'NoiseData', 0, 'NonlinearData', inf);
SecondCkt = read(rfckt.mixer, 'samplespur1.s2d');
ThirdCkt = rfckt.lcbandpasstee('L', [97.21 3.66 97.21]*1e-9, ...
'C', [1.63 43.25 1.63]*1.0e-12);
CascadedCkt = rfckt.cascade('Ckts', ...
{FirstCkt, SecondCkt, ThirdCkt});
This shows how the circuit index is assigned to the components in the cascade, based on its sequential position in the network.
Circuit index 0 corresponds to the cascade input.
Circuit index 1 corresponds to the LNA output.
Circuit index 2 corresponds to the mixer output.
Circuit index 3 corresponds to the filter output.
You create a spur plot for this cascade using the plot method with the second argument set to 'mixerspur', as shown in the following command:
plot(CascadedCkt,'mixerspur')
Within the three dimensional plot, the stem plot for each circuit index represents the power at that circuit index. This figure shows the mixer spur plot.
For more information on mixer spur plots, see the plot reference page.
You can use the RF toolbox to generate Polar plots and Smith charts. If you specify two or more parameters, the RF toolbox puts the parameters in a single plot.
The following table describes the Polar plot and Smith charts options, as well as the available parameters.
LS11, LS12, LS21, and LS22 are large-signal S-parameters. You can plot these parameters as a function of input power or as a function of frequency.
LS11, LS12, LS21, LS22 (Objects with data from a P2D file only)
Z Smith chart
smithplot with type argument set to 'z'
LS11, LS22 (Objects with data from a P2D file only)
Y Smith chart
smithplot with type argument set to 'y'
smithplot with type argument set to 'zy'
By default, the RF toolbox plots the parameter as a function of frequency. When you import block data from a .p2d or .s2d file, you can also plot parameters as a function of any operating condition from the file that has numeric values, such as bias.
The circle method lets you place circles on a Smith® Chart to depict stability regions and display constant gain, noise figure, reflection and immittance circles. For more information about this function, see the circle reference page or Designing Matching Networks for Low Noise Amplifiers example about designing matching networks.
The RF toolbox lets you compute and plot time-domain characteristics for RF components.
You use the s2tf function to convert 2-port S-parameters to a transfer function. The function returns a vector of transfer function values that represent the normalized voltage gain of a 2-port network.
The following code illustrates how to read a file data into a passive circuit object, extract the 2-port S-parameters from the object, and compute the transfer function of the data at the frequencies for which the data is specified. Here z0 is the reference impedance of the S-parameters, zs is the source impedance, and zl is the load impedance. See the s2tf reference page for more information on how these impedances are used to define the gain.
PassiveCkt = rfckt.passive('File','passive.s2p')
z0=50; zs=50; zl=50;
[SParams, Freq] = extract(PassiveCkt, 'S Parameters', z0);
TransFunc = s2tf(SParams, z0, zs, zl);
You use the rationalfit function to fit a rational function to the transfer function of a passive component. The rationalfit function returns an rfmodel object that represents the transfer function analytically.
The following code illustrates how to use the rationalfit function to create an rfmodel.rational object that contains a rational function model of the transfer function that you created in the previous example.
RationalFunc = rationalfit(Freq, TransFunc)
To find out how many poles the RF toolbox used to represent the data, look at the length of the A vector of the RationalFunc model object.
nPoles = length(RationalFunc.A)
The number of poles is important if you plan to use the RF model object to create a model for use in another simulator, because a large number of poles can increase simulation time. For information on how to represent a component accurately using a minimum number of poles, see Represent Circuit Object with Model Object.
Use the freqresp function to compute the frequency response of the fitted data. To validate the model fit, plot the transfer function of the original data and the frequency response of the fitted data.
Resp = freqresp(RationalFunc, Freq);
plot(Freq, 20*log10(abs(TransFunc)), 'r', ...
Freq, 20*log10(abs(Resp)), 'b--');
ylabel('Magnitude of H(s) (decibels)');
legend('Original', 'Fitting result');
title(['Rational fitting with ', int2str(nPoles), ' poles']);
You use the timeresp function to compute the time-domain response of the transfer function that RationalFunc represents. This code illustrates how to create a random input signal, compute the time-domain response of RationalFunc to the input signal, and plot the results.
SampleTime=1e-11;
NumberOfSamples=4750;
InputTime = double((1:NumberOfSamples)')*SampleTime;
InputSignal = ...
sign(randn(1, ceil(NumberOfSamples/OverSamplingFactor)));
[tresp,t]=timeresp(RationalFunc,InputSignal,SampleTime);
plot(t*1e9,tresp);
title('Fitting Time-Domain Response', 'fonts', 12);
ylabel('Response to Random Input Signal');
For more information about computing the time response of a model object, see the timeresp function.
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Hessenberg Form - Maple Help
Home : Support : Online Help : Mathematics : Linear Algebra : LinearAlgebra Package : Generic Subpackage : Hessenberg Form
compute the Hessenberg form of a square Matrix
HessenbergForm[F](A)
HessenbergForm[F](A,output=out)
a table or module, the domain of computation, a field
square Matrix of values in F
one of H, U or a list containing one or more of these names
HessenbergForm[F](A) returns the upper Hessenberg form H of A.
Given an n x n Matrix A of elements in a field F, the algorithm converts a copy of A into upper Hessenberg form H using O(n^3) operations in F. The algorithm requires that F be a field and should only be used if F is finite as there is severe expression swell in computing H.
\mathrm{with}\left(\mathrm{LinearAlgebra}[\mathrm{Generic}]\right):
Q[\mathrm{`0`}],Q[\mathrm{`1`}],Q[\mathrm{`+`}],Q[\mathrm{`-`}],Q[\mathrm{`*`}],Q[\mathrm{`/`}],Q[\mathrm{`=`}]≔0,1,\mathrm{`+`},\mathrm{`-`},\mathrm{`*`},\mathrm{`/`},\mathrm{`=`}:
A≔\mathrm{Matrix}\left([[2,-7,-3,4],[1,-3,-4,5],[-7,10,5,-7],[-7,10,5,-7]]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{-7}& \textcolor[rgb]{0,0,1}{-3}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{-3}& \textcolor[rgb]{0,0,1}{-4}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{-7}& \textcolor[rgb]{0,0,1}{10}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{-7}\\ \textcolor[rgb]{0,0,1}{-7}& \textcolor[rgb]{0,0,1}{10}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{-7}\end{array}]
H≔\mathrm{HessenbergForm}[Q]\left(A\right)
\textcolor[rgb]{0,0,1}{H}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{-14}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{-10}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-46}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{28}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}]
H,U≔\mathrm{HessenbergForm}[Q]\left(A,\mathrm{output}=['H','U']\right)
\textcolor[rgb]{0,0,1}{H}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{U}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{-14}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{-10}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-46}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{28}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \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}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{7}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-1}& \textcolor[rgb]{0,0,1}{1}\end{array}]
\mathrm{MatrixMatrixMultiply}[Q]\left(\mathrm{MatrixMatrixMultiply}[Q]\left(U,A\right),\mathrm{MatrixInverse}[Q]\left(U\right)\right)
[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{-14}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{-10}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-46}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{28}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}]
LinearAlgebra[Generic][HessenbergAlgorithm]
LinearAlgebra[Generic][MatrixMatrixMultiply]
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Create simple lossy transmission line model - MATLAB - MathWorks Italia
serdes.ChannelLoss
VoltageSwingIdeal
EnableCrosstalk
CrosstalkSpecification
FEXTICN
NEXTICN
Processing Ideal Sinusoid Using ChannelLoss Model
Creating Transmission Line Model
Create simple lossy transmission line model
The serdes.ChannelLoss System object™ constructs a lossy transmission line model for use in the SerDes Designer app and other exported Simulink® models in the SerDes Toolbox™. For more information, see Analog Channel Loss in SerDes System.
To construct the loss model from channel loss metric:
Create the serdes.ChannelLoss object and set its properties.
ChannelLoss = serdes.ChannelLoss
ChannelLoss = serdes.ChannelLoss(Name,Value)
ChannelLoss = serdes.ChannelLoss returns a ChannelLoss object that modifies an input waveform with a lossy printed circuit board transmission line model according to the method outlined in [1].
ChannelLoss = serdes.ChannelLoss(Name,Value) sets properties using one or more name-value pairs. Enclose each property name in quotes. Unspecified properties have default values.
Example: ChannelLoss = serdes.ChannelLoss('Loss',5,'TargetFrequency',14e9) returns a ChannelLoss object that has a channel loss of 5 dB at 14 GHz.
Loss — Channel power loss at target frequency
Channel loss at the target frequency, specified as a real scalar in dB.
TargetFrequency — Frequency of desired channel loss
1e10 (default) | positive real scalar
Frequency for the desired channel loss, specified as a positive real scalar in Hz.
dt — Sample interval
Sample interval in s, specified as a positive real scalar.
Zc — Differential characteristic impedance
Differential characteristic impedance of the channel, specified as a positive real scalar in ohms.
TxC — Single-ended capacitance of transmitter analog model
Single-ended capacitance of the transmitter analog model, specified as a nonnegative real scalar in farads.
RiseTime — 20%−80% rise time of stimulus input
VoltageSwingIdeal — Peak-to-peak voltage at input of transmitter analog model
EnableCrosstalk — Include crosstalk in simulation
Set EnableCrosstalk to true to include crosstalk in the simulation. By default, EnableCrosstalk is set to false.
CrosstalkSpecification — Specify magnitude of near and far end aggressors
CEI-28G-SR (default) | CEI-25G-LR | CEI-28G-VSR | 100GBASE-CR4 | Custom
fb — Baud rate for ICN calculation
Baud rate used for integrated crosstalk noise (ICN) calculation, specified as a positive real scalar in hertz. fb is the inverse of the symbol time.
This property is only available when EnableCrosstalk is set to true.
FEXTICN — Desired integrated noise level of far end aggressor
This property is only available when EnableCrosstalk is set to true and CrosstalkSpecification is set to Custom.
Aft — Amplitude factor of far end crosstalk aggressor
Amplitude factor of the far end crosstalk aggressor, specified as a positive real scalar in volts.
Tft — Rise time of far end crosstalk aggressor
Rise time of the far end crosstalk aggressor, specified as a positive real scalar in seconds.
NEXTICN — Desired integrated noise level of near end aggressor
Ant — Amplitude factor of near end crosstalk aggressor
Rise time of the near end crosstalk aggressor, specified as a positive real scalar in seconds.
Tnt — Rise time of near end crosstalk aggressor
y = ChannelLoss(x)
Estimated channel output that includes the effect of a lossy printed circuit board transmission line model according to the method outlined in Analog Channel Loss in SerDes System.
This example shows how to process an ideal sinusoidal input waveform with the ChannelLoss model and check that it modifies the amplitude of the waveform in a reasonable way.
Define the system parameters. Use a symbol time of 100 ps with 8 samples per symbol. The amplitude of the input signal is 1 V. The channel loss is 3 dB.
Calculate the sample interval. Define a time vector that is 30 symbols long.
t = (0:SamplesPerSymbol*30)*dt;
Create the sinusoidal input waveform.
F = 1/SymbolTime/2; %Fundamental frequency
inputWave = a0*sin(2*pi*F*t);
Create the channelModel object at the specified loss for near ideal transmitter and receiver termination.
channelModel = serdes.ChannelLoss('Loss',Loss,'dt',dt,...
'TargetFrequency',F,'TxR',50,'TxC',1e-14,...
'RxR',50,'RxC',1e-14);
Process the input waveform using the channelModel object.
outputWave = channelModel(inputWave);
Calculate the output amplitudes.
a1 = max(outputWave); %Output amplitude
aideal = a0*10^(-abs(channelModel.Loss)/20); %Theoretical output amplitude
Generate the frequency response.
s21 = channelModel.s21;
f = (0:length(s21)-1)*channelModel.dF;
Determine the loss at the target frequency of the frequency response.
f1 = find(f>channelModel.TargetFrequency,1,'first');
LossAtTarget = interp1(f(f1-1:f1),db(s21(f1-1:f1)),channelModel.TargetFrequency);
Plot the time and frequency response of the channel model.
tns = t*1e9;
thline = [tns(1),tns(end)];
fghz = f*1e-9;
plot(tns,outputWave,thline,aideal*[1 1],thline,a1*[1 1],'b--'),
xlabel('ns'),ylabel('Voltage')
title('Time Response of Channel Model')
legend('Output waveform',...
sprintf('Ideal amplitude: %g mV',round(aideal*1e3)),...
sprintf('Actual amplitude: %g mV',round(a1*1e3)),'Location','southwest')
plot(fghz,db(s21),...
channelModel.TargetFrequency*1e-9,LossAtTarget,'o')
title('Frequency Response of Channel Model')
legend('S_{21}(f)',sprintf('%g dB @ %g GHz',LossAtTarget,channelModel.TargetFrequency*1e-9))
xlabel('GHz')
\begin{array}{l}{W}_{ft}\left(f\right)=\left(\frac{{A}_{ft}^{2}}{4{f}_{b}}\right){\text{sinc}}^{2}\left(\frac{f}{{f}_{b}}\right)\left[\frac{1}{1+{\left(f/{f}_{ft}\right)}^{4}}\right]\left[\frac{1}{1+{\left(f/{f}_{ft}\right)}^{8}}\right]\\ {\sigma }_{fx}={\left(2\Delta f{\sum _{n}{W}_{ft}\left({f}_{n}\right)10}^{-MDFEX{T}_{loss}\left({f}_{n}\right)/10}\right)}^{1/2}\\ {\sigma }_{nx}={\left(2\Delta f{\sum _{n}{W}_{ft}\left({f}_{n}\right)10}^{-MDNEX{T}_{loss}\left({f}_{n}\right)/10}\right)}^{1/2}\\ {\sigma }_{x}=\sqrt{{\sigma }_{fx}^{2}+{\sigma }_{nx}^{2}}\end{array}
To obtain a lossy printed circuit board (PCB) transmission line (T-line) model with a given Loss at the TargetFrequency, two T-lines of length 100 mm and 150 mm are created and loss evaluated at the Target Frequency. These two data points are used to extrapolate to the transmission line length needed to achieve the requested loss. The transmission line model is a analytic equation based on the method described in [1].
This transmission line, with the requested loss is then combined with the Tx and Rx single ended termination resistance and capacitance as illustrated below:
{I}_{\text{FEXT}}\left(t\right)={k}_{\text{FEXT}}\frac{dI\left(t\right)}{dt}
{H}_{\text{FEXT}}\left(f\right)=ℱ\left[{I}_{\text{FEXT}}\left(t\right)\right]
The magnitude of the scale factor kFEXT is:
{k}_{\text{FEXT}}\left(f\right)=-\frac{IC{N}_{\text{FEXT}}}{\mathbb{I}ℂ{ℕ}_{\text{FEXT}}\left({H}_{\text{FEXT}}\left(f\right)\right)}
\mathbb{I}ℂℕ
is the integrated crosstalk noise operator.
{H}_{\text{NEXT}}\left(f\right)={k}_{\text{NEXT}}·{S}_{11}\left(f\right)
Then the scale factor kNEXT is:
{k}_{\text{NEXT}}=-\frac{IC{N}_{\text{NEXT}}}{\mathbb{I}ℂℕ\left({S}_{11}\left(f\right)\right)}
{I}_{\text{NEXT}}\left(t\right)={ℱ}^{-1}\left[{k}_{\text{NEXT}}·{S}_{11}\left(f\right)\right]
[1] IEEE 802.3bj-2014. "IEEE Standard for Ethernet Amendment 2: Physical Layer Specifications and Management Parameters for 100 Gb/s Operation Over Backplanes and Copper Cables." https://standards.ieee.org/standard/802_3bj-2014.html.
Analog Channel | Configuration | SerDes Designer
|
§ Exact sequence of pointed sets
This was a shower thought. I don't even if these form an abelian category. Let's assume we have pointed sets, where every set has a distinguished element
*
p
will be analogous to the zero of an abelian group. We will also allow multi-functions, where a function can have multiple outputs. Now let's consider two sets,
A, B
along with their 'smash union'
A \vee B
where we take the disjoint union of
A, B
with a smashed
*
. To be very formal:
A \vee B = \{0\} \times (A - \{ * \}) \cup \{1\}\times (B - \{ * \}) \cup \{ * \}
We now consider the exact sequence:
(A \cap B, *) \xrightarrow{\Delta} (A \vee B, *) \xrightarrow{\pi} (A \cup B, *)
with the maps as:
\begin{aligned} &ab \in A \cap B \xmapsto{\Delta} (0, ab), (1, ab) \in A \vee B \\ &(0, a) \in A \vee B \xmapsto{\pi} \begin{cases} * & \text{if } a \in B \\ a &\text{otherwise} \\ \end{cases} \\ &(1, b) \in A \vee B \xmapsto{\pi} \begin{cases} * & \text{if } b \in A \\ b &\text{otherwise} \\ \end{cases} \\ \end{aligned}
\Delta
is a multi-function, because it produces as output both
(0, ab)
(1, ab)
\ker(\pi) = \pi^{-1}(*) = \{ (0, a) : a \in B \} \cup \{ (1, b) : b \in A \}
Since it's tagged
(0, a)
a \in A
b \in B
Hence, write
\ker(\pi) = \{ (0, ab), (1, ab) : ab \in A \cap B \} = im(\Delta)
This exact sequence also naturally motivates one to consider
A \cup B - A \cap B = A \Delta B
, the symmetric difference. It also gives the nice counting formula
|A \vee B| = |A \cap B| + |A \cup B|
, also known as inclusion-exclusion. I wonder if it's possible to recover incidence algebraic derivations from this formuation?
§ Variation on the theme: direct product
This version seems wrong to me, but I can't tell what's wrong. Writing it down:
\begin{aligned} (A \cap B, *) \xrightarrow{\Delta} (A \times B, (*, *)) \xrightarrow{\pi} (A \cup B, *) \end{aligned}
\begin{aligned} &ab \in A \cap B \xmapsto{\Delta} (ab, ab) \in A \times B \\ &(a, b) \in A \times B \xmapsto{\pi} \begin{cases} * & \text{if } a = b \\ a, b &\text{otherwise} \\ \end{cases} \\ \end{aligned}
A \cap B \xrightarrow{\Delta} A \times B
is injective
A \cap B \xrightarrow{\pi} A \cup B
is surjective
ker(\pi) = \pi^{-1}(*) = \{ (a, b) : a \in A, b \in B, a = b \} = im(\Delta)
Note that to get the last equivalence, we do not consider elements like
\pi(a, *) = a, *
to be a pre-image of
*
, because they don't exact ly map into
*
[pun intended ].
|
Visualization of Different Flashback Mechanisms for H2/CH4 Mixtures in a Variable-Swirl Burner | J. Eng. Gas Turbines Power | ASME Digital Collection
Visualization of Different Flashback Mechanisms for H2/CH4 Mixtures in a Variable-Swirl Burner
Parisa Sayad,
Ole Römers Vägen1
e-mail: parisa.sayad@energy.lth.se
Alessandro Schönborn,
e-mail: alessandro.schonborn@energy.lth.se
e-mail: mao.li@energy.lth.se
e-mail: Jens.klingmann@energy.lth.se
Contributed by the Combustion and Fuels Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received July 24, 2014; final manuscript received July 29, 2014; published online October 7, 2014. Editor: David Wisler.
Sayad, P., Schönborn, A., Li, M., and Klingmann, J. (October 7, 2014). "Visualization of Different Flashback Mechanisms for H2/CH4 Mixtures in a Variable-Swirl Burner." ASME. J. Eng. Gas Turbines Power. March 2015; 137(3): 031507. https://doi.org/10.1115/1.4028436
Flame flashback from the combustion chamber to the premixing section is a major operability issue when using high H2 content fuels in lean premixed combustors. Depending on the flow-field in the combustor, flashback can be triggered by different mechanisms. In this work, three flashback mechanisms of H2/CH4 mixtures were visualized in an atmospheric variable-swirl burner using high speed OH* chemiluminescence imaging. The H2 mole fraction of the tested fuel mixtures varied between 0.1 and 0.9. The flow-field in the combustor was varied by changing the swirl number from 0.0 to 0.66 and the total air mass-flow rate from 75 to 200 SLPM (standard liters per minute). The following three types of flashback mechanism were observed: Flashback caused by combustion induced vortex breakdown (CIVB) occurred at swirl numbers
≥
0.53 for all of the tested fuel mixtures. Flashback in the boundary layer (BL) and flame propagation in the premixing tube caused by auto-ignition were observed at low swirl numbers and low total air mass-flow rates. The temporal and spatial propagation of the flame in the optical section of the premixing tube during flashback was studied and flashback speed for different mechanisms was estimated. The flame propagation speed during flashback was significantly different for the different mechanisms.
Combustion, Energy, Environmental, Experimental, Flame, Fuel cell applications, Fuel combustion, Fuels , Gas turbine technology, Swirling, Turbulence, Visualization, Combustion
Combustion chambers, Flames, Flow (Dynamics), Fuels, Ignition, Visualization
Intermittent Renewable Energy: The Only Future Source of Hydrogen
,” National Renewable Energy Laboratory, Golden, CO, Technical Report No. NREL/TP-5600-51995.
Establishing Operating Limits in a Commercial Lean Premixed Combustor Operating on Synthesis Gas Pertaining to Flashback and Blowout
Rapid Flame Propagation in a Vortex Tube in Perspective of Vortex Breakdown Phenomena
Flame Propagation in Swirling Flows-Effects of Local Extinction on the Combustion Induced Vortex Breakdown
Combustion, Flames and Explosion of Gases
Quenching, Flash-Back, Blow-Off-Theory and Experiment
An Experimental Investigation on the Stability of Turbulent Burner Flames
, “Flashback Phenomena Associated With Lean Premixed Syngas Combustion at Gas Turbine Like Conditions,” Processes and Technologies for a Sustainable Energy (
PTSE 2010
, Italy, June 27–30.10.4405/ptse2010.II3
Review of Flashback Reported in Prevaporizing/Premixing Combustors
OH*-Chemiluminescence During Autoignition of Hydrogen and Air in a Pressurised Flow Reactor
Combustion in Swirling Flows: A Review
|
§ Min cost flow (TODO)
Problem statement: Find a maximal flow with minimum cost.
Find max flow.
Find negative cost cycle in residual graph of max flow. Push flow around the negative cost cycle.
§ Relation between max flow and min cost circulation
Recall that min cost circulation asks to compute a circulation with minimum cost [no maximality constraint ].
Given a flow network
(V, E, s, t, C)
C
is capacity fn), create a new cost function
c: V \to \mathbb R
which assigns cost zero to all edges in the flow networ. Also add a new edge
t \to s
which has infinite capacity, cost
-1
A circulation with cost lower than zero will have to use the
t \to s
edge. To get minimum cost, it must send as much flow through this edge as possible. For it to be a circulation, the full flow in the network must be zero. So suppose we send
f
units of flow back from
to
s
. Then we must send
f
units of flow from
s
t
for it to be a circulation. Incrasing
f
(max flow) decreases the cost of the circluation! Thus, max flow is reduced to min cost circulation.
§ Min Cost Flow in general
First find max flow using whatever.
Next, we need to find negative cost cycle in the residual graph.
Use bellman ford, or SPFA to find negative cost cycles in
O(VE)
time [run edge relaxation
|V|
times ].
§ Minimum mean cycle
Which is best cycle to push flow around to reduce cost? The min cost cycle may not be best, since it may have very little capacity.
A negative cycle with max capacity may not have good cost.
Correct: total cost/number of edges --- that is, the mean cost.
§ shortest path as circulation.
Need to find single source shortest path in a graph (with possibly negative edges, no negative cycles).
We have a balance at each vertex
v
, which tells us how much extra flow must can have coming in versus going out. So,
\sum_u f(l \to v) - \sum_w f(v \to r) = b(v)
. Intuitively, the balance is stored in a tank at the vertex.
We need total balance to be zero.
We set the source
s
to have balance
1-v
(supply) and all the other nodes to have balance
1
(demand).
Let the cost of each edge be the distance, let the capacity of each edge be infinite.
Now, what is a min cost flow which obeys the demands?
Consider the shortest path tree. Imagine it as carrying a flow. Then the shortest path tree indeed obeys the flow constraints.
To convert this into circulation, add back edges from each node back to the source, with a capacity of 1, cost of zero.
This converts shortest path trees into flows/circulations.
§ Min cost circulation algorithms
Old algorithm: start with a circulation that obeys balance, then push more around (by using negative cycles)
New algorithm (successive shortest path): remove all negative cycles, then restore balance constraints.
how to remove negative cycles? We can just send flow down all negative edges. The resdiual graph will contain no negative cycles. (NOTE: we don't have a valid flow at this point!) This leaves us with resdiual balances at each vertex, about how much more flow we need to send.
Jeff E: algorithms video
|
Proximity relations for real rank one valuations dominating a local regular ring | EMS Press
We study 0-dimensional real rank one valuations centered in a regular local ring of dimension
n\geq 2
such that the associated valuation ring can be obtained from the regular ring by a sequence of quadratic transforms. We define two classical invariants associated to the valuation (the refined proximity matrix and the multiplicity sequence) and we show that are equivalent data of the valuation.
Ángel Granja, Cristina Rodríguez, Proximity relations for real rank one valuations dominating a local regular ring. Rev. Mat. Iberoam. 19 (2003), no. 2, pp. 393–412
|
Complex Numbers And Quadratic Equations, Popular Questions: Karnataka Class 11-commerce MATH, Mathematics Puc I - Meritnation
Jonathan Munluah asked a question
Find the square root of 1+2i
11vibha11... asked a question
If x^2 - 63x - 64 = 0 and p and n are integers such that p^n = x which of the following CANNOT be a value for p?
Aishani Pandey asked a question
Raghav Kejriwal asked a question
if a+b=10 then the value of a^3+b^3+30ab is
Prove that (1 + i)5 (1 + 1/i) 5 = 32
Jyoti Jha asked a question
determine real x and y for this equation- (3-4i)(x+yi)=1+0i
Elvin Kookabura asked a question
Varun Venkat asked a question
if z = x+iy and z^2 = a+ib where a,b,x,y are real numbers, show that 2x^2 = √a^2+b^2 + a
Venkata Ram asked a question
for what values of x and y are the nos. 3+ix^2y and x^2 + y+ 4i conjugate complexes? plz help..!!
Khilraj Solanki asked a question
ALL CHAPTERS CLASS 11 IMPORTANT FORMULAS
Plz solve these questions 23,24(b),27,29,30,31,32
Kumar Shree Rishikesh & 1 other asked a question
(3+i)x +(1-2i)y+7i =0
Riya Jeprish asked a question
Pls solve 23
{\mathrm{log}}_{a}x
{\mathrm{log}}_{a}3
= 0.4, then
{\mathrm{log}}_{3}x
Binalika Jidung asked a question
Mukul Agrawal asked a question
if x-iy=underroot [(a-ib)/(c-id)]
prove that(x2+y2)2= a2+b2/c2+d2
Chayanikabiswas asked a question
Prove that 2 complex numbers a+ib and c +id are equal if a =c and b=d.
Jashan Jot asked a question
|z1|=1 ,|z2|=2 ,|z3|=3 and |9z1z2 + 4z1z3 + z2z3|=12 then find |z1 + z2 + z3|
Find the value of a for which one root of the quadratic equation (a2-5a+3)x2+(3a-1)x+2=0 is twice as large as the other.How can we eliminate alpha here? Please solve the problem?
if z1,z2 are complex numbers such that |(z1-3z2)+(3-z1z2*)|=1 and |z2| is not equal to 1 then find |z1|{ Here * means conjugate of}Experts please don't give links and answer with detailed steps
Solve :- 2x2 - (3+7i) x - (3-9i) = 0
Find the modulus : (i^25)+(1+3i^3)
if (x+iy)(2-3i)=4+i then find (x,y)
if w is cube root of unity , then prove that
( x - y ) ( xw - y ) (xw2 - y ) = x3 - y3
Solve :- x2 -(7 - i) x + (18 - i) = 0 over C.
Prbiswajit asked a question
how is the multiplicative inverse of complex number z = a+ib is a/a2+b2 + ib/ a2b2
If x + iy = a + ib/a- ib, show that x2+y2=1
Jyoti Maurya asked a question
Find the square root of -48-14i.
Kanishk asked a question
If x = 2 + 22/3+ 21/3, then find the value of x3 - 6x2 + 6x.
if |z-(3+4i)|<=3 then find the complex number having least magnitude satisfying the above inequality
sreeejababu... asked a question
if Z is a complex number such that Z-1 / Z+1 is purely imaginary.prove that |Z|=1
Nandhitha K asked a question
Find the complex conjugate of i root-9+7i divided by1+root of root-1.
if the sum of the roots of the equation ax2 + bx + c = 0 is equal to the sum of the squares of their squares of their reciprocals , then show that bc2 , ca2 , ab2 are in A.P.
if a2 + b2 =1 , then find the value of 1+b+ia/1+b-ia
find solution of the equation x2 + 5=0 in complex no's
if a is not equal to b and a^2=5a -3 , b^2=5b-3, then form the equation whose roots are a/b and b/a
Convert into a+ib form 1/1+cos theta-i sin theta
let alpha and beta are the roots of x2-6x-2=0,with alpha beta . if an=alphan-betan for n=(greater than or equal to one)1,then value of a10-2a8/ 2a9 is??
Find the modulus of i^25 +(1+3i)^3.
Experts please dont give links and answer with detailed steps
if a+ib= c+i / c-i , where c is real part
a square + b square =1
and b / a =2c / c square-1
Paritosh Indalia asked a question
\left|\begin{array}{ccc}6\mathrm{i}& -3\mathrm{i}& 1\\ 2& 3& \mathrm{i}\\ 4& 3\mathrm{i}& -1\end{array}\right|=\mathrm{x}+\mathrm{iy}, \mathrm{then}
(1) x = y = 1 (2) x = y = 0
(3) x = 3, y = 1 (4) x = – 3, y = – 1
Mukul Agrawal & 1 other asked a question
Q. If p+iq = (a-i)2 / 2a-i , show that p2+q2 = (a2+1)2 / 4a2+1.
convert -16/1+iroot3 to polar form
For the quadratic equation ax2+bx+c+0, find the condition that
(i) one root is reciprocal of other root
(ii) one root is m times the other root
(iii) one root is square of the other root
(iv) one root is nth power of the other root
(v) the roots are in the ratio m:n
Vinayak Suresh asked a question
If (5+6i)/(3+4i)=a+ib find a and b
Krishna Kanahiya asked a question
Find two consecutive numbers, whose squares have the sum 85.
how to find the multiplicative inverse of 2-3i
find the complex numbers Z satisfying the equation:- |Z-4| / |Z-8|=1 & |Z-12 | /|Z-8i|=5/3
Express it in the polar form: (i-1) / (cospi/3) + (isin pi/3).Also Find the arguement and modulus.
(3-4i)(x+iy). Find the value of x and y
R. Vinodhini & 1 other asked a question
find the smallest positive integer n for which (1+i)^2n =(1-i)^2n
a,b,c are three distinct real numbers and they are in G.P. If a+b+c = xb, then prove that x<-1 or x>3.
If z = x+iy and w = 1 - i2 / z-i
Show that mod w = 1 implies z is purely real.
2x3+2x2-7x+72,when x=(3-5i)/2
if z= 3- 4i , then z4 - 3z3 +3z2+99z -95 is equal to
if a =cosA+isinA,find the value of (1+a)/(1-a)
[2+5w+2w2}6={2+2w+5w2}6=729
Pulkit Saboo asked a question
PROVE THAT A REAL VALUE OF x WILL SATISFY THE EQUATION 1 - ix / 1 + ix = a - ib , if a2 + b2 =1 ; where 'a' and 'b' are real.
cosx + cos2x +cos3x+cos4x / sinx+sin2x+sin3x+sin4x =cot5/2 x
Show that the roots of (x-b)(x-c) +(x-c)(x-a) +(x-a)(x-b) =0 are real, and that they cannot be equal unless a=b=c.
Find the value of {i}107 + {i]112 + {i}117 + {i}122
(107,112,117,122 are in the power of i )
simran_1997... asked a question
solve :(2 + i)x2 - (5 -i)x + 2(1-i) = 0
IF one root of x2 + px + q = 0 be square of the other, Then prove p3 + q2 +q = 3 pq
If (1+i/1-i)3 - (1-i/1+i)3 = x+iy, then find (x,y)
two roots of a biquadratic x4-18x3+kx2 +200x-1984=0
have their product equal to(-32).find the value of k.
Q. the value of 'b' for which equations
x2+bx-1=0
x2+x+b=0
have one root in common is ??
if |z1+z2|>|z1-z2| then prove that -pi/2<arg(z1/z2)<pi/2
Rivitha asked a question
if a+ib =( (x+i)2) / (2x2 + 1)
prove that a2 +b2 =((x2 + 1)2) / (2x2 +1)2
Harshvardhan Goplani asked a question
Represent the following complex number in polar form and Euler's form!
Harshitha & 1 other asked a question
Provide R.D Sharma solutions of class 11 as it is a common demand by many students
Amlan Moharana asked a question
arg(z conjugate)=-arg(z)
If the roots of the equation x2-2ax+a2+a-3 =0 are less than 3 then find the set of all possible values of a. (ans : -infinity, 2)
Express the following in the form of (a+ib):
1. (3+2i)(2+3i)/(1+2i)(2-i)
Express 5+i root 2 / 2i in the form of x+iy
Express (a+ib)3/a-ib - (a-ib)3/a+ib in the form a+ib.
Divyanshika Kumar asked a question
If a+ib = c+i / c-i, where c is real, prove that a2 + b2 =1 and b/a = 2c / c2 -1.
Monalisa Majumdar asked a question
Find the value of p for which the quadratic equation x^2 - px + p+ 3=0 has a) coincident roots b) real distinct roots c) one positive and one negative root. Please solve ☺😊😇😄
If iZ3+ Z2 - Z + i = 0 ' then show that mode of Z = 1
Arrange the following in the increasing order of their bulk modulus
Air Steel Water
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Quantization of Drinfeld Zastava in type $A$ | EMS Press
Quantization of Drinfeld Zastava in type
A
Leonid Rybnikov
Drinfeld Zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of the affine Lie algebra
\hat{sl}_n
. We introduce an affine, reduced, irreducible, normal quiver variety
Z
which maps to the Zastava space bijectively at the level of complex points. The natural Poisson structure on the Zastava space can be described on
Z
in terms of Hamiltonian reduction of a certain Poisson subvariety of the dual space of a (nonsemisimple) Lie algebra. The quantum Hamiltonian reduction of the corresponding quotient of its universal enveloping algebra produces a quantization
Y
of the coordinate ring of
Z
. The same quantization was obtained in the finite (as opposed to the affine) case generically in [4]. We prove that, for generic values of quantization parameters,
Y
is a quotient of the affine Borel Yangian.
Michael Finkelberg, Leonid Rybnikov, Quantization of Drinfeld Zastava in type
A
. J. Eur. Math. Soc. 16 (2014), no. 2, pp. 235–271
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25m2-110m+121/5m-11 - Maths - Factorisation - 8825985 | Meritnation.com
25m2-110m+121/5m-11
\frac{25{\mathrm{m}}^{2}-110\mathrm{m}+121}{5\mathrm{m}-11}=\frac{{\left(5\mathrm{m}\right)}^{2}-2×\left(5\mathrm{m}\right)×\left(11\right)+{\left(11\right)}^{2}}{5\mathrm{m}-11}\phantom{\rule{0ex}{0ex}} =\frac{{\left(5\mathrm{m}-11\right)}^{2}}{5\mathrm{m}-11} \left[\mathrm{Usedentity}, {\mathrm{a}}^{2}-2\mathrm{ab}+{\mathrm{b}}^{2}={\left(\mathrm{a}-\mathrm{b}\right)}^{2}\right]\phantom{\rule{0ex}{0ex}} =5\mathrm{m}-11\phantom{\rule{0ex}{0ex}}\mathrm{Hence}, \frac{\mathbf{25}{\mathbf{m}}^{\mathbf{2}}\mathbf{-}\mathbf{110}\mathbf{m}\mathbf{+}\mathbf{121}}{\mathbf{5}\mathbf{m}\mathbf{-}\mathbf{11}}\mathbf{=}\mathbf{5}\mathbf{m}\mathbf{-}\mathbf{11}
5m+11. Please tell me is it right or not.
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Universal objects in categories of reproducing kernels | EMS Press
We continue our earlier investigation on generalized reproducing kernels, in connection with the complex geometry of
C^*
- algebra representations, by looking at them as the objects of an appropriate category. Thus the correspondence between reproducing
(-*)
-kernels and the associated Hilbert spaces of sections of vector bundles is made into a functor. We construct reproducing
(-*)
-kernels with universality properties with respect to the operation of pull-back. We show how completely positive maps can be regarded as pull-backs of universal ones linked to the tautological bundle over the Grassmann manifold of the Hilbert space
\ell^2(\mathbb{N})
Daniel Beltiţă, José E. Galé, Universal objects in categories of reproducing kernels. Rev. Mat. Iberoam. 27 (2011), no. 1, pp. 123–179
|
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