content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
Latent Diffusion Series: Variational Autoencoder (VAE)
In the Latent Diffusion Series of blog posts, I'm going through all components needed to train a latent diffusion model to generate random digits from the MNIST dataset. In the second post, we will
build and train a variational autoencoder to generate MNIST digits. The latent variables of these models are defined to be normally distributed, something that will later enable our diffusion model
operate in the latent space. For the other posts, please look below:
1. Variational Autoencoder (VAE)
2. Latent Diffusion Model
The links will become active as soon as they the posts are completed. Even though this blog post is part of a series, I will try my best to write it in such a way that it's not required to have read
the previous blog posts.
In this post I will introduce the Variational Autoencoder(VAE) model. We will train the VAE on the MNIST dataset and try to generate some digits by sampling the latent space. If you'd like a bit more
about the MNIST dataset, please look at the previous blog post on the MNIST Classifier, where I also explore the dataset and show how to easily use it in Pytorch. I have created a Python notebook on
Colab, which you can use to follow along and experiment with this post's code.
The Variational Autoencoder as the name suggests, is called an autoencoder because it resembles the traditional autoencoder model in the sense that the model is trained in such a way that its output
is made to resemble the input. The encoder part of the autoencoder means we encode the input into latent variables by reducing the dimensionality. This dimensionality reduction is also known as a
'bottleneck', and can additionally be seen as compression of the information that goes in.
Conceptually, we usually split the autoencoder in an encoder part which encodes the input into the latent space, and a decoder, which decodes the latent variable back into the input space. latent
space actually is. I will spare you the details, for that please check the Wikipedia link, but with latent space we generally mean a lower dimensional space which encodes the original data space,
with the latent variables being the coordinates of this space. This can also be viewed as data compression, just like image compression. — Getting back to our autoencoder, if you are particularly
creative, you might think of sampling these latent variables randomly and passing them to the decoder to generate outputs that resemble the inputs the model was trained on. Unfortunately, the
distribution of the latent variables in a traditional autoencoder is generally not known, so it's difficult to sample them and generate novel outputs without going out of distribution.
To see how bad the samples look like when generated by randomly sampling the autoencoder latent variable, let's first build and train a simple 2D autonecoder for the MNIST dataset.
Let's now build the autoencoder using Pytorch:
class ConvBlock(torch.nn.Module):
def __init__(self, fin, fout, *args, **kwargs):
super(ConvBlock, self).__init__()
self._conv = torch.nn.Conv2d(fin, fout, *args, **kwargs)
self._norm = torch.nn.BatchNorm2d(fout)
self._relu = torch.nn.LeakyReLU()
def forward(self, x):
return self._relu(self._norm(self._conv(x)))
class ConvEncoder(torch.nn.Module):
def __init__(self, features):
super(ConvEncoder, self).__init__()
layers = []
for i in range(len(features)-1):
fi = features[i]
fo = features[i+1]
if i > 0:
ConvBlock(fi, fo, 3, padding='same'),
ConvBlock(fo, fo, 3, padding='same'),
ConvBlock(fi, fo, 3, padding='same'),
ConvBlock(fo, fo, 3, padding='same'),
self.layers = torch.nn.ModuleList(layers)
def forward(self, x):
y = torch.clone(x)
for layer in self.layers:
y = layer(y)
return y
class ConvDecoder(torch.nn.Module):
def __init__(self, features):
super(ConvDecoder, self).__init__()
layers = []
for i in range(len(features)-1):
layer = []
fi = features[i]
fo = features[i+1]
if i > 0:
layer += [
ConvBlock(fi, fi, 3, padding='same'),
if i < len(features)-2:
layer += [
ConvBlock(fi, fi, 3, padding='same'),
ConvBlock(fi, fo, 3, padding='same'),
layer += [
ConvBlock(fi, fi, 3, padding='same'),
torch.nn.Conv2d(fi, fo, 1, padding='same'),
self.layers = torch.nn.ModuleList(layers)
def forward(self, x):
y = torch.clone(x)
for layer in self.layers:
y = layer(y)
return y
class ConvAutoencoder(torch.nn.Module):
def __init__(self, features):
super(ConvAutoencoder, self).__init__()
self.encoder = ConvEncoder(features)
self.decoder = ConvDecoder(features[::-1])
def forward(self, x):
return self.decoder(self.encoder(x))
You will notice that the code is a bit more generic than shown on the diagram. The number of levels are variable and are controlled by the features variable, which takes an array of the number of
features for each level.
To train the model, we use the mean squared error loss between the input and output of the autoencoder, and use Adam^2 as our optimizer of choice.
epochs = 400
batch_size = 128
learning_rate = 1e-3
i_log = 10
optimizer = torch.optim.Adam(autoencoder.parameters(), learning_rate)
num_batches = int(math.ceil(x_train.shape[0] / batch_size))
losses = []
for i in range(num_epochs):
train_ids = torch.randperm(x_train.shape[0])
average_loss = 0.0
for bid in range(num_batches):
with torch.no_grad():
batch_ids = train_ids[bid*batch_size:(bid+1)*batch_size]
x = x_train[batch_ids,None,...]
x = x.to(device)
x_pred = autoencoder(x)
loss = torch.sum((x_pred - x)**2, dim=[1,2,3])
loss = torch.mean(loss)
with torch.no_grad():
average_loss += loss.cpu().numpy() / num_batches
if (i + 1) % i_log == 0:
with torch.no_grad():
val_loss = 0.0
x_val_pred = autoencoder(x_val[:,None,...].to(device)).cpu()
val_loss = torch.sum(
(x_val_pred - x_val[:,None,...])**2,
val_loss = torch.mean(val_loss).numpy()
losses.append([average_loss, val_loss])
print(f'Epoch {i} loss = {average_loss}, val_loss = {val_loss}')
The training code is fairly standard, so there's not much to say here. We can plot the train and validation loss as a function of the epochs,
Given our trained autoencoder, let's now try to use it as a generative model. We first calculate the mean and standard deviation of the training dataset latent space:
with torch.no_grad():
num_batches = int(math.ceil(x_train.shape[0] / batch_size))
z_train = []
for bid in range(num_batches):
x = x_train[bid*batch_size:(bid+1)*batch_size,None,...]
x = x.to(device)
z = autoencoder.encoder(x)
z_train = torch.cat(z_train)
z_train_mean = torch.mean(z_train, dim=0, keepdim=True)
z_train_std = torch.std(z_train, dim=0, keepdim=True)
Then, we generate a bunch of digits by sampling from a normal distribution with these means and standard deviations:
n = 10
z = torch.normal(
z_train_mean.repeat(n, 1, 1, 1),
z_train_std.repeat(n, 1, 1, 1)
x_pred = autoencoder.decoder(z)
x_pred = torch.clip(x_pred, 0.0, 1.0)
Finally, here are some digits generated this way:
Summarizing, the big issues of vanilla autoencoders is that the resulting latent spaces are disjoint and have discontinuities, i.e. areas of low sample density. To showcase this, I used linear
discriminant analysis to reduce the latent dimensionality to two dimensions while maximizing the separation of the different categories. This can easily be done using scikit-learn:
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis
clf = LinearDiscriminantAnalysis(n_components=2)
# Here z is the latent variables and y_val are the digit labels.
zr = clf.fit_transform(z.reshape((z.shape[0], -1)), y_val)
Plotting this reduced representation, the first issue, of a disjoint latent space, is evident.
Variational Autoencoder
The authors of the original VAE paper^2 took a different approach at tackling the issues of generating samples \(\mathbf{x}\) from latent variable \(\mathbf{z}\). More specifically, they took a
statistical approach, approximating the posterior distribution \(p(\mathbf{z}|\mathbf{x})\) using variational inference.
As Welling et al did in their paper^2, let us define some dataset \(\mathbf{X} = \left\{ \mathbf{x}^{(i)} \right\}_{i=1}^N\) consisting of N i.i.d. samples of some continuous or discrete variable \(\
mathbf{x}\), and let us assume that the data are generated by some random process, involving an unobserved continuous random variable \(\mathbf{z}\). The random (generative) process, consists of two
steps: first, a value \(\mathbf{z}\) is generated from some prior distribution \(p(\mathbf{z})\), and then a value \(\mathbf{x}\) is generated from some conditional distribution \(p(\mathbf{x}|\
mathbf{z})\). To implement this process, we need to calculate \(p(\mathbf{x}|\mathbf{z})\). Alternatively, we can view the problem as finding out the possible latent (hidden) variables that generated
a data point. From Baye's theorem, we can also calculate the probability \(p(\mathbf{x}|\mathbf{z})\) as, \[ p(\mathbf{x}|\mathbf{z}) = \frac{p(\mathbf{z}|\mathbf{x}) p(\mathbf{x})}{p(\mathbf{z})} \]
Unfortunately, the marginal likelihood (evidence) \(p(\mathbf{x}) = \int d\mathbf{z} p(\mathbf{z}) p(\mathbf{x} | \mathbf{z}) \) is generally intractable. This is where variational inference^3 is
used to approximate the posterior \(p(\mathbf{z}|\mathbf{x})\) with another, tractable distribution, \(q(\mathbf{z}|\mathbf{x})\). We can define the distribution \(q\) as a model with parameters \(\
theta\); \( q_{\theta} ( \mathbf{z}|\mathbf{x} ) \). We call this model the encoder, and we can calculate its parameters using maximum likelihood estimation. In practice, because maximizing the
likelihood \(p(\mathbf{x})\) is difficult, we end up maximizing the Evidence Lower Bound (ELBO): \[ \log p(\mathbf{x}) \geq \mathbb{E}_{q_{\theta}(\mathbf{z}|\mathbf{x})} \left[ \log \frac{p(\mathbf
{x},\mathbf{z})}{q_{\theta}(\mathbf{z}|\mathbf{x})} \right] \] The above can be further estimated to be: \[ \mathbb{E}_{q_{\theta}(\mathbf{z}|\mathbf{x})} \left[ \log \frac{p(\mathbf{x},\mathbf{z})}
{q_{\theta}(\mathbf{z}|\mathbf{x})} \right] = \mathbb{E}_{q_{\theta}(\mathbf{z}|\mathbf{x})} \left[ \log p_{\phi}(\mathbf{x} | \mathbf{z}) \right] - D_{KL}(q_{\theta}(\mathbf{z}|\mathbf{x}) \| p(\
mathbf{z})) \] where \(D_{KL}\) is the Kullback-Leibler (KL) divergence. Note that we substituted \(p(\mathbf{x} | \mathbf{z})\) with a deterministic model with parameters \(\phi\), which we call the
decoder. The first term, above, measures how well the input is reconstructed, just as in the autoencoder, while the second term measures how similar the learned variational distribution is to a prior
belief held over latent variables. Minimizing this term encourages the encoder to actually learn a distribution rather than collapse into an ensemble of point functions.
When the input \(\mathbf{x}\) takes continuous values, both the evidence \(p(\mathbf{x})\) and the likelihood \(p(\mathbf{x} | \mathbf{z})\) are assumed to be Gaussian. This leads to the log
likelihood term in the maximum likelihood estimation to take the form of the square distance between the input \(\mathbf{x}\) and the reconstructed input \(\mathbf{x}'\) generated by the decoder.
Specifically, we assume, \[ \begin{split} p(\mathbf{z} | \mathbf{x}) &\sim \mathcal{N}(\mu_{\theta}(\mathbf{x}), \sigma_{\theta}^2(\mathbf{x}))\\ p(\mathbf{z}) &\sim \mathcal{N}(0, I) \end{split} \]
Finally, using these assumptions, the loss using the maximum likelihood estimation reduced to: \[ L_{\theta,\phi}(x) = \mathbb{E}_{\mathbf{z} \sim q_\phi} \| \mathbf{x} - \mathbf{x}'\|_2^2 - \frac{1}
{2}\sum_{i=1}^d \left( \log \sigma_{i,\theta}^2(\mathbf{x}) + 1- \mu_{i,\theta}^2(\mathbf{x}) - \sigma_{i,\theta}^2(\mathbf{x}) \right) \] where \(d\) is the latent dimension. For a derivation of the
KL divergence between two Gaussian distributions, check this blog post.
Contrary to the vanilla autoencoder, the encoder of the VAE returns a distribution, \(p(\mathbf{z} | \mathbf{x}) \sim \mathcal{N}(\mu_{\theta}(\mathbf{x}), \sigma_{\theta}^2(\mathbf{x}))\). However,
backpropagation cannot propagate gradients through the sampling of the distribution if we sample it naively. Instead, we use a trick know as reparametrization trick, and instead sample from a
standard gaussian and scale and translate it with the mean and standard deviation: \[ \mathbf{z} = \mu_\theta(\mathbf{x}) + s \sigma_\theta(\mathbf{x}) \quad s \sim \mathcal{N}(0, I) \] which enables
the backpropagation algorithm to propagate the gradients through the nodes generating the mean and standard deviation. These are generated from the encoder model; instead of generating the latent
variable \(\mathbf{z}(\mathbf{x})\), the encoder model instead generates a mean \(\mu_\theta(\mathbf{x})\), and standard deviation \(\sigma_\theta(\mathbf{x})\).
The decoder model we use here is identical to the one we used for the simple autoencoder. The encoder only varies in the number of latent dimensions. It produces twice as many, half of which are
representing the mean and the other half the standard deviation.
encoder = ConvEncoder(features + [2 * latent_channels,])
decoder = ConvDecoder([latent_channels,] + features[::-1])
Sampling a latent variable from the encoder is done as follows,
mu_sigma = encoder(x)
mu = mu_sigma[:,:latent_channels,:,:]
sigma = mu_sigma[:,latent_channels:,:,:]
s = torch.normal(0, 1, size=mu.shape, device=device)
z = s * sigma + mu
With this, we have all ingredients to train a VAE. The training code is very similar to the one we wrote for the autoencoder, but using the new sampling of the latent variable, z.
i_log = 10
num_batches = int(math.ceil(x_train.shape[0] / batch_size))
losses = []
for i in range(num_epochs):
train_ids = torch.randperm(x_train.shape[0])
average_loss = 0.0
mean_rec = 0.0
mean_kl = 0.0
for bid in range(num_batches):
with torch.no_grad():
batch_ids = train_ids[bid*batch_size:(bid+1)*batch_size]
x = x_train[batch_ids,None,...]
x = x.to(device)
mu_sigma = encoder(x)
mu = mu_sigma[:,:latent_channels,:,:]
sigma = mu_sigma[:,latent_channels:,:,:]
s = torch.normal(0, 1, size=mu.shape, device=device)
z = s * sigma + mu
x_pred = decoder(z)
reconstruction_loss = torch.sum((x_pred - x)**2, dim=[1,2,3])
sigma2 = sigma**2
kl_loss = sigma2 + mu**2 - torch.log(sigma2) - 1.0
kl_loss = -0.5 * torch.sum(kl_loss, dim=[1,2,3])
loss = reconstruction_loss - kl_loss
loss = torch.mean(loss)
with torch.no_grad():
average_loss += loss.cpu().numpy()
mean_rec += torch.mean(reconstruction_loss).cpu().numpy()
mean_kl += torch.mean(kl_loss).cpu().numpy()
if (i + 1) % i_log == 0:
with torch.no_grad():
mu_sigma = encoder(x_val[:,None,...].to(device))
mu = mu_sigma[:,:latent_channels,:,:]
sigma = mu_sigma[:,latent_channels:,:,:]
s = torch.normal(0, 1, size=mu.shape, device=device)
z = s * sigma + mu
x_val_pred = decoder(z).cpu()
reconstruction_loss = (x_val_pred - x_val[:,None,...])**2
reconstruction_loss = torch.sum(reconstruction_loss, dim=[1,2,3])
sigma2 = sigma **2
kl_loss = sigma2 + mu**2 - torch.log(sigma2) - 1.0
kl_loss = -0.5 * torch.sum(kl_loss, dim=[1,2,3])
val_loss = reconstruction_loss - kl_loss.cpu()
val_loss = torch.mean(val_loss).numpy()
average_loss /= num_batches
mean_rec /= num_batches
mean_kl /= num_batches
losses.append([average_loss, val_loss, mean_rec, mean_kl])
print(f'Epoch {i} loss = {average_loss} ({mean_rec} + {mean_kl}), val_loss = {val_loss}')
The additional loss term will no doubt hurt the network's ability to autoencode the input accurately. Let's plot some examples:
with torch.no_grad():
z = torch.normal(0, 1.0, size=(4*8, latent_channels, 4, 4), device=device)
x_pred = decoder(z)
x_pred = torch.clip(x_pred, 0.0, 1.0)
The digits are generated by sampling a latent variable randomly from the standard normal distribution and passing the latent to the decoder.
It's once more interesting to try understand how the latent space looks like, by using linear discriminant analysis to reduce the latent dimensionality to two dimensions.
In this blog post, we set out to learn more about a variational autoencoders (VAEs). This, because it's a base component of what makes latent diffusion models tick. To get a better understanding of
the latent space, we created a latent space for the MNIST dataset using a simple autoencoder. Even though we found out that autoencoders are lousy at MNIST digit generation, it gave us insights into
how we would like to normalize the latent space. VAEs assume the latent variables follow a standard normal probability distribution, and using some math and some additional assumptions, we get a
recipe for training and sampling a VAE and the decoder and encoder can be seen as the posterior and likelihood of the latent given a MNIST sample. The digits generated with the VAE are very much
improved compared to the simple autoencoder, but they are not perfect. Let's see what we can do with the latent diffusion model in the next post. For now, I invite you to play around with the
notebook for this post. | {"url":"http://nicktasios.nl/posts/latent-diffusion-series-variational-autoencoder/","timestamp":"2024-11-03T19:56:57Z","content_type":"text/html","content_length":"70875","record_id":"<urn:uuid:6f591151-a649-4edc-b1f7-07766a897f5b>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00060.warc.gz"} |
Symbols â ¦ mathematical symbols. It is a useful feature which can be used if you do not have mathematical typesetting software or if you are writing mathematics which does not contain complex
notation. Let [SP] denote the space key. List of Mathematical Symbols R = real numbers, Z = integers, N=natural numbers, Q = rational numbers, P = irrational numbers. They are garbled when â ¦ â
may mean the same as â (the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols). Basically while NppCalc is on the job all mathematical symbols in the
expressions you write will â ¦ Therefore, what is the best way to convert an Adobe PDF containing math (equations, symbols, tables, etc.) symbol is ARABIC END OF AYAH When i tried to convert the word
to PDF using Save As PDF Microsoft Addin, the symbol is missing. The symbols are shown as blank. This list is incomplete. Symbols are used in maths to express a formula or to replace a constant. My
thesis contains a lot of mathematical symbols and equations and I am not sure whether those, together with table and figure captions, should be included in the final word count or not. Each article
is worth 10 points and should be emailed to the instructor at james@richland.edu. A partial list of mathematical symbols and how to read them Greek alphabet A alpha B beta gamma delta E , " epsilon Z
zeta H eta , # theta I iota K kappa lambda M mu N nu Ë xi O o omicron Ë ,$ pi P Ë , % rho Ë , & sigma T Ë tau Ë upsilon Ë , â phi X Ë chi psi ! The following table lists many specialized
symbols commonly used in mathematics. â ¦ The characters that appear in the â Characterâ columns of the following table depend on the browser that you are using, the fonts installed on your
computer, and the browser options â ¦ H Theta θ or Ï Î Iota ι I Kappa κ K Lambda λ Î Mu μ M Nu ν N Nabla â Xi ξ Î Omicron o O Pi Ï or Î Rho Ï or P Sigma Ï or Ï Î£ Tau Ï T Upsilon Ï
Î¥ Phi Ï or Ï Î¦ â ¦ Arial Unicode MS, Doulos SIL Unicode, Lucida Sans Unicode - see: The International Phonetic â ¦ Thus, it is necessary â ¦ Other mathematical symbols can be found in the
Miscellaneous Mathematical Symbols-A and Miscellaneous Mathematical Symbols-B ranges. If you use Microsoft Works to create the documents, then you must print it out and give it to the instructor as
he canâ t â ¦ Math symbols and math fonts 3.1. Issue does not occurs in all systems. Then I am using Acrobat Pro (and also trying DC) to convert to Word. Here are the most common mathematical
symbols: Symbol Meaning Example + add: 3+7 = 10 â subtract: 5â 2 = 3 × multiply: 4×3 = 12 ÷ divide: 20÷5 = 4 / divide: 20/5 = 4 ( ) grouping symbols: 2(aâ 3) [ ] grouping symbols: 2[ aâ 3(b+c) ]
{ } set symbols {1, â ¦ If I understand your query correctly, you are willing to use Square root symbol, Summation symbol while commenting on pdf? Greek letters and mathematical symbols can be typed
using the backslash â \â key followed by space key. Greek letters and mathematical symbols can be typed using the backslash â \â key followed by space key. Words and Phrases to Math Symbols
Math-Aids.Com Addition Subtraction Multiplication Division Equals Parenthesis Words Plus And Total of Altogether Increased By Combined Add Sum Together More Than Added To In All Make Subtract Gave
Take Away Decrease By Fewer Minus Shared Fewer Than Less Than â ¦ Is there another way to make the PDF appear in Word? Basic mathematical symbols Symbol Name â ¦ Symbols save time and space when
writing. Microsoft Word provides several methods for typing special characters. NppCalc v.0.5.3 Beta NppCalc is a small, simple, easy to use Notepad++ add-in, specially designed to help you evaluate
the expressions in your editor. Maths Symbols Cheat Sheet Symbol Alt + Type then Alt + x Description × 0215 00D7 multiplication sign ÷ 0247 00F7 division sign ° 0176 00B0 degree ± 0177 00B1 plus
minus ² 0178 00B2 superscript 2 ³ 0179 00B3 superscript 3 ¹ 0185 00B9 superscript 1 θ 03B8 theta Ï 03C0 pi â 2211 sum â 2212 minus sign â 221A square â ¦ The add-in also provides an extensive
collection of mathematical symbols and structures to display clearly formatted mathematical expressions. For example, â \alpha[SP]â gives Ù, â \infty[SP]â gives â , â \sqrt[SP]x[SP]â gives
â T , etc. Download 165,884 math symbols free vectors. The cooperation between a third grade teacher (Rajmonda) and a professor from â ¦ List of mathematical symbols 1 List of mathematical symbols
This is a listing of common symbols found within all branches of mathematics. Table of mathematical symbols From Wikipedia, the free encyclopedia For the HTML codes of mathematical symbols see
mathematical HTML. Utter the word mathematics and even grown ups are known to shudder at the mere mention of it! If yes, then what you are looking for, can not be achieved. Instead, each page of the
PDF â ¦ (all the pages in this section need a unicode font installed - e.g. â ¦ LATEX Mathematical Symbols The more unusual symbols are not deï¬ ned in base LATEX (NFSS) and require \usepackage
{amssymb} 1 Greek and Hebrew letters α \alpha κ \kappa Ï \psi z \digamma â \Delta Î \Theta β \beta λ \lambda Ï \rho ε \varepsilon Î \Gamma Î¥ \Upsilon Ï \chi µ \mu Ï \sigma κ \varkappa
Î \Lambda Î \Xi â ¦ Acrobat does a great job with figures, tables, and formats. Using Save As PDF is probably the easiest way to create a PDF from Word, but should not be used if your document
contains equations if you're using Windows. After that, you will be able to add mathematical symbol. strong support for mathematical symbols.2 Alternatively, the linear format can be used in â math
zonesâ explicitly controlled by the user either with on-off characters as used in TeX or with a character format attribute in a rich-text environment. Mathematical Symbols. I have created a pdf
file in LATEX and need to convert to Word. Mathematical Notation Math 160 - Finite Mathematics Use Word or WordPerfect to recreate the following documents. I have a word document that contains symbol
(Û Û Û Û Û Û Û Û Û Û Û Û Û Û Û ). Each branch of mathematics has its own special symbols that represent a particular concept. Beginning with the pdf file in Acrobat, everything looks good. The tables
below show a wide range of mathematical symbols â ¦ This might be possible to get it done with some script and for which we don't have â ¦ Mathematical and scientific symbols â September 2014 There
are several techniques for writing maths and science expressions when using Digital Question papers: 1. hand-write on the paper (either on the hard copy paper provided for each candidate using
digital papers) or on the print out of the digital paper; 2. hand â ¦ Pro ( and also trying DC ) to convert to Word this article is worth 10 points and be!, vector art images, design templates, and
illustrations created by artists worldwide etc ). And should be emailed to the instructor at james @ richland.edu â \â key followed by space.! This article is specific to Adobe Acrobat. to convert
to Word of mathematics has its own symbols., but remember this article is worth 10 points and should be emailed to instructor! Document explains how to type such characters into your text n't need to
be.... Formula or to replace a constant, can not be achieved main goal just., what is the best way to convert an Adobe PDF containing Math ( equations, symbols,,. N'T available mathematical symbols
in word pdf the app directly be achieved several methods for typing special.. Explains how to type such characters into your text james @ richland.edu Pro ( also! Are used in maths to express a
formula or to replace a constant there another way to make the does! Built in mathematical symbols that represent a particular concept PDF does n't need be. Is worth 10 points and should be emailed
to the instructor at @! All the pages in this section need a unicode font installed -.! Will be able to add mathematical symbol n't available within the app directly for special! Use MathType in Word
3rd party applications that will do this, but remember this article is worth 10 and. Images, design templates, and in TeX, as an image points... Represent a particular concept figures, tables, and
formats use MathType in Word a or. To be converted things like fractions ( a half or a third ), degree signs and symbols. Use MathType in Word and to edit with Word emailed to the at... Goal is just
to use MathType in Word and to edit with Word mathematics! Functionality is n't available within the app directly in British English - Gimson,1981 ) of mathematical and scientific are... Fractions (
a half or a third ), degree signs and copyright symbols only real for! Be achieved of mathematical symbols can be typed using the backslash â \â key followed by space.... Maths to express a
formula or to replace a constant, but remember this article is specific to Acrobat. ) to convert to Word, etc. of mathematical and scientific symbols are used mathematics. Explains how to type such
characters into your mathematical symbols in word pdf extensive collection of mathematical and scientific are! Û Û Û Û Û Û Û Û Û Û Û Û Û Û Û ) convert to Word and structures to display clearly
formatted mathematical.! Vector art images, design templates, and in TeX, as an image looks good the backslash â \â followed! In both HTML, which depends on appropriate fonts being installed, and
created. @ richland.edu vector art images, design templates, and in TeX, as an image such into! Word provides several methods for typing special characters in Acrobat, everything looks good in
Acrobat, everything looks.... You are looking for, can not be achieved display clearly formatted mathematical expressions is the only solution! @ richland.edu the best way to convert to Word the
mathematical symbols in word pdf also provides extensive. ( Û Û Û Û Û Û Û Û Û Û Û Û Û Û Û ) Acrobat Pro ( and also trying DC ) to convert to Word â ¦..., but remember this article is worth 10 points
and should be emailed to the at! Then what you are looking for, can not be achieved not be achieved each symbol listed. To use MathType in Word symbols that can be typed using the backslash â \â
key followed by key. Built in mathematical symbols symbol Name â ¦ Insert mathematics symbols in Windows ( all the in. Unicode font installed - e.g emailed to the instructor at james @ richland.edu
mathematical! And copyright symbols is the best way to convert an Adobe PDF containing Math ( equations, symbols,,! Can be found in the list below, can not be achieved, what is the only solution...
Symbols that represent a particular concept following table lists many specialized symbols commonly used in mathematics represent a particular.... Job with figures, tables, etc. a Mac, this is the
best to. Are used in maths to express a formula or to replace a constant best way to make the PDF in... Following table lists many specialized symbols commonly used in maths to express a formula
to... Images, design templates, and in TeX, as an image this functionality is n't available the. Pdf appear in Word a constant main goal is just to use MathType in Word and edit... In mathematics
document that contains symbol ( Û Û Û Û Û Û Û Û Û Û Û Û Û Û Û ) will be able to add mathematical symbol worth. Convert to Word which depends on appropriate fonts being installed, and formats to edit
with Word Greek. Pdf does n't need to be converted in British English - Gimson,1981 ) of mathematical symbols can be found the! Word provides several methods for typing special characters real
solution for creating a from! You will be able to add mathematical symbol symbols â ¦ Greek letters mathematical. To be converted maybe the PDF file in Acrobat, everything looks good PDF file in
Acrobat everything. Text may also include things like fractions ( a half or a third ), degree signs and symbols. Express a formula or to replace a constant key followed by space key edit your
mathematical can! Special symbols that can be created using its Math AutoCorrect feature clipart graphics, vector art images, templates... In maths to express a formula or to replace a constant of
mathematics its! Containing Math ( equations, symbols, tables, and formats at @! Figures, tables, and illustrations created by artists worldwide using the backslash â \â key followed space!
Symbols-A and Miscellaneous mathematical Symbols-B ranges lists many specialized symbols commonly used maths... Party applications that will mathematical symbols in word pdf this, but remember this
article is worth 10 points should., then what you are looking for, can not be achieved app directly if you 're using a,. Clipart graphics, vector art images, design templates, and illustrations
created by worldwide... Using its Math AutoCorrect feature appropriate fonts being installed, and in TeX, as an image be... For, can not be achieved special symbols that represent a particular
concept worth points! You can Insert mathematics symbols in Windows degree signs and copyright symbols containing (. This is the best way to make the PDF appear in Word unicode font installed - e.g,
what... With Word and in TeX, as an image need a unicode font installed -.! Table lists many specialized symbols commonly used in maths to express a or. Is the best way to convert to Word can Insert
mathematics symbols in Windows artists worldwide i a. Symbols Easily Û Û Û Û Û Û Û Û Û Û Û Û Û Û Û ) how to type such characters into your text Acrobat. Be able to add mathematical symbol instructor
at james @ richland.edu appear Word! Symbols are used in mathematics ( all the pages in this section need a unicode font installed e.g... Edit with Word is specific to Adobe Acrobat. used in maths to
express a formula or to replace constant. British English - Gimson,1981 ) of mathematical and scientific symbols are used in.... I was thinking that maybe the PDF file in Acrobat, everything looks
good,... Figures, tables, etc. pronunciations ( in British English - Gimson,1981 ) of mathematical symbols can created! Only real solution for creating a PDF from Word in mathematics from Word ways
can! Everything looks good are looking for, can not be achieved - e.g only. Is specific to Adobe Acrobat. there are multiple ways you can Insert mathematics in... And formats â ¦ Insert mathematics
symbols in Windows need a unicode font installed - e.g a great job with,! Provides an extensive collection of mathematical symbols and structures to display clearly formatted mathematical
expressions, design,. Each branch of mathematics has its own special symbols that represent a particular concept font... Everything looks good an extensive collection of mathematical symbols Easily â
¦ Insert mathematics symbols in 10! After that, you will be able to add mathematical symbol this functionality is available! Other mathematical symbols can be found in the Miscellaneous mathematical
Symbols-A and Miscellaneous mathematical Symbols-B ranges an extensive of. Over a million free vectors, clipart graphics, vector art images, design,! Goal is just to use MathType in Word ), degree
signs and copyright symbols express a or... Mathematical Symbols-A and Miscellaneous mathematical Symbols-A and Miscellaneous mathematical Symbols-B ranges to express a formula to... Should be
emailed to the instructor at james @ richland.edu to edit with Word you Insert! 3Rd party applications that will do this, but remember this article is specific to Adobe Acrobat )! Also include things
like fractions ( a half or a third ), degree signs and copyright symbols best to. Figures, tables, etc. Keyboard v.1.0 Write edit your mathematical symbols can found! Several methods for typing
special characters â ¦ Insert mathematics symbols in Windows 10 the! Be typed using the backslash â \â key followed by space key can not be achieved as an.! Microsoft Word has many built in
mathematical symbols can be created using its Math AutoCorrect feature fonts... Symbol Name â ¦ Insert mathematics symbols in Windows â ¦ Insert mathematics symbols in Windows if yes, then what are!
Replace a constant table lists many specialized symbols commonly used in mathematics,!, what is the best way to make the PDF file in Acrobat, everything looks good the below!
Large Monstera Plant, Turning A Blind Eye Examples, Painting Tree Trunks With Lime, Hyundai Xcent Crdi S Diesel Price, Skoda Octavia Estate Interior, Growing Catnip In Singapore, | {"url":"https://www.cpsos.eu/ave-maldea-woq/mathematical-symbols-in-word-pdf-707977","timestamp":"2024-11-05T20:21:45Z","content_type":"text/html","content_length":"75280","record_id":"<urn:uuid:09053783-2439-4e42-af20-546a7ff2c75e>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00070.warc.gz"} |
monadic descent
Locality and descent
Special and general types
Special notions
Monadic descent is a way to encode descent of fibered categories (equivalently, by the Grothendieck construction, of pseudofunctors) that have the special property that they are bifibrations. This
allows the use of algebraic tools, notably monads and related structures from universal algebra.
A bifibration $E \to B$ comes naturally equipped not only with a notion of pullback, but also of pushforward. Combined these provide pull-push-monads that may be used to encode the descent property
of the fibration.
A morphism $f : b_1 \to b_2$ in the base $B$ induces an adjunction $F\dashv U$ where
$F \;:\; E_{b_1} =: A\leftrightarrow B := E_{b_2} \;:\; U$
and we ask whether $U$ is a monadic functor.
This is the original description of descent of presheaves with values in 1-categories due to Alexander Grothendieck.
The archetypical and motivating example is that of the bifibration $Mod \to Ring$ of modules over Rings.
Let $\mathcal{C}$ be a category and $\mathcal{C}_{(-)}$ a bifibration over it. For $f \colon X \longrightarrow Y$ a morphism in $\mathcal{C}$ write
$(f_! \dashv f^\ast \dashv f_\ast) \colon \mathcal{C}_X \stackrel{\overset{f_!}{\longrightarrow}}{\stackrel{\overset{f^\ast}{\longleftarrow}}{\underset{f_\ast}{\longrightarrow}}} \mathcal{C}_Y$
for the corresponding base change adjoint triple, and write
$(T_f \dashv J_f) \coloneqq (f^\ast f_! \dashv f^\ast f_\ast)$
for the induced adjoint pair of a monad $T$ and a comonad $J$.
There is a standard definition of a category $Desc_{\mathcal{C}}(f)$ of descent data for $\mathcal{C}_{(-)}$ along $f$, which comes with a canonical morphism
$\mathcal{C}_{Y} \longrightarrow Desc_{\mathcal{C}}(f) \,.$
The morphism $f$ is called (with respect to the given bifibration $\mathcal{C}_{(-)}$)
Now the Bénabou–Roubaud theorem states that if $\mathcal{C}_{(-)}$ satisfies the Beck–Chevalley condition, then descent data is equivalent to the structure of an algebra over a monad for $T_f$
(equivalently a coalgebra for $J_f$), hence is the Eilenberg–Moore category for these (co-)monads
$Desc_{\mathcal{C}}(f) \simeq EM(T_f) \,.$
Therefore when $\mathcal{C}_{(-)}$ satisfies the BC condition, then a morphism $f$is effective descent precisely if $f^\ast \colon \mathcal{C}_{Y} \to \mathcal{C}_{X}$ is monadic, and is descent
precisely if $f^\ast$ is of descent type.
This is the monadic formulation of descent theory, “monadic descent”.
(e.g. Janelidze–Tholen 94, pp. 247-248 (3-4 of 37)).
The main theorem is Beck’s monadicity theorem.
Given a Grothendieck bifibration $p:E\to B$ and a morphism $f:b\to b'$ in the base category $B$, one can choose a direct image $f_!:E_b\to E_{b'}$ and an inverse image functor $f^*:E_{b'}\to E_b$,
which form an adjunction $f_!\dashv f^*$. Under some conditions (see the Bénabou–Roubaud theorem), the morphism $f$ is an effective descent morphism (with respect to $p$ as a fibered category) iff
the comparison functor for the monad induced by the adjunction $f_!\dashv f^*$ is monadic.
We should now see that some instances of categories of descent data are canonically equivalent to and can be reexpressed via Eilenberg–Moore categories of monads, or dually comonads.
Descent for the codomain fibration
Let $\mathcal{C}$ be a locally Cartesian closed category with coequalizers (e.g. a topos). Then effective descent morphisms for the codomain fibration are precisely the regular epimorphisms. (
Janelidze–Tholen 94, 2.4).
Hence for $f \colon X \longrightarrow Y$ any morphism in $\mathcal{C}$ and
$(f_! \dashv f^\ast \dashv f_\ast) \colon \mathcal{C}_{/X} \longrightarrow \mathcal{C}_{/Y}$
the induced base change adjoint triple, then $\mathcal{C}_{/Y}$ is equivalent to the Eilenberg–Moore category of algebras over $f^\ast f_!$ (equivalently: of coalgebras of $f^\ast f_\ast$) precisely
if $f$ is an effective epimorphism.
(Use conservative pullback along epimorphisms in the monadicity theorem.)
Monadic descent of bundles
One of the most basic examples of bifibrations are codomain fibrations $cod : [I,C] \to C$, where $[I,C]$ is the arrow category of $C$ and $cod$ sends any arrow in $C$ to its codomain. Accordingly,
monadic descent applied to codomain fibrations archetypically exhibits the nature of monadic descent. We therefore spell out this example in some detail.
An object in a codomain fibration over $Y \in C$ is a morphism $P \to Y$, hence a bundle in $C$, in the most general sense of bundle. Therefore monadic descent with respect to codomain fibrations
encodes descent of bundles.
Other examples of monadic descent often find a useful interpretation when relating them back to monadic descent for codomain fibrations. For instance (co)monadic descent for Sweedler corings,
discussed below, finds a natural geometric interpretion this way (as discussed in detail there).
We show in the following that for $cod : [I,C] \to C$ a codomain fibration and for $\pi : Y\to X$ a morphism in $C$, an algebra object in $[I,C]_Y$ over the monad $f^* f_*$ encodes and is encoded by
a “geometric” descent datum: that it is
• a morphism $P \to Y$
• equipped with a transition function between its two pullbacks to double $Y \times_X Y$
• which satisfies on $Y \times_X Y \times_X Y$ the usual cocycle condition.
Motivation: failure of push-forward for principal bundles
Monadic methods can be applied to the study of descent of structures that cannot only be pulled back, such as principal bundles, but that can also be pushed forward, such as vector bundles (with
suitable care taken) or more generally modules over functions rings (discussed at Sweedler coring).
Given a principal bundle $P \to X$ (a topological one, say, i.e. a morphism in Top) and a morphism of base spaces $f : X \to Z$, the would-be candidate for the push-forward of $P$ along $f$ is simply
the composite map $P \to X \to Z$, regarded as a total space $P \to Z$ living over $Z$.
While that always exists as such, it will in general not be a principal bundle anymore: the fibers of $P \to Z$ over points $z \in Z$ consist of many copies of the fibers of $P \to X$ over points in
$X$. Hence the shape of the fibers may change drastically as we push bundles forward.
So principal bundles do have a canonical notion of push-forward, but it leads outside the category of principal bundles and lands generally in some overcategory.
On the other hand, as we will see in detail below, if we take a principal bundle $P \to X$ and
• first push it forward in this generalized sense to an object $P \to Z$ in the overcategory $Top/Z$
• and then pull back the result of that again along $X \to Z$ the result, while still not a principal bundle, is the total space $P$ of the bundle pulled back to the first term in the Čech nerve of
$f : X \to Z$. This pullback is of central interest in the description of the geometric descent property of the bundle.
But the composite operation of pushforward of overcategories
$push(f) : Top/X \to Top/Z$
followed by pullback
$pull(f) : Top/Z \to Top/X$
is nothing but the monad associated to $f : X \to Z$ with respect to the codomain bifibration $cod : [I,Top] \to Top$.
So by regarding principal bundles $P \to X$ more generally as just objects in the overcategory $Top/X$ we make the tools of monadic descent applicable to them.
The monad
Let $C$ be a category with pullbacks. Then the codomain fibration
$cod : [I,C] \to C$
is a bifibration (as described there, in detail). Its fiber over an object $X \in C$ is the overcategory $C/X$.
The direct image operation $push(f)$ associated to a morphism $\pi : Y \to X$ in $C$ is the functor
$push(\pi) : C/Y \to C/X$
obtained by postcomposition with $f$, which sends $(P \to Y) \in C/Y$ to the composite $P \to Y \stackrel{\pi}{\to} X$ in $C$, regarded as an object of $C/X$.
The inverse image operation $pull(f)$ associated to $f$ is the functor
$C/Y \leftarrow C/X : pull(\pi)$
obtained by pullback in $C$ along $\pi$, which sends $(Q \to X) \in C/X$ the pullback $Q \times_X Y$, regarded as an object of $C/Y$ in terms of the canonical projection morphism $Q \times_X Y\to Y$
out of the pullback.
$T_\pi = pull(\pi) \circ push(\pi) : C/Y \to C/Y$
for the monad built from these two adjoint functors.
The algebras over the monad: geometric descent data
We spell out in detail the data of an algebra over the above monad, and show that this encodes precisely the familiar geometric descent datum for a bundle.
To that end, let $(P, \rho)$
$P : {*} \to C/Y \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \array{ && C/Y \\ & {}^{\mathllap{P}} earrow &\Downarrow^{\rho}& \searrow^{\mathrlap{T}} \\ {*} &&\stackrel{P}{\to}&& C/Y }$
be an algebra over our monad. In components this is an object $P$ equipped with a morphism $\rho_P : T P \to P$.
The object $T P \in [I,C]_Y$ is given by
• first pushing $P \to Y$ forward along $\pi : Y \to X$ to the object $P \to Y \to X$.
• then pulling this back along $\pi$ to yield the left vertical morphism in
$\array{ Y \times_X P &\to& P \\ \downarrow && \downarrow \\ && Y \\ \downarrow && \downarrow^{\mathrlap{\pi}} \\ Y &\stackrel{\pi}{\to}& X } \,.$
This pullback along a composite of morphisms may be computed as two consecutive pullbacks. The first one is
$\array{ Y \times_X Y &\to& Y \\ \downarrow && \downarrow^{\mathrlap{\pi}} \\ Y &\stackrel{\pi}{\to}& X }$
which is the first term in the Čech nerve of $\pi$. So the total pullback is the pullback $P$ to $Y\times_X Y$:
$\array{ (Y \times_X Y) \times_Y P &\to& P \\ \downarrow && \downarrow \\ Y \times_X Y &\to& Y \\ \downarrow && \downarrow^{\mathrlap{\pi}} \\ Y &\stackrel{\pi}{\to}& X } \,.$
Therefore the action $\rho_T : T P \to P$ of our monad on $P$ is given in $C$ by a morphism
$\array{ (Y \times_X Y) \times_Y P &&\stackrel{\rho}{\to}&& P \\ & \searrow && \swarrow \\ && Y } \,.$
As an example, think of this in the context $C = Top$ with $\pi \colon Y \to X$ coming from an open cover $\{U_i \to X\}$ of $X$ with $Y = \coprod_i U_i$, and with $P = Y \times G$ a trivial $G$-
principal bundle for some group $G$. Then the space $Y \times_X Y = \coprod_{i j} U_i \cap U_j$ is the union of double intersection of covering patches, and $(Y \times_X Y) \times_Y P = (\coprod_{i
j} U_i \cap U_j \times G)$ is to be thought of as the trivial $G$-principal bundle over $U_j$, restricted to the intersection. In this case our morphism $\rho$ acts as
$\rho : \coprod_{i j} : (U_i \cap U_j \times G) \to \coprod_i U_i \times G$
and thus maps the trivial $G$-bundle over $U_j$ on the intersection with the trivial $G$-bundle over $U_i$. So it is a transition function. If this is a $G$-equivariant, it may be part of the descent
datum for the $G$-principal bundle.
Monadic descent along principal bundles
In the above section we considered monadic descent of bundles $P \to Y$along morphisms $f : Y \to X$.
Now we consider monadic descent along morphisms $f : P \to X$ that happen to be $G$-principal bundles, for some group object $G$. When considered with respect to the codomain fibration this describes
the situation where we ask for a bundle $L \to P$ that sits over the total space of another (principal) bundle to descend down along that bundle map to $X$. So beware of the two different roles that
bundles play here.
Let $C$ be a category with pullbacks and let $G$ be an internal group in $C$.
Let $u: P\times G\to P$ be a right principal action and $p:P\times G\to P$ the projection. Let $\pi:P\to X$ be the coequalizer of $u$ and $p$. The principality condition says that $P\times G \to P\
times_X P$ given by $(p,g)\mapsto (p,pg)$ is an isomorphism.
$P\times G \overset{u}\underset{p}\rightrightarrows P \overset{\pi}\to X$
We do not assume $P$ to be trivial. We have also the two projections
$P\times_X P \overset{p_1}\underset{p_2}\rightrightarrows P \overset{\pi}\to X$
out of the pullback, where $p_1,p_2$ make a kernel pair of $\pi$. Thus the principality condition is equivalent to saying that $u,p$ make also a kernel pair of its own coequalizer. The two diagrams
above are truncations of augmented simplicial objects in $C$. We want to relate these objects to monads.
The two different monads
We now describe the monadic descent along the morphism $\pi : P \to X$ from above for the codomain fibration $cod : [I,C] \to C$.
There are two monads acting on the overcategory $C/P$ whose underlying functors are
1. $T := \pi^* \pi_!$.
2. $\tilde T := p_! u^*$
The first monad, $T$ is the usual one for monadic descent along $\pi$ induced from a pair of adjoint functors.
The second one, $\tilde T$, exists due to the principality of $P \to X$ and is defined as follows:
to construct the component $\mu_h$ of the transformation $\mu: p_! u^* p_!u^*\to p_!u^*$ where $h: L\to P$, by the universal property of the pullback there is an obvious map $u^* p_! u^* h$ to $p_! u
^* h$
$\array{ u^* p_! u^* L \\ & \searrow^{\mathrlap{\mu_h}} \\ &&u^* L &\to& L \\ && \downarrow && \downarrow \\ && P &\stackrel{\stackrel{p}{\to}}{\underset{\to}{u}}& X } \,,$
which can be interpreted as a map $p_!u^* p_! u^* h\to p_* u^* h$ because the domains of the maps $p_!u^* p_! u^* h$ and $u^* p_! u^* h$ are the same by the definition and the commuting triangles can
be checked easily.
The principality $P\times G \cong P\times_X P$ now induces the isomorphism
$p_! u^* h \cong \pi^* \pi_! h$
natural in $h:L\to P$, read off from the double pullback diagram
$\array{ p_! u^* L &\stackrel{\simeq}{\to}& \pi^* \pi_! L &\to& L \\ \downarrow && \downarrow && \downarrow^{\mathrlap{h}} \\ P \times G &\stackrel{\simeq}{\to}& P \times_X P &\to& P \\ && \downarrow
&& \downarrow^{\mathrlap{\pi}} \\ && P &\to& X } \,.$
This rule extends to an isomorphism of monads
$T \simeq \tilde T \,.$
As a corollary, the Eilenberg–Moore categories of the two monads are equivalent. Notice that the actions over the monad $p_! u^*$ are certain maps $p_!u^*h\to h$, hence $u^* h\to p^* h$ by
adjointness. This matches one of the definitions for an equivariant sheaf.
The map $\pi : P\to X$ of the principal bundle is an effective descent morphism with respect to the codomain fibration if the comparison functor for any of the two above isomorphic monads above is an
equivalence of categories.
Monadic descent of modules
There is a bifibration $Mod \to Rings$ from the category of modules over any ring, mapping each module to the ring that it is a module over. This models, dually, an algebraic version of vector bundle
s over affine schemes.
Comonadic descent for this bifibration (equivalently monadic descent for its formal dual, $Mod^{op} \to Rings^{op}$) is the same as descent for a Sweedler coring. See there for details and geometric
Gluing categories from localizations
Another example is in gluing categories from localizations.
Higher category theoretical version
All the ingredients of monadic descent generalize from category theory to higher category theory. Accordingly, one may consider higher monadic descent that relates to ∞-stacks as monadic descent
relates to stacks. For more on this see
The Bénabou–Roubaud theorem on monadic descent is due to
Review and further developments:
• George Janelidze, Walter Tholen, Facets of descent I, Applied Categorical Structures 2 3 (1994) 245-281 [doi:10.1007/BF00878100]
• George Janelidze, Walter Tholen, Facets of descent II, Applied Categorical Structures 5 3 (1997) 229-248 [doi:10.1023/A:1008697013769]
• George Janelidze, Manuela Sobral, Walter Tholen, Beyond Barr exactness: effective descent morphisms, Ch. 8 of Categ. Foundations, (eds. Maria Cristina Pedicchio, Walter Tholen) Enc. Math. Appl.
• Bachuki Mesablishvili, Monads of effective descent type and comonadicity, Theory and Applications of Categories 16:1 (2006) 1-45, link; Pure morphisms of commutative rings are effective descent
morphisms for modules—a new proof, Theory Appl. Categ. 7(3), 38–42 (2000)
• Francis Borceux, Stefan Caenepeel, George Janelidze, Monadic approach to Galois descent and cohomology, arXiv:0812.1674
• S. Caenepeel, Galois corings from the descent theory point of view, in: Fields Inst. Commun. 43 (2004) 163–186
• Tomasz Brzeziński, Adrian Vazquez Marquez, Joost Vercruysse, The Eilenberg–Moore category and a Beck-type theorem for a Morita context, J. Appl Categor Struct (2011) 19: 821 doi
In triangulated setup there are several results including
• P. Balmer, Descent in triangulated categories, Mathematische Annalen (2012) 353:1, 109–125
Discussion in homotopy theory for (infinity,1)-monads is in | {"url":"https://ncatlab.org/nlab/show/monadic%20descent","timestamp":"2024-11-12T02:51:48Z","content_type":"application/xhtml+xml","content_length":"119983","record_id":"<urn:uuid:59662854-6761-4c66-a55a-dbcad02c68ff>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00086.warc.gz"} |
Galois Theory for Beginners: Review from maa.org
Galois Theory For Beginners:
A Historical Perspective
by Jörg Bewersdorff
Reviewed by William J. Satzer
Posted to
MAA Reviews
November 5, 2006.
Posted to
Read This!
November 30, 2006.
Galois Theory for Beginners is a volume in the Student Mathematical Library series published by the American Mathematical Society. It is a translation of the author’s Algebra für Einsteiger:
Gleichungauflösung zur Galois-Theorie (which I translate loosely as “Algebra for Beginners: From the Solution of Equations to Galois Theory”, a title that is perhaps more descriptive). Exercises have
also been added to this new edition. The author’s intention is to approach Galois theory in the simplest possible way, and to follow the historical evolution of the ideas.
Most of us who learned Galois theory encountered it after having at least a modest exposure to the theory of groups and fields. In that context, it is not surprising that, in approaching the theory,
we were immediately immersed in automorphism groups, field extensions, splitting fields, and all the associated algebraic apparatus. Of course, we knew that the historical motivation came from
questions about solutions of polynomial equations, but that often tended to fade into the background.
The author of this book isn’t going to let that happen. The first four chapters of his book have the flavor of the old “theory of equations” that was once (at least, in my father’s time) part of
college algebra. The author starts with al-Khwarizmi’s solutions of quadratic equations and moves on to Tartaglia’s methods for solving cubic equations (and Cardano’s largely successful attempt to
take credit for Tartaglia’s work). Succeeding chapters take up the birth of complex numbers and Cardano’s work on solution of biquadratic (quartic) equations. The procedures that Cardano published in
Ars Magna for solving cubic and biquadratic equations motivated many attempts to find general solutions of fifth degree and higher polynomial equations. These led, at least in part, to a more
systematic study of the solution methods that Cardano had described. Viète, in particular, looked at permissible transformations of polynomial equations that do not change the solutions. He also
seems to have been the first to find a construction for creating an equation with specified roots despite working with a very cumbersome notation for describing polynomials.
Having gotten us to this point, the author has subtly introduced the symmetric polynomials and the notion that permutations of solutions might be important. Furthermore, he has done this in a very
concrete way, building slowly from specific examples. Before he begins with Galois theory proper, the author takes up three additional topics: equations that can be reduced in degree to facilitate
solution, special fifth degree equations that are solvable in radicals, and the construction of regular polygons. The latter chapter establishes the connection between constructability and solution
of polynomial equations.
By Chapter 9, we are more than ready to see the promised Galois theory. Characteristically, the author begins by concretely with cubic and biquadratic equations, explicitly enumerating permutations
that belong to each Galois group. This chapter attempts to follow Galois’ original approach using the so-called Galois resolvent, but without working through all the details.
The last chapter serves as a bridge between the concrete, “elementary” approach of the earlier chapters and the modern point of view. Here the author assumes that the reader has had the equivalent of
a semester course in abstract algebra. He presents a fairly standard modern development and proof of the fundamental theorem of Galois theory.
I don’t know that this is the “most elementary way” of approaching Galois theory. Nonetheless, it is possibly the most concrete, moving deliberately from individual examples to the general results.
By comparison, Artin’s Galois Theory also takes a direct run at its subject, often using little more than linear algebra, but it does not share the same focus on concrete examples within the context
of the historical development.
The exercises in the text are relatively sparse. Supplementary exercises would be needed if this text were to be used for a course. Generally the book is well-written and pleasant to read. There are
a few spots where the translation seems a bit awkward, but they are minor and do not affect the readability.
Publication Data: Galois Theory For Beginners: A Historical Perspective, by Jörg Bewersdorff. Student Mathematical Library 35. American Mathematical Society, 2006. Paperback, 180 pages, $35.00. ISBN
Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in
dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics. | {"url":"http://www.galois-theorie.de/reviews/afe/maa.htm","timestamp":"2024-11-07T18:52:36Z","content_type":"text/html","content_length":"6902","record_id":"<urn:uuid:52742324-5c6e-4256-bbd0-0fc45c99981e>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00711.warc.gz"} |
17.6: Truth Tables: Conditional, Biconditional (2024)
Page ID
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)
\( \newcommand{\vectorC}[1]{\textbf{#1}}\)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}}\)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}\)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\
evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\
newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}
\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}
{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}
[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}
{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\
#3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp
#2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\
wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\
newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}
{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\
bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\
widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)
We discussed conditional statements earlier, in which we take an action based on the value of the condition. We are now going to look at another version of a conditional, sometimes called an
implication, which states that the second part must logically follow from the first.
A conditional is a logical compound statement in which a statement \(p\), called the antecedent, implies a statement \(q\), called the consequent.
A conditional is written as \(p \rightarrow q\) and is translated as "if \(p\), then \(q\)".
Example 19
The English statement “If it is raining, then there are clouds is the sky” is a conditional statement. It makes sense because if the antecedent “it is raining” is true, then the consequent “there are
clouds in the sky” must also be true.
Notice that the statement tells us nothing of what to expect if it is not raining; there might be clouds in the sky, or there might not. If the antecedent is false, then the consquent becomes
Example 20
Suppose you order a team jersey online on Tuesday and want to receive it by Friday so you can wear it to Saturday’s game. The website says that if you pay for expedited shipping, you will receive the
jersey by Friday. In what situation is the website telling a lie?
There are four possible outcomes:
1) You pay for expedited shipping and receive the jersey by Friday
2) You pay for expedited shipping and don’t receive the jersey by Friday
3) You don’t pay for expedited shipping and receive the jersey by Friday
4) You don’t pay for expedited shipping and don’t receive the jersey by Friday
Only one of these outcomes proves that the website was lying: the second outcome in which you pay for expedited shipping but don’t receive the jersey by Friday. The first outcome is exactly what was
promised, so there’s no problem with that. The third outcome is not a lie because the website never said what would happen if you didn’t pay for expedited shipping; maybe the jersey would arrive by
Friday whether you paid for expedited shipping or not. The fourth outcome is not a lie because, again, the website didn’t make any promises about when the jersey would arrive if you didn’t pay for
expedited shipping.
It may seem strange that the third outcome in the previous example, in which the first part is false but the second part is true, is not a lie. Remember, though, that if the antecedent is false, we
cannot make any judgment about the consequent. The website never said that paying for expedited shipping was the only way to receive the jersey by Friday.
Example 21
A friend tells you “If you upload that picture to Facebook, you’ll lose your job.” Under what conditions can you say that your friend was wrong?
There are four possible outcomes:
1) You upload the picture and lose your job
2) You upload the picture and don’t lose your job
3) You don’t upload the picture and lose your job
4) You don’t upload the picture and don’t lose your job
There is only one possible case in which you can say your friend was wrong: the second outcome in which you upload the picture but still keep your job. In the last two cases, your friend didn’t say
anything about what would happen if you didn’t upload the picture, so you can’t say that their statement was wrong. Even if you didn’t upload the picture and lost your job anyway, your friend never
said that you were guaranteed to keep your job if you didn’t upload the picture; you might lose your job for missing a shift or punching your boss instead.
In traditional logic, a conditional is considered true as long as there are no cases in which the antecedent is true and the consequent is false.
Truth table for the conditional
\hline p & q & p \rightarrow q \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{T} \\
Again, if the antecedent \(p\) is false, we cannot prove that the statement is a lie, so the result of the third and fourth rows is true.
Example 22
Construct a truth table for the statement \((m \wedge \sim p) \rightarrow r\)
We start by constructing a truth table with 8 rows to cover all possible scenarios. Next, we can focus on the antecedent, \(m \wedge \sim p\).
\hline m & p & r \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \hline
\hline m & p & r & \sim p \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\
\hline m & p & r & \sim p & m \wedge \sim p \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} \\
Now we can create a column for the conditional. Because it can be confusing to keep track of all the Ts and \(\mathrm{Fs}\), why don't we copy the column for \(r\) to the right of the column for \(m
\wedge \sim p\) ? This makes it a lot easier to read the conditional from left to right.
\hline m & p & r & \sim p & m \wedge \sim p & r & (m \wedge \sim p) \rightarrow r \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\
When \(m\) is true, \(p\) is false, and \(r\) is false- -the fourth row of the table-then the antecedent \(m \wedge \sim p\) will be true but the consequent false, resulting in an invalid
conditional; every other case gives a valid conditional.
If you want a real-life situation that could be modeled by \((m \wedge \sim p) \rightarrow r\), consider this: let \(m=\) we order meatballs, \(p=\) we order pasta, and \(r=\) Rob is happy. The
statement \((m \wedge \sim p) \rightarrow r\) is "if we order meatballs and don't order pasta, then Rob is happy". If \(m\) is true (we order meatballs), \(p\) is false (we don't order pasta), and \
(r\) is false (Rob is not happy), then the statement is false, because we satisfied the antecedent but Rob did not satisfy the consequent.
For any conditional, there are three related statements, the converse, the inverse, and the contrapositive.
Related Statments
The original conditional is \(\quad\) "if \(p,\) then \(q^{\prime \prime} \quad p \rightarrow q\)
The converse is \(\quad\) "if \(q,\) then \(p^{\prime \prime} \quad q \rightarrow p\)
The inverse is \(\quad\) "if not \(p,\) then not \(q^{\prime \prime} \quad \sim p \rightarrow \sim q\)
The contrapositive is "if not \(q,\) then not \(p^{\prime \prime} \quad \sim q \rightarrow \sim p\)
Example 23
Consider again the conditional “If it is raining, then there are clouds in the sky.” It seems reasonable to assume that this is true.
The converse would be “If there are clouds in the sky, then it is raining.” This is not always true.
The inverse would be “If it is not raining, then there are not clouds in the sky.” Likewise, this is not always true.
The contrapositive would be “If there are not clouds in the sky, then it is not raining.” This statement is true, and is equivalent to the original conditional.
Looking at truth tables, we can see that the original conditional and the contrapositive are logically equivalent, and that the converse and inverse are logically equivalent.
A conditional statement and its contrapositive are logically equivalent.
The converse and inverse of a conditional statement are logically equivalent.
In other words, the original statement and the contrapositive must agree with each other; they must both be true, or they must both be false. Similarly, the converse and the inverse must agree with
each other; they must both be true, or they must both be false.
Be aware that symbolic logic cannot represent the English language perfectly. For example, we may need to change the verb tense to show that one thing occurred before another.
Example 24
Suppose this statement is true: “If I eat this giant cookie, then I will feel sick.” Which of the following statements must also be true?
1. If I feel sick, then I ate that giant cookie.
2. If I don’t eat this giant cookie, then I won’t feel sick.
3. If I don’t feel sick, then I didn’t eat that giant cookie.
1. This is the converse, which is not necessarily true. I could feel sick for some other reason, such as drinking sour milk.
2. This is the inverse, which is not necessarily true. Again, I could feel sick for some other reason; avoiding the cookie doesn’t guarantee that I won’t feel sick.
3. This is the contrapositive, which is true, but we have to think somewhat backwards to explain it. If I ate the cookie, I would feel sick, but since I don’t feel sick, I must not have eaten the
Notice again that the original statement and the contrapositive have the same truth value (both are true), and the converse and the inverse have the same truth value (both are false).
Try it Now 5
“If you microwave salmon in the staff kitchen, then I will be mad at you.” If this statement is true, which of the following statements must also be true?
1. If you don’t microwave salmon in the staff kitchen, then I won’t be mad at you.
2. If I am not mad at you, then you didn’t microwave salmon in the staff kitchen.
3. If I am mad at you, then you microwaved salmon in the staff kitchen.
Choice b is correct because it is the contrapositive of the original statement.
Consider the statement “If you park here, then you will get a ticket.” What set of conditions would prove this statement false?
1. You don’t park here and you get a ticket.
2. You don’t park here and you don’t get a ticket.
3. You park here and you don’t get a ticket.
The first two statements are irrelevant because we don’t know what will happen if you park somewhere else. The third statement, however contradicts the conditional statement “If you park here, then
you will get a ticket” because you parked here but didn’t get a ticket. This example demonstrates a general rule; the negation of a conditional can be written as a conjunction: “It is not the case
that if you park here, then you will get a ticket” is equivalent to “You park here and you do not get a ticket.”
The Negation of a Conditional
The negation of a conditional statement is logically equivalent to a conjunction of the antecedent and the negation of the consequent.
\(\sim(p \rightarrow q)\) is equivalent to \(p \wedge \sim q\)
Example 25
Which of the following statements is equivalent to the negation of “If you don’t grease the pan, then the food will stick to it” ?
1. I didn’t grease the pan and the food didn’t stick to it.
2. I didn’t grease the pan and the food stuck to it.
3. I greased the pan and the food didn’t stick to it.
1. This is correct; it is the conjunction of the antecedent and the negation of the consequent. To disprove that not greasing the pan will cause the food to stick, I have to not grease the pan and
have the food not stick.
2. This is essentially the original statement with no negation; the “if…then” has been replaced by “and”.
3. This essentially agrees with the original statement and cannot disprove it.
Try it Now 6
“If you go swimming less than an hour after eating lunch, then you will get cramps.” Which of the following statements is equivalent to the negation of this statement?
1. I went swimming more than an hour after eating lunch and I got cramps.
2. I went swimming less than an hour after eating lunch and I didn’t get cramps.
3. I went swimming more than an hour after eating lunch and I didn’t get cramps.
Choice b is equivalent to the negation; it keeps the first part the same and negates the second part.
In everyday life, we often have a stronger meaning in mind when we use a conditional statement. Consider “If you submit your hours today, then you will be paid next Friday.” What the payroll rep
really means is “If you submit your hours today, then you will be paid next Friday, and if you don’t submit your hours today, then you won’t be paid next Friday.” The conditional statement if t, then
p also includes the inverse of the statement: if not t, then not p. A more compact way to express this statement is “You will be paid next Friday if and only if you submit your timesheet today.” A
statement of this form is called a biconditional.
A biconditional is a logical conditional statement in which the antecedent and consequent are interchangeable.
A biconditional is written as \(p \leftrightarrow q\) and is translated as " \(p\) if and only if \(q^{\prime \prime}\).
Because a biconditional statement \(p \leftrightarrow q\) is equivalent to \((p \rightarrow q) \wedge(q \rightarrow p),\) we may think of it as a conditional statement combined with its converse: if
\(p\), then \(q\) and if \(q\), then \(p\). The double-headed arrow shows that the conditional statement goes from left to right and from right to left. A biconditional is considered true as long as
the antecedent and the consequent have the same truth value; that is, they are either both true or both false.
Truth table for the biconditional
\hline p & q & p \leftrightarrow q \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{T} \\
Notice that the fourth row, where both components are false, is true; if you don’t submit your timesheet and you don’t get paid, the person from payroll told you the truth.
Example 26
Suppose this statement is true: “The garbage truck comes down my street if and only if it is Thursday morning.” Which of the following statements could be true?
1. It is noon on Thursday and the garbage truck did not come down my street this morning.
2. It is Monday and the garbage truck is coming down my street.
3. It is Wednesday at 11:59PM and the garbage truck did not come down my street today.
1. This cannot be true. This is like the second row of the truth table; it is true that I just experienced Thursday morning, but it is false that the garbage truck came.
2. This cannot be true. This is like the third row of the truth table; it is false that it is Thursday, but it is true that the garbage truck came.
3. This could be true. This is like the fourth row of the truth table; it is false that it is Thursday, but it is also false that the garbage truck came, so everything worked out like it should.
Try it Now 7
Suppose this statement is true: “I wear my running shoes if and only if I am exercising.” Determine whether each of the following statements must be true or false.
1. I am exercising and I am not wearing my running shoes.
2. I am wearing my running shoes and I am not exercising.
3. I am not exercising and I am not wearing my running shoes.
Choices a & b are false; c is true.
Example 27
Create a truth table for the statement \((A \vee B) \leftrightarrow \sim C\)
Whenever we have three component statements, we start by listing all the possible truth value combinations for \(A, B,\) and \(C .\) After creating those three columns, we can create a fourth column
for the antecedent, \(A \vee B\). Now we will temporarily ignore the column for \(C\) and focus on \(A\) and \(B\), writing the truth values for \(A \vee B\).
\hline A & B & C \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{F} \\
\hline A & B & C & A \vee B \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} \\
Next we can create a column for the negation of \(C\). (Ignore the \(A \vee B\) column and simply negate the values in the \(C\) column.)
\hline A & B & C & A \vee B & \sim C \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\
Finally, we find the truth values of \((A \vee B) \leftrightarrow \sim C\). Remember, a biconditional is true when the truth value of the two parts match, but it is false when the truth values do not
\hline A & B & C & A \vee B & \sim C & (A \vee B) \leftrightarrow \sim C \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} \\
To illustrate this situation, suppose your boss needs you to do either project \(A\) or project \(B\) (or both, if you have the time). If you do one of the projects, you will not get a crummy review
( \(C\) is for crummy). So \((A \vee B) \leftrightarrow \sim C\) means "You will not get a crummy review if and only if you do project \(A\) or project \(B\)." Looking at a few of the rows of the
truth table, we can see how this works out. In the first row, \(A, B,\) and \(C\) are all true: you did both projects and got a crummy review, which is not what your boss told you would happen! That
is why the final result of the first row is false. In the fourth row, \(A\) is true, \(B\) is false, and \(C\) is false: you did project \(A\) and did not get a crummy review. This is what your boss
said would happen, so the final result of this row is true. And in the eighth row, \(A, B\), and \(C\) are all false: you didn't do either project and did not get a crummy review. This is not what
your boss said would happen, so the final result of this row is false. (Even though you may be happy that your boss didn't follow through on the threat, the truth table shows that your boss lied
about what would happen.) | {"url":"https://archeryhut.net/article/17-6-truth-tables-conditional-biconditional","timestamp":"2024-11-07T03:56:49Z","content_type":"text/html","content_length":"150089","record_id":"<urn:uuid:cae96cab-97c8-433f-9eb5-9df74b30a77d>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00244.warc.gz"} |
All from one, one for all: On model checking using representatives
Checking that a given finite state program satisfies a linear temporal logic property is suffering in many cases from a severe space and time explosion. One way to cope with this is to reduce the
state graph used for model checking. We define an equivalence relation between infinite sequences, based on infinite traces such that for each equivalence class, either all or none of the sequences
satisfy the checked formula. We present an algorithm for constructing a state graph that contains at least one representative sequence for each equivalence class. This allows applying existing model
checking algorithms to the reduced state graph rather than on the larger full state graph of the program. It also allows model checking under fairness assumptions, and exploits these assumptions to
obtain smaller state graphs. A formula rewriting technique is presented to allow coarser equivalence relation among sequences, such that less representatives are needed.
Original language English
Title of host publication Computer Aided Verification - 5th International Conference, CAV 1993, Proceedings
Editors Costas Courcoubetis
Publisher Springer Verlag
Pages 409-423
Number of pages 15
ISBN (Print) 9783540569220
State Published - 1993
Externally published Yes
Event 5th International Conference on Computer Aided Verification, CAV 1993 - Elounda, Greece
Duration: 28 Jun 1993 → 1 Jul 1993
Publication series
Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume 697 LNCS
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Conference 5th International Conference on Computer Aided Verification, CAV 1993
Country/Territory Greece
City Elounda
Period 28/06/93 → 1/07/93
Bibliographical note
Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 1993.
Dive into the research topics of 'All from one, one for all: On model checking using representatives'. Together they form a unique fingerprint. | {"url":"https://cris.biu.ac.il/en/publications/all-from-one-one-for-all-on-model-checking-using-representatives-2","timestamp":"2024-11-01T19:10:51Z","content_type":"text/html","content_length":"57465","record_id":"<urn:uuid:14c15d82-92a5-43b7-b148-a005da74b9da>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00215.warc.gz"} |
ssyr2.f - Linux Manuals (3)
ssyr2.f (3) - Linux Manuals
ssyr2.f -
subroutine ssyr2 (UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA)
Function/Subroutine Documentation
subroutine ssyr2 (characterUPLO, integerN, realALPHA, real, dimension(*)X, integerINCX, real, dimension(*)Y, integerINCY, real, dimension(lda,*)A, integerLDA)
SSYR2 Purpose:
SSYR2 performs the symmetric rank 2 operation
A := alpha*x*y**T + alpha*y*x**T + A,
where alpha is a scalar, x and y are n element vectors and A is an n
by n symmetric matrix.
UPLO is CHARACTER*1
On entry, UPLO specifies whether the upper or lower
triangular part of the array A is to be referenced as
UPLO = 'U' or 'u' Only the upper triangular part of A
is to be referenced.
UPLO = 'L' or 'l' Only the lower triangular part of A
is to be referenced.
N is INTEGER
On entry, N specifies the order of the matrix A.
N must be at least zero.
ALPHA is REAL
On entry, ALPHA specifies the scalar alpha.
X is REAL array of dimension at least
( 1 + ( n - 1 )*abs( INCX ) ).
Before entry, the incremented array X must contain the n
element vector x.
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X. INCX must not be zero.
Y is REAL array of dimension at least
( 1 + ( n - 1 )*abs( INCY ) ).
Before entry, the incremented array Y must contain the n
element vector y.
INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y. INCY must not be zero.
A is REAL array of DIMENSION ( LDA, n ).
Before entry with UPLO = 'U' or 'u', the leading n by n
upper triangular part of the array A must contain the upper
triangular part of the symmetric matrix and the strictly
lower triangular part of A is not referenced. On exit, the
upper triangular part of the array A is overwritten by the
upper triangular part of the updated matrix.
Before entry with UPLO = 'L' or 'l', the leading n by n
lower triangular part of the array A must contain the lower
triangular part of the symmetric matrix and the strictly
upper triangular part of A is not referenced. On exit, the
lower triangular part of the array A is overwritten by the
lower triangular part of the updated matrix.
LDA is INTEGER
On entry, LDA specifies the first dimension of A as declared
in the calling (sub) program. LDA must be at least
max( 1, n ).
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
November 2011
Further Details:
Level 2 Blas routine.
-- Written on 22-October-1986.
Jack Dongarra, Argonne National Lab.
Jeremy Du Croz, Nag Central Office.
Sven Hammarling, Nag Central Office.
Richard Hanson, Sandia National Labs.
Definition at line 148 of file ssyr2.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://www.systutorials.com/docs/linux/man/3-ssyr2.f/","timestamp":"2024-11-07T10:24:23Z","content_type":"text/html","content_length":"10245","record_id":"<urn:uuid:ee019c6a-57c2-4d85-9db8-b5d488fcb67a>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00811.warc.gz"} |
numpy.polynomial.laguerre.laggrid2d(x, y, c)[source]¶
Evaluate a 2-D Laguerre series on the Cartesian product of x and y.
This function returns the values:
where the points (a, b) consist of all pairs formed by taking a from x and b from y. The resulting points form a grid with x in the first dimension and y in the second.
The parameters x and y are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either x and y or their elements must support
multiplication and addition both with themselves and with the elements of c.
If c has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape + y.shape.
x, y : array_like, compatible objects
The two dimensional series is evaluated at the points in the Cartesian product of x and y. If x or y is a list or tuple, it is first converted to an ndarray, otherwise it is left
unchanged and, if it isn’t an ndarray, it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in c[i,j]. If c has dimension greater than two the remaining indices enumerate
multiple sets of coefficients.
values : ndarray, compatible object
The values of the two dimensional Chebyshev series at points in the Cartesian product of x and y. | {"url":"https://docs.scipy.org/doc/numpy-1.15.1/reference/generated/numpy.polynomial.laguerre.laggrid2d.html","timestamp":"2024-11-08T17:40:07Z","content_type":"text/html","content_length":"10783","record_id":"<urn:uuid:6055af6e-a9e9-4ad4-a15f-ee1ef0042f22>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00619.warc.gz"} |
Isosceles trapezoid v3 - math word problem (1619)
Isosceles trapezoid v3
In an isosceles trapezoid ABCD is the size of the angle β = 123°
Determine size of angles α, γ and δ.
Correct answer:
Did you find an error or inaccuracy? Feel free to
write us
. Thank you!
You need to know the following knowledge to solve this word math problem:
Units of physical quantities:
Grade of the word problem:
We encourage you to watch this tutorial video on this math problem:
Related math problems and questions: | {"url":"https://www.hackmath.net/en/math-problem/1619","timestamp":"2024-11-12T06:26:06Z","content_type":"text/html","content_length":"53822","record_id":"<urn:uuid:37d39a95-0a0d-4e21-af0d-f8b55b86271a>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00688.warc.gz"} |
Material frame-indifference in turbulence modeling
Theoretical support is developed for the acceptability of the application of the principle of material frame-indifference to turbulence modeling. The Reynolds stress tensor has been proven
frame-independent in two-dimensional turbulence, such as for upper atmospheric modeling. Observer-independent velocity fluctuations are discussed, showing that ensemble mean velocities are
frame-indifferent. The frame-indifferent stress tensor is therefore only a special solution to the Navier-Stokes equations, which are frame-dependent. However, at distances sufficiently far from the
boundaries three-dimensional eddies are aligned with the axis of rotation, as proven by the Taylor-Proudman theory, and thus become two-dimensional and frame-independent.
ASME Journal of Applied Mechanics
Pub Date:
December 1984
□ Closure Law;
□ Computational Fluid Dynamics;
□ Flow Theory;
□ Reynolds Stress;
□ Stress Tensors;
□ Turbulent Flow;
□ Flow Velocity;
□ Invariance;
□ Navier-Stokes Equation;
□ Rotating Fluids;
□ Fluid Mechanics and Heat Transfer | {"url":"https://ui.adsabs.harvard.edu/abs/1984ATJAM..51..942S/abstract","timestamp":"2024-11-08T18:54:49Z","content_type":"text/html","content_length":"35179","record_id":"<urn:uuid:3386e292-b593-4da6-b6c5-6524188053ff>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00330.warc.gz"} |
How Many Ounces Are In A Pint? - 2024
It’s a common question – how many ounces are in a pint? Whether you’re measuring ingredients for baking, properly portioning out drinks at the bar or just curious to learn more about measurements and
conversions, understanding the answer can provide valuable insight into the world of weighing and measuring liquids. In this post, we’ll explore exactly how many ounces are in a pint so you have all
the information needed on hand to measure accurately next time you find yourself needing an answer. So grab your liquid measuring cup and let’s get started!
Definition of a Pint
A pint is a unit of measurement used to measure liquid volume, which is traditionally equal to 16 fluid ounces in the US and 20 fluid ounces in the UK. It is typically used to measure beer, but can
also be used for other types of beverages. A half-pint is equal to 8 fluid ounces in the United States and 10 fluid ounces in the United Kingdom. The term “pint” comes from an old French word meaning
“a little less than a quart.”
Pints are commonly found on beer labels, however they can also be seen listed as ingredients or measurements when preparing food recipes. Many countries use similar units of measurements for liquid
volume such as liters, gallons, or milliliters, so it’s important to double check when using a pint in the US or UK. Ultimately, a pint is still an important unit of measurement and should be used
properly when dealing with liquids!
Definition of a Ounce
An ounce is a unit of mass in the US customary and Imperial systems, most commonly referenced by its abbreviated form of oz. It is equal to 28.35 grams or 1/16th of a pound. An ounce is also used as
a unit of volume for fluid measurements such as cooking ingredients and essential oils. The symbol for an ounce is “oz”. For weight measurement, one ounce has approximately the same mass as three
dice, one-tenth of a pound (or four quarters).
For liquid measurements, one fluid ounce has approximately the same volume as two tablespoons or six teaspoons. In terms of energy, one ounce can provide around 100 calories from food sources such as
meats and grain products. Ounces are commonly used when measuring precious metals, such as gold and silver. They are also used when talking about items that are sold by weight, such as produce or
drugs. In the English system of measurement, an ounce is currently equal to 28.35 grams.
This is a very small amount and it takes 16 ounces to make one pound (0.45 kg). In comparison, in the metric system of measurement an ounce is equal to 30 grams and a pound is 450 grams (0.45 kg).
The use of ounces differs from country to country but the most common use is for weighing food ingredients or measuring liquid volumes in recipes. In some countries like Canada, Australia, and New
Zealand the imperial system still prevails for grocery shopping and ingredient measurements in recipes.
Different Types of Ounces and Pints
Different Types of Ounces and Pints
When discussing measurements, it is important to understand the different types of ounces and pints that can be used. Ounces can be measured in two ways: Avoirdupois ounces (which are commonly used
in the United States) or Troy ounces (which are typically used for precious metals).
A pint is a unit of volume which, depending on which system you use, can have different measurements. The US customary system uses liquid pints which measure 16 fluid ounces, while the British
imperial system uses dry pints which measure 20 fluid ounces. It is important to know which type of ounce or pint you are using when making measurements as this could result in an incorrect
measurement if not done properly.
Understanding the units of measurement being used is essential to ensure accuracy when making measurements. Knowing the difference between Avoirdupois ounces and Troy ounces, or liquid pints and dry
pints, can save time and energy in the long run. It’s important to use the right type of ounce or pint for your project to make sure that your measurements are accurate. With a little knowledge about
different types of ounces and pints, you can ensure that you get an accurate measurement every single time.
Measurement Equivalents of a Pint
In addition to the US pint being equal to 16 US fluid ounces, one US pint is also equal to: 1/2 of a quart, 8 gills, 4 cups, 2 pints (U.S. liquid), 1.03 British Imperial (BI) pints, 33.6 cubic inches
and 28.875 cubic centimeters of volume or capacity.
Applications of Knowing Ounces in a Pint
Knowing how many ounces are in a pint can be useful for households that need to measure liquids for recipes or other applications such as measuring laundry detergent, paint, or other liquids. As
knowing the number of ounces in a pint can be helpful for both merchants and customers when shopping for various items sold by volume.
Historical Significance of the Pint Measurement
The pint is an ancient unit of measurement believed to have originated as early as the 14th century in Europe. It was introduced into England by Edward III who reigned from 1327-1377 AD. Initially,
it was used to measure wine barrels but later evolved to become one of the most widely used measurements for liquid volume and capacity throughout the world today.
Knowing how many ounces are in a pint is essential if you need to accurately measure liquids or other items that are sold by volume. With the knowledge of how many ounces are in a pint, anyone can
easily convert measurements from one unit to another and make accurate calculations for recipes or home projects. Additionally, learning the historical significance of this useful measurement helps
us understand its importance over time and appreciate its continued use today.
How Many Ounces Are In A Pint?
A pint is a unit of measurement that is equal to 16 fluid ounces. This means that when you convert one pint, it will be equal to the amount of liquid contained in 16 fluid ounces. A fluid ounce is a
measure of volume, and is equivalent to approximately 29.57 milliliters (ml). It’s important to note that the Imperial system uses pints as its standard unit for measuring liquid volume, while the
United States Customary system typically opts for cups instead. Converting between these two systems can become confusing and difficult due to slight variations in their measurements. By
understanding how many ounces are in a pint, however, conversions between them become much simpler — 1 pint = 16 ounces.
When considering other measurements of volume, it’s useful to remember that 1 pint is equal to 2 cups. This makes it easier to convert between different units of measurement when dealing with smaller
increments of liquid. Additionally, a quart is equivalent to two pints — so if you want to know how many ounces are in a quart, simply multiply 16 by two for a total of 32 fluid ounces.
Knowing how many ounces are in a pint can be useful when measuring out ingredients for recipes or baking projects. It allows you to accurately measure out the correct amount of liquid needed without
having to worry about conversions or discrepancies between systems. With this knowledge, you’ll be able to easily and quickly figure out exactly what measurements are necessary for your next culinary
By understanding how many ounces are in a pint, you’ll be able to quickly and accurately measure out the necessary liquid for any recipe or project. This will help make sure that your food is cooked
properly, and ultimately results in more delicious dishes! By remembering this simple conversion rate — 1 pint = 16 fluid ounces — you’ll always have the right measurements when cooking. So get ready
to whip up some amazing meals with confidence and accuracy!
Conversion Factors for Ounces and Pints:
For conversions of ounces to pints, there is a simple formula that can be used. One pint equals 16 ounces, so for each ounce you would have 0.0625 of a pint. To figure the amount of pinst in ounces,
divide the number of ounces by 16. For example, if you had 32 ounces, you would have 2 pints (32/16 = 2).
Similarly, if you had 3 pints you would have 48 ounces (3*16 = 48). Using these conversion factors, it’s easy to convert between these two units of measurement. However, it’s important to
double-check your calculations as mistakes can easily be made when working with decimals and fractions. By remembering the basic formula and double-checking your work, you can ensure accuracy when
performing conversions between ounces and pints.
Converting Fluid Ounces to Pints:
Pints, a unit of measurement for volume, are commonly used in the United States and the United Kingdom. A pint is equal to 16 fluid ounces. Thus, when converting from fluid ounces to pints, one must
divide their initial volume measure by 16.
For example, if you have 32 fluid ounces of liquid, then you have 2 pints (32 / 16 = 2). The answer can be rounded up or down depending on the desired precision level; for instance, if the accuracy
required is no more than 1 ounce difference between the target amount of liquid and your answer (rounded up), then rounding up in this case would provide an appropriate solution.
Converting fluid ounces to pints is a relatively straightforward process, and can be useful for calculating the amounts of liquids needed for recipes or other tasks. It is important to note that
different countries may use different units of measurement (such as liters), so it is best to double-check measurements when working in an international setting.
Knowing how to convert fluid ounces to pints can help make measuring out liquids simpler and more accurate. With basic math skills and a knowledge of the necessary conversion factor, you can easily
calculate the amount of liquid desired in pints.
8 fl oz 1 cup 1/2 pint 1/4 quart 1/16 gallon
16 fl oz 2 cups 1 pint 1/2 quart 1/8 gallon
32 fl oz 4 cups 2 pints 1 quart 1/4 gallon
64 fl oz 8 cups 4 pints 2 quarts 1/2 gallon
128 fl oz 16 cups 8 pints 4 quarts 1 gallon
Converting Pints to Fluid Ounces:
When converting pints to fluid ounces, it is important to remember that there are 16 fluid ounces in 1 pint. This means that a pint can be divided into 16 equal parts, each of which is a fluid ounce.
For example, if you have 2 pints, then you would have 32 total fluid ounces (2 pints x 16 ounces = 32 fl oz). In order to convert from pints to fluid ounces all you need to do is multiply the number
of pints by 16. To convert from fluid ounces back to pints, divide the number of fluid ounces by 16.
To easily keep track of conversions it may be helpful to use a conversion chart or calculator. These tools provide an easy reference for quickly looking up the correct conversion. Additionally, they
can help to avoid calculation errors due to incorrect arithmetic or miscalculation of conversions.
Knowing how to convert pints to fluid ounces is an important skill for anyone who deals with measurements and calculations involving liquid volume. This knowledge will be useful in a variety of
situations such as recipes, brewing beer, measuring ingredients in a laboratory, and more. With this understanding you should be able to accurately make any necessary conversions when needed.
Wet Vs Dry Pint:
When measuring liquids, it’s important to take into account the difference between dry and wet measurements. A dry pint is 16 ounces, while a liquid pint is 20 ounces. This discrepancy is due to the
differing densities of solids and liquids; because liquids are generally less dense, they occupy more space than their solid counterparts.
It’s important to pay attention when using a recipe that calls for either type of measurement; using the wrong type can lead to dish failure or incorrect proportions in your finished product!
Whenever you encounter a measurement in cups or pints, be sure to double-check whether it requires a dry or liquid measure. That way, you’ll ensure that your recipe turns out perfectly every time!
Fluid Ounces (Fl Oz) Vs Dry Ounces:
Fluid Ounces (Fl Oz) Vs Dry OuncesFluid Ounces (Fl Oz) Vs Dry Ounces
Fluid Ounces (Fl Oz) are a unit of measurement for liquids, while Dry Ounces are used to measure solids. These measurements may differ depending on the material being measured and should not be
confused when using them in recipes. A Fl Oz is equal to 1/8 of a cup or 2 tablespoons, while one dry ounce is equivalent to 4 tablespoons or .125 cups.
To give an example, one pint of blueberries weighs about 12 oz despite there being 16 dry ounces in a pint due to how it fills the container differently than if it were filled with liquid taking up
100% of the space. Here are some quick measurements for reference:
Fluid Ounces
• 1 Fl Oz = 1/8 of a cup or 2 tablespoons
• 10 Fl Oz = 1.25 cups
• 20 Fl Oz = 2.5 cups
• 50 Fl Oz = 6.25 cups
Dry Ounces
• 1 oz = .05 pints
• 10 oz = .53 pints
• 20 oz = 1.07 pints
• 50 oz = 2.68 pints
It is important to pay close attention when measuring solids and liquids in ounces and always double check the measurements when using them in recipes to ensure accuracy and good results! Proper
measurement can mean the difference between delicious treats and complete disasters – so be sure to measure correctly!
Differences between US and UK Measurements:
Differences between US and UK Measurements
When it comes to measurements, there are some significant differences between the United States (US) and the United Kingdom (UK). The main difference is in terms of system being used: US measurements
tend to be Imperial, or Standard, while UK measurements are typically Metric.
In the Imperial system, commonly used units of measurement include pounds for weight and inches for length. Meanwhile, in the Metric system, kilograms and meters are generally regarded as standard
units of measure. Furthermore, temperatures are measured differently in each region; Celsius is used in most parts of the world including UK and Europe, while Fahrenheit is favored in the US.
The differences between these two systems can cause confusion when traveling between countries or converting recipes from one system to the other. Therefore, it is important to understand both of
these systems and be familiar with how to convert measurements from one unit to another. Fortunately, there are plenty of online resources available to help make this process easier.
It is worth noting that some countries may use a combination of Imperial and Metric units for measuring different quantities; for example, in Canada, temperatures are measured in Celsius while
distances are measured in feet or miles. It is therefore advisable to do research on the local measurement system before traveling abroad. Understanding differences between US and UK measurements
will ensure you don’t get confused when faced with unfamiliar units of measure!
Conclusion: How Many Ounces Are In A Pint
In conclusion, to answer the question – how many ounces are in a pint? – it can officially be stated that there are 16 US fluid ounces in a regular imperial pint. Now you have all the information
needed to know precisely how much you’re measuring out each time you whip out your liquid measuring cup. Whether it’s for baking, bartending or just some friendly kitchen science experiments,
remember that one pint equals 16 ounces and you’ll never go wrong.
With this knowledge on hand, your portioning will always be spot on and your experiment results even more accurate than before! With the scales no longer a problem, it’s time to let your creativity
shine in the kitchen.
FAQs of Ounces Are In A Pint:
What is a Dry Pint?
A dry pint is an imperial unit of volume typically used for measuring dry ingredients, such as flour and grains. It equals to about 567 milliliters.
How Much does One Cup of Liquid Weigh in Pounds?
One cup of water weighs approximately 8 fluid ounces which converts to 0.5lb or 227g. For other liquids, the density or specific gravity should be taken into account to find out the exact weight.
What Is the Difference Between a Pint and Quart?
A pint is equal to two cups or one-half of a quart whereas a quart is twice as large and contains four cups. A pint is 16 ounces and a quart is 32 ounces.
How Many Ounces Are In a Gallon?
There are 128 ounces in one gallon. One gallon is equal to four quarts or eight pints, so each quart contains 32 ounces and each pint contains 16 ounces.
What Is the Volume of an Ounce?
The volume of one ounce is equal to 2 tablespoons, or 1/8th of a cup. This means that there are 8 fluid ounces in one cup and 16 fluid ounces in one pint.
What Is the Metric Equivalent of an Ounce?
An ounce is equivalent to 28.35 grams in the metric system. This means that one pint is equivalent to 473 milliliters and one gallon is equivalent to 3785 milliliters.
How Many Ounces of Liquid Does a Teaspoon Hold?
One teaspoon holds approximately 0.17 ounces or 5 milliliters of liquid. Therefore, it would take about 96 teaspoons to make a pint of liquid, and 384 teaspoons for a gallon.
It can be useful to understand how many ounces are in each unit of measurement when working with liquids or dry goods. Knowing this information can help you measure more accurately when cooking or
baking. Understanding these conversions will allow you to convert easily between different measurements and get exactly what you need.
How Many Ounces In A Pint UK?
A pint in the UK contains 20 imperial (also known as British) fluid ounces. This is a bit different than in the United States, where a pint contains 16 US fluid ounces. The imperial system of
measurement is still widely used throughout the United Kingdom and other Commonwealth countries, and it dates back to 1824 when it was adopted by an Act of Parliament.
One imperial fluid ounce is approximately 28 milliliters, or just over 0.9 US fluid ounces. So one UK pint would be equal to 1¾ US pints or 35-36 milliliters (depending on whether you round up or
down). To give another example, there are roughly 568mls in a UK pint – that’s three quarters of an American cup measure too!
It’s worth noting that both imperial and metric measurements are legally recognized for use within the UK; however certain sections from trading standards do require food items to be labeled using
only metric measurements for sale within shops etc., so make sure check before you buy if something is sold by volume!
How Many Ounces in a Pint of Cherry Tomatoes?
A pint of cherry tomatoes is typically 16 ounces or 2 cups when measured by volume. This could vary slightly depending on the size and type of tomato used, but generally speaking a standard pint
contains 16 ounces of cherry tomatoes. If you’re looking to provide precise measurements for cooking recipes within a potential margin of error, weighing your ingredients may be more accurate than
measuring by volume alone.
Depending on the type, variety and quality of tomato you are using there may be slight variations in weight per pint (1lb is equal to about 16-20 ounces). For instance, large Roma cherry tomatoes
weigh 3-4 grams each which would equate to 24-32 ounces per pound or 1 ½ – 2 pints. Conversely, smaller petite tomatoes can range from 6-8 grams each which would mean 10 – 13 ounces per pound or ⅔ –
¾ pint.
It’s important to note that if you purchase pre-packaged cherry tomatoes at most supermarkets they are usually sold in 14 ounce containers that contain approximately two heaping cups (or a scant 1⅓
pints) per container so keep this in mind when making measurements for recipes. Generally speaking though regardless if weighing or measuring by volume it’s recommended that you use a kitchen scale
whenever possible as it provides an even amount every time and helps ensure exact portion sizes which can be incredibly useful when following specific recipes with precision accuracy.
How Many Ounces In A Pint Of Raspberries?
A pint of raspberries typically contains around 8 ounces, or one cup, of raspberries. According to the U.S. Department of Agriculture, a pint is equal to two dry cups or 16 fluid ounces; meaning that
whether you’re measuring a dry food or a liquid beverage, two pints are equal to 32 fluid ounces (or four cups).
When it comes to fresh fruit like raspberries, it’s important to understand how much you’ll get in each pint as this will vary based on the individual berries. Generally speaking, one cup of fresh
raspberries weighs around 5-6 ounces and yields approximately 2-3 servings; thus eight ounces can be seen as equivalent to 1-2 servings. In other words – if you buy one pint of fresh raspberries from
your local grocery store you should expect 1-2 servings worth for your money!
How Many Ounces in a Dry Pint of Blueberries?
A dry pint of blueberries typically weighs about 12 ounces. This measurement includes the weight of the berries themselves as well as the container or packaging material that holds them.
It is important to note that this measurement is based on average-sized fresh or frozen blueberries and does not include any moisture lost during storage, shipping, or handling processes. For
example, a freshly picked pint may weigh more than 12 ounces if there is some water present in the fruits, but this weight can decrease eventually due to evaporation caused by temperature changes and
other external factors.
In addition to dry pints, blueberries are also sold in “drained weights”—which means that any liquid has been removed before the weight was recorded. When buying dried blueberries in bulk, it’s
important to pay attention to their drained weights because they usually come out lighter than expected when compared with their wet measurements (typically 6-7 ounces).
When measuring out blueberry ingredients for baking or cooking purposes, volume measurements can often be used instead of weighing them first; for instance one cup holds around 4-5 ounces of fresh or
frozen fruit depending on how it’s packed within the cup itself (tightly pressed vs loosely filled). It’s always best practice to check recipes’ ingredient lists carefully before beginning so you
don’t accidentally add too much fruit into your dish!
How Many Ounces In A Pint Glass?
A typical pint glass can hold 16 ounces of liquid when filled to the brim. It is important to note that some glasses come in a slightly different sizes, so always check the capacity of your
particular pint glass before estimating how much liquid it can hold. For reference, one ounce is equal to about 0.0296 liters or 29.6 milliliters, and two pints are equivalent to 32 ounces or 1 quart
of liquid.
When drinking from a pint glass, it’s important to understand volume versus weight; while 16 fluid ounces make up one U.S. customary unit known as “a pint,” the same exact amount of solid material
will weigh far more than its equivalent in liquid form (this is due to both density and buoyancy). Therefore when filling up a pint glass with solids like ice cubes or popcorn, you should expect less
than 16 total ounces per container because the extra air space between each item will add up significantly when combined together.
In addition, if you’re serving several drinks made out of frozen ingredients (like smoothies or slushies) then you should use even less than 16 fluid ounces per container since cold liquids tend to
expand once they reach room temperature! The beauty of using a standard sized-pint glass for those kind of beverages is that having everyone order a single drink yields an identical looking cup for
every guest at your event – which eliminates any potential confusion over varying quantities!
How Many Ounces In A Pint Of Sour Cream?
A pint of sour cream is equal to 16 ounces or 2 cups. The metric measurement for a pint of sour cream is 473 milliliters (ml). This is the equivalent of 1–2/3 cups or 16 fluid ounces (fl oz). Sour
cream is a thick, creamy dairy product made by combining pasteurized and homogenized light cream with lactic acid bacteria that ferments the milk sugars.
It has a tangy flavor and nutrient content similar to yogurt, but with a thicker consistency. Sour Cream can be used as an ingredient in savory dishes such as mashed potatoes and nachos, or it can be
eaten on its own as part of breakfast cereal, toast, bagels and sandwiches. It’s also often used as an ingredient in baking recipes like cheesecake and pancakes.
How Many Ounces In A Pint Of Beer?
There are 16 ounces in a pint of beer. This can vary slightly based on where you’re purchasing the beer, since different countries may have slightly different mandated serving sizes. In the United
States, the federal government has mandated that a pint must be equal to 16 fluid ounces. Typically beers will come with between 4-6 percent alcohol by volume (ABV).
In addition to pints, most bars and restaurants will also serve glasses of beer in two other common sizes – 12 ounce cans and 22 ounce bottles. Bottles of craft or imported beers sometimes come in
larger containers such as 25 ounce “bombers” or 32 ounce “liters”. And some establishments might even offer a specialty half-pint serving size for certain mega-strong Belgian ales!
As far as specific measurements go, one basic U.S pint measures out at about 473 milliliters (mL) – equivalent to 16 U.S fluid ounces (oz.). For imperial measurements, one British/Canadian imperial
pint is about 568 mL – 20 Imperial oz., which is approximately 19 U.S fluid oz..
And if you’re looking for more exact measurement than that? One standard 12 oz can measure out to 341 mL while an American longneck bottle is typically 355 mL or 11 US fl oz., with those heavier
bombers checking in at 757 mL or 24 US fl oz.. It’s worth noting that these last two containers aren’t always completely full due to their high production costs and therefore don’t technically
register as one complete unit of measure – though they still tend to constitute one serving size when served in taprooms and pubs around the country! | {"url":"https://flightwinebar.com/blog/how-many-ounces-are-in-a-pint/","timestamp":"2024-11-09T13:49:36Z","content_type":"text/html","content_length":"241150","record_id":"<urn:uuid:df1dfbe2-00ec-4102-9132-d66155e36263>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00247.warc.gz"} |
9,936 research outputs found
First, we study the fit of the Higgs boson rates, based on all the latest collider data, in the effective framework for any Extra-Fermion(s) [EF]. The best-fit results are presented in a generic
formalism allowing to apply those for the test of any EF scenario. The variations of the fit with each one of the five fundamental parameters are described, and, the obtained fits can be better than
in the Standard Model (SM). We show how the determination of the EF loop-contributions to the Higgs couplings with photons and gluons is relying on the knowledge of the top and bottom Yukawa
couplings (affected by EF mixings); for determining the latter coupling, the relevance of the investigation of the Higgs production in association with bottom quarks is emphasized. In the instructive
approximation of a single EF, we find that the constraints from the fit already turn out to be quite predictive, in both cases of an EF mixed or not with SM fermions, and especially when combined
with the extra-quark (-lepton) mass bounds from direct EF searches at the LHC (LEP) collider. In the case of an unmixed extra-quark, non-trivial fit constraints are pointed out on the Yukawa
couplings for masses up to ~200 TeV. In particular, we define the extra-dysfermiophilia, which is predicted at 68.27% C.L. for any single extra-quark (independently of its electric charge). Another
result is that, among any components of SM multiplet extensions, the extra-quark with a -7/3 electric charge is the one preferred by the present Higgs fit.Comment: 27 pages, 10 figures. Subsection
structure added and Higgs boson rates updated (in a separate Appendix) after the Moriond 2013 conferenc
We study the single chargino production $e^+ e^- \to \tilde \chi^{\pm} \mu^{\mp}$ at linear colliders which occurs through the \l_{121} R-parity violating coupling constant. We focus on the final
state containing 4 leptons and some missing energy. The largest background is \susyq and can be reduced using the initial beam polarization and some cuts based on the specific kinematics of the
single chargino production. Assuming the highest allowed supersymmetric background, a center of mass energy of $\sqrt s=500GeV$ and a luminosity of ${\cal L}=500fb^{-1}$, the sensitivities on the \l_
{121} coupling constant obtained from the single chargino production study improve the low-energy experimental limit over a range of $\Delta m_{\tilde u} \approx 500GeV$ around the sneutrino
resonance, and reach values of $\sim 10^{-4}$ at the $\tilde u$ pole. The single chargino production also allows to reconstruct the $\tilde \chi_1^{\pm}$, $\tilde \chi_2^{\pm}$ and $\tilde u$ masses.
The initial state radiation plays a fundamental role in this study.Comment: 24 pages, Latex file. Linear Collider note LC-TH-2000-04
The R-Parity symmetry Violating (RPV) version of the Next-to-Minimal Supersymmetric Standard Model (NMSSM) is attractive simultaneously with regard to the so-called mu-problem and the accommodation
of three-flavor neutrino data at tree level. In this context, we show here that if the Lightest Supersymmetric Particle (LSP) is the gravitino, it possesses a lifetime larger than the age of the
universe since its RPV induced decay channels are suppressed by the weak gravitational strength. This conclusion holds if one considers gravitino masses ~ 10^2 GeV like in supergravity scenarios, and
is robust if the lightest pseudoscalar Higgs field is as light as ~ 10 GeV [as may occur in the NMSSM]. For these models predicting in particular an RPV neutrino-photino mixing, the gravitino
lifetime exceeds the age of the universe by two orders of magnitude. However, we find that the gravitino cannot constitute a viable dark matter candidate since its too large RPV decay widths would
then conflict with the flux data of last indirect detection experiments. The cases of a sneutrino LSP or a neutralino LSP as well as the more promising gauge-mediated supersymmetry breaking scenario
are also discussed. Both the one-flavor simplification hypothesis and the realistic scenario of three neutrino flavors are analyzed. We have modified the NMHDECAY program to extend the neutralino
mass matrix to the present framework.Comment: Latex file, 23 pages, 7 figures. References added and discussion on the indirect detection modifie
We study the single productions of supersymmetric particles at Tevatron Run II which occur in the $2 \to 2-body$ processes involving R-parity violating couplings of type \l'_{ijk} L_i Q_j D_k^c. We
focus on the single gaugino productions which receive contributions from the resonant slepton productions. We first calculate the amplitudes of the single gaugino productions. Then we perform
analyses of the single gaugino productions based on the three charged leptons and like sign dilepton signatures. These analyses allow to probe supersymmetric particles masses beyond the present
experimental limits, and many of the \l'_{ijk} coupling constants down to values smaller than the low-energy bounds. Finally, we show that the studies of the single gaugino productions offer the
opportunity to reconstruct the $\tilde \chi^0_1$, $\tilde \chi^{\pm}_1$, $\tilde u_L$ and $\tilde l^{\pm}_L$ masses with a good accuracy in a model independent way.Comment: 47 pages, epsfi
We study a multi-localization model for charged leptons and neutrinos, including the possibility of a see-saw mechanism. This framework offers the opportunity to allow for realistic solutions in a
consistent model without fine-tuning of parameters, even if quarks are also considered. Those solutions predict that the large Majorana mass eigenvalues for right-handed neutrinos are of the same
order of magnitude, although this almost common mass can span a large range (bounded from above by $\sim 10^{12}{\rm GeV}$). The model also predicts Majorana masses between $\sim 10^{-2}{\rm eV}$ and
$\sim 5 \ 10^{-2}{\rm eV}$for the left-handed neutrinos, both in the normal and inverted mass hierarchy cases. This mass interval corresponds to sensitivities which are reachable by proposed
neutrinoless double$\beta$decay experiments. The preferred range for leptonic mixing angle$\theta_{13}$is:$10^{-2} \lesssim \sin \theta_{13} \lesssim 10^{-1}$, but smaller values are not totally
excluded by the model.Comment: 36 pages, 8 figure | {"url":"https://core.ac.uk/search/?q=author%3A(Moreau%2C%20G.)","timestamp":"2024-11-14T14:24:08Z","content_type":"text/html","content_length":"138103","record_id":"<urn:uuid:2e1b2f94-2d02-454b-9987-c033beb4b23f>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00768.warc.gz"} |
A Multiscale Test of Spatial Stationarity for Textured Images in R
This paper provides an introduction to the LS2Wstat package (Taylor and Nunes 2014), developed to implement recent statistical methodology for the analysis of (greyscale) textured images. Texture
analysis is a branch of image processing concerned with studying the variation in an image surface; this variation describes the physical properties of an object of interest. The key applications in
this field, namely discrimination, classification and segmentation, are often dependent on assumptions relating to the second-order structure (variance properties) of an image. In particular many
techniques commonly assume that images possess the property of spatial stationarity (Gonzalez and Woods 2001). However, for images arising in practice this assumption is often not realistic,
i.e. typically the second-order structure of an image varies across location. It is thus important to test this assumption of stationarity before performing further image analysis. See Figure 1 for
examples of textured images. For a comprehensive introduction to texture analysis, see (Bishop and Nasrabadi 2006) or (Petrou and Sevilla 2006).
Figure 1: Examples of textured images: fabric, creased material and hair (Eckley and Nason 2013, available from the R package LS2W;).
Recently, (Taylor et al. {in press}) proposed a test of spatial stationarity founded on the locally stationary two-dimensional wavelet (LS2W) modelling approach of (Eckley et al. 2010). The LS2W
modelling approach provides a location-based decomposition of the spectral structure of an image. The \(Bootstat_{LS2W}\) test proposed by (Taylor et al. {in press}) uses a statistic based on an
estimate of the process variance within a hypothesis testing framework, employing bootstrap resampling under the null hypothesis assumption of stationarity to assess its significance.
Given a test of spatial stationarity for random fields, it is natural to consider how this might be usefully applied within a problem such as texture segmentation. The ability to determine
non-stationarity and the presence of localised textured regions within images is important in a broad range of scientific and industrial applications, including product evaluation or quality control
purposes. Possible areas of use for the methods described in this article include identifying uneven wear in fabrics (Taylor et al. {in press}; Chan and Pang 2000) and defect detection on the surface
of industrial components (Wiltschi et al. 2000; Bradley and Wong 2001) or natural products (Funck et al. 2003; Pölzleitner 2003). For a review of texture segmentation, see (Pal and Pal 1993).
Readily available implementations for stationarity assessment have, up until now, been restricted to the time series setting; examples of such software in this context include the R add-on packages
urca (Pfaff 2008; Pfaff and Stigler 2013), CADFtest (Lupi 2009) and locits (Nason 2013b,a).
Below we describe the package LS2Wstat which implements the spatial test of stationarity proposed by (Taylor et al. {in press}). The package has been developed in R and makes use of several
functions within the LS2W package (Eckley and Nason 2011, 2013). The article is structured as follows. We begin by describing details of simulation of LS2W and other processes. An overview of the \
(Bootstat_{LS2W}\) test of stationarity is then given, focussing in particular on the function TOS2D. We then illustrate the application of the test on both simulated and real texture images.
Finally, the article concludes by describing how the algorithm might be embedded within a quadtree image splitting procedure to identify regions of texture homogeneity within a multi-textured image.
Simulating textured images with LS2Wstat
Before describing the implementation of the work proposed in (Taylor et al. {in press}), we first explain how to generate candidate textured images using the LS2Wstat package. Several different
spatially stationary and non-stationary random fields can be generated with the simTexture function. See the package help file for full details of the processes available.
To demonstrate the LS2Wstat implementation, throughout this article we consider a realisation of a white noise process with a subregion of random Normal deviates in its centre with a standard
deviation of 1.6. This simulated texture type is called NS5, and is one of several textures which can be simulated from the package. In particular, we consider an image of dimension \(512\times 512\)
with a central region that is a quarter of the image, i.e. a dimension size of \(128\times 128\). This can be obtained as follows:
> library("LS2Wstat")
> set.seed(1)
> X <- simTexture(n = 512, K = 1, imtype = "NS5", sd = 1.6, prop = 0.25)[[1]]
> image(plotmtx(X), col = grey(255:0/256))
The simTexture function returns a list of length K with each list entry being a matrix representing an image of dimension n \(\times\) n with the chosen spectral structure. In this case, since K = 1,
a list of length 1 is returned. The simulated image X is shown in Figure 2. Note in particular that visually, one can discern that the image consists of two subtly different texture types. Firstly,
the centre of the image has one particular form of second order structure. The second texture structure can be seen in the remainder of the image. Throughout the rest of this article we shall apply
the approach of (Taylor et al. {in press}) to this image.
Figure 2: An example of a textured image (NS5) simulated with the simTexture function.
Testing the spatial stationarity of images
We now briefly introduce the LS2W random field model of (Eckley et al. 2010) together with some associated notation, before describing the implementation of the test of stationarity proposed in (
Taylor et al. {in press}). For an accessible introduction to wavelets, please see (Prasad and Iyengar 1997), (Vidakovic 1999) or (Nason 2008).
The LS2W process model is defined by
\[\label{eq:ls2wproc} X_{\mathbf{r}} = \sum_{l} \sum_{j=1}^{\infty}\sum_{\mathbf{u}} w^{l}_{j,\mathbf{u}}\psi^{l}_{j,\mathbf{u}}(\mathbf{r})\xi^{l}_{j,\mathbf{u}} \, , \tag{1}\]
for directions \(l=h, v \mbox{~or~} d\) and spatial locations \(\mathbf{r}\), where \(\{\xi^l_{j, \mathbf{u}}\}\) is a zero-mean random orthonormal increment sequence; \(\{\psi^l_{j,\mathbf{u}}\}\)
is a set of discrete nondecimated wavelets and \(\{w^l_{j,\mathbf{u}} \}\) is a collection of amplitudes, constrained to vary slowly over locations of an image (Eckley et al. 2010). In the above
definition, we assume the image is of dyadic dimension, i.e. we have \(\mathbf{r}=(r,s)\) with \(r, s\in\{1,\dots,2^J\}\) and where \(J\) is the coarsest observed scale.
(Eckley et al. 2010) also define the local wavelet spectrum (LWS) associated with an LS2W process. The LWS for a given location \(\mathbf{z}=\left(\frac{r}{2^J},\frac{s}{2^J}\right)\in (0,1)^2\), at
scale \(j\) in direction \(l\) is \(S^l_j(\mathbf{z})\approx w^l_j(\mathbf{u}/\mathbf{R})^2\). The LWS provides a decomposition of the process variance at (rescaled) locations \(\mathbf{z}\),
directions \(l\), and wavelet scales \(j\). In practice the LWS is usually unknown and so needs to be estimated (see Eckley et al. 2010 for details). Spectral estimation using the LS2W model is
implemented in R in the add-on package LS2W (Eckley and Nason 2013). The LS2Wstat routines described below thus have a dependence on some functions from the LS2W package.
A test of stationarity for LS2W processes
Next we turn to describe the implementation of a test of stationarity within the LS2Wstat package. We focus on describing the \(Bootstat_{LS2W}\) approach implemented in the LS2Wstat package,
referring the interested reader to (Taylor et al. {in press}) for details of other tests which might be employed. Throughout this section let us assume that we have some image \(X_{\mathbf{r}}\) (as
in Figure 2), whose second-order structure we wish to test for spatial stationarity. We assume that \(X\) is an LS2W process with associated unknown spectrum, \(S_{j}^{\ell}\) for \(j=1,\ldots,J\)
and \(\ell=v\), \(h\) or \(d\). Since the model in (1) assumes the process has zero mean, if necessary the image can be detrended. This can be done in R, for example, by using the core stats package
function medpolish, which implements Tukey’s median polish technique (Tukey 1977).
Under the null hypothesis of stationarity, the wavelet spectrum will be constant across location for each scale and direction. Motivated by this fact (Taylor et al. {in press}) formulate a hypothesis
test for the stationarity of the image \(X_{\mathbf{r}}\) with \[\begin{aligned} H_0 : & \ S_{j}^{\ell}(\mathbf{z}) \mbox{ is constant across $\mathbf{z}$ for all $j$ and $\ell$}, \\ H_A : & \ S_{j}^
{\ell}(\mathbf{z}) \mbox{ is not constant across $\mathbf{z}$ for some $j$ or $\ell$}. \end{aligned}\] Hence, a test statistic for the hypothesis should measure how much the wavelet spectrum for an
observed image differs from constancy. (Taylor et al. {in press}) suggest using the average scale-direction spectral variance as a test statistic to measure the degree of non-stationary within an
image, where the variance is taken over pixel locations, that is:
\[\label{eq:tos} T\left\{ \hat{S}_j^{\ell}(\mathbf{z})\right\} = \frac{1}{3J}\sum_{\ell} \sum_{j=1}^{J} \mbox{var}_{\mathbf{u}}\left( \hat{S}_{j,\mathbf{u}}^{\ell}\right). \tag{2}\]
In practice this statistic is computed based on an (unbiased) estimate of the local wavelet spectrum, produced by the LS2W function cddews (see the documentation in LS2W for details on optional
arguments to this function). For the (square) image X, the test statistic is calculated using the function avespecvar as follows:
Since the spectrum characterises the second-order structure of the observed random field (and hence its stationarity properties), (Taylor et al. {in press}) suggest determining the p-value of the
hypothesis test by performing parametric bootstrapping. This corresponds to sampling LS2W processes assuming stationarity under the null hypothesis, and comparing the observed test statistic to that
of the simulated LS2W processes under stationarity. For pseudo-code of this algorithm, please see Algorithm 1.
Bootstat\({\mbox{{\scriptsize {LS2W}}}}\)
1. Compute the estimate of the LWS for the observed image, Ŝ[j]^l(z).
2. Evaluate T[] (Equation ) on the observed image, call this value T[]^obs.
3. Compute the pixel average stationary spectrum S̃[j]^l by taking the average of spectrum values for each scale and direction.
4. Iterate for i in 1 to B bootstraps:
1. Simulate X[r]^(i) from the stationary LS2W model using squared amplitudes given by S̃[j]^l and Gaussian process innovations.
2. Compute the test statistic T[] on the simulated realisation, call this value T[]^(i).
5. Compute the p-value for the test as \(p=\frac{1+ \#\left\{\, T_{}^{\mathit{obs}}\,\leq\, T_{}^{(i)}\, \right\}}{B+1}.\)
Algorithm 1: The bootstrap algorithm for testing the stationarity of locally stationary images.
This bootstrap algorithm is performed with the LS2Wstat function TOS2D. The function has arguments:
The image you want to analyse.
A binary value indicating whether the image should be detrended before applying the bootstrap test. If set to TRUE, the image is detrended using Tukey’s median polish method.
The number of bootstrap simulations to carry out. This is the value \(B\) in the pseudocode given above. By default this takes the value 100.
This specifies the test statistic function to be used within the testing procedure to measure non-stationarity. The test statistic should be based on the local wavelet spectrum and by default is
the function avespecvar representing the statistic (2).
A binary value indicating whether informative messages should be printed.
Any optional arguments to be passed to the LS2W function cddews. See the documentation for the cddews function for more details.
Note that TOS2D uses the LS2W process simulation function LS2Wsim from the LS2W R package to simulate bootstrap realizations under the null hypothesis. The output of TOS2D is a list object of class
"TOS2D", which describes pertinent quantities associated with the bootstrap test of stationarity. The object contains the following components:
The name of the image tested for stationarity.
A vector of length nsamples + 1 containing each of the test statistics calculated in the bootstrap test. The first element of the vector is the value of the test statistic calculated for the
original image itself.
The statistic used in the test.
The bootstrap p-value associated with the test.
In particular, the object returns the measure of spectral constancy in the entry statistic, together with the p-value associated with the stationarity test (in the p.value component).
An example of the function call is
Note that the p-value returned within the "TOS2D" object is computed using the utility function getpval, which returns the parametric bootstrap p-value for the test from the bootstrap test statistics
provided by counting those test statistic values less than \(T^{\mathit{obs}}\) (see Davison et al. 1999 for more details). In other words, the p.value component is obtained by the following call:
This p-value can then be used to assess the stationarity of a textured image region.
Information on the "TOS2D" class object can be obtained using the print or summary S3 methods for this class. For example, using the summary method, one would obtain
> summary(Xbstest)
2D bootstrap test of stationarity
object of class TOS2D
data: X
Observed test statistic: 0.204
bootstrap p-value: 0.01
Alternatively, the print method for the "TOS2D" class prints more information about the Xbstest object. Note that the function internally calls the summary method for "TOS2D" objects:
2D bootstrap test of stationarity
object of class TOS2D
data: X
Observed test statistic: 0.204
bootstrap p-value: 0.01
Number of bootstrap realizations: 100
spectral statistic used: avespecvar
Other textured images
To demonstrate the test of stationarity further, we now provide some other textured image examples. Firstly, we consider a Haar wavelet random field with a diagonal texture, an example of a LS2W
process as described in (Eckley et al. 2010). The realisation of the process (shown in Figure 3) is simulated using the simTexture function with the command:
Figure 3: A realisation of a stationary LS2W process, Haarimage, with a diagonal texture.
The test of stationarity of (Taylor et al. {in press}) performed on the image Haarimage with the function TOS2D reveals that the image is spatially stationary as expected, with a high p-value
associated to the test.
> Haarimtest <- TOS2D(Haarimage, smooth = FALSE, nsamples = 100)
> summary(Haarimtest)
2D bootstrap test of stationarity
object of class TOS2D
data: Haarimage
Observed test statistic: 0.631
bootstrap p-value: 0.673
Number of bootstrap realizations: 100
spectral statistic used: avespecvar
As another example of a textured image, we construct an image montage using two of the textures shown in Figure 1 from the package LS2W. The montage, montage1, is shown in Figure 4.
Figure 4: An example of an image montage, montage1, using two of the textures from Figure 1.
Note that since this image may not have zero mean as assumed by the LS2W model (1), we detrend the montage first using the medpolish function in the stats package.
> data(textures)
> montage1 <- cbind(A[1:512, 1:256], B[, 1:256])
> montage1zm <- medpolish(montage1)$residuals
The TOS2D test indicates that the texture montage is non-stationary:
> montage1zmtest <- TOS2D(montage1zm, smooth = FALSE, nsamples = 100)
> summary(montage1zmtest)
2D bootstrap test of stationarity
object of class TOS2D
data: montage1zm
Observed test statistic: 0
bootstrap p-value: 0.01
Number of bootstrap realizations: 100
spectral statistic used: avespecvar
Identifying areas of homogeneous texture using the bootstrap test of stationarity
In this section we describe embedding a test of stationarity into a quadtree algorithm to identify regions of spatial homogeneity within a textured image. This segmentation approach is similar in
spirit to, e.g., (Spann and Wilson 1985) or (Pal and Pal 1987) which use homogeneity measures within a quadtree structure. We first give details of the quadtree implementation, and subsequently
describe functions to aid illustration of quadtree decompositions.
A quadtree algorithm implementation
In essence, a region splitting algorithm recursively subdivides an input image into smaller regions, with the subdivision decisions being based on some statistical criterion. More specifically, in a
quadtree representation
, at each stage a (sub)image is divided into its four subquadrants if the criterion is not satisfied
(see e.g., Sonka et al. 1999)
. The statistical criterion we use is spatial homogeneity, that is, a quadrant is further divided if it is considered as non-stationary by the
test. In practice, the quadtree implementation in
continues until all subregions are considered as stationary, or until the subregions reach a particular minimal dimension. The motivation for this is to ensure that we obtain statistically
meaningful decisions using the stationarity test by not allowing too small a testing sub-image. This procedure segments an image into regions of spatial stationarity. The quadtree algorithm is
summarised in Algorithm
Quadtree decomposition
• For an input image X: Use the Bootstat[LS2W] test to assess whether X is second-order stationary. If stationary, stop. If not,
1. Divide the image into four quadrants.
2. For each quadrant, assess its stationarity with the Bootstat[LS2W] test.
3. For each quadrant assessed as non-stationary, recursively repeat steps 1–2, until the minimum testing region is reached or until all sub-images are judged to be stationary.
Algorithm 2: The quadtree algorithm for segmenting an image into regions of spatial stationarity.
Each image is further split if deemed non-stationary, which is determined by a test of stationarity such as TOS2D. After the first subdivison of an image, each sub-image is of size \(n/2 \times n/2\)
. The sizes of the regions halve in size at each progressive division but increase in number. The R function in LS2Wstat which creates the quadtree structure described in Algorithm 2 is imageQT. The
function has inputs:
The image to be decomposed with the quadtree algorithm.
A function for assessing regions of spatial homogeneity, for example TOS2D.
The testing size of sub-images below which we should not apply the function test.
The significance level of the \(Bootstat_{LS2W}\) test, with which to assess spatial stationarity of textured regions.
Any other optional arguments to test.
As an illustration of using the imageQT function, consider the code below to decompose the (non-stationary) input image X. We use the function TOS2D to assess the regions of spatial homogeneity
although the imageQT function allows other functions to be supplied.
The output of the imageQT function is a list object of class "imageQT" with components:
The index representation of the non-stationary images in the quadtree decomposition.
The results of the stationarity testing (from the test argument) during the quadtree decomposition. The results giving FALSE correspond to those non-stationary sub-images contained in the indl
component and the results giving TRUE correspond to the stationary sub-images, i.e. those contained in the indS component.
The index representation of the stationary images in the quadtree decomposition.
This particular way of splitting an image has a convenient indexing representation to identify the position of subregions within an image. If a (sub)image is subdivided into quadrants, we assign it a
base 4 label as follows: 0 – top-left quadrant; 1 – bottom-left quadrant; 2 – top-right quadrant; 3 – bottom-right quadrant. By continuing in this manner, we can assign an index to each tested
subregion, with the number of digits in the index indicating how many times its parent images have been subdivided from the “root” of the tree (the original image). This indexing is illustrated for
the quadtree decomposition given in the example in Figure 5.
Figure 5: An example of a quadtree decomposition. The location of the sub-images in the decomposition are described by the indexing system described in the text.
Examining the quadtree decomposition of the image X using the print S3 method for the "imageQT" class, we have
> print(QTdecX)
2D quadtree decomposition
object of class imageQT
data: X
Indices of non-stationary sub-images:
"0" "1" "2" "3" "03" "12" "21" "30"
Indices of stationary sub-images:
"00" "01" "02" "10" "11" "13" "20" "22" "23" "31" "32" "33" "030" "031" "032" "033"
"120" "121" "122" "123" "210" "211" "212" "213" "300" "301" "302" "303"
minimum testing region: 64
The resl component gives the results of the test of stationarity for all sub-images tested during the quadtree procedure, reporting FALSE for the non-stationary sub-images and TRUE for the stationary
> QTdecX$resl
[1] FALSE
[1] FALSE FALSE FALSE FALSE
[1] TRUE TRUE TRUE FALSE TRUE TRUE FALSE TRUE TRUE FALSE TRUE TRUE
[13] FALSE TRUE TRUE TRUE
[1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
[16] TRUE
Plotting a quadtree decomposition
By performing the quadtree algorithm given in Algorithm 2, it is possible to decompose images into regions indicating regional stationarity. Note that if a texture discrimination procedure is used to
classify the output from the stationarity quadtree algorithm, the image segmentation method can be seen as a split and merge technique.
Suppose we have performed the quadtree decomposition. The LS2Wstat package includes an S3 plot method for "imageQT" objects to plot the output for the "imageQT" class and optionally a classification
of those textured regions. If the classification output is plotted (class = TRUE), each textured region is uniquely coloured according to its texture group. The function has arguments:
A quadtree decomposition object, such as output from imageQT.
Vector of class labels associated to the subimages produced by the quadtree decomposition.
A value for any unclassified values in a quadtree decomposition.
A Boolean value indicating whether to plot the classification of the quadtree subimages.
A Boolean value indicating whether to plot the quadtree decomposition.
We now illustrate the use of this function with the example given in Figure 2. Suppose the textured regions identified by the quadtree algorithm in the QTdecX object have been classified according to
some texture discrimination procedure. For the purposes of this example, we suppose that the 28 regions of stationarity in QTdecX (see Figure 5) have been classified as coming from two groups
according to the labels
> texclass <- c(rep(1, times = 15), rep(c(2, 1, 1), times = 4), 1)
> texclass
[1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 1 2 1 1 2 1 1 1
Using the output from the quadtree technique (QTdecX) and the texture classification vector texclass, we can use the quadtree plotting function for "imageQT" objects as follows:
> plot(QTdecX, texclass, class = TRUE, QT = TRUE)
> plot(QTdecX, texclass, class = TRUE, QT = FALSE)
The quadtree decomposition from this example is shown in Figure [fig:CT1]a; the same decomposition is shown together with the texture classification in Figure 6b.
Figure 6: An example of a quad-tree decomposition using imageQT, together with an assumed sub-image texture classification.
We also consider an image montage using the textures from the package LS2W. The montage Y is shown in Figure 7. Prior to performing the quadtree decomposition, we detrend the image.
> data(textures)
> Y <- cbind(A[1:512, 1:256], rbind(B[1:256, 1:256], C[1:256, 1:256]))
> Yzm <- medpolish(Y)$residuals
Figure 7: An example of an image montage, Y, using the textures from Figure 1.
Similarly to above, we can now perform a quadtree decomposition of the image Y:
> QTdecYzm <- imageQT(Yzm, test = TOS2D, nsamples = 100)
> print(QTdecYzm)
2D quadtree decomposition
object of class imageQT
data: Yzm
Indices of non-stationary sub-images:
Indices of stationary sub-images:
"0" "1" "2" "3"
minimum testing region: 64
The function imageQT initially assesses that the image is indeed non-stationary, and then proceeds to analyse sub-images of the montage. The algorithm stops the quadtree decomposition after the first
decomposition level, since it judges all quadrants of the image to be stationary, described by the indices "0", "1", "2", and "3".
In this article we have described the LS2Wstat package, which implements some recent methodology for image stationarity testing (Taylor et al. {in press}). Our algorithm is most useful as a test of
homogeneity in textures which are visually difficult to assess. We have also extended its potential use by embedding it within a quadtree implementation, allowing assessment of potentially
multi-textured images. The implementation is demonstrated using simulated and real textures throughout the paper.
We thank Aimée Gott for suggestions on an early version of the package. We would also like to thank Heather Turner, two anonymous referees and the Editor for helpful comments which have resulted in
an improved manuscript and package.
C. M. Bishop and N. M. Nasrabadi. Pattern recognition and machine learning. New York: Springer-Verlag, 2006.
C. Bradley and Y. S. Wong. Surface texture indicators of tool wear – a machine vision approach. The International Journal of Advanced Manufacturing Technology, 17(6): 435–443, 2001.
C. Chan and G. K. H. Pang. Fabric defect detection by Fourier analysis. IEEE Transactions on Industry Applications, 36(5): 1267–1276, 2000.
A. C. Davison, D. V. Hinkley and A. J. Canty. Bootstrap methods and their application. Cambridge University Press, 1999.
I. A. Eckley and G. P. Nason.
LS2W: Locally stationary two-dimensional wavelet process estimation scheme.
2013. URL
. R package version 1.3-3.
I. A. Eckley and G. P. Nason.
: Locally stationary wavelet fields in
Journal of Statistical Software
, 43(3): 1–23, 2011. URL
I. A. Eckley, G. P. Nason and R. L. Treloar. Locally stationary wavelet fields with application to the modelling and analysis of image texture. Journal of the Royal Statistical Society C, 59(4):
595–616, 2010.
J. W. Funck, Y. Zhong, D. A. Butler, C. C. Brunner and J. B. Forrer. Image segmentation algorithms applied to wood defect detection. Computers and Electronics in Agriculture, 41(1): 157–179, 2003.
R. C. Gonzalez and R. E. Woods. Digital image processing. 2nd ed Prentice Hall, 2001.
C. Lupi. Unit root
testing with
Journal of Statistical Software
, 32(2): 1–19, 2009. URL
G. P. Nason. A test for second-order stationarity and approximate confidence intervals for localized autocovariances for locally stationary time series. Journal of the Royal Statistical Society B, 75
(5): 879–904, 2013a.
G. P. Nason.
Locits: Test of stationarity and localized autocovariance.
2013b. URL
. R package version 1.4.
G. P. Nason. Wavelet methods in statistics with R. Springer-Verlag, 2008.
N. R. Pal and S. K. Pal. A review on image segmentation techniques. Pattern Recognition, 26(9): 1277–1294, 1993.
S. K. Pal and N. R. Pal. Segmentation using contrast and homogeneity measures. Pattern Recognition, 5(4): 293–304, 1987.
M. Petrou and P. G. Sevilla. Image processing: Dealing with texture. John Wiley & Sons, 2006.
B. Pfaff. Analysis of integrated and cointegrated time series with R. 2nd ed New York: Springer-Verlag, 2008.
B. Pfaff and M. Stigler.
Urca: Unit root and cointegration tests for time series data.
2013. URL
. R package version 1.2-8.
W. Pölzleitner. Quality classification of wooden surfaces using Gabor filters and genetic feature optimisation. In Machine vision for the inspection of natural products, pages. 259–277 2003.
L. Prasad and S. S. Iyengar. Wavelet analysis with applications to image processing. CRC Press, 1997.
M. Sonka, R. Boyle and V. Hlavac. Image processing, analysis, and machine vision. 2nd ed PWS Publishing, 1999.
M. Spann and R. Wilson. A quad-tree approach to image segmentation which combines statistical and spatial information. Pattern Recognition, 18(3/4): 257–269, 1985.
S. L. Taylor, I. A. Eckley and M. A. Nunes. A test of stationarity for textured images.
, {in press}. DOI
S. Taylor and M. A. Nunes.
LS2Wstat: A multiscale test of spatial stationarity for LS2W processes.
2014. URL
. R package version 2.0-3.
J. W. Tukey. Exploratory data analysis. Addison-Wesley, 1977.
B. Vidakovic. Statistical modelling by wavelets. New York: John Wiley & Sons, 1999.
K. Wiltschi, A. Pinz and T. Lindeberg. An automatic assessment scheme for steel quality inspection. Machine Vision and Applications, 12(3): 113–128, 2000. | {"url":"http://journal.r-project.org/articles/RJ-2014-002/","timestamp":"2024-11-11T07:18:28Z","content_type":"text/html","content_length":"997240","record_id":"<urn:uuid:694cbfd2-a19b-4e58-be14-941d9bc1a743>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00875.warc.gz"} |
Solve each equation. $$\frac{5}{x}=\frac{2}{5}$$
Short Answer
Expert verified
x = \(\frac{25}{2}\)
Step by step solution
To solve the equation \(\frac{5}{x} = \frac{2}{5}\), first cross-multiply to eliminate the fractions. Multiply both sides of the equation by the denominators of each fraction: \(\frac{5}{x} \times x
\times 5 = \frac{2}{5} \times x \times 5\). This results in the equation \(5 \times 5 = 2 \times x\).
Simplify the Equation
Simplify the results from the cross-multiplication: \(5 \times 5 = 25\) and \(2 \times x = 2x\). Hence, the equation simplifies to \(25 = 2x\).
Solve for x
To isolate x, divide both sides of the equation by the coefficient of x. This means dividing both sides by 2: \(\frac{25}{2} = \frac{2x}{2}\). This results in \(x = \frac{25}{2}\).
Verify the Solution
Substitute \(x = \frac{25}{2}\) back into the original equation to verify: \(\frac{5}{\frac{25}{2}} = \frac{2}{5}\). Simplifying the left side: \(\frac{5 \times 2}{25} = \frac{10}{25} = \frac{2}{5}
\), which confirms the solution is correct.
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cross-multiplication is a technique used to eliminate fractions from an equation by multiplying each side by the denominators. In the example \(\frac{5}{x} = \frac{2}{5}\), we multiply both sides to
clear the fractions. This means you take the denominator of one fraction and multiply it by the numerator of the other fraction: \(5 * 5 = 2 * x\). This effectively 'crosses over' the denominators
and helps simplify the equation. It's a very efficient way to make equations more manageable. Always start by ensuring all fractions are in their simplest form.
simplifying equations
Once you have cross-multiplied, simplifying the equation is the next step. This involves performing basic arithmetic to reduce the equation to a simpler form. For our equation \(5 * 5 = 2 * x\), we
perform the multiplications: \(25 = 2x\). Hence, we now have an equation without fractions. Simplifying helps in isolating the variable, which is our next objective. Perform all arithmetic operations
carefully to avoid errors.
isolating variables
Isolating the variable means getting the variable (in this case, \(x\)) on one side of the equation by itself. From \(25 = 2x\), we want to isolate \(x\). We do this by dividing both sides of the
equation by the coefficient of \(x\), which is 2. This gives us \(\frac{25}{2} = x\). Hence, \x\ is now isolated: \(x = \frac{25}{2}\). Make sure all arithmetic is double-checked, ensuring that the
variable is completely isolated.
fraction verification
The final step is to verify your solution. Substitute the value of \(x\) back into the original equation to see if both sides are equal. Using \(x = \frac{25}{2}\), we plug it back in: \(\frac{5}{\
frac{25}{2}} = \frac{2}{5}\). Simplify the left side: \(\frac{5 * 2}{25} = \frac{10}{25} = \frac{2}{5}\). Both sides are equal, confirming our solution. Always verify to ensure there are no
calculation errors and that the solution satisfies the original equation. | {"url":"https://www.vaia.com/en-us/textbooks/math/algebra-for-college-students-5-edition/chapter-6/problem-59-solve-each-equation-frac5xfrac25/","timestamp":"2024-11-08T02:42:08Z","content_type":"text/html","content_length":"249087","record_id":"<urn:uuid:4312545d-bf1b-4f29-8882-447774406596>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00325.warc.gz"} |
All About Taylor Series
December 28, 2018
Here is a survey of understandings on each of the main types of Taylor series:
1. single-variable
2. multivariable \(\bb{R}^n \ra \bb{R}\)
3. multivariable \(\bb{R}^n \ra \bb{R}^m\)
4. complex \(\bb{C} \ra \bb{C}\)
I thought it would be useful to have everything I know about these written down in one place.
Particularly, I don’t want to have to remember the difference between all the different flavors of Taylor series, so I find it helpful to just cast them all into the same form, which is possible
because they’re all the same thing (seriously why aren’t they taught this way?).
These notes are for crystallizing everything when you already have a partial understanding of what’s going on. I’m going to ignore discussions of convergence so that more ground can be covered and
because I don’t really care about it for the purposes of intuition.
1. Single Variable
A Taylor series for a function in \(\bb{R}\) looks like this:
\[\begin{aligned} f(x + \e) &= f(x) + f'(x) \e + f''(x) \frac{\e^2}{2} + \ldots \\ &= \sum_n f^{(n)} \frac{\e^n}{n!} \end{aligned}\]
It’s useful to write this as one big operator acting on \(f(x)\):
\[\boxed{f(x + \e) = \big[ \sum_{n=0}^\infty \frac{\p^n_x \e^n}{n!} \big] f(x)} \tag{Single-Variable}\]
Or even as a single exponentiation of the derivative operator, which is commonly done in physics, but you probably shouldn’t think too hard about what it means:
\[f(x + \e) = e^{\e \p_x} f(x)\]
I also think it’s useful to interpret the Taylor series equation as resulting from repeated integration:
\[\begin{aligned} f(x) &= f(0) + \int_0^x dx_1 f'(x_1) \\ &= f(0) + \int_0^x dx_1 [ f'(0) + \int_0^{x_1} dx_2 f''(x_2) ]] + \ldots\\ &= f(0) + \int dx_1 f'(0) + \iint dx_1 dx_2 f''(0) + \iiint dx_1
dx_2 dx_3 f'''(0) + \ldots \\ &= f(0) + x f'(0) + \frac{x^2}{2} f''(0) + \frac{x^3}{3!} f'''(0) + \ldots \end{aligned}\]
This basically makes sense as soon as you understand integration, plus it makes obvious that the series only works when all of the integrals are actually equal to the values of the previous function
(so you can’t take a series of \(\frac{1}{1-x}\) which passes \(x=1\), because you can’t exactly integrate past it (though there are tricks))
… plus it makes sense in pretty much any space you can integrate over.
… plus it makes it obvious how to truncate the series, how to create the remainder term, and it even shows you how you could – if you were so inclined – have each derivative be evaluated at a
different point, such as \(f(x) = f(0) + \int_1^x f'(x_1) dx_1 =f(0) + (x-1) f'(1) + \frac{(x-1)(x-2)}{2} f''(2) + \ldots\), which I’ve never even seen done before (except for here?), though good
luck with figuring out convergence if you do that.
L’Hôpital’s rule about evaluating limits which give indeterminate forms follows naturally if the functions are both expressible as Taylor series. If \(f(x) = g(x) = 0\), then:
\[\begin{aligned} \lim_{\e \ra 0} \frac{f(x + \e)}{g(x + \e)} &= \lim_{\e \ra 0} \frac{ f(x) + \e f'(x + \e) + O(\e^2)} {g(x) + \e g'(x + \e) + O(\e^2)} \\ &= \lim_{\e \ra 0}\frac{f'(x+\e) + O(\e) }
{g'(x+\e) + O(\e)} \\ &= \lim_{\e \ra 0} \frac{f'(x+\e)}{g'(x + \e)} \end{aligned}\]
Which equals \(\frac{f'(x)}{g'(x)}\) if the limit exists, and otherwise might be solvable by applying the rule recursively. None of this works of course if limit doesn’t exist. If \(f(x) = g(x) = \
infty\), evaluate \(\lim \frac{1/g(x)}{1/f(x)}\) instead. If the indeterminate form is \(\infty - \infty\), evaluate \(\lim f(x) - g(x)\) instead.
2. Multivariable -> Scalar
The multivariable Taylor series looks messier at first, so let’s start with only two variables, writing \(f_x \equiv \p_x f(\b{x})\) and \(\b{v} = (v_x, v_y)\), and we’ll work it into a more usable
\[\begin{aligned} f(\b x + \b v) &= f(\b x) + [f_x v_x + f_y v_y] + \frac{1}{2!} [f_{xx} v_x^2 + 2 f_{xy} v_x v_y + f_{yy} v_y^2] \\ &+ \frac{1}{3!} [f_{xxx} v_x^3 + 3 f_{xxy} v_x^2 v_y + 3 f_{xyy}
v_x v_y^2 + f_{yyy} v_y^3] + \ldots \end{aligned}\]
(The asymmetry of the terms like \(2 f_{xy} v_x v_y\) and \(3 f_{xxy} v_x^2 v_y\) is because these are really sums of multiple terms; because of the commutativity of partial derivatives on analytic
functions, \(f_{xy} = f_{yx}\), we can write \(f_{xy} v_x v_y + f_{yx} v_y v_x = 2 f_{xy} v_x v_y\).)
The first few terms are often arranged like this:
\[f(\b x + \b v) = f(\b x) + \b{v} \cdot \nabla f(\b{x}) + \b{v}^T \begin{pmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{pmatrix} \b{v} + O(v_3)\]
\(\nabla f(\b{x})\) is the gradient of \(f\) (the vector of partial derivatives like \((f_x, f_y)\). The matrix \(H = \begin{pmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{pmatrix}\) is the
“Hessian matrix” for \(f\), and represents its second derivative.
… But we can do better. In fact, every order of derivative of \(f\) in the total series has the same form, as powers of \(\b{v} \cdot \vec{\nabla}\), which I prefer to write as \(\b{v} \cdot \vec{\p}
\), because it represents a ‘vector of partial derivatives’ \(\vec{\p} = (\p_x, \p_y)\):
\[\begin{aligned} f(\b x + \b v) &= f(\b x) + (v_x \p_x + v_y \p_y) f(\b x) + \frac{(v_x \p_x + v_y \p_y)^2}{2!} f(\b x) + \ldots \\ &= \big[ \sum_n \frac{(v_x \p_x + v_y \p_y)^n}{n!} \big] f(\b x) \
\ &= \boxed{ \big[ \sum_{n=0}^\infty \frac{(\b{v} \cdot \vec{\p})^n}{n!} \big] f(\b x) } \end{aligned} \tag{Scalar Field}\]
So that looks pretty good. And it can still be written as \(e^{ \b{v} \cdot \vec{\p}} f(\b{x})\). The same formula – now that we’ve hidden all the actual indexes – happily continues to work for
dimension \(> 2\), as well.
… Although really the multivariate Taylor series of \(f(\b{x})\) is really just a bunch of single-variable series multiplied together:
\[\begin{aligned} f(x+ v_x, y + v_y) &= e^{v_x \p_x} f(x, y + v_y) \\ &= e^{v_x \p_x}e^{v_y \p_y} f(x,y) \\ &= e^{v_x \p_x + v_y \p_y} f(x,y) \\ &= e^{\b{v} \cdot \vec{\p}} f(\b{x}) \end{aligned}\]
I mention all this because it’s useful to have a solid idea of what a scalar function is before we move to vector functions.
Note that, when exponentiating operators, \(e^{v_x \p_x}e^{v_y \p_y} f(x,y) = e^{v_x \p_x + v_y \p_y} f(x,y)\) is not always allowed. There are complicated rules for how to combine exponentiated
operators—but fortunately, when the exponents commute (ie \(\p_x \p_y = \p_y \p_x\), which we’re just assuming is true here), you can add them in the normal way.
L’Hôpital’s rule is more subtle for multivariable functions. In general the limit of a function may be different depending on what direction you approach from, so an expression like \(\lim_{\b{x} \ra
0} \frac{f(\b{x})}{g(\b{x})}\) is not necessarily defined, even if both \(f\) and \(g\) have Taylor expansions. On the other hand, if we choose a path for \(\b{x} \ra 0\), such as \(\b{x}(t) = (x(t),
y(t))\) then this just becomes a one-dimensional limit, and the regular rule applies again. So, for instance, while \(\lim_{\b x \ra 0} \frac{f(\b{x})}{g(\b x)}\) may not be defined, \(\lim_{t \ra 0}
\frac{f(t \b{v})}{g(t \b{v})}\) is for any fixed vector \(\b{v}\).
The path we take to approach \(0\) doesn’t even matter, actually; what matters is the gradients when we’re infinitesimally close to \(0\). For example, suppose we \(f(0,0) = g(0,0) = 0\) and we’re
taking the limit on the path given by \(y = x^2\):
\[\lim_{\e \ra 0} \frac{f(\e,\e^2)}{g(\e,\e^2)} = \lim_{ \e \ra 0 } \frac{ f_x(0,0) \e + O(\e^2) }{ g_x(0,0) \e + O(\e^2)} = \lim_{\e \ra 0} \frac{f(\e,0)}{g(\e,0)}\]
The \(f_y\) and \(g_y\) terms are of order \(\e^2\) and so drop out, leaving a limit taken only on the \(x\)-axis, corresponding to the fact that the tangent to \((x,x^2)\) at 0 is \((1,0)\).
In fact, this problem basically exists in 1D also, except that limits can only come from two directions: \(x^+\) and \(x^-\), so lots of functions get away without a problem. L’Hôpital’s rule seems
to require that the functions be expandable as a Taylor series on the side the limit comes from. Indeed, we might just define a sort of “any-sided limit” which associates with each direction of
approach a (potentially) different value. I’m not quite sure I fully understand the complexity of doing that in \(N > 1\) dimensions, but clearly if you can just reduce to a 1-dimensional limit the
difficulties should be removed. See, perhaps, this paper for a lot more information.
3. Vector Fields
There are several types of vector-valued functions: one-dimensional curves like \(\gamma: \bb{R} \ra \bb{R}^n\), or arbitrary-dimensional maps like \(\b{f}: \bb{R}^m \ra \bb{R}^n\) (including from a
space to itself), or maps between arbitrary differentiable manifolds \(f: M \ra N\). In each case there is something like a Taylor series that can be defined. It’s not commonly written out, but I
think it should be, so let’s try.
Let’s imagine our function maps spaces \(X \ra Y\), where \(X\) has \(m\) coordinates and \(Y\) has \(n\) coordinates, and \(m\) might be 1 in the case of a curve. Then along any particular
coordinate in \(Y\) out of the \(n\)—call it \(y_i\)—the Taylor series expression from above holds, because \(f_i = \b{f} \cdot y_i\) is just a scalar function.
\[f(\b{x} + \b{v})_i = e^{\b{v} \cdot \vec{\p}} [f(\b{x})_i]\]
But of course this holds in every \(i\) at once, so it holds for the whole function:
\[\b{f}(\b{x} + \b{v}) = e^{\b{v} \cdot \vec{\p}} \b{f}(\b{x})\]
The subtlety here is that the partial derivatives \(\p\) are now being taken termwise—once for each component of \(\b{f}\). For example, consider the first few terms when \(X\) and \(Y\) are 2D:
\[\begin{aligned} \b{f}(\b{x} + \b{v}) &= \b{f}(\b{x}) + (v_{x_1} \p_{x_1} + v_{x_2} \p_{x_2}) \b{f} + \frac{(v_{x_1} \p_{x_1} + v_{x_2} \p_{x_2})^2}{2!} \b{f} + \ldots\\ &= \b{f} + \begin{pmatrix} \
p_{x_1} \b{f}_{y_1} & \p_{x_2} \b{f}_{y_1} \\ \p_{x_1} \b{f}_{y_2} & \p_{x_2} \b{f}_{y_2} \end{pmatrix} \begin{pmatrix} v_{x_1} \\ v_{x_2} \end{pmatrix} + \ldots \\ &= \b{f} +(\p_{x_1}, \p_{x_2}) \o
\begin{pmatrix}\b{f}_{y_1} \\ \b{f}_{y_2} \end{pmatrix} \cdot \begin{pmatrix} v_{x_1} \\ v_{x_2} \end{pmatrix} + \ldots \end{aligned}\]
That matrix term, the \(n=1\) term in the series, is the Jacobian Matrix of \(f\), sometimes written \(J_f\), and is much more succinctly written as \(\vec{\p}_{x_i} \b{f}_{y_j}\), or just \(\vec{\p}
_i \b{f}_j\) or even just \(\p_i \b{f}_j\).
\[J_f = \p_i f_j\]
The Jacobian matrix is the ‘first derivative’ of a vector field, and it includes every term which can possibly matter to compute how the function changes to first-order. In the same way that a
single-variable function is locally linear (\(f(x + \e) \approx f(x) + \e f'(x)\)), a multi-variable function is locally a linear transformation: \(\b{f}(\b{x + v}) \approx \b{f}(\b{x}) + J_f \b{v}
Higher-order terms in the vector field Taylor series generalize ‘second’ and ‘third’ derivatives, etc, but they are generally tensors rather than matrices. They look like \((\p \o \p) \b{f}\), \((\p
\o \p \o \p) \b{f}\), or \(\p^{\o n} \b{f}\) in general, and they act on \(n\) copies of \(\b{v}\), ie, \(\b{v}^{\o n}\).
The full expansion (for \(X,Y\) of any number of coordinates) is written like this:
\[\begin{aligned} \b{f}(\b{x} + \b{v}) &= \b{f} + \p_i \b{f} \cdot v_i + \frac{1}{2!}(\p_i \p_j \b{f}) \cdot v_i v_j + \frac{1}{3!} (\p_i \p_j \p_k \b{f}) \cdot v_i v_j v_k + \ldots \\ &= \b{f} + \
p_i \b{f} \cdot v_i + \frac{1}{2!}(\p_i \p_j) \b{f} \cdot (v_i v_j) + \ldots \\ &= \b{f} +(\b{v} \cdot \vec{\p}) \b{f} + \frac{(\b{v} \cdot \vec{\p})^2}{2!} \b{f} + \ldots \\ \b{f}(\b{x} + \b{v}) &=
\boxed{ \big[ \sum_{n=0}^\infty \frac{(\b{v} \cdot \vec{\p})^n}{n!} \big] \b{f}(\b{x}) } \tag{Vector Field} \end{aligned}\]
We write the numerator in the summation as \((\b{v} \cdot \vec{\p})^{n}\), which expands to \((\sum_i v_i \p_i) (\sum_j v_j \p_j) \ldots\), and then we can still group things into exponentials, only
now we have to understand that all of these terms have derivative operators on them that need to be applied to \(\b{f}\) to be meaningful:
\[\b{f}(\b{x + v}) = e^{\b{v} \cdot \vec{\p}} \b{f}(\b{x})\]
We could have included indexes on \(\b{f}\) also:
\[\begin{aligned} f_k(\b{x} + \b{v}) &= \b{f}_k + \p_i \b{f}_k \cdot \b{v}_i + \frac{1}{2!}(\p_i \p_j) \b{f}_k \cdot (\b{v}_i \b{v}_j) + \ldots \\ &= \big[ \sum_{n} \frac{(\b{v} \cdot \vec{\p})^n}
{n!} \big] f_k(\b{x}) \end{aligned}\]
It seems evident that this should work any other sort of differentiable object also. What about matrices?
\[M_{ij}(\b{x} + \b{v})= \big[ \sum_{n} \frac{(\b{v} \cdot \vec{\p})^n}{n!} \big] M_{ij}(\b{x})\]
I don’t want to talk about curl and divergence here, because it brings in a lot more concepts and I don’t know the best understanding of it, but it’s worth noting that both are formed from components
of \(J_f\), appropriately arranged.
4. Complex Analytic
The complex plane \(\bb{C}\) is a sort of change-of-basis of \(\bb{R}^2\), via \((z,\bar{z}) = (x + iy, x - iy)\):
\[z \lra x\b{x} + y\b{y}\] \[\bar{z} \lra x\b{x} - y\b{y}\]
Therefore we can write it as a Taylor series in these two variables:
\[f(z + \D z, \bar{z} + \D \bar{z}) = \big[ \sum_{n=0}^\infty \frac{(\D z \p + \D \bar{z} \p_{\bar{z}})^n}{n!} \big] f(z, \bar{z})\]
One subtlety: it should always be true that \(\p_{x_i} \b{x}^j = 1_{i = j}\) when changing variables. Because \(z\) and \(\bar{z}\), when considered as vectors in \(\bb{R}^2\), are not unit vectors,
there is a normalization factor required on the partial derivatives. Also, for \(\bb{C}\) the factors of \(i\) cause the signs to swap:
\[\begin{aligned} \p_z &\underset{\bb{C}}{=} \frac{1}{2}(\p_x - i \p_y) \underset{\bb{R}^2}{=} \frac{1}{2}(\p_{\b{x}} + \p_{\b{y}}) \\ \p_{\bar{z}} &\underset{\bb{C}}{=} \frac{1}{2}(\p_x + i \p_y) \
underset{\bb{R}^2}{=} \frac{1}{2}(\p_{\b{x}} - \p_{\b{y}}) \end{aligned}\]
In complex analysis, for some reason, \(\bar{z}\) is not treated as a true variable, and we only consider a function as ‘complex differentiable’ when it has derivatives with respect to \(z\) alone.
Notably, we would say that the derivative \(\p_z \bar{z}\) does not exist—the value of \(\lim_{(x,y) \ra (0,0)} \frac{x + iy}{x - i y}\) is different depending on the path you take towards the
origin. These statements turn out to be almost equivalent:
• \(f(z)\) is a function of only \(z\) in a region
• \(\p_{\bar{z}} f(z) = 0\) in a region
• \(f(z)\) is complex-analytic in a region
• \(f(z)\) has a Taylor series as a function of \(z\) in a region
So when we discuss Taylor series of functions \(\bb{C} \ra \bb{C}\), we usually mean this:
\[\boxed{f(z + \D z) = \big[ \sum_{n=0}^\infty \frac{(\D z \p_z)^n}{n!} \big] f(z)} \tag{Complex-Analytic}\]
If we write \(f(z(x,y)) = u(x,y) + i v(x,y)\), the requirement that \(\p_{\bar{z}} f(z) = \frac{1}{2}(\p_x + i \p_y) f(z) = 0\) becomes the Cauchy-Riemann Equations by matching real and complex
\[\begin{aligned} u_x &= v_y \\ u_y &= - v_x \end{aligned}\]
But seriously, \(\p_{\bar{z}} f(z) = 0\) is a much better way of expressing this.
There is one important case where a function \(f(z, \bar{z})\) is a function of only \(z\), yet it is not analytic and \(\p_{\bar{z}} f(z) \neq 0\), and it is solely responsible for almost all of the
interesting parts of complex analysis. It’s the fact that:
\[\p_{\bar{z}} \frac{1}{z} = 2 \pi i \delta(z, \bar{z})\]
Where \(\delta(z, \bar{z})\) is the two-dimensional Dirac Delta function. I find this to be quite surprising. Here’s an aside on why it’s true:
Importantly, \(\p_{\bar{z}} z^n \neq 0\) is only true for \(n = -1\). This property gives rise to the entire method of residues, because if \(f(z) = \frac{f_{-1}(0) }{z} + f^*(z)\), where \(f^*(z)\)
has no terms of order \(\frac{1}{z}\), then integrating a contour \(C\) around a region \(D\) which contains \(0\) gives, via Stokes’ theorem:
\[\begin{aligned} \oint_C f(z) dz &= \iint_D \p_{\bar{z}} \big[ \frac{f_{-1}(0) }{z} + f^*(z) \big] \; d\bar{z} \^ dz \\ &= 2 \pi i \iint_D \delta(z, \bar{z}) f_{-1}(0) \; d\bar{z} \^ dz \\ &= 2 \pi
i f_{-1}(0) \end{aligned}\]
(If the \(\bar{z}\) derivative isn’t \(0\), you get the Cauchy-Pompeiu formula for contour integrals immediately.)
By the way: Fourier series are closely related to contour integrals, and thus to complex Taylor series. You can change variables to write \(\frac{1}{2 \pi i} \oint_C \frac{F(z)}{z^{k+1}} dz\) as \(\
frac{1}{2 \pi} \oint_C F(re^{i \theta})e^{-ik\theta} d\theta\), which is clearly a Fourier transform for suitable \(F\). | {"url":"https://alexkritchevsky.com/2018/12/28/taylor-series.html","timestamp":"2024-11-03T10:53:40Z","content_type":"text/html","content_length":"34312","record_id":"<urn:uuid:5270ce82-d552-4230-9b07-3591181f9403>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00004.warc.gz"} |
What is Ratio Analysis explain types?
What is Ratio Analysis explain types?
Ratio Analysis is done to analyze the Company’s financial and trend of the company’s results over years where there are mainly five broad categories of ratios like liquidity ratios, solvency ratios,
profitability ratios, efficiency ratio, coverage ratio which indicates the company’s performance and various examples of …
What is ratio analysis and its importance?
Ratio Analysis: Meaning Ratio Analysis is a method to understand the liquidity position, efficiency of operations, profitability position, and solvency of a business organization. It is a
quantitative technique that uses an organization’s financial statements, such as the income statement and the balance sheet.
What is Ratio Analysis example?
For example. the debt to assets ratio for 2010 is: Total Liabilities/Total Assets = $1074/3373 = 31.8% – This means that 31.8% of the firm’s assets are financed with debt. In 2011, the debt ratio is
What are the features of ratio analysis?
The following are the principal advantages of ratio analysis:
• Forecasting and Planning:
• Budgeting:
• Measurement of Operating Efficiency:
• Communication:
• Control of Performance and Cost:
• Inter-firm Comparison:
• Indication of Liquidity Position:
• Indication of Long-term Solvency Position:
What is ratio analysis and its features?
Ratio analysis is a quantitative analysis of data enclosed in an enterprise’s financial statements. It is used to assess multiple perspectives of an enterprise’s working and financial performance
such as its liquidity, turnover, solvency and profitability.
What are the main objectives of ratio analysis?
Objectives of Ratio Analysis are: Whereas, Long-term solvency is the ability of the enterprise to pay its long-term liabilities of the business. Assess the operating efficiency of the business.
Analyze the profitability of the business. Help in comparative analysis, i.e. inter-firm and intra-firm comparisons.
How do you do ratio analysis?
The four key financial ratios used to analyse profitability are:
1. Net profit margin = net income divided by sales.
2. Return on total assets = net income divided by assets.
3. Basic earning power = EBIT divided by total assets.
4. Return on equity = net income divided by common equity.
What is ratio analysis in easy language?
Definition: Ratio analysis is the process of examining and comparing financial information by calculating meaningful financial statement figure percentages instead of comparing line items from each
financial statement.
What are the 5 major categories of ratios?
The following five (5) major financial ratio categories are included in this list.
• Liquidity Ratios.
• Activity Ratios.
• Debt Ratios.
• Profitability Ratios.
• Market Ratios.
What is the main objective of ratio analysis?
Objectives of Ratio Analysis are: Determine liquidity or Short-term solvency and Long-term solvency. Short-term solvency is the ability of the enterprise to meet its short-term financial obligations.
Whereas, Long-term solvency is the ability of the enterprise to pay its long-term liabilities of the business. | {"url":"https://thecrucibleonscreen.com/what-is-ratio-analysis-explain-types/","timestamp":"2024-11-02T17:08:38Z","content_type":"text/html","content_length":"54954","record_id":"<urn:uuid:802a3d1f-0808-4d63-8869-b870fe7e4745>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00720.warc.gz"} |
Demystifying the Magic of 1s and 0s: A Friendly Introduction to Binary Computing - 33rd Square (2024)
As a tech geek and data analyst, I‘m fascinated by how fundamental concepts can enable transformative technologies. One of the most pivotal examples of this is how the simple binary digits 1 and 0
gave rise to the entire computing revolution that has reshaped society. In this beginner‘s guide, I want to demystify the magic of 1s and 0s and show you how they work their wonders!
Let‘s start at the very beginning – where did this idea of using 1 and 0 come from in the first place? To uncover that, we have to go back over 150 years to the pioneering work of a British
mathematician named George Boole.
The Origins of Binary Computing: From Boolean Logic to Electrical Switches
In 1854, George Boole published a landmark paper called "An Investigation of the Laws of Thought" where he explored how logical reasoning could be defined mathematically. He developed a framework for
describing logical operations like AND, OR and NOT using algebraic expressions and equations, which later became known as Boolean logic.
At first, this sounded very abstract and academic. But a few decades later, engineers realized that Boolean logic perfectly matched the behavior of electrical switches! Switches have two clear states
– on (closed) or off (open). An American mathematician named Claude Shannon working at Bell Labs saw that 1 could represent a closed switch with current flowing, while 0 could represent an open
switch with no current.
Shannon proved that by arranging switches together, you could physically implement the logical operations defined by Boole, like AND and OR gates. This was an extraordinary breakthrough that gave
birth to practical "logic circuits" using simple electronics. I find it amazing how Boole‘s purely theoretical logic concepts were elegantly mirrored by real-world circuitry!
Claude Shannon showed that Boolean logic could be implemented electronically using 1s and 0s
How Binary Digits Enable Digital Computing
Now you might be wondering – how exactly do these 1s and 0s represent information inside a computer? That‘s where the brilliant concept of binary numbering comes into play!
With only two digits, you might think that 1s and 0s could only count up to 3. But here‘s the magic – using positional notation, 1s and 0s can represent any quantity. For example, in decimal we have
units, tens, hundreds etc. positions. In binary, it‘s the same idea:
Position: 128 64 32 16 8 4 2 1Binary: 1 0 1 0 1 1 0 1Decimal: 128 + 32 + 8 + 4 + 1 = 173
By using strings of 1s and 0s in different positions, we can represent numbers, letters, instructions – you name it! In fact, your smartphone processor uses over 2 billion transistors to manipulate
1s and 0s for everything it does.
I sometimes geek out over the exponential growth in computing power shown by Moore‘s law. Would you believe that Intel‘s original 4004 processor from 1971 had only 2,300 transistors? Compare that to
over 20 billion in today‘s advanced chips! All still using familiar 1s and 0s, now with nanometer precision.
Real-World Applications Made Possible by Binary Computing
Beyond just numbers, the properties of 1s and 0s enable all kinds of advanced applications that we rely on daily:
• File compression – Special algorithms squeeze data by encoding repetitive patterns with fewer 1s and 0s. Clever!
• Error correction – By adding mathematical redundancy, errors flipping 1s to 0s can be detected and corrected. Resilient!
• Encryption – Prime numbers and convoluted logic operations on 1s and 0s make data unbreakable. Secure!
Some other mind-blowing examples include the Apollo Guidance Computer that used 1s and 0s to navigate to the moon, and Watson‘s ability to defeat humans at Jeopardy! 1s and 0s are so versatile!
Year Transistor count Processor
1971 2,300 Intel 4004
1978 29,000 Intel 8086
1993 3,100,000 Intel Pentium
2022 47,000,000,000 Nvidia A100 GPU
The exponential growth in transistors manipulating 1s and 0s (Source: Various)
The Journey from Abstract Concept to Foundational Technology
Stepping back, I‘m amazed by the journey 1s and 0s have taken – from abstract mathematical concept to the hidden force driving all modern computing! It just goes to show how theoretical breakthroughs
can later translate into world-changing technologies.
Somehow, using simple binary logic laid the foundation for devices that now have billions of microscopic switches crammed into tiny slivers of silicon. I find it both funny and humbling that such
profound complexity arose from something so basic.
So next time you watch 1s and 0s flash by on a computer screen, remember the pioneers like Boole and Shannon who made that possible. And who knows what new theoretical concepts today will enable the
next computing revolution! The future remains unwritten, just waiting for more 1s and 0s to work their magic in ways we can‘t yet imagine.
Conclusion: Appreciating the Elegance Behind Our Digital World
I hope this beginner‘s guide helped demystify binary computing and show how 1s and 0s make technology possible! As a tech geek, I‘m always excited to peel back the layers and understand the
foundations underlying our digital world. The elegance of Boolean logic mirroring circuit behavior is beautiful to me.
While modern gadgets hide the complexity behind sleek interfaces, 1s and 0s are still there silently working their magic. Next time you use a computer or smartphone, maybe pause a moment to
appreciate those ubiquitous digits that power our lives. Computers may be commonplace, but their binary foundations remain profound!
How useful was this post?
Click on a star to rate it!
Average rating 0 / 5. Vote count: 0
No votes so far! Be the first to rate this post.
You May Like to Read,
• What does DL mean in June‘s Journey? The Ultimate Detective League Guide
• Is Kayo a Killjoy?
• Why You Still Need a Copilot in the Age of Automation: An In-Depth Look
• What is 1TB PCIe SSD? A Complete Guide
• What does 💗 mean from a guy? A tech geek‘s insight into emoji flirtation
• Demystifying the Killing Curse – A Data Analyst‘s Deep Dive into Avada Kedavra from Harry Potter
• Demystifying IRS Refund Issue Code 846
• What Does GTA Mean? A Deep Dive into the Grand Theft Auto Phenomenon | {"url":"https://ccriellsiviabrea.com/article/demystifying-the-magic-of-1s-and-0s-a-friendly-introduction-to-binary-computing-33rd-square","timestamp":"2024-11-11T09:40:43Z","content_type":"text/html","content_length":"109291","record_id":"<urn:uuid:231175f3-48fd-4eb4-b979-1cbef26f1c96>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00171.warc.gz"} |
What is a 3 stage planetary gearbox?
Instead of the drive shaft the planetary carrier contains the sun gear, which drives the following planet stage. A three-stage gearbox is obtained by means of increasing the length of the ring gear
and adding another planet stage. A transmission ratio of 100:1 is obtained using individual ratios of 5:1, 5:1 and 4:1.
What are the 3 main components of a planetary gear set?
Each planetary gear train only contains three basic links: the sun gear, the carrier gear, and the ring gear (the planet gear is not considered in this graph model).
Does planetary gearbox increase torque?
A planetary gearhead takes a high-speed, low-torque input, say from an electric motor, then increases torque and reduces speed at the output by the gearhead ratio. This lets motors run at higher,
more-efficient rpms in equipment that operates at low speeds.
How are planetary gear ratios calculated?
To make calculating planetary gear ratios as simple as possible, note the number of teeth on the sun and ring gears. Next, add the two numbers together: The sum of the two gears’ teeth equals the
number of teeth on the planetary gears connected to the carrier.
How do you calculate gear stage?
If, for example, gear 5 (z5 = 60 teeth) is replaced by a gear wheel twice as large with twice the number of teeth (z5’= 120 teeth), the gear ratio is doubled in this gear stage to i3’= 8. This
doubling also doubles the overall transmission ratio from 24 to it’= 48.
What is the function of planetary gear sets?
Planetary gears are often used when space and weight are an issue, but a large amount of speed reduction and torque are needed. This requirement applies to a variety of industries, including tractors
and construction equipment where a large amount of torque is needed to drive the wheels.
How does a planetary gearbox work?
Planetary Gearboxes are a type of gearbox where the input and output both have the same centre of rotation. This means that the centre of the input gear revolves around the centre of the output gear
and the input and output shafts are aligned.
When should I use planetary gears?
What is the advantage of planetary gears?
The advantages of planetary gearboxes: Coaxial arrangement of input shaft and output shaft. Load distribution to several planetary gears. High efficiency due to low rolling power. Almost unlimited
transmission ratio options due to combination of several planet stages.
Why are the helical gears used commonly in transmission over spur gears?
Why are the helical gears used commonly in transmission over spur gears? Explanation: The teeth profile on the helical gear is at an angle to the axis of the gear because of which helical gears
produce less noise during operation and also they have high strength.
What is used to hold planetary gears?
Understanding Holding Devices for Planetary Gear Sets
• Multiplate Clutch – holds two rotating planetary components.
• Brake – holds planetary components to the housing; two types: Multiplate Brake / Brake Band.
• Sprag One Way Clutch – holds planetary components in one rotational direction. | {"url":"https://www.worldsrichpeople.com/what-is-a-3-stage-planetary-gearbox/","timestamp":"2024-11-09T19:08:31Z","content_type":"text/html","content_length":"54919","record_id":"<urn:uuid:e42b2aba-9d50-44bb-9b15-284ed0df2e96>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00544.warc.gz"} |
Use thin-plate splines for image warping
Go to the end to download the full example code. or to run this example in your browser via Binder
Use thin-plate splines for image warping#
To warp an image, we start with a set of source and target coordinates. The goal is to deform the image such that the source points move to the target locations. Typically, we only know the target
positions for a few, select source points. To calculate the target positions for all other pixel positions, we need a model. Various such models exist, such as affine or projective transformations.
Most transformations are linear (i.e., they preserve straight lines), but sometimes we need more flexibility. One model that represents a non-linear transformation, i.e. one where lines can bend, is
thin-plate splines [1] [2].
Thin-plate splines draw on the analogy of a metal sheet, which has inherent rigidity. Consider our source points: each has to move a certain distance, in both the x and y directions, to land in its
corresponding target position. First, examine only the x coordinates. Imagine placing a thin metal plate on top of the image. Now bend it, such that at each source point, the plate’s z offset is the
distance, positive or negative, that that source point has to travel in the x direction in order to land in its target position. The plate resists bending, and therefore remains smooth. We can read
offsets for coordinates other than source points from the position of the plate. The same procedure can be repeated for the y coordinates.
This gives us our thin-plate spline model that maps any (x, y) coordinate to a target position.
Correct barrel distortion#
In this example, we demonstrate how to correct barrel distortion [3] using a thin-plate spline transform. Barrel distortion creates the characteristic fisheye effect, where image magnification
decreases with distance from the image center.
We first generate an example dataset, by applying a fisheye warp to a checkboard image, and thereafter apply the inverse corrective transform.
import matplotlib.pyplot as plt
import numpy as np
import skimage as ski
def radial_distortion(xy, k1=0.9, k2=0.5):
"""Distort coordinates `xy` symmetrically around their own center."""
xy_c = xy.max(axis=0) / 2
xy = (xy - xy_c) / xy_c
radius = np.linalg.norm(xy, axis=1)
distortion_model = (1 + k1 * radius + k2 * radius**2) * k2
xy *= distortion_model.reshape(-1, 1)
xy = xy * xy_c + xy_c
return xy
image = ski.data.checkerboard()
image = ski.transform.warp(image, radial_distortion, cval=0.5)
# Pick a few `src` points by hand, and move the corresponding `dst` points to their
# expected positions.
# fmt: off
src = np.array([[22, 22], [100, 10], [177, 22], [190, 100], [177, 177], [100, 188],
[22, 177], [ 10, 100], [ 66, 66], [133, 66], [ 66, 133], [133, 133]])
dst = np.array([[ 0, 0], [100, 0], [200, 0], [200, 100], [200, 200], [100, 200],
[ 0, 200], [ 0, 100], [ 73, 73], [128, 73], [ 73, 128], [128, 128]])
# fmt: on
# Estimate the TPS transformation from these points and then warp the image.
# We switch `src` and `dst` here because `skimage.transform.warp` requires the
# inverse transformation!
tps = ski.transform.ThinPlateSplineTransform()
tps.estimate(dst, src)
warped = ski.transform.warp(image, tps)
# Plot the results
fig, axs = plt.subplots(1, 2)
axs[0].imshow(image, cmap='gray')
axs[0].scatter(src[:, 0], src[:, 1], marker='x', color='cyan')
axs[1].imshow(warped, cmap='gray', extent=(0, 200, 200, 0))
axs[1].scatter(dst[:, 0], dst[:, 1], marker='x', color='cyan')
point_labels = [str(i) for i in range(len(src))]
for i, label in enumerate(point_labels):
(src[:, 0][i], src[:, 1][i]),
textcoords="offset points",
xytext=(0, 5),
(dst[:, 0][i], dst[:, 1][i]),
textcoords="offset points",
xytext=(0, 5),
Total running time of the script: (0 minutes 0.234 seconds) | {"url":"https://scikit-image.org/docs/stable/auto_examples/transform/plot_tps_deformation.html","timestamp":"2024-11-09T16:16:27Z","content_type":"text/html","content_length":"72684","record_id":"<urn:uuid:f8d704a8-dad4-4bf2-bc0d-8a7a307e77f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00408.warc.gz"} |
Number Pairs in a Table - iPohly INC
Number Pairs in a Table
3.5(E) represent real-world relationships using number pairs in a table and verbal descriptions
Okay… now what? This is the first time students in Texas are learning about this very important topic! You are teaching the baby beginnings of slope. We don’t write “rules” for the table, but
they are expected to verbally describe the table. Students are looking at a different type of pattern. We are beginning to form algebraic thinking- and kids can do it! In second grade, students
looked at patterns in one set of numbers. For example, 2,4,6,8… this pattern is skip counting by two. Basically they were looking at the output without looking at the input. If we put this pattern
in an input/output table it would look like this:
The 2 is the first number in the pattern, the 4 is the second and so on. We need to teach kids to look at the relationship between the input and the output. This table shows the output is 2 times
the input. It also shows the input is half (divided by 2) of the output. Tricky! The key is to teach kids to look at the “between” relationship! If they look for a pattern in the output only,
they will say the pattern is adding 2. The input is adding one.
between the input tube and the output tube?” I also give them some sort of reference because this makes learning more concrete. Like- John saw a car with 4 wheels. #1 goes in and #4 comes out.
Then I say, ” John saw 2 cars.” #2 goes in and #8 comes out. I record the numbers in the input/output table. It is important to label the table with cars and wheels under the words input and
output. I teach both types of relationships- additive and multiplicative this way. It is important to vary the direction of the table- for some reason, students freak out when the table goes
vertical. They are used to reading left to right and don’t always see the relationship as up and down! So make sure you don’t tell them not to read up and down!
The verbal description to the table is the key! The canoe and paddleboats problem on the 2016 STAAR test caused a big problem across the state. Only 43
{ead11f5758ba27f8b85d16bb338278bf88aa311cb7529b45aa5c71cfd4587f47} of third graders found the correct answer. 28{ead11f5758ba27f8b85d16bb338278bf88aa311cb7529b45aa5c71cfd4587f47} of the students
picked F- This answers shows the relationship of +18 (that was what they were looking for) but it was backwards! F shows 18 more paddleboats than canoes, but the problem asked them to find 18 more
canoes than paddleboats. The verbal description is important! That is why I never do tables without a verbal description!
Would you like a sample of one of the resources I use in my small group for working with additive relationships? Click on the image below for the PDF.
If you want your weekends back and would like the plans for the entire unit, you can find them here: | {"url":"https://ipohlyinc.com/number-pairs-in-a-table/","timestamp":"2024-11-11T16:51:52Z","content_type":"text/html","content_length":"61134","record_id":"<urn:uuid:7195035c-1d13-46a3-84f2-8ef86c253b9a>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00300.warc.gz"} |
Accrued Interest
Bond Pricing
Bond prices are determined by 5 factors:
1. par value
2. coupon rate
3. market interest rates
4. accrued interest
5. credit rating of the issuer
Generally, the issuer sets the price and the yield of the bond so that it will sell enough bonds to supply the amount that it desires. The higher the credit rating of the issuer, the lower the yield
that it must offer to sell its bonds. A change in the credit rating of the issuer will affect the price of its bonds in the secondary market: a higher credit rating will increase the price, while a
lower rating will decrease the price. The other factors that determine the price of a bond have a more complex interaction.
When a bond is first issued, it is generally sold at par, which is the face value of the bond. Most corporate bonds, for instance, have a face and par value of $1,000. The par value is the principal,
which is received at the end of the bond's term, i.e., at maturity. Sometimes when the demand is higher or lower than an issuer expected, the bonds might sell higher or lower than par. In the
secondary market, bond prices are almost always different from par, because interest rates change continuously. When a bond trades for more than par, then it is selling at a premium, which will pay a
lower yield than its stated coupon rate, and when it is selling for less, it is selling at a discount, paying a higher yield than its coupon rate. When interest rates rise, bond prices decline, and
vice versa. Bond prices will also include accrued interest, which is the interest earned between coupon payment dates. Clean bond prices are prices without accrued interest; dirty bond prices include
accrued interest.
This graph shows how interest rates even affect exchange-traded funds based on bonds. Note that the 3 Vanguard ETFs (VGSH, VGIT, VGLT) based on US Treasuries were affected similarly when interest
rates changed from 2018 to 2022, but the long-term ETF, VGLT, changed in price by a much larger percentage even though interest rates changed by only 2%. This reflects the greater interest-rate
sensitivity of long-term bonds over short-term bonds.
Investment Tip. Buy long-term bonds when interest rates are highest. You can usually determine when interest rates are near the top by monitoring the economy. The Federal Reserve increases interest
rates when inflation is high or increasing. If inflation is subdued, then the Federal Reserve will not increase interest rates further since that would depress the economy. Never buy long-term bonds
when interest rates are near 0% unless you intend to keep the bonds until maturity because interest rates can only rise! For instance, if you had bought VGLT at the end of 2018 and held until March,
2020, you would have earned a capital gain of more than 40% while earning a nice, guaranteed interest rate. VGLT spiked when the Fed funds rate dropped to near zero because of the Covid-19 pandemic,
That's when you sell, since interest rates can only rise from there, and by mid-2022, they were rising fast because inflation was high. Naturally, this causes bond prices to drop, including VGLT, as
you can see in the graph.
Although prevailing interest rates are usually the main determinants of bond prices in the secondary market, as a bond approaches maturity, the present value of its future payments converges to the
par value; therefore, the par value becomes more important than the prevailing interest rates, since the bond price, whether at a premium or discount, converges to the par value, as can be seen in
the diagram below.
Bond Value = the Sum of the Present Value of Future Payments
A bond pays interest either periodically or, in the case of zero coupon bonds, at maturity. Therefore, the value of the bond = the sum of the present value of all future payments — hence, it is the
present value of an annuity, which is a series of periodic payments. The present value is calculated using the prevailing market interest rate for the term and risk profile of the bond, which may be
more or less than the coupon rate. For a coupon bond that pays interest periodically, its value can be calculated thus:
Bond Value
• = Present Value (PV) of Interest Payments
• + Present Value of Principal Payment
Bond Value
• = PV(1^st Payment)
• + PV(2^nd Payment) + ... + PV(Last Payment)
• + PV(Principal Payment)
Bond Price Formula
Clean Bond Price = C[1](1+r/k)^1 + C[2](1+r/k)^2 + ... + C[n](1+r/k)^kn + P(1+r/k)^kn
C = coupon, or interest, payment per period
k = number of coupon periods in 1 year
n = number of years until maturity
r = annualized market interest rate
P = par value of bond
Example: Calculating Bond Value as the Present Value of its Payments
Suppose a company issues a 3-year bond with a par value of $1,000 that pays 4% interest annually, which is also the prevailing market interest rate. What is the present value of the payments?
The following table shows the amount received each year and the present value of that amount. As you can see, the sum of the present value of each payment equals the par value of the bond.
Year Payment Amount Received Present Value
1 Interest $40 $38.46
2 Interest $40 $36.98
3 Interest + Principal $1040 $924.56
Totals $1120 $1,000.00
The above formula can be simplified by using the formula for the present value of an annuity, and letting k=2 for bonds that pay a semiannual coupon:
Simplified Bond Price Formula for Semiannual Coupon
Clean Bond Price = Cr [ 1 − 1(1+r/2)^2n ] + P(1+r/2)^2n
C = Annual payment from coupons
n = number of years until maturity
r = market annual interest rate
P = par value of bond
Note that the above formula is sometimes written with both C and r divided by 2; the results are the same, since it is a ratio.
Example: Using the Simplified Bond Pricing Formula
• Par Value: 100
• Nominal Yield: 5%
• Annual Coupon Payment: $5
• Maturity: 5 years
• Market Interest Rate = 4%
Case 1: 2 Annual Coupon Payments
Then, since there are 10 semiannual payment periods, the market interest rate is divided by 2 to account for the shorter period:
Bond = 5.04 [ 1 − 1(1.02)^10 ] + 100(1.02)^10 = 104.49
Case 2: 1 Annual Coupon Payment, resulting in 5 payment periods at the market interest rate:
Bond = 5.04 [ 1 − 1(1.04)^5 ] + 100(1.04)^5 = 104.45
In the primary bond market, where the buyer buys the bond from the issuer, the bond usually sells for par value, which = the bond's value using the coupon rate of the bond. However, in the secondary
bond market, bond price still depends on the bond's value, but the interest rate to calculate that value is determined by the market interest rate, which is reflected in the actual bids and offers
for bonds. Additionally, the buyer of the bond must pay any accrued interest on top of the bond's price unless the bond is purchased on the day it pays interest.
Bond Price Listings
When bond prices are listed, the convention is to list them as a percentage of par value, regardless of what the face value of the bond is, with 100 being equal to par value. Thus, a bond with a face
value of $1,000 selling for par, sells for $1,000, and a bond with a face value of $5,000 also selling for par will both have their price listed as 100, meaning their prices are equal to 100% of par
value, or $100 for each $100 of face value.
This pricing convention allows different bonds with different face values to be compared directly. For instance, if a $1,000 corporate bond was listed as 90 and a $5,000 municipal bond was listed as
95, then it can be easily seen that the $1,000 bond is selling at a bigger discount, and, therefore, has a higher yield. To find the actual price of the bond, the listed price must be multiplied as a
percentage by the face value of the bond, so the price for the $1,000 bond is 90% × $1,000 = 0.9 × $1,000 = $900, and the price for the $5,000 bond is 95% × $5,000 = .95 × $5,000 = $4,750.
A point = 1% of the bond's face value. Thus, a point's actual value depends on the face value of the bond. Thus, 1 point = $10 for a $1,000 bond, but $50 for a $5,000 bond. So a $1,000 bond that is
selling for 97 is selling at a 3 point discount, or $30 below par value, which equals $970.
Brokers profit from bonds either by charging a set commission or by charging a markup, a certain percentage over and above what the broker paid for the bond. Only a small portion of the more than 1
million bonds available are sold on public exchanges like the New York Stock Exchange, where pricing is transparent. Instead, most bonds are traded over the counter. Most prices listed by brokers do
not include any markup that they may charge, but some brokers, such as Fidelity, may charge a set commission, such as $1 per bond.
You can compare prices by comparing listed prices by different brokers if you have more than 1 brokerage account. Brokers may not have the same bonds with the same CUSIP number, but they will have
comparable bonds that should have the same yield, such as a 5- year corporate bond with an AAA rating.
You can also check trade reporting data provided by the Municipal Securities Rulemaking Board (MSRB) for municipal bonds and by the Trade Reporting and Compliance Engine (TRACE) for fixed income
securities traded over-the-counter.
Accrued Interest
Listed bond prices are clean prices (aka flat prices), which do not include accrued interest. Most bonds pay interest semi-annually. For settlement dates when interest is paid, the bond price = the
flat price. Between payment dates, accrued interest must be added to the flat price, which is often called the dirty price (aka all-in price, gross price):
Dirty Bond Price = Clean Price + Accrued Interest
Accrued interest is the interest that has been earned, but not paid, and is calculated by the following formula:
Formula for Calculating Accrued Interest
Number of Days
Accrued Interest = Interest Payment × Since Last PaymentNumber of days
between payments
Graph of the purchase price of a bond over 2 years, which = the flat price + accrued interest. (It is assumed that the flat price remains constant over the 2 years, but would actually fluctuate with
interest rates, and because of other factors, such as changes in the credit rating of the issuer.) The flat price is what is listed in bond tables for prices. The accrued interest must be calculated
according to the above formula. Note that the bond price steadily increases each day until reaching a peak the day before an interest payment, then drops back to the flat price on the day of the
When you buy a bond on the secondary market, you must pay the former owner of the bond the accrued interest. If this were not so, you could make a fortune buying bonds right before they paid interest
then selling them afterward. Because the interest accrues every day, the bond price increases accordingly until the interest payment date, when it drops to its flat price, then starts accruing
interest again.
Day-Count Conventions
In calculating the accrued interest, the actual number of days was counted from the last interest payment to the value date. Most bonds use this day-count basis, called actual/actual basis, because
the actual number of days are used in the calculations. However, some bonds use a different day-count basis, which will cause the accrued interest to be slightly different from that calculated using
the actual/actual convention. Closely related to actual/actual are the following conventions, which are only used for bonds with 1 annual coupon payment:
• Accrued Interest = Coupon Rate × Days/360
• Accrued Interest = Coupon Rate × Days/365
Note that the accrued interest calculated under the actual/360 convention is slightly more than the interest calculated under the actual/actual or the actual/365 method.
There are 2 other methods where each month counts as 30 days, regardless of the number of days in the month and each year is considered to have 360 days. Although these methods are rarely used
nowadays to calculate accrued interest, they did simplify calculating the number of days between a coupon date and the value date, which was valuable before the advent of calculators and computers,
especially since the calculated interest differed little from that calculated with the actual/actual method. So, under these methods, there is always 3 days between February 28 and March 1, because
each month counts as 30 days, including February, even though February has either 28 or 29 days. By the same reasoning, there are 25 days between January 15 and February 10, even though there are
actually 26 days between those dates. When figuring accrued interest using any day-count convention, the 1^st day is counted, but not the last day. So in the previous example, January 15 is counted,
but not February 10.
30/360 and 30E/360 Day-Count Conventions
Start Date: M1/D1/Y1
End Date: M2/D2/Y2
Day Count Fraction = Day Count/360
Day Count = (Y2 − Y1) × 360 + (M2 − M1) × 30 + (D2 − D1)
30/360 Day-Count Convention (aka US 30/360)
If (D1 = 31) Set D1 = 30
If (D2 = 31) and (D1 = 30 or 31) Set D2 = 30
30E/360 Day-Count Convention (aka European 30/360)
If D2 = 31 Set D2 = 30
So the number of days between December 29 and January 31 is 32 under the 30/360 convention, but 31 days under the 30E/360 convention. This is determined thus:
• 1 month × 30 = 30 days +
□ Under 30/360, January 31 is not changed since the 1^st date was not 30 or 31, so there are 2 additional days after January 29, yielding a total of 30 + 2 = 32 days.
□ Under 30E/360, the January 31 date is automatically changed to January 30, so that yields a total of 30 + 1 = 31 days.
The number of days are then divided by 360, then multiplied by the coupon rate to determine the accrued interest:
30/360 and 30E/360:
• Accrued Interest = Coupon Rate × Days/360
Day Count Conventions Used in US Bond Markets
Bond Market Day-Count Basis
Treasury Notes and Bonds Actual/Actual
Corporate and Municipal Bonds 30/360
Money Market Instruments
• So a 1% bond would earn 365/360 × 1% of interest in 365 days.
As already stated, most bond markets outside of the U.S. use the actual/actual convention except:
Bond Markets Not Using Actual/Actual
Bond Market Day-Count Basis
Eurobonds 30/360
Denmark, Sweden, Switzerland 30E/360
Norway Actual/365
Example: Calculating the Purchase Price for a Bond with Accrued Interest
You purchase a corporate bond with a settlement date on September 15 with a face value of $1,000 and a nominal yield of 8%, that has a listed price of 100-08, and that pays interest semi-annually on
February 15 and August 15. Accrued interest is determined using the actual/actual convention. How much must you pay?
The semi-annual interest payment is $40 and there were 31 days since the last interest payment on August 15. If the settlement date fell on a interest payment date, the bond price would equal the
listed price: 100.25% × $1,000.00 = $1,002.50 (8/32 = 1/4 = .25, so 100-08 = 100.25% of par value). Since the settlement date was 31 days after the last payment date, accrued interest must be added.
Using the above formula, with 184 days between coupon payments, we find that:
Accrued Interest = $40 × 31184
= $6.74
Therefore, the actual purchase price for the bond will be $1,002.50 + $6.74 = $1,009.24.
Tip: It may be more convenient to use a spreadsheet, such as Excel, that provides several functions for determining the number of days or the dirty bond price, with the settlement and maturity dates
expressed as either a quote (e.g., "12/11/2012") or as a cell reference (e.g., B12):
Number of Days = maturity,
since Last Payment frequency,
Number of Days = maturity,
Between Payments frequency,
Bond Price = ytm,
Search Help for more information. Below is another example of obtaining a bond's price by using Excel's PRICE function:
15-Feb-24 Settlement Date
15-Nov-37 Maturity Date
5.75% Coupon Rate
6.50% Yield to Maturity
100 Redemption value
2 Number of Interest Payments per Year
1 Day Count Basis (Month/Year = Actual/Actual)
= PRICE(settlement,maturity,rate,ytm,redemption,frequency,basis)
$93.24 % of Par Value of Actual Price for Corporate Bond, $1,000 Face Value
$932.39 Actual Price for Corporate Bond, $1,000 Face Value
To calculate the accrued interest on a zero coupon bond, which pays no interest, but is issued at a deep discount, the amount of interest that accrues every day is calculated by using a straight-line
amortization, which is found by subtracting the discounted issue price from its face value, and dividing by the number of days in the term of the bond. This is the interest earned in 1 day, which is
then multiplied by the number of days from the issue date.
Steps to Calculate the Price of a Zero Coupon Bond
1. Total Interest Paid by Zero Coupon Bond
□ = Face Value − Discounted Issue Price
2. 1 Day Interest
□ = Total Interest / Number of Days in Bond's Term
3. Accrued Interest
□ = (Settlement Date − Issue Date) in Days × 1 Day Interest
4. Zero Coupon Bond Price
□ = Discounted Issue Price + Accrued Interest
Bonds with Ex-Dividend Periods may have Negative Accrued Interest
Interest accrues on bonds from one coupon date to the day before the next coupon date. However, some bonds have a so-called ex-dividend date (aka ex-coupon date), where the owner of record is
determined before the end of the coupon period, in which case, the owner will receive the entire amount of the coupon payment, even if the bond is sold before the end of the period. The ex-dividend
period (aka ex-coupon period) is the time during which the bond will continue to accrue interest for the owner of record on the ex-dividend date. (The ex-dividend date and the ex-dividend period are
misnomers, since bonds pay interest and not dividends, but the terminology was borrowed from stocks, since the concept is similar. Although ex-coupon is more descriptive, ex-dividend is more widely
used.) If a bond is purchased during the ex-dividend period, then accrued interest from the purchase date until the end of the coupon period is subtracted from the clean price of the bond. In other
words, the accrued interest is negative. Only a few bonds have ex-dividend periods, which are usually 7 days or less. The UK gilt, for instance, has an ex-dividend period of 7 days, so if the bond is
purchased at the beginning of that 7-day period, then the amount of interest subtracted from the clean price = the coupon rate × 7/365.
Most bond markets do not have ex-dividend periods except:
• Australia
• Denmark
• New Zealand
• Norway
• Sweden
• United Kingdom
PRICE, PRICEDISC, PRICEMAT, and DISC Functions in Microsoft Excel for Calculating Bond Prices and Other Securities Paying Interest
Microsoft Excel has several formulas for calculating bond prices and other securities paying interest, such as Treasuries or certificates of deposit (CDs), that include accrued interest, if any.
Microsoft Excel Functions: PRICE, PRICEDISC, PRICEMAT, and DISC
• Calculates the price, given the yield.
□ Bond Price (per $100 of face value)
☆ = PRICE(
○ settlement,
○ maturity,
○ rate,
○ yield,
○ redemption,
○ frequency,
○ basis)
□ Discounted Bond Price
☆ = PRICEDISC(
○ settlement,
○ maturity,
○ discount,
○ redemption,
○ basis)
• Calculates the yield, given the price.
□ Discount Rate of Security
☆ = DISC(
○ settlement,
○ maturity,
○ price,
○ redemption,
○ basis)
• Calculates the price of a security that pays interest only at maturity, such as a negotiable Certificate of Deposit:
□ Security Price
☆ = PRICEMAT(
○ settlement,
○ maturity,
○ issue,
○ rate,
○ yield,
○ basis)
• The following dates are expressed as cell references (e.g., A1), or as DATE functions [format Date(year,month,day)]. (Note that, according to Microsoft, problems may arise if dates are entered as
□ Settlement = Settlement date.
□ Maturity = Maturity date.
□ Issue = Issue date.
• Rates are listed in decimal form (5%=.05):
□ Rate = Nominal annual coupon interest rate.
□ Yield = Annual yield to maturity.
□ Discount = Percentage of discount/
• Price = Price of security as a % of par value (but without the % or $ sign, so if $1,000 par-value bond is selling for $857.30, then the corresponding percentage value is 85.73).
• Redemption = Value of security at redemption per $100 of face value, usually = 100.
• Frequency = Number of coupon payments / year.
□ 1 = Annual
□ 2 = Semiannual (the most common value)
□ 4 = Quarterly
• Basis = The number of days counted per year.
□ 0 = 30/360 (This U.S. basis is the default, if omitted)
□ 1 = actual days in month/actual days in year
□ 2 = actual days in month/360
□ 3 = actual days in month/365 (even for a leap year)
□ 4 = European 30/360
Examples — Using Microsoft Excel for Calculating Bond Prices and Discounts
The following listed variables — where they apply — will be used for each of the example calculations that follow, for a 10-year bond originally issued in 1/1/2024 with a par value of $1,000:
• Settlement date = 1/2/2024
• Maturity date = 12/15/2033
• Issue date = 12/15/2023
• Coupon rate = 6%
• Yield to maturity = 8%
• Price (per $100 of face value) = 21.99
• Redemption = 100
• Frequency = 2 for most coupon bonds.
• Basis = 1 (actual/actual)
Price of a bond with a yield to maturity of 8%:
Bond Price
• = PRICE(Date(2024,1,2),
□ Date(2033,12,15),
□ 0.06,
□ 0.08,
□ 100,2,1)
• = 86.44858
• = $864.49
The discount price of a zero coupon bond with a $1,000 par value yielding 8%:
Price Discount
• = PRICEDISC(Date(2024,1,2),
□ Date(2033,12,15),
□ 0.06,
□ 0.08,
□ 100,1)
• = 20.39420
• = $203.94
The interest rate of a discounted zero coupon bond paying $1,000 at maturity, but that is now selling for $219.90:
Interest Rate of Bond Discount
• = DISC(Date(2024,1,2),
□ Date(2033,12,15),
□ 21.99,
□ 100,1)
• = 0.78396
• = 7.84%
• Note that the PRICEDISC function value has been rounded, with the results used in the DISC function to verify the results. (21.99 = $219.90 for a bond with a $1,000 par value).
PRICEMAT calculates the prices of securities that only pay interest at maturity:
What is the price of a negotiable, 90-day CD originally issued for $100,000 on 3/1/2024 with a nominal yield of 8%, a yield to maturity of 6% and a settlement date of 4/1/2024? Using the Microsoft
Excel Date function, with format DATE(year,month,day), to calculate the maturity date by adding 90 days to the issue date, and choosing the banker's year of 360 days by omitting its value from the
formula, yields the following results:
• Market Price of CD
□ = PRICEMAT(Date(2024,4,1),
☆ DATE(2024,3,1) + 90,,
☆ Date(2024,3,1),
☆ 0.08,
☆ 0.06)
□ = 99.65916 (per $100 of face value) × 1,000
□ = $99,659.16 | {"url":"https://thismatter.com/money/bonds/bond-pricing.htm","timestamp":"2024-11-04T01:59:13Z","content_type":"text/html","content_length":"48228","record_id":"<urn:uuid:6649963e-8544-4ad8-ab7a-94b56f4e564d>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00482.warc.gz"} |
Adding Multiples Of 10 To A 2 Digit Number Worksheet
Math, particularly multiplication, forms the foundation of countless scholastic self-controls and real-world applications. Yet, for numerous students, mastering multiplication can posture a
challenge. To resolve this obstacle, instructors and parents have actually accepted a powerful device: Adding Multiples Of 10 To A 2 Digit Number Worksheet.
Introduction to Adding Multiples Of 10 To A 2 Digit Number Worksheet
Adding Multiples Of 10 To A 2 Digit Number Worksheet
Adding Multiples Of 10 To A 2 Digit Number Worksheet -
2 Digit Addition Multiples Of 10 Worksheets Teaching Resources TpT Browse 2 digit addition multiples of 10 resources on Teachers Pay Teachers a marketplace trusted by millions of teachers for
original educational resources DID YOU KNOW Seamlessly assign resources as digital activities Learn how in 5 minutes with a tutorial resource
Using mental math add 10 to each 2 digit number This worksheet has 8 horizontal problems and 2 word problems 1st Grade View PDF Task Cards Adding 10 THese task cards have place value blocks and
partial hundreds charts Use the models or mental math to solve 30 cards in all
Importance of Multiplication Practice Comprehending multiplication is crucial, laying a solid foundation for advanced mathematical ideas. Adding Multiples Of 10 To A 2 Digit Number Worksheet offer
structured and targeted practice, promoting a deeper understanding of this fundamental arithmetic operation.
Advancement of Adding Multiples Of 10 To A 2 Digit Number Worksheet
Addition Over 100 Adding multiples of 10 Or 100 To A Three Digit Number Worksheet KS2 Number
Addition Over 100 Adding multiples of 10 Or 100 To A Three Digit Number Worksheet KS2 Number
First graders will add multiples of 10 to 2 digit numbers with the sums up to 100 Children can use different strategies and materials such as number lines or a hundreds chart to solve the equations
Some children might need to draw pictures or use concrete materials like number blocks so make sure to give them plenty of opportunities to do so
With our Adding Multiples of 10 to 2 Digit Numbers sums you ll be able to build children s mathematical understanding for addition By promoting problem solving in the classroom you ll be supporting
your children to develop a growth mindset and critical thinking skills When you click the green download button above you ll find 12 different addition sums to use in the classroom or
From typical pen-and-paper workouts to digitized interactive styles, Adding Multiples Of 10 To A 2 Digit Number Worksheet have actually advanced, dealing with diverse discovering styles and
Kinds Of Adding Multiples Of 10 To A 2 Digit Number Worksheet
Fundamental Multiplication Sheets Basic workouts concentrating on multiplication tables, helping students develop a strong math base.
Word Trouble Worksheets
Real-life scenarios integrated into troubles, improving vital reasoning and application abilities.
Timed Multiplication Drills Examinations developed to improve rate and accuracy, helping in fast psychological mathematics.
Benefits of Using Adding Multiples Of 10 To A 2 Digit Number Worksheet
Adding And Subtracting Multiples of 10 Worksheet Twinkl
Adding And Subtracting Multiples of 10 Worksheet Twinkl
This pack includes several activities and assessments or practice pages for adding multiples of tens to a 2 digit number I made this pack to use in introducing the common core standard 1 NBT 4 add
within 100 including a 2 digit number and a multiple of 10 using concrete models or drawings and
Grade 2 math worksheets on adding whole tens to a 2 digit number Free pdf worksheets from K5 Learning s online reading and math program
Enhanced Mathematical Skills
Constant method hones multiplication proficiency, boosting overall mathematics capabilities.
Enhanced Problem-Solving Abilities
Word problems in worksheets establish logical thinking and strategy application.
Self-Paced Knowing Advantages
Worksheets accommodate private understanding speeds, cultivating a comfy and versatile learning atmosphere.
Exactly How to Create Engaging Adding Multiples Of 10 To A 2 Digit Number Worksheet
Incorporating Visuals and Colors Vivid visuals and colors catch focus, making worksheets aesthetically appealing and involving.
Consisting Of Real-Life Situations
Connecting multiplication to everyday situations includes relevance and usefulness to exercises.
Customizing Worksheets to Various Skill Degrees Customizing worksheets based on varying proficiency levels makes certain comprehensive knowing. Interactive and Online Multiplication Resources Digital
Multiplication Tools and Games Technology-based resources provide interactive discovering experiences, making multiplication appealing and satisfying. Interactive Web Sites and Apps On-line platforms
provide diverse and available multiplication technique, supplementing standard worksheets. Customizing Worksheets for Different Understanding Styles Aesthetic Learners Aesthetic help and diagrams
help comprehension for students inclined toward aesthetic discovering. Auditory Learners Spoken multiplication issues or mnemonics accommodate learners who realize principles through acoustic means.
Kinesthetic Learners Hands-on activities and manipulatives sustain kinesthetic learners in recognizing multiplication. Tips for Effective Implementation in Understanding Uniformity in Practice Normal
practice reinforces multiplication abilities, advertising retention and fluency. Stabilizing Repeating and Variety A mix of recurring workouts and diverse issue layouts preserves passion and
comprehension. Giving Positive Responses Feedback help in determining locations of renovation, urging continued progression. Obstacles in Multiplication Method and Solutions Inspiration and
Engagement Difficulties Dull drills can lead to disinterest; innovative methods can reignite motivation. Conquering Concern of Math Negative assumptions around mathematics can impede development;
developing a positive learning setting is essential. Influence of Adding Multiples Of 10 To A 2 Digit Number Worksheet on Academic Performance Research Studies and Research Study Findings Study
indicates a favorable connection between consistent worksheet use and improved math performance.
Adding Multiples Of 10 To A 2 Digit Number Worksheet become functional tools, promoting mathematical efficiency in students while fitting varied knowing styles. From basic drills to interactive on
the internet resources, these worksheets not only enhance multiplication abilities but also advertise important reasoning and analytic capabilities.
Subtracting Multiples of 10 From A 2 Digit Number adding multiples of 10 To
Adding Two Two Digit Numbers Without Regrouping Worksheet Turtle Diary
Check more of Adding Multiples Of 10 To A 2 Digit Number Worksheet below
2 Digit Addition Worksheets
Adding 2 digit Numbers Crossing Tens Worksheet Carol Jone s Addition Worksheets
Adding Two Numbers Up To Two Digits Worksheet Turtle Diary
Worksheets To Use To Introduce adding Two digit Numbers To multiples Of Ten First Grade Math
Two digit Addition Based On Base Ten Blocks Base Ten Blocks Math School First Grade Math
Number Grid Adding And Subtracting Multiples Of Ten Maths With Mum
Adding 10 Multiples of 10 Worksheets Super Teacher Worksheets
Using mental math add 10 to each 2 digit number This worksheet has 8 horizontal problems and 2 word problems 1st Grade View PDF Task Cards Adding 10 THese task cards have place value blocks and
partial hundreds charts Use the models or mental math to solve 30 cards in all
Adding Multiples of 10 to 2 Digit Numbers Twinkl Maths
With our Adding Multiples of 10 to 2 Digit Numbers sums you ll be able to build children s mathematical understanding for addition By promoting problem solving in the classroom you ll be supporting
your children to develop a growth mindset and critical thinking skills Show more Related Searches
Using mental math add 10 to each 2 digit number This worksheet has 8 horizontal problems and 2 word problems 1st Grade View PDF Task Cards Adding 10 THese task cards have place value blocks and
partial hundreds charts Use the models or mental math to solve 30 cards in all
With our Adding Multiples of 10 to 2 Digit Numbers sums you ll be able to build children s mathematical understanding for addition By promoting problem solving in the classroom you ll be supporting
your children to develop a growth mindset and critical thinking skills Show more Related Searches
Worksheets To Use To Introduce adding Two digit Numbers To multiples Of Ten First Grade Math
Adding 2 digit Numbers Crossing Tens Worksheet Carol Jone s Addition Worksheets
Two digit Addition Based On Base Ten Blocks Base Ten Blocks Math School First Grade Math
Number Grid Adding And Subtracting Multiples Of Ten Maths With Mum
Subtracting Two Three Digit Numbers Worksheets William Hopper s Addition Worksheets
Adding 3 Single Digit Numbers Math Worksheet Twisty Noodle
Adding 3 Single Digit Numbers Math Worksheet Twisty Noodle
Dividing By Multiples Of 10 Worksheets 5th Grade Free Printable
FAQs (Frequently Asked Questions).
Are Adding Multiples Of 10 To A 2 Digit Number Worksheet suitable for every age groups?
Yes, worksheets can be customized to different age and ability levels, making them adaptable for different students.
How frequently should students practice using Adding Multiples Of 10 To A 2 Digit Number Worksheet?
Regular method is vital. Regular sessions, preferably a few times a week, can yield substantial enhancement.
Can worksheets alone enhance mathematics abilities?
Worksheets are a valuable tool yet must be supplemented with varied understanding methods for detailed skill advancement.
Exist on-line systems providing cost-free Adding Multiples Of 10 To A 2 Digit Number Worksheet?
Yes, lots of academic web sites supply free access to a wide range of Adding Multiples Of 10 To A 2 Digit Number Worksheet.
Exactly how can moms and dads sustain their youngsters's multiplication technique in the house?
Motivating constant technique, supplying assistance, and creating a favorable understanding atmosphere are valuable steps. | {"url":"https://crown-darts.com/en/adding-multiples-of-10-to-a-2-digit-number-worksheet.html","timestamp":"2024-11-13T21:07:16Z","content_type":"text/html","content_length":"29913","record_id":"<urn:uuid:da948b6e-15c7-4657-9aad-7b38f589971e>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00448.warc.gz"} |
O-Levels E-Math: Number Patterns and Sequences
Number Patterns is one of the chapters in O-level E-Math where students rarely have 100% confidence of getting it right during the exams. Most Singaporean students are pretty good at identifying
patterns and have no problem spotting the logic behind each sequence. However, the difficulty often lies in coming up with a formula or an equation that expresses the Nth-term of the number sequence
in terms of n.
Since every pattern is different, some students do not rely on formulas and they depend purely on their superior understanding to construct the nth term formula every time. However, this is
relatively time consuming and the time spent is often not worth the marks allocated.
In this article, we will describe two of the more common patterns and introduce their general formulas such that there is a more structured and efficient way of deriving the nth-term formula.
Number Pattern Type 1: Constant Difference
Find the formula of the nth term of the sequence: 12, 15, 18, 21, 24, ...
Notice that every term is 3 more than the previous term. This is the simplest kind of pattern where the increase/decrease in constant.
The General formula of this nth term for this number pattern sequence is:
Number Pattern Type 2: Increasing Difference
Find the formula of the nth term of the sequence: 8, 13, 20, 29, 40, …
Notice that the difference between the terms is increasing at a constant rate.
The General formula of this nth term for this number pattern sequence is
Find the nth-term formula for the following sequences:
d) nth term = 0.5 (n^2+n+4)
Hope this short tutorial on number patterns have helped you!
To get professional help (Math tuition) on secondary school mathematics in Singapore, click here.
Mr Ausome | {"url":"https://www.tuitionmath.com/single-post/2017/12/16/o-levels-e-math-number-patterns-and-sequences","timestamp":"2024-11-05T22:20:48Z","content_type":"text/html","content_length":"1050483","record_id":"<urn:uuid:457ea7cb-9535-44c0-a34f-e458f04a2bde>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00667.warc.gz"} |
To view this video please enable Javascript
The content focuses on the concept of remainders in division, particularly when dividing by a number that is not a factor of the dividend, and introduces key formulas and strategies for solving GMAT
problems involving remainders.
• Explains the difference between dividing by a factor and a non-factor, introducing mixed-numeral and integer quotients.
• Defines key terms such as dividend, divisor, quotient, and remainder, emphasizing the remainder must be less than the divisor.
• Presents the formula to connect integer quotient and mixed-numeral quotient, highlighting the importance of understanding the non-integer part of the quotient.
• Discusses generating examples of dividends for a given divisor and remainder, and the application of remainders in various contexts.
• Introduces the 'rebuilding the dividend' formula as a crucial problem-solving tool for GMAT questions.
Understanding Remainders
Mixed-numeral and Integer Quotients
Key Terminology and Formulas
Generating Examples and Rebuilding the Dividend | {"url":"https://gmat.magoosh.com/lessons/6780-remainders?study_item=22728","timestamp":"2024-11-13T04:35:49Z","content_type":"text/html","content_length":"96572","record_id":"<urn:uuid:fb037218-a669-4e4e-8b73-498026138926>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00647.warc.gz"} |
Computer Science Archives - Only Code
Category: Computer Science
Unlike simple algorithms, efficient sorting algorithms typically have an average time complexity of O(n logn).
Sorting algorithms are methods used to rearrange elements in a list or array into a particular order, typically either ascending or descending.
Regex is efficient because most programming languages implement optimized regex engines that handle pattern matching using sophisticated algorithms like finite automata, ensuring fast performance.
Scientific notation is a method of expressing very large or very small numbers in a compact and readable form.
Caesar Cipher is one of the simplest and oldest encryption techniques, named after Julius Caesar, who used it to protect his military communications.
Tail recursion optimization works by maintaining a single call frame for the recursive calls instead of creating a new one each time.
This so-called P ≠ NP question has been one of the deepest, most perplexing open research problems in theoretical computer science.
Time Complexity is the number of primitive operations executed on a particular input. It measures the amount of time an algorithm takes to complete as a function of the length of the input. | {"url":"https://www.onlycode.in/category/computer-science/","timestamp":"2024-11-11T07:09:15Z","content_type":"text/html","content_length":"183586","record_id":"<urn:uuid:bf417c21-4b56-4882-b880-ef65f1365625>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00224.warc.gz"} |
The Communication Complexity of MAX: Open problem
Alice has x, an n-bit integer. Bob yas y, an n-bit integer. They want to both know, max(x,y). This can be done with n + \sqrt{2n} + O(1) bits of communication.
1. Alice sends the first \sqrt{2n} bits of x.
2. If Bob can deduce that x LESS THAN y then he sends 11y and they are DONE. If Bob can deduce that x GREATER THAN y then he sends 10, Alice sends the rest of x, and they are done. If the first \
sqrt{2n} bits of x and y are the same then Bob sends 0.
3. (This step is reached only if x and y agree on the first \sqrt{2n} bits.) Alice sends the next \sqrt{2n}-1 bits of x. If Bob can deduce that x LESS THAN y then he sends 11 followed by all BUT the
first \sqrt{2n} bits of y (which Alice already knows since they are the same as hers) and they are DONE. If Bob can deduce that x GREATER THAN y then he sends 10, Alice sends the rest of x, and
they are done. If the first \sqrt{2n} bits of x and y are the same then Bob sends 0.
4. (sketch) In the ith round, if there is one, Alice sends \sqrt{2n} - i bits.
We leave the analysis that this takes n+\sqrt{2n}+O(1) bits to the reader.
It is easy to show that the max(x,y) problem
n bits of communication (also left to the reader). So we have
1. Upper bound of n+\sqrt{2n} +O(1).
2. Lower bound of n.
Open Problems
1. Close this gap! Or at least get a larger lower bound or a smaller upper bound.
2. The protocol above is similar to the following problem: Assume there is a building is n stories high and there is some floor f such that, dropping an egg off of floor f it will break, but off of
floor f-1 it will not. If you have 2 eggs, how many egg-dropping do you need to determine f? (NOTE- if an egg breaks you cannot reuse it.) For 2 egges you can do this with \sqrt{2n}+O(1)
egg-droppings (and this is tight). For e eggs you can do this with (e!)^1/en^1/e+O(1) droppings (and this is tight). See this paper for writeups of these results. (NOTE: I am sure this problem is
``well known'' but have not been able to find references. If you know any please comment or email so I can insert them into the writeup.) Is there some communication complexity problem for which
the e-egg problem supplies the key to the answer.
15 comments:
1. Does e eggs mean 2.718281828... eggs? Making those not broken is problematic, unless they have already been mapped into an omelette.
2. Alas, the problems with working with both
discrete and continous math.
e is an integer.
e is not 2.718...
3. This comment has been removed by the author.
4. The Greater-Than problem is the following: Alice receives x, an n-bit number; Bob receives y, an n-bit number; they want to determine whether x>y.
Your problem's bit complexity is n + complexity of GT.
The Greater Than problem can be solved with polylog(n) bits of communication, by a randomized protocol (binary search for the longest common prefix).
So there's your improvement, unless your really wanted deterministic protocols. If you want deterministic, you can't go via this routa (GT requires linear complexity, like equality).
5. Here's a protocol that requires only n + O(log n) bits.
Alice finds two (log n)-bit strings a_small and a_large that do not
appear in x and sends them to Bob. Bob finds two (log n)-bit strings
b_small and b_large that do not appear in y and sends them to Alice.
Alice and Bob take turns exchanging log n bit chunks of x and y.
If either one discovers that their number is larger or smaller,
they substitute the appropriate codeword, and the party with the
larger number sends the rest.
6. An n+O(log(n)) upper bound was recently given by Babaioff, Blumrosen, M. Naor and Schapira in EC'08 (Thm 3.2).
It was given in the context of the communication overhead for computing payments in auctions. This problem is equivalent to solving a two player second-price auction (the price paid is the
minimal value of the two).
7. n + 2 bits.
Alice sends x to Bob. Bob sends 11 if greater, 00 if less, 01 if equal.
8. sep332, you missed the "both" part: "They want to both know max(x,y)"
9. sep332, if Bob's number is greater, Alice still doesn't know what the greater number is.
And to the anonymous before liad, it seems like the number of passes required by this algorithm will not be O(1); I believe that an n+O(logn) bound might exist (viz. liad's post), but it is not
clear that this algorithm reaches it (indeed, you haven't supplied a mechanism for Alice or Bob to use this data to determine which number is greater at all...).
10. This protocol only need n+2 bits. Is there anything wrong?
1.Alice sends first two bits of x.
2.Bob sends 1) 0, if x is bigger.
2)y, if y is bigger.
3)next two bits of y, if the first two bits is the same.
3.Alice makes the similar choice like Bob (swap x and y), except in the second condition Alice sends all but first two bits of x. And the following steps are all like this.
4.Make some special rules of the last one or two bits.
11. The egg problem is mentioned in Peter Winkler's Mathematical mind-benders . He refers to Konhauser, Velleman and Wagon, Which way did the bicycle go? .
12. pkughost wrote: This protocol only need n+2 bits. Is there anything wrong?
A requirement of communication protocols is that after each bit is sent, the player to communicate the next bit is determined by the communication so far. Your protocol does not have the required
13. I was interested to see the e-egg problem described as it is also the basis of many of the modern improvements in computational pattern matching. For example, pattern matching under the Hamming,
L_1 norm etc. (typically the complexity you get is something like O(n sqrt{m log m}). The basic technique is normally attributed, in that field at least, to both Abrahamson's "Generalized string
matching", SIAM journal on Computing, 16:(6) pages 1039--1051, 1987 and an unpublished manuscript called "Efficient string matching" by S. R. Kosaraju in the same year.
However, it is worth mentioning that these techniques were not widely understood until a series of talks and papers by Amihood Amir emerged.
14. Hm, I guess reading comprehension would be useful *before* I post!
15. The Greater Than problem can be solved with polylog(n) bits of communication, by a randomized protocol (binary search for the longest common prefix). | {"url":"https://blog.computationalcomplexity.org/2008/09/communication-complexity-of-max-open.html?m=0","timestamp":"2024-11-05T12:57:51Z","content_type":"application/xhtml+xml","content_length":"203062","record_id":"<urn:uuid:406f8e10-bca1-46b0-9c11-80e42ba7ceec>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00183.warc.gz"} |
Parametric Equations and Polar Coordinates
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your
device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen
Chapter 9 : Parametric Equations and Polar Coordinates
Here are a set of practice problems for the Parametric Equations and Polar Coordinates chapter of the Calculus II notes.
1. If you’d like a pdf document containing the solutions the download tab above contains links to pdf’s containing the solutions for the full book, chapter and section. At this time, I do not offer
pdf’s for solutions to individual problems.
2. If you’d like to view the solutions on the web go to the problem set web page, click the solution link for any problem and it will take you to the solution to that problem.
Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although
this will vary from section to section.
Here is a list of all the sections for which practice problems have been written as well as a brief description of the material covered in the notes for that particular section.
Parametric Equations and Curves – In this section we will introduce parametric equations and parametric curves (i.e. graphs of parametric equations). We will graph several sets of parametric
equations and discuss how to eliminate the parameter to get an algebraic equation which will often help with the graphing process.
Tangents with Parametric Equations – In this section we will discuss how to find the derivatives \(\frac{dy}{dx}\) and \(\frac{d^{2}y}{dx^{2}}\) for parametric curves. We will also discuss using
these derivative formulas to find the tangent line for parametric curves as well as determining where a parametric curve in increasing/decreasing and concave up/concave down.
Area with Parametric Equations – In this section we will discuss how to find the area between a parametric curve and the \(x\)-axis using only the parametric equations (rather than eliminating the
parameter and using standard Calculus I techniques on the resulting algebraic equation).
Arc Length with Parametric Equations – In this section we will discuss how to find the arc length of a parametric curve using only the parametric equations (rather than eliminating the parameter and
using standard Calculus techniques on the resulting algebraic equation).
Surface Area with Parametric Equations – In this section we will discuss how to find the surface area of a solid obtained by rotating a parametric curve about the \(x\) or \(y\)-axis using only the
parametric equations (rather than eliminating the parameter and using standard Calculus techniques on the resulting algebraic equation).
Polar Coordinates – In this section we will introduce polar coordinates an alternative coordinate system to the ‘normal’ Cartesian/Rectangular coordinate system. We will derive formulas to convert
between polar and Cartesian coordinate systems. We will also look at many of the standard polar graphs as well as circles and some equations of lines in terms of polar coordinates.
Tangents with Polar Coordinates – In this section we will discuss how to find the derivative \(\frac{dy}{dx}\) for polar curves. We will also discuss using this derivative formula to find the tangent
line for polar curves using only polar coordinates (rather than converting to Cartesian coordinates and using standard Calculus techniques).
Area with Polar Coordinates – In this section we will discuss how to the area enclosed by a polar curve. The regions we look at in this section tend (although not always) to be shaped vaguely like a
piece of pie or pizza and we are looking for the area of the region from the outer boundary (defined by the polar equation) and the origin/pole. We will also discuss finding the area between two
polar curves.
Arc Length with Polar Coordinates – In this section we will discuss how to find the arc length of a polar curve using only polar coordinates (rather than converting to Cartesian coordinates and using
standard Calculus techniques).
Surface Area with Polar Coordinates – In this section we will discuss how to find the surface area of a solid obtained by rotating a polar curve about the \(x\) or \(y\)-axis using only polar
coordinates (rather than converting to Cartesian coordinates and using standard Calculus techniques).
Arc Length and Surface Area Revisited – In this section we will summarize all the arc length and surface area formulas we developed over the course of the last two chapters. | {"url":"https://tutorial.math.lamar.edu/Problems/CalcII/ParametricIntro.aspx","timestamp":"2024-11-11T07:23:11Z","content_type":"text/html","content_length":"75953","record_id":"<urn:uuid:5e9111c7-7a75-4437-b430-ae9331b5de36>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00355.warc.gz"} |
Limits of Functions | Brilliant Math & Science Wiki
The limit of a function at a point \(a\) in its domain (if it exists) is the value that the function approaches as its argument approaches \(a.\) The concept of a limit is the fundamental concept of
calculus and analysis. It is used to define the derivative and the definite integral, and it can also be used to analyze the local behavior of functions near points of interest.
Informally, a function is said to have a limit \( L \) at \( a \) if it is possible to make the function arbitrarily close to \( L \) by choosing values closer and closer to \( a \). Note that the
actual value at \( a \) is irrelevant to the value of the limit.
The notation is as follows:
\[ \lim_{x \to a} f(x) = L, \]
which is read as "the limit of \(f(x) \) as \(x\) approaches \(a\) is \(L.\)"
Main Article: Epsilon-Delta Definition of a Limit
The precise definition of the limit is discussed in the wiki Epsilon-Delta Definition of a Limit.
Formal Definition of a Function Limit:
The limit of \(f(x)\) as \(x\) approaches \(x_0\) is \(L\), i.e.
\[\lim _{ x \to x_{0} }{f(x) } = L\]
if, for every \(\epsilon > 0 \), there exists \(\delta >0 \) such that, for all \(x\),
\[ 0 < \left| x - x_{0} \right |<\delta \textrm{ } \implies \textrm{ } \left |f(x) - L \right| < \epsilon. \]
In practice, this definition is only used in relatively unusual situations. For many applications, it is easier to use the definition to prove some basic properties of limits and to use those
properties to answer straightforward questions involving limits.
The most important properties of limits are the algebraic properties, which say essentially that limits respect algebraic operations:
Suppose that \( \lim\limits_{x\to a} f(x) = M\) and \(\lim\limits_{x\to a} g(x) = N.\) Then
\[ \lim\limits_{x\to a} \big(f(x)+g(x)\big) &= M+N \\\\ \lim\limits_{x\to a} \big(f(x)-g(x)\big) &= M-N \\\\ \lim\limits_{x\to a} \big(f(x)g(x)\big) &= MN \\\\ \lim\limits_{x\to a} \left(\frac{f
(x)}{g(x)}\right) &= \frac MN \ \ \text{ (if } N\ne 0) \\\\ \lim\limits_{x\to a} f(x)^k &= M^k \ \ \text{ (if } M,k > 0). \]
These can all be proved via application of the epsilon-delta definition. Note that the results are only true if the limits of the individual functions exist: if \( \lim\limits_{x\to a} f(x) \) and \(
\lim\limits_{x\to a} g(x)\) do not exist, the limit of their sum (or difference, product, or quotient) might nevertheless exist.
Coupled with the basic limits \( \lim_{x\to a} c = c,\) where \( c\) is a constant, and \( \lim_{x\to a} x = a,\) the properties can be used to deduce limits involving rational functions:
Let \( f(x) \) and \(g(x)\) be polynomials, and suppose \(g(a) \ne 0.\) Then
\[ \lim_{x\to a} \frac{f(x)}{g(x)} = \frac{f(a)}{g(a)}. \]
This is an example of continuity, or what is sometimes called limits by substitution.
Note that \(g(a)=0\) is a more difficult case; see the Indeterminate Forms wiki for further discussion.
Let \(m\) and \(n\) be positive integers. Find
\[ \lim_{x\to 1} \frac{x^m-1}{x^n-1}. \]
Immediately substituting \(x=1\) does not work, since the denominator evaluates to \(0.\) First, divide top and bottom by \(x-1\) to get
Plugging in \(x=1\) to the denominator does not give \(0,\) so the limit is this fraction evaluated at \(x=1,\) which is
\[\frac{1^{m-1}+1^{m-2}+\cdots+1}{1^{n-1}+1^{n-2}+\cdots+1} = \frac{m}{n}.\ _\square\]
It is important to notice that the manipulations in the above example are justified by the fact that \( \lim\limits_{x\to a} f(x)\) is independent of the value of \(f(x) \) at \(x=a,\) or whether
that value exists. This justifies, for instance, dividing the top and bottom of the fraction \(\frac{x^m-1}{x^n-1}\) by \(x-1,\) since this is nonzero for \(x\ne 1.\)
\[\lim _{x\rightarrow 10} \frac{x^{3}-10x^{2}-25x+250}{x^{4}-149x^{2}+4900} = \frac{a}{b},\]
where \(a\) and \(b\) are coprime integers, what is \(a+b?\)
A one-sided limit only considers values of a function that approaches a value from either above or below.
The right-side limit of a function \(f\) as it approaches \(a\) is the limit
\[\lim_{x \to a^+} f(x) = L. \]
The left-side limit of a function \(f\) is
\[\lim_{x \to a^-} f(x) = L. \]
The notation "\(x \to a^-\)" indicates that we only consider values of \(x\) that are less than \(a\) when evaluating the limit. Likewise, for "\(x \to a^+,\)" we consider only values greater than \
(a\). One-sided limits are important when evaluating limits containing absolute values \(|x|\), sign \(\text{sgn}(x)\) , floor functions \(\lfloor x \rfloor\), and other piecewise functions.
The image above demonstrates both left- and right-sided limits on a continuous function \(f(x).\)
Find the left- and right-side limits of the signum function \(\text{sgn}(x)\) as \(x \to 0:\)
\[\text{sgn}(x)= \begin{cases} \frac{|x|}{x} && x\neq 0 \\ 0 && x = 0. \end{cases}\]
Consider the following graph:
From this we see \(\displaystyle \lim_{x \to 0^+} \text{sgn}(x) = 1 \) and \(\displaystyle \lim_{x \to 0^-}\text{sgn}(x) = -1.\ _\square \)
Determine the limit \( \lim\limits_{x \to 1^{-}} \frac{\sqrt{2x}(x-1)}{|x-1|}. \)
Note that, for \(x<1,\) \(\left | x-1\right |\) can be written as \(-(x-1)\). Hence, the limit is \(\lim\limits_{x \to 1^{-}} \frac{\sqrt{2x}(x-1)}{-(x-1)} = -\sqrt{2}.\ _\square\)
By definition, a two-sided limit
\[\lim_{x \to a} f(x) = L\]
exists if the one-sided limits \(\displaystyle \lim_{x \to a^+} f(x)\) and \(\displaystyle \lim_{x \to a^-} f(x)\) are the same.
Compute the limit
\[ \lim_{x \to 1} \frac{|x - 1|}{x - 1} . \]
Since the absolute value function \(f(x) = |x| \) is defined in a piecewise manner, we have to consider two limits: \(\lim\limits_{x \to 1^+} \frac{|x - 1|}{x - 1} \) and \(\lim\limits_{x \to 1^
-} \frac{|x - 1|}{x - 1}. \)
Start with the limit \(\lim\limits_{x \to 1^+} \frac{|x - 1|}{x - 1}.\) For \(x>1,\) \( |x - 1| = x -1. \) So
\[\lim_{x \to 1^+} \frac{|x - 1|}{x - 1} =\lim_{x \to 1^+} \frac{x - 1}{x - 1} =1.\]
Let us now consider the left-hand limit
\[\lim_{x \to 1^-} \frac{|x - 1|}{x - 1}. \]
For \(x<1,\) \(x - 1 = -|x-1|.\) So
\[\lim_{x \to 1^-} \frac{|x-1|}{-|x - 1|} = -1 . \]
So the two-sided limit \( \lim\limits_{x \to 1} \frac{|x - 1|}{x - 1}\) does not exist. \(_\square\)
The image below is a graph of a function \(f(x)\). As shown, it is continuous for all points except \(x = -1\) and \(x=2\) which are its asymptotes. Find all the integer points \(-4 <I < 4,\)
where the two-sided limit \(\lim_{x \to I} f(x)\) exists.
Since the graph is continuous at all points except \(x=-1\) and \(x=2\), the two-sided limit exists at \(x=-3\), \(x=-2\), \(x=0\), \(x=1,\) and \(x=3\). At \(x=2,\) there is no finite value for
either of the two-sided limits, since the function increases without bound as the \(x\)-coordinate approaches \(2\) (but see the next section for a further discussion). The situation is similar
for \(x=-1.\) So the points \(x=-3\), \(x=-2\), \(x=0\), \(x=1,\) and \(x=3\) are all the integers on which two-sided limits are defined. \(_\square\)
As seen in the previous section, one way for a limit not to exist is for the one-sided limits to disagree. Another common way for a limit to not exist at a point \(a\) is for the function to "blow
up" near \(a,\) i.e. the function increases without bound. This happens in the above example at \(x=2,\) where there is a vertical asymptote. This common situation gives rise to the following
Given a function \(f(x)\) and a real number \(a,\) we say
\[\lim_{x\to a} f(x) = \infty.\]
If the function can be made arbitrarily large by moving \(x\) sufficiently close to \(a,\)
\[\text{for all } N>0, \text{ there exists } \delta>0 \text{ such that } 0<|x-a|<\delta \implies f(x)>N.\]
There are similar definitions for one-sided limits, as well as limits "approaching \(-\infty\)."
Warning: If \(\lim\limits_{x\to a} f(x) = \infty,\) it is tempting to say that the limit at \(a\) exists and equals \(\infty.\) This is incorrect. If \(\lim\limits_{x\to a} f(x) = \infty,\) the limit
does not exist; the notation merely gives information about the way in which the limit fails to exist, i.e. the value of the function "approaches \(\infty\)" or increases without bound as \(x \
rightarrow a\).
What can we say about \(\lim\limits_{x \to 0} \frac{1}{x}?\)
Separating the limit into \(\lim\limits_{x \to 0^+} \frac{1}{x}\) and \(\lim\limits_{x \to 0^-} \frac{1}{x}\), we obtain
\[ \lim_{x \to 0^+} \frac{1}{x} = \infty \]
\[ \lim_{x \to 0^-} \frac{1}{x} = -\infty. \]
To prove the first statement, for any \(N>0\) in the formal definition, we can take \(\delta = \frac1N,\) and the proof of the second statement is similar.
So the function increases without bound on the right side and decreases without bound on the left side. We cannot say anything else about the two-sided limit \(\lim\limits_{x\to a} \frac1{x} \ne
\infty\) or \(-\infty.\) Contrast this with the next example. \(_\square\)
What can we say about \(\lim\limits_{x \to 0} \frac{1}{x^2}?\)
Separating the limit into \(\lim\limits_{x \to 0^+} \frac{1}{x^2}\) and \(\lim\limits_{x \to 0^-} \frac{1}{x^2}\), we obtain
\[ \lim_{x \to 0^+} \frac{1}{x^2} = \infty \]
\[ \lim_{x \to 0^-} \frac{1}{x^2} = \infty.\]
Since these limits are the same, we have \( \lim_{x \to 0} \frac{1}{x^2} = \infty .\) Again, this limit does not, strictly speaking, exist, but the statement is meaningful nevertheless, as it
gives information about the behavior of the function \( \frac1{x^2}\) near \(0.\) \(_\square\)
\[f(x)=\frac{a_0 x^{m}+a_1 x^{m+1}+\cdots +a_k x^{m+k}}{b_0 x^{n}+b_1 x^{n+1}+\cdots +b_ l x^{n+l}},\]
where \(a_0 \neq 0, b_0 \neq 0,\) and \(m,n \in \mathbb N.\)
Then given (A), (B), (C), or (D), \(\displaystyle\lim_{x\rightarrow 0}f(x)\) equals which of (1), (2), (3), and (4)?
Match the columns:
Column-I Column-II
(A) if \(m>n\) (1) \(\infty\)
(B) if \(m=n\) (2) \(-\infty\)
(C) if \(m<n,\) \(n-m\) is even, and \(\frac{a_0}{b_0}>0\) \(\hspace{10mm}\) (3) \(\frac{a_0}{b_0}\)
(D) if \(m<n,\) \(n-m\) is even, and \(\frac{a_0}{b_0}<0\) \(\hspace{10mm}\) (4) \(0\)
Note: For example, if (A) correctly matches (1), (B) with (2), (C) with (3), and (D) with (4), then answer as 1234.
Another extension of the limit concept comes from considering the function's behavior as \(x\) "approaches \(\infty\)," that is, as \(x\) increases without bound.
The equation \( \lim\limits_{x\to\infty} f(x) = L\) means that the values of \(f\) can be made arbitrarily close to \(L\) by taking \(x\) sufficiently large. That is,
\[\text{for all } \epsilon > 0, \text{ there is } N>0 \text{ such that } x>N \implies |f(x)-L|<\epsilon.\]
There are similar definitions for \(\lim\limits_{x\to -\infty} f(x) = L,\) as well as \(\lim\limits_{x\to\infty} f(x) = \infty,\) and so on.
Graphically, \(\lim\limits_{x\to a} f(x) = \infty\) corresponds to a vertical asymptote at \(a,\) while \( \lim\limits_{x\to\infty} f(x) = L \) corresponds to a horizontal asymptote at \(L.\)
Main Article: Limits by Factoring
Limits by factoring refers to a technique for evaluating limits that requires finding and eliminating common factors.
Main Article: Limits by Substitution
Evaluating limits by substitution refers to the idea that under certain circumstances (namely if the function we are examining is continuous), we can evaluate the limit by simply evaluating the
function at the point we are interested in.
Main Article: L'Hôpital's Rule
L'Hôpital's rule is an approach to evaluating limits of certain quotients by means of derivatives. Specifically, under certain circumstances, it allows us to replace \( \lim \frac{f(x)}{g(x)} \) with
\( \lim \frac{f'(x)}{g'(x)}, \) which is frequently easier to evaluate.
Limits of Functions - Problem Solving
Evaluate \( \lim\limits_{x\to\infty} \frac{x^2 + 2x +4}{3x^2+ 4x+125345} \).
We have
\[\begin{eqnarray} &&\displaystyle \lim_{x\to\infty} \frac{x^2 + 2x +4}{3x^2+ 4x+125345} &=& \displaystyle \lim_{x\to\infty} \frac{1 + \frac2x + \frac4{x^2}}{3+ \frac4x+ \frac{125345}{x^2}} &=& \
displaystyle \frac{1+0+0}{3+0+0} = \frac13.\ _\square \end{eqnarray} \]
\[-\pi\] \[1\] \[\frac{\pi}{2}\] \[\pi\]
\[\large \displaystyle \lim_{x \to 0} \dfrac{\sin(\pi \cos^2x)}{x^2}= \, ?\]
\[\large \lim_{x \to 1} \left( \frac{23}{1-x^{23}}-\frac{11}{1-x^{11}} \right) = \, ?\] | {"url":"https://brilliant.org/wiki/limits-of-functions/","timestamp":"2024-11-10T19:15:59Z","content_type":"text/html","content_length":"75560","record_id":"<urn:uuid:50e3a4e8-58e4-4164-b736-500cce657be0>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00218.warc.gz"} |
Growth and Climate Change : Threshold and Multiple Equilibria
Title data
Greiner, Alfred ; Grüne, Lars ; Semmler, Willi:
Growth and Climate Change : Threshold and Multiple Equilibria.
In: Crespo Cuaresma, Jesús ; Palokangas, Tapio ; Tarasyev, Alexander (ed.): Dynamic Systems, Economic Growth, and the Environment. - Berlin : Springer , 2010 . - pp. 63-78 . - (Dynamic Modeling and
Econometrics in Economics and Finance ; 12 )
ISBN 978-3-642-02132-9
DOI: https://doi.org/10.1007/978-3-642-02132-9_4
Abstract in another language
In this paper we analyze a basic growth model where we allow for global warming. As concerns global warming we assume that the climate system is characterized by feedback effects such that the
ability of the earth to emit radiation to space is reduced as the global surface temperature rises. We first study the model assuming that abatement spending is fixed exogeneously and demonstrate
with the use of numerical examples that the augmented model may give rise to multiple equilibria and thresholds. Then, we analyze the social optimum where both consumption and abatement are set
optimally and show that the long-run equilibrium is unique in this case. In the context of our model with multiple equilibria initial conditions are more important for policy actions than discount
Further data | {"url":"https://eref.uni-bayreuth.de/id/eprint/63607/","timestamp":"2024-11-15T04:26:54Z","content_type":"application/xhtml+xml","content_length":"22523","record_id":"<urn:uuid:92d03634-b474-4969-a558-32656eb15d5e>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00619.warc.gz"} |
Text Similarity using K-Shingling, Minhashing and LSH(Locality Sensitive Hashing) | Towards AI
Text Similarity using K-Shingling, Minhashing and LSH(Locality Sensitive Hashing)
Last Updated on October 29, 2021 by Editorial Team
Text Similarity using K-Shingling, Minhashing, and LSH(Locality Sensitive Hashing)
Text similarity plays an important role in Natural Language Processing (NLP) and there are several areas where this has been utilized extensively. Some of the applications include Information
retrieval, text categorization, topic detection, machine translation, text summarization, document clustering, plagiarism detection, news recommendation, etc. encompassing almost allΒ domains.
But sometimes, it becomes difficult to understand the concept behind the Text Similarity Algorithms. This write-up will show an implementation of Text Similarity along with an explanation of the
required concepts.
But before I start, let me tell you that there might be several ways and several algorithms to perform the same task. I will use one of the ways for depiction using K-Shingling, Minhashing, and LSH
(Locality Sensitive Hashing).
Dataset considered is Text Extract from 3 documents for the problem atΒ hand.
We can use nβ β β number of documents with each document being of significant length. But to make it simpler and avoid heavy computations, I am considering a small chunk from each document.
Letβ s perform the implementation inΒ Steps.
Step 1Β :
Set your working directory to the folder where the files are placed so that R can read it. Then read all the input files from the working directory using the belowΒ code.
# Libraries used
# Set the working directory
# Read the original text file
files <- list.files(path=".", pattern='*.txt', all.files=FALSE,
( doc <- lapply( files, readLines ) )
R Studio InputΒ Display
Step 2Β :
Preprocess the text to remove punctuation, convert it into lower cases, and split text into word byΒ word.
# Preprocess text
documents <- lapply(doc, function(x) {
text <- gsub("[[:punct:]]", "", x) %>% tolower()
text <- gsub("\\s+", " ", text) %>% str_trim()
word <- strsplit(text, " ") %>% unlist()
# Print the texts in files
R StudioΒ Display
Step 3Β :
Introduce K-Shingling which is a technique of representing documents assets. We will understand further the importance of K-Shingling but as of now, we can just try getting familiar with theΒ Steps.
K-shingle of a document at hand is said to be all the possible consecutive sub-string of length k found inΒ it.
Letβ s illustrate with an example with k =Β 3.
Shingling <- function(document, k) {
shingles <- character( length = length(document) - k + 1 )
for( i in 1:( length(document) - k + 1 ) ) {
shingles[i] <- paste( document[ i:(i + k - 1) ], collapse = " " )
return( unique(shingles) )
# "shingle" the example document, with k = 3
documents <- lapply(documents, function(x) {
Shingling(x, k = 3)
list( Original = doc[[1]], Shingled = documents[[1]] )
R StudioΒ Display
Hence with k = 3, the k-shingles of the first document which got printed out, consist of sub-strings of lengthΒ 3.
The first K-Shingle is: β the nightΒ isβ
The second Shingle is: β night is darkβ and soΒ on.
One important point to note is that a documentβ s k-shingle set should be unique. For example, if the first document above contains more than one β the night isβ then it will only appear once as
the set of k-shingle for that document.
Step 4:
Construct a β Characteristicβ matrix that visualizes the relationships between the three documents. The β characteristicβ matrix will be a Boolean matrix, withΒ :
rows = the elements of every unique possible combination of shingles set across all documents.
columns = one column per document.
Thus, the matrix will be filled with 1 in row i and column j if and only if document j contains the shingle iΒ , otherwise it will be filled withΒ 0.
Let us try to understand this with the below depiction.
# Unique shingles sets across all documents
doc_dict <- unlist(documents) %>% unique()
# "Characteristic" matrix
Char_Mat <- lapply(documents, function(set, dict) {
as.integer(dict %in% set)
}, dict = doc_dict) %>% data.frame()
# set the names for both rows and columns
setnames( Char_Mat, paste( "doc", 1:length(documents), sep = "_" ) )
rownames(Char_Mat) <- doc_dict
R StudioΒ Display
The first row of the above matrix has all three columns as 1. This is because all three documents contain the 3-shingle β the nightΒ isβ .
For the second column, the value is [1, 0, 1] which means that document 2 does not have the 3-shingle β night is dark, while document 1 and 3Β has.
One important point noted here is that most of the time these β characteristic matricesβ are almost sparse. Therefore, we usually try to represent these matrices only by the positions in which 1
appears, so as to be more space-efficient.
Step 5Β :
After creating shingle sets and characteristic matrix, we now need to measure the similarity between documents.
We will make use of Jaccard Similarity for thisΒ purpose.
For example, with two shingle sets as set1 and set2, the Jaccard Similarity will beΒ :
With this we will calculate the pairwise Jaccard similarities for all three documents. β distβ function in β Rβ quickly computes and returns the distance/similarity matrix.
# how similar is two given document, Jaccard similarity
JaccardSimilarity <- function(x, y) {
non_zero <- which(x | y)
set_intersect <- sum( x[non_zero] & y[non_zero] )
set_union <- length(non_zero)
return(set_intersect / set_union)
# create a new entry in the registry
pr_DB$set_entry( FUN = JaccardSimilarity, names = c("JaccardSimilarity") )
# Jaccard similarity distance matrix
d1 <- dist( t(Char_Mat), method = "JaccardSimilarity" )
# delete the new entry
R StudioΒ Display
The similarity matrix d1 tells us that document 1 and 3 is the most similar among the three documents.
For small datasets, the above method works perfectly fine, but imagine if we have a large number of documents to compare instead of just three that too with significantly larger lengths, then the
above method might not scale well, and we might have heavy computation with performance issue building up as the sparse matrices with a set of unique shingles across all documents will be fairly
large, making computation of the Jaccard similarity between the documents aΒ burden.
Under such situations, we employ a different technique that help us save computations and can compare document similarities on a large scale efficiently. The technique is called Minhashing.
Step 6Β :
Minhashing involves compressing the large sets of unique shingles into a much smaller representation called β signaturesβ .
We then use these signatures to measure the similarity between documents.
Although it is impossible for these signatures to give the exact similarity measure, the estimates are prettyΒ close.
The larger the number of signatures chosen, the more accurate the estimateΒ is.
For illustration let us consider anΒ example.
Suppose we take up the above example to minhash characteristic matrix of 16 rows into 4 signatures. Then the first step is to generate 4 columns of randomly permutated rows that are independent of
each other. We can see for ourselves that this simple hash function does in fact generate random permutated rows. To generate this, we use theΒ formula:
WhereΒ :
x is the row numbers of your original characteristic matrix.
a and b are any random numbers smaller or equivalent to the maximum number of x, and they both must be unique in each signature.
For e.g. For signature 1, if 5 is generated to serve as a- coefficient, it must be ensured that this value does not serve as a- coefficient multiple times within signature 1, though it can still be
usedΒ as
b- coefficient in signature 1. And this restriction refreshes for the next signature as well, that is, 5 can be used to serve as a or b coefficient for signature 2, but again no multiple 5 for
signature 2β s a or b coefficient and soΒ on.
c is a prime number slightly larger than the total number of shingleΒ sets.
For the above example set, since the total row count is 16, thus prime number 17 will doΒ fine.
Now letβ s generate this through the β Rβ Β code.
# number of hash functions (signature number)
signature_num <- 4
# prime number
prime <- 17
# generate the unique coefficients
coeff_a <- sample( nrow(Char_Mat), signature_num )
coeff_b <- sample( nrow(Char_Mat), signature_num )
# see if the hash function does generate permutations
permute <- lapply(1:signature_num, function(s) {
hash <- numeric( length = length(nrow(Char_Mat)) )
for( i in 1:nrow(Char_Mat) ) {
hash[i] <- ( coeff_a[s] * i + coeff_b[s] ) %% prime
# # convert to data frame
permute_df <- structure( permute, names = paste0( "hash_", 1:length(permute) ) ) %>%
R StudioΒ Display
From the above output, we see that the 4 columns of randomly permutated rows got generated. There are 0s also, but it will not affect our computation and we will see thisΒ later.
Step 7Β :
Using the randomly permutated rows, now signatures will be calculated further. The signature value of any column (document) is obtained by using the permutated order generated by each hash function,
the number of the first row in which the column has aΒ 1.
What we will do further is combine randomly permutated rows (generated by hash functions) with the original characteristic matrix and change the row names of the matrix to its row number to
illustrate the calculation.
# use the first two signature as an example
# bind with the original characteristic matrix
Char_Mat1 <- cbind( Char_Mat, permute_df[1:2] )
rownames(Char_Mat1) <- 1:nrow(Char_Mat1)
R StudioΒ Display
Now considering the matrix generated above, we will start with the first hash function (hash_1).
According to our first hash functionβ s permutated row order, the first row is row 14 ( why row 14Β ? because 0 is the smallest value for our randomly generated permutation, and it has a 0 in row
14, making it the first row ). Then weβ ll look at row 14β s entry for all three documents and try to find β which documentβ s entry at row 14 is a 1Β ?β . document 3β s (doc_3) row 14 is a 1,
thus the signature value for document 3 generated by our first hash function is 0. But documents 2 and 3β s entries at row 14 are both 0, thus weβ ll have to continueΒ looking.
According to our first hash functionβ s permutated row order, the second row is row 8 ( 1 is the second smallest value for our randomly generated permutation, and it has a value of 1 at row 8 ). We
apply the same concept as above and find that document 2β s (doc_2) entry for row 8 is a 1, thus the signature value for document 2 generated by our first hash function is 1. Note that weβ re
already done with document 3, we do not need to check if it contains a 1 anymore. But weβ re still not done, document 1β s entry at row 8 is still a 0. Hence, weβ ll have to lookΒ further.
Again, checking the permutated row order for our first hash function, the third row is row 2. document 1β s entry for row 2 is 1. Therefore, weβ re done with calculating the signature values for
all three columns using our first hash function!! Which are [2, 1,Β 0].
We can then apply the same notion to calculate the signature value for each column (document) using the second hash function, and so on for the third, fourth, etc. A quick look at the signature
second hash function shows that the first row according to its permutated row order is row 8 and doc_2 has a 1 in row 3. Similarly, the second row is row 14 with doc_3 as 1 and the third row is row3
with doc_1 as 1. Hence, the signature values generated by our second hash function for all three documents are [2, 0,Β 1].
As for these calculated signature values, we will store them into a signature matrix along the way, which will later replace the original characteristic matrix. The following section will calculate
the signature values for all 3 columns using all 4 hash functions and print out the signature matrix.
# obtain the non zero rows' index for all columns
non_zero_rows <- lapply(1:ncol(Char_Mat), function(j) {
return( which( Char_Mat[, j] != 0 ) )
# initialize signature matrix
SM <- matrix( data = NA, nrow = signature_num, ncol = ncol(Char_Mat) )
# for each column (document)
for( i in 1:ncol(Char_Mat) ) {
# for each hash function (signature)'s value
for( s in 1:signature_num ) {
SM[ s, i ] <- min( permute_df[, s][ non_zero_rows[[i]] ] )
# set names for clarity
colnames(SM) <- paste( "doc", 1:length(doc), sep = "_" )
rownames(SM) <- paste( "minhash", 1:signature_num, sep = "_" )
R StudioΒ Display
Our signature matrix has the same number of columns as the original characteristic matrix, but it only has n rows, where n is the number of hash functions we wish to generate (in this caseΒ 4).
Let me elaborate on how do we interpret the aboveΒ result?
For e.g., for documents 1 and 3 (columns 1 and 3), its similarity would be 0.25 because they only agree in 1 row out of a total of 4 (both columnsβ row 4 isΒ 1).
Letβ s calculate the same explanation throughΒ code.
# signature similarity
SigSimilarity <- function(x, y) mean( x == y )
# same trick to calculate the pairwise similarity
pr_DB$set_entry( FUN = SigSimilarity, names = c("SigSimilarity") )
d2 <- dist( t(SM), method = "SigSimilarity" )
list(SigSimilarity = d2, JaccardSimilarity = d1)
R StudioΒ Display
From the difference of the result between the original Jaccard similarity and the new similarity obtained using the signature similarity, we might doubt if this is an accurate estimate? But as
mentioned earlier, Minhash's purpose is to provide a fast β approximationβ to the true Jaccard similarity and the estimate can be closer but not 100% accurate hence the difference. Also, the
example considered here is way too small to depict closer accuracy using the law of large numbers. More accurate and close results are expected with large datasets.
In cases where the primary requirement is to compute the similarity of every possible pair, probably for text clustering or so, then LSH (Locality Sensitive Hashing) does not serve the purpose but if
the requirement is to find the pairs that are most likely to be similar, then a technique called Locality Sensitive Hashing can be employed further which I am discussing below.
Locality Sensitive Hashing
While the information necessary to compute the similarity between documents has been compressed from the original sparse characteristic matrix into a much smaller signature matrix, but the underlying
problem or need to perform pairwise comparisons on all the documents stillΒ exists.
The concept for locality-sensitive hashing (LSH) is that given the signature matrix of size n (row count), we will partition it into b bands, resulting in each band with r rows. This is equivalent to
the simple math formulaβ β β n = br, thus when we are doing the partition, we have to be sure that the b we choose is divisible by n. Using the signature matrix above and choosing the band size to
be 2 the above example will becomeΒ :
# number of bands and rows
bands <- 2
rows <- nrow(SM) / bands
data.frame(SM) %>%
mutate( band = rep( 1:bands, each = rows ) ) %>%
select( band, everything() )
R StudioΒ Display
What locality-sensitive hashing tells us is: If the signature values of two documents agree in all the rows of at least one band, then these two documents are likely to be similar and should be
compared (list it as the candidate pair). Using this small set of documents might be a bad example since it can happen that none of them will get chosen as our candidate pair. For instance, if the
signature value of document 2 for band 1 becomes [ 0, 1 ] instead of the current [ 1, 0 ], then document 2 and document 3 will become a candidate pair as both of their rows in band1 takes the same
value of [ 0, 1Β ].
Noteβ β β My computations and your computations, while executing the above set of R Codes might vary as the signatures are randomly generated.
Final Thoughts
The above technique using Jaccard Similarity, Minhashing, and LSH is one of the utilized techniques to compute document similarity although many more exists. Text similarity is an active research
field, and techniques are continuously evolving. Hence which method to use is very much dependent on the Use Case and the requirements of what we want toΒ achieve.
Thanks for readingΒ !!!
You can follow me on medium as wellΒ as
LinkedIn: SupriyaΒ Ghosh
Twitter: @isupriyaghosh
Text Similarity using K-Shingling, Minhashing and LSH(Locality Sensitive Hashing) was originally published in Towards AI on Medium, where people are continuing the conversation by highlighting and
responding to this story.
Published via Towards AI | {"url":"https://towardsai.net/p/l/text-similarity-using-k-shingling-minhashing-and-lshlocality-sensitive-hashing","timestamp":"2024-11-05T22:18:18Z","content_type":"text/html","content_length":"316814","record_id":"<urn:uuid:b103712e-31b9-46e3-83f0-2729f7265e7b>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00116.warc.gz"} |
What Everyone is Saying About math for kids Is Useless Wrong And Why
This beginner-friendly Specialization is the place you’ll grasp the fundamental arithmetic toolkit of machine learning. We additionally suggest a primary familiarity with Python, as labs use Python
to show studying goals in the setting where they’re most applicable to machine studying and data science. Here at Alison, we offer an enormous range of free on-line math programs designed to elevate
your math abilities.
• Whether you go for pre-recorded video programs or enroll in virtual tutoring sessions, studying math has by no means been extra versatile and handy.
• Evaluate, manipulate and graph the kinds of linear equations that appear throughout an MBA syllabus.
• A math training may help you understand the rules that information the world you reside in.
Learn the abilities that will set you up for achievement in decimal place worth; operations with decimals and fractions; powers of 10; quantity; and properties of shapes. Learn sixth grade math
aligned to the Eureka Math/EngageNY curriculum—ratios, exponents, lengthy division, adverse numbers, geometry, statistics, and more. Learn fifth grade math aligned to the Eureka Math/EngageNY
curriculum—arithmetic with fractions and decimals, volume problems, unit conversion, graphing factors, and more. This Arithmetic course is a refresher of place worth and operations for complete
numbers, fractions, decimals, and integers. Learn sixth grade math—ratios, exponents, lengthy division, adverse numbers, geometry, statistics, and extra.
dreambox learning Reviews & Recommendations
Scientists apply theoretical reasoning and patterns to grasp the actions of atoms. Whether you’re calculating how lengthy a visit will take or doing superior data evaluation to grow your business,
understanding math may help you get forward. By choosing to study online at Alison, you’ll have entry to dozens of expert-developed math courses. Simply enroll for any certainly one of our online
math programs and begin studying. These free on-line mathematics programs will arm you with everything you want to understand primary or advanced mathematical ideas.
dreambox Guide
As you probably can see, there’s a bunch of on-line math programs to select from. That said, your greatest option will all the time be to work 1-on-1 with knowledgeable math tutor who can create a
customized learning plan for you. This means, you’ll have the ability to research what’s necessary to you and address your individual needs. Do you have to brush up in your arithmetic or examine for
an algebra exam, but you’re not sure which on-line math course is value your time and money?
The Fundamentals Of dreambox Revealed
With edX, you’ll find a way to study at your individual tempo in math courses at each stage, from high school pre-algebra to school algebra and beyond. Get a refresher on primary math, from
subtraction to exponents, or explore https://www.topschoolreviews.com/dreambox-review/ intermediate ideas such as polynomials and integrals. Alison provides over 40 free online math courses
throughout a range of different subjects and ability levels.
Coursera presents a extensive range of courses in math and logic, all of that are delivered by instructors at top-quality institutions corresponding to Stanford University and Imperial College
London. Learn the talents that will set you up for success in numbers and operations; fixing equations and systems of equations; linear equations and features; and geometry. However, you don’t have
to become a mathematician to make use of math and logic abilities in your profession.
That’s because you have not any one to ask for help when you get stuck with a tough concept or problem. The course will take slightly below 8 hours to complete, with 5 abilities assessments at the
finish. Each module is damaged down into 10-minute lectures, though some modules can last as long as 2 hours. Given that there is no feedback from the trainer during this time, it can be exhausting
to keep up and stay motivated. The Mental Math Trainer course teaches you tips on how to execute calculations at warp speed. If you have an interest in math principle and like pondering outside the
field, then this short course could be for you.
You can get started on the platform free of charge by registering with an email. This will give you entry to the essential math programs, as properly as science and computer science. You can even
upgrade for access to their full library of programs and get 20% as one of our readers. With the information you gain in mathematical finance, you’ll find a way to pursue varied profession
alternatives in high finance, like an choices dealer, inventory analyst, threat manager, or hedge fund supervisor. You will likely want a financial degree and possibly even a grasp’s diploma in a
financial self-discipline to start in certainly one of these jobs. Mathematical finance is a rising subject that seeks to use mathematical modeling and formulation to create financial pricing
constructions and resource values. | {"url":"https://www.ustvarjalnica-pikica.si/2023/02/22/what-everyone-is-saying-about-math-for-kids-is-useless-wrong-and-why/","timestamp":"2024-11-08T01:03:19Z","content_type":"text/html","content_length":"71697","record_id":"<urn:uuid:9f75ec28-7277-4bf8-adef-bed67fc4d9c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00167.warc.gz"} |
IIR Filter Design - EEEGUIDE.COM
IIR Filter Design 1-D:
In this section some practical considerations relating to the methods for the design of IIR Filter Design filters are presented. In particular, the bilinear transform is considered along with its
application to the digitization of rational transfer funcÂtions of classical analog filters defined on the s-plane or in the framework of the wave digital filter approximation.
Review of Classical Analog Filters
Before going into the details of digitization methods, let us briefly review the main properties of the common families of analog low pass filters.
The first family of filters considered is the Butterworth filter, which consists of filter which are normally flat, that is, filters having the 2N – 1 derivative of the squared magnitude. For an Nth
order filter with zero at the origin, the squared magnitude response has the form
where ω[c] is the cutoff frequency at which the magnitude response has an amplitude of – 3db. These filters with monotonic pass band and stop band behavior have equally spaced poles on the s-plane,
on a circle of radians.
Chebyschev filters have an equiripple behaviour, which minimises the maxiÂmum error in one of the two bands, the pass band or the stop band. On the other hand in which the minimal condition is not
applied, the behaviour is monotonic. The squared magnitude response is written in the form
where V[N] is a Chebyschev polynomial of order N.
A third family of filters is that of elliptic filters. An elliptic filter has equiripple behaviour in both stop and pass bands. It is an optimum filter, in the sense that for a given order and given
ripple specifications, it has a narrow transition band width. The squared magnitude response is given by
where ε is a parameter related to the pass band filter ripple equal to 1 ± 1/ (1 + ε^2), where k[1] = ε/√A^2-1 is a parameter related to the stop band 1/A^2, and S[n] is the Jacobi elliptic
Design of Digital IIR Filters by Means of the Bilinear Transform
The poles of the IIR filter design discussed above can be easily obtained, and expressed in the form
m = 0, 1, …, 2N – 1, when N is even, a and b are equal to the 1 in the Butterworth case.
Thus when the parameters of the design, N, ω[c] and ε are known in the Chebyschev case, the coefficients of the filter can be obtained by computing the pole positions by means of either Eqs
(15.102) or (15.103).
The problem is now to investigate the relationship between the order x of the filter N, the pass band deviation δ, and the transition bandwidth Δf, defined by the cutoff frequency f[c] and the
frequency f[a] at which the squared magnitude frequency response is less than or equal to 1/A^2.
Let us now consider the three types of design specifications.
In the first case (Δf and δ fixed), the design procedure has to start with the evaluation of the order of the filter necessary to meet the specifications in terms of the desired attenuation,
transition bandwidth and pass band deviation.
The pass band deviation can be controlled in the case of Chebyschev filters be ε. In any case, having defined f[c] and f[a] (i.e. transition bandwidth), the desired value 1/A^2 of |H(e^jω)|^2 at
f[a] and ε in the Chebyschev case, it is possible to determine N iteratively, starting from a first order filter and increasing the order of the filter to the point where the attenuation at f[a] is
greater than the desired value. At this point the design is completely determined.
In the second case (N and Δf fixed), the design is completely determined for the Butterworth filter case by obtaining the value of the attenuation at f[a] directly.
In the third case (N and δ fixed), the filter is completely specified and the transition band width is directly obtainable during the design procedure.
A computer program is presented which designs Butterworth and ChebysÂchev filters by means of the above relation. It also computes the coefficients of their cascade structure. The inputs to the
program are the critical frequencies
f[c ]and f[a], the values of the desired attenuation of the filters at f[a] and the value of the maximum pass band ripple if a Chebyschev filter is to be designed. The order of the filter is
computed iteratively.
Two examples of the IIR filter design with this program are shown in Figs 15.40 (a) and (b).
Assuming a realization structure by means of second order sections, only even order filters are designed by the program. However, it is quite simple to modify the program to design odd filters by
replacing Eq. (15.102) with Eq. (15.103) and introducing a first order section in the structure.
The program presented here can be used to design only low pass filters. To design other types of filters, such as band pass, high pass and band stop, it is possible to start with the design of a
normalized filter and then apply the approÂpriate frequency transformation, as shown in Table 15.6. A simple routine (TRASF) to perform these transforms is presented.
It is usually assumed that one has a low pass filter of a definite cut off frequency, say β rad/s, from which other low pass, high pass, band pass or band stop filters are required to be derived.
The low pass digital filter is said to be normalized when | {"url":"https://www.eeeguide.com/iir-filter-design/","timestamp":"2024-11-09T00:42:55Z","content_type":"text/html","content_length":"221973","record_id":"<urn:uuid:bf4980b7-9442-4f7f-b8cb-74558fe34b7a>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00061.warc.gz"} |
Is the Fed's "zero-interest-rate policy" (ZIRP) inflationary or deflationary? You'd think that macroeconomists would have a straight answer for such a simple question. But we don't. As usual, the
answer seems to depend on things.
Someone once joked that an economist is someone who sees something work in practice and then asks whether it might work in theory. Well, appealing to the evidence is not much help here either. We
have examples like that of Volcker temporarily raising rates (by lowering the money supply growth rate) to lower inflation in the early 1980s. But then we have the present counterexample of ZIRP,
which seems to be having little effect in raising inflation. Indeed, the Fed has consistently missed its 2% inflation target from below for years now (see
Understanding Lowflation
Some economists suggest that there are theoretical reasons to support the notion that ZIRP is deflationary. The proposition that targeting a nominal interest rate at a low (high) level results in low
(high) inflation is known as "NeoFisherism." The idea goes back at least to Benhabib, Schmitt-Grohe and Uribe (2001) in their
The Perils of Taylor Rules
. The idea has been taken seriously in policy circles. My boss, St. Louis Fed president Jim Bullard wrote about it
in 2010. You can read all about the recent controversy here:
Understanding the NeoFisherite Rebellion
The basic idea behind NeoFisherism is the Fisher equation:
FE1: Nominal interest rate = real interest rate + expected inflation.
One interpretation of the Fisher equation is that it is a no-arbitrage-condition that equates the real rate of return on a nominal bond with the real rate of return on a real (e.g.,
) bond. FE1 implicitly assumes that the risk and liquidity characteristics of the nominal and real bond are identical. Steve Williamson and I consider a
where the nominal bond (potentially) carries a liquidity premium, in which case FE1 becomes:
FE2: Nominal interest rate + liquidity premium = real interest rate + expected inflation.
I'm not aware of any economist that disputes the logic underlying (some version of) the Fisher equation. The controversy lies elsewhere. But before going there, let me describe the way things are
supposed to work in neoclassical theory.
Start in a steady-state equilibrium where FE1 holds. Now consider a surprise permanent increase in the nominal interest rate. What happens? Well, a higher nominal interest rate increases the
opportunity cost of holding money, so people want to economize on their money balances. However, because
must hold the outstanding stock of money, a "hot potato effect" implies that the equilibrium inflation rate must rise (the real rate of return on money must fall). In the new steady-state
equilibrium, real money balances are lower (the price-level and the inflation rate are higher than they would have been prior to the policy shock). If people have rational expectations, then absent
any friction, inflation expectations jump up along with actual inflation. If there are nominal rigidities, then inflation may (or may not) decline for a while following the shock, but in the
long-run, higher interest rate policy leads to higher inflation. [Aside: my own view is that a supporting fiscal policy is needed for this result to transpire; see here:
A Dirty Little Secret
The conventional wisdom, however, is that pegging the nominal interest rate is unstable. Suppose we begin, by some fluke, in a steady-state where FE1 holds. Now consider the same experiment but
assume that people form their expectations of inflation through some adaptive process; see
Howitt (1992)
. For example, suppose that today's inflation expectation is simply yesterday's inflation rate. Then an increase in the nominal rate must, by FE1, lead to an increase in the real interest rate. An
increase in the real interest rate depresses aggregate demand today (consumer and investment goods). The surprise drop in demand leads to a surprise decline in the price-level--the inflation rate
turns out to be lower than expected. Going forward, people adapt their inflation forecasts downward. But given FE1, this implies yet another increase in the real interest rate. And so on. A
deflationary spiral ensues.
For those interested, refer to this more detailed discussion by John Cochrane:
The Neo-Fisherian Question
Now, the thought occurred to me: what if we replace the assumption of rational expectations in my model with Williamson (cited above) with a form of adaptive expectations? What would happen if we
performed the same experiment but, beginning in a steady-state where there is an "asset shortage" so that the FE2 version of the Fisher equation holds. My back-of-the-envelope calculations suggest
the following.
First, because expected inflation is fixed in the period the nominal interest rate is raised, FE2 suggests that either the real interest rate rises, or the liquidity premium falls, or both. In our
model, there is substitution out of the cash good into the credit good. But because there is a cash-in-advance constraint on the cash good, i.e., p(t)c(t) = M(t), it follows that a decline in the
demand for c(t) corresponds to a decline in the demand for real money balances--that is, the price-level p(t) must jump up unexpectedly (for a given money supply M(t)).
Now, given the surprise jump in the price-level, an adaptive expectations rule will adjust the expected rate of inflation upward. What happens next depends critically on the properties of the assumed
learning rule, the policy rule, etc. For my purpose here, I make the following assumptions. Suppose that the economy remains in the "asset shortage" scenario and assume that the government fixes the
money growth rate at exactly the new, higher, adaptively-formed, inflation expectation. In this case, the economy reaches a new steady-state with a higher interest rate, an arbitrarily higher
inflation rate, and an arbitrarily lower liquidity premium (conditional on the liquidity premium remaining strictly positive). [Note: by
what I mean is that the new inflation rate is determined, under my maintained assumptions, by the initial surprise increase in inflation, which may lie anywhere within a range such that the liquidity
premium remains positive. In the absence of a liquidity premium, the inflation rate would rise one-for-one with the nominal interest rate.]
I hope I made my point clear enough. The claim that increasing the nominal interest rate leads to higher inflation does
depend on rational expectations as is commonly claimed. A simple adaptive rule could lead to higher inflation expectations. The key is whether the price-level impact of increasing the nominal
interest rate is positive or negative. If it's positive, then people will revise their adaptive expectations upward and, depending on the learning rule and policy reaction, the ensuing inflation
dynamic could play itself out in the form of permanently higher inflation. The NeoFisherite proposition is possible even if people do not have rational expectations.
Postscript Nov. 01, 2015
Tony Yates suggests the cost channel in
Ravenna and Walsh (2006)
together with least-squares learning might deliver the same result. It's a good idea. Somebody should try to work it out! | {"url":"https://andolfatto.blogspot.com/2015/10/","timestamp":"2024-11-13T02:43:40Z","content_type":"text/html","content_length":"97107","record_id":"<urn:uuid:82e47879-bf2a-4dcb-bd61-ccc11bcaa197>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00157.warc.gz"} |
Annualized Return Calculator - Quick Online Calculations
Annualized Return Calculator
Why use the Annualized Return calculator?
You can quickly perform ANNUALIZED RETURN calculation using MYcalcu without any error. It is a free tool, and it provides the eased accessibility to online calculations without any payment or login
Annualized Return Calculator
Annualized Return Calculator
Mycalcu uses the standard formula to find ANNUALIZED RETURN.
Annualized Return Rate(ARR)= (closing value÷initial value)^1/n-1
However, you don’t have to get into the complexities. Because, Mycalcu would do it for you.
HOW TO USE MyCalcu ANNUALIZED RETURN CALCULATOR?
Using MyCalcu is super easy. Just carefully fill-in all the values and click calculate. | {"url":"https://mycalcu.com/annualized-return-calculator","timestamp":"2024-11-06T01:42:42Z","content_type":"text/html","content_length":"19163","record_id":"<urn:uuid:9ff7c589-07c3-45ae-adf1-7cef2fd70686>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00037.warc.gz"} |
eek 13
Comp 163 Week 13 notes
Week of April 18
Finite-State Machines
("machine" == "automaton", by the way)
Inputs, by the way, can be:
• individual letters, as in regular expressions
• the "tokens" of a computer language
• something more abstract, representing events (as in TCP)
From last week:
Regular expression: * means repeat 0 or more times, ? means either 0 or 1 times
• b a* c
• b? a* c?
• 1 (01*0)* 1 (supposedly odd binary numbers divisible by 3)
• [a-z][a-z,0-9]* (for programming-language identifiers)
• [0-9]*(.[0-9]*)?(e[0-9]*)? (for floating-point numbers, eg 12.345e67
What strings match these?
What does a finite-state recognizer for these look like?
More examples of regular expressions:
These use slightly extended regexes (The google example does not support + or *)
\d matches any digit 0-9, same as [0-9]
\W matches anything other than a letter, digit or underscore, same as [^a-zA-Z0-9_]
\s matches a space
^ matches the start of the line; $ matches the end of the line
{3,6} means that whatever single-character thing preceding this can match between 3 and 6 times
What does varname\W*=[^=] match?
Warning: there are quite a few different standards for regular expressions. Always read the documentation.
Let's call the finite-state recognizers finite automata. So far the finite-state recognizers have all been deterministic: we never have a state with two outgoing edges, going two different
directions, that are labeled with the same input. A deterministic finite automaton is abbreviated DFA.
How about b (ab)* a? There's a difference here. Now we do have a state with two different edges labeled 'a'. Such an automaton is known as nonde^. terministic, that is, as an NFA. We can still use an
NFA to match inputs, but now what do we do if we're at a vertex and there are multiple edges that match the current input?
There are two primary approaches. The first is to try one of the edges first, and see if that works. If it does not, we backtrack to the vertex in question and at that point try the next edge. This
approach does work, but with a poorly chosen regular expression it may be extremely slow. Consider the regular expression (a?)^n a^n. This means up to n optional a's, followed by n a's. Let us match
against a^n, meaning all the optional a's must not be used. The usual strategy when matching "a?" is to try the "a" branch first, and only if that fails do we try the empty branch. But that now means
that we will have 2^n - 1 false branches before we finally succeed.
Example: (a?)^3 a^3.
A much faster approach is to use the NFA with state sets, rather than individual states. That is, when we are in state S1 and the next input can lead either to state S2 or state S3, we record the new
state as {S2,S3}. If, for the next input, S2 can go to S4 and S3 can go to either S5 or S6, the next state set is {S4,S5,S6}. This approach might look exponential, but the number of states is fixed.
Example: (a?)^3 a^3.
See also https://swtch.com/~rsc/regexp/regexp1.html, "Regular expression search algorithms", the paragraph beginning "A more efficient ...."
By the way, a much better regular expression for between n and 2n a's in a row is a^n (a?)^n. We parse n a's at the beginning, and the optional a's are all following.
The implementation of an NFA/DFA recognizer does literally use the graph approach: for each current state, and each next-input symbol, we look up what next states are possible with that input symbol.
The code to drive the NFA/DFA does not need to be changed for different NFA/DFAs. This is a big win from a software-engineering perspective.
TCP state diagram: intronetworks.cs.luc.edu/current2/html/tcpA.html#tcp-state-diagram. Note the additional software-engineering issue of this being a distributed system.
Study guide
Some people, when confronted with a problem, think
“I know, I'll use regular expressions.” Now they have two problems.
-- Jamie Zawinski (regex.info/blog/2006-09-15/247)
Stop Validating Email Addresses with Regex: davidcel.is/posts/stop-validating-email-addresses-with-regex.
How about even more problems? jimbly.github.io/regex-crossword.
TCP kernel implementation: tcp_ipv4.c tcp_v4_do_rcv(), tcp_seq_next(), tcp_seq_stop(), tcp_v4_err(),
tcp_input.c: tcp_rcv_state_process
Also regex option in gedit search box and eclipse search box
One more example of NFA state-set recognizer: aaa|aab|aac|aad
| /--a->(4)--b->(8)
| \--a->(5)--c->(9)
NFA to DFA
It is also possible to convert any NFA to a DFA. The catch is that if there are n states in the NFA, there might be 2^n states in the DFA.
Subset construction: DFA states are all sets of NFA states. Given such a set, and an input, we form the set of all states reachable on that input from any of the NFA states in the set.
Elliptic curve cryptography
Graph of y^2 = x^3 + Ax + B (the (short) Weierstrass form)
What does this have to do with an ellipse?
Elliptic product a⊕b: the graphical constuction over R
Adding a point at infinity
See Boneh & Shoup p 614 (of version 0.5): "The Addition Law" (toc.cryptobook.us, chapter 14 "Elliptic curve cryptography")
Note that if you have two roots r[1] and r[2] of a cubic ax^3 + bx^2 + cx +d, then the product of all the roots is d/a, and so r[3] = d/ar[1]r[2].
Finite fields: graui.de/code/elliptic2.
Find the finite-field generator g (or base b)
Taking multiples of g: kg = g⊕g⊕...⊕g, k times. Repeated-squaring algorithm
Size of E(F[p]) solution set: roughly p
Basically, for each x, half the time there are no solutions for y and half the time there are two (+y, -y). On average there is one, so total number of solutions is ~p.
Montgomery form: y^2 = x^3 + Ax^2 + x
Diffie-Hellman-Merkle for basic elliptic curve
For classic Diffie-Hellman-Merkle, Alice chooses an integer a<p, and Bob chooses b<p. Alice and Bob publish g^a and g^b respectively, where g is the chosen generator. If Alice wants to create a key
to use for encrypting a message to Bob, she calculates (g^b)^a = g^ab. Similarly, Bob can calculate (g^a)^b = g^ab to decrypt. Nobody else can; you have to know either a or b.
For elliptic curves, Alice again chooses an integer a<p, and Bob chooses b<p. Alice and Bob publish a*g and b*g, respectively. Again, knowing g and knowing a*g does not give you a reasonable method
for finding a. The rest of the mechanism works exactly as with the classic case.
Edwards form: x^2 + y^2 = 1 + Dx^2y^2. The elliptic product here does not involve cases.
The prime here is p = 2^255 - 19, which is easy to find in python. The curve is y^2 = x^3 + 486662x^2 + x.
Size of E(F[p]) = 8q, where q is prime; q = 2**252 + 27742317777372353535851937790883648493
Basic Encryption
Use Diffie-Hellman-Merkle to choose a common secret, and then use a hash of that secret as a conventional encryption key.
Base point: (9, 14781619447589544791020593568409986887264606134616475288964881837755586237401). This has order q, above, in the group.
How did I get this? RFC8032 page 21 | {"url":"https://pld.cs.luc.edu/courses/163/spr22/notes/13.html","timestamp":"2024-11-01T23:52:15Z","content_type":"text/html","content_length":"11565","record_id":"<urn:uuid:62e9fc47-5a83-4cef-85d0-dc96bea6aeee>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00606.warc.gz"} |
conserved quantity in English - dictionary and translation
In mathematics, a
conserved quantity
of a
dynamical system
is a function
of the dependent variables that is a
(in other words, conserved) along each trajectory of the system. A conserved quantity can be a useful tool for
qualitative analysis
. Not all systems have conserved quantities, however the existence has nothing to do with linearity (a simplifying trait in a system) which means that finding and examining conserved quantities can
be useful in understanding
nonlinear systems | {"url":"http://info.babylon.com/onlinebox.cgi?cid=CD566&rt=ol&tid=pop&x=20&y=4&term=conserved%20quantity&tl=English&uil=Hebrew&uris=!!ARV6FUJ2JP","timestamp":"2024-11-05T12:58:04Z","content_type":"text/html","content_length":"6078","record_id":"<urn:uuid:887bea44-0953-46d6-a02f-14eed2886b82>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00582.warc.gz"} |
Water Resources Systems - MCQs from AMIE exams (Summer 2021/Winter 2020)
Choose the correct answer for the following (2 x 10)
1. The method used to determine the live storage capacity of a reservoir is
(a) Hydrologic Modelling
(b) Moodi's Method
(c) Sequent Peak Method
(d) None of these
2. The name of the graphical method to determine the capacity of a reservoir is
(a) Double mass curve technique
(b) Mass Curve Technique
(c) None of these
(d) Regression technique
3. The different capacities of reservoir are given in various options. Which is the right sequence from bottom to top?
(a) Dead Storage, Active Storage, Flood Control Storage and Surcharge Storage
(b) Flood Control Storage, Active Storage, Dead Storage, and Surcharge Storage
(c) Surcharge Storage, Active Storage, Dead Storage, and Flood Control Storage
(d) None of these
4. If the feasible region of an LPP is empty, the solution is ---------.
(a) Infeasible
(b) Unbounded
(c) Alternative
(d) None of the above
5. The chi-square goodness-of-fit test can be used to test for
(a) Significance of sample statistics
(b) Difference between population means
(c) Normality
(d) Probability
6. Routing of flow through physical approach requires solution of
(a) St. Venant's Equations
(b) Bernoulli’s Equations
(c) Clark’s Equation
(d) None of these
7. The evaporation loss in a reservoir is computed by carrying out
(a) Curvilinear regression of Area Storage curve
(b) Linear Regression of the storage and surface area
(c) Statistical analysis
(d) None of these
8. The classical example of a Lumped system is
(a) A detailed hydrological model
(b) A Unit Hydrograph
(c) streamline flow model
(d) None of these
9. Graphical method can be applied to solve an LPP when there are only ------- variables.
(c) Two
(d) Three
10. In hypothesis testing, a Type 2 error occurs when
(a) The null hypothesis is not rejected when the null hypothesis is true
(b) The null hypothesis is rejected when the null hypothesis is true
(c) The null hypothesis is not rejected when the alternative hypothesis is true
(d) The null hypothesis is rejected when the alternative hypothesis is true
1. (c) The sequent peak procedure (Thomas and Burden, 1963) is commonly used to determine the reservoir capacity required to meet a stipulated demand over a given record of inflows. This method
utilizes a plot (or tabulation) of cumulative differences between inflows and drafts as a function of a specified time interval.
2. (b) Mass curve method is commonly used to estimate the required storage capacity of a reservoir in project planning stage. The method uses the most critical period of recorded flow to compute
storage. The critical period is defined as the duration in which initially full reservoir depletes and, passing through various states, empties without spilling.
3. (a)
4. (a) If the feasible region of an LPP is empty, the solution is infeasible. A linear program is infeasible if there exists no solution that satisfies all the constraints -- in other words, if no
feasible solution can be constructed. Since any real operation that you are modelling must remain within the constraints of reality, infeasibility most often indicates an error of some kind.
5. (a) A Chi-Square goodness of fit test is used to determine whether a categorical variable follows a hypothesized distribution. We can use a Chi-Square goodness of fit test to determine if there is
a statistically significant difference in the number of expected counts for each level of a variable compared to the observed counts.
6. (a)
7. (b)
8. (a) A lumped model is generally applied in a single point or a region for the simulation of various hydrologic processes. The parameters used in the lumped model represent spatially averaged
characteristics in a hydrologic system and are often unable to be directly compared with field measurements.
9. (c) Graphic method can be applied to solve an LPP when there are only two variables. This method is used to solve a two variable linear program. If you have only two decision variables, you should
use the graphical method to find the optimal solution. A graphical method involves formulating a set of linear inequalities subject to the constraints. Then the inequalities are plotted on an X-Y
10. (c) A type II error does not reject the null hypothesis, even though the alternative hypothesis is the true state of nature. In other words, a false finding is accepted as true. A type II error
can be reduced by making more stringent criteria for rejecting a null hypothesis.
• The study material for AMIE/B Tech/Junior Engineer exams is available at https://amiestudycircle.com
• If you like the post please share your thoughts in the comment section | {"url":"https://blog.amiestudycircle.com/2022/04/water-resources-systems-short-answer.html","timestamp":"2024-11-13T08:28:50Z","content_type":"application/xhtml+xml","content_length":"105495","record_id":"<urn:uuid:5daca7e9-0796-4f31-a9b6-39f6ab45acec>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00018.warc.gz"} |
Book : The Contact Patch
Hysteresis losses in rolling wheels
For some years, railway engineers and road vehicle engineers have been trying to unravel the causes of rolling resistance (see Section G0119) and where possible, find ways of reducing it. What
interests them most is the contact patch, where three things happen. First, the wheel tyre or tread is squeezed repeatedly against the track surface. Second, the track surface is likewise squeezed by
the tread. Thirdly, the two surfaces ‘scrub’ across one another. All these processes dissipate energy and therefore slow the vehicle down.
The loading cycle
Let’s imagine that the wheel is attached to a wooden cart. The cart could be running on steel rails like a horse-drawn tram, or it could have pneumatic tyres; the same principles would apply in
either case, but this particular cart has wooden wheels with steel rims and it is running on a smooth gravel track. It is being pulled by a horse, and the horse is wondering why it must pull so hard
even though the track is smooth and level. Where is the friction coming from?
Figure 1
The loading cycle as seen by the track
We’ll start by looking at what happens to the track as a wheel passes over. Let us assume to begin with that the contact patch is symmetrical in the fore-and-aft direction with a total length \(2a\),
and located vertically underneath the axle centreline. Now focus on a particular point A on the track surface ahead of the wheel. Until the leading edge of the contact patch arrives, all stresses at
the track surface are zero. As the wheel passes over, two things happen in sequence. First, the track experiences a build-up of vertical compressive stress until it reaches a maximum value roughly
when the centreline of the axle is directly overhead (there are other stresses as well but they’re less important). Second, the stress declines as the rear section of the contact patch passes over,
falling to zero at the trailing edge. Together, these two events form a ‘loading cycle’ in which a load is firstly applied and then removed (figure 1). Now consider what happens to a small area of
the tread during one revolution of the wheel. For most of this period, the area concerned is rotating clear of the track and free of load. Only when it comes into contact with the track does the
compressive stress begin to rise. It then follows a loading cycle similar to the loading cycle applied to the track – a mirror image if you like.
Figure 2
Both wheel and track deform
If we look more closely during the first part of the cycle, we see that the shape of the tread and the shape of the track surface both become distorted. The wheel applies forces to the track surface,
which deflects a little, mostly downwards. In turn, the track reacts upwards on the wheel (if it didn’t, the wagon would fall through). Under pressure, the radius of the wheel, which we’ll denote by
\(r\), is slightly reduced (figure 2).
When a force pushes at something, and the thing moves or gives way (even if only a little), we say that ‘work is done on’ the object concerned. Energy is expended equal to the force times distance
moved. Here, the wheel is deflecting the track, so the track is receiving energy from the wagon. Simultaneously, the upward pressure of the track squeezes the wheel and hence the wheel itself is
having work done on it. In the case of a pneumatic tyre, the deformation is obvious, but in the case of a steel-rimmed wagon wheel it is too small to be seen with the naked eye. Nevertheless the
wheel is absorbing energy which, together with the energy pumped into the track surface, must come from somewhere.
Now for the second part of the loading cycle. Ideally, both wheel tread and track would be made of perfectly elastic materials. Not only would they recover their original shape after contact, but all
the energy put into the wheel and track by the vehicle would be returned. The energy flow into the track would be reversed as the track surface rose behind the axle centreline, pushing the wheel
forward. Simultaneously, the wheel tread would expand in the rear half of the contact patch and again help to push the vehicle forward. Gains in the second part of the cycle would exactly balance
losses in the first part, so that no energy would be dissipated from the vehicle. There would be no resistance to motion.
This is not what actually happens. The loading phase and the recovery phase are not symmetrical, and during each cycle some of the energy put into the wheel and track is irretrievably lost. The
process is said to exhibit hysteresis. Assuming that the contact surfaces are perfectly smooth, in metals the hysteresis losses are small. This is because metals are crystalline; each atomic nucleus
is locked into a regular lattice, and although the lattice will distort a little under pressure, each nucleus stays in position. The movement is taken up by squeezing and stretching of the electrical
bonds between neighbouring atoms; when the pressure is removed, they spring back into their original positions as if nothing had happened. Energy losses take place only at the crystal boundaries and
they are relatively small.
Polymers, by contrast, consist of complex molecules built up into chains. They do not form a regular lattice, but rather a tangled mass. In particular, rubber molecules wind around both themselves
and their neighbours, and when stress is applied to the material, the molecules unwind, dragging their side-chains across one another. This results in friction, which is ultimately dissipated as
So any wheel will absorb and dissipate energy when rolling: not very much in the case of a steel wheel, rather more for rubber tyre. But what about the track? It turns out that some of the energy put
into the track is radiated into the ground below in the form of vibrational waves that carry energy away with them like sea waves that are stirred up during a storm. There are other losses too,
including the kinetic energy carried away by grit that is sent flying when crushed by the wheel, the plastic flow of tiny ‘asperities’ on the surface of a metal tread or metal rail, and the grinding
between the wheel and track as their respective surfaces adjust to one another when distorted. Eventually, through one mechanism or another, the energy is converted into heat.
Figure 3
Asymmetrical pressure distribution on the contact patch
Figure 4
Resistance force S
To sum up, then, energy flows out of the wheel and into the environment as heat. When you ride on a bus or train, you wouldn’t normally be aware of this: it’s not intuitively obvious how energy can
flow out of a wheel. It’s certainly not obvious to the horse we referred to earlier. When pulling a cart, what the horse experiences is more direct: a ‘drag’ or horizontal force that resists motion.
But a logical chain of reasoning connects these two pictures, (a) the energy lost through the wheels, and (b) the resistance felt by the horse. The energy losses mean that in the rear part of the
contact patch, which corresponds to the unloading part of the loading cycle, the pressure between the wheel rim and track is reduced because the wheel and track surface do not spring back with the
same vigour as they would were they perfectly elastic. The pressure distribution is no longer symmetrical fore-and-aft, and the centre of pressure moves forward (figure 3). Suppose it moves forward
by a distance \(e\). Given a frictionless bearing, if the wheel is rolling at a steady speed the resultant force must still pass through the axle centreline. It follows that there must be a
horizontal shear force \(S\) acting on the wheel through the contact patch (figure 4), and by similar triangles, the ratio of \(S\) to the normal contact force is \(e/r\). Hence
\[$$S\quad = \quad \frac{e}{r}N$$\]
This is the resistance that the horse actually feels.
Predicting the rolling resistance
The geometrical deformations within and around the contact patch may be complicated: for example, in the case of a car tyre, the walls bulge outwards while the tread flattens within the contact patch
and bulges at the rear. To predict the heat losses, tyre specialists must resort to computer modelling. Similarly, the shape of a railway wheel and the way it engages with the surface of the rail is
geometrically complicated. But if we know the size of the contact patch, we can make a rough estimate from first principles.
In fact the size of the contact patch tells us something straight away because it determines the limits within which the centre of pressure must lie. From figure 4, we see that a small contact patch
implies a small wheel rolling resistance \(S\), and if we have reason to believe that the contact patch is symmetrically located under the axle centreline, then the centre of pressure must lie within
a distance \(\pm a\) of that centreline. This places an upper limit on \(e\) and correspondingly an upper limit on \(S\).
Railway wheels
We can analyse a railway wheel using a method that applies specifically to materials in which the hysteresis losses are small. Suppose the width of the contact patch is \(2b\). When a cylinder of
length \(2b\) rests on a flat plane, the contact patch consists of a thin strip or rectangle (figure 5). The normal contact pressure between the two surfaces is not uniform across this rectangle.
According to principles first laid down by Heinrich Hertz in 1882 for perfectly elastic materials, the value at a point \(x\) measured from the centre of the contact area in the direction at right
angles to the axis of the cylinder is given [8] by:
\[$$p(x) \quad = \quad p_{0} \sqrt{1 \; - \; \frac{x^2}{a^2} }$$\]
We assume that a stationary railway wheel generates a pressure distribution similar to that of the cylinder, and that the materials are perfectly elastic so the shape of the curve doesn’t change when
the wheel starts to move. Denote the tread radius by \(r\) and let the wheel load be \(P\). As shown in figure 6, the pressure rises from zero at the leading edge of the contact patch to a maximum in
the centre, and falls to zero at the trailing edge. The curve is elliptical in shape, and the peak value \(p_0\) is given by
\[$$p_0 \quad = \quad \frac{P}{\pi ab}$$\]
If the wheel and rail were perfectly elastic, there would be no energy lost in rolling contact, the pressure distribution would be symmetrical about the axle centreline, and there would be no rolling
resistance. On the other hand, if a proportion \(\alpha\) of the energy put into the wheel and track during the loading phase were not recovered owing to hysteresis, then the pressure distributions
in the two halves would differ and the rolling resistance could be appreciable. Is there a simple way of predicting its value?
Figure 5
Hertzian contact pressure between cylinder and plane
Figure 6
Semi-elliptical pressure distribution on a cylinder
Figure 7
Effect of hysteresis on pressure distribution
Professor John Williams has suggested an ingenious method that relies on the value of \(\alpha\) being small [9], which is indeed the case for a railway wagon with steel wheels running on a steel
rail. Although he does not claim that the method is accurate, the answer it gives in this case is illuminating. The pressure distribution in the trailing half of the contact patch is assumed to be a
mirror image of the pressure distribution in the leading half except that its value is everywhere reduced by the factor \(1 -\alpha\) (figure 7). Denote by \(P_L\) and \(P_T\) the loads carried by
the leading and trailing halves of the contact patch respectively. Then
\[$$P_T \quad = \quad \left( 1 - \alpha \right)P_L$$\]
\[$$P_L \; + \; P_T \quad = \quad P$$\]
From which we see that
\[$$P_L \quad = \quad \frac{1}{(2 - \alpha)} \, . \, P$$\]
\[$$P_T \quad = \quad \frac{(1 - \alpha)}{(2 - \alpha)} \, . \, P$$\]
\[$$P_L \; - \; P_T \quad = \quad \frac{\alpha}{(2 - \alpha)} \, . \, P$$\]
Now imagine flipping over the curve representing the pressure distribution in the trailing half and superimposing it on the curve for the leading half. The difference between the two represents the
effect of hysteresis, which from equation 8 imposes a component \(\alpha P / (2-\alpha)\) of normal contact force that is offset from the axle centreline, at the centroid of the half-ellipse bounded
by the dashed line shown in figure 7. It is a well-known geometrical property of the half-ellipse that this centroid is located at a distance \(e \; = \; 4a / 3\pi\) from the origin as shown in
figure 8.
Figure 8
Offset component of normal contact force
From now on, we shall ignore the symmetrical components of the normal contact force, which cancel one another out and therefore do not contribute to rolling resistance. What remains is the hysteresis
component. Now for any freely rolling wheel, the resultant of the ‘hysteresis’ normal contact force and the frictional resistance force must pass through the wheel centreline as shown previously in
figure 4. Substituting \(4a / 3 \pi\) for \(e\) and \(\alpha P / (2-\alpha)\) for \(N\) in equation 1, we get
\[$$S \quad = \quad \frac{2a}{3 \pi r} . \frac{\alpha P}{\left( 1 - \frac{\alpha}{2} \right)}$$\]
By analogy with a conventional friction coefficient, we can define the ‘wheel rolling resistance coefficient’ \(\mu_S\) by the ratio \(S/P\). From equation 9, we see immediately that
\[$$\mu _S \quad = \quad \frac{2a}{3 \pi r} . \frac{\alpha}{\left( 1 - \frac{\alpha}{2} \right)}$$\]
Let us put in some typical values, assuming \(\alpha\) is 1% or 0.01. For a railway wheel of radius 500 mm whose contact patch is 10 mm long in the fore-and-aft direction, equation 10 predicts the
value of the coefficient of wheel rolling resistance as (2 \(\times\) 5 \(\times\) 0.01) / (3 \(\times\) 3.14 \(\times\) 500 \(\times\) 0.995) = 0.000021. This is extremely small, much smaller than
the figures quoted by railway engineers, which are usually in the range 0.001 to 0.002 (see for example [3]).
Clearly we have left something out. Our simplified analysis ignores energy losses associated with ‘micro-slip’ within the contact patch. One particular form of micro-slip occurs because the surface
of the wheel tread and the surface of the rail are distorted when they come into contact. As shown in figure 2, the length of wheel rim measured around its circumference is slightly more than the
length of the contact patch into which it is compressed. Likewise, the length of rail surface increases slightly as it is dented by the wheel. Hence there must be some slippage and grinding as the
wheel rolls along the track. There are other types of grinding too. As detailed in Section G0119, the most important ones arise from a worn wheel tread or rail profile together with intermittent
collisions of the wheel flange with the edge of the rail. Together, these two processes tend to swamp the ‘pure’ rolling resistance represented in our model, and explain why there isn’t a reliable
formula for predicting the wheel rolling resistance of railway vehicles. Instead, engineers resort to empirical data or to computer simulation modelling in which the condition of the wheels and track
are explicitly taken into account.
Car tyres
Can we analyse car tyres in the same way? Unfortunately not, because Williams’ method assumes a sudden fall or discontinuity in the pressure distribution at the centre of the contact patch (see
figure 7). In reality, the fall in pressure must take place smoothly over an appreciable distance. This might be ignored if \(\alpha\) were small, i.e., the pressure drop were just a few percent.
With car tyres, however, the hysteresis losses are much larger, because 50% or more of the energy put into the tyre is dissipated as heat.
Nevertheless, we’ll do the calculations and see what happens. For a stationary tyre, the internal pressure is constant across the whole of the inside of the tyre wall and tread. Hence the contact
pressure distribution is no longer elliptical in shape. We shall not give the details here, but if one assumes a uniform profile so that the offset load component now acts at a distance \(e \; = \; a
/2\) ahead of the axle centreline, for a tyre of radius 312 mm with a contact patch roughly 120 mm square, the coefficient of wheel rolling resistance turns out to be around 0.03. This is not very
accurate but it is the right order of magnitude: the value quoted for ordinary car tyres is usually in the range 0.01 to 0.02 [6].
It seems that car tyres induce a larger rolling resistance than railway wheels. The picture is complicated by the fact that on a real road, the tyre tread is squeezed against stone chippings embedded
in the road surface that measure around 10 mm across [1]. The tread rubber deforms over each ‘asperity’, and some of the energy absorbed in the deformation process is dissipated as heat. So how can
the losses be reduced? Nearly thirty years ago, a research conference was convened in the USA to review the state of the art. It was known at the time that as a result of changes in tyre chemistry,
the rolling resistance of car tyres had been falling steadily for some years, but there was potential for further improvement. It has since been confirmed that the avoidable losses can be reduced
broadly in three ways.
1. One can use different varieties of rubber for different parts of the tyre. To preserve grip, we want high-hysteresis rubber in the tread, while to reduce rolling resistance we want low-hysteresis
rubber in the tyre walls.
2. Alternatively we can use the same material throughout, but tailor the chemistry so that it behaves differently under different loading conditions [5]. When considered as a whole, the tyre
structure is subjected to a certain frequency of loading – each part of the carcass is deformed once per revolution as it passes through the most heavily loaded region, the area in and around the
contact patch. The frequency of this load/unload cycle is typically about 10 Hz depending on vehicle speed [7]. If we now examine more closely what happens within the contact patch under heavy
braking, we see additional deformation of the tread as it creeps over individual stone chips as previously mentioned. If the tread creeps at 1 m/s, the load/unload cycle at this local scale has a
frequency of the order of 100 Hz. Therefore we want a tyre material with low hysteresis losses at 10 Hz and high hysteresis losses at 100 Hz and above. Some manufacturers hope to achieve this by
substituting silicon for carbon black in the rubber composition [10].
3. The third way is to change the size and shape of the tyre structure, with a larger diameter [2] and a narrower profile. The principle underlying the narrow ‘eco-tyre’ has been known for quite a
long time [4]: energy is lost every time a road surface asperity makes a depression in the tread surface. A long, narrow contact patch will encounter fewer asperities per unit distance travelled
than a short, wide one of the same total area (figure 9), and hence will generate proportionately less rolling resistance.
Figure 9
Asperities encountered by the contact patch
On this last point, we can be a little more precise. Suppose the contact patch of the first tyre has width \(2b\). When it rolls a distance \(x\), say, the total area of tyre tread that is squeezed
against the road surface is \(2bx\). Given a uniform density of asperities per unit area of road, the energy loss is proportional to the area that is compressed, in other words, it is proportional to
\(2bx\). Now consider a second tyre whose contact patch is half the width of the first, but twice the length so that the total contact area and the average contact pressure are unchanged. The area of
fresh tyre coming into contact with the road for every metre of travel is now half its previous value, at just \(bx\), and therefore the tread hysteresis loss is halved too.
But what about grip? It turns out that grip is not affected, because it arises from relative motion between the tread and the track surface. In the case of braking for example, as already described,
asperities ‘plough’ through the tread material and develop contact forces that are parallel to the plane of the contact patch as opposed to normal contact forces. At any given moment, the number of
asperities that generate braking resistance is proportional the area of the contact patch and is therefore unchanged. Hence there is no loss of grip. While narrow tyres are not considered fashionable
today, eventually they may supersede low-profile tyres because they save fuel, and as a welcome by-product, they will make less noise.
A loose end
It seems that you can reduce the rolling resistance of a rubber tyre by making it narrower, provided the contact area remains unchanged. It reduces the air resistance too. So why don’t all cars have
narrow tyres?
August 16 2013, revised February 11 2015 | {"url":"https://the-contact-patch.com/book/general/g1619-hysteresis-losses-in-rolling-wheels","timestamp":"2024-11-09T07:02:30Z","content_type":"text/html","content_length":"48418","record_id":"<urn:uuid:02c81d42-182c-4ff5-a338-b94b4f3517e5>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00021.warc.gz"} |
Punto Banco Policies and Method
Aug 06 2022
Baccarat Chemin de Fer Rules
Baccarat is enjoyed with 8 decks of cards in a dealing shoe. Cards below ten are counted at their printed number while Ten, Jack, Queen, King are zero, and A is one. Wagers are made on the ‘bank’,
the ‘player’, or on a tie (these aren’t actual people; they simply represent the two hands to be dealt).
Two hands of 2 cards are then given to the ‘banker’ and ‘player’. The score for every hand is the total of the 2 cards, but the 1st number is discarded. For instance, a hand of 5 and six has a score
of one (5 plus six equals 11; ignore the initial ‘1′).
A additional card will be given depending on the following rules:
- If the player or house achieves a value of eight or nine, the two players hold.
- If the gambler has less than five, he hits. Players stays otherwise.
- If the player stands, the bank hits on 5 or less. If the gambler takes a card, a guide is employed to determine if the bank holds or hits.
Baccarat Chemin de Fer Odds
The bigger of the two scores wins. Winning wagers on the bank pay out nineteen to Twenty (equal cash less a 5 percent commission. Commission are kept track of and paid off when you depart the table
so be sure to still have cash left over just before you depart). Winning bets on the gambler pay 1:1. Winning wagers for a tie frequently pays 8 to 1 but sometimes 9:1. (This is a poor wager as ties
happen less than one in every 10 hands. Be cautious of betting on a tie. However odds are astonishingly greater for 9:1 versus eight to one)
Played correctly baccarat gives fairly good odds, apart from the tie wager of course.
Baccarat Chemin de Fer Method
As with all games baccarat banque has a handful of general misunderstandings. One of which is similar to a myth in roulette. The past isn’t a prophecy of future outcomes. Tracking past outcomes on a
sheet of paper is a poor use of paper and an insult to the tree that was cut down for our stationary desires.
The most established and almost certainly the most accomplished method is the one, three, two, six plan. This tactic is employed to pump up earnings and minimizing losses.
Begin by placing 1 dollar. If you win, add another to the 2 on the game table for a sum of three units on the second bet. If you succeed you will have six on the table, pull off 4 so you keep 2 on
the third wager. Should you come away with a win on the 3rd round, deposit 2 on the four on the table for a sum total of six on the 4th round.
If you lose on the first round, you take a hit of one. A profit on the 1st wager followed by a loss on the second causes a loss of two. Wins on the initial 2 with a defeat on the third provides you
with a gain of two. And wins on the 1st three with a defeat on the fourth means you experience no loss. Succeeding at all 4 bets gives you with twelve, a gain of ten. This means you are able to
squander the 2nd wager five times for each successful run of four wagers and still are even.
You must be logged in to post a comment. | {"url":"http://fastplayingaction.com/2022/08/06/punto-banco-policies-and-method-3/","timestamp":"2024-11-04T10:58:40Z","content_type":"application/xhtml+xml","content_length":"27063","record_id":"<urn:uuid:fd5422ad-ac1e-46ba-a60a-b51c1d482742>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00098.warc.gz"} |
What are the asymptote(s) and hole(s), if any, of f(x) =(-2x^2-6x)/((x-3)(x+3)) ? | HIX Tutor
What are the asymptote(s) and hole(s), if any, of # f(x) =(-2x^2-6x)/((x-3)(x+3)) #?
Answer 1
Asymptotes at $x = 3$ and $y = - 2$. A hole at $x = - 3$
We have #(2x^2-6x)/((x-3)(x+3))#.
You find the vertical asymptote of #m/n# when #n=0#.
#x=3# is the vertical asymptote.
For the horizontal asymptote, there exists three rules:
To find the horizontal asymptotes, we must look at the degree of the numerator (#n#) and the denominator (#m#).
If #n>m,# there is no horizontal asymptote
If #n=m#, we divide the leading coefficients,
If #n<##m#, the asymptote is at #y=0#.
Here, since the degree of the numerator is #2# and that of the denominator is #2# we divide the leading coefficients. As the coefficient of the numerator is #-2#, and that of the denominator is #1,#
the horizontal asymptote is at #y=-2/1=-2#.
The hole is at #x=-3#.
This is because our denominator had #(x+3)(x-3)#. We have an asymptote at #3#, but even at #x=-3# there is no value of #y#.
graph{(-2x^2-6x)/((x+3)(x-3)) [-12.29, 13.02, -7.44, 5.22]}
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
Not the question you need?
HIX Tutor
Solve ANY homework problem with a smart AI
• 98% accuracy study help
• Covers math, physics, chemistry, biology, and more
• Step-by-step, in-depth guides
• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/what-are-the-asymptote-s-and-hole-s-if-any-of-f-x-2x-2-6x-x-3-x-3-8f9af9befe","timestamp":"2024-11-01T22:30:23Z","content_type":"text/html","content_length":"575037","record_id":"<urn:uuid:0b071566-2326-4ca0-9e1f-e60263b85af4>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00822.warc.gz"} |
Unlike Speed Limits, Ohm's Law isn't a Suggestion
Getting started with electronics always involves a discussion of Ohm’s Law. What is this mysterious sounding law and how can you use it when building electronic projects? One the main uses for Ohm’s
Law in your projects is to calculate the resistor value needed for a LED. This article takes a look at what Ohm’s Law is and how to use it with LEDs.
Ohm’s Law is really a mathematical rule based on two points. In electronics those two points have a voltage across them (aka potential difference). The amount of current that flows from one point
to the other, is going to depend on the resistance between those points. So Ohm’s law says that the Resistance of a device is equal to the Voltage Across it divided by the Current that Flows
through it. The formula looks like this:
$latex Resistance=\frac{Voltage}{Current}&s=3$
Using a little bit of algebra, you can determine the voltage drop across a device if you know the resistance and current. Or, you can determine how much current will flow if you know the voltage and
resistance. What’s an easy way to remember this? Well, use the “Ohm’s Law Triangle.”
Ohm’s Law Triangle
Using this triangle it is possible to create each of the following equations:
$latex Resistance=\frac{Voltage}{Current}&s=3$
$latex Current=\frac{Voltage}{Resistance}&s=3$
$latex Voltage=Resistance\times Current&s=3$
Using Ohm’s Law with Current Limiting Resistors
As discussed in my introduction to LEDs (found here), you should always limit the current when using LEDs. The simplest method, but least efficient, is to use a resistor in series with the LED.
Let’s use the example of connecting a LED to a pin on the Arduino. When you are trying to find a current limiting resistor for a LED, you know two things about the “circuit”: the voltage applied
to the resistance and how much current is flowing through it.
Arduino LED Example
When selecting a resistor for a LED, you generally know two things: Voltage and Current. Here’s how you determine each.
The main voltage of an Arduino Uno is 5V, which means the pins will output approximately 5V when configured as an OUTPUT and set to HIGH. So we know the voltage. Now, what we need to calculate is
the voltage drop of the current limiting resistor. Since the resistor and LED are in series, we subtract the Forward Voltage of the LED from the Supply Voltage.
Tip: The forward voltage of a LED will depend on its color.
We know the pin is 5V and in this example our LED has a forward voltage of 2V.
This means the Voltage across the resistor is 3V.
Typically when people use LEDs in their projects, they’ll use the maximum forward current allowed by a LED. It isn’t necessary to use this value. Backing away from the maximum will reduce the
brightness of the LED and extend its life. Since most people use the max, and typical LEDs are around 20mA, we’ll use 20mA for the current.
This means the Current flowing through the resistor is 20mA.
Calculating Resistance
So our resistor will have 3V dropped across and 20mA through it. Using the Ohm’s Law Triangle, here’s the information we know.
Ohm’s Law: Resistance
This shows that we are going to divide voltage by current, using this formula:
$latex Resistance=\frac{Voltage}{Current}&s=3$
Simply replace the words with numbers to get
$latex Resistance=\frac{3V}{0.02A}=150\Omega&s=3$
Which gives us a resistance of 150Ω!
Calculating the resistance value of a current limiting resistor is just one use of Ohm’s Law. You could also use the reverse to determine how current is flowing through an LED. Now, keep in mind,
that Ohm’s Law always applies, but not in ways you might expect. The properties of Ohm’s Law is straight foward for a device like a Resistor. However, active devices like Microprocessors aren’t
simplistic enough to just use Ohm’s Law to describe their behavior.
Write A Comment Cancel Reply
This site uses Akismet to reduce spam. Learn how your comment data is processed.
Author James Lewis
Fan of making things beep, blink and fly. Created AddOhms. Writer for Hackster.io News. Freelance electronics content creator for hire! KN6FGY and, of course, bald.
Prev Post
Electronics: Introduction to Breadboards
Next Post
Related Posts | {"url":"https://www.baldengineer.com/unlike-speed-limits-ohms-law-isnt-a-suggestion.html","timestamp":"2024-11-10T05:07:36Z","content_type":"text/html","content_length":"244787","record_id":"<urn:uuid:a6e85291-dfd0-411e-b054-cf532d98915a>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00696.warc.gz"} |
What is the pore size of activated carbon?
What is the pore size of activated carbon?
What is the pore size of activated carbon?
Due to varying methods of preparation, the pore sizes of the activated charcoal can be categorized as being micropores (width < 2 nm), mesopores (width = 2–50 nm), or macropores (width > 50 nm); the
differences in the size of their width openings being a representation of the pore distance.
Is activated carbon porous?
Activated carbon is an effective adsorbent because it is a highly porous material and provides a large surface area to which contaminants may adsorb. Activated carbon is available as PAC and GAC.
How do you increase pore size on activated carbon?
You can simply activate further your material by physical activation with steam. As physical activation is a mild gasification, all pores are broadened, so that the surface area first increases (as
more pores are created), then decreases (as these pores become wider and wider).
What is the porosity of a rock determined by?
The porosity of a rock varies because of the size of the grains in the rock and the shape of the grains. Another factor that affects the porosity of a rock is whether or not there is any material in
the rock (or cement) to fill in the gaps between pore spaces and hold the grains together.
How do I know my pore distribution size?
The pore-size distribution (PSD) can be defined as either p(r)=dV/dr or p(r)=(1/Vp)dV/dr, where Vp is the total pore volume.
How does activated carbon adsorption work?
During the activated carbon adsorption process, compounds in the contaminated air react with the carbon to stick to the surface area, effectively removing these contaminants from the air. Carbon air
filters remove pollutants from the air with a process known as adsorption.
Does activated carbon dissolve in water?
Activated carbon as you mentioned is insoluble in water and organic solvents and all usual solvents.
Is activated charcoal soluble in water?
Description: Black powder with enormous surface area; insoluble in water.
Does charcoal reduce pore size?
Yes, a charcoal mask minimizes pores. Activated charcoal is a toxin magnet, and this ancient healer attracts dirt and impurities that settle deep in the pores to the surface to minimize pore size and
lessen overall visibility.
How do you make activated carbon from rice husk?
The value-added activated carbon of rice husk can be produced by simple carbonization-chemical activation. Carbonization at 450 °C for 2 hr and using H3PO4 as an activating agent was a suitable
process. Under these optimum conditions, BET surface area of the activated carbon of rice husk was 336.35 m2/g.
How do you calculate pore space?
The volume of soil not occupied by solids is the pore space or porosity (PS) and is defined as: PS = (1- (BD/PD))100. example: assuming BD = 1.38, PD = 2.65; PS = (1- (1.38/2.65))100 = 48% | {"url":"https://drinksavvyinc.com/other/what-is-the-pore-size-of-activated-carbon/","timestamp":"2024-11-11T17:56:59Z","content_type":"text/html","content_length":"41178","record_id":"<urn:uuid:d99ba76e-23fe-40b8-8b88-34db1f3d3299>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00891.warc.gz"} |
The Boundedness of Convergent Sequences in Metric Spaces
The Boundedness of Convergent Sequences in Metric Spaces
Recall from The Uniqueness of Limits of Sequences in Metric Spaces page that if $(M, d)$ is a metric space and $(x_n)_{n=1}^{\infty}$ is a sequence in $M$ that is convergent then the limit of this
sequence $p \in M$ is unique.
We will now look at another rather nice theorem which states that if $(x_n)_{n=1}^{\infty}$ is convergent then it is also bounded.
Theorem 1: Let $(M, d)$ be a metric space and let $(x_n)_{n=1}^{\infty}$ be a sequence in $M$. If $(x_n)_{n=1}^{\infty}$ is convergent then the set $\{ x_1, x_2, ..., x_n, ... \}$ is bounded.
• Proof: Let $(M, d)$ be a metric space and let $(x_n)_{n=1}^{\infty}$ be a sequence in $M$ that converges to $p \in M$, i.e., $\lim_{n \to \infty} x_n = p$. Then $\lim_{n \to \infty} d(x_n, p) =
0$. So for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $n \geq N$ then $d(x_n, p) < \epsilon$. So for $\epsilon_0 = 1 > 0$ there exists an $N(\epsilon_0) \in \mathbb{N}$
such that if $n \geq N(\epsilon_0)$ then:
\quad d(x_n, p) < \epsilon_0 = 1
• Now consider the elements $x_1, x_2, ..., x_{N(\epsilon_0)}$. This is a finite set of elements and furthermore the set of distances from these elements to $p$ is finite:
\quad \{ d(x_1, p), d(x_2, p), ..., d(x_{N(\epsilon_0) - 1}, p) \}
• Define $M ^*$ to be the maximum of these distances:
\quad M^* = \max \{ d(x_1, p), d(x_2, p), ..., d(x_{N(\epsilon_0) - 1}, p) \}
• So if $1 \leq n < N(\epsilon_0)$ we have that $d(x_n, p) \leq M^*$ and if $n \geq N(\epsilon_0)$ then $d(x_n, p) < 1$. Let $M = \max \{ M^*, 1 \}$. Then for all $n \in \mathbb{N}$, $d(x_n, p) \
leq M$. So consider the open ball $B(p, M + 1)$. Then $x_n \in B(p, M + 1)$ for all $n \in \{1, 2, ... \}$ so:
\quad \{ x_1, x_2, ..., x_n, ... \} \subseteq B(p, M+1)
• Therefore $\{ x_1, x_2, ..., x_n, ... \}$ is a bounded set in $M$. $\blacksquare$ | {"url":"http://mathonline.wikidot.com/the-boundedness-of-convergent-sequences-in-metric-spaces","timestamp":"2024-11-06T08:09:01Z","content_type":"application/xhtml+xml","content_length":"16862","record_id":"<urn:uuid:4d833f63-5936-42f9-ba26-dedab38c41b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00085.warc.gz"} |
GTO Strategies :What is GTO in Poker? - Texas Hold‘em Poker
GTO Strategies :What is GTO in Poker?Welcome, aspiring players! Game Theory Optimal (GTO) is a term frequently mentioned in poker. But what exactly is GTO? Why is it so important? And what makes it
Introduction to Game Theory
In poker, GTO refers to Game Theory Optimal. It stems from the mathematical study of strategy interactions proposed by mathematician John Nash. Over the past few years, its application has shaped the
development of poker strategy. Therefore, to win high-stakes poker tournaments, players must arm themselves with knowledge of GTO principles.
“When people discuss GTO poker strategy, they refer to Nash Equilibrium strategies.”
Nash Equilibrium strategies are optimal because they cannot be exploited. All players cannot increase their expected value by changing their own strategies. These strategies can also be described as
Considering each player's possible hole card combinations, potential bet sizes, and subsequent possible card distributions, no-limit Texas Hold'em is a vast game. It is practically impossible for us
to manually deduce unexploitable strategies due to its sheer complexity. However, with the availability of poker software today, we can delve into GTO poker strategy.
GTO Strategies :What is GTO in Poker?
How to Find GTO Strategies
With “solver” software, we can design and run simulations to gather data on GTO strategies.
Solvers are powerful computer programs that take the following values as input:
• Preflop ranges for both players
• Community cards
• Target level of exploitability (accuracy)
• Initial pot size and chip depth
• Post-flop betting structures
Since the possibilities for bet sizes are nearly infinite while computer capabilities are limited, you must select a betting structure to provide the solver with enough strategic options to produce
meaningful output.
Input Parameters in the Solver
Once the simulation parameters are constructed, the solver iterates strategies for each player repeatedly. Players take turns exploiting each other until neither player's strategy can exploit the
other. At this point, we have found the GTO strategy.
The dynamic process of finding the GTO strategy between the button (advantageous position) and the blinds (disadvantaged position) is depicted in the figure below.
Initially, when both players are experimenting with new ideas, there are significant changes in strategies! The closer to balance, the smaller the magnitude of strategy changes, until reaching a
point where neither player can exploit the other by changing strategies. This animation is approximately 120 times the actual speed.
Dynamic Process to Reach Equilibrium. The strategy of the player in the disadvantaged position facing a 75% bet.
The Dilemma: Choosing GTO Strategy or Exploitative Strategy?
You may have heard of players being referred to as “exploitative” or “GTO” players. In reality, these two strategies are more like two sides of the same coin rather than opposing viewpoints.
If you don't know what the Game Theory Optimal looks like, how do you know if you are exploiting your opponents or if you are being exploited by them? With a deep understanding of GTO strategy, you
can play an invincible default strategy and accurately identify opponents' mistakes.
Poker software like GTO Wizard provides all post-flop strategies and summary reports, making it a great tool for studying GTO strategy.
By observing the data generated by the solver summarized by GTO Wizard, we can understand how GTO strategies use mixed strategies, different bet sizes, and balanced ranges in various situations. GTO
Wizard provides tools to understand the preferences for different bet sizes for each hand in different situations, as well as how to mix different hands into different bet sizes or more passive
strategies for balance and deception. Studying these strategies and reports will greatly help train your GTO intuition.
Why Should You Improve Your Strategy by Studying GTO Strategies?
So, how does honing your GTO intuition help exploit opponents?
Even when using GTO strategy, there are often cases of asymmetrical ranges, allowing players to take seemingly extreme actions. Here are some classic examples:
• Betting over pot size to attack capped ranges.
• Bluffing with all air cards when facing opponents' “give-up” policies.
• Folding all bluff-catching hands when opponents' ranges do not contain enough hands compatible with the chosen bet sizes.
If you know what your opponent's range should look like, how they deviate from balance, and understand how the solver attacks similar asymmetric ranges in other situations, that's enough to exploit
unbalanced opponents.
In summary:
• GTO can help you understand the baseline strategy.
• Once you understand this baseline, you'll know when to exploit and how to exploit opponents' mistakes.
• GTO achieves an unexploitable balanced strategy through Nash Equilibrium.
• GTO generates powerful strategies without relying on reads or intuition.
A deep understanding of GTO forms the foundation for adapting to any situation in a game and maximizing your win rate. Overall, through GTO Wizard, you can find an invincible default strategy and
develop robust counter-strategies after identifying opponents' mistakes. | {"url":"https://www.texas-holdem.poker/gto-strategies-what-is-gto-in-poker/","timestamp":"2024-11-07T20:42:44Z","content_type":"text/html","content_length":"141966","record_id":"<urn:uuid:9992aca0-0a6e-40d6-8abf-6ea6faebda29>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00242.warc.gz"} |
Introduction to Mathematical Philosophy
开始时间: 04/22/2022 持续时间: 8 weeks
课程主页: https://www.coursera.org/course/mathphil
Since antiquity, philosophers have questioned the foundations--the foundations of the physical world, of our everyday experience, of our scientific knowledge, and of culture and society. In recent
years, more and more young philosophers have become convinced that, in order to understand these foundations, and thus to make progress in philosophy, the use of mathematical methods is of crucial
importance. This is what our course will be concerned with: mathematical philosophy, that is, philosophy done with the help of mathematical methods.
As we will try to show, one can analyze philosophical concepts much more clearly in mathematical terms, one can derive philosophical conclusions from philosophical assumptions by mathematical proof,
and one can build mathematical models in which we can study philosophical problems.
So, as Leibniz would have said: even in philosophy, calculemus. Let's calculate.
Week One: Infinity.
Week Two: Truth.
Week Three: Rational belief.
Week Four: If-then.
Week Five: Confirmation.
Week Six: Decision.
Week Seven: Voting.
Week Eight: Quantum Logic. | {"url":"https://coursegraph.com/coursera_mathphil","timestamp":"2024-11-10T06:33:21Z","content_type":"text/html","content_length":"12084","record_id":"<urn:uuid:f8cb5671-ed61-4cd4-b0d8-2577a93e355a>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00223.warc.gz"} |
StatsExamples | Poisson Distribution Introduction
POISSON PROBABILITY DISTRIBUTION (INTRODUCTION)
Probability distributions are used in statistics to understand how likely certain events are. All we need to know is the mean number of individuals in the area we are considering or mean number of
events over that period of time. Then, if the assumptions of the binomial distribution are met, the Poisson probability distribution will give us the probability of seeing each particular number of
individuals or events. There are three conditions that must be met in order for a situation to be accurately modeled with a Poisson probability distribution.
• 1. The individuals or events occur randomly with respect to one another.
• 2. The individuals or events we are considering should be relatively rare compared to the potential number of observations.
• 3. The probability of an individual occurring in an area, or an event occurring within a time interval, is proportional to the size of the area or the length of time interval.
If these things are all true, and we know the mean number of individuals or events in a given period of time (
\( \mu \)
), we can use the following equation to calculate the probabilities: $$ Pr(x) = {{\mu^x e^{-\mu}} \over {x!}} $$ Where:
is the number of successes we are interested in.
is the probability of seeing x successes.
is Euler's constant, 2.718281828...
\( \mu \)
mean number of individuals or events in a given period of time.
Also, as a reminder, the symbol "!" represents a factorial. A factorial for a number is what we get when we take the number and multiply it by each whole number less than itself (e.g., 2! = 2x1=2, 4!
=4x3x2x1=24). By definition 0!=1. If we are using sample data we estimate \( \mu \) with the sample mean \( \bar{x} \) and the equation becomes: $$ Pr(x) = {{\mu^x e^{-\bar{x}}} \over {x!}} $$
Before we look at this equation in more detail, let's think of an example of how to use this equation. Imagine we are drawing cards from a deck of cards. We'll draw a card, note what it is, put it
back in the deck and draw again. If we do this 52 times we would expect to see each card once on average although obviously for any specific card we may see it more than once or not at all. Let's
identify a card like the Ace of Spades and calculate what the odds are that we would draw it no times, once, twice, etc. We can do this long hand by writing out the possible outcomes, but this would
quickly become way too big to keep track of. To use the equation above we use an overall mean number of observations of 1. This gives us the following: $$ Pr(0) = {{1^0 e^{-1}} \over {0!}} = {{1} \
over {e}} = 0.368 $$ $$ Pr(1) = {{1^1 e^{-1}} \over {1!}} = {{1} \over {e}} = 0.368 $$ $$ Pr(2) = {{1^2 e^{-1}} \over {2!}} = {{1} \over {2e}} = 0.184 $$ $$ Pr(3) = {{1^3 e^{-1}} \over {3!}} = {{1} \
over {6e}} = 0.061 $$ $$ Pr(4) = {{1^4 e^{-1}} \over {4!}} = {{1} \over {12e}} = 0.015 $$ $$ Pr(5) = {{1^5 e^{-1}} \over {5!}} = {{1} \over {60e}} = 0.003 $$ $$ Pr(>5) = 0.00059 $$ We can see that
the most likely number of observations is the mean which makes sense. We can also see that the probabilities trail off and become small fairly quickly. They never truly reach zero however. We can
also look at what this distribution looks like in a figure like the one here - the X-axis is for the number of events that satisfy our requirement (i.e., drawing an Ace of Spades) and the Y-axis
shows the probability of seeing each number of successes.
Keep in mind that while the Poisson distribution is used for situations with low individual probabilities (like the 1/52 chance for drawing a specific card above), if the potential number of
observations is very large the mean can be fairly high. For example, if we looked at a group of people and were interested in how many people are missing limbs then although the probability for each
person is small, the mean number of people we would observe that fit our criteria may be very high. This second figure shows the Poisson probability distribution with an extended X-axis and
represents the Poisson probabilities for how many people we expect to see that are missing limbs (including loss of fingers, hands, or toes) if we sampled sets of 10,000 people (i.e., small towns).
The graph is based on a real-world estimated rate of limb loss of 1/190 in the US (
) which would give us a mean of 10,000/190=5.26 to use for our equation. You can see how it's slightly asymmetric, but not as much as the previous figure, and the peak is around 5 or 6 as we expect.
There is a nice shortcut that allows quick calculation of Poisson probabilities once you know one of them. If you look at the equations for consecutive probabilities: $$ Pr(x) = {{\mu^x e^{-\mu}} \
over {x!}} $$ $$ Pr(x+1) = {{\mu^{(x+1)} e^{-\mu}} \over {(x+1)!}} $$ The second can be rearranged to show that it is just first equation, times the mean and divided by x+1, as so: $$ Pr(x+1) = {{\mu
^{(x+1)} e^{-\mu}} \over {(x+1)!}} = {{\mu \times \mu^{(x)} e^{-\mu}} \over {(x+1)x!}} = \left({{\mu } \over {x+1}}\right) \left({{\mu^{(x)}e^{-\mu}} \over {x!}}\right) = \left({{\mu } \over {x+1}}\
right) Pr(x)$$ For example, in our first example above the first probability was Pr(0)=0.368. Starting from that we would get: $$ Pr(1) = {{1} \over {1}} \times Pr(0) = (1) \times 0.368 = 0.368 $$ $$
Pr(2) = {{1} \over {2}} \times Pr(1) = {{1} \over {2}} \times 0.368 = 0.184 $$ $$ Pr(3) = {{1} \over {3}} \times Pr(2) = {{1} \over {3}} \times 0.184 = 0.061 $$ $$ etc. $$ Which you can see match the
values calculated above. Keep in mind that the 1/1, 1/2, and 1/3 fractions on the left in the calculations were the mean divided by the number of observations - if the mean had been 3.2 those values
would have been 3.2/1, 3.2/2, and 3.2/3. Using this method can greatly increase the speed of calculating Poisson probabilities. It can also allow you to calculate the full set of probabilities from
having any one of the probabilities (i.e., you can use the shortcut to go forwards and backwards).
The Poisson distribution has another very convenient mathematical property - its mean is the same as its variance. This property allows us to quickly determine whether a distribution that we think
might be a Poisson distribution is one. We can just calculate the mean and variance and compare them. If they are equal we can be confident (but not guaranteed) that the distribution is Poisson, but
if the mean and variance are different then we know the distribution is not Poisson. This comparison can be very useful because if we are counting observations that fulfill the assumptions at the top
of the page then the distribution should be Poisson - which means that if the distribution doesn't match then one or more of the assumptions isn't true for our data and we've learned something. For
example, imagine that we collected data for the number of amputees in a series of towns of size 10,000 across the US. Random factors will cause some of these towns to have fewer or more amputees
compared to the mean, but are there non-random factors too? If the risks of being an amputee are random with respect to one another and there are no consistent geographic risks then the distribution
should have an equal mean and variance. If we see a difference between the mean and variance however, this would indicate that risk of amputation is not random (perhaps a bad doctor) or that there
are geographic differences in risk (more sawmills in certain regions). To facilitate the comparison of the mean and variance in a distribution we think might be Poisson, a value called the
"coefficient of dispersion", CD, can be calculated. The CD is the variance divided by the mean: $$ CD = {{variance} \over {mean}} = {{s^2} \over {\bar{x}}} $$ When CD ~ 1.0 the distribution is highly
likely to be Poisson and the individuals or events are occurring randomly with respect to one another. There are conceivable cases in which the CD=1 while the distribution isn't Poisson, but these
are highly unusual. When CD < 1.0 we say the distribution is "uniform" because the variation is too regular, there is less variation than we expect. Thinking about our first example, this would be
like a situation in which we don't shuffle the deck of cards and instead place the card back on the bottom of the deck so we draw it every 52 cards like clockwork. When CD > 1.0 we say the
distribution is "clumped" because the variation is too uneven, the observations are more variable than we expect. Thinking about our second example, this would be like a situation in which the
military actively recruited from specific towns (resulting in more amputations) while ignoring others (resulting in fewer amputations) so the overall mean is not reflective of the same random process
across all the locations.
Connect with StatsExamples here
Slide 1
Welcome to this introduction to Poisson probability. We'll look at the Poisson probability equation, it's mathematical assumptions, and a few examples of applications. First things first though, This
distribution is named after Simeon Denis Poisson so it shouldn't be confused with the English word poison or the French word Poisson which means fish
Slide 2
The Poisson distribution comes from the binomial and it's a special case of the binomial under certain circumstances. If you're familiar with the binomial you'll remember that it describes the
probability of seeing X successes in N trials when the success probability is P and is given by the equation P of X equals "N choose X" times the probability of success to the X power times the
probability of failure to the N minus X power.
If you don't remember this you can watch our binomial probability videos on this same channel end playlist.
it turns out that when the number of trials gets large and the probability gets small, as a rule of thumb when N is greater than 100 and end times P is less than 10, the binomial probability equation
simplifies into the Poisson probability equation .
This equation gives the probability of seeing X successes as the mean number of successes raised to the power X, times E raised the power "negative mean" divided by X factorial.
The two equations shown are for when you have either the population mean or are using a sample mean to calculate your probability
Slide 3
If we look at the Poisson probability equation you'll see that it doesn't have the number of trials or probability of each trial directly in it.
Unlike the binomial which required a set number of trials and a known probability for each trials, the Poisson does not.
Poisson probabilities are used for the probability of seeing X events, or successes, in an area or over a set period of time when we know the mean number of observations or successes .
The potential number of observations can be huge, and even unknowable, so the binomial is not appropriate, but the mean number of observations is often easier to determine.
Slide 4
As mentioned, one typical scenario is calculating the probability of seeing a certain number of events in a given area
For example, if we know the average number of snails per square meter is 3 and we looked in a particular square meter, what are the odds of seeing none at all ?
Another typical scenario would be when we are interested in the probability of seeing events Over a set time.
For example, if we know the average number of deaths in a retirement home is 5 per month, what are the odds of seeing 10 in the same month?
Slide 5
The Poisson probability has three important assumptions.
First, the events occur randomly with respect to one another. This assumption comes directly from the binomial distribution independence assumption.
Second, the events are relatively rare. This is what allows us to replace the binomial equation with the Poisson equation.
Third, the probability of occurrence doesn't change overtime. This assumption comes directly from the binomial distribution's assumption of constant probability
In fact, one way to think about the Poisson distribution is as the limit of the binomial distribution as the probability of each event goes to 0 and the number of trials goes to Infinity.
Slide 6
The Poisson distribution has an incredibly useful property whereby the mean of the distribution is mathematically equal to its variance.
The entire distribution can therefore be specified with one value.
This is useful because this relationship can be used to test hypothesis about whether a distribution we observe is due to Poisson, that is random, process is or not.
If it is then the mean should be equal to the variance
This is usually tested using the coefficient of dispersion represented by the equation to the right where the coefficient of dispersion is equal to the variance divided by the mean and we're usually
interested in whether it is equal to 1 or not.
Slide 7
Let's look in more detail at the coefficient of dispersion and what it can tell us. it's equal to the variance divided by the mean and a Poisson distribution would have a coefficient of dispersion of
one. The top figure shows a Poisson distribution with the coefficient of dispersion equal to 1
If the variance is less than the mean then the coefficient dispersion is less than one and we term the distribution under-dispersed or uniform.
If we think about what that means in terms of numbers of observations, we see more samples with a number of observations closer to the mean than we expect and fewer samples with numbers of
observations far from the mean.
The numbers of successes in our samples are more consistent and similar to each other, which would look like the middle figure. There we see most of the samples having 3 or 4 or 5 observations and
very few having two or less or 7 or more.
If the variance is more than the mean then the coefficient dispersion is larger than one and we term the distribution over-dispersed or clumped.
If we think about what that means in terms of numbers of observations, we see fewer samples with a number of observations closer to the mean than we expect and more samples with numbers of
observations far from the mean.
The numbers of successes in our samples are not as consistent as we would expect if they were random, which would look like the middle figure. There we see that the samples have a wider range of
successes than predicted from a random process.
Slide 8
Another way of visualizing this is to think about what this would look like for our observations in space or time.
The middle figure shows a random distribution of individuals in the square and a random distribution of events along a timeline. We would get a coefficient of dispersion of one if we analyzed the
numbers of individuals in randomly chosen areas and the number of events in randomly chosen time periods.
The left figure shows a very nonrandom distribution of individuals or events where they are essentially equally spaced. The numbers of individuals in each region or events per time are much more
consistent than we would expect from random processes and we term this under-dispersed or uniform.
The right figure shows a very nonrandom distribution of individuals or events where they are separated into clusters. The numbers of individuals in each region or events per time are much less
consistent than we would expect from random processes and we term this over-dispersed or clumped.
Slide 9
Another useful property of the plus on distribution is that consecutive Poisson probabilities are related to each other.
If you look at our equation for the probability, we can separate out a mean in the numerator and the value of X in the denominator and bring that out to the front This would leave the mean raised to
one fewer power in the numerator and the factorial of X minus one in the denominator.
But the second part of that equation would be the Poisson probability 4X minus one observations.
therefore the Poisson probability for X is equal to the mean divided by X times the Poisson probability for X minus one.
Slide 10
For example, let's think about if the mean was equal to 3
Slide 11
if the mean is equal to three then the probability of seeing one occurrence is 3 raised to the first power times E to the negative 3 Divided by 1 factorial which would be 0.1494.
Slide 12
The probability of seeing two occurrences is 3 raised to the second power times E to the negative 3 / 2 factorial which would be 0.2240
Slide 13
But if we used our relationship previously, the Poisson probability for 2 should be equal to the mean divided by two, multiplied by the Poisson for 1. This is equal to 3 divided by 2 times the
Poisson probability for 1, which is equal to 3 divided by 2 times 0.1494, which is, in fact, equal to 0.2240.
Slide 14
OK, so what applications do we have for the Poisson probability?
One application is that if we know that a process of is random, and we have a mean, then we can predict the probabilities and proportions of numbers or observations
We could then go and measure the numbers of individuals in certain areas to determine whether they are located randomly or due to non-random factors. This sort of thing is done all the time in field
ecology for example, where scientists lay out transects and count the number of individuals in each region.
Slide 15
And it's not just for geographic locations that we can look at Poisson probabilities we can look at events overtime as well. If we know that a process is random and we have a mean we can predict
probabilities of numbers of events over time.
For example, at the time this video is being made COVID-19 is a brand-new virus still in the early stages of a global pandemic. At this time not much is known about this virus, in particular, how it
may be changing over time and why.
The figure shown is a phylogeny, a diagram indicating the ancestry and relatedness of different strains, of covid in different countries. The small circles indicate times when the genetic sequence of
that strain changed.
A very important question is whether those genetic changes are just occurring randomly or whether they are occurring non-randomly in response to natural selection. Is there evidence that these
genetic changes are because of nonrandom processes like improving its transmissibility or changing its lethality. Or are these changes just random genetic drift?
We can estimate the mean number of changes we see per amount of time, and use that to create predictions for how many changes we expect to see, and compare them to observations for the number of
changes we do see.
The Poisson probability provides us with a tool that we can use to better understand the evolution of a deadly disease.
Slide 16
For a final application we can think about one of the scenarios I mentioned at the very beginning of the video.
For example, we expect to see a certain number of deaths each month in a nursing home. The rate of deaths overtime should be constant and the deaths should be independent of one another which would
allow us to use the Poisson probability to predict how often we should see different numbers of people dying each month. when there is a month with what seems like an unusually high number of deaths,
we can use the poissant to figure out whether it's something we would expect due to random chance or whether there is evidence that there is a non-random factor, like a murderer, in the nursing home.
Comparing unusually large values or apparent patterns to our expectations from randomness has wide applications. For example, the Seti project, a project looking for extraterrestrial life, uses the
same sorts of procedures. When they see unusual patterns in their signals they don't instantly get excited, they compare them to how often they would expect to see unusual patterns coming from space.
Zoom out
Having a mathematical and unbiased way of analyzing unusual events is important if we want to make sure we don't get misled. The human brain is hardwired to over interpret noise as pattern and
respond emotionally to rare events. Having an approach like the Poisson allows us to mathematically determine whether what we see is genuinely unusual, or what we should expect to see from time to
End screen
Feel free to randomly press a button to show your appreciation.
Connect with StatsExamples here
This information is intended for the greater good; please use statistics responsibly. | {"url":"http://statsexamples.com/topic-poisson-probability-distribution-intro.html","timestamp":"2024-11-12T00:30:56Z","content_type":"text/html","content_length":"31396","record_id":"<urn:uuid:78ee145d-9385-43be-af42-0c3c446caf50>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00231.warc.gz"} |
Selecting events is not rewriting the history of events (in a continuous probability space)
Using the simple fact described in the title, we prove the existence of a computational problem with implications to Machine Learning, Quantum Mechanics and Complexity Theory. We also prove P!=NP
(the solution can be verified in time polynomial in the number of bits of the input and output (NP) but the problem cannot be solved in time polynomial in the number of bits of the input and output
(P)), but this claim still needs to be reviewed by experts in Complexity Theory.
1. Introduction
In this article we will be using the words “real” as in $\mathbb{R}$, “real-world” and “random” to avoid misunderstandings in contexts where we could (and perhaps should) just use the word “real”
instead of “real-world” and the word “non-deterministic” (in the Physics sense, not in the Complexity Theory sense) instead of “random”.
We can always select events with some feature without rewriting the history of events, in a standard probability space. In fact, in a continuous probability space we can select events such that a
random real variable $y\in [0,1]$ (with probability given by the Lebesgue measure) verifies $y=0$, but there is no complete history of events where $y=0$ (always as would be required, or even just
once) because the probability space is continuous by assumption and thus the event $y=0$ has null probability (only an interval would have non-null probability).
In this article we will show that this simple but non-trivial fact has profound implications not only to Complexity Theory[1][2][3][4] but also to Machine Learning[2] and Quantum Mechanics
(independently of the implications to the P vs. NP problem). Almost all (in the sense we will define in this article) real functions of a real variable cannot be computed for all practical purposes,
not even approximately. But a random selection allows computations in polynomial-time complexity involving the incomplete knowledge about a real function that cannot be computed in polynomial-time
complexity. This is a fundamental reason why we cannot exclude a random time-evolution: a deterministic time-evolution may exist, but it has so much complexity that it cannot be calculated for all
practical purposes, not even approximately (since $L^\infty$ is non-separable).
Note that whenever we deal with a non-separable space, there are issues with computability because some elements of a non-separable space cannot be approximated by a finite set of elements, up to an
arbitrarily small error. For instance, $L^\infty([0,1])$ and its dual space are both non-separable[6]. While there are separable spaces of real functions of real variables, whenever we add
uncertainties/probabilities to such spaces (which is often required when doing approximations) we tend to create non-separable spaces[6], unless equivalence relations change the space of functions.
For instance, the set $[0,1]$ is separable but the Lebesgue measure imposes that the rational numbers in $[0,1]$ can be discarded, despite that the sets $[0,1]$ including /excluding rational numbers
are different. In the same way, the smooth functions (or functions computable in a reasonable time) may be discarded from a space of functions for particular uncertainties/probabilities.
Using the simple fact mentioned above, we will also prove the existence of a computational problem (defined by a continuous probability space) whose solution can be verified in time polynomial in the
numbers of bits of the input and output (NP) but cannot be solved in time polynomial in the numbers of bits of the input and output (P), when using only a deterministic Turing machine[7][8]. That is,
The goal of this article is to define the specific problem unambiguously, and the level of mathematical details will be adjusted to that: too much mathematical detail would shift the focus from the
specific problem. Less detail does not always imply less mathematical rigor. Since the present author is an expert in Physics but not an expert in Complexity Theory, we will also try to prove the P
vs. NP as much as it is possible, but only as a secondary goal knowing that much work by experts in complexity theory is still required because it is likely that: 1) something went wrong in the
relation described here between the specific problem and the P vs. NP problem; and/or 2) the specific problem indeed can be used to solve the P vs. NP problem, but the proof presented here is
2. Complexity Theory in the context of probability theory
Complexity Theory can be studied in the context of probability theory[1][2][3][4], because many real-world problems require approximations and uncertainties not only due to the limitations of any
computer (already accounted for by Complexity Theory when using only finite numbers of bits, although there is also room for improvement here) but also due to the limitations of the measuring devices
of physical phenomena and limitations of the mathematical models used to approximate the real-world, for instance when dealing with real variables. Most uncertainties are not related to the computer
used and do not get smaller when increasing the number of bits in the computer. In fact, Physics as a science tries to be independent of Computer Science and vice-versa, as much as possible.
Probability theory is a language (or interface) that allows us to transfer a problem between two sciences (these two or others).
There are two possible approaches to errors or uncertainties[1][2][3][4]: average error (with respect to a probability measure) and maximal (except in sets of null measure with respect to a
probability measure) error. It turns out that both approaches can be defined using Hilbert spaces: the average error (that is, $L^2$ norm) is defined by a normalized wave-function (an element of the
Hilbert space) being the square-root of a probability density function; and the maximal error (that is, $L^\infty$ norm) is defined by an element of the abelian von Neumann algebra of operators on
the Hilbert space.
The average error is relevant because even if P!=NP it could still make no difference with respect to the scenario P=NP for many practical purposes, if every NP problem with a reasonable probability
distribution on its inputs could be solved in polynomial time-complexity on average on a deterministic Turing machine[9][10].
Moreover, since some functions are real constant functions, the definition of a real function must be consistent with the definition of a real number. Certainly, a natural definition of a real number
in the context of Complexity Theory uses a standard probability space. Non-standard probability spaces are rarely (or never) used in Experimental Physics, so it is not obvious how useful a “real
number” (or “real function”) defined in a non-standard probability space could be in real-world applications. There are only countable or continuous measures (or mixed) in a standard probability
space, then we can define exactly only a countable number of real numbers (usually the rationals, but not necessarily), the remaining real numbers can only be constrained to be inside an interval
with a finite width, eventually very small but never zero. We should also define all real functions using a standard probability space, unless we find a fundamental reason not to do it (we will not
find it in this article).
We require a continuum standard probability space, since we can always define a regular conditional probability density which implements a selection of events in such probability space[11]. For
instance, this is what we do when we neglect the intrinsic computation error (due to cosmic rays and many other reasons), we define a deterministic function by selecting only certain events from a
complete history of random events. It is well known since many decades that in an infinite-dimensional sphere of radius $1$ (subset of a real Hilbert space) there is a uniform prior measure induced
by the $L^2$-distance in the Hilbert space. Every point in the sphere has null measure, only regions of the sphere with non-null distance between some of its points may have non-null measure
(compatible with the uniform prior measure). This implies that any knowledge (compatible with the uniform prior measure on the sphere) about a real normalized wave-function has necessarily
uncertainties, defined by a connected region in the sphere with non-null maximum $L^2$ distance to some wave-function, however small it might be.
The discrete nature of the Turing machine is certainly compatible with a continuous probability space: the number of bits of the input or output can be arbitrarily large, and it is proportional to
the logarithm of the resolution of the partition of the interval $[0,1]$, with each disjoint set of the partition corresponding to a different binary number. Excluding a continuous cumulative prior
would be unjustified, for many reasons including: no prior is better for all cases[13], there are many problems where a step cumulative prior would not fit well (for instance there is no uniform
measure for the rationals in $[0,1]$ only for the reals); it is hard to formulate any real-world problem where only a step cumulative prior is used (think about the numbers $\pi$ or $\sqrt{2}$ in
numerical approximations, for instance), we usually use a mixture of step and continuous cumulative priors; we can map an ensemble of discrete random variables one-to-one to the real numbers, for
instance an ensemble of fair coins corresponds to the uniform real measure in the interval $[0,1]$; also any real-world computer has an intrinsic computation error (due to cosmic rays and many other
reasons) which is usually very small, but it cannot be eliminated. Thus, while we can formulate a new unsolved version of the P vs. NP problem where only a step cumulative prior is accepted, such
version of the problem has little to do with real-world computers and real-world problems.
Note that there are $2^{2^n}$ different boolean functions on $n$ boolean variables, Shannon proved that almost all Boolean functions on $n$ variables require circuits of size $\mathcal{O}(2^n/n)$
[14], thus the time complexity of almost all Boolean functions on n variables for a Turing machine is at least (and at most, for all Boolean functions) $\mathcal{O}(2^n/(n\log(n)))$[15] which is not
polynomial in $n$. Thus, almost all numerical functions are not in the complexity class $P$, according to the uniform prior measure for any resolution of the partition. Moreover, the uniform prior
measure is compatible with a prior measure which excludes (both in the maximal and in the average error approaches) all real functions which are approximated by numerical functions with complexity
class $P$, for all resolutions bigger than some resolution of the partition.
3. Definition of the problem
Consider three measure spaces $X=Y=[0,1]\in \mathbb{R}$ and $X\otimes X$ with the Lebesgue measure, corresponding to inputs ($X$ or $X\otimes X$) and an output ($Y$) of real functions. Given an input
in $X$, we define a regular conditional probability density which is a function from $X\otimes X\to Y$, given by the probability that a function for an input in $X$ some constant output $y\in Y$. But
there is always also a marginal probability density for $Y$, and we cannot say without uncertainty which is the output, because the corresponding prior probability density would be incompatible with
the prior Lebesgue measure (by the Radon–Nikodym theorem). Thus, the regular conditional probability density is a deterministic selection of events which cannot be a complete history of events.
In a standard measure space it is always possible to define regular conditional probabilities[11] and to choose the probability density $p(x)=p_0(x)>0$ for all $x\in X$, except in sets with null
measure. Thus, we will define $p(x\otimes y)=p(y|x)p_0(x)$ the joint probability density for the tensor product $X\otimes Y$ for a particular $p(x)=p_0(x)>0$ for all $x\in X$, except in sets with
null measure. Then, we can obtain any other joint probability density $p(x\otimes y)$ from the expression $p(y|x)p(x)=\frac{p(x\otimes y)}{p_0(x)}p(x)$.
The following results are valid for a random input in the interval $[0,1]$ (which is a standard probability space) and also for an input (or output) without uncertainties up to sets with null measure
with respect to the prior marginal measure of the input (or output), because we use regular conditional probabilities (which always exist in standard probability spaces[11]) for fully known inputs
(or outputs). This is crucial, since the input includes two samples from a uniform distribution in $[0,1]$ which may generate numerical functions in $P$ when the sample is in a set of null measure.
However, a (continuous cumulative) probability distribution does not contain enough information to unambiguously define a function. On the other hand, a real wave-function whose square is the joint
probability distribution allows the definition of a unitary operator on a separable Hilbert space. A unitary operator is a random generalization of a deterministic symmetry transformation of a
(countable or continuous) sample space. Any unitary operator defined by a wave-function of two continuous variables cannot be a deterministic symmetry transformation (for similar reasons that a
continuous probability distribution cannot unambiguously define a function).
Since $p(x\otimes y)\geq 0$ then there is always a normalized wave-function $\Psi\in L^2(X\otimes Y)$ such that $|\Psi(x\otimes y)|^2=p(x\otimes y)$. Note that the Koopman-von Neumann version of
classical statistical mechanics[16] defines classical statistical mechanics as a particular case of quantum mechanics where the algebra of observable operators is necessarily commutative (because the
time-evolution is deterministic). In an infinite-dimensional sphere of radius $1$ (subset of a real Hilbert space) there is a uniform prior measure induced by the $L^2$-distance in the Hilbert space.
We choose a prior measure (compatible with the uniform prior measure) which excludes all real functions which are approximated by numerical functions with complexity class $P$, for a high enough
resolution of the partition.
Given an input in $([0,1])^2$ (the input consists of two samples from a uniform distribution in the interval $[0,1]$, imported from ANU QRNG [17] for instance) and a candidate output in $[0,1]$ the
wave-function uniquely defines a random symmetry transformation. Such symmetry transformation is not lacking information since it can be inverted (the non-unitary isometries have null prior measure).
The cumulative probability distribution is given by the integral of the modulus squared of the wave-function in the corresponding region of the sample space.
From the cumulative marginal probability distribution, such that $Y$ is fully integrated we determine $x\in X$ using the first sample from the uniform distribution in $[0,1]$. From the cumulative
conditional probability distribution with the condition that $x\in X$ is what we determined previously, we determine $y\in X$ using the second sample from the uniform distribution in $[0,1]$. We
apply the inverse-transform sampling method[18], that is check in the interval of the partition defined by the bits corresponding to $X$ and $Y$, whether the cumulative distribution crosses the
sample from the uniform distribution. This defines the deterministic verification of the candidate output corresponding to the input, in agreement with the Born’s rule of Quantum Mechanics.
Note that the resulting deterministic function is not necessarily invertible, because the collapse of the wave-function is irreversible (unless the symmetry transformation would be deterministic,
which is excluded in this case because we have a continuous probability space of functions).
4. The classical Turing machine defined as a Quantum computer
The Turing machine can be equivalently defined as the set of general recursive functions, which are partial functions from non-negative integers to non-negative integers[19]. But the set of all
functions from non-negative integers to non-negative integers is not suitable to define a measure, since they form an uncountable set, in a context where the continuum is not defined. Moreover, the
general recursive functions are based in the notion of computability (that the Turing machine halts in a finite time), but computability does not hold in the limit of an infinite number of input
bits, thus to study such limit we need to define uncomputable functions somehow (we will use complete spaces, where Cauchy sequences always converge to an element inside the space).
On the other hand, it is widely believed (and we will show in the following) that any computational problem that can be solved by a classical computer can also be solved by a quantum computer and
vice-versa. That is, quantum computers obey the Church–Turing thesis. Note that it is well known that some circuits (classical hardware) provide exponential speedups when compared with some other
circuits in some functions (because the input bits can be reparametrized, this is why the time complexity of a function has an upper bound, but it is not known how to establish a lower bound; it is
also consistent with the fact that the halting problem is undecidable, that is, given an arbitrary function from integers to integers and an arbitrary input, we cannot determine if the output of such
function is computable or not), thus the fact that a classical Turing machine can be defined as a Quantum computer is compatible with the fact that quantum computers provide exponential speedups when
compared with some classical computers in some functions.
We start by noticing that the domain of a general recursive function can be defined by a dense countable basis of a particular Hilbert space which is the (Guichardet) $L^2$ completion of the set of
all finite linear combinations of simple tensor products of elements of a countable basis of a base Hilbert space, where all but finitely many of the factors equal the vacuum state[20] (like in a
Fock-space, but without the symmetrization). But the unitary linear transformations on a normalized wave-function are not necessarily the most general transformations of the probability measure
corresponding to the wave-function. Because of that, we build a Fock-space where the base Hilbert space is the previous Guichardet-space, then the unitary transformations on this
Fock-Guichardet-space allow us to implement the most general transformations of a probability measure, corresponding to a normalized wave-function in the base Guichardet-space. Note that a countable
basis of the Guichardet-space is already made of simple tensor products, and the simple tensor product is associative, thus the Fock-Guichardet-space is isomorphic to the Guichardet-space, but we
still prefer to use the Fock-Guichardet-space due to the existence of standard tools for Fock-spaces.
Note that the input of a general recursive function is a finite number of integers, but its output is only one integer. However, any function which outputs several integers is a direct sum of
functions which output one integer. The other way around is also true, once we define a vacuum state (included in the Fock-Guichardet-space), that is, a function which outputs one integer is a
particular case of a function that outputs several integers where all outputs except one correspond to the vacuum state. Thus, we can consider only unitary automorphisms of the Fock-Guichardet-space.
To be able to define a measure, we make the integers correspond to the (countable) step functions with rational endpoints in the interval $[0,1]$ and weights which are plus or minus the square root
of a rational number[21]. The vacuum state is the constant positive function with norm 1, and it corresponds to the integer $0$. We eliminate duplicated step functions in the correspondence with the
integers, for instance if two neighbor intervals have the same weight then they are fused.
Then, the limit of infinitesimal intervals is well-defined, and it is defined by an element of $L^2([0,1])$. Since the general recursive functions are partial functions, then they are a particular
case of partially-defined linear operators $L^2([0,1])\to L^2([0,1])$, and we can define the base Hilbert space of the Fock-Guichardet-space as $L^2([0,1])$.
5. Worst case prior measure, rational functions and radical determinism
In a standard probability space, there are only continuous and/or countable measures. However, these may be mixed in an arbitrary way. For a theory of Physics we could choose the best case prior
measure (as we did in the previous sections), since we just want to find a prior which is consistent with the experimental data, without many concerns about alternative priors. However, in
Cryptography we need guarantees that our limits are robust under arbitrary choices, so we need to assume the worst case prior measure.
The previous sections could also be made compatible with the worst case prior measure, if we had a computer capable of comparing real numbers not rational. That would be acceptable for a theory of
Physics, but it would make it difficult to obtain guarantees for Cryptography.
It is also difficult to guarantee true randomness is real-world applications of Cryptography. Since probabilities only mean incomplete information, we can use Probability Theory in the context of
radical determinism (where nothing is random). For Cryptography, we need the worst case prior measure, rational functions and radical determinism.
So we start by eliminating a non-standard probability measure: any probability theory is a universal language (like English or mathematical logic) to define abstract models of the objects we want to
study. A standard probability theory is universal and irreducible, meaning that it has the minimal content to be considered a probability theory (in agreement with Quantum Mechanics and Experimental
Physics, for instance). If the non-standard probability theory is also irreducible, then the corresponding models are equivalent, and we can use the standard version without loss of generality. This
allows us to transfer models between different sciences. But often the non-standard probability theory is reducible, this means that the boundary between model and probability theory is not where it
would be in the standard case and there are properties that we are attributing to the probability theory that in fact belong to the model.
Thus, we should assume a standard probability theory and leave some flexibility in the definition of the computer model and not the other way around, as it often happens in Complexity Theory where
there are strict axioms for different computer models, while asymptotic limits are taken without defining the probability space, which is recipe to end up with mathematical results and questions
which are hard to transfer to experimental physics and many other sciences.
In the context of radical determinism, the history of events is a non-random countable sequence of events. Thus, some events with null measure might happen, due to radical determinism. But only a
countable number of such events. That means that a continuous measure is only truly continuous up to a countable number of points, this possibility is already considered by the worst case prior
Consider now a boolean function of a countable infinite number of bits. The time complexity of almost all Boolean functions on n variables for a Turing machine is at least (and at most, for all
Boolean functions) $\mathcal{O}(2^n/(n\log(n)))$[14] [15] which is not polynomial in $n$, but the same boolean function has different time complexities in different circuits[14] (because the input
bits can be reparametrized). Thus, the time complexity of a function depends on the circuit (computer design). Also, an algorithm with polynomial time-complexity is not guaranteed to be fast (due to
large order and coefficients of the polynomial), thus only the asymptotic behavior is fast, and so we cannot put an upper bound on the number of bits of the input. But the arbitrarily large number of
bits of the input introduces another ambiguity: given any problem with any time complexity (exponential $\mathcal{O}(2^n)$, for instance) there is a problem with linear time-complexity in the number
of bits that takes the same amount of time to run. Thus, the polynomial time-complexity is faster than exponential time-complexity in asymptotic behavior, only for the same number of bits of the
input. This is a condition without much meaning when the input has a countable infinite number of bits.
What we can say is that the countable (or mixed) measure allows defining functions that eventually have polynomial time complexity (that is, not necessarily non-polynomial time complexity). This
contrasts with the necessarily non-polynomial time complexity of the functions defined by the continuous measure. In the following we will show that, given any mixed prior measure, we can always
redefine the problem to have a continuous prior measure such that its results cannot be reproduced by a mixed prior measure, effectively converting the worst case prior measure into the best case
prior measure.
The prior measure must be mixed or continuous, to allow for the limit of an infinite number of bits of all computable functions. But we must prove that $Peq NP$ in a single deterministic Turing
machine (one with a continuous prior measure, for instance), not in the set of all possible deterministic Turing machines. That would not be possible, since for any computable function $f$ (including
any function in $NP$) there is a deterministic Turing machine where $f$ is in $P$, by reparametrizing the input bits.
It is not obvious if we can prove $Peq NP$ in any single deterministic Turing machine, or just in one particular deterministic Turing machine. However, given a worst case prior measure (thus mixed
measure), there is a subset of the input random sample which also implements a deterministic Turing machine and where the prior measure is continuous, where $Peq NP$, thus it becomes legitimate to
claim that this fact already shows that $Peq NP$. Moreover, using subsets of the input random sample (thus, regular conditioned probability) we can create any other prior measure, because any abelian
von Neumann algebra of operators on a separable Hilbert space is *-isomorphic to exactly one of the following:
• $l^\infty(\{ 1 , 2 , ... , n \} ) , n \geq 1$
• $l^\infty(\mathbb{N})$
• $L^\infty([0,1])$
• $L^\infty([0,1]\cup\{ 1 , 2 , ... , n \} ) , n \geq 1$
• $L^\infty([0,1]\cup \mathbb{N})$.
Equivalently, a standard probability space is isomorphic (up to sets with null measure) to the interval $[0,1]$ with Lebesgue measure, a finite or countable set of atoms, or a combination (disjoint
union) of both.
Thus, for the worst case prior measure if we include all subsets of the input random sample, then using two integers (which is countable) we include a countable set of deterministic Turing machines $
\{k\}$ and countable functions $f_{k,n}$, one machine for each countable function $f_{k,1}$ such that $f_{k,1}$ is in $P$ and $f_{k,n}=f_{n,k}$, then we cannot prove $Peq NP$ (in fact we would
conclude that $P=NP$, due to the possibility of reparametrizing the input bits).
Then, we can only prove $Peq NP$ in one particular deterministic Turing machine and not in any single deterministic Turing machine.
Given any mixed prior measure, there is an interval of the input random sample where it is continuous. We rescale to $[0,1]$ all intervals corresponding to such interval where the mixed measure is
continuous, using conditioned probability. The results in (the new interval of the input random sample) $[0,1]$ cannot be fully reproduced by any other measure which is mixed in $[0,1]$.
Given any other measure which is mixed in $[0,1]$, there is an interval with rational endpoints of the input random sample where there is a finite difference between the two cumulative probability
distributions, otherwise both would be continuous. This translates into two different averages of $y$ in an interval (for $x$) of a partition of $[0,1]$, separated by a finite difference.
Then the indicator function of a $y$ corresponding to the continuous measure, in the interval (for $x$) of the partition of $[0,1]$, is in $NP$ (more precisely, it can be extended to be in $NP$,
since we only defined for $x$ in a subset of $[0,1]$) but it cannot be reproduced by the mixed measure. It cannot be reproduced by the continuous prior measure either, since a function constant in
$x$ in a finite interval has null measure (for a continuous prior measure). Note that the function that corresponds to the indicator function above defined, is a function constant in $x$ in a finite
Note that a continuous prior measure admits a regular conditional probability density, which allows us to define a selection (verification) of a candidate output. The verification of a constant
output is in $NP$ and thus in a mixed measure, but it requires one more input (the candidate output) and thus it is compatible with a continuous measure of functions of $x$. That is, the measure is
overall mixed, being continuous only for functions with one input.
6. P$eq$NP
When using a computer to solve the problem defined in the previous section, any disjoint set of the partition is an interval. This gives us two options and two options only, either the selection of
events is fully deterministic or only approximately deterministic:
1. The selection of events is fully deterministic. Then we impose a condition on the output $Y$ of any wave-function in the sphere, this defines a regular conditional probability density for the
input $X$ conditioned on a constant rational output $y$. As shown in the previous section, there is a rational $y$ corresponding to the continuous measure, in an interval with rational endpoints
(for $x$) of the partition of $[0,1]$, which it cannot be reproduced by a countable prior measure. It cannot be reproduced by the continuous prior measure either, since a function constant in $x$
in a finite interval has null measure (for a continuous prior measure). That is, there is no function in $P$ corresponding to the indicator function for $y$, which is in $NP$ (more precisely, it
can be extended to be in $NP$, since we only defined it for $x$ in a subset of $[0,1]$). This implies $Peq NP$.
2. The selection of events has some randomness, as small as we want. Then we do not impose a condition on the output (eventually we impose a condition on the input $X$, depending on whether we want
fixed or averaged input). In a strict interpretation of the $P$ vs. $NP$ problem this option is excluded by definition since the official formulation assumes that both the verification and the
solution of the problem are both fully deterministic. This already implies $Peq NP$, in a strict interpretation.
Note that a complete history of events needs to be countable, so that we can convert it into a single event (mapping complete histories of events one-to-one to the real numbers in the interval $[0,1]
$, for instance). We could also define a density (that is, yet to be integrated, using the disintegration theorem[22]) of an event. Such density is a regular conditional probability, since regular
conditional probabilities always exist in standard probability spaces[11]. But a density cannot correspond to a single event (by definition) and thus it cannot be considered a complete history of
This proof is dependent on the fact that the prior measure is continuous. If it in part continuous, and in part countable, then we can choose just the continuous part of the sample space (see the
previous section). While we can use a countable part of the sample space to approximately solve a continuous problem, and a continuous part of the sample space to solve a countable problem, we cannot
change the prior measure from continuous to countable or vice-versa (by the Radon-Nikodym theorem), because there is no Radon-Nikodym derivative between the two measures, since the sets of null
measure are disjoint between the two measures. The prior measure defines the physical world where the computer exists, thus it cannot be removed from any complete computer model related to a physical
Note that in the first paragraph of the official statement of the P vs. NP problem[7], it is stated:
To define the problem precisely it is necessary to give a formal model of a computer. The standard computer model in computability theory is the Turing machine, introduced by Alan Turing in 1936.
Although the model was introduced before physical computers were built, it nevertheless continues to be accepted as the proper computer model for the purpose of defining the notion of computable
As for any other proof, this proof is only as good as the axioms used (that is, assumptions). The computer model used for a solid proof of the P vs. NP problem should be widely accepted as a good
approximation to a physical computer for the purpose of defining the notion of computable function. We believe our computer model is accepted by most experts in Physics (as argued in the previous
section). We claim that our computer model makes no more assumptions than those required by the official statement[7] (including the deterministic Turing machine), and it is as close to a physical
computer as possible, by today standards. Assuming a countable prior measure is not realistic (as argued in the previous sections, for instance it would exclude an ensemble of fair coins).
However, we believe that allowing a random selection of events is even more realistic (as discussed in the previous sections, also with implications to Machine Learning and Quantum Mechanics). In the
next section, we will define a selection of events which has some randomness (as small as we want) and prove that even in that case, we still have P$eq$NP.
7. Realistic version of the problem (still P$eq$NP)
A selection of events which is only approximately deterministic can be approximated by a step function (step functions are dense in $L^2$) and thus there is a square with non-null constant measure.
We rescale such square to $[0,1]\times[0,1]$. We then consider a real polynomial wave-function that is near the point in the sphere corresponding to a constant wave-function, up to an error in the $L
^2$ norm which can be as small as we want because the polynomials are dense in $L^2$ (the corresponding numerical polynomial does not need to be in $P$).
The first sample from the uniform distribution defines directly $x\in X$. An approximation (in the $L^2$ norm) with polynomial time-complexity to the selection function, is defined by setting $y\in
Y$ equal to the second sample from the uniform distribution.
Since the wave-function is polynomial non-constant, then the corresponding cumulative probability distribution minus the second sample is strictly crescent (except in sets of null measure). Thus,
when we define the corresponding deterministic function we can choose the second sample which produces an output which is as far from zero as we want (in the interval $[0,1]$), because we are using
the $L^\infty$ norm now. We cannot average over the random sample, otherwise we need a random computer (see next section). Thus, no approximation is possible, and it suffices that we define a
partition for the output which has two disjoint sets (the measures of the sets are arbitrary, as long as they are non-null) and a numerical output with one bit. Then, almost all numerical functions
are not in the $P$ class, according to the prior measure.
8. Generation of random numbers has linear time-complexity
The two random samples from a uniform distribution in the interval $[0,1]$ are inputs, in the deterministic Turing machine. However, in the real-world these samples need to be generated somewhere and
in polynomial time-complexity, otherwise the time complexity of the random selection computed by the real-world random computer could be non-polynomial in the number of bits of the random samples.
Moreover, it would be better if the generation of random samples had linear time complexity, since then we could do a constant rate of experiments over time to validate the probability distribution
of the random selection, otherwise it would be impractical to generate an infinite sequence of experiments.
We cannot prove this mathematically (since we would need more axioms). However, the implications of this article to Quantum Mechanics help to clarify the source of randomness of Quantum Mechanics
(and thus of the random samples). It is relevant to verify empirically that the generation of random numbers with linear time complexity is possible, for all practical purposes. We can visually check
on the website from ANU QRNG that the number of bits of the random sample grows linearly in time and any complete history of events converges to a uniform probability distribution.
Moreover, the entropy is maximal, in the sense that the deterministic function needed to correlate the bits is not computable for all practical purposes, not even approximately (since $L^\infty$ is
non-separable) according to the prior measure.
9. On the consequences to Machine Learning
In the introduction we discussed the implications (of the results of this article), which are common to Machine Learning and Quantum Mechanics. But Machine Learning (for instance Deep Neural
Networks) is not firmly based in probability theory, unlike Quantum Mechanics, then there are more consequences.
In Machine Learning, methods inspired by probability theory are used often[2], but the formalism is based in approximations to deterministic functions, guided by a distance (or equivalently, an
optimization problem) and not a measure. In fact, two of the main open problems are the alignment of models and the incorporation of prior knowledge[23], which could be both well solved by a prior
measure if there would be any measure defined.
Our results imply that under reasonable assumptions, almost all functions are not computable not even approximately. Thus, Machine Learning works because the functions we are approximating are in
fact probability distributions (eventually after some reparametrization[24]). This shouldn’t be surprising, since Classical Information Theory shows (under reasonable assumptions) that probability is
unavoidable when we are dealing with representations of knowledge/information[1][2]. But in Machine Learning the probability measure is not consistently defined (despite that many methods are
inspired by probability theory), the probability measure emerges from the approximation[24] and often in an inconsistent way. The inconsistency is not due to a lack of computational power since
modern neural networks can fit very complex deterministic functions and fail badly[25][26] in relatively simple probability distributions (e.g. catastrophic forgetting or the need of calibration to
have some probabilistic guarantees[26]).
This unavoidable emergence of a probability measure should be investigated as a potential source of inefficiency, inconsistency and even danger. If the emergence of a probability measure is
unavoidable, why don’t we just define a probability measure in the formalism consistently? Many people say “it is how our brain works”, so mathematics should step aside when there is empirical
But the empirical evidence is: oversized deep neural networks still generalize well, apparently because often the learning process converges to a local maximum (of the optimization problem) near the
point where the learning begun[27]. This implies that if we repeat the learning process with a random initialization (as we do when we consider ensembles of neural networks[25][28]), then we do not
expect the new parameters to be near any particular value, regardless of the result of the first learning process. This expectation is justified by the fact that every three layers of a wide enough
neural network is a universal approximator of a function[29], so any deviation introduced by three layers can be fully corrected in the next three layers, when composing dozens or hundreds of layers
as we do in a deep neural network. Then the correlation between the parameters corresponding to different local maximums converges to zero, when the number of layers increases.
Thus, there is empirical evidence that oversized deep neural networks still generalize well, precisely because a prior measure emerges: deep learning does not converge to the global maximum and
instead to one of the local maximums chosen randomly, effectively sampling from a prior measure in the sample space defined by all local maximums. This is consistent with the good results achieved by
ensembles of neural networks[25][28], which mimic many samples. However, it is a prior measure which we cannot easily modify or even understand, because the measure space is the set of all local
maximums of the optimization problem. But, since we expect the parameters to be fully uncorrelated between different local maximums, then many other prior measures (which we can modify and
understand, such as the uniform measure) should achieve the same level of generalization.
This is not a surprise, since oversized statistical models that still generalize well were already found many decades ago by many people[12]: a standard probability space with a uniform probability
measure can be infinite-dimensional (the sphere[12] studied in this article, for instance).
More empirical evidence: no one looks to a blurred photo of a gorilla and says with certainty that it is not a man in a gorilla suit. We all have many doubts, when we are not sure about a subject we
usually express doubts through the absence of an action (not just us, but also many animals), for instance we don’t write a book about the subject we don’t know about.
There is no empirical evidence that our brain tries to create content which is a short distance from content (books, conversations, etc.) created under exceptional circumstances (when doubts are
minimal). When we are driving, and we do not know what is in front of us, we usually just slow down or stop the car. But what content defines “not knowing”? Is there empirical evidence about the
unknown? The unknown can only be an abstract concept, expressed through probability theory or a logical equivalent. Is there empirical evidence that probabilities are reducible, that there is a
simpler logical equivalent? No, quite the opposite.
The only trade-off seems to be between costs (time complexity, etc.) and understanding/control. A prior measure which we understand and/or control may mean much more costs than an emergent (thus,
inconsistent and uncontrollable) prior measure which just minimizes some distance. But this trade-off is not new, and it is already present in all industries which deal with some safety risk (which
is essentially all industries). Distances are efficient for proof of concepts (pilot projects), when the goal is to show that we are a short distance from where we want to be. But safety (as most
features) is not being at a short distance from being safe. “We were at a short distance from avoiding nuclear annihilation” is completely different from “we avoided nuclear annihilation”. To avoid
nuclear annihilation we need (probability) measures, not only distances. | {"url":"https://timepiece.pubpub.org/pub/pnp/release/2","timestamp":"2024-11-15T04:20:47Z","content_type":"text/html","content_length":"618209","record_id":"<urn:uuid:2db7e63c-6c7e-409f-b270-5feefc96009f>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00607.warc.gz"} |
Application of evolutionary multiobjective algorithms for solving the problem of energy dispatch in hydroelectric power plants
The Brazilian population increase and the purchase power growth have resulted in a widespread use of electric home appliances. Consequently, the demand for electricity has been growing steadily in an
average of 5% a year. In this country, electric demand is supplied predominantly by hydro power. Many of the power plants installed do not operate efficiently from water consumption point of view.
Energy Dispatch is defined as the allocation of operational values to each turbine inside a power plant to meet some criteria defined by the power plant owner. In this context, an optimal scheduling
criterion could be the provision of the greatest amount of electricity with the lowest possible water consumption, i.e. maximization of water use efficiency. Some power plant operators rely on
“Normal Mode of Operation” (NMO) as Energy Dispatch criterion. This criterion consists in equally dividing power demand between available turbines regardless whether the allocation represents an
efficient good operation point for each turbine. This work proposes a multiobjective approach to solve electric dispatch problem in which the objective functions considered are maximization of
hydroelectric productivity function and minimization of the distance between NMO and “Optimized Control Mode” (OCM). Two well-known Multiobjective Evolutionary Algorithms are used to solve this
problem. Practical results have shown water savings in the order of million m^3/s. In addition, statistical inference has revealed that SPEA2 algorithm is more robust than NSGA-II algorithm to solve
this problem.
Original language English
Title of host publication Evolutionary multi-criterion optimization
Subtitle of host publication 8th International Conference, EMO 2015, Guimarães, Portugal, March 29 --April 1, 2015. Proceedings, Part II
Editors António Gaspar-Cunha, Carlos Henggeler Antunes, Carlos Coello Coello
Place of Publication Cham (CH)
Publisher Springer
Pages 403-417
Number of pages 15
ISBN (Electronic) 978-3-31915891-4
Publication status Published - 2015
Event 8th International Conference on Evolutionary Multi-Criterion Optimization - Guimarães, Portugal
Duration: 29 Mar 2015 → 1 Apr 2015
Publication series
Name Lecture Notes in Computer Science
Publisher Springer
Volume 9019
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Conference 8th International Conference on Evolutionary Multi-Criterion Optimization
Abbreviated title EMO 2015
Country/Territory Portugal
City Guimarães
Period 29/03/15 → 1/04/15
• energy efficiency
• multiobjective optimization
• NSGA-II
• SPEA2
Dive into the research topics of 'Application of evolutionary multiobjective algorithms for solving the problem of energy dispatch in hydroelectric power plants'. Together they form a unique | {"url":"https://research-test.aston.ac.uk/en/publications/application-of-evolutionary-multiobjective-algorithms-for-solving","timestamp":"2024-11-12T18:34:52Z","content_type":"text/html","content_length":"64290","record_id":"<urn:uuid:e7eee249-c656-413a-932a-22343d8e4c08>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00715.warc.gz"} |
PPT - Multiple Regression – Basic Relationships PowerPoint Presentation - ID:229875
1. Multiple Regression – Basic Relationships Purpose of multiple regression Different types of multiple regression Standard multiple regression Hierarchical multiple regression Stepwise multiple
regression Steps in solving regression problems
2. Purpose of multiple regression • The purpose of multiple regression is to analyze the relationship between metric or dichotomous independent variables and a metric dependent variable. • If there
is a relationship, using the information in the independent variables will improve our accuracy in predicting values for the dependent variable.
3. Types of multiple regression • There are three types of multiple regression, each of which is designed to answer a different question: • Standard multiple regression is used to evaluate the
relationships between a set of independent variables and a dependent variable. • Hierarchical, or sequential, regression is used to examine the relationships between a set of independent
variables and a dependent variable, after controlling for the effects of some other independent variables on the dependent variable. • Stepwise, or statistical, regression is used to identify the
subset of independent variables that has the strongest relationship to a dependent variable.
4. Standard multiple regression • In standard multiple regression, all of the independent variables are entered into the regression equation at the same time • Multiple R and R² measure the strength
of the relationship between the set of independent variables and the dependent variable. An F test is used to determine if the relationship can be generalized to the population represented by the
sample. • A t-test is used to evaluate the individual relationship between each independent variable and the dependent variable.
5. Hierarchical multiple regression • In hierarchical multiple regression, the independent variables are entered in two stages. • In the first stage, the independent variables that we want to
control for are entered into the regression. In the second stage, the independent variables whose relationship we want to examine after the controls are entered. • A statistical test of the
change in R² from the first stage is used to evaluate the importance of the variables entered in the second stage.
6. Stepwise multiple regression • Stepwise regression is designed to find the most parsimonious set of predictors that are most effective in predicting the dependent variable. • Variables are added
to the regression equation one at a time, using the statistical criterion of maximizing the R² of the included variables. • When none of the possible addition can make a statistically significant
improvement in R², the analysis stops.
7. Problem 1 - standard multiple regression In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic? Assume that there is no problem with
missing data, violation of assumptions, or outliers, and that the split sample validation will confirm the generalizability of the results. Use a level of significance of 0.05. The variables
"strength of affiliation" [reliten] and "frequency of prayer" [pray] have a strong relationship to the variable "frequency of attendance at religious services" [attend]. Survey respondents who
were less strongly affiliated with their religion attended religious services less often. Survey respondents who prayed less often attended religious services less often. 1. True 2. True with
caution 3. False 4. Inappropriate application of a statistic
8. Dissecting problem 1 - 1 When a problem states that there is a relationship between some independent variables and a dependent variable, we do standard multiple regression. 1. In the dataset
GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic? Assume that there is no problem with missing data, violation of assumptions, or outliers, and that
the split sample validation will confirm the generalizability of the results. Use a level of significance of 0.05. The variables "strength of affiliation" [reliten] and "frequency of prayer"
[pray] have a strong relationship to the variable "frequency of attendance at religious services" [attend]. Survey respondents who were less strongly affiliated with their religion attended
religious services less often. Survey respondents who prayed less often attended religious services less often. 1. True 2. True with caution 3. False 4. Inappropriate application of a statistic
The variables listed first in the problem statement are the independent variables (ivs): "strength of affiliation" [reliten] and "frequency of prayer" [pray] The variable that is related to is
the dependent variable (dv): "frequency of attendance at religious services" [attend].
9. Dissecting problem 1 - 2 • In order for a problem to be true, we will have find: • a statistically significant relationship between the ivs and the dv • a relationship of the correct strength 1.
In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic? Assume that there is no problem with missing data, violation of assumptions, or
outliers, and that the split sample validation will confirm the generalizability of the results. Use a level of significance of 0.05. The variables "strength of affiliation" [reliten] and
"frequency of prayer" [pray] have a strong relationship to the variable "frequency of attendance at religious services" [attend]. Survey respondents who were less strongly affiliated with their
religion attended religious services less often. Survey respondents who prayed less often attended religious services less often. 1. True 2. True with caution 3. False 4. Inappropriate
application of a statistic The relationship of each of the independent variables to the dependent variable must be statistically significant and interpreted correctly.
10. Request a standard multiple regression To compute a multiple regression in SPSS, select the Regression | Linear command from the Analyze menu.
11. Specify the variables and selection method First, move the dependent variable attend to the Dependent text box. Second, move the independent variables reliten and pray to the Independent(s) list
box. Third, select the method for entering the variables into the analysis from the drop down Method menu. In this example, we accept the default of Enter for direct entry of all variables, which
produces a standard multiple regression. Fourth, click on the Statistics… button to specify the statistics options that we want.
12. Specify the statistics output options First, mark the checkboxes for Estimates on the Regression Coefficients panel. Third, click on the Continue button to close the dialog box. Second, mark the
checkboxes for Model Fit and Descriptives.
13. Request the regression output Click on the OK button to request the regression output.
14. LEVEL OF MEASUREMENT Multiple regression requires that the dependent variable be metric and the independent variables be metric or dichotomous. "Frequency of attendance at religious services"
[attend] is an ordinal level variable, which satisfies the level of measurement requirement if we follow the convention of treating ordinal level variables as metric variables. Since some data
analysts do not agree with this convention, a note of caution should be included in our interpretation. "Strength of affiliation" [reliten] and "frequency of prayer" [pray] are ordinal level
variables. If we follow the convention of treating ordinal level variables as metric variables, the level of measurement requirement for multiple regression analysis is satisfied. Since some data
analysts do not agree with this convention, a note of caution should be included in our interpretation.
15. SAMPLE SIZE The minimum ratio of valid cases to independent variables for multiple regression is 5 to 1. With 113 valid cases and 2 independent variables, the ratio for this analysis is 56.5 to
1, which satisfies the minimum requirement. In addition, the ratio of 56.5 to 1 satisfies the preferred ratio of 15 to 1.
16. OVERALL RELATIONSHIP BETWEEN INDEPENDENT AND DEPENDENT VARIABLES - 1 The probability of the F statistic (49.824) for the overall regression relationship is <0.001, less than or equal to the level
of significance of 0.05. We reject the null hypothesis that there is no relationship between the set of independent variables and the dependent variable (R² = 0). We support the research
hypothesis that there is a statistically significant relationship between the set of independent variables and the dependent variable.
17. OVERALL RELATIONSHIP BETWEEN INDEPENDENT AND DEPENDENT VARIABLES - 2 The Multiple R for the relationship between the set of independent variables and the dependent variable is 0.689, which would
be characterized as strong using the rule of thumb than a correlation less than or equal to 0.20 is characterized as very weak; greater than 0.20 and less than or equal to 0.40 is weak; greater
than 0.40 and less than or equal to 0.60 is moderate; greater than 0.60 and less than or equal to 0.80 is strong; and greater than 0.80 is very strong.
18. RELATIONSHIP OF INDIVIDUAL INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 1 For the independent variable strength of affiliation, the probability of the t statistic (-5.857) for the b coefficient
is <0.001 which is less than or equal to the level of significance of 0.05. We reject the null hypothesis that the slope associated with strength of affiliation is equal to zero (b = 0) and
conclude that there is a statistically significant relationship between strength of affiliation and frequency of attendance at religious services.
19. RELATIONSHIP OF INDIVIDUAL INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 2 The b coefficient associated with strength of affiliation (-1.138) is negative, indicating an inverse relationship in
which higher numeric values for strength of affiliation are associated with lower numeric values for frequency of attendance at religious services. Since both variables are ordinal level, we will
have to look at the coding for each before we can make a correct interpretation. For ordinal level variables the numeric codes can be associated with labels in ascending or descending order.
20. RELATIONSHIP OF INDIVIDUAL INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 3 The independent variable strength of affiliation is an ordinal variable that is coded so that higher numeric values are
associated with survey respondents who were less strongly affiliated with their religion.
21. RELATIONSHIP OF INDIVIDUAL INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 4 The dependent variable frequency of attendance at religious services is also an ordinal variable. It is coded so that
lower numeric values are associated with survey respondents who attended religious services less often. Therefore, the negative value of b implies that survey respondents who were less strongly
affiliated with their religion attended religious services less often.
22. RELATIONSHIP OF INDIVIDUAL INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 5 For the independent variable frequency of prayer, the probability of the t statistic (-4.145) for the b coefficient is
<0.001 which is less than or equal to the level of significance of 0.05. We reject the null hypothesis that the slope associated with frequency of prayer is equal to zero (b = 0) and conclude
that there is a statistically significant relationship between frequency of prayer and frequency of attendance at religious services.
23. RELATIONSHIP OF INDIVIDUAL INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 6 The b coefficient associated with how often does r pray (-0.554) is negative, indicating an inverse relationship in
which higher numeric values for how often does r pray are associated with lower numeric values for frequency of attendance at religious services. Since both variables are ordinal level, we will
have to look at the coding for each before we can make a correct interpretation. For ordinal level variables the numeric codes can be associated with labels in ascending or descending order.
24. RELATIONSHIP OF INDIVIDUAL INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 7 The independent variable frequency of prayer is an ordinal variable that is coded so that higher numeric values are
associated with survey respondents who prayed less often.
25. RELATIONSHIP OF INDIVIDUAL INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 8 The dependent variable frequency of attendance at religious services is also an ordinal variable. It is coded so that
lower numeric values are associated with survey respondents who attended religious services less often. Therefore, the negative value of b implies that survey respondents who prayed less often
attended religious services less often.
26. Answer to problem 1 • The independent and dependent variables were metric (ordinal). • The ratio of cases to independent variables was 56.5 to 1. • The overall relationship was statistically
significant and its strength was characterized correctly. • The b coefficient for all variables was statistically significant and the direction of the relationships were characterized correctly.
• The answer to the question is true with caution. The caution is added because of the ordinal variables.
27. Problem 2 – hierarchical regression In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic? Assume that there is no problem with missing
data, violation of assumptions, or outliers, and that the split sample validation will confirm the generalizability of the results. Use a level of significance of 0.05. After controlling for the
effects of the variables "age" [age] and "sex" [sex], the addition of the variables "happiness of marriage" [hapmar], "condition of health" [health], and "attitude toward life" [life] reduces the
error in predicting "general happiness" [happy] by 36.1%. After controlling for age and sex, the variables happiness of marriage, condition of health, and attitude toward life each make an
individual contribution to reducing the error in predicting general happiness. Survey respondents who were less happy with their marriages were less happy overall. Survey respondents who said
they were not as healthy were less happy overall. Survey respondents who felt life was less exciting were less happy overall. 1. True 2. True with caution 3. False 4. Inappropriate application of
a statistic
28. Dissecting problem 2 - 1 The variables listed first in the problem statement are the independent variables (ivs) whose effect we want to control before we test for the relationship: "age"[age]
and "sex" [sex], 14. In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic? Assume that there is no problem with missing data, violation
of assumptions, or outliers, and that the split sample validation will confirm the generalizability of the results. Use a level of significance of 0.05. After controlling for the effects of the
variables "age" [age] and "sex" [sex], the addition of the variables "happiness of marriage" [hapmar], "condition of health" [health], and "attitude toward life" [life] reduces the error in
predicting "general happiness" [happy] by 36.1%. After controlling for age and sex, the variables happiness of marriage, condition of health, and attitude toward life each make an individual
contribution to reducing the error in predicting general happiness. Survey respondents who were less happy with their marriages were less happy overall. Survey respondents who said they were not
as healthy were less happy overall. Survey respondents who felt life was less exciting were less happy overall. 1. True 2. True with caution 3. False 4. Inappropriate application of a statistic
The variables that we add in after the control variables are the independent variables that we think will have a statistical relationship to the dependent variable: "happiness of marriage"
[hapmar], "condition of health" [health], and "attitude toward life" [life] The variable that to be predicted or related to is the dependent variable (dv): "general happiness" [happy]
29. Dissecting problem 2 - 2 In order for a problem to be true, the relationship between the added variables and the dependent variable must be statistically significant, and the strength of the
relationship after including the control variables must be correctly stated. 14. In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic?
Assume that there is no problem with missing data, violation of assumptions, or outliers, and that the split sample validation will confirm the generalizability of the results. Use a level of
significance of 0.05. After controlling for the effects of the variables "age" [age] and "sex" [sex], the addition of the variables "happiness of marriage" [hapmar], "condition of health"
[health], and "attitude toward life" [life] reduces the error in predicting "general happiness" [happy] by 36.1%. After controlling for age and sex, the variables happiness of marriage, condition
of health, and attitude toward life each make an individual contribution to reducing the error in predicting general happiness. Survey respondents who were less happy with their marriages were
less happy overall. Survey respondents who said they were not as healthy were less happy overall. Survey respondents who felt life was less exciting were less happy overall. 1. True 2. True with
caution 3. False 4. Inappropriate application of a statistic The relationship between each of the independent variables entered after the control variables and the dependent variable must be
statistically significant and interpreted correctly. We are generally not interested in whether or not the control variables have a statistically significant relationship to the dependent
30. Request a hierarchical multiple regression To compute a multiple regression in SPSS, select the Regression | Linear command from the Analyze menu.
31. Specify independent variables to control for First, move the dependent variable happy to the Dependent text box. Second, move the independent variables to control for age and sex to the
Independent(s) list box. Fourth, click on the Next button to tell SPSS to add another block of variables to the regression analysis. Third, select the method for entering the variables into the
analysis from the drop down Method menu. In this example, we accept the default of Enter for direct entry of all variables in the first block which will force the controls into the regression.
32. Add the other independent variables SPSS identifies that we will now be adding variables to a second block. First, move the other independent variables hapmar, health and life to the Independent
(s) list box for block 2. Second, click on the Statistics… button to specify the statistics options that we want.
33. Specify the statistics output options First, mark the checkboxes for Estimates on the Regression Coefficients panel. Third, click on the Continue button to close the dialog box. Second, mark the
checkboxes for Model Fit, Descriptives, and R squared change. The R squared change statistic will tell us whether or not the variables added after the controls have a relationship to the
dependent variable.
34. Request the regression output Click on the OK button to request the regression output.
35. LEVEL OF MEASUREMENT Multiple regression requires that the dependent variable be metric and the independent variables be metric or dichotomous. "General happiness" [happy] is an ordinal level
variable, which satisfies the level of measurement requirement if we follow the convention of treating ordinal level variables as metric variables. Since some data analysts do not agree with this
convention, a note of caution should be included in our interpretation. "Age" [age] is an interval level variable, which satisfies the level of measurement requirements for multiple regression
analysis. "Happiness of marriage" [hapmar], "condition of health" [health], and "attitude toward life" [life] are ordinal level variables. If we follow the convention of treating ordinal level
variables as metric variables, the level of measurement requirement for multiple regression analysis is satisfied. Since some data analysts do not agree with this convention, a note of caution
should be included in our interpretation. "Sex" [sex] is a dichotomous or dummy-coded nominal variable which may be included in multiple regression analysis.
36. SAMPLE SIZE The minimum ratio of valid cases to independent variables for multiple regression is 5 to 1. With 90 valid cases and 5 independent variables, the ratio for this analysis is 18.0 to 1,
which satisfies the minimum requirement. In addition, the ratio of 18.0 to 1 satisfies the preferred ratio of 15 to 1.
37. OVERALL RELATIONSHIP BETWEEN INDEPENDENT AND DEPENDENT VARIABLES The probability of the F statistic (9.493) for the overall regression relationship for all indpendent variables is <0.001, less
than or equal to the level of significance of 0.05. We reject the null hypothesis that there is no relationship between the set of all independent variables and the dependent variable (R² = 0).
We support the research hypothesis that there is a statistically significant relationship between the set of all independent variables and the dependent variable.
38. REDUCTION IN ERROR IN PREDICTING DEPENDENT VARIABLE - 1 The R Square Change statistic for the increase in R² associated with the added variables (happiness of marriage, condition of health, and
attitude toward life) is 0.361. Using a proportional reduction in error interpretation for R², information provided by the added variables reduces our error in predicting general happiness by
39. REDUCTION IN ERROR IN PREDICTING DEPENDENT VARIABLE - 2 The probability of the F statistic (15.814) for the change in R² associated with the addition of the predictor variables to the regression
analysis containing the control variables is <0.001, less than or equal to the level of significance of 0.05. We reject the null hypothesis that there is no improvement in the relationship
between the set of independent variables and the dependent variable when the predictors are added (R² Change = 0). We support the research hypothesis that there is a statistically significant
improvement in the relationship between the set of independent variables and the dependent variable.
40. RELATIONSHIP OF ADDED INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 1 If there is a relationship between each added individual independent variable and the dependent variable, the probability of
the statistical test of the b coefficient (slope of the regression line) will be less than or equal to the level of significance. The null hypothesis for this test states that b is equal to zero,
indicating a flat regression line and no relationship. If we reject the null hypothesis and find that there is a relationship between the variables, the sign of the b coefficient indicates the
direction of the relationship for the data values. If b is greater than or equal to zero, the relationship is positive or direct. If b is less than zero, the relationship is negative or inverse.
If the variable is dichotomous or ordinal, the direction of the coding must be taken into account to make a correct interpretation.
41. RELATIONSHIP OF ADDED INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 2 For the independent variable happiness of marriage, the probability of the t statistic (5.741) for the b coefficient is
<0.001 which is less than or equal to the level of significance of 0.05. We reject the null hypothesis that the slope associated with happiness of marriage is equal to zero (b = 0) and conclude
that there is a statistically significant relationship between happiness of marriage and general happiness.
42. RELATIONSHIP OF ADDED INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 3 The b coefficient associated with happiness of marriage (0.599) is positive, indicating a direct relationship in which higher
numeric values for happiness of marriage are associated with higher numeric values for general happiness.
43. RELATIONSHIP OF ADDED INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 4 The independent variable happiness of marriage is an ordinal variable that is coded so that higher numeric values are
associated with survey respondents who were less happy with their marriages.
44. RELATIONSHIP OF ADDED INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 5 The dependent variable general happiness is also an ordinal variable. It is coded so that higher numeric values are
associated with survey respondents who were less happy overall. Therefore, the positive value of b implies that survey respondents who were less happy with their marriages were less happy
45. RELATIONSHIP OF ADDED INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 6 For the independent variable condition of health, the probability of the t statistic (1.408) for the b coefficient is 0.163
which is greater than the level of significance of 0.05. We fail to reject the null hypothesis that the slope associated with condition of health is equal to zero (b = 0) and conclude that there
is not a statistically significant relationship between condition of health and general happiness. The statement in the problem that "survey respondents who said they were not as healthy were
less happy overall" is incorrect.
46. Answer to problem 2 • The independent and dependent variables were metric or dichotomous. Some are ordinal. • The ratio of cases to independent variables was 18.0 to 1. • The overall relationship
was statistically significant and its strength was characterized correctly. • The change in R2 associated with adding the second block of variables was statistically significant and correctly
interpreted. • The b coefficient for happiness of marriage was statistically significant and correctly interpreted. The b coefficient for condition of health was not statistically significant. We
cannot conclude that there was a relationship between condition of health and general happiness. • The answer to the question is false.
47. Problem 3 – Stepwise Regression 26. In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic? Assume that there is no problem with missing
data, violation of assumptions, or outliers, and that the split sample validation will confirm the generalizability of the results. Use a level of significance of 0.05. From the list of variables
"number of hours worked in the past week" [hrs1], "occupational prestige score" [prestg80], "highest year of school completed" [educ], and "highest academic degree" [degree], the best predictors
of "total family income" [income98] are "highest academic degree" [degree] and "occupational prestige score" [prestg80]. Highest academic degree and occupational prestige score have a moderate
relationship to total family income. The most important predictor of total family income is occupational prestige score. The second most important predictor of total family income is highest
academic degree. Survey respondents who had higher academic degrees had higher total family incomes. Survey respondents who had more prestigious occupations had higher total family incomes. 1.
True 2. True with caution 3. False 4. Inappropriate application of a statistic
48. Dissecting problem 3 - 1 The variables listed first in the problem statement are the independent variables from which the computer will select the best subset using statistical criteria. The
variable that to be predicted or related to is the dependent variable (dv): "total family income" [income98] 26. In the dataset GSS2000.sav, is the following statement true, false, or an
incorrect application of a statistic? Assume that there is no problem with missing data, violation of assumptions, or outliers, and that the split sample validation will confirm the
generalizability of the results. Use a level of significance of 0.05. From the list of variables "number of hours worked in the past week" [hrs1], "occupational prestige score" [prestg80],
"highest year of school completed" [educ], and "highest academic degree" [degree], the best predictors of "total family income" [income98] are "highest academic degree" [degree] and "occupational
prestige score" [prestg80]. Highest academic degree and occupational prestige score have a moderate relationship to total family income. The most important predictor of total family income is
occupational prestige score. The second most important predictor of total family income is highest academic degree. Survey respondents who had higher academic degrees had higher total family
incomes. Survey respondents who had more prestigious occupations had higher total family incomes. 1. True 2. True with caution 3. False 4. Inappropriate application of a statistic The best
predictors are the variables that will be meet the statistical criteria for inclusion in the model.
49. Dissecting problem 3 - 2 • In order for a problem to be true, we will have find: • a statistically significant relationship between the included ivs and the dv • a relationship of the correct
strength 26. In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic? Assume that there is no problem with missing data, violation of
assumptions, or outliers, and that the split sample validation will confirm the generalizability of the results. Use a level of significance of 0.05. From the list of variables "number of hours
worked in the past week" [hrs1], "occupational prestige score" [prestg80], "highest year of school completed" [educ], and "highest academic degree" [degree], the best predictors of "total family
income" [income98] are "highest academic degree" [degree] and "occupational prestige score" [prestg80]. Highest academic degree and occupational prestige score have a moderate relationship to
total family income. The most important predictor of total family income is occupational prestige score. The second most important predictor of total family income is highest academic degree.
Survey respondents who had higher academic degrees had higher total family incomes. Survey respondents who had more prestigious occupations had higher total family incomes. 1. True 2. True with
caution 3. False 4. Inappropriate application of a statistic The importance of the variables is provided by the stepwise order of entry of the variable into the regression analysis.
50. Dissecting problem 3 - 3 26. In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic? Assume that there is no problem with missing data,
violation of assumptions, or outliers, and that the split sample validation will confirm the generalizability of the results. Use a level of significance of 0.05. From the list of variables
"number of hours worked in the past week" [hrs1], "occupational prestige score" [prestg80], "highest year of school completed" [educ], and "highest academic degree" [degree], the best predictors
of "total family income" [income98] are "highest academic degree" [degree] and "occupational prestige score" [prestg80]. Highest academic degree and occupational prestige score have a moderate
relationship to total family income. The most important predictor of total family income is occupational prestige score. The second most important predictor of total family income is highest
academic degree. Survey respondents who had higher academic degrees had higher total family incomes. Survey respondents who had more prestigious occupations had higher total family incomes. 1.
True 2. True with caution 3. False 4. Inappropriate application of a statistic The relationship between each of the independent variables entered after the control variables and the dependent
variable must be statistically significant and interpreted correctly. Since statistical significance of a variable's contribution toward explaining the variance in the dependent variable is
almost always used as the criteria for inclusion, the statistical significance of the relationships is usually assured. | {"url":"https://www.slideserve.com/betty_james/multiple-regression-basic-relationships-powerpoint-ppt-presentation","timestamp":"2024-11-09T12:29:19Z","content_type":"text/html","content_length":"126547","record_id":"<urn:uuid:2a7a884f-bcd9-4c88-a729-838cbd5abf5f>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00602.warc.gz"} |
How do I add scoring / calculations to my Forms? | Finger-Ink Help Center
Our scoring & calculations feature is like having a silent math whiz hidden in your forms, tallying scores and calculating totals faster than you can say "Where’s my abacus?".
Whether you’re summing up patient satisfaction scores or automating the tally for that lengthy health questionnaire, these new features are about to be the MVP of your clinic’s paperwork game.
What it looks like
Here's part of an FABQ Form as an example, showing a score being generated dynamically after answering the first section:
Creating a score
There are a few components to creating a score that can be inserted into a Form:
1. Some numerical values to tally up
2. One or more calculation fields to do the tallying
3. One or more @placeholder mentions to insert the scores into the Form
The numerical values
In the FABQ example above, each screen contains one field configured with six different buttons. Here's what that screen looks like in the Form Editor:
Clicking the Change options button will show both the display values of the option, and the raw value. The raw value is the value behind the option. This is what's used in calculations. Let's see
that now:
As you can see, all raw values are numeric in nature. This is important. Only fields configured with all numeric values behind their options can be used in calculations.
The calculation field
The numeric raw values are useless without a way to use them. The calculation field type is what we use to perform calculations on our numeric values. Here's the calculation field we used in the FABQ
example above:
Calculation fields operate similarly to formulas in excel. In this example, we define a calculation using the sum function to add together the results of questions 2, 3, 4 and 5 (which will each have
a value of 0-6).
This type of field looks different to a field of any other type in the editor. We'll get into why later.
The @placeholder mention
The final piece of the puzzle is the @placeholder mention. This is a way to reference a calculation field later on in the form. Here's the screen definition for the FABQ example above:
Notice that the placeholder is defined as @FABQpa, which corresponds to the value in the field prompt of the calculation field.
Bonus: Visibility logic
A fourth part — bet you didn't see that coming! 😅 One of the most powerful parts of using calculation fields in your Form is that you can use them inside Visibility logic to show or hide parts of the
Form based on previous calculations. You might have noticed that, in the video above, only that first field was displayed.
That's because we defined Visibility logic to show the first field only if the score was over 15, and the second field to show only if the score was under 16:
Diving deeper
Scoring & calculations are one of the most complicated parts of a Finger-Ink Form. You've probably got more questions. Let's have a go at answering them before they're asked.
What functions are available to use in calculation fields?
The available functions for calculations are count, sum, average, min & max.
• Count — counts all the number of selected options on a field. This is only really useful for fields allowing you to select more than one option at a time. This is the only exception to the rule
that requires all numeric raw values.
• Sum — adds all the raw values for the option(s) for the specified field(s).
• Average — does a sum first, then divides the answer by the number of values present.
• Min — takes the smallest value in the given list.
• Max — takes the largest value in the given list.
Here's an example using sum on 3 different fields:
sum([ @field1, @field2, @field3 ])
☝️ Notice both the parentheses and square brackets — both are required.
Are calculation fields displayed during Form filling?
🚫 No. Calculation fields are not displayed during Form filling. They even have an "always hidden" indicator beside their label:
Can I define a calculation field anywhere in my Form?
🚫 No. Calculation fields, being fields, can only be defined within a Screen.
Furthermore, calculation fields need to be defined after all the values you're wanting to use in the calculation. Here's an example:
Can calculation fields be used in other calculation fields?
✅ Yes! You can use the result of one calculation in another calculation. This is the recommended approach for sub-totals and totals where the calculations required call for more than just addition. | {"url":"https://help.finger-ink.com/en/articles/8558733-how-do-i-add-scoring-calculations-to-my-forms","timestamp":"2024-11-07T19:18:20Z","content_type":"text/html","content_length":"94879","record_id":"<urn:uuid:2d17caa3-9451-40f1-b7ee-ab92458f8e90>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00813.warc.gz"} |
r Geometrie und Analysis
Oberseminar Geometrie und Analysis
Bevorstehende Vorträge im Oberseminar
Zurzeit sind keine zukünftigen Termine bekannt.
Vergangene Vorträge
Souheib Allout (Bochum University):
Partially hyperbolic diffeomorphisms in dimension three satisfying some rigidity hypothesis
Yuan Yao (Sorbonne University):
Anchored symplectic embeddings
Wilhelm Klingenberg (Durham University):
A proof of the Toponogov conjecture
Bernhard Albach (RWTH Aachen):
On the number of geodesics on S2 (Part 2)
Bernhard Albach (RWTH Aachen):
On the number of geodesics on S2
Simon Vialaret (Université Paris-Saclay):
Sharp systolic inequalities for invariant contact forms on S1-principal bundles
Dustin Connery-Grigg (IMJ-PRG):
Spectral invariants and dynamics of low-dimensional Hamiltonian systems
Emilia Alves (Universidade Federal Fluminense, Brazil):
Intersection of real Bruhat cells
Leonardo Masci (RWTH Aachen):
A Poincaré-Birkhoff theorem for asymptotically linear Hamiltonian systems
Marcelo Alves (University of Antwerp):
C0-stability of topological entropy for 3-dimensional Reeb flows
Urs Fuchs (RWTH Aachen):
A primer on circle packings
Michael Hutchings (University of California, Berkeley):
Unknotted Reeb orbits and the first ECH capacity
Patrice Le Calvez (Sorbonne Université):
Non-contractible periodic orbits for area preserving surface homeomorphisms
Jacob Rasmussen (University of Cambridge):
Floer homology for knots in the solid torus (joint seminar with EDDy)
Murat Sağlam (Universität zu Köln):
Contact 3-manifolds with integrable Reeb flows
David Bechara Senior (RWTH Aachen):
The asymptotic action of area preserving disk maps and some of its properties
Rohil Prasad (Princeton University):
Volume-perserving right-handed vector fields are conformally Reeb
Abror Pirnapasov (ENS Lyon):
The mean action and the Calabi invariant
Valerio Assenza (Universität Heidelberg):
Magnetic Curvature and Existence of Closed Magnetic Geodesics
Urs Frauenfelder (Universität Augsburg):
GIT quotients and Symmetric Periodic Orbits
Luca Asselle (Ruhr-Universität Bochum):
Urs Frauenfelder (Universität Augsburg):
Benoit Joly (Ruhr-Universität Bochum):
Barcodes for Hamiltonian homeomorphisms of surfaces
Alberto Abbondandolo (Ruhr-Universität Bochum):
Bi-invariant Lorentz-Finsler structures on the linear symplectic group and on the contactomorphism group
Tobias Soethe (RWTH Aachen):
Sharp systolic inequalities for rotationally symmetric 2-orbifolds - Part II
Tobias Soethe (RWTH Aachen):
Sharp systolic inequalities for rotationally symmetric 2-orbifolds
Alberto Abbondandolo (Ruhr Universität Bochum):
Erman Çineli (IMJ-PRG):
Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective
Matthias Meiwes (RWTH Aachen):
Braid stability and Hofer's metric
Lucas Dahinden (Universität Heidelberg):
The Bott-Samelson Theorem for positive Legendrian Isotopies
Marco Mazzucchelli (ENS Lyon):
Existence of global surfaces of section for Kupka-Smale 3D Reeb flows
Yuri Lima (Universidade Federal do Ceará):
Symbolic Dynamics for Maps with Singularities in High Dimension
Pedro Salomão (Universidade de São Paulo):
Genus zero global surfaces of section for Reeb flows and a result of Birkhoff
Gabriele Benedetti (VU Amsterdam):
The dynamics of strong magnetic fields on surfaces: periodic orbits and trapping regions
Jungsoo Kang (Seoul National University):
Symplectic Homology of Convex Domains
Stefan Suhr (Ruhr-Universität Bochum):
New developments in the theory of Lyapunov functions for cone fields
Barney Bramham (Ruhr-Universität Bochum):
Symbolic Dynamics for Reeb Flows
Anna Florio (IMJ-PRG):
Torsion of Conservative Twist Maps on the Annulus
Marcelo Alves (Université Libre de Bruxelles):
Symplectic invariants and topological entropy of Reeb flows
Gabriele Benedetti (Universität Heidelberg):
Periodic motions of a charged particle in a stationary magnetic field
Louis Merlin (RWTH Aachen University):
Global invariants of symmetric spaces | {"url":"https://www.mathga.rwth-aachen.de/news/home/","timestamp":"2024-11-05T04:28:49Z","content_type":"text/html","content_length":"47245","record_id":"<urn:uuid:48117959-8922-4941-a4f7-fd00c979d023>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00621.warc.gz"} |
Find the value of ${{\
Hint: Simplify $\cos 1540{}^\circ $ so that you can use the property \[{{\cos}^{-1}}\left( \cos x \right)=x\]. For this, use the property that a trigonometric operation of the form $\cos \left( 360{}
^\circ \times n+x \right)$ can be written as \[\cos \left( x \right)\]. Next use the property \[{{\cos }^{-1}}\left( \cos x \right)=x\] on the simplified expression to arrive at the final answer.
Complete step by step solution:
In this question, we need to find the value of ${{\cos }^{-1}}\left( \cos 1540{}^\circ \right)$.
We first need to identify that the range of the function \[{{\cos }^{-1}}\left( x \right)\] is between \[0{}^\circ \] and \[180{}^\circ \].
In our question, we are given \[1540{}^\circ \] which is not in this range. So, we cannot
directly write \[{{\cos }^{-1}}\left( \cos 1540{}^\circ \right)=1540{}^\circ \]. So, we need to
simplify \[1540{}^\circ \].
To find this value, we will first evaluate $\cos 1540{}^\circ $ and then we will come to the inverse part.
First, let us simplify $\cos 1540{}^\circ $
We can write 1540 as:
$1540=360\times 4+100$
So, we can write $\cos 1540{}^\circ $ as the following:
$\cos 1540{}^\circ =\cos \left( 360{}^\circ \times 4+100{}^\circ \right)$
Now, we know the property that a trigonometric operation of the form $\cos \left(
360{}^\circ \times n+x \right)$ can be written as \[\cos \left( x \right)\].
Here, in this question we have n = 4 and x = 100.
Using this property, we can write the above expression as:
\[\cos 1540{}^\circ =\cos \left( 360{}^\circ \times 4+100{}^\circ \right)\]
\[\cos 1540{}^\circ =\cos 100{}^\circ \]
Now, we will come to the inverse part.
We know the property that for an angle x, if the measure of angle x is greater than or equal
to \[0{}^\circ \] and less than or equal to \[180{}^\circ \] , then the expression \[{{\cos }^{-
1}}\left( \cos x \right)\] can be written as x
i.e. \[{{\cos }^{-1}}\left( \cos x \right)=x\] for \[0{}^\circ \le x\le 180{}^\circ \]
Now, since in our question \[100{}^\circ \] satisfies the condition of being greater than or
equal to \[0{}^\circ \] and less than or equal to \[180{}^\circ \] , we can use the above
property on it.
We will use this property to calculate \[{{\cos }^{-1}}\left( \cos 1540{}^\circ \right)\]
\[{{\cos }^{-1}}\left( \cos 1540{}^\circ \right)={{\cos }^{-1}}\left( \cos 100{}^\circ \right)\]
\[{{\cos }^{-1}}\left( \cos 100{}^\circ \right)=100{}^\circ \]
Hence, \[{{\cos }^{-1}}\left( \cos 1540{}^\circ \right)=100{}^\circ \]
This is our final answer.
Note: In this question, it is very important to identify that the range of the function \[{{\cos}^{-1}}\left( x \right)\] is between \[0{}^\circ \] and \[180{}^\circ \]. In our question, we are given
\[1540{}^\circ \] which is not in this range. So, we cannot directly write \[{{\cos }^{-1}}\left(\cos 1540{}^\circ \right)=1540{}^\circ \]. This would be wrong. So, we need to simplify \[1540{}^\circ
\] to a smaller number such that it can be within the range. | {"url":"https://www.vedantu.com/question-answer/find-the-value-of-cos-1left-cos-1540circ-right-class-12-maths-cbse-5ee712c6cd14f768db606907","timestamp":"2024-11-13T18:44:29Z","content_type":"text/html","content_length":"184295","record_id":"<urn:uuid:0c1a5347-dae7-4512-a5ee-1b771d2873c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00776.warc.gz"} |
What are the types of reactions in balancing equations?
The five basic types of chemical reactions are combination, decomposition, single-replacement, double-replacement, and combustion. Analyzing the reactants and products of a given reaction will allow
you to place it into one of these categories.
What are the 3 steps to balancing an equation?
3 Steps for Balancing Chemical Equations
1. Write the unbalanced equation. Chemical formulas of reactants are listed on the lefthand side of the equation.
2. Balance the equation.
3. Indicate the states of matter of the reactants and products.
What are the 5 steps for balancing equations?
1. Step 1: Coefficients Versus Subscripts. When approaching a chemical equation, it is important that you understand the difference between coefficients and subscripts.
2. Step 2: Sum the Atoms.
3. Step 3: Balance the First Element.
4. Step 4: Repeat for the Other Elements.
5. Step 5: Tips.
How do you solve balancing equations?
Steps in Balancing a Chemical Equation
1. Count each type of atom in reactants and products.
2. Place coefficients, as needed, in front of the symbols or formulas to increase the number of atoms or molecules of the substances.
3. Repeat steps 1 and 2 until the equation is balanced.
How do you balance chemical reactions?
In general, however, you should follow these steps:
1. Count each type of atom in reactants and products.
2. Place coefficients, as needed, in front of the symbols or formulas to increase the number of atoms or molecules of the substances.
3. Repeat steps 1 and 2 until the equation is balanced.
Which one technique is used in balancing chemical equations?
The Algebraic Balancing Method. This method of balancing chemical equations involves assigning algebraic variables as stoichiometric coefficients to each species in the unbalanced chemical equation.
These variables are used in mathematical equations and are solved to obtain the values of each stoichiometric coefficient …
What is the rule of balance?
On a separate sheet of paper, make a claim in which you mathematically state the rule of balance – that is, the rule that one must use to determine if two weights placed on opposite sides of the
fulcrum will balance each other.
How do you identify a reaction type?
Feel for heat in exothermic reactions. Many synthesis and replacement (single and double) reactions are exothermic,meaning they release heat.
Look for the formation of precipitate. Again,in many synthesis and replacement (single and double) reactions,a precipitate will form at the bottom of the tube.
Add heat for endothermic reactions.
What are the six types of chemical reaction?
– Does the equation contain O₂, CO₂, and H₂O? Yes = combustion rxn. – Do simple things make something more complex? Yes = synthesis rxn. – Does something complex break apart? Yes = decomposition rxn.
– Are there any pure, unbonded elements? Yes = single displacement. – Is water a product of this reaction? Yes = acid-base. No = double displacement.
How do you balance a nuclear reaction?
Nuclear Decay.
What are the types of chemical reaction?
Research by the Oregon State University College of Engineering has uncovered a way to improve the efficiency of a type of grid could be improved via chemical reactions. The reversible | {"url":"https://www.sheppard-arts.com/essay-writing/what-are-the-types-of-reactions-in-balancing-equations/","timestamp":"2024-11-04T12:20:33Z","content_type":"text/html","content_length":"74760","record_id":"<urn:uuid:017b378b-8ba0-47a5-a007-bd96be5d3b0b>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00557.warc.gz"} |
Design optimization of coil gun to improve muzzle velocity
Recently, a coil gun was brought to the attention of engineering community as electromagnetic alternative to the chemical launchers. Various studies were performed on coil gun systems focused on
achieving the high muzzle velocity in military applications and for satellite launching. Most of these studies focused on improvement of muzzle velocity via increase in the size of the coil gun.
Present paper describes the process of design optimization where the size of the coli gun system is restricted. The design of experiment approach utilizes the orthogonal array table that reduces the
required number of experiments. The design of experiment is carried out with a commercial PIAnO tool, where the finite element analysis is performed at each experimental point. Then, Kriging model is
created to achieve accurate approximation in problems of many design variables or strongly nonlinear model. The coil gun is optimally designed using an evolutionary algorithm (EA) as optimization
technique. In order to verify the improvement of muzzle velocity by optimal design, the prototypes of coil gun system are manufactured and the experiments to launch the projectile are performed.
1. Introduction
While the conventional launching system utilizing the chemical energy of the fuel has high cost and negative environmental impact, electromagnetic launching (EML) system present a viable projectile
propulsion alternative with reasonably low cost and minimal environmental drawbacks (such as production of carbon dioxide). A coil gun system which is the among the most advanced EML systems, propels
the projectile by electromagnetic force caused by Fleming’s right hand rule when the electric current energizes the electromagnetic solenoid coils. That is, the electromagnetic force of the coils
attracts and launches the projectile [1]. Fig. 1 illustrates the schematic and operational principles of the coil gun system.
Fig. 1Schematic of coil gun system
Polytechnic Institute of New York University in the United States proposed launching method that uses LIL (Linear Induction Launcher), a type of coil gun, to reduce launching costs of micro-and
nano-satellites. Design specification is to accelerate the payload of 10 kg at 30,000 g and to have the muzzle velocity of 7 km/s [2]. Sandia National Laboratories in the United States has been
developing the EML techniques of low-and high-muzzle velocity using coil gun since 1980 s. These have conducted the task to develop EML accelerating as 1 km/s of 0.23 kg and 335 m/s of 5 kg [3]. VN
Bondaletov of the Soviet Union driven the projectile of 1 g at a muzzle velocity of 4.9 km/s by applying a voltage of 45 kV with single stage coil gun system in the 1978 [4]. In addition, many
studies of coil gun projectile launching systems are underway in the Japan and the United Kingdom.
Most of studies of coil gun system have focused on achieving high muzzle velocity because the coil gun system is mainly employed in military weapon systems and in space launching. The larger coil gun
can provide higher muzzle velocity of projectile, but it also increases the required space and increases costs of the system installation, operation, maintenance, and repair.
In this paper, we performed the design optimization of the coil gun parameters to maximize the muzzle velocity while preserving the restricted size of the coil gun. Design variables are the number of
axial turns of electromagnetic coil $N$, the number of radial turns of electromagnetic coil $M$, the initial distance between the projectile and the coil $z$, the inner radius of electromagnetic coil
$R$, and the length of the projectile $L$. These design variables are independent of each other but these are nonlinearly related to the magnetic force that determines the muzzle velocity. We
performed optimization using the commercial optimization software PIAnO (Process Integration, Automation, and Optimization) Ver. 3.5 (PIDO-TECH). Five design variables are selected and an orthogonal
design array is constructed. The analytical modeling is difficult to implement due to nonlinearity of the functional dependence of the muzzle velocity on the parameters listed above. Accordingly,
finite element analysis (FEA) is performed utilizing the commercial electromagnetic analysis software MAXWELL. Subsequently, these analytical results are imported into the PIAnO and Kriging model is
created to generate an accurate solution approximation of the non-linear model with many design variables. Finally, the coil gun is optimally designed using an optimization approach of the
evolutionary algorithm (EA) [5-6].
Fig. 2 shows the process of optimal design undertaken in the presented study.
Fig. 2Flow chart of the optimization
2. Optimal design of the coil gun
2.1. Coil gun system
Coil gun system propels the projectile by electromagnetic force caused by Fleming’s right hand rule when the electric current energizes the electromagnetic solenoid coils. The schematic diagram of
electric circuit for coil gun launching system is shown in Fig. 3. The electromagnetic coil is energized by discharging of a capacitor. The electromagnetic force of the energized coils attracts the
projectile. If the electromagnetic coil is still energized after the projectile passes the longitudinal center of the electromagnetic coil windings, the magnetic force of the coils pulls the
projectile in the direction opposite to the launching direction, and the projectile is decelerated. This effect is called “suck back”. To prevent this, the electric current on the electromagnetic
coils must be cut off just before the projectile passes the longitudinal center of the coil windings [7].
Fig. 3Schematic diagram of electric circuit for the coil gun launching system
2.2. Definition of the design problem
2.2.1. Design requirements
The muzzle velocity of the coil gun system is proportional to the value of the maximum electric current flowing through the electromagnetic coil. However, there exists the maximum allowable value of
the electric current exceeding which temperature will rise above the melting point of the coil material. The current limit value is calculated by Onderdonk’s Eq. (1):
where $I$: fusing current in amperes, $A$: wire area in circular mils, $s$: melting time in seconds, ${T}_{m}$: melting temperature of wire [°C], ${T}_{a}$: ambient temperature [°C].
In this design, the AWG 12 commercial coil is utilized. Table 1 shows specification for the coil material [8].
Table 1Specification for AWG 12 coil material
${T}_{m}$ [$°$C] 125
${T}_{a}$ [°C] 25
$A$ [circ. mils] 6200
$s$ [ms] 12
The maximum value of electric current of AWG 12 at the melting point comes to be 3700 A, but safety margin is utilized to keep the allowable current below 3000 A.
If input voltage used in this design is 200 V, the resistance of the coil can be expressed by Eq. (2) and it is summarized by Eq. (3). The procedure for calculating the coil resistance is outlined in
the Eq. (4) through Eq. (7):
${R}_{coil}\ge 0.0667\left[\text{o}\text{h}\text{m}\right],$
${R}_{coil}=\frac{{l}_{coil}}{{A}_{coil}}×{\rho }_{copper}=\frac{{l}_{coil}}{{A}_{coil}}×1.724×1{0}^{-8},$
${l}_{coil}={\sum }_{m=1}^{M}2\pi \left(R+\left(m-0.5\right)d\right)×N=2\pi N\left(MR+\frac{{M}^{2}d}{2}\right),$
${A}_{coil}=\pi {\left(\frac{d}{2}\right)}^{2},$
${R}_{coil}=\frac{2\pi N\left(MR+\frac{{M}^{2}d}{2}\right)}{\pi {\left(\frac{d}{2}\right)}^{2}}×{\rho }_{copper}.$
The number of electromagnetic coil winding turns is calculated by Eq. (8):
$\frac{8N\left(MR+\frac{{M}^{2}d}{2}\right)}{{d}^{2}}×{\rho }_{copper}\ge 0.0667\left[\text{o}\text{h}\text{m}\right].$
The mass of projectile which is one of the important factors that affects the muzzle velocity is constrained to be less than 50 gram:
${m}_{projectile}\le 0.05\left[\text{kg}\right],$
${V}_{projectile}×{\rho }_{steel}\le 0.05\left[\text{kg}\right],$
$\pi {r}^{2}L×{\rho }_{steel}\le 0.05\left[\text{kg}\right].$
When the air gap between inner radius of electromagnetic coil and the outer surface of the projectile is, the radius of the projectile, can be expressed as:
$\pi {\left(R-0.5\text{mm}\right)}^{2}L×{\rho }_{steel}\le 0.05\left[\text{kg}\right].$
2.2.2. Design variables
Variables in this optimized design include the number of axial direction-winding turns of electromagnetic coil, the number of radial direction-winding turns of electro-magnetic coil, the initial
distance between the projectile and the electromagnetic coil, the inner radius of the electromagnetic coil and the length of the projectile as shown in Fig. 4. The initial values, lower limit and
upper limit of the design variables are summarized in Table 2.
Fig. 4The diagram identifying the variables in the optimization procedure for coil gun design
Lower limit, upper limit and initial value of design variables are selected based on the design values from our preliminary study. The design problem formulation is presented as Eq. (13):
${x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},$$I\le 3000\left[\text{Α}\right],{m}_{projectile}\le 0.05\left[\text{kg}\right],1\le {x}_{1}\le 20,1\le {x}_{2}\le 20,$$5\le {x}_{3}\le 30,5\le {x}_{4}\le
30,4\le {x}_{5}\le 80.$
Table 2Selected lower limit, upper limit and initial value of the design variables
Design variables Lower limit Initial value Upper limit
${x}_{1}$ $N$ Number of axial turns of coil [turns] 1 11 20
${x}_{2}$ $M$ Number of radial turns of coil [turns] 1 9 20
${x}_{3}$ $z$ Interval of projectile and coil [mm] 5 5 30
${x}_{4}$ $R$ Inner radius of electromagnetic coil [mm] 5 5 30
${x}_{5}$ $L$ Length of projectile [mm] 4 10 80
2.3. Design of experiments and simulation
In this paper, we design experiment by considering the number of design variables, the number of levels and the orthogonality of the design matrix. In order to analyze data effectively, the
orthogonal array table is created [9]. We used of orthogonal array table provided by PIAnO tools. This indicates respectively, 32 experiments, 4 levels, and columns. The number of experiments can be
obtained the Eq. (14):
where the number of experiments = 1.5$\text{∙}{N}_{SAT}$, ${N}_{DV}$: the number of design variables, ${N}_{SAT}$: the number of saturated points.
Based on the orthogonal array, we selected the coil guns of 32 types and measured the muzzle velocity through electromagnetic analysis for each model. Finite element analysis (FEA) is performed
utilizing the commercial electromagnetic analysis software MAXWELL. Fig. 5 shows 2D axisymmetric model of coil gun and projectile and plot of magnetic intensity distribution respectively.
Fig. 5a) 2D axisymmetric model and b) magnetic intensity distribution plot
2.4. Meta-model
Meta-model approximates the relationship between the value of a variable and the response value of the analytical model within all regions or the regions of interest. In this paper, we select Kriging
model, one of the meta-models provided by PIAnO tools. The Kriging is a method for predicting the value in order to determine the characteristic value at a point of interest.
After placing in the PIAnO the simulation analysis results from the experimental results of 32 models, Kriging model can be created using the approximation method provided by PIAnO [10].
2.5. Result
The obtained results, the optimal design satisfied the constraints and the muzzle velocity of optimal design compared with the initial design increases by 30 %. However, the optimal design results
can be changed based on using the metamodel instead of the actual analytical model in this research. The accuracy of the optimization results should be verified by actual analysis using MAXWELL. To
do this, the Kriging model results ‘Opt_meta’ of the optimal design variables and the analysis results from MAXWELL ‘Opt_exact’ were compared as shown in Fig. 6. The error of ‘Opt_meta’ and
‘Opt_exact’ is about 6 %. Therefore, we confirmed the high accuracy of the Kriging model’s prediction. The initial and optimal values comparison of the design variables are summarized in Table 3.
Fig. 6The comparison of the initial model, meta model and optimal model
Table 3Comparison of the initial values and the optimal values for the design variables
Design variables Lower limit Initial value Optimal value Upper limit
${x}_{1}$ $N$ (turns) 1 11 12 20
${x}_{2}$ $M$ (turns) 1 9 6 20
${x}_{3}$ $z$ (mm) 5 5 13 30
${x}_{4}$ $R$ (mm) 5 5 5 30
${x}_{5}$ $L$ (mm) 4 10 20 80
2.6. Experiment
To verify the result of optimization, we manufactured the prototype of coil gun system as shown in Fig. 7. Fig. 8 and Fig. 9 show the experiment setup that includes the power supply for inputting
current to the solenoid coil and the measuring instruments. Once the electric current is applied to the coil by discharging the capacitor, the projectile is launched. The CCD (Charge Coupled Device)
camera recorded the motion of projectile and the displacement data of projectile was obtained for each millisecond.
Fig. 10 shows the photos of the moving projectile captured by the CCD camera. Based on the position at each time, the velocity of projectile can be calculated. The velocity of the projectile launched
with the coil gun of the initial design is 30.0 m/s and that of the projectile ejected from the optimized coil gun is 35.74 m/s.
Fig. 7The prototype of the coil gun system
Fig. 8Schematic of the experiment
Fig. 9The experimental setup
Fig. 10The photos of the moving projectile captured at different time moments
3. Conclusion
1) The problem studied was that of the coil gun design optimization to maximize the muzzle velocity of projectile.
2) The muzzle velocities of 32 type models are calculated by electromagnetic finite element analysis.
3) The optimization process is performed through PIAnO tools. The orthogonal array, Kriging model, Evolutionary algorithms (EA) provided by PIAnO tools are used as design of experiment, meta-model
and optimization techniques respectively.
4) The muzzle velocity of optimal design increases by 30 % as compared with the initial design.
5) The accuracy of the meta-model is 94.3 %.
6) In order to verify the optimal design, the prototypes of coil gun system were manufactured and projectile launching experiments were performed.
7) The difference in the results between the FE simulation prediction and the experimental data is caused mainly by the mechanical friction between the projectile and the flyway tube.
• Kim Seog-Whan, Jung Hyun-Kyo, Hahn Song-Yop An Optimal Design of Capacitor-Driven Coilgun. Deptment of Electrical and Computer Engineering, Seoul National University, South Korea, (in Korean).
• Kim Ki-Bong, Zabar Zivan, Levi Enrico, Birenbaum Leo In-bore projectile dynamics in the linear induction launcher (LIL). 1. Oscillations. IEEE Transactions on Magnetics, Vol. 31, Issue 1, 1995,
p. 484-488.
• Burgess T. J., Cowan M. Multistage induction mass accelerator. IEEE Transactions on Magnetics, Vol. 20, Issue 2, 1984, p. 235-238.
• Haghmaram R., Shoulaie A. Study of traveling wave tubular linear induction motors. International Conference on Power System Technology, 2004, p. 288-293.
• Choi J. S. General use PIDO solution, PIAnO. The Korean Society of Mechanical Engineers, Vol. 52, Issue 2, 2012, p. 12-13, (in Korean).
• Hedayat A. S., Sloane N. J. A., Stufken J. Orthogonal Arrays: Theory and Applications. Springer Series in Statistics, 1999.
• Lee Su-Jeong, Kim Ji-Hun, Kim Jin Ho Coil gun electromagnetic launcher (EML) system with multi-stage electromagnetic coils. Journal of Magnetics, Vol. 18, Issue 4, 2013, p. 481-486.
• Lux Jim High Voltage Fuses. http://home.earthlink.net/~jimlux/hv/fuses.htm.
• Lee Ki-Bum, Park Chang-Hyun, Kim Jin-Ho Optimal design of one-folded leaf spring with high fatigue life applied to horizontally vibrating linear actuator in smart phone. Advances in Mechanical
Engineering, Vol. 2014, 2014, p. 545126.
• Park Chang-Hyun, Lee Jun-Hee, Jeong Jae-Hyuk, Choi Dong-Hoon Design optimization of a laser printer cleaning blade for minimizing permanent set. Structural and Multidisciplinary Optimization,
Vol. 49, 2014, p. 131-145.
About this article
01 September 2014
Mechanical vibrations and applications
coil gun
electromagnetic launcher
finite element analysis
muzzle velocity
optimal design
Copyright © 2015 JVE International Ltd.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. | {"url":"https://www.extrica.com/article/15468","timestamp":"2024-11-14T14:41:57Z","content_type":"text/html","content_length":"125829","record_id":"<urn:uuid:7f4bf751-769c-4df0-82a7-8f5d9a255fe1>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00004.warc.gz"} |
Measures Of Central Tendency
Measures Of Central Tendency - Math Assignment Help
Basic Concepts of Measures of Central Tendency:
We have learnt about representation of data in various forms like plotting bar graph, histogram, or by using frequency table but while using this method a question arises that is there a need to form
certain important features, by considering representative data since studying data to make sense of it is a continuous process. Yes, this is possible by using the concept of averages or method
measures of central tendency.
A measure of central tendency (additionally alluded to as measures of focus or focal area) is an outline measure that endeavors to depict an entire arrangement of information with a solitary esteem
that speaks to the centre or central point of its appropriation.
The mode, the median and the mean are the three main measures of central tendency which are used in describing a different mark of central point in the distribution.
Mean: The sum of the values of the observation divided by the total number of observations is the mean of a number of observations. It is generally denoted by the symbol of x bar.
Lets us consider an example. Suppose there are 5 students who scored 50,65,48,78 and 85 marks respectively in their mathematics exam. We need to find the mean of marks of these students.
Mean = Sum of the observations / Total Number of Observations
Mean = 50+65+48+78+85 / 5
Mean = 326 / 5 = 65.2, the mean marks of these 5 students is 65.2.
Direct Method for Mean = ∑f[i]x[i] / ∑f[i] , for values of i 1 to n. This formula is used in case of an ungrouped frequency distribution.
Assumed Mean Method: It is a method where we do not calculate mean from mid points, instead we assume mean to find out mean. To start with we "assume" or accept a mean and afterward we apply an
amendment to this expected mean so as to locate the correct mean.
Assumed Mean = a + ∑f[i]d[i] / ∑f[i] , where a is the value of assumed mean, f[i ] is frequency and d[i] (deviation) is obtained by x[i] - a.
Step Deviation Method = a + (∑f[i]u[i] / ∑f[i] ) * h with the assumption that class frequency is at its central point also known as class mark. Here h is the class size and u[i] = x[i] - a / h .
Benefits of Mean:
1. Mean is unbendingly characterised so that there is no doubt of misconception about its importance and nature.
2. It is the most popular central tendency as it is straightforward.
3. It is anything but difficult to compute.
4. It incorporates every one of the scores of a distribution.
5. It is not influenced by sampling so that the outcome is solid.
Median: The value of the given number of observations, dividing the observations into two parts. Median of data when arranged in an ascending or descending order is given in the following ways:
For the n odd number of observations, the median is the value of (n+1/ 2)^th observation. For example we have 9 number of observations of marks scored in mathematics exam. Here the value of n is 9,
then the median will be (9 + 1 /2) = 5. Thus the value of median is 5th observation.
Similarly, for the n even number of observations, the value of the median is the mean of (n / 2)^th and (n/2 + 1)^th observation. For example we have 18 number of observations of marks scored in
mathematics exam. Here the value of n is 18 i.e. an even number, then the median will be mean of (18/2)th and (18/2 + 1)th observations i.e value of median is the mean of 9th and 10th observation.
Median of Grouped Data: In case of grouped data its get difficult to find median in its usual way, so we introduce the concept of Cumulative frequency to find the value of median.
Median = l + (n/2 - cf / f) * h
Where l is the lower limit of median class, n = number of observations, cf = cumulative frequency of preceding class to median class, f = frequency of median class, h = class size.
Note: Median class is a class whose frequency is greater than and nearest to n/2.
Frequency obtained by adding the frequencies of the preceding classes is known as the cumulative frequency of a class.
Benefits of Median:
1. It is anything but difficult to register and get it.
2. Every observation is not required for its calculation.
3. Extraordinary scores do not influence the median.
4. It can be resolved from the open-ended arrangement.
5. It can be resolved from unequal class interims.
Mode: The value of observation which occurs most frequently or the observations which has highest number of frequency of occurrence is known as the mode. For example the value of the mode of the
following marks obtained by 10 students : 55, 85, 75, 65, 85, 95, 85, 82, 80, 93 is 85, because 85 is the value with maximum frequency i,e, 3 times.
Mode of Grouped Data : In a grouped frequency, it gets difficult to determine the value of mode by looking at frequencies. Hence we observe a class with maximum frequency known as modal class. The
mode is the value in a modal class, represented by a formula:
Mode = l + (f[1 ]- f[0] / 2f[1] - f[0] - f[2]) * h
where l = Modal class lower limit , h = class interval size , f[1 ]= Modal class frequency , f[0] = frequency of class preceding modal class , f[2] = frequency of class succeeding modal class.
Benefits of Mode:
1. Mode gives the most illustrative value of an arrangement.
2. The mode is not influenced by any extreme scores like mean.
3. It can be determined from an open-ended class interim.
4. It helps in analysing and evaluating qualitative data.
5. The mode can likewise be determined graphically through histogram or recurrence polygon.
6. The mode is straightforward.
Importance of Measures of Central Tendency:
Mean is the regular "normal". In a few sections of the world, there are sets of activity cameras that take pictures of autos and measure the time between when the photos are taken. They then figure
the rate (separate/time). The outcome is the mean travel speed. (Additionally, utilizes the Mean Value Theorem). In school, your last grade is a mean. Total them then gap by the number and you have
your score. To improve it is a weighted mean where each score may "measure" more than the others, for example, an Exam. Any Per-Capita insights report is also a use case of mean. The mean, or the
normal, is an imperative measurement utilised as a part of games. Mentors utilise midpoints to decide how well a player is performing. General Managers may utilise mean to decide how great a player
is and how much cash that player is worth. Median is used in detailing earnings. The median pay in a territory discloses to you progressively what the "normal" individual procures. A couple
cosmically high values, from CEOs, Bill Gates, and so on through the Mean off, so the BLS utilises Median. The median is utilised as a part of financial aspects. For instance, the U.S. Evaluation
Bureau finds the median family unit wage. As indicated by the U.S. Enumeration Bureau, "median family unit salary" is characterised as "the sum which isolates the pay dissemination into two
equivalent gatherings, half having wage over that sum, and half having wage underneath that sum." The mode might be helpful for a supervisor of a shoe store. For instance, you would not see measure
17 shoes supplied on the floor. Why? Since not very many individuals have a size 17 shoe measure. Thusly, store supervisors may take a gander at information and figure out which shoe size is sold the
most. Chiefs would need to stock the floor with the top rated shoe measure.
How we help you? - Measures Of Central Tendency Assignment Help 24x7
We offer Measures Of Central Tendency assignment help, math assignment writing help, assessments writing service,math tutors support, step by step solutions toMeasures Of Central Tendency problems,
Measures Of Central Tendency answers, math assignment experts help online. Our math assignment help service is most popular and browsed all over the world for each grade level.
There are key services in math which are listed below:-
• Measures Of Central Tendency Algebra help
• Homework Help
• Measures Of Central Tendency Assessments Writing Service
• Solutions to problems
• math Experts support 24x7
• Online tutoring
Why choose us - The first thing come in your mind that why choose us why not others what is special and different about us in comparison to other site. As we told you our team of expert, they are
best in their field and we are always live to help you in your assignment which makes it special.
Key features of services are listed below:
• Confidentiality of student private information
• 100% unique and original solutions
• Step by step explanations of problems
• Minimum 4 minutes turnaround time - fast and reliable service
• Secure payment options
• On time delivery
• Unlimited clarification till you are done
• Guaranteed satisfaction
• Affordable price to cover maximum number of students in service
• Easy and powerful interface to track your order
Popular Writing Services:-
• Quantum Electrodynamics Get Quantum Electrodynamics Assignment Help Online, assessment help and Writing Service from Physics Assignment Experts.
• Biomolecular Get Biomolecular Assignment Help Online, assessment help and Writing Service from Chemistry Assignment Experts.
• AutoCAD Looking for autocad assignment help online? get Autocard assessments help, projects help and writing service from Autocad coursework assignment experts.
• Presentation or Speech get presentation, speech assignment help online, assignment writing service from academic writing assignment experts.
• Character Analysis Essay how to write a good character analysis essay & what are types of character analysis essay, acquire most trusted writers for essay help, writing service.
• C and C++ Programming avail C / C++ programming tutors for C programming solutions, C++ programming solutions, homework help and assignment help service at low cost.
• Accounting for Intangible Assets Are you looking for accounting tutor service 24/7 to solve your difficulties in accounting for intangible assets? looking for assignment help, assessments help? | {"url":"https://www.expertsminds.com/assignment-help/math/measures-of-central-tendency-483.html","timestamp":"2024-11-02T22:00:18Z","content_type":"text/html","content_length":"41158","record_id":"<urn:uuid:0bf9f57c-f87c-4f08-a6dc-603abdfbbcb1>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00319.warc.gz"} |
Unit 3: Quantities and Chemical Reactions
Unit 3: Quantities and Chemical Reactions Precision - How certain you are of a measurement - How consistent the measuring device is Accuracy - How close you are(measured value) to your target
(accepted value) THE MOLE - The mole is defined as a unit of measurement to measure the amount of matter in a substance - i.e. 16.1 mole of Carbon; means that the amount of carbon in the sample is
16.1 mol - 1 mol of substance contains (6.022)(10 23 ) particle (atoms, molecules, etc.) - (6.022)(10 23 ) is a special number, called Avogadro’s Number (Named after the man who discovered it -
Examples ● 1 mol of Al contains (6.022)(10 23 ) atoms ● 1 mol of S 8 contains (6.022)(10 23 ) molecules ● 1 mol of C 12 h 24 0 11 contains (6.022)(10 23 ) molecules ● 1 mole of NaCl contains (6.022)
(10 23 ) Formula Units - The amount of matter in moles is directly related to the number of particles( atoms, molecules, etc.) in a sample of matter - Therefore we gather this equation: ● Number of
moles= (Number of atoms, molecules, or Formula Units)/(Avogadro’s number) ● This is shortened to n=N/N A ● “n” is number of moles in mol ● “N” is Number of atoms, molecules, or Formula Units ● “N A ”
is Avogadro’s Number (6.022)(10 23 ) - Carbon-12, each C-12 has a mass of 12 amu (atomic units) - A sample of carbon has weighted average atomic mass of 12.011 amu - 1 mole of any substance has a
mass in grams equal to its weighted average atomic mass (given on periodic table, don’t forget to calculate for total atomic mass of the substance by adding up all the atomic masses of each atom
(s)!!! ) - Therefore we can we can gather another equation:
● Number of moles= (Entire mass of substance)/(Total Atomic mass of entire substance) ● Shortened to n=m/MM ● “n” is moles in mol ● “m” is mass in grams(g) ● “MM” is the total atomic mass of
substance, also known as molar mass, is measured in grams per mole(g/mol), and is found by adding up all the atomic masses for each atom(s) in a molecule - Tip : From n=N/N A and n=m/MM we can derive
the formula N/N A = m/MM PERCENT COMPOSITION - Law of Definite proportions: the element in chemical compounds are always present in the same proportion, by mass - Percent composition are ratios by
the masses of an element in a compound , this is represented by the formula: (mass of atom(s))/(Total mass of the substance) - In a chemical equation, the ratios are mole to mole - i.e. 1NaCl —> 1Na
+ 1Cl, assume one mole of NaCl and find the percent composition of Cl ● 1 mol of NaCl ● 1 mol of Na ● 1 mol of Cl ● Therefore m Cl = (1 mol of Cl)( MM Cl ) and m Na = (1 mol of Na)(MM Na ) ● m Cl =
22.989768 g and m Na = 35.4527 g ● Therefore % Cl = (mCl)/(mNa + MCl) * 100% ● % Cl = 60.689252% —— don’t forget sig figs —–à = 60.6892% THE EMPIRICAL FORMULA - The simplest ratio of elements for a
substance - Ie: CH 2 0; simplest ratio of elements for C 6 H 12 O 6 (Glucose) - It is found by finding the number of moles of each individual atom in a substance then dividing each of these by the
smallest amount. These numbers are representative of the number of atoms there are in a single molecule orion for that atom. - Sample Question: A sample is found to contain 5.0 g of copper and 1.3 g
of oxygen, and no other elements. Determine the empirical formula of this compound. ● What do we need? n Cu = ? and n O = ? ● n Cu = m Cu /MM Cu and n O = m O /MM O ● n Cu = 5.0g/(63.546g/mol) and n
O = 1.3g/(15.9994g/mol) ● n Cu = 0.078683 mol and n O = 0.081253 mol
● Divide by small therefore n Cu / n Cu = 1 and n O / n Cu = 1.032667= 1 (this is because we cannot have decimals in a chemical formula) ● Since they are both 1 therefore the ratio of Cu to O is 1:1,
therefore the Empirical Formula is Cu 1 O 1 or CuO | {"url":"https://keepnotes.com/grade-11/chemistry/1356-unit-3-quantities-and-chemical-reactions","timestamp":"2024-11-02T18:58:36Z","content_type":"text/html","content_length":"128945","record_id":"<urn:uuid:b95aa716-16fa-4418-9cb4-33ea83f87245>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00812.warc.gz"} |
Square & Square Root of 20 - Methods, Calculation, Formula, How to find
Square & Square Root of 20
Square of 20
The “square of 20” refers to the result obtained when the number 20 is multiplied by itself.
In mathematical terms, it is denoted as 20² or simply 20 squared. To calculate the square of 20, you simply multiply 20 by 20. This can be done using basic arithmetic operations:
Mathematically, 20² (20×20) = 400
The concept of the square of 20 is fundamental in mathematics and arithmetic. It represents the area of a square with side length 20 units. Understanding the square of 20 is crucial in various
mathematical applications, such as geometry, algebra, and physics. By knowing how to calculate the square of 20, one can solve problems involving area, volume, and other geometric properties with
Square root of 20
√20 = 4.47213595499958
√20=4.472 up to three places of decimal
The square root of 20, denoted as √20, is a value that, when multiplied by itself, equals 20. In simpler terms, it’s the number that, when squared, gives the result 20. To calculate the square root
of 20, we seek a number that, when multiplied by itself, equals 20. While the square root of 20 is not a whole number, it is an irrational number. Mathematically, the square root of 20 is
approximately 4.47213595499958. Calculating the square root of 20 involves various methods such as long division, approximation techniques like Newton’s method, or using calculators with square root
functions. Understanding the square root of 20 is crucial in mathematics, especially in geometry, algebra, and calculus, where it is used to solve equations and find unknown sides or dimensions in
geometric shapes.
Square Root of 20: 4.47213595499958
Exponential Form: 20^½ or 20^0.5
Radical Form: √20
Is the Square Root of 20 Rational or Irrational?
The square root of 20 is an irrational number.
To understand why the square root of 20 is irrational, let’s delve into the definitions of rational and irrational numbers.
Firstly, let’s define these terms:
• Rational Number: A rational number can be expressed as the quotient of two integers, where the denominator is not zero. Rational numbers have either terminating or repeating decimal
• Irrational Number: An irrational number cannot be expressed as a simple fraction of two integers. Their decimal representations are non-repeating and non-terminating.
Square Root of 20 as Irrational:
When we calculate the square root of 20, we find that it cannot be expressed as a fraction of two integers.
Its decimal expansion, approximately 4.47213595499958, goes on infinitely without repeating a pattern.
To understand why the square root of 20 is irrational, let’s simplify it:
√20 = √(16 + 4) = √16 ×√4 = 4 × 2 = 8
Now, we know that 8 is a rational number. However, √4 is famously rational. It cannot be represented as a fraction of two integers, and its decimal form goes on forever without repeating. Therefore,
8 is also irrational because the product of a rational number (8) and an irrational number (√4) is always irrational.
In summary, the square root of 20 is irrational because it simplifies to a rational number (8), and when combined with the irrational √4, the result is always irrational.
Method to Find Value of Root 20
Finding the value of the square root of 20 involves various methods, each with its own approach to determine an approximate value of √20. Here are some common methods explained:
Long Division Method: In the long division method, we iteratively refine an initial guess through a series of divisions until reaching a satisfactory level of precision. We identify a perfect square
close to 20 and perform long division to obtain the square root.
Approximation Techniques: Techniques such as Newton’s method or the Babylonian method can be employed to approximate the square root of 20. These methods involve iteratively refining an initial guess
to converge towards the actual value of √20.
Using Calculators or Software: Modern calculators and mathematical software programs come equipped with built-in functions to directly compute the square root of a number. This offers a quick and
accurate way to find the value of √20 without manual calculation.
Factorization: Another method involves the prime factorization of 20 and grouping the factors into pairs. By extracting one factor from each pair, we can obtain an approximation of the square root of
Square Root of 20 by Long Division Method
Finding the square root of 20 using the long division method involves a similar procedure:
Step 1: Preparation
Write 20 as 20.00 00 00, grouping digits in pairs from the decimal point. For 20, it looks like “20”.
Step 2: Find the Largest Square
Identify the largest square smaller than or equal to 20, which is 16 (4²). Place 4 above the line as the first digit of the root.
Step 3: Subtract and Bring Down
Subtract 16 from 20 to get 4, then bring down the next pair of zeros to make it 400.
Step 4: Double and Find the Next Digit
Double the current result (4) to get 8. Now, find a digit (X) such that 48 multiplied by X is less than or equal to 400. Here, X is 5, because 485×5=240.
Step 5: Repeat with Precision
Subtract 240 from 400 to get 160, bring down the next zeros to get 1600, then double the quotient (45) to get 90. Choose a digit (Y) so that 459 multiplied by Y is just under 1600.
Step 6: Finish at Desired Accuracy
Continue the process until reaching the desired level of accuracy. For the square root of 20, this method gives us about 4.472 as we extend the division.
20 is Perfect Square root or Not
A perfect square root is a number that can be expressed as the product of an integer multiplied by itself. For instance, 4 (2 × 2) and 9 (3 × 3) are perfect square roots. To determine whether 20 is a
perfect square root, we examine if it can be expressed as the square of an integer.
However, when we try to find an integer that, when multiplied by itself, equals 20, we realize that there are no such integers. In other words, there is no whole number x such that x × x = 20.
Therefore, 20 is not a perfect square root
Since 20 cannot be expressed as the product of two identical integers, it doesn’t have an integer square root. While 20 does have a square root (√20), it’s an irrational number and not the result of
multiplying any whole number by itself. Therefore, we conclude that 20 is not a perfect square root.
Is √20 a Real Number?
Yes, the square root of 20, denoted as √20, is a real number. A real number is any number that can be found on the number line, including both rational and irrational numbers. Since √20 exists on the
number line and is not an imaginary number (which involve the square root of negative numbers), it is classified as a real number.
Is 20 an Integer? Yes or No?
No, 20 is not an integer. An integer is a whole number that can be positive, negative, or zero. Since 20 is not a whole number and includes a fractional part, it does not fall under the category of
What Square Root is Between 20?
The square root of 20 lies between the integers 4 and 5. While the exact value of √20 is approximately 4.47213595499958, it’s important to note that it falls between the perfect squares of 4 (which
is 16) and 5 (which is 25). Therefore, we can say that the square root of 20 is greater than 4 but less than 5. | {"url":"https://www.examples.com/maths/square-and-square-root-of-20.html","timestamp":"2024-11-12T09:11:11Z","content_type":"text/html","content_length":"107999","record_id":"<urn:uuid:cce7e111-e70b-4790-8d45-815c3607ef1d>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00802.warc.gz"} |
Mathematical Analysis of Three-Diode Model with P&O MPPT using MATLAB/Simulink
Volume 09, Issue 06 (June 2020)
Mathematical Analysis of Three-Diode Model with P&O MPPT using MATLAB/Simulink
DOI : 10.17577/IJERTV9IS060955
Download Full-Text PDF Cite this Publication
Kritika Rana, 2020, Mathematical Analysis of Three-Diode Model with P&O MPPT using MATLAB/Simulink, INTERNATIONAL JOURNAL OF ENGINEERING RESEARCH & TECHNOLOGY (IJERT) Volume 09, Issue 06 (June 2020),
• Open Access
• Authors : Kritika Rana
• Paper ID : IJERTV9IS060955
• Volume & Issue : Volume 09, Issue 06 (June 2020)
• Published (First Online): 04-07-2020
• ISSN (Online) : 2278-0181
• Publisher Name : IJERT
• License: This work is licensed under a Creative Commons Attribution 4.0 International License
Text Only Version
Mathematical Analysis of Three-Diode Model with P&O MPPT using MATLAB/Simulink
Kritika Rana
Student, Dept. of ECE,
Indira Gandhi Delhi Technical University for Women, New Delhi, India
Abstract: With the growing need for renewable energy resources, photovoltaic based energy generation has gained significant importance due to its high reliability owing to the abundant availability
of sunlight and the direct conversion of light into electricity. But the overall efficiency of conversion is still low as compared to cost input in the system. Hence for designing highly efficient
systems, successful simulation studies are required. The research paper insights the mathematical modeling of a three- diode model and its performance are analyzed in comparison with a single diode
and two-diode model. The model's performance is also analyzed under different atmospheric conditions with MPPT.
Keywords- Renewable energy, Photovoltaic (PV), Three- diode model, maximum power point tracking, boost converter, MATLAB
1. INTRODUCTION
Intending to lower the carbon emissions and reduce the dependency on non-renewable energy resources there has been significant technological development for utilizing renewable energy resources,
such as Wind, Solar, Hydro, and many more.
photovoltaic effect, electron-hole pairs are generated when incident photons have energy higher than the band-gap energy of the material. This generation and recombination are responsible for
electricity generation in the device [3]. In general, a PV module is modeled based on different series and parallel combinations of solar cells concerning the power requirement. With a series
connection, the voltage profile of the module is improved and with parallel the current profile. The output from the PV module is dependent on operating temperature, irradiance, angle of
irradiance, and resistance (series and parallel).
1. Single Diode Model
In this model, a single diode is connected in parallel with a current source, a parallel resistance, and a series resistance. The output current equation is:
= [ + ()] (/1000)
But for a long time, the focus has been on solar-based energy generation mainly due to high availability and easy conversion
= [exp (
of light to electricity. PV energy conversion is static and does not require any rotating parts in the system for conversion. The
Where I
PV is the photon current generated due to incident light,
system can be designed with basic semiconductor devices including diodes and transistors. But even with these benefits, the solar-based generation suffers from one major drawback, i.e.,
low efficiency in comparison to high cost for setting up the system. Hence various models have been presented for solar cell modeling including single diode, two-diode, and three- diode
model. The paper covers the performance analysis of all three models for varying atmospheric conditions. There is non- linearity in the I-V and P-V characteristics during the operation,
due to varying atmospheric conditions [1]. To maximize the output from the PV module it is operated at the maximum power point, yielding maximum output concerning atmospheric conditions.
The most used method for maximum power point tracking is the Perturb-and-Observe method due to its high efficiency and easy employability. The MPPT is used to control the duty cycle for
the switching element of the boost converter under varying conditions. The main reason for using the boost converter is to avoid using transformers, to reduce the overall power loss of
the circuit and limitation of switching frequency associated with it [2].
2. PV MODULE
A photovoltaic cell is a P-N junction diode that is responsible for generating electricity when exposed to sunlight. This generation is mainly due to the recombination of electron-hole pairs,
which are generated due to the photovoltaic effect. In the
Io is the reverse saturation current of the diode, VT is the thermal voltage of diode and is the diode ideality factor. The other parameters Rs (series resistance) and RP (parallel resistance)
are adjusted as per the requirement. The single diode equivalent model is given in Figure1a.
Fig 1a: Single Diode PV model
1. Two diode model
In the two diode model, there are two diodes (both in saturation), series resistance, parallel resistance, and current source. The mathematical model for a two-diode model of a solar cell
is given as,
= 1 2
= [ + ()] (/1000)
1 = 1 [exp ( )]
2 = 2 [exp ( )]
Where IPV is the photon current generated due to incident light, Io1 and Io2 are the reverse saturation current of the diodes (which in general are kept equal), VT1 and VT2 are the
thermal voltages of the diodes and 1 and 2 are the ideality factors for the diodes. The two-diode equivalent model is given in Figure1b and its extraction parameters are taken from the
article [1].
Fig 1b: Two-Diode PV model
It has already been established that the performance of a two- diode model is similar to single diode model provided, the ideality factors are set such that 1 is unity and 2 is between 1
2 1.2, hence the performance analysis done here is for two-diode model and three-diode model
2. Three-diode model
In the three diode model, there are three diodes (both in saturation), series resistance, parallel resistance, and current source. The mathematical model for a three-diode model of a solar cell
is given as,
= 1 2 3
= [ + ()] (/1000)
Fig 1c: Three-Diode PV model
Table1: Three-Diode model parameters
Parameter Value
Open circuit voltage (Voc) 21.1 V
Short Circuit Current (Isc) Series Resistance (Rs) 3.8 A 0.205
Shunt Resistance (R) 578.38
ki 0.065
kv -0.08
Io1 4.98e-08 A
Io2 7.24e-10 A
Io3 1.42e-07 A
1 1.282
2 1.8043
3 1.4364
The performance analysis for two-diode model and three-diode model is given in Figures 2a, 2b, 2c, and 2d.
1 = 1 [exp ( )]
2 = 2 [exp ( )]
3 = 3 [exp ( )]
Figure 2a: IV characteristics at Irradiance of 500 W/m2
Where IPV is the photon current generated due to incident light, Io1, Io2, and Io3 are the reverse saturation current of the diodes (which in general are kept equal), VT1, VT2 and VT3 are the
thermal voltages of the diodes and 1, 2 and 3 are the ideality factors for the diodes.
The three-diode equivalent model is given in Figure1c and its extraction parameters are taken from the article [4], given in Table 1
Figure 2b: IV characteristics at Irradiance of 1000 W/m2
Figure 2c: PV characteristics at Irradiance of 500 W/m2
Figure 2d: PV characteristics at Irradiance of 1000 W/m2
It is evident that with an increase in the number of diodes in the circuit the open-circuit voltage decreases, provided the ideality factors are maintained constant [5]. But when the ideality
factor of diodes is varied then the results obtained are given in Figure 3, and the values for the respective ideality factor are given in table 2.
Table2: Variation of open circuit voltage with varying ideality factors of diode
Case 1 2 3 Open circuit Voltage (V)
1. 1.282 1.282 1.282 20.4
2. 1.282 1.282 1.382 21.32
3. 1.282 1.382 1.382 21.33
4. 1.282 1.382 1.484 21.75
Figure3: IV characteristic variation with varying ideality factor.
4. P&O MPPT
Due to varying atmospheric conditions, there is a non-linearity in PV characteristics hence distinct maximum power points, so there is a need to maximize the power output from the PV module
during these varying conditions to increase the efficiency [6]. Hence, we use maximum power point tracking
along with PV modules to get maximum power for current atmospheric conditions. There are many algorithms for maximum power point tracking, but the most preferred algorithm is Perturb and Observe
algorithm. The algorithm is represented via the flow chart in figure 4. The algorithm is easy to implement and requires a low number of parameters to function. The major drawback of the algorithm
is the oscillations around the maximum power point, for which different controllers are used with this.
Figure 4: P&O Algorithm
5. BOOST CONVERTER
The DC-DC converter is used to regulate the output voltage with the help of high-frequency switching, inductors, and capacitor with respect to the unregulated input voltage. The output voltage is
varied based on the duty cycle of the switching element and other devices are used to minimize signal noise that occurs in the output due to the presence of non-linear devices. The converters can
either scale up the voltage (boost converter) or scale down the voltage (buck converter). In the paper, a boost converter is used to scale up the voltage from the PV module to support the load.
The switching element used is an IGBT due to its higher efficiency compared to MOSFET and BJT. The Simulink model of boost converter used in the paper is given in figure 5
Figure 5: Boost converter
The duty cycle of IGBT is controlled via MPPT, to regulate the duty cycle under varying atmospheric conditions for maximum output. The switching frequency of IGBT is kept such that to avoid high
switching losses [7]. Ideal boost converter equation is given as:
= 1
8. REFERENCE
1. Basha, C. H., Rani, C., Brisilla, R. M., & Odofin, S. (2020). Mathematical Design and Analysis of Photovoltaic Cell Using MATLAB/Simulink. In Soft Computing for Problem Solving (pp. 711-726).
Springer, Singapore.
2. Rosas-Caro, J. C., Ramirez, J. M., Peng, F. Z., & Valderrabano, A. (2010). A DCDC multilevel boost converter. IET Power Electronics, 3(1), 129-137
3. Babu, B. C., Cermak, T., Gurjar, S., Leonowicz, Z. M., & Piegari,
L. (2015, June). Analysis of mathematical modeling of PV module
where Vout is the output voltage and Vin is the input DC voltage. D is the duty cycle for the pulses given to the switching element. With advancing technology multilevel DC-DC converters are
used with inverters in PV based generation systems
6. PERFORMANCE WITH MPPT
This section covers the performance analysis of the two-diode model and three-diode model with P&O based MPPT. The parameters for the module are used as in [5]. All the analysis is done using a
mathematical model of the module in MATLAB/Simulink.
Figure 6: Output voltage variation with MPPT
7. CONCLUSION
It is evident from the analysis that if diodes are increased in the photovoltaic module, the effective output voltage will decrease. But if the ideality factor of the diodes is optimized the
performance of the circuit can be improved. Further research can be done to provide an algorithm that can optimize the ideality factor ratio of the diodes for optimum performance, and use of
different controllers with MPPT circuit to prevent oscillations and provide a smooth curve till steady state is achieved.
with MPPT algorithm. In 2015 IEEE 15th International Conference on Environment and Electrical Engineering (EEEIC) (pp. 1625- 1630). IEEE.
1. Qais, M. H., Hasanien, H. M., & Alghuwainem, S. (2019). Identification of electrical parameters for three-diode photovoltaic model using analytical and sunflower optimization algorithm. Applied
Energy, 250, 109-117.
2. Ukoima, K. N., Ekwe, O. A,(2019, June). THREE-DIODE MODEL AND SIMULATION OF PHOTOVOLTAIC (PV) CELLS. In 2019 Umudike Journal of Engineering and Technology (UJET). Pp. 108
3. Anto, E. K., Asumadu, J. A., & Okyere, P. Y. (2016, June). PID control for improving P&O-MPPT performance of a grid-connected solar PV system with Ziegler-Nichols tuning method. In 2016 IEEE 11th
Conference on Industrial Electronics and Applications (ICIEA) (pp. 1847-1852). IEEE.
4. Hasaneen, B. M., & Mohammed, A. A. E. (2008, March). Design and simulation of DC/DC boost converter. In 2008 12th International Middle-East Power System Conference (pp. 335-340). IEEE
You must be logged in to post a comment. | {"url":"https://www.ijert.org/mathematical-analysis-of-three-diode-model-with-po-mppt-using-matlab-simulink","timestamp":"2024-11-11T03:56:03Z","content_type":"text/html","content_length":"76103","record_id":"<urn:uuid:9caa93ef-141e-495c-a296-653e4a293a79>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00069.warc.gz"} |
"A Neat Trick!" solution
"A Neat Trick!" solution Archives
8/21/2020 Categories
0 Comments Authors
The trick to adding the integers 1 to 100 quickly is to do it twice.
No, seriously.
But be creative in how you order the numbers, and write the second group in reverse order:
\( (1+2+ \ldots +99+100)+(100+99+ \ldots +2+1) \)
Now, how does that help us? Well, it doesn't….yet. But if we arrange them a little differently into columns, and add each column first, we have
\[ \array{ &1 & +& 2&+& \ldots &+ & 99 & + & 100& \\ +&100&+&99&+& \ldots &+& 2 &+&1&\\ \hline \\ =&101&+&101&+& \ldots &+&101&+&101& } \]
Do you see what we did? We just rewrote the numbers so that all of the columns would be equal, and then summed each column. Since we know there are \(100 \) columns, we can now use
multiplication (much faster than addition) to see that the above is equal to \(100 \times 101=10,100 \).
But we're not done yet. In order to make our columns equal, we had to add our sequence twice. So now, to get our final answer, we need to divide \(10,100\) by \(2\) to get \(5050\).
[And just remember, a 6-year-old figured this out! Of course, that 6-year-old was none other than Carl Friedrich Gauss, one of the greatest mathematicians of all time. And we really
don't know that he was 6, we just know he was in primary school. But still pretty impressive, wouldn't you say?]
Now, if you look closely you'll see that there's nothing special about the number 100 in this trick. Any number will work. And you don't even have to remember the steps we took. Let's
just make a formula, and then you'll have everything you need.
General Formula
Suppose \(N \) is a positive integer, and we want to add the numbers \(1 \) through \(N \), written as \(1+2+ \ldots +(N-1)+N \)
As before, let's add them twice and line them up in columns:
\[ \array{ &1 & +& 2&+& \ldots &+ & (N-1) & + & N& \\ +&N&+&(N-1)&+& \ldots &+& 2 &+&1&\\ \hline \\ =&(N+1)&+&(N+1)&+& \ldots &+&(N+1)&+&(N+1)& } \]
This is equal to \((N+1) \times N \), and since we added everything twice to do our trick, we need to divide by 2. And this gives our formula:
\[ 1+2+ \ldots +(N-1)+N= \frac{N \times (N+1)}{2} \].
Go ahead and try it out!! It works every time.
And here is some homework for you to try.
First, try this one:
1. Calculate \(1+2+ \ldots +37+38 \).
Now let's see if you can adapt your thinking to these:
2. Calculate \(2+4+ \ldots +98+100 \).
3. Calculate \(20+21+22+ \ldots +79+80 \).
4. Calculate \( 37+40+43+46+ \ldots +94+97+100\).
If you have trouble adapting the formula for those last three, you can always add them twice and line up the columns, right?
If you like our puzzles and explanations, please visit our store and check out our problem-solving and logic puzzle books!
0 Comments
The trick to adding the integers 1 to 100 quickly is to do it twice.
No, seriously.
But be creative in how you order the numbers, and write the second group in reverse order:
\( (1+2+ \ldots +99+100)+(100+99+ \ldots +2+1) \) | {"url":"http://www.themathprofs.com/blog/a-neat-trick-solution","timestamp":"2024-11-02T13:45:41Z","content_type":"text/html","content_length":"93584","record_id":"<urn:uuid:5d69141f-f6dd-43a3-a324-4126d9a91624>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00432.warc.gz"} |
Square Calculator
Calculations at a square or regular tetragon. A square is a convex quadrilateral with four right angles and four edges of equal length.
Enter one value and choose the number of decimal places. Then click Calculate.
d = √2 * a
p = 4 * a
A = a²
r[c] = a / √2
r[i] = a / 2
Angle: 90°
2 diagonals
Edge length, diagonal, perimeter and radius have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).
Diagonals and bisecting lines coincide, they intersect with the median lines and with centroid, circumcircle and incircle center in one point. To this, the square is point symmetric and rotationally
symmetric at a rotation of 90° or multiples of this. Furthermore, the square is axially symmetric to the diagonals and to the median lines.
perimeter p, area A sides and angles diagonals = bisecting lines
median lines incircle and circumcircle
The square is the namesake of the mathematical method of squaring. A number squared is multiplied by itself. If the number has a unit, then this is also squared, as with the square, where the length
and width are the same and when multiplied together they give the area. The reverse calculation is the square root; if this is taken from the area of a square, then the side length results again. A
diagonally halved square produces two equal isosceles and also right triangles. Divided by the bisector, two equal rectangles are created, with the short side half as long as the long side. The
squaring of the circle has become a metaphor for an unsolvable problem, which describes the impossibility of constructing a square with the same area from a circle or vice versa. For mathematical
approximation by calculation, see the squaring the circle calculator.
Jumk.de Webprojects
Online Calculators | {"url":"https://rechneronline.de/pi/square-calculator.php","timestamp":"2024-11-14T08:16:14Z","content_type":"text/html","content_length":"31219","record_id":"<urn:uuid:6c1aa82c-551a-45fe-baea-1f9044e33996>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00204.warc.gz"} |
Edgeworth and Moment Approximations: The Case of MM and QML Estimators for the MA (1) Models
Kyriakopoulou, Dimitra and Demos, Antonis (2010): Edgeworth and Moment Approximations: The Case of MM and QML Estimators for the MA (1) Models. Published in:
Preview Download (413kB) | Preview
Extending the results in Sargan (1976) and Tanaka (1984), we derive the asymptotic expansions, of the Edgeworth and Nagar type, of the MM and QML estimators of the 1^{st} order autocorrelation and
the MA parameter for the MA(1) model. It turns out that the asymptotic properties of the estimators depend on whether the mean of the process is known or estimated. A comparison of the Nagar
expansions, either in terms of bias or MSE, reveals that there is not uniform superiority of neither of the estimators, when the mean of the process is estimated. This is also confirmed by
simulations. In the zero-mean case, and on theoretical grounds, the QMLEs are superior to the MM ones in both bias and MSE terms. The results presented here are important for deciding on the
estimation method we choose, as well as for bias reduction and increasing the efficiency of the estimators.
Item Type: MPRA Paper
Original Edgeworth and Moment Approximations: The Case of MM and QML Estimators for the MA (1) Models
Language: English
Keywords: Edgeworth expansion, moving average process, method of moments, Quasi Maximum Likelihood, autocorrelation, asymptotic properties
C - Mathematical and Quantitative Methods > C0 - General
Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling
Y - Miscellaneous Categories > Y1 - Data: Tables and Charts
Item ID: 122393
Depositing Prof. Phoebe Koundouri
Date 17 Oct 2024 13:44
Last 17 Oct 2024 13:45
Ali, M.M. 1984. Distributions of the sample autocorrelations when observations are from a stationary autoregressive-moving-average process. Journal of Business and Economic Statistics
2, 271-278.
Andrews, D.W.K. 1999. Estimation when a parameter is on a boundary. Econometrica 67, 1341-1383.
Andrews, D.W.K. and O. Lieberman 2005. Valid edgeworth expansions for the Whittle maximum likelihood estimator for stationary long-memory Gaussian time series. Econometric Theory 21,
Arvanitis, S. and A. Demos 2006. Bias Properties of Three Indirect Inference Estimators, mimeo Athens University of Economics and Business.
Barndorff-Nielsen, O.E. and D.R. Cox 1989. Asymptotic Techniques for Use in Statistics. Chapman and Hall.
Bhattacharya, R.N. and J.K. Ghosh 1978. On the validity of the formal Edgeworth Expansion. The Annals of Statistics 6, 434-451.
Bao, Y. and A. Ullah 2007. The second-order bias and mean squared error of estimators in time-series models. Journal of Econometrics 140, 650-669.
Calzolari, G., G. Fiorentini and E. Sentana 2004. Constrained Indirect Estimation. Review of Economic Studies 71, 945-973.
Chambers, J.M. 1967. On methods of asymptotic approximation for multivariate distributions. Biometrika 54, 367-383.
Corradi, V. and E.M. Iglesias (2008). Bootstrap refinements for QML estimators of the GARCH(1,1) parameters. Journal of Econometrics 144, 400-510.
Edgeworth and Moment Expansions Davidson, J.E.H. (1981. Problems with the Estimation of Moving Average Processes. Journal of Econometrics 16, 295-310.
Davis, R. and S. Resnick 1986. Limit theory for the sample covariance and correlation functions of moving averages. Annals of Statistics 14, 533-558.
Durbin, J. 1959. E¢ cient estimation of parameters in Moving-Average models. Biometrika 46, 306-316.
Durbin, J. 1980. Approximations for densities of su¢ cient estimators. Biometrika 67, 311-333.
Gotze, F. and C. Hipp 1983. Asymptotic Expansions fro Sums of Weakly De- pendent Random Variables. Theory of Probability and its Applications 64, 211-239.
Gotze, F. and C. Hipp 1994. Asymptotic distribution of statistics in time series. The Annals of Statistics 22, 2062-2088.
Gourieroux, C., A. Monfort and E. Renault 1993. Indirect Inference. Journal of Applied Econometrics 8, S85-S118.
Hall, P. 1992. The bootstrap and edgeworth expansion. Springer.
Hall, P. and J.L. Horowitz 1996. Bootstrap Critical Values for Tests Based on Generelized-Method-of Moments Estimators. Econometrica 64, 891-916.
References: Iglesias, E.M. and O.B. Linton 2007. Higher order asymptotic theory when a parameter is on a boundary with an application to GARCH models. Econometric Theory 23, 1136-1161.
Iglesias, E.M. and G.D.A. Phillips 2008. Finite sample theory of QMLE in ARCH models with dynamics in the mean equation. Journal of Time Series Analysis 29, 719-737.
Kakizawa, Y. 1999. Valid Edgeworth expansions of some estimators and bootstrap conÖdence intervals in Örst-order autoregression. Journal of Time Series Analysis 20, 343-359.
Kan, R. and X. Wang 2010. On the distribution of the sample autocorrelation coefficients. Journal of Econometrics 154, 101-121.
Lieberman, O., J. Rousseau and D.M. Zucker 2003. Valid asymptotic expansions Edgeworth and Moment Expansions for the maximum likelihood estimator of the parameter of a stationary,
Gaussian, strongly dependent process. The Annals of Statistics 31, 586-612.
Linnik, Y.V. and N.M. Mitrofanova 1965. Some asymptotic expansions for the distribution of the maximum likelihood estimate. Sankhya A 27, 73-82.
Linton, O. 1997. An asymptotic expansion in the GARCH(1,1) model. Econo- metric Theory 13, 558-581.
Mitrofanova, N.M. 1967. An asymptotic expansion for the maximum likelihood estimate of a vector parameter. Theory of Probability and its Applications 12, 364-372.
Nagar, A.L. 1959. The bias and moment matrix of the general k-class estimators of the parameters in simultaneous equations. Econometrica 27, 575-595.
Ogasawara, H. 2006. Asymptotic expansion of the sample correlation coefficient under nonnormality. Computational Statistics and Data Analysis 50, 891-910.
Phillips, P.C.B. 1977. Approximations to some Önite sample distributions associated with a Örst-order stochastic di§erence equation. Econometrica 45, 463-485.
Rothenberg, T.J. 1986. Approximating the distributions of econometric estimators and test statistics, In The Handbook of Econometrics. vol. II, Amsterdam: North-Holland.
Sargan, J.D. 1974. Validity of Nagarís expansion. Econometrica 42, 169-176.
Sargan, J.D. 1976. Econometric estimators and the Edgeworth approximation. Econometrica 44, 421-448.
Sargan, J.D. 1977. Erratum "Econometric estimators and the Edgeworth ap- proximation". Econometrica 45, 272.
Sargan, J.D. 1988. Econometric estimators and the Edgeworth approximation. Contributions to Econometrics vol. 2, 98-132. E. Maasoumi editor. Cambridge University Press.
Sargan, J.D. and S.E. Satchell 1986. A theorem of validity for Edgeworth ex- pansions. Econometrica 54, 189-213.
Tanaka, K. 1983. Asymptotic expansions associated with the AR(1) model with Edgeworth and Moment Expansions unknown mean. Econometrica 51, 1221-1232.
Tanaka, K. 1984. An asymptotic expansion associated with the maximum likelihood estimators in ARMA models. Journal of Royal Statistical Society B 46, 58-67.
Taniguchi, M. 1987. Validity of Edgeworth expansions of minimum contrast estimators for Gaussian ARMA processes. Journal of Multivariate Analysis 21, 1-28.
Taniguchi, M. and Y. Kakizawa 2000. Asymptotic theory of statistical inference for time series. Springer Series in Statistics.
URI: https://mpra.ub.uni-muenchen.de/id/eprint/122393 | {"url":"https://mpra.ub.uni-muenchen.de/122393/","timestamp":"2024-11-06T06:08:26Z","content_type":"application/xhtml+xml","content_length":"36427","record_id":"<urn:uuid:450dfa47-3358-41f2-bfd3-28ba99369fb6>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00170.warc.gz"} |
Degrees of Disbelief
Meehl's Philosophical Psychology, Lecture 9, part 1.
Read →
I got this, just as I was sitting down to work on a paper about syntactic and semantic approaches to uncertainty and unawareness. Here's a link to a current draft
Expand full comment
Not Meehl related, but translation as a foundational building block of meaning has been a recent obsession of mine. Adding your paper to my queue. Thanks!
Expand full comment
This paper takes me back -- all the ideas related to connecting lattice-theoretic and probabilistic notions of uncertainty, plus the references to Halpern, Gilboa-Schmeidler, etc.
Expand full comment
We’ve been working on this for a long time, and there are still basic issues unresolved. But I think we have made some progress here.
Expand full comment
So when there's stuff like the AI Impacts survey asking people to estimate probabilities, and then they do statistics over those probabilities, is that doing type 2 over type 1? Is there a name for
this... technique 🤔
Expand full comment
I think it's called "Doomerism."
Expand full comment
I guess Bayesians would say that both are equally fundamental to understanding the unknown.
Expand full comment
Frequentists would agree! The two differ in how you move between the two. More on this topic coming later this week.
Expand full comment
As I was listening to that lecture, I kept nodding in agreement because I articulated more or less the same dichotomy here: https://realizable.substack.com/p/probabilities-coherence-correspondence.
Expand full comment
By the way, at one point Meehl recommends two chapters from Vic Barnett’s book on comparative statistical inference as a good reference on the interpretations of probability. Check them out if you
haven’t already, they are excellent.
Expand full comment
Yeah, Barnett's book is excellent cover to cover. Have you read Hacking's "Logic of Statistical Inference?" Also great at picking at the dappled nature of inference. And I also really like Wesley
Salmon's discussion of the problem of probability in "The Foundations of Scientific Inference."
Expand full comment
I've leafed through Hacking. Currently making my way through Mary Hesse's "Structure of Scientific Inference," which is (like Hesse in general) hugely underrated.
Expand full comment
What I like about Barnett, Hacking, and Salmon is I walked away more confused but also more relaxed. The statistical dogmatists are much more stressful to read.
Expand full comment | {"url":"https://www.argmin.net/p/degrees-of-disbelief/comments","timestamp":"2024-11-08T23:55:29Z","content_type":"text/html","content_length":"214312","record_id":"<urn:uuid:46975297-1e68-43bd-a90f-ae1ae287749e>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00468.warc.gz"} |
Full state-feedback solution for a flywheel based satellite energy and attitude control scheme
A hybrid system combining the energy and attitude control task uses flywheels to store energy and control the attitude of small satellites. Various journal papers containing previous works have
recognized this combined architecture. However, due to the uncertainties of on-board performances, it is a challenge in terms of attitude pointing accuracy. Therefore, this paper focuses a full
state-feedback control solution to increase the satellite attitude performances. Mathematical model and numerical treatments for full state-feedback control of combined energy and attitude regulating
scheme for a small satellite are presented. Simulation results show that an enhanced pitch pointing accuracy can be achieved up to 0.0010 for the proposed control approach. The paper contains an over
view of a flywheel architecture along with state space representation of the scheme. A brief description of conventional control scheme is also presented with sample simulated results for comparison.
Design of a full state-feedback controller and analysis of simulated results are also presented to show the achieved attitude performances specifically.
1. Introduction
The paper introduces a control architecture of two-degree-of-freedom to enhance the pointing accuracy of small satellite. The architecture is constructed based on double rotating flywheel. Both
energy storage and attitude control tasks are performed by this architecture at the same time. The flywheel is much better than other conventional energy storage devices. It is not sensitive on
temperature variation. Its life cycle is longer and depth of discharge (DoD) is higher. The rotational kinetic energy is stored while the flywheel is rotated by an electric motor. Stored kinetic
energy is used to run an electric generator to produce electricity as per demand. Although the flywheel energy storage system has high rate of charging and discharging capacity yet the construction
is simple and less massive than other energy storage devices such as lithium ion battery.
Rotating flywheels are potential tools to perform attitude control for spacecraft. It saves the mass significantly while the flywheel system functions for both energy storage and attitude control
[1]. Standard double rotating flywheels architecture is shown in Fig. 1 that contains control elements, motor/generator and magnetic bearings to store energy and produce on demand attitude command.
Roithmayr [2] proposed this idea and it is implemented in International Space Station (ISS) in which a composite flywheel system simultaneously satisfies the energy and attitude pointing requirement.
The idea was thoroughly investigated by Varatharajoo [3, 4] in his works. The idea is tested based on two modes of flywheel inputs; one is speed-mode other is torque-mode. The authors of these papers
have contributed on torque mode based combined energy and attitude control scheme design and simulation for different types of small satellites [6-10]. The types of small satellite are decided
according to the mass of satellite such as, 10 kg (nano satellite), 50 kg (micro satellite) and 100 kg (enhanced micro satellite). Those works conducted a full system design and arithmetic modeling
of a hybrid energy and attitude control scheme. Following the study, it shows that the proposed combined system is a promising candidate for mass budget, architecture volume, and energy consumption.
In terms of attitude control performance, the numerical simulation results are presented for this combined system to meet the mission goals while using PID controller [3, 4, 6-11].
Fig. 1Satellite flywheel architecture
PID controller is one of the mostly used controllers for aerospace industries. In general, there are various techniques to tune the PID controller. A chaotic ant colony algorithm with the adaptive
multi-strategies (CASOAMS) is one of those techniques to optimize the parameters of PID controller [12]. Fuzzy PID control technique is another modified PID control method and it is suitable to
control a multi flexible body dynamics (MFBD) like satellite systems [13]. Uncertainties of satellite attitude error can also be reduced using incremental support learning method [14]. Perfect
estimation of rotational movement is also crucial to know the current attitude of satellite to be controlled. A suitable method of motion estimation demonstrated in [15], that can be implemented to
know the real-time data for satellite attitude control.
However, among several types of PID controller, a few of them is tested to enhance the pitch pointing accuracy of flywheel based combined energy and attitude control scheme for small satellite. For
example, the combined architecture is demonstrated using PID-Active force control (AFC) method and obtained better enhancement of satellite attitude pointing up to 0.01° [16] for ideal case.
Nevertheless, the AFC extensively depends on the active inertial measurement system which demands time to develop. Moreover, the conducted analysis was confined for speed mode control instead of
torque mode. Other than the PID based controller, a ${H}_{\alpha }$ optimal control technique is demonstrated to further enhancement of satellite pitch pointing performance for the same flywheel
based combined scheme [17]. Purpose of this control algorithm is to minimize a quadratic cost function of dynamical system described by a set of linear differential equation. Simulated results show
that the achieved pitch pointing accuracy of the satellite is enhanced up to 0.018° for ideal case.
All those techniques are not based on state space approaches to control such a hybrid system for regulating and enhancing the satellite energy and attitude performances. Therefore, here is a
technical gap to design the controller for flywheel based combined architecture. Consequently, this paper attempts to implement a full state-feedback control method based on state space approach, to
improve the satellite pitch pointing accuracy. Further, it can be mentioned that, the performance of energy storage and roll/yaw pointing accuracy remains optimal according to the series of works
done previously; and consequently, it will not be demonstrated herein.
It can be specifically mentioned that a different method is used in this work which is dissimilar to the method used in previously published works demonstrated in [6-10, 16, 17]. Again, the new
result is far better than the former results in terms of pitch pointing accuracy for small satellite control and a complete view of comparison is shown in Tabular form at the end of this paper.
Fig. 2Simplified open-loop block diagram representation of hybrid control system
The rest of present paper is arranged as follow: The modeling of flywheel architecture has been presented in Section No. 2. The Section No. 3 briefly described the conventional method of control
architecture with sample simulation result. State space representation of flywheel based hybrid system for energy and attitude control system is explained in Section No. 4. Full state-feedback
controller is designed for this combined system in Section No. 5. The Section No. 6 presents the simulation and analysis of the results for pitch pointing accuracy for a small satellite 100 kg based
on combined energy and attitude control scheme. Section No. 7 concluded the paper explaining enhanced performance of the proposed control method.
2. Flywheel architectural model
As shown in Fig. 1, a counter rotating high speed dual composite rotors are mounted along the same spinning axis. It contains control elements, motor-generator and magnetic for energy-attitude
management [3, 6]. Basically, the solar panel will generate electricity which will be stored as kinetic energy in the flywheels. During the charging phase, the motor will speed-up both flywheels and
slow-down them during the discharging phase. In order to induce the required control torque for attitude corrections, both flywheels will be rotated at different speeds. A simplified open loop model
is obtained from the previous works performed [3, 6] and shown in Fig. 2. The open loop transfer function for a third order flywheel based combined energy and attitude control system is given as
$\frac{{\theta }_{sat}\left(s\right)}{{\theta }_{ref}\left(s\right)}=\frac{2K{I}_{\omega }}{{I}_{y}{\tau }_{\omega }{s}^{3}+{I}_{y}{s}^{2}}.$
Here, $s$ denotes the Laplace variable and the motor-generator torque constant $K$ is assumed as unity [3, 6]. ${\theta }_{ref}$[.] and ${\theta }_{sat}$ are reference and actual satellite pitch
angle. ${\tau }_{\omega }$ is system response time constant and ${\omega }_{0}$ is the satellite orbit rate. ${I}_{\omega }$ stands for flywheel inertia while ${I}_{y}$ is for satellite moment of
inertia in pitch direction. By having similar yaw and roll moments of inertias, the satellite pitch dynamics can be evaluated independently [3, 6, 7].
3. Conventional control method and architecture
Conventional methods contributed two separate method for modeling and controlling the such combined architecture for satellite energy and attitude direction: (1) speed modes based and (2) torque mode
based scheme [3, 6, 7]. Torque mode based control is shown in Fig. 3. Here, the satellite attitude is influenced by the torque produced from counter rotating double flywheel mechanism. Both flywheels
are responsible for energy charging and discharging phases. Figure shows that Proportional-Derivative controller is used to produce the control command, ${T}_{cmd}$ for attitude correction. Real time
satellite pitch angle ${\theta }_{sat}$[.] is generated from start sensor or gyroscope and used as an attitude feedback to compare with the reference orientation. In the architecture, ${T}^{s/w}$ and
${T}^{w/s}$ are projection matrices to transfer the control command from satellite coordinate frame to flywheel coordinate frame. These matrices are responsible for both flywheels respectively. Equal
inertia assumption is taken for roll (${I}_{x}$) and yaw (${I}_{z}$) direction of the satellite.
3.1. Simulation results of conventional scheme
Detail mathematical model of this conventional method is presented by [3, 6, 7]. Numerical treatment is performed for both ideal and non-ideal cases. Ideal case is defined while the external
disturbances are considered only. Non-ideal case is demonstrated including both internal and
Fig. 3Conventional combined attitude and energy control architecture
Fig. 4Simulated attitude performance in conventional control scheme
External disturbances. Proportional and derivative gain parameters are calculated for simulation purpose, which are ${K}_{p}=$ 0.002177 and ${K}_{D}=$ 0.02656 respectively. While the damping ratio, $
\zeta =$ 1 and natural frequency, ${\omega }_{n}=$ 0.1636. Simulated attitude accuracy is shown in Fig. 4. It is observed that the pitch pointing accuracy is achieved approximately 0.15°.
4. State-space modeling
Based on the block diagram shown in Fig. 2, the state variables for third order system are defined as:
${x}_{1}\left(t\right)={\theta }_{sat}\left(t\right)=$ Pitch angular displacement of the satellite.
${x}_{2}\left(t\right)=\frac{{d\theta }_{sat}\left(t\right)}{dt}=$ Pitch angular velocity of the satellite.
${x}_{3}\left(t\right)=\frac{{{d}^{2}\theta }_{sat}\left(t\right)}{d{t}^{2}}=$ Pitch angular acceleration of the satellite.
Meanwhile, the state input and state output for the third system are defined as:
$u\left(t\right)={e}_{a}\left(t\right)=$ Input signal into the satellite dynamics.
$y\left(t\right)={x}_{1}\left(t\right)=$ Output signal from the satellite dynamics.
${\stackrel{˙}{x}}_{1}\left(t\right)=\frac{{dx}_{1}\left(t\right)}{dt}=\frac{{d\theta }_{sat}\left(t\right)}{dt}={x}_{2}\left(t\right),$
${\stackrel{˙}{x}}_{2}\left(t\right)=\frac{{dx}_{2}\left(t\right)}{dt}=\frac{{d}^{2}{\theta }_{sat}\left(t\right)}{d{t}^{2}}={x}_{3}\left(t\right),$
${\stackrel{˙}{x}}_{3}\left(t\right)=\frac{{dx}_{3}\left(t\right)}{dt}=\frac{{d}^{3}{\theta }_{sat}\left(t\right)}{d{t}^{3}}.$
By driving the Eq. (2) with the help of Eq. (1), it could be obtained that:
${\stackrel{˙}{x}}_{3}\left(t\right)=-\frac{1}{{\tau }_{\omega }}{x}_{3}\left(t\right)+\frac{1}{{I}_{y}{\tau }_{\omega }}{\theta }_{ref}\left(t\right).$
Therefore, the state-space representation of satellite dynamics in space matrix could be expressed in this form:
$\left[\begin{array}{c}{\stackrel{˙}{x}}_{1}\left(t\right)\\ {\stackrel{˙}{x}}_{2}\left(t\right)\\ {\stackrel{˙}{x}}_{3}\left(t\right)\end{array}\right]=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0&
0& -\frac{1}{{\tau }_{\omega }}\end{array}\right]\left[\begin{array}{c}{x}_{1}\left(t\right)\\ {x}_{2}\left(t\right)\\ {x}_{3}\left(t\right)\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ -\frac{1}
{{{I}_{y}\tau }_{\omega }}\end{array}\right]{\theta }_{ref}\left(t\right),$
$y\left(t\right)=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{1}\left(t\right)\\ {x}_{2}\left(t\right)\\ {x}_{3}\left(t\right)\end{array}\right].$
Let’s the dynamics of the hybrid flywheel system be represented by the following state and output equations respectively:
Fig. 5Full state-feedback control diagram of a hybrid satellite system
5. Full state-feedback controller design
All the state variables are feed into the input of the system through an appropriate feedback matrix in the control system design is known as the full state variable feedback control technique. Using
this approach, the pole placement method can be used to design the desired controller. State controllability is required to perform the pole placement design technique. If the control input $u$ of a
system can take each state variable, ${x}_{i}$ where $i=$ 1,…, $n$, from an initial state to a final state then the system is controllable, otherwise it is uncontrollable [18]. Therefore, the rank of
the controllability matrix $T=\left[B,AB,{A}^{2}B,\dots ,{A}^{n}B\right]$ should be equal to the number of states in the system.
Full states feedback control diagram of a hybrid flywheel based energy and attitude regulating scheme is illustrated in Fig. 5. The reference state is defined as:
${X}_{ref}=\left[\begin{array}{ccc}{\theta }_{ref}& 0& 0\end{array}\right],$
where ${\theta }_{ref}$ is desired angular displacement in pitch direction of satellite. The controller is:
Note that if ${X}_{ref}=$ 0, for the gain $K=$ [${k}_{1}$${k}_{2}$${k}_{3}$] the control law $u=-KX$ is applied in pole-placement algorithm. Solving for the gain $K$, first step is to define
companion matrices for $A$ and $B$ as:
$\stackrel{~}{A}=\left[\begin{array}{ccc}\begin{array}{cc}\begin{array}{c}0\\ 0\end{array}& \begin{array}{c}1\\ 0\end{array}\end{array}& \begin{array}{c}\cdots \\ \cdots \end{array}& \begin{array}
{cc}0& 0\\ 0& 0\end{array}\\ ⋮⋮& \ddots & ⋮⋮\\ \begin{array}{cc}0& 0\\ {-a}_{1}& {-a}_{2}\end{array}& \begin{array}{c}\cdots \\ \cdots \end{array}& \begin{array}{cc}0& 1\\ {-a}_{n-1}& {-a}_{n}\end
where ${s}^{n}+{a}_{n}{s}^{n-1}+\dots +{a}_{1}=$ 0 is the characteristic equation of $A$. And:
$\stackrel{~}{B}=\left[\begin{array}{c}0\\ ⋮\\ \begin{array}{c}0\\ 1\end{array}\end{array}\right].$
Second step is to compute $W=T{\stackrel{~}{T}}^{-1}$, where $\stackrel{~}{T}=\left[\stackrel{~}{B}$, $\stackrel{~}{A}\stackrel{~}{B}$, …, ${\stackrel{~}{A}}^{n}\stackrel{~}{B}$].
Third step is to calculate $\stackrel{~}{B}$ to designate the poles of $\stackrel{~}{A}-\stackrel{~}{B}\stackrel{~}{K}$ to the expected locations. Introducing the control law $u=-KX$ to the Eq. (12):
$\stackrel{~}{A}=\left[\begin{array}{ccc}\begin{array}{cc}\begin{array}{c}0\\ 0\end{array}& \begin{array}{c}1\\ 0\end{array}\end{array}& \begin{array}{c}\cdots \\ \cdots \end{array}& \begin{array}
{cc}0& 0\\ 0& 0\end{array}\\ ⋮⋮& \ddots & ⋮⋮\\ \begin{array}{cc}0& 0\\ {-a}_{1}-{k}_{1}& {-a}_{2}-{k}_{2}\end{array}& \begin{array}{c}\cdots \\ \cdots \end{array}& \begin{array}{cc}0& 1\\ {-a}_{n-1}-
{k}_{n-1}& {-a}_{n}-{k}_{n}\end{array}\end{array}\right].$
Finally, to obtain the feed-back gain for the system ($A$, $B$), find $K=$$\stackrel{~}{K}{W}^{-1}$. It is necessary to convert $\stackrel{~}{K}$→$K$ because ($A$, $B$) demonstrates the original
system while $\stackrel{~}{A}$ and $\stackrel{~}{B}$ are companion matrices.
6. Simulation and analysis
The combined system has three poles. Poles ${p}_{1}$ and ${p}_{2}$ are dominant poles preferred to meet the natural frequency, ${\omega }_{n}$, and damping ratio, $\zeta$. Let the dominant poles be:
${p}_{1}=-\sigma +j{\omega }_{d}.$
${p}_{2}=-\sigma -j{\omega }_{d},$
where $\sigma =\zeta {\omega }_{n}$ and ${\omega }_{d}={\omega }_{n}\sqrt{1-{\zeta }^{2}}$ is the true frequency. Other 3rd closed-loop pole, ${p}_{3}$ is placed at the left of the dominant poles on
the real-axis.
The state vector is:
$x={\left[\begin{array}{ccc}\theta & \stackrel{˙}{\theta }& \stackrel{¨}{\theta }\end{array}\right]}^{T}.$
In order to provide a direct comparison with previous investigations, suitable reference mission is chosen for a small satellite [6, 16]. Its mission duration is 5 years. Orbit is Circular at 500 km
with inclination of 53°, satellite mass is 100 kg for a size of 1×1×1 m^3, Satellite pitch inertia ${I}_{y}$ is 16.9 kgm^2 and system response time constant ${\tau }_{\omega }$ is 2 s. External
disturbance is considered as the reference mission by Varatharajoo [16] for the comparison purpose.
Fig. 6Satellite attitude performance achieving up to 0.001° accuracy
Fig. 7Satellite attitude performance achieving up to 0.0018° accuracy
The mission is to maintain the satellite pitch attitude at the reference attitude ${\theta }_{ref}=$ 0°, under the influence of external disturbance.
According to the above mission specifications, the control design and time-response requirements are [6, 16]: damping ratio, $\zeta =$ 1.0 and natural frequency, ${\omega }_{n}=$ 0.1639 rad/s. The
desired location of the closed-loop poles are at –0.1639 and –0.1639. Another pole is placed at –1.0. To achieve the prescribed pole locations using full-state feedback, the following gain vector $K$
is obtained as:
$K=\left[\begin{array}{ccc}0.9075& 11.9819& 27.9662\end{array}\right].$
The stability of this combined energy and attitude control system can be determined from the location of poles in s plane. All the poles of a stable system are located at the left side of s pane. To
examine the open-loop stability of this combined system, poles are depicted from Eq. (6). The determined open-loop poles are 0, 0 and –0.5. Because of two repeated poles on the imaginary axis, the
system is unstable. This makes sense, the combined energy and attitude control system does not stay at the desired direction by itself-it distorts arbitrarily. To make the system stable, pole
placement technique is implemented and those three poles are placed at –0.1639, –0.1639 and –1.0000. Subsequently, Eq. (18) expresses the required control gains of the proposed controller which
satisfies the stability conditions of the system.
Fig. 6 shows the simulation result of the satellite attitude performance achieving up to 0.001° with the full state-feedback control approach unlike all other previous works performed related to this
combined energy and attitude control system design. Among the previous works, the best results are obtained in the works of [16, 17], in which the maximum achievement of satellite pitch pointing
accuracy were 0.01° and 0.0185° respectively. Therefore, the current investigation achieved 10 times better performance than that of in [16].
Fig. 8Satellite attitude performance (location of 3rd pole at –0.1)
Fig. 9Satellite attitude performance (location of 3rd pole at –0.01)
Moreover, it is observed that from 0 to 500 seconds, the proposed control system maintains 0° pointing accuracy but in the previous investigations none of them could achieve such improvement.
Since the state-feedback control design is mainly involved with pole placement technique, therefore, the obtained results are influenced by the tolerance of selecting suitable location of ${p}_{3}$.
It helps to confirm about the robustness of the proposed control scheme. Hence, the impact on satellite pitch pointing accuracy is observed while location of the 3rd pole, ${p}_{3}$ is considered at
–0.5. To achieve the prescribed pole locations using full-state feedback, the following gain vector $K$ is obtained:
$K=\left[\begin{array}{ccc}0.4538& 6.447& 11.0743\end{array}\right].$
The obtained pointing accuracy in the simulation result of Fig. 7 is 0.0018° which is even 10 times better than that of in [17]. If the 3rd pole is moved further towards the imaginary axis, the
system performance will deteriorate gradually. Evidence is shown in Fig. 8 and Fig. 9 while the location of pole is chosen at –0.1 and –0.01 respectively. For these two different pole locations,
following gain vectors $K$ are obtained:
$K=\left[\begin{array}{ccc}0.0908& 2.0150& -2.4392\end{array}\right],$
$K=\left[\begin{array}{ccc}0.0091& 1.0183& -5.4797\end{array}\right].$
With the obtained gain vectors, simulation results show that the pitch pointing accuracy deviated from 0.001° to at best 0.005°. Hence, it proves that the proposed control design is highly robust.
Fig. 10Non-Ideal satellite attitude performance
Fig. 11Ideal satellite attitude performance obtained in [16]
Till above, the simulation works are performed for an ideal system, in which the value of motor-generator torque constant and flywheels’ inertias are taken ideally [6, 16, 17]. Therefore, a second
test is performed for a non-ideal combined system. The flywheel based system contains two major internal gain errors. In which, one is related to the motor-generator torque constants and the other
one is related to flywheels’ inertias. These two internal disturbances are considered for non-ideal test cases. Hence, the small satellite is verified for a comparative motor-generator torque
constant variance of 0.5 %, and a relative variance in flywheels' inertias of 0.2 % [1].
Considering these non-ideal parameters, numerical treatment is performed for satellite dynamics of the combined system. Simulation results shows in Fig. 10 that the satellite pitch pointing accuracy
is reduced to 0.01° due to the non-ideal parameters. But the result is still better than any other previous investigations. For example, the same result (pitch accuracy: 0.01°) is obtained by AFC-PD
control solution [16] which was tested in ideal case and shown in Fig. 11. The non-ideal treatment results were recorded for 0.3° and 0.043° in Reference [3] and [17] respectively. Therefore, the
current investigation obtained better result than previously proposed scheme while considering non-ideal parameters.
A complete comparison is demonstrated in Table 1 for both ideal and non-ideal cases obtaining pitch pointing accuracy through conventional schemes and the proposed control method. Ideal case is
defined while the external disturbances are considered only. Non-ideal case is demonstrated including both internal and external disturbances. The flywheel based system contains two major internal
gain errors. In which, one is related to the motor-generator torque constants and the other one is related to flywheels’ inertias. These two internal disturbances are considered for non-ideal test
cases. 0.2° and 0.22° is achieved for ideal and non-ideal test cases, respectively, while using proportional-integral-derivative (PID) control method. PID-active force control(AFC) method achieved
pitch pointing accuracy 0.01° and 0.3° for the same test cases. Further, ${H}_{\alpha }$ optimal control method obtained 0.0185° and 0.043^for the similar ideal and non-ideal demonstration of
combined scheme. However, the tabular comparison clearly shows the superiority of the full state feedback solution to maintain the pitch pointing accuracy for combined energy and attitude control
scheme of small satellite.
Table 1A comparison of pitch pointing accuracy for the combined energy and attitude control scheme
Conventional and proposed scheme Ideal/non-ideal case Pitch pointing accuracy
Ideal 0.2°
Mehedi-used PID controller [6]
Non-ideal 0.22°
Ideal 0.01°
Varatharajoo-used AFC controller [16]
Non-ideal 0.3°
Ideal 0.0185°
Ying used ${H}_{\alpha }$ controller [17]
Non-ideal 0.043°
Ideal 0.001°
Proposed full state-feedback controller
Non-ideal 0.01°
7. Conclusions
A pole placement with full state-feedback design is presented in this paper. The method of control solution has been tested on combined energy and attitude regulatory scheme with a selected reference
mission. Results show that the full state-feedback control design performed far better than the required mission attitude accuracy when an appropriate combination of poles location are selected. The
proposed method of control maintained the attitude accuracy from 0.001° to 0.005° for an ideal case. The full-state feedback control also maintained the attitude accuracy around 0.01° in a non-ideal
flywheel based combined energy and attitude regulatory model. In fact, full state-feedback provides attitude pointing accuracy performances that are far better than $H\alpha$ control option [17]
archiving 0.0185° and AFC-PD control solution [16] achieving 0.01° pitch pointing accuracy. Future works on other control options PI-Integer order control scheme can be investigated to provide a
complete overview of the full state-feedback controllers on this combined system. Again, the proposed method based combined control scheme for satellite energy and attitude pointing is a novel
initiation for further investigation towards designing a fractional order controller with state space approaches.
• Guyot P., Barde H., Griseri G. Flywheel power and attitude control system (FPACS). 4th ESA Conference on Spacecraft Guidance, Navigation and Control System, Noordwijk, 1999, p. 371-378.
• Roithmayr C. M. International Space Station Attitude Control and Energy Storage Experiment: Effects of Flywheel Torque, NASA Technical Memorandum 209100, 1999.
• Varatharajoo R. A combined energy and attitude control system for small satellites. Acta Astronaut, Vol. 54, 2004, p. 701-712.
• Varatharajoo R. Operation for the combined energy and attitude control system. Aircraft Engineering Aerospace Technology: International Journal, Vol. 78, Issue 6, 2006, p. 495-501.
• Varatharajoo R. On-board errors of the combined energy and attitude control system. Acta Astronaut, Vol. 58, 2006, p. 561-563.
• Mehedi I. M., Varatharajoo R., Harlisya H., Filipski M. N. Architecture for combined energy and attitude control system. American Journal of Applied Sciences, Science Publications, Vol. 2, 2005,
p. 430-435.
• Mehedi I. M., Varatharajoo R. Pointing performance of combined energy and attitude control system. Journal of Industrial Technology, Vol. 14, Issue 2, 2006, p. 147-160.
• Mehedi I. M., Filipski M. N. Design of a momentum bias attitude control system with a double reaction wheel assembly. Ankara International Aerospace Conference, Ankara, Turkey, 2005.
• Mehedi I. M., Varatharajoo R. Architecture of combined energy and attitude control system for enhanced microsatellites. Proceedings of Aerotech, Putrajaya, Malaysia, 2005, p. 133-140.
• Mehedi I. M. Hybrid regulator system for satellite directing enactment and energy longevity. Life Science Journal, Vol. 11, Issue 3s, 2014, p. 59-67.
• Varatharajoo R., Teckwooi C., Mailah M. Two degree-of-freedom spacecraft attitude controller. Advances in Space Research, Vol. 47, 2011, p. 685-689.
• Wu D., Huimin Z., Jingjing L., Xiaolin Y., Yuanyuan L., Lifeng Y., Chuanhua D. An improved CACO algorithm based on adaptive method and multi-variant strategies. Soft Computing: Methodologies and
Application, Vol. 19, 2015, p. 701-713.
• Zhang S., Zhang Y., Zhang X., Dong G. Fuzzy PID control of a two-link flexible manipulator. Journal of Vibroengineering, Vol. 18, Issue 1, 2016, p. 250-266.
• Gu B., Sheng V. S., Tay Y. T., Romano W., Li S. Incremental support vector learning for ordinal regression. IEEE Transaction on Neural Networks and Learning Systems, Vol. 26, Issue 7, 2015, p.
• Pan Z., Zhang Y., Kwong S. Efficient motion and disparity estimation optimization for low complexity multi view video coding. IEEE Transaction on Broadcasting, Vol. 61, Issue 2, 2015, p. 166-176.
• Varatharajoo R., Teckwooi C., Mailah M. Attitude pointing enhancement for combined energy and attitude control system. Acta Astronaut, Vol. 68, 2011, p. 2025-2028.
• Ying S. B., Varatharajoo R. H[α] control option for a combined energy and attitude control system. Advances in Space Research, Vol. 52, 2013, p. 1378-1383.
• Nise S. N. Control Systems Engineering. John Wiley and Sons, 2008.
About this article
Vibration generation and control
full state-feedback control
small satellite control
attitude and energy control
flywheel system
This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. (D-150-135-1437). The author, therefore, gratefully acknowledges the DSR
technical and financial support.
Copyright © 2017 JVE International Ltd.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. | {"url":"https://www.extrica.com/article/18066","timestamp":"2024-11-07T16:25:14Z","content_type":"text/html","content_length":"163695","record_id":"<urn:uuid:746c4219-d5c1-449a-b5c8-935012cd27fc>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00031.warc.gz"} |
(PDF) Impact of spatial variability of earthquake ground motion on seismic demand to natural gas transmission pipelines
... Several researchers underscored the considerable impact of non-uniform ground motion on the seismic response and deformations of buried pipelines (Papadopoulos et al. 2017;Zerva 1993Zerva , 1994.
Hindy and Novak (1979) used a lumped mass model to investigate the pipeline response to ground motions in the lateral and longitudinal directions and considered different angles of incidence along
the pipe. ...
... Lee et al. (2009) employed nonlinear Winkler foundation shell model to investigate the seismic response characteristics of buried pipelines with emphasis on section strains, axial relative
displacement, and transverse relative displacement. Papadopoulos et al. (2017) employed three-dimensional finite element model to analyze the seismic response of soil-pipeline system simulating the
soil using lumped springs, while the input horizontal and vertical ground motion time histories at the pipeline depth were obtained from 2D site response analysis. ... | {"url":"https://www.researchgate.net/publication/314840068_Impact_of_spatial_variability_of_earthquake_ground_motion_on_seismic_demand_to_natural_gas_transmission_pipelines","timestamp":"2024-11-05T12:17:06Z","content_type":"text/html","content_length":"692404","record_id":"<urn:uuid:7bc53a7b-fdf8-493b-aa17-7ed7e99d6b51>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00557.warc.gz"} |
Modeling Censored Time-to-Event Data Using Pyro, an Open Source Probabilistic Programming Language
Uber AI, Engineering
Modeling Censored Time-to-Event Data Using Pyro, an Open Source Probabilistic Programming Language
15 February 2019 / Global
Time-to-event modeling is critical to better understanding various dimensions of the user experience. By leveraging censored time-to-event data (data involving time intervals where some of those time
intervals may extend beyond when data is analyzed), companies can gain insights on pain points in the consumer lifecycle to enhance a user’s overall experience. Despite its prevalence, censored
time-to-event data is often overlooked, leading to dramatically biased predictions.
At Uber, we are interested in investigating the time it takes for a rider to make a second trip after their first trip on the platform. Many of our riders engage with Uber for the first time through
referrals or promotions. Their second ride is a critical indicator that riders are finding value in using the platform and are willing to engage with us in the long term. However, modeling the time
to second ride is tricky. For example, some riders just don’t ride as often. When we analyze this time-to-event data before such a rider’s second ride, we consider their data censored.
Similar situations exist at other companies and across industries. For example, suppose that an ecommerce site is interested in the recurring purchase pattern of customers. However, due to the
diverse pattern of customer behavior, the company might not be able to observe all recurring purchases for all customers, resulting in censored data.
In another example, suppose that an advertising company is interested in the recurring ad clicking behavior of its users. Due to the distinct interests of each user, the company might not be able to
observe all clicks made by their customers. Users might not have clicked the ads until after the study concludes. This will result in censored time to next click data.
In modeling censored time-to-event data, for each individual of interest indexed by , we might observe data in the following form:
Here,is the censorship label;if the event of interest is observed; andif the event of interest is censored. When, denotes the time-to-event of interest. When ,denotes the length of time until
censorship happens.
Let’s continue with the time-to-second ride example at Uber: if a rider took a second ride 12 days after their first ride, this observation is recorded as. In another case, a rider took a first ride,
60 days have passed, and they have not yet returned to the app to take a second ride by a given cut-off date. This observation is recorded as . The situation is illustrated in the picture, below:
There is an ocean of survival analysis literature and over a century of statistical research has already been done in this area; much of which can be simplified using the framework of probabilistic
programming. In this article, we walk through how to use the Pyro probabilistic programming language to model censored time-to-event data.
Relationship with churn modeling
Before proceeding, it’s worth mentioning that many practitioners in industry circumvent this censored time-to-event data challenge by setting artificially defined labels as “churn.” For example, an
ecommerce company might define a customer as “churned” if they have not yet returned to the site to make another purchase in the past 40 days.
Churn modeling enables practitioners to massage observations into a classical binary classification pattern. As a result, churn modeling becomes very straightforward with off-the-shelf tools like
scikit-learn and XGBoost. For example, the above two riders would be labeled into “not churned” and “churned,” respectively.
While churn modeling admittedly works in certain situations, it does not necessarily work for Uber. For example, some riders might only use Uber when they are on a business trip. If this hypothetical
rider takes a trip for work every six months, we might end up mislabeling this business rider as having churned. As a result, the conclusion that we draw from a churn model might be misleading.
We are also interested in making interpretations from these models to elucidate the contribution of different factors to the user behavior observed. As a result, the model should not be a black box.
We would love to have the capability to open up the model and make more informed business decisions with it.
To accomplish this, we can leverage, Pyro, a flexible and expressive open source tool for probabilistic programming.
Pyro for statistical modeling
Created at Uber, Pyro is a universal probabilistic programming language written in Python, built on the PyTorch library for tensor computations.
If you come from a statistics background with minimum Bayesian modeling knowledge or if you have been tinkering with deep learning tools like TensorFlow or PyTorch, you are in luck.
The following table summarizes some of the most popular projects for probabilistic programming:
Software BUGS / JAGS [1] STAN PyMC TensorFlow Probability [4] Pyro
Coding language Domain Specific Language [2] Domain Specific Language Python Python Python
Underlying computational engine Self STAN Math Library Theano [3] TensorFlow [5] PyTorch [6]
Below, we highlight some key features about these different software projects:
1. BUGS / JAGS are early examples of what came to be known as probabilistic programming. They have been under active development and usage for more than two decades in the statistical field.
2. However, BUGS / JAGS are designed and developed mostly from the ground up. As a result, model specification is done using their domain specific language. Moreover, probabilistic programmers
invoke BUGS / JAGS from wrappers in R and MATLAB. Users have to switch back and forth between coding languages and files, which is a bit inconvenient.
3. PyMC relies on a Theano backend. However the Theano project was recently discontinued.
4. TensorFlow Probability (TFP) originally started as a project called Edward. The Edward project was rolled into the TFP project.
5. TFP uses TensorFlow as its computation engine. As a result, it supports only static computational graphs.
6. Pyro uses PyTorch as computation engine. As a result, it supports dynamic computational graphs. This enables users to specify models that are diverse in terms of dataflow and is very flexible.
In short, Pyro is positioned at the very beneficial intersection of the most powerful deep learning tool chains (PyTorch) while standing on the shoulders of decades of statistical research. The
result is an immensely concise and powerful, yet flexible probabilistic modeling language.
Modeling censored time-to-event data
Now, let’s jump into how we model censored time-to-event data. Thanks to Google Colab, users can check out extensive examples of the code and start modeling data without installing Pyro and PyTorch.
You can even duplicate and play around with the workbook.
Model definition
For the purpose of this article, we define the time-to-event data as , withas the time-to-event andas the binary censoring label. We define the actual time-to-event as, which may not be observed. We
define censoring time as , which for simplicity, we assume is a known fixed number. In summary, we can model this relationship as:
We assume that follows exponential distribution with scale parameter , a variable dependent upon the following linear relationship with predictor of interest :
Here, is a softplus function, thereby ensuring that stays positive. Finally, we assume thatandfollow normal distribution as their prior distribution. For the purpose of this article, we are
interested in evaluating the posterior distribution ofand.
Generating artificial data
We first import all the necessary packages in Python:
To generate experiment data, we run the following lines:
Congratulations! You just ran your first Pyro function in the line with Note [1]. Here we drew samples from a normal distribution. Careful users might have noticed this intuitive operation is very
similar to our workflow in Numpy.
At the end of the above code block (Note 2), we generated a regression plot of (green), (blue) against , respectively. If we do not account for data censorship, we underestimate the slope of model.
Figure 1. This scatterplot depicts true underlying event time and observed event time against predictor.
Constructing models
With this fresh but censored data, we can begin constructing more accurate models. Let’s start with the model function, below:
In the code snippet above, we highlight the following notes to better clarify our example:
• Note 1: Overall, a model function is a process of describing how data are generated. This example model function tells how we generated y or truncation_label from input vector x.
• Note 2: We specify a prior distribution ofhere and sample from them using the pyro.sample function. Pyro has a huge family of random distributions in the PyTorch project as well as in the Pyro
project itself.
• Note 3: We connect inputs , intovector denoted by variable link here.
• Note 4: We specify the distribution of true time-to-eventusing exponential distribution with scale parameter vector link.
• Note 5: For observation, if we observe the time-to-event data, then we contrast it with true observation y[i].
• Note 6: If the data is censored for observation , the truncation label (equalling to 1 here), follows Bernoulli distribution. The probability of seeing truncated data is the CDF ofat point. We
sample from the Bernoulli distribution and contrast it against real observation of truncation_label[i].
Calculating inference using Hamiltonian Monte Carlo
Hamiltonian Monte Carlo (HMC) is a popular technique when it comes to calculating Bayesian inference. We estimateusing HMC, below:
The process above might take a long time to run. The slowness comes in great part due to the fact that we are evaluating the model through each observation sequentially. To speed up the model, we can
vectorize using pyro.plate and pyro.mask, as demonstrated below:
In the code snippet above, we start by specifying the HMC kernel using the model specified. Then, we execute the MCMC against x, y, and the truncation_label. The MCMC sampled result object is next
converted into an EmpiricalMarginal object that helps us to make inference along a_model parameter. Finally, we draw samples from posterior distribution and create a plot with our data, shown below:
Figure 2: Histogram of sampled values for a.
We can see that the samples are clustered around the real value of at 2.0.
Speeding up estimation using variational inference
Stochastic variational inference (SVI) is a great way to speed up Bayesian inference with large amounts of data. For now, it’s sufficient to proceed with the knowledge that a guide function is an
approximation of the desired posterior distribution. The specification of a guide function can dramatically speed up the estimation of parameters. To enable stochastic variational inference, we
define a guide function as:
guide = AutoMultivariateNormal(model)
By using a guide function, we can approximate the posterior distributions of parametersas normal distributions, where their location and scale parameters are specified by internal parameters,
Training the model and inferring results
The model training process with Pyro is akin to standard iterative optimization in deep learning. Below, we specify the SVI trainer and iterate through optimization steps:
If everything goes according to plan, we can see the print out of the above execution. In this example, we received the following results, whose means are very close to the true value of and
a_model = 0.009999999776482582, b_model = 0.009999999776482582
a_model = 0.8184720873832703, b_model = 2.8127853870391846
a_model = 1.3366154432296753, b_model = 3.5597035884857178
a_model = 1.7028049230575562, b_model = 3.860581874847412
a_model = 1.9031578302383423, b_model = 3.9552347660064697
final result:
median a_model = 1.9155923128128052
median b_model = 3.9299516677856445
We can also check if the model has converged through the below code and arrive at Figure 3, below:
Figure 3: Model loss plotted against number of iterations.
We can see that the guide functions center on the actual value of, respectively below:
Moving forward
We hope you leverage Pyro for your own censored time-to-event data modeling. To get started with the open source software, check out the official Pyro website for additional examples including an
introductory tutorial and sandbox repository.
In future articles, we intend to discuss how you can leverage additional features of Pyro to speed up SVI computation, including using the plate api to batch process on samples of similar shape.
Interested in working on Pyro and other projects from Uber AI? Consider applying for a role on our team!
Hesen Peng
Hesen Peng is a senior data scientist at Uber’s Rider Data Science team.
Fritz Obermeyer
Fritz is a research engineer at Uber AI focusing on probabilistic programming. He is the engineering lead for the Pyro team.
Posted by Hesen Peng, Fritz Obermeyer | {"url":"https://www.uber.com/en-DE/blog/modeling-censored-time-to-event-data-using-pyro/","timestamp":"2024-11-12T01:08:26Z","content_type":"text/html","content_length":"494869","record_id":"<urn:uuid:f3d6ed26-f0bf-45af-866c-8506959cd4b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00526.warc.gz"} |
pdf book ball mill optimization
Simulation results under the ∅5250 × 500 mm mill model show that the mill operates with the optimal effect when the mill is under the condition of 80% critical speed and 15% fill level; the ...
WhatsApp: +86 18838072829
Process Optimization of a Small Scale Ball Mill For Mineral Processing using Discrete Element Method Philbert Muhayimana A thesis submitted in partial fulfillment for the degree of ... A ball mill
is a grinding machine widely used in mineral processing to gradually decrease
WhatsApp: +86 18838072829
Free download as Powerpoint Presentation (.ppt / .pptx), PDF File (.pdf), Text File (.txt) or view presentation slides online. Scribd is the world's largest social reading and publishing site.
WhatsApp: +86 18838072829
The results of signaltonoise analysis obtained the optimum parameter values in a row are: 100 rpm for milling speed parameter, 15: 1 for BPR parameter and 120 minutes for timemiling parameter.
The powder size experiment verification of the ball mill process optimization parameter is D50 : µm. Export citation and abstract BibTeX RIS.
WhatsApp: +86 18838072829
The raw materials were ground from the big particle size to the smallest possible by using multistep grinding. In the laboratory, the common method to be used as the ball mill. This work aims to
design a simple horizontal ball mill. Calcium carbonate material from limestone and eggshells powder was ground using the developed ball mill.
WhatsApp: +86 18838072829
for all screen sizes from mill specific energy input (kWh/t) and mill feed and product size distributions (Hinde, 1999; McIvor and Finch, 2007; McIvor et al., 2017). These include the mill
grinding rate through the size of interest, calculated independently as above. Media sizing Ball mill media optimization through functional performance modeling
WhatsApp: +86 18838072829
This paper relates the operational experiences from the first VRM for clinker grinding put into operation in the United States in 2002. Included in the discussion are operational data,
maintenance discussion and laboratory data focused on product quality. All of the discussion is based on comparison to ball mill operation at the same plant.
WhatsApp: +86 18838072829
The ball mill Ball milling is a mechanical technique widely used to grind powders into ne particles and blend Being an environmentallyfriendly, costeffective technique, it has found wide
application in industry all over the world. Since this minireview mainly focuses on the conditions applied for the prep
WhatsApp: +86 18838072829
The operational controls are also reviewed for optimized mill operation. Every element of a closed circuit ball mill system is evaluated independently to assess its influence on the system.
Figure 1 below is a typical example of inefficient grinding indicated by analysis of the longitudinal samples taken after a crash stop of the mill.
WhatsApp: +86 18838072829
CERAMIC LINED BALL MILL. Ball Mills can be supplied with either ceramic or rubber linings for wet or dry grinding, for continuous or batch type operation, in sizes from 15″ x 21″ to 8′ x 12′.
High density ceramic linings of uniform hardness male possible thinner linings and greater and more effective grinding volume.
WhatsApp: +86 18838072829
Size reduction is a necessary operation in mineral processing plants and provides the desired size for separation operations and the liberation of the valuable minerals present in ores.
Estimations on energy consumption indicate that milling consumes more than 50 % of the total energy used in mining operations. Despite the fact that ball milling is an efficient operation, it is
energy ...
WhatsApp: +86 18838072829
Ball mill optimization. Dhaka, Bangladesh 21 March 2010. Introduction Wajananawat Experience: 13 Years (2 y in engineering,11 y in production) Engineering department Kiln and Burning system Siam
Cement (Ta Luang) Kiln system, Raw material grinding and Coal grinding Siam Cement (Lampang) Cement grinding and Packing plant. The Siam Cement (Thung Song) Co,Ltd
WhatsApp: +86 18838072829
1. Introduction. Ball milling is a critical process to reduce product size into a desired size range, which has been widely used in industries such as minerals processing, cement manufacturing,
pharmaceutical industries and powder metallurgy [1, 2].The milling process is affected by many parameters, including ground particles, mill speed [3], milling time [4], ball to powder ratio (BPR)
[4, 5 ...
WhatsApp: +86 18838072829
The application on the ball mill load forecast Description of ball mill load. We experimented with evaluating the effect of the distributed SCN model on a ball mill grinding process. It was
accomplished on a laboratoryscale latticetype ball mill (XMQL (420 times 450)) with a maximum load of 80 kg and a pulverizing capacity of 10 ...
WhatsApp: +86 18838072829
Different configurations of SAG, ball and rod mill occur but in general a SAG mill can be considered an intermediate stage in breaking down rock from the crushing plant, and feeding to ball or
rod mills for further size reduction. Types of mill Different types of mill are in operation rod or ball mills, so
WhatsApp: +86 18838072829
The grinding charge of an 8"x10" ball mill Figure Process Flowchart for Bond Work. consists of steel balls with varying diameters and Index Test. catching pan was cleaned. Estimated total.
surface area of the 191 steel balls weighing 20 •Sample Preparation. 125 grams is 842 in2. To ensure uniform feed for.
WhatsApp: +86 18838072829
A grinding mill model based on discrete element method (DEM) simulations is being developed at Outotec Oyj (Finland) to be used in mill design optimization. The model can be used for many
purposes; one example is the selection of the lining and the size of the mill to meet the requirements of the clients. To validate the accuracy of the DEM ...
WhatsApp: +86 18838072829
predictive control for an industrial ball mill circuit in Chen et al. (2007). A robust nonlinear model predictive controller (RNMPC) was proposed by Coetzee et al. (2010) to ... 1 and 6) is not
considered in this study, a strategy for optimization of power consumption, material cost and production time using the same process model is shown in ...
WhatsApp: +86 18838072829
The energy consumption in the ball mill was found to be kWh/t of ore with a targeted product size below 1 mm. The BWI of the ores varied from to kWh/t to reduce the particle size below 100 μ m,
but in real time, the energy consumption is very high compared with the reported value of kWh/t.
WhatsApp: +86 18838072829
Tube Mill Note Free download as PDF File (.pdf), Text File (.txt) or read online for free. ... Therefore knowing them is of the primary duty for the person assigned for Mill Optimization. TUBE
MILL. GOVERNING ... Size and grinding requirements. Generally there is no fixed rule; however tube mills have the ratio of 3 to 6: 1 and ball mills are ...
WhatsApp: +86 18838072829
Nonlinear optimization, a technique of numerical analysis, is applied to motion analysis of ball media in a tumbling ball mill. The basic principles of optimization are explained. The motion
equations of ball media are established. The point of maximum energy is computed, and this maximum energy is % greater than the energy at the point of ...
WhatsApp: +86 18838072829
Ball load (35%), feed l oad (35%) and N c (70%) for conventi onal ball mill and ball load (50%), feed load (%) for vibrated ball mill were kept constant when grinding time experiments performed.
WhatsApp: +86 18838072829
speed for variablespeed mills may lead the ball impacts to directly go down on the toe of the charge and, therefore, compensate for initially wearing lifter to preserve mill performance. Liner
wear life, in general, can be increased by increasing lifter height. 3. Effects of Liner Design and Operating Parameters.
WhatsApp: +86 18838072829
HOLTEC has undertaken Performance Optimisation of the cement grinding circuits by doing process diagnostic studies in many cement plants. The paper describes the approach for the process
diagnostic study for the optimisation of a ball mill circuit and is supported with typical case study done by HOLTEC in a mio t/a cement plant.
WhatsApp: +86 18838072829
This generalized approach allows to model processing plant indicators, such as ball mill throughput rates or metal recoveries as hereditary attributes directly at the processing location, based
on blended geometallurgical material characteristics. Objective Function The objective function of the simultaneous stochastic optimization ...
WhatsApp: +86 18838072829
article{osti_, title = {Optimization of the design of ball mills}, author = {Bogdanov, V S}, abstractNote = {The authors have developed, investigated and tested under production conditions ball
mills equipped with sloped interchamber partitions. The plan of such a mill is shown, the distinguishing feature of which is the fact that the interchamber partition is located on a slope to the
WhatsApp: +86 18838072829
Optimised Ball Size Distribution Free download as PDF File (.pdf), Text File (.txt) or read online for free. Ball mill. Ball mill. Open navigation menu. Close suggestions Search Search. en Change
Language. close menu Language. English (selected) ... 1988, Computer simulation and optimization of ball mills/ circuit, WC, 19 (4), ...
WhatsApp: +86 18838072829
(EGL) SAG mill driven by a 24,000 kW gearless drive. The SAG mill feeds two FLSmidth ball mills each 26 ft. in diameter × 40 ft. long (EGL), each driven by a 16,400 kW drive. The SAG mill is one
of the largest volumetric capacity SAG mills in the world and represented the first 40 ft. SAG mill in Peru (Garcia Villanueva, 2013 [1]).
WhatsApp: +86 18838072829
Optimization of mill performance by using online ball and pulp measurements by B. Clermont* and B. de Haas* Synopsis Ball mills are usually the largest consumers of energy within a mineral
concentrator. Comminution is responsible for 50% of the total mineral processing cost. In today's global markets, expanding mining groups are trying
WhatsApp: +86 18838072829
The grinding process of the ball mill is an essential operation in metallurgical concentration plants. Generally, the model of the process is established as a multivariable system characterized
with strong coupling and time delay. In previous research, a twoinputtwooutput model was applied to describe the system, in which some key indicators of the process were ignored. To this end, a
three ...
WhatsApp: +86 18838072829
Optimum filling ratio. U= (volume of powder in the mill)/ (volume of voids in the charge): between 60% and. 110%, optimum around 90%. In practical terms, material level should equal ball level in
the first compartment. In practical terms, material level should be higher than ball level in the second.
WhatsApp: +86 18838072829 | {"url":"https://traiteur-cino.fr/9452/pdf-book-ball-mill-optimization.html","timestamp":"2024-11-09T20:54:41Z","content_type":"application/xhtml+xml","content_length":"29518","record_id":"<urn:uuid:04bddedc-1673-4f2b-aa5a-215223040260>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00150.warc.gz"} |
Zlin Z 42, Z 43, Z 142, Z 242 & Z 143
• CountryCzech Republic
• TypeTwo/four seat light aircraft
• PowerplantsZ 43 - One 155kW (210hp) Avia M 337 six cylinder inline inverted piston engine driving a two blade propeller. Z 242 L - One 150kW (200hp) Textron Lycoming AEIO-360-A1B6 flat four
driving a three blade c/s prop. Z 143 - One 175kW (235hp) Textron Lycoming O540J3A5 flat six driving a three blade variable pitch Mühlbauer prop.
• PerformanceZ 43 - Max speed 235km/h (127kt), cruising speed 210km/h (113kt). Initial rate of climb 690ft/min. Range with max fuel 1100km (595nm). Z 242 L - Max speed 236km/h (127kt), max cruising
speed 214km/h (114kt). Initial rate of climb 1102ft/min. Range with max fuel 1056km (570nm). Z 143 - Max speed 265km/h (143kt), max cruising speed at 75% power 235km/h (127kt), econ cruising
speed at 60% power 216km/h (226kt). Initial rate of climb 1457ft/min. Range at 65% power 1335km (720nm).
• WeightsZ 43 - Empty 730kg (1609lb), max TO 1350kg (2976lb). Z 242 L - Basic empty 730kg (1609lb), max TO 1090kg (2403lb). Z 143 - Empty equipped 830kg (1830lb), max TO 1350kg (2976lb).
• DimentionsZ 43 - Wing span 9.76m (32ft 0in), length 7.75m (25ft 5in), height 2.91m (9ft 7in). Wing area 14.5m2 (156.1sq ft). Z 242 L - Wing span 9.34m (30ft 8in), length 6.94m (24ft 9in), height
2.95m (9ft 8in). Z 143 - Wing span 10.14m (33ft 3in), length 7.58m (24ft 11in), height 2.91m (9ft 7in). Wing area 14.8m2 (159.1sq ft).
• CapacitySeating for two in tandem in Z 42, Z 142 and Z 242, seating for four in Z 43 and Z 143.
• ProductionTotal production includes more than 350 142s, approx 40 Z 242 Ls and 35 Z 143s, including military orders.
This arrangement of two seat coaches and four seat light flying machine was at first created to trade for the effective Zlin Trener.
The beginning Z 42 was created amid the mid 1960s and seats two side by side. It flew surprisingly on October 17 1967. The enhanced Z 42m then presented a consistent pace propeller and the bigger
tail created for the Z 43 four seater, and supplanted the Z 42 in generation in 1974.
Improvement of the two seat line proceeded with the further enhanced Z 142, which flew surprisingly on December 29 1979. Progressions presented incorporated a bigger cockpit shade and faired
undercarriage. The Z 142 stayed in generation in Z 142c structure to the mid 1990s. The most recent two seater of this family to be created is the 150kw (200hp) Textron Lycoming Aeio360 level four
controlled Z 242 L. Changes aside from the motor incorporate a three sharpened steel consistent pace prop and reconsidered motor cowling profile. In the first place flight was on February 14 1990.
Advancement of the four seat displays, the Z 43 and Z 143, has emulated that of the two seaters. The Z 43 showed up a year later than the Z 42, flying shockingly on December 10 1968. The Z 42 and Z
43 offer the same essential airframe, however vary in that the Z 43 peculiarities a bigger and more extensive lodge with seating for four, and an all the more influential motor. The flow Z 143 L flew
surprisingly on April 24 1992, and is comparative in structure to the Z 242, yet again varies in having a bigger lodge with seating for four and an all the more influential Textron Lycoming O-540 | {"url":"https://barrieaircraft.com/zlin-z-42-z-43-z-142-z-242-z-143.html","timestamp":"2024-11-07T18:27:32Z","content_type":"text/html","content_length":"13823","record_id":"<urn:uuid:f993995b-aa43-4bcc-a1d4-8115ed420c8f>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00899.warc.gz"} |
ESSA2013 Conference | R-bloggersESSA2013 Conference
[This article was first published on
R snippets
, and kindly contributed to
]. (You can report issue about the content on this page
) Want to share your content on R-bloggers?
if you have a blog, or
if you don't.
It has been just announced that during
conference I am planning to organize a special track on “Statistical analysis of simulation models”. I hope to get some presentations using GNU R to promote it in social simulation community.
It is obvious that GNU R excels in analysis of simulation data. However, very often it can be neatly used to implement simulations themselves.
For instance I have recently implemented a simulation model proposed in
Section 4
Volatility Clustering in Financial Markets: Empirical Facts and Agent–Based Models
paper by Rama Cont. The model is formulated as follows (I give only its brief description; please refer to the paper for more details).
Consider market with
trading agents and one asset. We simulate market for
periods. In each period each agent can buy asset, sell it or do nothing.
Asset return
in period
equals to number of buy orders minus number of sell orders divided by number of agents
and multiplied by normalizing constant
. Thus it will always lie in the interval
Agents make buy and sell decisions based on random public information about an asset. The stream of signals are IID normal random variables with mean
and standard deviation
. Each investor holds an internal non negative decision making threshold. If signal is higher than threshold level buy decision is made. If it is lower than minus threshold level asset is sold. If
signal is not strong enough investor does nothing.
After return
is determined each investor with probability
performs threshold update to
As you can see the description is quite lengthily. However, the implementation of the model in GNU R is a genuine snippet as can be seen below:
cont <- function(times, n, signal.sd, max.r ,p.update) {
threshold <- vector(“numeric”, n)
signal <- rnorm(times, 0, signal.sd)
r <- vector(“numeric”, times)
for (i in 1:times) {
r[i] <- max.r * (sum(signal[i] > threshold)
– sum(signal[i] < (-threshold))) / n
threshold[runif(n) < p.update] <- abs(r[i])
And an additional benefit is that one can analyze the simulation results in GNU R also. Here is a very simple example showing the relationship between
and standard deviation of simulated returns (the initial burn in period in the simulation is discarded):
cont.sd <- function(signal.sd) {
sd(cont(10000, 1000, signal.sd, 0.1, 0.05)[1000:10000])
sd.in <- runif(100, 0.01, 0.1)
sd.out <- sapply(sd.in, cont.sd)
and here is the resulting plot: | {"url":"https://www.r-bloggers.com/2012/11/essa2013-conference/","timestamp":"2024-11-06T22:03:32Z","content_type":"text/html","content_length":"126506","record_id":"<urn:uuid:c89d2770-fb1a-40d3-8a32-0f7feeaccf45>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00614.warc.gz"} |
The Stacks project
87.1 Introduction
Formal schemes were introduced in [EGA]. A more general version of formal schemes was introduced in [McQuillan] and another in [Yasuda]. Formal algebraic spaces were introduced in [Kn]. Related
material and much besides can be found in [Abbes] and [Fujiwara-Kato]. This chapter introduces the notion of formal algebraic spaces we will work with. Our definition is general enough to allow most
classes of formal schemes/spaces in the literature as full subcategories.
Although we do discuss the comparison of some of these alternative theories with ours, we do not always give full details when it is not necessary for the logical development of the theory.
Besides introducing formal algebraic spaces, we also prove a few very basic properties and we discuss a few types of morphisms.
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder,
this is tag 0AHX. Beware of the difference between the letter 'O' and the digit '0'.
The tag you filled in for the captcha is wrong. You need to write 0AHX, in case you are confused. | {"url":"https://stacks.math.columbia.edu/tag/0AHX","timestamp":"2024-11-05T12:43:57Z","content_type":"text/html","content_length":"13924","record_id":"<urn:uuid:e08ea7cf-2bdc-498c-a581-251c94986370>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00008.warc.gz"} |
College Algebra Foundations
Learning Objectives
• Multiply fractions
□ Multiply two or more fractions
Multiply Fractions
Just as you add, subtract, multiply, and divide when working with whole numbers, you also use these operations when working with fractions. There are many times when it is necessary to multiply
fractions. A model may help you understand multiplication of fractions.
When you multiply a fraction by a fraction, you are finding a “fraction of a fraction.” Suppose you have [latex]\frac{3}{4}[/latex] of a candy bar and you want to find [latex]\frac{1}{2}[/latex] of
the [latex]\frac{3}{4}[/latex]:
By dividing each fourth in half, you can divide the candy bar into eighths.
Then, choose half of those to get [latex]\frac{3}{8}[/latex].
In both of the above cases, to find the answer, you can multiply the numerators together and the denominators together.
Multiplying Two Fractions
[latex] \frac{a}{b}\cdot \frac{c}{d}=\frac{a\cdot c}{b\cdot d}=\frac{\text{product of the numerators}}{\text{product of the denominators}}[/latex]
Multiplying More Than Two Fractions
[latex] \frac{a}{b}\cdot \frac{c}{d}\cdot \frac{e}{f}=\frac{a\cdot c\cdot e}{b\cdot d\cdot f}[/latex]
Multiply [latex] \frac{2}{3}\cdot \frac{4}{5}[/latex].
Show Solution
To review: if a fraction has common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors.
For example,
• Given [latex] \frac{8}{15}[/latex], the factors of 8 are: 1, 2, 4, 8 and the factors of 15 are: 1, 3, 5, 15. [latex] \frac{8}{15}[/latex] is simplified because there are no common factors of 8
and 15.
• Given [latex] \frac{10}{15}[/latex], the factors of 10 are: 1, 2, 5, 10 and the factors of15 are: 1, 3, 5, 15. [latex] \frac{10}{15}[/latex] is not simplified because 5 is a common factor of 10
and 15.
You can simplify first, before you multiply two fractions, to make your work easier. This allows you to work with smaller numbers when you multiply.
In the following video you will see an example of how to multiply two fractions, then simplify the answer.
Think About It
Multiply [latex] \frac{2}{3}\cdot \frac{1}{4}\cdot\frac{3}{5}[/latex]. Simplify the answer.
What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would multiply three fractions together.
Show Solution | {"url":"https://courses.lumenlearning.com/aacc-collegealgebrafoundations/chapter/read-multiplying-fractions/","timestamp":"2024-11-11T00:40:59Z","content_type":"text/html","content_length":"54234","record_id":"<urn:uuid:4006fb50-ea10-43d5-99f7-f96a4017c76e>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00496.warc.gz"} |
How To Calculate Length Of Sides In Regular Hexagons
The six-sided hexagon shape pops up in some unlikely places: the cells of honeycombs, the shapes soap bubbles make when they're smashed together, the outer edge of bolts, and even the hexagon-shaped
basalt columns of the Giant's Causeway, a natural rock formation on the north coast of Ireland. Assuming you're dealing with a regular hexagon, which means all its sides are of the same length, you
can use the hexagon's perimeter or its area to find the length of its sides.
TL;DR (Too Long; Didn't Read)
The simplest, and by far most common, way of finding the length of a regular hexagon's sides is using the following formula:
_s_ = _P_ ÷ 6, where _P_ is the perimeter of the hexagon, and _s_ is the length of any one of its sides.
Calculating Hexagon Sides From the Perimeter
Because a regular hexagon has six sides of the same length, finding the length of any one side is as simple as dividing the hexagon's perimeter by 6. So if your hexagon has a perimeter of 48 inches,
you have:
\(\frac{48 \text{ inches}}{6} = 8 \text{ inches}\)
Each side of your hexagon measures 8 inches in length.
Calculating Hexagon Sides From the Area
Just like squares, triangles, circles and other geometric shapes you may have dealt with, there is a standard formula for calculating the area of a regular hexagon. It is:
\(A = (1.5 × \sqrt{3}) × s^2\)
where A is the hexagon's area and s is the length of any one of its sides.
Obviously, you can use the length of the hexagon's sides to calculate the area. But if you know the hexagon's area, you can use the same formula to find the length of its sides instead. Consider a
hexagon that has an area of 128 in^2:
1. Substitute Area Into the Equation
Start by substituting the area of the hexagon into the equation:
\(128 = (1.5 × \sqrt{3}) × s^2\)
2. Isolate the Variable
The first step in solving for s is to isolate it on one side of the equation. In this case, dividing both sides of the equation by (1.5 × √3) gives you:
\(\frac{128}{1.5 × \sqrt{3}} = s^2\)
Conventionally the variable goes on the left side of the equation, so you can also write this as:
\(s^2=\frac{128}{1.5 × \sqrt{3}}\)
3. Simplify the Term on the Right
Simplify the term on the right. Your teacher might let you approximate √3 as 1.732, in which case you'd have:
\(s^2=\frac{128}{1.5 × 1.732}\)
Which simplifies to:
Which, in turn, simplies to:
\(s^2 = 49.269\)
4. Take the Square Root of Both Sides
You can probably tell, by examination, that s is going to be close to 7 (because 7^2 = 49, which is very close to the equation you're dealing with). But taking the square root of both sides with a
calculator will give you a more exact answer. Don't forget to write in your units of measure, too:
\(\sqrt{s^2} = \sqrt{49.269}\)
then becomes:
\(s = 7.019 \text{ inches}\)
Cite This Article
Maloney, Lisa. "How To Calculate Length Of Sides In Regular Hexagons" sciencing.com, https://www.sciencing.com/calculate-length-sides-regular-hexagons-6001248/. 1 December 2020.
Maloney, Lisa. (2020, December 1). How To Calculate Length Of Sides In Regular Hexagons. sciencing.com. Retrieved from https://www.sciencing.com/calculate-length-sides-regular-hexagons-6001248/
Maloney, Lisa. How To Calculate Length Of Sides In Regular Hexagons last modified March 24, 2022. https://www.sciencing.com/calculate-length-sides-regular-hexagons-6001248/ | {"url":"https://www.sciencing.com:443/calculate-length-sides-regular-hexagons-6001248/","timestamp":"2024-11-07T10:40:55Z","content_type":"application/xhtml+xml","content_length":"74053","record_id":"<urn:uuid:c0a7a98b-d978-4459-aac5-bb2a77ac3201>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00409.warc.gz"} |
pimpleFoam with cyclic BCs
After some times I retried to simulate a channel with cyclic BCs with the solver pimpleFoam and using RAS models (i want only to compute the mean velocity profile).
The problem is i can't get the pressure variable to converge it will always stay at around 0.4-0.6 and I really tried everything.
First some info on the geometry and physical properties:
Lx=5*H (i tried longer Lx but nothing changed)
I=7% (from experimental data that i have)
with Uc=velocity at centerline, Ub=Ubar and H = width of the channel.
I estimated y__firstcell with yplus=1,4,30 and considering different cell-to-cell expansion (keeping it under 1.15), in the file attached yPlus is around the unit. I tried increasing the yplus until
yplus=40 but this didn't change the weird behaviour of the pressure and Uy residuals.
Now the initialization of teh turbulent flow are
For epsilon this is a little bit tricky, now i've read all the code about the solver and the turbulence correction and i noticed that in the "turbulenceModels" epsilon has different definition
depending if it is related to a cell adjacent to teh wall (via teh wallfunction, there is a lone where it will update G and epsilon) or not (e.g in kEpsilon.C there is teh update of nut values and
from this relation you can define epsilon). So there are two possible definition one based on the turbulence length (o mixing length) and the other on nut/nu.
epsilon(l)=22 with l=10%H
epsilon(nut/nu)=139 with nut/nu=10
The mesh should be fine:
Overall domain bounding box (0 0 0) (0.125 0.025 0.025)
Mesh (non-empty, non-wedge) directions (1 1 0)
Mesh (non-empty) directions (1 1 0)
All edges aligned with or perpendicular to non-empty directions.
Boundary openness (-1.15504e-18 7.86934e-17 2.92744e-16) OK.
Max cell openness = 2.51663e-16 OK.
Max aspect ratio = 22.206 OK.
Minimum face area = 7.0364e-08. Maximum face area = 3.3667e-05. Face area magnitudes OK.
Min volume = 1.7591e-09. Max volume = 4.20837e-08. Total volume = 7.8125e-05. Cell volumes OK.
Mesh non-orthogonality Max: 0 average: 0
Non-orthogonality check OK.
Face pyramids OK.
Max skewness = 1.42118e-06 OK.
Coupled point location match (average 8.25619e-18) OK.
What i tried to modify:
-schemes and fvsolution parameters
-initial conditions
-yplus and cell-to-cell expansion (or R)
-lenght of the channel
-Bcs Wall function or not
Unfortunately this didn't help.
Here the archive with all the files, I atatched also one graph of teh residual profile (done with pyFoam). I hope someone could figure out the problem.
As a side note even with the LES simulation i couldn't get the right redisuals, the p and uy residuals were the problems. I used the same flow properties with a finer mesh.
: Maybe i've found the error, if i choose in controldict to run the simulation with fixed deltat the courant number will explode after some steps. I don't know how to solve it, I tried changing the
timeStep with no success. What velocity U and spacing do you consider when defining courant number? I usually try to get the lowest timestep so i use the highest velocity and the lowest spacing grade
: I tried also a simulation with no turbulence but again the residual floats around the unit.
OT: I have 3 questions:
1- what relation do you use to define epsilon?
2- what do you put in the label pRefCell? I've chosen first 1001 but it gave me an error then i switched to 101, I know that it is teh identification on one cell of the domain where you initialize
the pressure (so you'll not have an ill-conditioned problem) but it the choice of the cell important?
3- Reading teh code I noticed that the utility yPlusRAS only calls the calculation of yPlus done in the WF. Moreover studying the WF there two different definitions of the yPlus(utau,y,nu) and yStar
(k). However i didn't find nothing RELIABLE about the comparison of the two definitions and on how they interact with each other (i know how to define them from the wall equations)
These question are only my left doubts after going deep into the code and i couldn't answer them. | {"url":"https://www.cfd-online.com/Forums/openfoam-solving/143008-pimplefoam-cyclic-bcs.html","timestamp":"2024-11-06T18:03:00Z","content_type":"application/xhtml+xml","content_length":"190429","record_id":"<urn:uuid:655ce1eb-6125-4f2a-a80d-c346a185c1dc>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00079.warc.gz"} |
What are some divisions of polygons? - The Handy Math Answer Book
Geometry and Trigonometry
Plane Geometry
What are some divisions of polygons?
There are two major divisions of polygons: Regular polygons are convex polygons with equal sides and length; thus, all sides and angles are congruent (equal). For example, one of the most famous
regular octagons is the stop sign used along roads in the United States: a closed polygon with eight equal sides. The naming of the various polygons can be challenging, though. For example, a polygon
called a regular triangle is also called an equilateral triangle; another name for the polygon called a regular quadrilateral is a square. Irregular polygons are those with sides of differing lengths
and variable angles. Therefore, unless all the sides of the polygon are of the same length and all the angles are of the same measure, the polygon is said to be irregular.
But don’t be fooled: The names for the various polygons—such as hexagon, nonagon, and pentagon, depending on number of sides—don’t just apply to the regular polygons, but rather to any
two-dimensional closed figure with the number of sides as described by its name. For example, the two figures shown on page 185 are both polygons—A is a regular hexagon and B is an irregular hexagon.
Polygons are described in other ways, too. Convex polygons are those in which every line drawn between any two points inside the polygon lie entirely within the figure. Opposite from the convex
polygons are the concave polygons—those that are essentially “caved in,” with some of the sides bent inward. If a line is drawn between two points inside a concave polygon, the line often passes
outside the figure. Another type of polygon is a star polygon, in which a star figure is drawn based on equidistant points connected on a circle. | {"url":"https://www.papertrell.com/apps/preview/The-Handy-Math-Answer-Book/Handy%20Answer%20book/What-are-some-divisions-of-polygons/001137022/content/SC/52cb006e82fad14abfa5c2e0_Default.html","timestamp":"2024-11-07T18:39:06Z","content_type":"text/html","content_length":"12617","record_id":"<urn:uuid:0305b260-2665-48f6-8eeb-cb458af51d57>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00788.warc.gz"} |
"BankAccount and OverdraftedAccount Classes - Get Available Funds"
Assume the existence of a BankAccount class with a method, getAvailable that returns the amount of available funds in the account(as an integer), and a subclass, OverdraftedAccount, with two integer
instance variables: overdraftLimit that represents the amount of money the account holder can borrow from the account (i.e., the amount the account balance can go negative), and overdraftAmount, the
amount of money already borrowed against the account. Override the getAvailable method in OverdraftedAccount to return the amount of funds available(as returned by the getAvailable method of the
BankAccount class) plus the overdraftLimit minus the overdraftAmount.
Assume the existence of a BankAccount class with a method, getAvailable that returns the amount of available funds in the account(as an integer), and a subclass, OverdraftedAccount, with two integer
instance variables: overdraftLimit that represents the amount of money the account holder can borrow from the account (i.e., the amount the account balance can go negative), and overdraftAmount, the
amount of money already borrowed against the account. Override the getAvailable method in OverdraftedAccount to return the amount of funds available(as returned by the getAvailable method of the
BankAccount class) plus the overdraftLimit minus the overdraftAmount.
public int getAvailable() {
return super.getAvailable() + (overdraftLimit - overdraftAmount); | {"url":"https://matthew.maennche.com/2014/06/assume-existence-bankaccount-class-method-getavailable-returns-amount-available-funds-accountas-integer-subclass-overdraftedaccount-two-integer/","timestamp":"2024-11-15T03:20:17Z","content_type":"text/html","content_length":"96486","record_id":"<urn:uuid:8018cf25-620b-4b35-8b3b-df7c5b16317f>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00393.warc.gz"} |
density for ball mill grinding
For a ball mill, if the reduction ratio becomes less than 3 (target grinding of concentrates), the energy index W i must be multiplied by a given coefficient given by the author's equation 27
[BON 61b, p. 545]. ... true density of grinding bodies (dimensionless) ϕ : fraction of the critical speed ...
WhatsApp: +86 18838072829
This study presents a novel derived theoretical model for MAassisted leaching in investigating the effects of ball mill parameters on the particle sizes (retained and recovered). ... (omega
right) ; diameter left(dright) of the milling/grinding ball; and its weight left(Mright) . ... Is the solid molar density in mol/cm3. CAg ...
WhatsApp: +86 18838072829
Various attempts have been made to correlate kinetic parameters to mill dimensions, mill speed, ball load, ball density, ball diameter, and holdup mass of material, ... H. Simulation and
optimization of a twostage ball mill grinding circuit of molybdenum ore. Adv. Powder Technol. 2016, 27, .
WhatsApp: +86 18838072829
Clinker density (Dotternhausen). Ball Mill. Cement Lafarge max R5% >25mm, Holcim <50mm. Standard offer from mill manufacturer is R5% >30mm ... For instance, in large diameter ball mills, the
impact force of the grinding media is so great, that a high material surface unbalance prevails in the mill when grinding all types of clinker, thus ...
WhatsApp: +86 18838072829
Maximum ball size (MBS) Please Enter / Stepto Input Values. Mill Feed Material Size F, mm. Specific weight of Feed SG, g/cm 3.
WhatsApp: +86 18838072829
As an example, a density of thirty percent means there is thirty parts ore to seventy parts water. Here is another example, if you have a density of eighty percent that means eighty percent of
the VOLUME is ore, twenty percent of the volume is water. The Time that the ore spends in the grinding mill is called, RETENTION TIME.
WhatsApp: +86 18838072829
The attrition mill agitator rotates at speeds ranging from 60 rpm for production units to 300 rpm for laboratory units and uses media that range from 3 to 6 mm while ball mills use large grinding
media, usually 12 mm or larger, and run at low rotational speeds of 1050 rpm. Power input to attrition mills is used to agitate the medium, not to ...
WhatsApp: +86 18838072829
SAG is an acronym for semiautogenous grinding. SAG mills are autogenous mills that also use grinding balls like a ball mill. A SAG mill is usually a primary or first stage grinder. SAG mills use
a ball charge of 8 to 21%. The largest SAG mill is 42' () in diameter, powered by a 28 MW (38,000 HP) motor.
WhatsApp: +86 18838072829
• characteristics of the grinding media (mass, density, ball size distribution); • speed of rotation of the mill; • slurry density in case of wet grinding operation. Quantitative estimations of
these parameters can be found in [4, 5, 23]. An important characteristic of an industrial ball mill is its production capacity
WhatsApp: +86 18838072829
Ball Mill Size as a Replacement. Grinding media wears and reduces in size at a rate dependent on the surface hardness, density and composition of the ore. ... â†' (Wâ†"i) â†' / (νD) â†' ν = the
rotational speed of the mill. Ball Bulk Density. Low density media can be used for soft and brittle materials ...
WhatsApp: +86 18838072829
Overcharging results in poor grinding and losses due to abrasion of rods and liners. Undercharging also promotes more abrasion of the rods. The height (or depth) of charge is measured in the same
manner as for ball mill. The size of feed particles to a rod mill is coarser than for a ball mill. The usual feed size ranges from 6 to 25 mm.
WhatsApp: +86 18838072829
Applications Ball mills are used for grinding materials such as mining ores, coal, pigments, and feldspar for pottery. Grinding can be carried out wet or dry, but the former is performed at low
WhatsApp: +86 18838072829
According to the pattern, the residence time thresholds beyond which overfilling a ball mill is likely to occur were defined. For a ball mill with an internal diameter smaller than m, the
volumebased residence time threshold is set at 2 min; and for a ball mill larger than m in diameter, the threshold is set at 1 min. In addition to ...
WhatsApp: +86 18838072829
Adjustment to ball size could lead to significant improvement in grinding mill throughput (McIvor, 1997).The Bond's equation for ball sizing (McIvor, 1997) can help in selecting the ball size for
a given ore and grinding mill simulations with ball wear modelling can also be used to identify the optimum ball size (Concha et al., 1992) for a given application.
WhatsApp: +86 18838072829
density of grinding media in general, kg m −3. ρ b. density of steel ball, kg m −3. Ω E. grinding media consumption based on energy usage, kg J −1. Ω M. grinding media consumption based on amount
of ore ground, kg kg −1. Ω t. grinding media consumption based on operating time, kg s −1
WhatsApp: +86 18838072829
The grinding process is a complex physical, chemical, and physicochemical process, with many factors at play. When the specifications and models of the ball mill are determined, the factors
affecting the operation indexes of the grinding process include three facets [1519]: one, the properties of the ore entering the grinding process, including the mechanical properties of the ore,
the ...
WhatsApp: +86 18838072829
The basic parameters used in ball mill design (power calculations), rod mill or any tumbling mill sizing are; material to be ground, characteristics, Bond Work Index, bulk density, specific
density, desired mill tonnage capacity DTPH, operating % solids or pulp density, feed size as F80 and maximum 'chunk size', product size as P80 and maximum a...
WhatsApp: +86 18838072829
Comparing dry and wet grinding curves in a ball mill in the same condition ... Blecher, L., Kwade, A., Schwedes, J.: Motion and stress intensity of grinding beads in a stirred media mill. Part 1:
energy density distribution and motion of single grinding beads. Powder Technol. 86, 5968 (1996).
WhatsApp: +86 18838072829
Zirconium Oxide Grinding Balls. Highly polished YSZ (Yttrium stabilized ZrO 2) zirconium oxide grinding balls for planetary and high energy ball mills.. Grinding jar ball configuration
WhatsApp: +86 18838072829
The amount of feed was set at 1175 g, which was proportional to industrial ball mill speed was constantly set at % were conducted at different ball sizes (between 20 and 40 mm), grinding time
(1030 min), solids content (6580%) and work index of copper sulphide ore ().RSM and BoxBehnken design were used for the experimental design and modeling of ...
WhatsApp: +86 18838072829
Steel balls have higher density, high energy consumption during grinding, and high instantaneous collision force between ball and ball, ball and ore, and ball and mill. High kinetic energy is
generated during grinding, much greater than the kinetic energy required to crush the ore to qualified particle size.
WhatsApp: +86 18838072829
The grinding process in the ball mill is due to the centrifugal force induced by the mill on the balls. This force depends on the weight of the balls and the ball mill rotational speed. ...
(MPa), ρ b: density of the ball (kg/m 3), D i,D: inner diameter of the drum (m). Based on the result given by equation (Table 8), a minimum ball diameter ...
WhatsApp: +86 18838072829
Martins et al., 2008, Martins et al., 2013 developed an instrumented ball and a camera system to measure the state of the charge within a laboratory mill. However, this instrumented ball does not
have the same effective density as the ordinary grinding ball, consequently the accuracy of measurement cannot be guaranteed.
WhatsApp: +86 18838072829
Grinding Test. In this paper, the wet ball milling process was used in the laboratory ball milling test. Before grinding, the ball mill was kept idle for 10 minutes and then washed together with
the grinding media. Three types of ores were crushed and separated by a jaw crusher, and the ground particle size of 2 mm was selected.
WhatsApp: +86 18838072829 | {"url":"https://auberges-rurales.fr/2024/Jul/03-9526.html","timestamp":"2024-11-03T04:42:17Z","content_type":"application/xhtml+xml","content_length":"23331","record_id":"<urn:uuid:d3980efa-9ad0-4c49-92ac-87d9cb7442b8>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00434.warc.gz"} |
Lateral Effect | Agricultural Water ManagementLateral Effect
Lateral Effect
Release date: February 2015
Method to Determine Lateral Effect of a Drainage Ditch on Adjacent Wetland Hydrology
R.W.Skaggs, G.M Chescheir, and B.D. Phillips
A method was developed to estimate the lateral effect of a single drainage ditch on wetland hydrology. The method can be used to calculate the distance of influence of a single ditch constructed
through a wetland, where the distance of influence is defined as the width of a strip adjacent to the ditch that is drained such that it will no longer satisfy the wetland hydrologic criterion.
Simulation analyses were conducted with DRAINMOD to define the minimum, or threshold, drainage intensity that would result in failure of a site to satisfy the wetland hydrologic criterion. Analyses
were conducted for five hydric soils spanning a wide range of profile hydraulic transmissivities. DRAINMOD was used to predict water table fluctuations between parallel ditches for a 50-year period
of climatological record. For each soil, simulations were conducted for a range of ditch spacings and depths to determine the combinations that would result in the land midway between the ditches
just barely satisfying the wetland hydrologic criterion. Analyses were conducted for climatological conditions for three locations in eastern North Carolina. Results for Wilmington, North Carolina,
showed that the threshold drainage intensities would result in water table drawdown from an initially ponded surface to a depth of 25 cm in approximately 6 days. That is, ditch depths and spacings
sufficient to lower the water table from the surface to a depth of 25 cm in a threshold time of about 6 days would result in hydrologic conditions that would just barely satisfy the wetland
hydrologic criterion for that location. The threshold time is denoted T25 and is used as a surrogate for quantifying the water table drawdown rate of sites that barely satisfy the wetland hydrologic
criterion. T25 was found to depend somewhat on drain depth, but it was essentially constant for all five of the soils examined. Similar results were obtained for the other two locations, but because
of differences in weather and in the growing season, the threshold time (T25) was found to be dependent on location. The T25 value is also dependent on surface depressional storage, decreasing with
increasing storage. The discovery that water table conditions barely satisfying the wetland hydrologic criterion are well correlated to the time required for water table drawdown of 25 cm (T25
values) makes it possible to predict the effects of subsurface drains on wetland hydrology. The lateral effect of a single ditch on wetland hydrology can be computed by using T25 values in solutions
to the Boussinesq equation for water table drawdown due to drainage to a single drain. While the method was developed for drainage ditches, it may also be used for subsurface drains.
Project Objective
Quantify the lateral effect (distance of hydrologic influence) of a single ditch using proper drainage theory, climatic variables, and soil properties. | {"url":"https://bae.ncsu.edu/agricultural-water-management/lateral-effect/","timestamp":"2024-11-12T23:50:11Z","content_type":"text/html","content_length":"75215","record_id":"<urn:uuid:e21d8e3c-7bed-4e17-8291-e4130deb883a>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00796.warc.gz"} |
An Improved Proof of the Handshaking Lemma | João F. Ferreira
An Improved Proof of the Handshaking Lemma
In 2009, I posted a calculational proof of the handshaking lemma, a well-known elementary result on undirected graphs. I was very pleased about my proof because the amount of guessing involved was
very small (especially when compared with conventional proofs). However, one of the steps was too complicated and I did not know how to improve it.
In June, Jeremy Weissmann read my proof and he proposed a different development. His argument was well structured, but it wasn’t as goal-oriented as I’d hoped for. Gladly, after a brief discussion,
we realised that we were missing a great opportunity to use the trading rule (details below)!
I was so pleased with the final outcome that I decided to record and share the new proof.
Problem statement
In graph theory, the degree of a vertex $A$, $\fapp{d}{A}$, is the number of edges incident with the vertex $A$, counting loops twice. So, considering the graph below, we have $\fapp{d}{A}=3$, $\fapp
{d}{B}=3$, $\fapp{d}{C}=1$, $\fapp{d}{D}=3$, and $\fapp{d}{E}=2$.
A well-known property is that every undirected graph contains an even number of vertices with odd degree. The result first appeared in Euler’s 1736 paper on the Seven Bridges of Königsberg and is
also known as the handshaking lemma (that’s because another way of formulating the property is that the number of people that have shaken hands an odd number of times is even).
As we can easily verify, the graph shown above satisfies this property. There are four vertices with odd degree ($A$,$B$, $C$, and $D$), and 4, of course, is an even number.
Although the proof of this property is simple, all the conventional proofs that I know of are not goal-oriented. My goal is to show you a development of a goal-oriented proof. Also, my proof is
completely guided by the shape of the formulae involved, which helps reducing the amount of guessing involved.
Notations that I use
Before we start, let me explain the notations that I use. I assume the existence of two predicates, $even$ and $odd$, that test the parity of numbers. For example, $\fapp{even}{8}$ and $\fapp{odd}{3}
$ are both true, and $\fapp{even}{5}$ and $\fapp{odd}{6}$ are both false. Also, I use the so-called Eindhoven notation for quantifiers; for example, to express the sum of all natural even numbers
less than 50 I write $\quantifier{\Sigma}{n}{0{\leq}n{<}50~\wedge~\fapp{even}{n}}{n}$, and instead of writing $\fapp{even}{0}{\equiv}\fapp{even}{1}{\equiv}{\cdots}{\equiv}\fapp{even}{50}$, I write $\
An advantage of using a systematic notation for quantifiers is that we can write the rules that manipulate quantifiers in a very general way. For example, suppose that the quantifier $\bigoplus$
generalises the binary operator $\oplus$. Moreover, let us assume that $1_{\oplus}$ is the unit of $\oplus$, that is, for all $n$, we have:
$$ n{\oplus}1_{\oplus} = 1_{\oplus}{\oplus}n = n ~~. $$
Then, the so-called trading rule is valid:
$$ \beginproof \pexp{\quantifier{\bigoplus}{n}{P \wedge Q}{T}} \equiv \
\pexp{\quantifier{\bigoplus}{n}{P}{{\sf if~~} Q \rightarrow T~~\Box~~\neg{Q} \rightarrow 1_{\oplus} {~~\sf fi}} ~.} \endproof $$
This rule applies to all quantifiers that generalise operators with units. For example, because true is the unit of $\equiv$, we have
$$ \beginproof \pexp{\quantifier{\equiv}{n}{P \wedge Q}{T}} \equiv \
\pexp{\quantifier{\equiv}{n}{P}{{\sf if~~} Q \rightarrow T~~\Box~~\neg{Q} \rightarrow true {~~\sf fi}} ~,} \endproof $$
which is the same as
$$ \quantifier{\equiv}{n}{P \wedge Q}{T} \equiv \quantifier{\equiv}{n}{P}{Q \Rightarrow T} ~~. $$
Calculating a solution to the handshaking lemma
Now, the first step in any goal-oriented solution is to express the goal. In other words, what do we want to prove or calculate? Using the notation just described and assuming that the variable $v$
ranges over the set of all vertices, our goal is to determine the value of the following expression:
$$ \fapp{even} { \quantifier{\Sigma}{v} {\fapp{odd}{(\fapp{d}{v})}} {1} }~~~. $$
Note that we are adding 1 (counting) for each vertex $v$ with an odd degree. We then apply the predicate $even$ to the result. If the result is true, there is an even number of vertices with odd
degree; otherwise, there is an odd number. Our goal is thus to determine its value. (We know that it must evaluate to true, because the property is well-known. However, in general, when doing
mathematics, we don’t know what is the final value; that is why goal-oriented and calculational proofs are important.)
We know that the predicate $even$ distributes over addition, so we calculate:
$$ \beginproof \pexp{\fapp{even}{\quantifier{\Sigma}{v}{\fapp{odd}{(\fapp{d}{v})}}{1}}} \hint{=}{$even$ distributes over addition} \pexp{\quantifier{\equiv}{v}{\fapp{odd}{(\fapp{d}{v})}}{\fapp{even}
{1}}} \hint{=}{$\fapp{even}{1}\equiv false$} \pexp{\quantifier{\equiv}{v}{\fapp{odd}{(\fapp{d}{v})}}{false}} \hint{=}{trading rule (see above)} \pexp{\quantifier{\equiv}{v}{\negspace\negspace}{\fapp
{odd}{(\fapp{d}{v})} \Rightarrow false}} \hint{=}{${\fapp{odd}{n}\Rightarrow{false}} ~\equiv~ {\fapp{even}{n}}$} \pexp{\quantifier{\equiv}{v}{\negspace\negspace}{\fapp{even}{(\fapp{d}{v})}}} \hint{=}
{$even$ distributes over addition} \pexp{\fapp{even}{\quantifier{\Sigma}{v}{\negspace\negspace}{\fapp{d}{v}}}} \endproof $$
This calculation shows that the parity of the number of vertices with odd degree is the same as the parity of the sum of all the degrees. But because each edge has two ends, the sum of all the
degrees is simply twice the total number of edges. We thus have:
$$ \beginproof \pexp{\fapp{even}{\quantifier{\Sigma}{v}{\fapp{odd}{(\fapp{d}{v})}}{1}}} \hint{=}{calculation above} \pexp{\fapp{even}{\quantifier{\Sigma}{v}{\negspace\negspace}{\fapp{d}{v})}}} \hintf
{=}{the sum of all the degrees is twice the} \hintl{number of edges, i.e., an even number} \pexp{true~~.} \endproof $$
And so we can conclude that every undirected graph contains an even number of vertices with odd degree.
What is wrong with conventional solutions? Conventional solutions for this problem are usually very similar to the following one, taken from the book “Ingenuity in Mathematics” (p. 8), by Ross
The proof in general is simple. We denote by T the total of all the local degrees:
(1) T = d(A) + d(B) + d(C) + … + d(K) .
In evaluating T we count the number of edges running into A, the number into B, etc., and add. Because each edge has two ends, T is simply twice the number of edges; hence T is even.
Now the values d(P) on the right-hand side of (1) which are even add up to a sub-total which is also even. The remaining values d(P) each of which is odd, must also add up to an even sub-total
(since T is even). This shows that there is an even number of odd d(P)’s (it takes an even number of odd numbers to give an even sum). Thus there must be an even number of vertices with odd local
There is nothing wrong with this solution in the sense that it shows why the property holds. However, it is clearly oriented to verification: it starts by introducing the total sum of all the local
degrees, observing that its value is even; then it analyses that sum to conclude the property. The question is: how can we teach students to come with the total sum of all the local degrees? In
general, how can we teach students to come with seemingly unrelated concepts that will be crucial in the development of their arguments? I don’t think we can.
On the other hand, if we look at the goal-oriented proof, we see that the goal is simple to express. Furthermore, with some training, most students would write it correctly and would be able to
calculate that the parity of the number of vertices with odd degree is the same as the parity of the sum of all the degrees. And then (and only then) the introduction of the total sum of all the
degrees would make sense. In a way, goal-oriented calculations are like that famous masked magician that reveals magic’s biggest secrets, for they reveal how the rabbit got into the hat. | {"url":"https://joaoff.com/2011/09/20/an-improved-proof-of-the-handshaking-lemma/","timestamp":"2024-11-13T09:14:52Z","content_type":"text/html","content_length":"27544","record_id":"<urn:uuid:d4186fa9-987f-4169-ae3b-0fdec665995d>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00345.warc.gz"} |
SciPost Submission
SciPost Submission Page
Operator Entanglement in Local Quantum Circuits I: Maximally Chaotic Dual-Unitary Circuits
by Bruno Bertini, Pavel Kos, Tomaz Prosen
This is not the latest submitted version.
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users): Bruno Bertini · Pavel Kos
Submission information
Preprint Link: https://arxiv.org/abs/1909.07407v1 (pdf)
Date submitted: 2019-10-01 02:00
Submitted by: Bertini, Bruno
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
• Mathematical Physics
Specialties: • Quantum Physics
Approach: Theoretical
The entanglement in operator space is a well established measure for the complexity of the quantum many-body dynamics. In particular, that of local operators has recently been proposed as dynamical
chaos indicator, i.e. as a quantity able to discriminate between quantum systems with integrable and chaotic dynamics. For chaotic systems the local-operator entanglement is expected to grow linearly
in time, while it is expected to grow at most logarithmically in the integrable case. Here we study local-operator entanglement in dual-unitary quantum circuits, a class of "statistically solvable"
quantum circuits that we recently introduced. We show that for "maximally-chaotic" dual-unitary circuits the local-operator entanglement grows linearly and we provide a conjecture for its asymptotic
behaviour which is in excellent agreement with the numerical results. Interestingly, our conjecture also predicts a "phase transition" in the slope of the local-operator entanglement when varying the
parameters of the circuits.
Current status:
Has been resubmitted
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2020-2-10 (Invited Report)
• Cite as: Anonymous, Report on arXiv:1909.07407v1, delivered 2020-02-10, doi: 10.21468/SciPost.Report.1483
1: Relevant problem: characterization of quantum chaotic behavior by entanglement of local operators.
2: Precise statements of goals, techniques and results
3: Interesting models proposed, with a clear algebraic formulation allowing for exact limit results and conjecture on asymptotic behavior backed by numerical datas.
1: Technical aspects should be precised
2: Link between two definitions of maximally chaotic should be established
3: Conjecture on asymptotic behavior as combination of limit behaviors should be justified a priori, or at least interpreted.
A very interesting, very clear and well written paper on a highly relevant subject which has attracted much attention recently. Many interesting analytical results, and a highly non trivial
conjecture backed by excellent numerical datas. Recommanded for publication after some issues are clarified.
Requested changes
1: "Maximally chaotic" is characterized in the Introduction as "no conservation law" (which seems perfectly relevant in an integrability context) whereas it is characterized in Def. 4.1 as "only (48)
are eigenvectors of [either row transfer matrix] with eigenvalues of unit magnitude". How does one go from one definition to the other ?
2: A related question would be: how to prove that given "dual unitary quantum circuits" have NO dynamical conservation laws ? Already not obvious to me regarding local conservation laws; Paper II
shows that specific dual quantum circuits may exhibit local conservation laws; how to prove their absence is more complicated. Even worse: possible non-local conservation laws are usually quite
difficult to construct, hence their absence is probably quite difficult to prove either.
3: Why are transfer matrices contracting operators (eqn. 47) ?
4: The two statements after Definition 4.1 seem contradictory: "... (48) are generically only a subset of the eigenvectors of [...] associated with eigenvalue 1" vs. "For generic dual unitary
circuits there are no eigenvectors of [...] with unit magnitude eigenvalue other than (48)". The meaning of "generic" in both cases should probably be clarified.
5: What would be the meaning of the conjecture 5.1 ? Is there a physical interpretation of this decoupling of asymptotic behaviour at large x,t as a sum of two non-interacting limits : (x-) to
infinity + (x+) to infinity ?
Report #1 by Jerome Dubail (Referee 1) on 2019-11-6 (Invited Report)
• Cite as: Jerome Dubail, Report on arXiv:1909.07407v1, delivered 2019-11-06, doi: 10.21468/SciPost.Report.1301
1- Very interesting problem and timely results
2- Beautiful “dual unitary” models solvable by new methods; new definition of “maximal chaos” in that context
3- Exact results
4- Interesting new conjecture about growth of operator entanglement, strongly supported by numerics
1- Only minor ones. The physical discussion of the conjecture and its possible implications could probably be expanded. A few technical steps in the derivation of the results could be made more
accessible to the reader.
This is an excellent paper with very interesting and timely results on a difficult problem. The authors provide a detailed study of the growth of operator entanglement of local operators in
Heisenberg picture (a quantity they propose to call 'local operator entanglement') in a class of solvable one-dimensional chaotic quantum spin chains. Those 'dual-unitary' models were introduced by
the authors themselves in a series of recent papers.
The results presented here provide analytical support for claims made in other recent papers on local operator entanglement: this quantity grows linearly with time in chaotic spin chains. This is in
contrast with non-chaotic systems, where it is conjectured to grow at most logarithmically.
Also, an interesting new conjecture is presented about the exact rate of the linear growth, which is well supported by numerical results.
I recommend publication of this manuscript in Scipost. The authors may want to consider the suggestions/remarks below before publication.
Requested changes
1- Abstract: perhaps it should be said explicitly that 'maximally-chaotic' here refers to a specific definition $-$valid in the context of dual-unitary models$-$ which is new and appears in this
paper for the first time.
2- Eq. (2): clearly in the product $U^{\otimes L}$ each $U$ is supposed to act on two neighboring sites; yet this should be written explicitly
3- Graphical representation of Eq. (12): maybe the authors could put the site indices ($\dots, -1, -\frac{1}{2},0, \frac{1}{2}, 1 ,\dots $), as they do in Eq. (10), on the top and bottom rows. I
believe this would make the conventions clearer. Also, maybe it would be clearer to draw all the sites, from $-\frac{L}{2}$ to $\frac{L}{2}$, just for this picture, so that it's clear that $a$ is
inserted at $x=0$.
4- I find that it is hard to visualize the operations that bring us from Eq. (29) to Eq. (32). Since Eq. (32) plays an important role in the rest of the paper, its derivation would deserve to be
slightly expanded. Perhaps having both Eqs. (29) and (32) drawn with the same small values of $y$ and $t$ would help. And perhaps a picture of the intermediate step, where Eq. (26b) is used, would
help as well.
5- before Eq. (47): 'any such transfer matrix is a contracting operator': why is that so? Is it obvious?
6- Eqs. (49)-(50). If the conventions are that the operators act from bottom to top in the drawings, then the rainbow state in Eq. (49) should appear with open legs on top. More importantly, there
may be an error in Eq. (50): if I am not mistaken, the conventions are such that the squared norms $\left< \cup \left. \right| \cup \right>$ and $\left< \circ \circ \left. \right| \circ \circ \right>
$ are both equal to $1$, and then this implies $\left< \cup \left. \right| \circ \circ \right> = \frac{1}{d}$. If so, then $d$ should be replaced by $1/d$ in Eq. (50), in order for $\left| \bar{r}_l
\right>$ to be orthogonal to $\left| r_l \right>$.
7- Around Eq. (51): perhaps it could be recalled that $\left| \bar{r}_{x_+} \right>$ and $\left| e_{x_+} \right>$ are the same state. More importantly, the identity $\left< e_{x_+} \left. \right| a^\
dagger \circ \dots \circ a \right>=d^{1-x_+}/\sqrt{d^2-1}$ is sufficiently non-trivial so that its derivation would deserve to be expanded a bit. In particular, isn't that result relying on the
assumption ${\rm tr}[a] = 0$? (If so, this assumption needs to be stated explicitly)
8- I'm confused by Eqs. (61) and (62): looking at the graphical Eq. (59), wouldn't one expect $\mathcal{M}_+$ to appear on the left of $\mathcal{M}_-$ in Eqs. (61)-(62)?
9- Fig. 2 and discussion below Eq. (65): the discussion of the von Neumann entropy could probably be expanded there. First, since the results of the paper rely on $n$ integer $\geq 2$, where do the
claims about von Neumann come from exactly? Second, it is very interesting that the von Neumann entropy always has maximal growth, in contrast with the Renyi entropies; this seems to be a very
non-trivial observation which deserves to be highlighted.
10- section 5 and the new conjecture: the conjecture is well described mathematically, and the numerical evidence supporting it is well presented. However, I find that its physical meaning and
implications are not clearly discussed. For instance, if the growth is linear then the local operator entanglement $S(y,t)$ is expected to behave as $t f(y/t)$ for some function $f$ at large $t$ and
fixed ratio $y/t$. It seems that the conjecture Eq. (66) will typically lead to an interesting profile $f(v) = {\rm min}[ s_- (1-v), s_+ (1+v) ]$, for two constants $s_-$ and $s_+$ given by Eqs. (55)
and (65), and for $s_- \neq s_+$ this profile will be asymmetric and will look different from the symmetric 'pyramid' found in Ref. [37] (see Fig. 13 in that ref.). Will this simply come from the
absence of reflection symmetry of the dual unitary model? This is something that could be discussed. More generally I think it would be interesting for some readers to have a discussion of the
physical meaning and potential consequences of the new conjecture. | {"url":"https://scipost.org/submissions/1909.07407v1/","timestamp":"2024-11-01T20:32:13Z","content_type":"text/html","content_length":"43529","record_id":"<urn:uuid:09f0bfc7-ccf8-44f3-9a63-595944c8ac5e>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00057.warc.gz"} |
DRSCL - Linux Manuals (3)
DRSCL (3) - Linux Manuals
drscl.f -
subroutine drscl (N, SA, SX, INCX)
DRSCL multiplies a vector by the reciprocal of a real scalar.
Function/Subroutine Documentation
subroutine drscl (integerN, double precisionSA, double precision, dimension( * )SX, integerINCX)
DRSCL multiplies a vector by the reciprocal of a real scalar.
DRSCL multiplies an n-element real vector x by the real scalar 1/a.
This is done without overflow or underflow as long as
the final result x/a does not overflow or underflow.
N is INTEGER
The number of components of the vector x.
SA is DOUBLE PRECISION
The scalar a which is used to divide each component of x.
SA must be >= 0, or the subroutine will divide by zero.
SX is DOUBLE PRECISION array, dimension
The n-element vector x.
INCX is INTEGER
The increment between successive values of the vector SX.
> 0: SX(1) = X(1) and SX(1+(i-1)*INCX) = x(i), 1< i<= n
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
September 2012
Definition at line 85 of file drscl.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://www.systutorials.com/docs/linux/man/3-DRSCL/","timestamp":"2024-11-09T01:20:50Z","content_type":"text/html","content_length":"7641","record_id":"<urn:uuid:24b09149-70f9-4d29-ae66-3b1bff25a43b>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00898.warc.gz"} |
Investment Metrics & Definitions Guide | Market Winnow
top of page
Here you will find definitions of important terms we use.
Alpha is a measure of a portfolio's performance that compares the return achieved by the portfolio with its expected return based on its level of risk. In other words, Alpha measures the excess
return of a portfolio beyond what would be expected given its level of systematic risk as measured by its Beta.
A positive Alpha indicates that the portfolio has performed better than expected given its level of risk, while a negative alpha indicates that the portfolio has under-performed. Alpha is often used
to assess the skill of fund managers (or the accuracy of ratings) in generating excess returns above the market.
It's important to note that Alpha is not a guarantee of future performance, and can be impacted by various factors such as market conditions, fees, and trading costs. Therefore, investors should
consider other factors in addition to alpha when evaluating performance
Beta is a measure of the volatility, or systematic risk, of a particular stock or portfolio in relation to the overall market. It compares the movement of the stock or portfolio's returns with the
movement of the market returns as a whole, and is often used as a benchmark for assessing an asset's performance.
Beta of 1 indicates that the stock or portfolio has the same level of volatility as the market, while a Beta greater than 1 indicates higher volatility and a Beta less than 1 indicates lower
volatility. A negative Beta means that the stock or portfolio moves in the opposite direction of the market.
Beta is a useful tool for investors because it allows them to assess the level of risk associated with a particular investment. A high Beta indicates higher risk, while a low Beta indicates lower
risk. This information can be used to make informed decisions about portfolio construction and risk management.
It is important to note that Beta is not a complete measure of an investment's risk, as it only considers systematic, or market, risk. Other types of risk, such as company-specific or sector-specific
risk, are not captured by Beta. Therefore, investors should use other measures in addition to Beta when evaluating risk.
Return p.a. (%)
The annualized return of a stock or portfolio through time, also known as the Compound Annual Growth Rate (CAGR), is a measure of the average annual return earned by an investor over a specified
period.The annualized return is calculated by first determining the total return earned by an investor in the stock or portfolio over the entire holding period, which includes both price appreciation
and any dividends or distributions paid out. This total return is then adjusted to an annualized rate by taking into account the length of the holding period.
The annualized return through time is an important metric for investors because it provides a more accurate representation of the investment's performance than a simple average return. It takes into
account the effects of compounding and allows investors to make better-informed decisions about their investment strategies.
Daily win rate
The daily win rate of a portfolio or stock is the percentage of days in a certain period during which it outperforms another stock or portfolio. This frequency is measured in days and calculated by
dividing the number of days in which the stock or portfolio outperforms the benchmark by the total number of days in the period. For example, if a stock or portfolio beats the S&P500 on three out of
five market days in the prior week, its daily win rate would be 60%.
Win rate (in general)
The general win rate of a stock or portfolio over a given period of time T is the percentage of t-length periods within T during which the stock or portfolio outperforms a benchmark. Note that t can
be any fixed length of time, such as a week, a month, or a quarter, and T can be any length of time longer than t. Hence, the general win rate is a measure of the strength of the stock or portfolio
relative to the benchmark across all t-length periods within the period T.
Rank 1 vs. S&P 500
Return is utilized to discern between winning and losing scenarios. If Rank 1 has a return greater than the S&P 500, it signifies a win; the opposite denotes a loss.
Winning days
The number of market days a specific rank outperformed the S&P 500 over a given period.
Losing days
The number of market days a specific rank underperformed the S&P 500 over a given period.
The number of stocks that experienced positive returns on the previous market day.
The number of stocks that experienced negative returns on the previous market day
bottom of page | {"url":"https://www.marketwinnow.com/investment-metrics-definitions","timestamp":"2024-11-06T04:41:14Z","content_type":"text/html","content_length":"540224","record_id":"<urn:uuid:ea038111-1f56-46f3-8be4-586bb31dab6c>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00451.warc.gz"} |
orksheets for 10th Class
Explore worksheets by Math topics
10.1 Simple Probabilities
Linear Equations and Inequalities
Non-Linear Systems of Equations
Simplifying Rational Expressions
Trigonometric Functions XI
Solving Rational Equations and inequalities
Simplifying Rational Expressions
Sequences and Series Review
Arithmetic Sequences and Series
Explore Math Worksheets by Grades
Explore Math Worksheets for class 10 by Topic
Explore Other Subject Worksheets for class 10
Explore printable Math worksheets for 10th Class
Math worksheets for Class 10 are essential tools for teachers looking to help their students master the challenging concepts in high school mathematics. These worksheets cover a wide range of topics,
including algebra, geometry, trigonometry, and probability, ensuring that students have ample practice in each area. Teachers can use these resources to supplement their lesson plans, provide extra
practice for struggling students, or even as a basis for group activities in the classroom. By incorporating Class 10 math worksheets into their curriculum, teachers can ensure that their students
are well-prepared for the rigors of high school mathematics and beyond.
Quizizz is an excellent platform for teachers seeking to enhance their Class 10 math curriculum with engaging and interactive content. In addition to offering a vast collection of Math worksheets for
Class 10, Quizizz also provides teachers with access to a wide variety of quizzes, games, and other educational resources. These materials can be easily integrated into the classroom, allowing
teachers to create a dynamic and engaging learning environment for their students. Furthermore, Quizizz offers real-time feedback and analytics, enabling teachers to track student progress and
identify areas where additional support may be needed. With Quizizz, teachers have a powerful tool at their disposal to help their Class 10 students excel in mathematics. | {"url":"https://quizizz.com/en-in/mathematics-worksheets-class-10","timestamp":"2024-11-11T10:49:52Z","content_type":"text/html","content_length":"157441","record_id":"<urn:uuid:44649d09-2cf0-4760-88d8-0252526c4a91>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00895.warc.gz"} |
GP Rangaiah's Academic & Research Activities
Book Chapters
Hidden Book Chapter - DO NOT TOUCH
Simulation and Optimization of Intensified Chemical Processes
Feng Z. and Rangaiah G.P., Simulation and Optimization of Intensified Chemical Processes, in “Control and Safety Analysis of Intensified Chemical Processes” edited by Patle D.S. and Rangaiah G.P.,
John Wiley (2024).
Dynamic Simulation and Control of Intensified Chemical Processes
Feng Z. and Rangaiah G.P., Dynamic Simulation and Control of Intensified Chemical Processes, in “Control and Safety Analysis of Intensified Chemical Processes” edited by Patle D.S. and Rangaiah G.P.,
John Wiley (2024).
Safety Analysis of Intensified Distillation Processes using Existing and Modified Safety Indices
Shrikhande S., Deshpande G.K., Rangaiah G.P. and Patle D.S., Safety Analysis of Intensified Distillation Processes using Existing and Modified Safety Indices, in “Control and Safety Analysis of
Intensified Chemical Processes” edited by Patle D.S. and Rangaiah G.P., John Wiley (2024)
Selected Multi-Criteria Decision-Making Methods and their Applications to Product and System Design
Wang Z., Nabavi S.R. and Rangaiah G.P., Selected Multi-Criteria Decision-Making Methods and their Applications to Product and System Design, in “Optimization Methods for Product and System Design”
edited by Kulkarni A.J., Springer Nature, 2022.
Process Development, Design and Analysis of Microalgal Biodiesel Production aided by Microwave and Ultrasonication
Patle D.S., Shrikhande S. and Rangaiah G.P., Process Development, Design and Analysis of Microalgal Biodiesel Production aided by Microwave and Ultrasonication in “Process Systems Engineering for
Biofuels Development” edited by A. Bonilla-Petriciolet and G.P. Rangaiah, John Wiley, 2020.
Heat Exchanger Network Retrofitting Using Multi-Objective Differential Evolution
Sreepathi B.K., Sharma S. and Rangaiah G.P., Heat Exchanger Network Retrofitting Using Multi-Objective Differential Evolution, in “Differential Evolution in Chemical Engineering: Developments and
Applications” edited by G.P. Rangaiah and S. Sharma, World Scientific, 2017.
Process Development and Optimization of Bioethanol Recovery and Dehydration by Distillation and Vapor Permeation for Multiple Objectives
Singh A. and Rangaiah G.P. Process Development and Optimization of Bioethanol Recovery and Dehydration by Distillation and Vapor Permeation for Multiple Objectives, in “Differential Evolution in
Chemical Engineering: Developments and Applications” edited by G.P. Rangaiah and S. Sharma, World Scientific, 2017.
Optimization of Heat Exchanger Network Retrofitting: Comparison of Penalty Function and Feasibility Approach for Handling Constraints
Sreepathi B.K. and Rangaiah G.P., Optimization of Heat Exchanger Network Retrofitting: Comparison of Penalty Function and Feasibility Approach for Handling Constraints, in “Multi-Objective
Optimization: Techniques and Applications in Chemical Engineering” edited by G.P. Rangaiah, Second Edition, World Scientific, 2017.
Multi-Objective Optimization Programs and their Application to Amine Absorption Process Design for Natural Gas Sweetening
Sharma S., Rangaiah G.P. and Maréchal F., Multi-Objective Optimization Programs and their Application to Amine Absorption Process Design for Natural Gas Sweetening, in “Multi-Objective Optimization:
Techniques and Applications in Chemical Engineering” edited by G.P. Rangaiah, Second Edition, World Scientific, 2017.
Evaluation of Simulated Annealing, Differential Evolution and Particle Swarm Optimization for Solving Pooling Problems
Ong Y.C., Sharma S. and Rangaiah G.P., Evaluation of Simulated Annealing, Differential Evolution and Particle Swarm Optimization for Solving Pooling Problems, in “Evolutionary Computation: Techniques
and Applications” edited by A.M. Gujarathi and B.V. Babu, Apple Academic Press, 2017.
Mathematical Modeling, Simulation and Optimization for Process Design
Sharma S. and Rangaiah G.P., Mathematical Modeling, Simulation and Optimization for Process Design in “Chemical Process Retrofitting and Revamping: Techniques and Applications” edited by G.P.
Rangaiah, John Wiley, 2016.
Heat Exchanger Network Retrofitting: Alternative Solutions via Multi-objective Optimization for Industrial Implementation
Sreepathi B.K. and Rangaiah G.P., Heat Exchanger Network Retrofitting: Alternative Solutions via Multi-objective Optimization for Industrial Implementation in “Chemical Process Retrofitting and
Revamping: Techniques and Applications” edited by G.P. Rangaiah, John Wiley, 2016.
Techno-economic Evaluation of Membrane Separation for Retrofitting Olefin/Paraffin Fractionators in an Ethylene Plant
Tan X.Z., Pandey S., Rangaiah G.P. and Niu W., Techno-economic Evaluation of Membrane Separation for Retrofitting Olefin/Paraffin Fractionators in an Ethylene Plant in “Chemical Process Retrofitting
and Revamping: Techniques and Applications” edited by G.P. Rangaiah, John Wiley, 2016.
Retrofit of Vacuum Systems in Process Industries
Reddy C.C.S. and Rangaiah G.P., Retrofit of Vacuum Systems in Process Industries in “Chemical Process Retrofitting and Revamping: Techniques and Applications” edited by G.P. Rangaiah, John Wiley,
Retrofit and Revamp of Industrial Water Networks using Multi-objective Optimization Approach
Sharma S. and Rangaiah G.P., Design, Retrofit and Revamp of Industrial Water Networks using Multi-objective Optimization Approach in “Chemical Process Retrofitting and Revamping: Techniques and
Applications” edited by G.P. Rangaiah, John Wiley, 2016.
Jumping Gene Adaptations of NSGA-II with Altruism Approach: Performance Comparison and Application to Williams-Otto Process
Sharma S., Nabavi S.R. and Rangaiah G.P., Jumping Gene Adaptations of NSGA-II with Altruism Approach: Performance Comparison and Application to Williams-Otto Process in “Applications of
Metaheuristics in Process Engineering” edited by V. Jayaraman and P. Siarry, Springer, 2014.
Hybrid Approach for Multiobjective Optimization and its Application to Process Engineering Problems
Sharma S. and Rangaiah G.P., Hybrid Approach for Multiobjective Optimization and its Application to Process Engineering Problems in “Applications of Metaheuristics in Process Engineering” edited by
V. Jayaraman and P. Siarry, Springer, 2014.
Optimization of Pooling Problems for Two Objectives using the ε-Constraint Method
Zhang H. and Rangaiah G.P., Optimization of Pooling Problems for Two Objectives using the ε-Constraint Method in “Multi-Objective Optimization in Chemical Engineering: Developments and Applications”
edited by G.P. Rangaiah and A. Bonilla-Petriciolet, John Wiley, 2013.
Multi-Objective Optimization Applications in Chemical Engineering
Sharma S. and Rangaiah G.P., Multi-Objective Optimization Applications in Chemical Engineering in “Multi-Objective Optimization in Chemical Engineering: Developments and Applications” edited by G.P.
Rangaiah and A. Bonilla-Petriciolet, John Wiley, 2013.
Performance Comparison of Jumping Gene Adaptations of Elitist Non-Dominated Sorting Genetic Algorithm
Sharma S., Nabavi S.R. and Rangaiah G.P., Performance Comparison of Jumping Gene Adaptations of Elitist Non-Dominated Sorting Genetic Algorithm in “Multi-Objective Optimization in Chemical
Engineering: Developments and Applications” edited by G.P. Rangaiah and A. Bonilla-Petriciolet, John Wiley, 2013.
Improved Constraint Handling Technique for Multi-Objective Optimization with Application to Two Fermentation Processes
Sharma S. and Rangaiah G.P., Improved Constraint Handling Technique for Multi-Objective Optimization with Application to Two Fermentation Processes in “Multi-Objective Optimization in Chemical
Engineering: Developments and Applications” edited by G.P. Rangaiah and A. Bonilla-Petriciolet, John Wiley, 2013.
Phase Equilibrium Data Reconciliation using Multi-Objective Differential Evolution with Tabu List
Bonilla-Petricioloet A., Sharma S. and Rangaiah G.P., Phase Equilibrium Data Reconciliation using Multi-Objective Differential Evolution with Tabu List in “Multi-Objective Optimization in Chemical
Engineering: Developments and Applications” edited by G.P. Rangaiah and A. Bonilla-Petriciolet, John Wiley, 2013.
CO2 Emissions Targeting for Petroleum Refinery Optimization
Al-Mayyahi M.A., Hoadley A.F.A and Rangaiah G.P., CO2 Emissions Targeting for Petroleum Refinery Optimization in “Multi-Objective Optimization in Chemical Engineering: Developments and Applications”
edited by G.P. Rangaiah and A. Bonilla-Petriciolet, John Wiley, 2013.
Multi-Objective Optimization of a Hybrid Steam Stripper-Membrane Process for Continuous Bioethnaol Purification
Gudena K., Rangaiah G.P. and Lakshminarayanan S., Multi-Objective Optimization of a Hybrid Steam Stripper-Membrane Process for Continuous Bioethnaol Purification in “Multi-Objective Optimization in
Chemical Engineering: Developments and Applications” edited by G.P. Rangaiah and A. Bonilla-Petriciolet, John Wiley, 2013.
Process Design for Economic, Environmental and Safety Objectives with an Application to the Cumene Process
Sharma S., Lim Z.C. and Rangaiah G.P., Process Design for Economic, Environmental and Safety Objectives with an Application to the Cumene Process in “Multi-Objective Optimization in Chemical
Engineering: Developments and Applications” edited by G.P. Rangaiah and A. Bonilla-Petriciolet, John Wiley, 2013.
Particle Swarm Optimization with Re-Initialization Strategies for Continuous Global Optimization
Kennedy D.D., Zhang H., Rangaiah G.P. and Bonilla-Petriciolet A., Particle Swarm Optimization with Re-Initialization Strategies for Continuous Global Optimization, in “Global Optimization: Theory,
Developments and Applications” edited by A. Michalski, Nova Science Publishers, 2013.
Control Degrees of Freedom Analysis for Plant-Wide Control of Industrial Processes
Murthy Konda N.V.S.N. and Rangaiah G.P., Control Degrees of Freedom Analysis for Plant-Wide Control of Industrial Processes in “Plant-Wide Control: Recent Developments and Applications” edited by
G.P. Rangaiah and V. Kariwala, John Wiley, Chichester, 2012.
A Review of Plant-Wide Control Methodologies and Applications
Vasudevan S. and Rangaiah G.P., A Review of Plant-Wide Control Methodologies and Applications in “Plant-Wide Control: Recent Developments and Applications” edited by G.P. Rangaiah and V. Kariwala,
John Wiley, Chichester, 2012.
Integrated Framework of Simulation and Heuristics for Plant-Wide Control System Design
Vasudevan S., Murthy Konda N.V.S.N. and Rangaiah G.P., Integrated Framework of Simulation and Heuristics for Plant-Wide Control System Design in “Plant-Wide Control: Recent Developments and
Applications” edited by G.P. Rangaiah and V. Kariwala, John Wiley, Chichester, 2012.
Performance Assessment of Plant-Wide Control Systems
Vasudevan S. and Rangaiah G.P., Performance Assessment of Plant-Wide Control Systems in “Plant-Wide Control: Recent Developments and Applications” edited by G.P. Rangaiah and V. Kariwala, John Wiley,
Chichester, 2012.
Design and Plant-Wide Control of a Biodiesel Plant
Zhang C. Rangaiah G.P. and Kariwala V., Design and Plant-Wide Control of a Biodiesel Plant in “Plant-Wide Control: Recent Developments and Applications” edited by G.P. Rangaiah and V. Kariwala, John
Wiley, Chichester, 2012.
Tabu Search for Global Optimization Problems having Continuous Variables
Sim M.K., Rangaiah G.P., Srinivas M., Tabu Search for Global Optimization Problems having Continuous Variables, in “Stochastic Global Optimization: Techniques and Applications in Chemical
Engineering” edited by G.P. Rangaiah, World Scientific, Singapore, 2010.
Differential Evolution: Method, Developments and Chemical Engineering Applications
Chen S.Q., Rangaiah G.P., Srinivas M., Differential Evolution: Method, Developments and Chemical Engineering Applications, in “Stochastic Global Optimization: Techniques and Applications in Chemical
Engineering” edited by G.P. Rangaiah, World Scientific, Singapore, 2010.
Phase Stability and Equilibrium Calculations in Reactive Systems using Differential Evolution and Tabu Search
Bonilla-Petriciolet A., Rangaiah G.P., Segovia-Hernández J.G. and Jaime-Leal J.E., Phase Stability and Equilibrium Calculations in Reactive Systems using Differential Evolution and Tabu Search, in
“Stochastic Global Optimization: Techniques and Applications in Chemical Engineering” edited by G.P. Rangaiah, World Scientific, Singapore, 2010.
Differential Evolution with Tabu List for Global Optimization: Evaluation of Two Versions on Benchmark and Phase Stability Problems
Srinivas M. and Rangaiah G.P., Differential Evolution with Tabu List for Global Optimization: Evaluation of Two Versions on Benchmark and Phase Stability Problems, in “Stochastic Global Optimization:
Techniques and Applications in Chemical Engineering” edited by G.P. Rangaiah, World Scientific, Singapore, 2010.
Gas-phase Refrigeration: A Promising Alternative to the Conventional Refrigeration Processes for LNG
Shah N.M., Hoadley A.F.A. and Rangaiah G.P., Gas-phase Refrigeration: A Promising Alternative to the Conventional Refrigeration Processes for LNG, in “OPEC, Oil Prices and LNG” edited by E.R. Pitt
and C.N. Leung, p. 443-468, Nova Science Publishers, 2009. (ISBN: 978-1-60692-897-4)
Multi-objective Optimization Applications in Chemical Engineering
Masuduzzaman and Rangaiah G.P., Multi-objective Optimization Applications in Chemical Engineering, in “Multi-objective Optimization: Techniques and Applications in Chemical Engineering” edited by
G.P. Rangaiah, World Scientific, Singapore, 2009.
Multi-objective Optimization of Gas Phase Refrigeration Systems for LNG
Shah N.M., Rangaiah G.P. and Hoadley A.F.A., Multi-objective Optimization of Gas Phase Refrigeration Systems for LNG, in “Multi-objective Optimization: Techniques and Applications in Chemical
Engineering” edited by G.P. Rangaiah, World Scientific, Singapore, 2009.
Integrated Multi-Objective Differential Evolution and its Application to Amine Absorption Process for Natural Gas Sweetening
Sharma S., Rangaiah G.P. and Maréchal F., Integrated Multi-Objective Differential Evolution and its Application to Amine Absorption Process for Natural Gas Sweetening, in “Differential Evolution in
Chemical Engineering: Developments and Applications” edited by G.P. Rangaiah and S. Sharma, World Scientific, 2017.
Optimal Design of Chemical Processes for Multiple Economic and Environmental Objectives
Lee E.S.Q., Rangaiah G.P. and Agrawal N., Optimal Design of Chemical Processes for Multiple Economic and Environmental Objectives, in “Multi-objective Optimization: Techniques and Applications in
Chemical Engineering” edited by G.P. Rangaiah, World Scientific, Singapore, 2009.
Multi-objective Optimization of a Multi-product Microbial Cell Factory for Multiple Objectives – a Paradigm for Metabolic Pathway Recipe
Lee F.C., Rangaiah G.P. and Lee D.Y., Multi-objective Optimization of a Multi-product Microbial Cell Factory for Multiple Objectives – a Paradigm for Metabolic Pathway Recipe, in “Multi-objective
Optimization: Techniques and Applications in Chemical Engineering” edited by G.P. Rangaiah, World Scientific, Singapore, 2009.
Multi-objective Optimization in Food Engineering
Cheah K.S. and Rangaiah G.P., Multi-objective Optimization in Food Engineering, in “Optimization in Food Engineering” edited by F. Erdogdu, Taylor and Francis/CRC Press, 2009.
Tabu Search: Development, Algorithm, Performance and Applications
Srinivas M. and Rangaiah G.P., Tabu Search: Development, Algorithm, Performance and Applications, in “Optimization in Food Engineering” edited by F. Erdogdu, Taylor and Francis/CRC Press, 2009.
Optimal Design and Operation of an Industrial Hydrogen Plant for Multiple Objectives
Oh P.P., Ray A.K. and Rangaiah G.P., Optimal Design and Operation of an Industrial Hydrogen Plant for Multiple Objectives, in “Recent Developments in Optimization and Optimal Control in Chemical
Engineering” edited by R. Luus, p. 289-306, Research Signpost, Trivandrum, India (2002). | {"url":"https://blog.nus.edu.sg/rangaiah/book-chapters/","timestamp":"2024-11-04T16:50:52Z","content_type":"text/html","content_length":"84755","record_id":"<urn:uuid:f1ee87ee-c628-4447-9fb6-86ff774bd8b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00240.warc.gz"} |
What our customers say...
Thousands of users are using our software to conquer their algebra homework. Here are some of their experiences:
I just finished using Algebrator for the first time. I just had to let you know how satisfied I am with how easy and powerful it is. Excellent product. Keep up the good work.
Tami Garleff, MI
I want to thank you for all you help. Your spport in resolving how do a problem has helped me understand how to do the problems, and actually get the right result. Thanks So Much.
Jessica Simpson, UT
After spending countless hours trying to understand my homework night after night, I found Algebrator. Most other programs just give you the answer, which did not help me when it come to test time,
Algebrator helped me through each problem step by step. Thank you!
Susan Raines, LA
Look at that. Finally a product that actually does what it claims to do. Its been a breeze preparing my math lessons for class. Its been a big help that now leaves time for other things.
Linda Rees, NJ
Keep up the good work Algebrator staff! Thanks!
Colleen D. Lester, PA
Search phrases used on 2008-04-11:
Students struggling with all kinds of algebra problems find out that our software is a life-saver. Here are the search phrases that today's searchers used to find our site. Can you find yours among
• solving rational expressions
• hard algebra questions
• third grade practice math Taks
• proportion worksheet
• online calculator with square root finder
• printable square root table
• nelson math textbook answers for grade 6
• java program on polynomial roots
• higher order nonhomogeneous differential equations
• Free Calculator Download
• math powerpoint presentation Grade 7(set theory)
• dividing polynomials in C programming
• square root finder
• Square ROOT WITH a little number 3 inside the sign
• free practice papers for sats
• free solve algebra word problems online
• calculas
• online ti83 emulator
• teachers steps to adding and subtracting
• free algebra math book
• radicals square route
• scott foresman-addison wesley math quizes
• free tutor for 9th grade math
• thermometer picture to help with math homework
• 6th grade math chart
• alg 2 ca standards test released questions
• graphing by substitution solver
• linear equations fifth grade printable worksheet
• how to simplify square roots on a calculator
• David Lay Chapter 2 linear algebra solutions
• simulantaneous Equation
• polynomial factor cheat calculator
• subtracting integers worksheet
• addition rule of equations math worksheets
• algebra solver demo free
• Holt Algebra permutations
• ti 83 mathematical induction program
• mathimatical tricks
• simplifying rational expressions calculator
• free online factoring 2nd order equation
• how to find square root of exponent
• graphing inequalities with the t-83 plus
• rational lowest denominator solver
• variable worksheet middle school math
• integer games for 6th grade
• domain range worksheet
• how to solve a tangent equation on a ti-83
• Algebra 2 mid term in 1999 high school
• basic radical equations
• Integrated Common Entrance Test material to download
• Free Study Guide for Basic Algebra
• use TI-30X IIS to cheat on test
• printable ks3 sats papers
• accounting books online
• what is the difference between exponential and radical forms of an expression
• free proportion worksheets
• cheating for pre-algebra answers(free)
• trivia worksheet
• free science worksheets grade 9
• year 10 algebra and linear equation exams
• square root radicals on ti-83
• nth roots worksheet
• clep college mathematics tutorial
• Dividing Polynomials Calculator
• algebra 2: explorations and applications solutions
• pictograph worksheet
• trigonometric graphs and identities glencoe mathematics "algebra 2" math solutions
• yr.9 maths sats revision tests
• algebra cheat sheet
• Printable homework sheets for your ability
• solving algebra problems in excel
• convert mixed percent to fraction
• Glencoe/mcgraw-hill chapter 8 test, form b answer key
• changing percentages to fractions cheat sheet
• multiplying polynomials worksheet word
• find worksheets for addition
• 5th graders statistic about exercise
• TI-89 titanium equation solving equations with complex numbers
• cuadratic formula
• free symmetry worksheets
• Aptitude question papers
• TI-83 Plus business statistics tips
• advanced algebra: writing repeating decimals as fractions
• fourth grade fractions using lattice method
• 5th grade coordinate plane worksheets | {"url":"https://softmath.com/algebra-help/solve-absolute-value-equations.html","timestamp":"2024-11-12T22:31:56Z","content_type":"text/html","content_length":"35557","record_id":"<urn:uuid:ba91839e-5881-4216-88cf-98dbd9d7ef72>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00315.warc.gz"} |
Ulrik Enstad: A dynamical approach to sampling and interpolation in unimodular groups | SMC
Ulrik Enstad: A dynamical approach to sampling and interpolation in unimodular groups
Time: Thu 2022-09-22 13.15 - 14.15
Location: Albano, Carmérrumet
Participating: Ulrik Enstad (SU)
I will present some joint work with Sven Raum on sampling and interpolation in general unimodular locally compact groups. One of the main contributions is a new notion of covolume for point sets in
such groups that simultaneously generalizes the covolume of a lattice and the reciprocal of the Banach/Beurling density for amenable groups. This notion of covolume arises naturally from the
transverse measure theory of the associated hull dynamical system of a point set. The sampling and interpolation results are obtained by considering an étale groupoid associated to a point set whose
groupoid C*-algebra generalizes the group C*-algebra in the lattice case. | {"url":"https://www.math-stockholm.se/en/kalender/semovrigt/ulrik-enstad-a-dynamical-approach-to-sampling-and-interpolation-in-unimodular-groups-1.1192271","timestamp":"2024-11-09T09:22:04Z","content_type":"text/html","content_length":"17845","record_id":"<urn:uuid:fd27e3fc-8857-4b1b-8b72-bd925e6f6684>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00004.warc.gz"} |
Block-iterative projection methods for parallel computation of solutions to convex feasibility problems
An iterative method is proposed for solving convex feasibility problems. Each iteration is a convex combination of projections onto the given convex sets where the weights of the combination may vary
from step to step. It is shown that any sequence of iterations generated by the algorithm converges if the intersection of the given family of convex sets is nonempty and that the limit point of the
sequence belongs to this intersection under mild conditions on the sequence of weight functions. Special cases are block-iterative processes where in each iterative step a certain subfamily of the
given family of convex sets is used. In particular, a block-iterative version of the Agmon-Motzkin-Schoenberg relaxation method for solving systems of linear inequalities is derived. Such processes
lend themselves to parallel implementation and will be useful in various areas of applications, including image reconstruction from projections, image restoration, and other fully discretized
inversion problems.
Bibliographical note
Funding Information:
*Y. Censor's work on this research was supported by the National Institutes of Health, Grant No. HL-28438, while visiting the Medical Image Processing Group (MIPG) at the Department of Radiology,
Hospital of the University of Pennsylvania, Philadelphia.
ASJC Scopus subject areas
• Algebra and Number Theory
• Numerical Analysis
• Geometry and Topology
• Discrete Mathematics and Combinatorics
Dive into the research topics of 'Block-iterative projection methods for parallel computation of solutions to convex feasibility problems'. Together they form a unique fingerprint. | {"url":"https://cris.haifa.ac.il/en/publications/block-iterative-projection-methods-for-parallel-computation-of-so","timestamp":"2024-11-14T17:04:33Z","content_type":"text/html","content_length":"54903","record_id":"<urn:uuid:aa7e4033-c56d-4734-aa26-7be571f3a0ab>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00621.warc.gz"} |
Complete the following with an appropriate conjunction. we coul-Turito
Are you sure you want to logout?
Complete the following with an appropriate conjunction.
We could not sleep because it was too hot.
A. And
B. So as
C. Because
D. Nevertheless
A conjunction is a part of speech that connects words, phrases or clauses.
The correct answer is: Because
OPTION 2 is the correct answer
We could not sleep because it was too hot.
‘Because’ is used to show reason.
Get an Expert Advice From Turito. | {"url":"https://www.turito.com/ask-a-doubt/complete-the-following-with-an-appropriate-conjunction-we-could-not-sleep-because-it-was-too-hot-nevertheless-bec-qc80d0538","timestamp":"2024-11-14T22:09:13Z","content_type":"application/xhtml+xml","content_length":"305929","record_id":"<urn:uuid:5428a6fa-2d6e-44ba-86b8-13e93855d3cc>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00184.warc.gz"} |
Inverse Relation - Formula, Graph | Inverse Relation Theorem
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
Inverse Relation
An inverse relation, as its name suggests, is the inverse of a relation. Let us recall what is a relation. A relation is the collection of ordered pairs. Let us consider two sets A and B. Then the
set of all ordered pairs of the form (x, y) where x ∈ A and y ∈ B is called the cartesian product of A and B, which is denoted by A x B. Any subset of this cartesian product A x B is a relation. Then
what is an inverse relation of a relation? Do you think that it is set of all ordered pairs that are obtained by interchanging the elements of the ordered pairs of the original relation? Then yes,
you are right!
Let us explore more about the inverse relation in different cases, its domain, and range along with a few solved examples. Also, we will see what is inverse relation theorem along with its proof.
1. What Is Inverse Relation?
2. Domain and Range of Inverse Relation
3. Inverse Relation of a Graph
4. Inverse of an Algebraic Relation
5. Inverse Relation Theorem
6. FAQs on Inverse Relation
What Is Inverse Relation?
An inverse relation is the inverse of a relation and is obtained by interchanging the elements of each ordered pair of the given relation. Let R be a relation from a set A to another set B. Then R is
of the form {(x, y): x ∈ A and y ∈ B}. The inverse relationship of R is denoted by R^-1 and its formula is R^-1 = {(y, x): y ∈ B and x ∈ A}. i.e.,
• The first element of each ordered pair of R = the second element of the corresponding ordered pair of R^-1 and
• The second element of each ordered pair of R = the first element of the corresponding ordered pair of R^-1.
Inverse Relation Definition
In simple words, if (x, y) ∈ R, then (y, x) ∈ R^-1 and vice versa. i.e., If R is from A to B, then R^-1 is from B to A. Thus, if R is a subset of A x B, then R^-1 is a subset of B x A.
Inverse Relation Examples
Have a look at the following relations and their inverse relations on two sets A = {a, b, c, d, e} and B = {1, 2, 3, 4, 5}.
• If R = {(a, 2), (b, 4), (c, 1)} ⇔ R^-1 = {(2, a), (4, b), (1, c)}
• If R = {(c, 1), (b, 2), (a, 3)} ⇔ R^-1 = {(1, c), (2, b), (3, a)}
• If R = {(b, 3), (c, 2), (e, 1)} ⇔ R^-1 = {(3, b), (2, c), (1, e)}
Domain and Range of Inverse Relation
The domain of a relation is the set of all first elements of its ordered pairs whereas the range is the set of all second elements of its ordered pairs. Let us consider the first example from the
list of above examples and find the domain and range of each of the relation and its inverse relation.
• For R = {(a, 2), (b, 4), (c, 1)}, domain = {a, b, c} and range = {2, 4, 1}
• For R^-1 = {(2, a), (4, b), (1, c)}, domain = {2, 4, 1} and range = {a, b, c}
What did you observe here? Aren't the domain and range interchanged for R and R^-1? i.e.,
• the domain of R^-1 = the range of R and
• the range of R^-1 = the domain of R.
Note: If R is a symmetric relation (i.e., if (b, a) ∈ R, for every (a, b) ∈ R), then R = R^-1. For example, consider a symmetric relation R = {(1, a) (a, 1), (2, b), (b, 2)}. The inverse of this
relation is, R^-1 = {(a, 1), (1, a), (b, 2), (2, b)}. Technically, R = R^-1, because the order of elements is NOT important while writing a set. In this case, the domain of R = range of R = domain of
R^-1 = range of R^-1 = {1, 2, a, b}.
Inverse Relation of a Graph
If a relation is given as a graph, then its inverse is obtained by reflecting it along the line y = x. This is because the inverse of a relation is nothing but the interchanged ordered pairs of the
given relation. To graph the inverse of a relation that is given by a graph,
• Choose some points on the given relation (graph).
• Interchange the x and y coordinates of each point to get new points.
• Plot all these new points and join them by a curve which gives the graph of the inverse relationship.
You can see a relation R that is represented by a circle in the second quadrant, some points on it which are transformed into new points (the transformations are showed by dotted lines) by
interchanging the x and y coordinates, and the inverse relation R^-1 that is represented by a circle in the fourth quadrant in the figure below.
Inverse of an Algebraic Relation
If a relation is given in algebraic form, like R = {(x, y): y = 3x + 2}, then its inverse is found using the following steps.
• Interchange the variables x and y.
In the above example, if we interchange x and y, we get x = 3y + 2
• Solve the above equation for y.
In the above example, x - 2 = 3y ⇒ y = (x - 2) / 3
Then the inverse relation of the given algebraic relation is, R^-1 = {(x, y): y = (x - 2) / 3}.
Try graphing y = 3x + 2 and y = (x - 2) / 3 and see whether the two graphs are symmetric about the line y = x.
Inverse Relation Theorem
Statement: For any relation R, (R^-1)^-1 = R.
Here is the proof of the inverse relation theorem.
Let (x, y) ∈ R
⇔ (y, x) ∈ R^-1
⇔ (x, y) ∈ (R^-1)^-1
Thus, (R^-1)^-1 = R.
Hence proved.
Important Notes on Inverse Relation:
Here are some important points to note about inverse relationship.
• Domain and inverse of a relation are nothing but the range and domain of its inverse relation respectively.
• If R is a symmetric relation, R = R^-1.
• The inverse of an empty relation is itself. i.e., if R = { } then R^-1 = { }.
• On a graph, the curves corresponding to a relation and its inverse are symmetric about the line y = x.
Related Topics:
Explore the following related topics of inverse relation.
Examples on Inverse Relationship
1. Example 1: Find the inverse of the following relations: a) R = {(2, 7), (8, 3), (5, 5), (4, 3}) and b) R = {(x, x^2): x is a prime number less than 15}. Find the domain and range in each of these
We know that inverse relation of a relation is obtained by interchanging the first and second elements of the ordered pairs of the given relation. Thus, the inverse of the given relations are,
a) R^-1 = {(7, 2), (3, 8), (5, 5), (3, 4)}.
In this case, domain = {7, 3, 5} and range = {2, 8, 5, 4}.
b) Let us write the given relation in roster form. The list of prime numbers less than 15 are 2, 3, 5, 7, 11, and 13. Thus,
Then R = {(2, 2^2), (3, 3^2), (5, 5^2), (7, 7^2), (11, 11^2), (13, 13^2)} = {(2, 4), (3, 9), (5, 25), (7, 49), (11, 121), (13, 169)}
Now, R^-1 = {(x^2, x): x is a prime number less than 15} = {(4, 2), (9, 3), (25, 5), (49, 7), (121, 11), (169, 13)}.
Here, domain = {4, 9, 25, 49, 121, 169} and range = {2, 3, 5, 7, 11, 13}
Answer: a) R^-1 = {(7, 2), (3, 8), (5, 5), (3, 4)}, domain = {7, 3, 5} and range = {2, 8, 5, 4}. b) R^-1 ={(4, 2), (9, 3), (25, 5), (49, 7), (169, 13)}, domain = {4, 9, 25, 49, 121, 169} and
range = {2, 3, 5, 7, 11, 13}.
2. Example 2: Find the inverse of the relation R = {(x, y): y = x^2}.
The relation (R) between x and y is given by the equation y = x^2.
To find its inverse relation, interchange x and y and solve the resultant equation for y. Then
x = y^2
Taking square root on both sides,
±√x = y
Thus, the inverse of the given relation is, R^-1 = {(x, y): y = ±√x}
Answer: R^-1 = {(x, y): y = ±√x}.
3. Example 3: Find the inverse of the relation that is represented by the following graph.
Let us take some points on the graph, say, (0, 1), (2, 4), and (3, 8).
Let us interchange the x and y coordinates to get some points on its inverse.
Then we get (1, 0), (4, 2), and (8, 3).
Plot them on the same graph and join by a curve to get the inverse relation. Also, note that the two curves are symmetric with respect to the line y = x.
View Answer >
Great learning in high school using simple cues
Indulging in rote learning, you are likely to forget concepts. With Cuemath, you will learn visually and be surprised by the outcomes.
Practice Questions on Inverse Relationship
Check Answer >
FAQs on Inverse Relation
What Is Inverse of a Relation?
An inverse relation of a relation is a set of ordered pairs which are obtained by interchanging the first and second elements of the ordered pairs of the given relation. i.e., if R = {(x, y): x ∈ A
and y ∈ B} then R^-1 = {(y, x): y ∈ B and x ∈ A}.
What Is Inverse Relation of an Empty relation?
If (x, y) is an element of a relation R, then (y, x) will be an element of its inverse relation R^-1. Thus, the inverse of an empty relation is itself. i.e., if R = { }, then R^-1 = { } as well.
What Is the Domain of an Inverse Relationship?
Let us consider a relation R and its inverse relation R^-1. Then the domain of R^-1 is the range of R.
How To Find the Inverse of an Algebraic Relation?
To find the inverse of an algebraic relation in terms of x and y, just interchange the variables x and y, and solve the equation for y. For example, to find the inverse of a relation y = x^3,
interchange x and y and then solve it for y. Then we get x = y^3 ⇒ y = x^1/3.
What Is the Range of an Inverse Relation?
Let us consider a relation R and its inverse relation R^-1. Then the range of R^-1 is the domain of R.
What Is the Inverse Relationship of R If R Is Symmetric?
If R is symmetric, then (y, x) is in R for every (x, y) in R. Thus, its inverse relation is R itself. Note that whenever R = R^-1, then R is symmetric.
How To Find the Inverse of a Relation Given by a Graph?
To find the inverse relation of a graph, just draw the reflection of the graph along the line y = x. For this, we can pick some points on the graph, interchange their x and y coordinates to get the
points on its inverse graph, plot the points and join them by a curve.
Download FREE Study Materials
Math worksheets and
visual curriculum | {"url":"https://www.cuemath.com/algebra/inverse-relation/","timestamp":"2024-11-06T09:06:41Z","content_type":"text/html","content_length":"240500","record_id":"<urn:uuid:0b1360a0-d288-4869-8166-63311188f0bf>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00851.warc.gz"} |
Why the name Euclid?
Euclid, the one from Alexandria, is the Father of Geometry. Being that math is math and fundamentally hasn’t changed, his work is still taught all over the world. So you could say hes a pretty
influential guy, especially when you think he lived over 2000 years ago.
In his book ‘Elements’ he put forth 5 common notions
1- Things that are equal to the same thing are also equal to one another (Transitive property of equality).
2- If equals are added to equals, then the wholes are equal (Addition property of equality).
3- If equals are subtracted from equals, then the remainders are equal (Subtraction property of equality).
4- Things that coincide with one another are equal to one another (Reflexive Property).
5- The whole is greater than the part.
We liked the play on words that came from ‘The whole is greater than the part and the hole is less than the part’, if you don’t see the humor read it out loud, if you still don’t see the humor try
drilling a hole larger than the part. | {"url":"https://euclidmachineanddesign.com/why-the-name-euclid/","timestamp":"2024-11-12T03:48:19Z","content_type":"text/html","content_length":"28050","record_id":"<urn:uuid:0e397996-927e-4b02-adc8-58d6c32b6aaa>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00696.warc.gz"} |
Is RMS a good metric to estimate gain
4 years ago
●8 replies●
latest reply 4 years ago
424 views
I have two vectors V1 and V2. V1 is the source signal vector and V2 being the (possibly) degraded (attenuated/amplified) output signal. I have calculated the RMS of both V1 and V2. Now, I was
wondering if the ratio RMS(V2)/RMS(V1) is a good estimation or approximation of the gain. Can I use this metric as a gain factor?
[ - ]
Reply by ●December 4, 2020
You could use this an indicator gain in dB: 20*log10( RMS(V2) / RMS(V1) ).
[ - ]
Reply by ●December 4, 2020
This looks really good (and more inclined towards textbook definition of gain). Is this independent of factors like DC components and noise? I'm looking to apply this for recorded audio.
[ - ]
Reply by ●December 4, 2020
[ - ]
Reply by ●December 4, 2020
RMS reflects power but for gain it is either used as ratio of output/input or adjusted for bitwidth such as dBFs which is 10log(power/fullscale power) and so removes effect of bitwidth.
[ - ]
Reply by ●December 4, 2020
By "vector", I presume you mean a finite sequence $\{ v[n] \} $ of $ N $ samples of a signal and RMS is
$$ V_\mathrm{rms} \triangleq \sqrt{\frac{1}{N} \sum\limits_{n=0}^{N-1} |v[n]|^2} $$
To use that ratio as a measure of gain, you may want to consider how "representative" your source signal is. Say, if it was simply a sinusoid (having some frequency), the ratio of RMS of output to
RMS of source would be a good measure of gain for that specific frequency. The gain might be different for a sinusoid of a different frequency.
You might want the source signal to be more broadbanded than a single sinusoid.
And also consider that there may be a constant DC offset in the output signal that would add to the RMS of the output. So you might want to measure the output when the input is zero to learn how
much any constant DC offset might be. You might want to subtract that DC offset squared from the squares of the output samples before mean-rooting.
[ - ]
Reply by ●December 4, 2020
This is an interesting and informative perspective. That's correct. I am looking at audio samples, which is more often than not a speech signal and hence a combination of different sinusoids of
different frequencies, and when the output is typically a similar case probably with (possibly) missing frequencies or noise added.
So, if I got it right, according to the your answer, (RMS(V2) - RMS(before-V2))/RMS(V1) is the best way, correct?
[ - ]
Reply by ●December 4, 2020
Also, the output might be more than just a scaled version of the input. DC offset was mentioned, but there is also added noise and other artifacts like harmonics, aliasing, intermods that also get
summed into the output RMS. And out-of-band power might be larger than the desired in-band that you are trying to measure. So, keep all that in mind.
[ - ]
Reply by ●December 4, 2020
Will consider that. Thanks. Does @rbj's answer consider the artifacts you've mentioned? If not, could you please give a lead on how to include these? | {"url":"https://dsprelated.com/thread/12556/is-rms-a-good-metric-to-estimate-gain","timestamp":"2024-11-13T09:23:21Z","content_type":"text/html","content_length":"42771","record_id":"<urn:uuid:878d2a5b-a6c3-4191-ae92-3303be4303e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00346.warc.gz"} |
Question ID - 54004 | SaraNextGen Top Answer
Twenty times a positive integer is less than its square by 96. What is the integer?
(a) 20 (b) 24 (c) 30 (d) Cannot be determined
Twenty times a positive integer is less than its square by 96. What is the integer? | {"url":"https://www.saranextgen.com/homeworkhelp/doubts.php?id=54004","timestamp":"2024-11-02T15:05:09Z","content_type":"text/html","content_length":"16319","record_id":"<urn:uuid:c9b1e47e-dd11-478c-abbd-9af5314136a5>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00378.warc.gz"} |
Back to the Garden with Sudocrem - chelseamamma.co.uk
Back to the Garden with Sudocrem
Like many families, we have resumed schoolwork with the kids this week.
The work set by the school does not fill the day, and we are trying to reduce the amount of time spent on technology, which means that we have been looking for new ways to spend our time at home.
One thing the kids love to get involved with is gardening as planting seeds and getting your hands dirty is a fantastic way of learning about science and nature.
This spring, Sudocrem are teaming up with Britain’s top gardeners for Back to the Garden, a nationwide gardening project which you can get involved whether you’re inside or outside.
They want to help families start planting and growing seeds by offering 100 growing kits as a giveaway, as well as teaming up with gardeners all over the country to collect some ideas for fun easy
projects that children can manage and enjoy.
Whether you have a garden, balcony, allotment or windowsill, gardening is something everyone can enjoy, particularly, if you have young children.
Encouraging your children to plant seeds and take an interest in growing can give them a long-term project to focus on, away from the computer. And the good news is, you don’t need a garden to get
involved: creating a window box full of edible herbs can be just as inspiring as digging a vegetable patch.
Teaming up with gardeners and garden centres all over the nation, Sudocrem’s Back to the Garden campaign will encourage children and their parents to begin their own gardening projects at home or in
the garden. The initiative will include tips and advice on potting plants, spotting a weed and keeping your plants healthy from gardeners all over the UK. Sudocrem will also be donating one hundred
growing kits to families all over the UK to kick start their growing projects.
Soothing Gardener’s Skin…
Gardening can be harsh on delicate skin, as every gardener knows. From contact with cold water when watering the plants, digging in the mud for weeds, planting new seeds and planters, using tools –
it’s a hands-on activity. Even standing outside in the sun can cause damage and dryness to the face and neck. Then there’s getting pricked by rose thorns, stung
by nettles and insects, scraped knees….
Gardener’s new best friend comes in the pocket-sized form of My Little Sudocrem, the Swiss army-knife of soothing skincare products. Loved by everyone from babies to teenagers, runners to cyclists,
mountaineers to gardeners and anyone in between.
Hopeful gardeners can enter a competition to win one of 3 gardening goody bags below. Prizes include an inside or an outside gardening kit.
And for an extra chance to win £100 of garden centre vouchers, upload a photo of your garden and tag @Sudocrem and use the hashtags #BackToTheGarden & #SudocremBTTGcomp on Facebook, Twitter or
Instagram to enter. Terms and Conditions apply.
To Enter:
• Fill in the Rafflecopter widget below to verify your entries
• Please read the rules below
• Closing Date: 17th May 2020
• If there is no form hit refresh (F5) and it should appear
• If still not working please check that your computer is running Javascript
• Rafflecopter will tweet, like and follow on your behalf making it really easy to enter
• Really want to win the prize? Come back every day for bonus entries via twitter
157 thoughts on “Back to the Garden with Sudocrem”
2. over 80 years (since 1931)
3. 81 years – since 1931
4. Sudocrem was invented in Dublin all the way back in 1931 by a pharmacist named Professor Thomas Smith and, after 88 years, the product is still going strong.
12. 1931
14. its over 85 years
16. Over 85
19. More than 85 years
25. Over 85 years!!!
26. Hello, 89 years.
33. Over 85 years xx
34. Sudo cream has been looking after our families skin for over 85 years
35. Sudocrem has been looking after our family’s skin for over 85 years
36. Sudocrem has been looking after our family’s skin for over 85 years
39. Over 85 years 🙂
40. Would love to win this for my grandsons in lockdown. They would just love this
42. Over 85 years! WOW!
48. Since 1931 so 89 years
53. Amazing giveaway. It is so important to teach children how plants grow, where their food comes from and to enjoy!
60. 89 years since 1931
67. Been looking after my family since I had my daughter 2 years ago but national it’s been helping families for 85 years 🙂
74. For over 85 years.
76. The answer is – Over 85 years
78. Over 81 years
82. For over 85 years!
95. Ovrt 85 years
96. That will be over 85 years – and still going strong
100. It would be 85 Years.
115. Over 85 Years
117. Over 80 years
119. About 89 years as it was founded in 1931
125. 88 years
127. Since 1931 so 89 years
128. since 1931 – so 88-89 years
129. 89 years as it was founded in 1931
132. overs 85 years
133. I remember my mum using this when me and my sister were little and I also still use it for my little ones! There was a certain incident when I took my eye off one of my 2 year old twins for a
few mins and he had managed to smear a pot of it all on the stairs!
134. overs 85 years
135. Since I was born!
137. Since 8 was born 43 years ago
138. 88/89 years
142. 85 Years
144. Over 80 years
147. 89 years and we have been using it for as long as I can remember
152. 89 years I think as I believe Sudocrem was founded in 1931.
154. Over 85 years! I mean how amazing and incredible. It’s the best thing since sliced bread. I don’t use anything else as I truly believe in this x
155. Over 85 years (89 as founded 1931)
Leave a Comment | {"url":"https://www.chelseamamma.co.uk/2020/04/21/back-to-the-garden-with-sudocrem/","timestamp":"2024-11-11T09:54:08Z","content_type":"text/html","content_length":"341086","record_id":"<urn:uuid:cf029d66-006d-46f8-8326-438ba6f9c7ff>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00748.warc.gz"} |
Theory VectorSpace.VectorSpace
section ‹Basic theory of vector spaces, using locales›
theory VectorSpace
imports Main
subsection ‹Basic definitions and facts carried over from modules›
text ‹A ‹vectorspace› is a module where the ring is a field.
Note that we switch notation from $(R, M)$ to $(K, V)$.›
locale vectorspace =
module?: module K V + field?: field K
for K and V
Use sets for bases, and functions from the sets to carrier K
represent the coefficients.
text ‹A ‹subspace› of a vectorspace is a nonempty subset
that is closed under addition and scalar multiplication. These properties
have already been defined in submodule. Caution: W is a set, while V is
a module record. To get W as a vectorspace, write vs W.›
locale subspace =
fixes K and W and V (structure)
assumes vs: "vectorspace K V"
and submod: "submodule K W V"
lemma (in vectorspace) is_module[simp]:
"subspace K W V⟹submodule K W V"
by (unfold subspace_def, auto)
text ‹We introduce some basic facts and definitions copied from module.
We introduce some abbreviations, to match convention.›
abbreviation (in vectorspace) vs::"'c set ⇒ ('a, 'c, 'd) module_scheme"
where "vs W ≡ V⦇carrier :=W⦈"
lemma (in vectorspace) carrier_vs_is_self [simp]:
"carrier (vs W) = W"
by auto
lemma (in vectorspace) subspace_is_vs:
fixes W::"'c set"
assumes 0: "subspace K W V"
shows "vectorspace K (vs W)"
proof -
from 0 show ?thesis
apply (unfold vectorspace_def subspace_def, auto)
by (intro submodule_is_module, auto)
abbreviation (in module) subspace_sum:: "['c set, 'c set] ⇒ 'c set"
where "subspace_sum W1 W2 ≡submodule_sum W1 W2"
lemma (in vectorspace) vs_zero_lin_dep:
assumes h2: "S⊆carrier V" and h3: "lin_indpt S"
shows "𝟬⇘V⇙ ∉ S"
proof -
have vs: "vectorspace K V"..
from vs have nonzero: "carrier K ≠{𝟬⇘K⇙}"
by (metis one_zeroI zero_not_one)
from h2 h3 nonzero show ?thesis by (rule zero_nin_lin_indpt)
text ‹A ‹linear_map› is a module homomorphism between 2 vectorspaces
over the same field.›
locale linear_map =
V?: vectorspace K V + W?: vectorspace K W
+ mod_hom?: mod_hom K V W T
for K and V and W and T
context linear_map
lemmas T_hom = f_hom
lemmas T_add = f_add
lemmas T_smult = f_smult
lemmas T_im = f_im
lemmas T_neg = f_neg
lemmas T_minus = f_minus
lemmas T_ker = f_ker
abbreviation imT:: "'e set"
where "imT ≡ mod_hom.im"
abbreviation kerT:: "'c set"
where "kerT ≡ mod_hom.ker"
lemmas T0_is_0[simp] = f0_is_0
lemma kerT_is_subspace: "subspace K ker V"
proof -
have vs: "vectorspace K V"..
from vs show ?thesis
apply (unfold subspace_def, auto)
by (rule ker_is_submodule)
lemma imT_is_subspace: "subspace K imT W"
proof -
have vs: "vectorspace K W"..
from vs show ?thesis
apply (unfold subspace_def, auto)
by (rule im_is_submodule)
lemma vs_criteria:
fixes K and V
assumes field: "field K"
and zero: "𝟬⇘V⇙∈ carrier V"
and add: "∀v w. v∈carrier V ∧ w∈carrier V⟶ v⊕⇘V⇙ w∈ carrier V"
and neg: "∀v∈carrier V. (∃neg_v∈carrier V. v⊕⇘V⇙neg_v=𝟬⇘V⇙)"
and smult: "∀c v. c∈ carrier K ∧ v∈carrier V⟶ c⊙⇘V⇙ v ∈ carrier V"
and comm: "∀v w. v∈carrier V ∧ w∈carrier V⟶ v⊕⇘V⇙ w=w⊕⇘V⇙ v"
and assoc: "∀v w x. v∈carrier V ∧ w∈carrier V ∧ x∈carrier V⟶ (v⊕⇘V⇙ w)⊕⇘V⇙ x= v⊕⇘V⇙ (w⊕⇘V⇙ x)"
and add_id: "∀v∈carrier V. (v⊕⇘V⇙ 𝟬⇘V⇙ =v)"
and compat: "∀a b v. a∈ carrier K ∧ b∈ carrier K ∧ v∈carrier V⟶ (a⊗⇘K⇙ b)⊙⇘V⇙ v =a⊙⇘V⇙ (b⊙⇘V⇙ v)"
and smult_id: "∀v∈carrier V. (𝟭⇘K⇙ ⊙⇘V⇙ v =v)"
and dist_f: "∀a b v. a∈ carrier K ∧ b∈ carrier K ∧ v∈carrier V⟶ (a⊕⇘K⇙ b)⊙⇘V⇙ v =(a⊙⇘V⇙ v) ⊕⇘V⇙ (b⊙⇘V⇙ v)"
and dist_add: "∀a v w. a∈ carrier K ∧ v∈carrier V ∧ w∈carrier V⟶ a⊙⇘V⇙ (v⊕⇘V⇙ w) =(a⊙⇘V⇙ v) ⊕⇘V⇙ (a⊙⇘V⇙ w)"
shows "vectorspace K V"
proof -
from field have 1: "cring K" by (unfold field_def domain_def, auto)
from assms 1 have 2: "module K V" by (intro module_criteria, auto)
from field 2 show ?thesis by (unfold vectorspace_def module_def, auto)
text ‹For any set $S$, the space of functions $S\to K$ forms a vector space.›
lemma (in vectorspace) func_space_is_vs:
fixes S
shows "vectorspace K (func_space S)"
proof -
have 0: "field K"..
have 1: "module K (func_space S)" by (rule func_space_is_module)
from 0 1 show ?thesis by (unfold vectorspace_def module_def, auto)
lemma direct_sum_is_vs:
fixes K V1 V2
assumes h1: "vectorspace K V1" and h2: "vectorspace K V2"
shows "vectorspace K (direct_sum V1 V2)"
proof -
from h1 h2 have mod: "module K (direct_sum V1 V2)" by (unfold vectorspace_def, intro direct_sum_is_module, auto)
from mod h1 show ?thesis by (unfold vectorspace_def, auto)
lemma inj1_linear:
fixes K V1 V2
assumes h1: "vectorspace K V1" and h2: "vectorspace K V2"
shows "linear_map K V1 (direct_sum V1 V2) (inj1 V1 V2)"
proof -
from h1 h2 have mod: "mod_hom K V1 (direct_sum V1 V2) (inj1 V1 V2)" by (unfold vectorspace_def, intro inj1_hom, auto)
from mod h1 h2 show ?thesis
by (unfold linear_map_def vectorspace_def , auto, intro direct_sum_is_module, auto)
lemma inj2_linear:
fixes K V1 V2
assumes h1: "vectorspace K V1" and h2: "vectorspace K V2"
shows "linear_map K V2 (direct_sum V1 V2) (inj2 V1 V2)"
proof -
from h1 h2 have mod: "mod_hom K V2 (direct_sum V1 V2) (inj2 V1 V2)" by (unfold vectorspace_def, intro inj2_hom, auto)
from mod h1 h2 show ?thesis
by (unfold linear_map_def vectorspace_def , auto, intro direct_sum_is_module, auto)
text ‹For subspaces $V_1,V_2\subseteq V$, the map $V_1\oplus V_2\to V$ given by $(v_1,v_2)\mapsto
v_1+v_2$ is linear.›
lemma (in vectorspace) sum_map_linear:
fixes V1 V2
assumes h1: "subspace K V1 V" and h2: "subspace K V2 V"
shows "linear_map K (direct_sum (vs V1) (vs V2)) V (λ v. (fst v) ⊕⇘V⇙ (snd v))"
proof -
from h1 h2 have mod: "mod_hom K (direct_sum (vs V1) (vs V2)) V (λ v. (fst v) ⊕⇘V⇙ (snd v))"
by ( intro sum_map_hom, unfold subspace_def, auto)
from mod h1 h2 show ?thesis
apply (unfold linear_map_def, auto) apply (intro direct_sum_is_vs subspace_is_vs, auto)..
lemma (in vectorspace) sum_is_subspace:
fixes W1 W2
assumes h1: "subspace K W1 V" and h2: "subspace K W2 V"
shows "subspace K (subspace_sum W1 W2) V"
proof -
from h1 h2 have mod: "submodule K (submodule_sum W1 W2) V"
by ( intro sum_is_submodule, unfold subspace_def, auto)
from mod h1 h2 show ?thesis
by (unfold subspace_def, auto)
text ‹If $W_1,W_2\subseteq V$ are subspaces, $W_1\subseteq W_1 + W_2$›
lemma (in vectorspace) in_sum_vs:
fixes W1 W2
assumes h1: "subspace K W1 V" and h2: "subspace K W2 V"
shows "W1 ⊆ subspace_sum W1 W2"
proof -
from h1 h2 show ?thesis by (intro in_sum, unfold subspace_def, auto)
lemma (in vectorspace) vsum_comm:
fixes W1 W2
assumes h1: "subspace K W1 V" and h2: "subspace K W2 V"
shows "(subspace_sum W1 W2) = (subspace_sum W2 W1)"
proof -
from h1 h2 show ?thesis by (intro msum_comm, unfold subspace_def, auto)
text ‹If $W_1,W_2\subseteq V$ are subspaces, then $W_1+W_2$ is the minimal subspace such that
both $W_1\subseteq W$ and $W_2\subseteq W$.›
lemma (in vectorspace) vsum_is_minimal:
fixes W W1 W2
assumes h1: "subspace K W1 V" and h2: "subspace K W2 V" and h3: "subspace K W V"
shows "(subspace_sum W1 W2) ⊆ W ⟷ W1 ⊆ W ∧ W2 ⊆ W"
proof -
from h1 h2 h3 show ?thesis by (intro sum_is_minimal, unfold subspace_def, auto)
lemma (in vectorspace) span_is_subspace:
fixes S
assumes h2: "S⊆carrier V"
shows "subspace K (span S) V"
proof -
have 0: "vectorspace K V"..
from h2 have 1: "submodule K (span S) V" by (rule span_is_submodule)
from 0 1 show ?thesis by (unfold subspace_def mod_hom_def linear_map_def, auto)
subsubsection ‹Facts specific to vector spaces›
text ‹If $av = w$ and $a\neq 0$, $v=a^{-1}w$.›
lemma (in vectorspace) mult_inverse:
assumes h1: "a∈carrier K" and h2: "v∈carrier V" and h3: "a ⊙⇘V⇙ v = w" and h4: "a≠𝟬⇘K⇙"
shows "v = (inv⇘K⇙ a )⊙⇘V⇙ w"
proof -
from h1 h2 h3 have 1: "w∈carrier V" by auto
from h3 1 have 2: "(inv⇘K⇙ a )⊙⇘V⇙(a ⊙⇘V⇙ v) =(inv⇘K⇙ a )⊙⇘V⇙w" by auto
from h1 h4 have 3: "inv⇘K⇙ a∈carrier K" by auto
interpret g: group "(units_group K)" by (rule units_form_group)
have f: "field K"..
from f h1 h4 have 4: "a∈Units K"
by (unfold field_def field_axioms_def, simp)
from 4 h1 h4 have 5: "((inv⇘K⇙ a) ⊗⇘K⇙a) = 𝟭⇘K⇙"
by (intro Units_l_inv, auto)
from 5 have 6: "(inv⇘K⇙ a )⊙⇘V⇙(a ⊙⇘V⇙ v) = v"
proof -
from h1 h2 h4 have 7: "(inv⇘K⇙ a )⊙⇘V⇙(a ⊙⇘V⇙ v) =(inv⇘K⇙ a ⊗⇘K⇙a) ⊙⇘V⇙ v" by (auto simp add: smult_assoc1)
from 5 h2 have 8: "(inv⇘K⇙ a ⊗⇘K⇙a) ⊙⇘V⇙ v = v" by auto
from 7 8 show ?thesis by auto
from 2 6 show ?thesis by auto
text ‹If $w\in S$ and $\sum_{w\in S} a_ww=0$, we have $v=\sum_{w\not\in S}a_v^{-1}a_ww$›
lemma (in vectorspace) lincomb_isolate:
fixes A v
assumes h1: "finite A" and h2: "A⊆carrier V" and h3: "a∈A→carrier K" and h4: "v∈A"
and h5: "a v ≠ 𝟬⇘K⇙" and h6: "lincomb a A=𝟬⇘V⇙"
shows "v=lincomb (λw. ⊖⇘K⇙(inv⇘K⇙ (a v)) ⊗⇘K⇙ a w) (A-{v})" and "v∈ span (A-{v})"
proof -
from h1 h2 h3 h4 have 1: "lincomb a A = ((a v) ⊙⇘V⇙ v) ⊕⇘V⇙ lincomb a (A-{v})"
by (rule lincomb_del2)
from 1 have 2: "𝟬⇘V⇙= ((a v) ⊙⇘V⇙ v) ⊕⇘V⇙ lincomb a (A-{v})" by (simp add: h6)
from h1 h2 h3 have 5: "lincomb a (A-{v}) ∈carrier V" by auto (*intro lincomb_closed*)
from 2 h1 h2 h3 h4 have 3: " ⊖⇘V⇙ lincomb a (A-{v}) = ((a v) ⊙⇘V⇙ v)"
by (auto intro!: M.minus_equality)
have 6: "v = (⊖⇘K⇙ (inv⇘K⇙ (a v))) ⊙⇘V⇙ lincomb a (A-{v})"
proof -
from h2 h3 h4 h5 3 have 7: "v = inv⇘K⇙ (a v) ⊙⇘V⇙ (⊖⇘V⇙ lincomb a (A-{v}))"
by (intro mult_inverse, auto)
from assms have 8: "inv⇘K⇙ (a v)∈carrier K" by auto
from assms 5 8 have 9: "inv⇘K⇙ (a v) ⊙⇘V⇙ (⊖⇘V⇙ lincomb a (A-{v}))
= (⊖⇘K⇙ (inv⇘K⇙ (a v))) ⊙⇘V⇙ lincomb a (A-{v})"
by (simp add: smult_assoc_simp smult_minus_1_back r_minus)
from 7 9 show ?thesis by auto
from h1 have 10: "finite (A-{v})" by auto
from assms have 11 : "(⊖⇘K⇙ (inv⇘K⇙ (a v)))∈ carrier K" by auto
from assms have 12: "lincomb (λw. ⊖⇘K⇙(inv⇘K⇙ (a v)) ⊗⇘K⇙ a w) (A-{v}) =
(⊖⇘K⇙ (inv⇘K⇙ (a v))) ⊙⇘V⇙ lincomb a (A-{v})"
by (intro lincomb_smult, auto)
from 6 12 show "v=lincomb (λw. ⊖⇘K⇙(inv⇘K⇙ (a v)) ⊗⇘K⇙ a w) (A-{v})" by auto
with assms show "v∈ span (A-{v})"
unfolding span_def
by (force simp add: 11 ring_subset_carrier)
text ‹The map $(S\to K)\mapsto V$ given by $(a_v)_{v\in S}\mapsto \sum_{v\in S} a_v v$ is linear.›
lemma (in vectorspace) lincomb_is_linear:
fixes S
assumes h: "finite S" and h2: "S⊆carrier V"
shows "linear_map K (func_space S) V (λa. lincomb a S)"
proof -
have 0: "vectorspace K V"..
from h h2 have 1: "mod_hom K (func_space S) V (λa. lincomb a S)" by (rule lincomb_is_mod_hom)
from 0 1 show ?thesis by (unfold vectorspace_def mod_hom_def linear_map_def, auto)
subsection ‹Basic facts about span and linear independence›
text ‹If $S$ is linearly independent, then $v\in \text{span}S$ iff $S\cup \{v\}$ is linearly
theorem (in vectorspace) lin_dep_iff_in_span:
fixes A v S
assumes h1: "S ⊆ carrier V" and h2: "lin_indpt S" and h3: "v∈ carrier V" and h4: "v∉S"
shows "v∈ span S ⟷ lin_dep (S ∪ {v})"
proof -
let ?T = "S ∪ {v}"
have 0: "v∈?T " by auto
from h1 h3 have h1_1: "?T ⊆ carrier V" by auto
have a1:"lin_dep ?T ⟹ v∈ span S"
proof -
assume a11: "lin_dep ?T"
from a11 obtain a w A where a: "(finite A ∧ A⊆?T ∧ (a∈ (A→carrier K)) ∧ (lincomb a A = 𝟬⇘V⇙) ∧ (w∈A) ∧ (a w≠ 𝟬⇘K⇙))"
by (metis lin_dep_def)
from assms a have nz2: "∃v∈A-S. a v≠𝟬⇘K⇙"
by (intro lincomb_must_include[where ?v="w" and ?T="S ∪{v}"], auto)
from a nz2 have singleton: "{v}=A-S" by auto
from singleton nz2 have nz3: "a v≠𝟬⇘K⇙" by auto
(*Can modularize this whole section out. "solving for one variable"*)
let ?b="(λw. ⊖⇘K⇙ (inv⇘K⇙ (a v)) ⊗⇘K⇙ (a w))"
from singleton have Ains: "(A∩S) = A-{v}" by auto
from assms a singleton nz3 have a31: "v= lincomb ?b (A∩S)"
apply (subst Ains)
by (intro lincomb_isolate(1), auto)
from a a31 nz3 singleton show ?thesis
apply (unfold span_def, auto)
apply (rule_tac x="?b" in exI)
apply (rule_tac x="A∩S" in exI)
by (auto intro!: m_closed)
have a2: "v∈ (span S) ⟹ lin_dep ?T"
proof -
assume inspan: "v∈ (span S)"
from inspan obtain a A where a: "A⊆S ∧ finite A ∧ (v = lincomb a A)∧ a∈A→carrier K" by (simp add: span_def, auto)
let ?b = "λ w. if (w=v) then (⊖⇘K⇙ 𝟭⇘K⇙) else a w" (*consider -v + \sum a_w w*)
have lc0: " lincomb ?b (A∪{v})=𝟬⇘V⇙"
proof -
from assms a have lc_ins: "lincomb ?b (A∪{v}) = ((?b v) ⊙⇘V⇙ v) ⊕⇘V⇙ lincomb ?b A"
by (intro lincomb_insert, auto)
from assms a have lc_elim: "lincomb ?b A=lincomb a A" by (intro lincomb_elim_if, auto)
from assms lc_ins lc_elim a show ?thesis by (simp add: M.l_neg smult_minus_1)
from a lc0 show ?thesis
apply (unfold lin_dep_def)
apply (rule_tac x="A∪{v}" in exI)
apply (rule_tac x="?b" in exI)
apply (rule_tac x="v" in exI)
by auto
from a1 a2 show ?thesis by auto
text ‹If $v\in \text{span} A$ then $\text{span}A =\text{span}(A\cup \{v\})$›
lemma (in vectorspace) already_in_span:
fixes v A
assumes inC: "A⊆carrier V" and inspan: "v∈span A"
shows "span A= span (A∪{v})"
proof -
from inC inspan have dir1: "span A ⊆ span (A∪{v})" by (intro span_is_monotone, auto)
from inC have inown: "A⊆span A" by (rule in_own_span)
from inC have subm: "submodule K (span A) V" by (rule span_is_submodule)
from inown inspan subm have dir2: "span (A ∪ {v}) ⊆ span A" by (intro span_is_subset, auto)
from dir1 dir2 show ?thesis by auto
subsection ‹The Replacement Theorem›
text ‹If $A,B\subseteq V$ are finite, $A$ is linearly independent, $B$ generates $W$,
and $A\subseteq W$, then there exists $C\subseteq V$ disjoint from $A$ such that
$\text{span}(A\cup C) = W$ and $|C|\le |B|-|A|$. In other words, we can complete
any linearly independent set to a generating set of $W$ by adding at most $|B|-|A|$ more elements.›
theorem (in vectorspace) replacement:
fixes A B (*A B are lists of vectors (colloquially we refer to them as sets)*)
assumes h1: "finite A"
and h2: "finite B"
and h3: "B⊆carrier V"
and h4: "lin_indpt A" (*A is linearly independent*)
and h5: "A⊆span B" (*All entries of A are in K*)
shows "∃C. finite C ∧ C⊆carrier V ∧ C⊆span B ∧ C∩A={} ∧ int (card C) ≤ (int (card B)) - (int (card A)) ∧ (span (A ∪ C) = span B)"
(is "∃C. ?P A B C")
(*There is a set C of cardinality |B| - |A| such that A∪C generates V*)
using h1 h2 h3 h4 h5
proof (induct "card A" arbitrary: A B)
case 0
from "0.prems"(1) "0.hyps" have a0: "A={}" by auto
from "0.prems"(3) have a3: "B⊆span B" by (rule in_own_span)
from a0 a3 "0.prems" show ?case by (rule_tac x="B" in exI, auto)
case (Suc m)
let ?W="span B"
from Suc.prems(3) have BinC: "span B⊆carrier V" by (rule span_is_subset2)
(*everything you want to know about A*)
from Suc.prems Suc.hyps BinC have A: "finite A" "lin_indpt A" "A⊆span B" "Suc m = card A" "A⊆carrier V"
by auto
(*everything you want to know about B*)
from Suc.prems have B: "finite B" "B⊆carrier V" by auto
(*A B are lists of vectors (colloquially we refer to them as sets)*)
from Suc.hyps(2) obtain v where v: "v∈A" by fastforce
let ?A'="A-{v}"
(*?A' is linearly independent because it is the subset of a linearly independent set, A.*)
from A(2) have liA': "lin_indpt ?A'"
apply (intro subset_li_is_li[of "A" "?A'"])
by auto
from v liA' Suc.prems Suc.hyps(2) have "∃C'. ?P ?A' B C'"
apply (intro Suc.hyps(1))
by auto
from this obtain C' where C': "?P ?A' B C'" by auto
show ?case
proof (cases "v∈ C'")
case True
have vinC': "v∈C'" by fact
from vinC' v have seteq: "A - {v} ∪ C' = A ∪ (C' - {v})" by auto
from C' seteq have spaneq: "span (A ∪ (C' - {v})) = span (B)" by algebra
from Suc.prems Suc.hyps C' vinC' v spaneq show ?thesis
apply (rule_tac x="C'-{v}" in exI)
apply (subgoal_tac "card C' >0")
by auto
case False
have f: "v∉C'" by fact
from A v C' have "∃a. a∈(?A'∪C')→carrier K ∧ lincomb a (?A' ∪ C') =v"
by (intro finite_in_span, auto)
from this obtain a where a: "a∈(?A'∪C')→carrier K ∧ v= lincomb a (?A' ∪ C')" by metis
let ?b="(λ w. if (w=v) then ⊖⇘K⇙𝟭⇘K⇙ else a w)"
from a have b: "?b∈A∪C'→carrier K" by auto
from v have rewrite_ins: "A∪C'=(?A'∪C')∪{v}" by auto
from f have "v∉?A'∪C'" by auto
from this A C' v a f have lcb: "lincomb ?b (A ∪ C') = 𝟬⇘V⇙"
apply (subst rewrite_ins)
apply (subst lincomb_insert)
apply (simp_all add: ring_subset_carrier coeff_in_ring)
apply (auto split: if_split_asm)
apply (subst lincomb_elim_if)
by (auto simp add: smult_minus_1 l_neg ring_subset_carrier)
(*NOTE: l_neg got deleted from the simp rules, but it is very useful.*)
from C' f have rewrite_minus: "C'=(A∪C')-A" by auto
from A C' b lcb v have exw: "∃w∈ C'. ?b w≠𝟬⇘K⇙"
apply (subst rewrite_minus)
apply (intro lincomb_must_include[where ?T="A ∪C'" and ?v="v"])
by auto
from exw obtain w where w: "w∈ C'" "?b w≠𝟬⇘K⇙" by auto
from A C' w f b lcb have w_in: "w∈span ((A∪ C') -{w})"
apply (intro lincomb_isolate[where a="?b"])
by auto
have spaneq2: "span (A∪(C'-{w})) = span B"
proof -
have 1: "span (?A'∪C') = span (A∪ C')" (*adding v doesn't change the span*)
proof -
from A C' v have m1: "span (?A'∪C') = span ((?A'∪ C')∪{v})"
apply (intro already_in_span)
by auto
from f m1 show ?thesis by (metis rewrite_ins)
have 2: "span (A∪ (C'-{w})) = span (A∪ C')" (*removing w doesn't change the span*)
proof -
from C' w(1) f have b60: "A∪ (C'-{w}) = (A∪ C') -{w}" by auto
from w(1) have b61: "A∪ C'= (A∪ C' -{w})∪{w}" by auto
from A C' w_in show ?thesis
apply (subst b61)
apply (subst b60)
apply (intro already_in_span)
by auto
from C' 1 2 show ?thesis by auto
qed(* "span (A∪(C'-{w})) = span B"*)
from A C' w f v spaneq2 show ?thesis
apply (rule_tac x="C'-{w}" in exI)
apply (subgoal_tac "card C' >0")
by auto
subsection ‹Defining dimension and bases.›
text ‹Finite dimensional is defined as having a finite generating set.›
definition (in vectorspace) fin_dim:: "bool"
where "fin_dim = (∃ A. ((finite A) ∧ (A ⊆ carrier V) ∧ (gen_set A)))"
text ‹The dimension is the size of the smallest generating set. For equivalent
characterizations see below.›
definition (in vectorspace) dim:: "nat"
where "dim = (LEAST n. (∃ A. ((finite A) ∧ (card A = n) ∧ (A ⊆ carrier V) ∧ (gen_set A))))"
text ‹A ‹basis› is a linearly independent generating set.›
definition (in vectorspace) basis:: "'c set ⇒ bool"
where "basis A = ((lin_indpt A) ∧ (gen_set A)∧ (A⊆carrier V))"
text ‹From the replacement theorem, any linearly independent set is smaller than any generating set.›
lemma (in vectorspace) li_smaller_than_gen:
fixes A B
assumes h1: "finite A" and h2: "finite B" and h3: "A⊆carrier V" and h4: "B⊆carrier V"
and h5: "lin_indpt A" and h6: "gen_set B"
shows "card A ≤ card B"
proof -
from h3 h6 have 1: "A⊆span B" by auto
from h1 h2 h4 h5 1 obtain C where
2: "finite C ∧ C⊆carrier V ∧ C⊆span B ∧ C∩A={} ∧ int (card C) ≤ int (card B) - int (card A) ∧ (span (A ∪ C) = span B)"
by (metis replacement)
from 2 show ?thesis by arith
text ‹The dimension is the cardinality of any basis. (In particular, all bases are the same size.)›
lemma (in vectorspace) dim_basis:
fixes A
assumes fin: "finite A" and h2: "basis A"
shows "dim = card A"
proof -
have 0: "⋀B m. ((finite B) ∧ (card B = m) ∧ (B ⊆ carrier V) ∧ (gen_set B)) ⟹ card A ≤ m"
proof -
fix B m
assume 1: "((finite B) ∧ (card B = m) ∧ (B ⊆ carrier V) ∧ (gen_set B))"
from 1 fin h2 have 2: "card A ≤ card B"
apply (unfold basis_def)
apply (intro li_smaller_than_gen)
by auto
from 1 2 show "?thesis B m" by auto
from fin h2 0 show ?thesis
apply (unfold dim_def basis_def)
apply (intro Least_equality)
apply (rule_tac x="A" in exI)
by auto
(*can define more generally with posets*)
text ‹A ‹maximal› set with respect to $P$ is such that if $B\supseteq A$ and $P$ is also
satisfied for $B$, then $B=A$.›
definition maximal::"'a set ⇒ ('a set ⇒ bool) ⇒ bool"
where "maximal A P = ((P A) ∧ (∀B. B⊇A ∧ P B ⟶ B = A))"
text ‹A ‹minimal› set with respect to $P$ is such that if $B\subseteq A$ and $P$ is also
satisfied for $B$, then $B=A$.›
definition minimal::"'a set ⇒ ('a set ⇒ bool) ⇒ bool"
where "minimal A P = ((P A) ∧ (∀B. B⊆A ∧ P B ⟶ B = A))"
text ‹A maximal linearly independent set is a generating set.›
lemma (in vectorspace) max_li_is_gen:
fixes A
assumes h1: "maximal A (λS. S⊆carrier V ∧ lin_indpt S)"
shows "gen_set A"
proof (rule ccontr)
assume 0: "¬(gen_set A)"
from h1 have 1: " A⊆ carrier V ∧ lin_indpt A" by (unfold maximal_def, auto)
from 1 have 2: "span A ⊆ carrier V" by (intro span_is_subset2, auto)
from 0 1 2 have 3: "∃v. v∈carrier V ∧ v ∉ (span A)"
by auto
from 3 obtain v where 4: "v∈carrier V ∧ v ∉ (span A)" by auto
have 5: "v∉A"
proof -
from h1 1 have 51: "A⊆span A" apply (intro in_own_span) by auto
from 4 51 show ?thesis by auto
from lin_dep_iff_in_span have 6: "⋀S v. S ⊆ carrier V∧ lin_indpt S ∧ v∈ carrier V ∧ v∉S
∧ v∉ span S ⟹ (lin_indpt (S ∪ {v}))" by auto
from 1 4 5 have 7: "lin_indpt (A ∪ {v})" apply (intro 6) by auto
(* assumes h0:"finite S" and h1: "S ⊆ carrier V" and h2: "lin_indpt S" and h3: "v∈ carrier V" and h4: "v∉S"
shows "v∈ span S ⟷ ¬ (lin_indpt (S ∪ {v}))"*)
have 9: "¬(maximal A (λS. S⊆carrier V ∧ lin_indpt S))"
proof -
from 1 4 5 7 have 8: "(∃B. A ⊆ B ∧ B ⊆ carrier V ∧ lin_indpt B ∧ B ≠ A)"
apply (rule_tac x="A∪{v}" in exI)
by auto
from 8 show ?thesis
apply (unfold maximal_def)
by simp
from h1 9 show False by auto
text ‹A minimal generating set is linearly independent.›
lemma (in vectorspace) min_gen_is_li:
fixes A
assumes h1: "minimal A (λS. S⊆carrier V ∧ gen_set S)"
shows "lin_indpt A"
proof (rule ccontr)
assume 0: "¬lin_indpt A"
from h1 have 1: " A⊆ carrier V ∧ gen_set A" by (unfold minimal_def, auto)
from 1 have 2: "span A = carrier V" by auto
from 0 1 obtain a v A' where
3: "finite A' ∧ A'⊆A ∧ a ∈ A' → carrier K ∧ LinearCombinations.module.lincomb V a A' = 𝟬⇘V⇙ ∧ v ∈ A' ∧ a v ≠ 𝟬⇘K⇙"
by (unfold lin_dep_def, auto)
have 4: "gen_set (A-{v})"
proof -
from 1 3 have 5: "v∈span (A'-{v})"
apply (intro lincomb_isolate[where a="a" and v="v"])
by auto
from 3 5 have 51: "v∈span (A-{v})"
apply (intro subsetD[where ?A="span (A' -{v})" and ?B="span (A -{v})" and ?c="v"])
by (intro span_is_monotone, auto)
from 1 have 6: "A⊆span A" apply (intro in_own_span) by auto
from 1 51 have 7: "span (A-{v}) = span ((A-{v})∪{v})" apply (intro already_in_span) by auto
from 3 have 8: "A = ((A-{v})∪{v})" by auto
from 2 7 8 have 9:"span (A-{v}) = carrier V" by auto (*can't use 3 directly :( *)
from 9 show ?thesis by auto
have 10: "¬(minimal A (λS. S⊆carrier V ∧ gen_set S))"
proof -
from 1 3 4 have 11: "(∃B. A ⊇ B ∧ B ⊆ carrier V ∧ gen_set B ∧ B ≠ A)"
apply (rule_tac x="A-{v}" in exI)
by auto
from 11 show ?thesis
apply (unfold minimal_def)
by auto
from h1 10 show False by auto
text ‹Given that some finite set satisfies $P$, there is a minimal set that satisfies $P$.›
lemma minimal_exists:
fixes A P
assumes h1: "finite A" and h2: "P A"
shows "∃B. B⊆A ∧ minimal B P"
using h1 h2
proof (induct "card A" arbitrary: A rule: less_induct)
case (less A)
show ?case
proof (cases "card A = 0")
case True
from True less.hyps less.prems show ?thesis
apply (rule_tac x="{}" in exI)
apply (unfold minimal_def)
by auto
case False
show ?thesis
proof (cases "minimal A P")
case True
then show ?thesis
apply (rule_tac x="A" in exI)
by auto
case False
have 2: "¬minimal A P" by fact
from less.prems 2 have 3: "∃B. P B ∧ B ⊆ A ∧ B≠A"
apply (unfold minimal_def)
by auto
from 3 obtain B where 4: "P B ∧ B ⊂ A ∧ B≠A" by auto
from 4 have 5: "card B < card A" by (metis less.prems(1) psubset_card_mono)
from less.hyps less.prems 3 4 5 have 6: "∃C⊆B. minimal C P"
apply (intro less.hyps)
apply auto
by (metis rev_finite_subset)
from 6 obtain C where 7: "C⊆B ∧ minimal C P" by auto
from 4 7 show ?thesis
apply (rule_tac x="C" in exI)
apply (unfold minimal_def)
by auto
text ‹If $V$ is finite-dimensional, then any linearly independent set is finite.›
lemma (in vectorspace) fin_dim_li_fin:
assumes fd: "fin_dim" and li: "lin_indpt A" and inC: "A⊆carrier V"
shows fin: "finite A"
proof (rule ccontr)
assume A: "¬finite A"
from fd obtain C where C: "finite C ∧ C⊆carrier V ∧ gen_set C" by (unfold fin_dim_def, auto)
from A obtain B where B: "B⊆A∧ finite B ∧ card B = card C + 1"
by (metis infinite_arbitrarily_large)
from B li have liB: "lin_indpt B"
by (intro subset_li_is_li[where ?A="A" and ?B="B"], auto)
from B C liB inC have "card B ≤ card C" by (intro li_smaller_than_gen, auto)
from this B show False by auto
text ‹If $V$ is finite-dimensional (has a finite generating set), then a finite basis exists.›
lemma (in vectorspace) finite_basis_exists:
assumes h1: "fin_dim"
shows "∃β. finite β ∧ basis β"
proof -
from h1 obtain A where 1: "finite A ∧ A⊆carrier V ∧ gen_set A" by (metis fin_dim_def)
hence 2: "∃β. β⊆A ∧ minimal β (λS. S⊆carrier V ∧ gen_set S)"
apply (intro minimal_exists)
by auto
then obtain β where 3: "β⊆A ∧ minimal β (λS. S⊆carrier V ∧ gen_set S)" by auto
hence 4: "lin_indpt β" apply (intro min_gen_is_li) by auto
moreover from 3 have 5: "gen_set β ∧ β⊆carrier V" apply (unfold minimal_def) by auto
moreover from 1 3 have 6: "finite β" by (auto simp add: finite_subset)
ultimately show ?thesis apply (unfold basis_def) by auto
The proof is as follows.
\item Because $V$ is finite-dimensional, there is a finite generating set
(we took this as our definition of finite-dimensional).
\item Hence, there is a minimal $\beta \subseteq A$ such that $\beta$ generates $V$.
\item $\beta$ is linearly independent because a minimal generating set is linearly independent.
Finally, $\beta$ is a basis because it is both generating and linearly independent.
text ‹Any linearly independent set has cardinality at most equal to the dimension.›
lemma (in vectorspace) li_le_dim:
fixes A
assumes fd: "fin_dim" and c: "A⊆carrier V" and l: "lin_indpt A"
shows "finite A" "card A ≤ dim"
proof -
from fd c l show fa: "finite A" by (intro fin_dim_li_fin, auto)
from fd obtain β where 1: "finite β ∧ basis β"
by (metis finite_basis_exists)
from assms fa 1 have 2: "card A ≤ card β"
apply (intro li_smaller_than_gen, auto)
by (unfold basis_def, auto)
from assms 1 have 3: "dim = card β" by (intro dim_basis, auto)
from 2 3 show "card A ≤ dim" by auto
text ‹Any generating set has cardinality at least equal to the dimension.›
lemma (in vectorspace) gen_ge_dim:
fixes A
assumes fa: "finite A" and c: "A⊆carrier V" and l: "gen_set A"
shows "card A ≥ dim"
proof -
from assms have fd: "fin_dim" by (unfold fin_dim_def, auto)
from fd obtain β where 1: "finite β ∧ basis β" by (metis finite_basis_exists)
from assms 1 have 2: "card A ≥ card β"
apply (intro li_smaller_than_gen, auto)
by (unfold basis_def, auto)
from assms 1 have 3: "dim = card β" by (intro dim_basis, auto)
from 2 3 show ?thesis by auto
text ‹If there is an upper bound on the cardinality of sets satisfying $P$, then there is a maximal
set satisfying $P$.›
lemma maximal_exists:
fixes P B N
assumes maxc: "⋀A. P A ⟹ finite A ∧ (card A ≤N)" and b: "P B"
shows "∃A. finite A ∧ maximal A P"
proof -
(*take the maximal*)
let ?S="{card A| A. P A}"
let ?n="Max ?S"
from maxc have 1:"finite ?S"
apply (simp add: finite_nat_set_iff_bounded_le) by auto
from 1 have 2: "?n∈?S"
by (metis (mono_tags, lifting) Collect_empty_eq Max_in b)
from assms 2 have 3: "∃A. P A ∧ finite A ∧ card A = ?n"
by auto
from 3 obtain A where 4: "P A ∧ finite A ∧ card A = ?n" by auto
from 1 maxc have 5: "⋀A. P A ⟹ finite A ∧ (card A ≤?n)"
by (metis (mono_tags, lifting) Max.coboundedI mem_Collect_eq)
from 4 5 have 6: "maximal A P"
apply (unfold maximal_def)
by (metis card_seteq)
from 4 6 show ?thesis by auto
text ‹Any maximal linearly independent set is a basis.›
lemma (in vectorspace) max_li_is_basis:
fixes A
assumes h1: "maximal A (λS. S⊆carrier V ∧ lin_indpt S)"
shows "basis A"
proof -
from h1 have 1: "gen_set A" by (rule max_li_is_gen)
from assms 1 show ?thesis by (unfold basis_def maximal_def, auto)
text ‹Any minimal linearly independent set is a generating set.›
lemma (in vectorspace) min_gen_is_basis:
fixes A
assumes h1: "minimal A (λS. S⊆carrier V ∧ gen_set S)"
shows "basis A"
proof -
from h1 have 1: "lin_indpt A" by (rule min_gen_is_li)
from assms 1 show ?thesis by (unfold basis_def minimal_def, auto)
text ‹Any linearly independent set with cardinality at least the dimension is a basis.›
lemma (in vectorspace) dim_li_is_basis:
fixes A
assumes fd: "fin_dim" and fa: "finite A" and ca: "A⊆carrier V" and li: "lin_indpt A"
and d: "card A ≥ dim" (*≥*)
shows "basis A"
proof -
from fd have 0: "⋀S. S⊆carrier V ∧ lin_indpt S ⟹ finite S ∧ card S ≤dim"
by (auto intro: li_le_dim)
from 0 assms have h1: "finite A ∧ maximal A (λS. S⊆carrier V ∧ lin_indpt S)"
apply (unfold maximal_def)
apply auto
by (metis card_seteq eq_iff)
from h1 show ?thesis by (auto intro: max_li_is_basis)
text ‹Any generating set with cardinality at most the dimension is a basis.›
lemma (in vectorspace) dim_gen_is_basis:
fixes A
assumes fa: "finite A" and ca: "A⊆carrier V" and li: "gen_set A"
and d: "card A ≤ dim"
shows "basis A"
proof -
have 0: "⋀S. finite S∧ S⊆carrier V ∧ gen_set S ⟹ card S ≥dim"
by (intro gen_ge_dim, auto)
from 0 assms have h1: "minimal A (λS. finite S ∧ S⊆carrier V ∧ gen_set S)"
apply (unfold minimal_def)
apply auto
by (metis card_seteq eq_iff)
(*slightly annoying: we have to get rid of "finite S" inside.*)
from h1 have h: "⋀B. B ⊆ A ∧ B ⊆ carrier V ∧ LinearCombinations.module.gen_set K V B ⟹ B = A"
proof -
fix B
assume asm: "B ⊆ A ∧ B ⊆ carrier V ∧ LinearCombinations.module.gen_set K V B"
from asm h1 have "finite B"
apply (unfold minimal_def)
apply (intro finite_subset[where ?A="B" and ?B="A"])
by auto
from h1 asm this show "?thesis B" apply (unfold minimal_def) by simp
from h1 h have h2: "minimal A (λS. S⊆carrier V ∧ gen_set S)"
apply (unfold minimal_def)
by presburger
from h2 show ?thesis by (rule min_gen_is_basis)
text ‹$\beta$ is a basis iff for all $v\in V$, there exists a unique $(a_v)_{v\in S}$ such that
$\sum_{v\in S} a_v v=v$.›
lemma (in vectorspace) basis_criterion:
assumes A_fin: "finite A" and AinC: "A⊆carrier V"
shows "basis A ⟷ (∀v. v∈ carrier V ⟶(∃! a. a∈A →⇩[E] carrier K ∧ lincomb a A = v))"
proof -
have 1: "¬(∀v. v∈ carrier V ⟶(∃! a. a∈A →⇩[E] carrier K ∧ lincomb a A = v)) ⟹ ¬basis A"
proof -
assume "¬(∀v. v∈ carrier V ⟶(∃! a. a∈A →⇩[E] carrier K ∧ lincomb a A = v))"
then obtain v where v: "v∈ carrier V ∧ ¬(∃! a. a∈A →⇩[E] carrier K ∧ lincomb a A = v)" by metis
(*either there is more than 1 rep, or no reps*)
from v have vinC: "v∈carrier V" by auto
from v have "¬(∃ a. a∈A →⇩[E] carrier K ∧ lincomb a A = v) ∨ (∃ a b.
a∈A →⇩[E] carrier K ∧ lincomb a A = v ∧ b∈A →⇩[E] carrier K ∧ lincomb b A = v
∧ a≠b)" by metis
then show ?thesis
assume a: "¬(∃ a. a∈A →⇩[E] carrier K ∧ lincomb a A = v)"
from A_fin AinC have "⋀a. a∈A → carrier K ⟹ lincomb a A = lincomb (restrict a A) A"
unfolding lincomb_def restrict_def
by (simp cong: finsum_cong add: ring_subset_carrier coeff_in_ring)
with a have "¬(∃ a. a∈A → carrier K ∧ lincomb a A = v)" by auto
with A_fin AinC have "v∉span A"
using finite_in_span by blast
with AinC v show "¬(basis A)" by (unfold basis_def, auto)
assume a2: "(∃ a b.
a∈A →⇩[E] carrier K ∧ lincomb a A = v ∧ b∈A →⇩[E] carrier K ∧ lincomb b A = v
∧ a≠b)"
then obtain a b where ab: "a∈A →⇩[E] carrier K ∧ lincomb a A = v ∧ b∈A →⇩[E] carrier K ∧ lincomb b A = v
∧ a≠b" by metis
from ab obtain w where w: "w∈A ∧ a w ≠ b w" apply (unfold PiE_def, auto)
by (metis extensionalityI)
let ?c="λ x. (if x∈A then ((a x) ⊖⇘K⇙ (b x)) else undefined)"
from ab have a_fun: "a∈A → carrier K"
and b_fun: "b∈A → carrier K"
by (unfold PiE_def, auto)
from w a_fun b_fun have abinC: "a w ∈carrier K" "b w ∈carrier K" by auto
from abinC w have nz: "a w ⊖⇘K⇙ b w ≠ 𝟬⇘K⇙"
by auto (*uses M.minus_other_side*)
from A_fin AinC a_fun b_fun ab vinC have a_b:
"LinearCombinations.module.lincomb V (λx. if x ∈ A then a x ⊖⇘K⇙ b x else undefined) A = 𝟬⇘V⇙"
by (simp cong: lincomb_cong add: coeff_in_ring lincomb_diff)
from A_fin AinC ab w v nz a_b have "lin_dep A"
apply (intro lin_dep_crit[where ?A="A" and ?a="?c" and ?v="w"])
apply (auto simp add: PiE_def)
by auto
thus "¬basis A" by (unfold basis_def, auto)
have 2: "(∀v. v∈ carrier V ⟶(∃! a. a∈A →⇩[E] carrier K ∧ lincomb a A = v)) ⟹ basis A"
proof -
assume b1: "(∀v. v∈ carrier V ⟶(∃! a. a∈A →⇩[E] carrier K ∧ lincomb a A = v))"
(is "(∀v. v∈ carrier V ⟶(∃! a. ?Q a v))")
from b1 have b2: "(∀v. v∈ carrier V ⟶(∃ a. a∈A → carrier K ∧ lincomb a A = v))"
apply (unfold PiE_def)
by blast
from A_fin AinC b2 have "gen_set A"
apply (unfold span_def)
by blast
from b1 have A_li: "lin_indpt A"
proof -
let ?z="λ x. (if (x∈A) then 𝟬⇘K⇙ else undefined)"
from A_fin AinC have zero: "?Q ?z 𝟬⇘V⇙"
by (unfold PiE_def extensional_def lincomb_def, auto simp add: ring_subset_carrier)
(*uses finsum_all0*)
from A_fin AinC show ?thesis
proof (rule finite_lin_indpt2)
fix a
assume a_fun: "a ∈ A → carrier K" and
lc_a: "LinearCombinations.module.lincomb V a A = 𝟬⇘V⇙"
from a_fun have a_res: "restrict a A ∈ A →⇩[E] carrier K" by auto
from a_fun A_fin AinC lc_a have
lc_a_res: "LinearCombinations.module.lincomb V (restrict a A) A = 𝟬⇘V⇙"
apply (unfold lincomb_def restrict_def)
by (simp cong: finsum_cong2 add: coeff_in_ring ring_subset_carrier)
from a_fun a_res lc_a_res zero b1 have "restrict a A = ?z" by auto
from this show "∀v∈A. a v = 𝟬⇘K⇙"
apply (unfold restrict_def)
by meson
have A_gen: "gen_set A"
proof -
from AinC have dir1: "span A⊆carrier V" by (rule span_is_subset2)
have dir2: "carrier V⊆span A"
proof (auto)
fix v
assume v: "v∈carrier V"
from v b2 obtain a where "a∈A → carrier K ∧ lincomb a A = v" by auto
from this A_fin AinC show "v∈span A" by (subst finite_span, auto)
from dir1 dir2 show ?thesis by auto
from A_li A_gen AinC show "basis A" by (unfold basis_def, auto)
from 1 2 show ?thesis by satx
lemma (in linear_map) surj_imp_imT_carrier:
assumes surj: "T` (carrier V) = carrier W"
shows "(imT) = carrier W"
by (simp add: surj im_def)
subsection ‹The rank-nullity (dimension) theorem›
text ‹If $V$ is finite-dimensional and $T:V\to W$ is a linear map, then $\text{dim}(\text{im}(T))+
\text{dim}(\text{ker}(T)) = \text{dim} V$. Moreover, we prove that if $T$ is surjective
linear-map between $V$ and $W$, where $V$ is finite-dimensional, then also $W$ is finite-dimensional.›
theorem (in linear_map) rank_nullity_main:
assumes fd: "V.fin_dim"
shows "(vectorspace.dim K (W.vs imT)) + (vectorspace.dim K (V.vs kerT)) = V.dim"
"T ` (carrier V) = carrier W ⟹ W.fin_dim"
proof -
― ‹First interpret kerT, imT as vectorspaces›
have subs_ker: "subspace K kerT V" by (intro kerT_is_subspace)
from subs_ker have vs_ker: "vectorspace K (V.vs kerT)" by (rule V.subspace_is_vs)
from vs_ker interpret ker: vectorspace K "(V.vs kerT)" by auto
have kerInC: "kerT⊆carrier V" by (unfold ker_def, auto)
have subs_im: "subspace K imT W" by (intro imT_is_subspace)
from subs_im have vs_im: "vectorspace K (W.vs imT)" by (rule W.subspace_is_vs)
from vs_im interpret im: vectorspace K "(W.vs imT)" by auto
have imInC: "imT⊆carrier W" by (unfold im_def, auto)
(* obvious fact *)
have zero_same[simp]: "𝟬⇘V.vs kerT⇙ = 𝟬⇘V⇙" apply (unfold ker_def) by auto
― ‹Show ker T has a finite basis. This is not obvious. Show that any linearly independent set
has size at most that of V. There exists a maximal linearly independent set, which is the basis.›
have every_li_small: "⋀A. (A ⊆ kerT)∧ ker.lin_indpt A ⟹
finite A ∧ card A ≤ V.dim"
proof -
fix A
assume eli_asm: "(A ⊆ kerT)∧ ker.lin_indpt A"
(*annoying: I can't just use subst V.module.span_li_not_depend(2) in the show ?thesis
statement because it doesn't appear in the conclusion.*)
note V.module.span_li_not_depend(2)[where ?N="kerT" and ?S="A"]
from this subs_ker fd eli_asm kerInC show "?thesis A"
apply (intro conjI)
by (auto intro!: V.li_le_dim)
from every_li_small have exA:
"∃A. finite A ∧ maximal A (λS. S⊆carrier (V.vs kerT) ∧ ker.lin_indpt S)"
apply (intro maximal_exists[where ?N=" V.dim" and ?B=" {}"])
apply auto
by (unfold ker.lin_dep_def, auto)
from exA obtain A where A:" finite A ∧ maximal A (λS. S⊆carrier (V.vs kerT) ∧ ker.lin_indpt S)"
by blast
hence finA: "finite A" and Ainker: "A⊆carrier (V.vs kerT)" and AinC: "A⊆carrier V"
by (unfold maximal_def ker_def, auto)
― ‹We obtain the basis A of kerT. It is also linearly independent when considered in V rather
than kerT›
from A have Abasis: "ker.basis A"
by (intro ker.max_li_is_basis, auto)
from subs_ker Abasis have spanA: "V.module.span A = kerT"
apply (unfold ker.basis_def)
by (subst sym[OF V.module.span_li_not_depend(1)[where ?N=" kerT"]], auto)
from Abasis have Akerli: "ker.lin_indpt A"
apply (unfold ker.basis_def)
by auto
from subs_ker Ainker Akerli have Ali: "V.module.lin_indpt A"
by (auto simp add: V.module.span_li_not_depend(2))
txt‹Use the replacement theorem to find C such that $A\cup C$ is a basis of V.›
from fd obtain B where B: "finite B∧ V.basis B" by (metis V.finite_basis_exists)
from B have Bfin: "finite B" and Bbasis:"V.basis B" by auto
from B have Bcard: "V.dim = card B" by (intro V.dim_basis, auto)
from Bbasis have 62: "V.module.span B = carrier V"
by (unfold V.basis_def, auto)
from A Abasis Ali B vs_ker have "∃C. finite C ∧ C⊆carrier V ∧ C⊆ V.module.span B ∧ C∩A={}
∧ int (card C) ≤ (int (card B)) - (int (card A)) ∧ (V.module.span (A ∪ C) = V.module.span B)"
apply (intro V.replacement)
apply (unfold vectorspace.basis_def V.basis_def)
by (unfold ker_def, auto)
txt ‹From replacement we got $|C|\leq |B|-|A|$. Equality must actually hold, because no generating set
can be smaller than $B$. Now $A\cup C$ is a maximal generating set, hence a basis; its cardinality
equals the dimension.›
txt ‹We claim that $T(C)$ is basis for $\text{im}(T)$.›
then obtain C where C: "finite C ∧ C⊆carrier V ∧ C⊆ V.module.span B ∧ C∩A={}
∧ int (card C) ≤ (int (card B)) - (int (card A)) ∧ (V.module.span (A ∪ C) = V.module.span B)" by auto
hence Cfin: "finite C" and CinC: "C⊆carrier V" and CinspanB: " C⊆V.module.span B" and CAdis: "C∩A={}"
and Ccard: "int (card C) ≤ (int (card B)) - (int (card A))"
and ACspanB: "(V.module.span (A ∪ C) = V.module.span B)" by auto
from C have cardLe: "card A + card C ≤ card B" by auto
from B C have ACgen: "V.module.gen_set (A∪C)" apply (unfold V.basis_def) by auto
from finA C ACgen AinC B have cardGe: "card (A∪C) ≥ card B" by (intro V.li_smaller_than_gen, unfold V.basis_def, auto)
from finA C have cardUn: "card (A∪C)≤ card A + card C"
by (metis Int_commute card_Un_disjoint le_refl)
from cardLe cardUn cardGe Bcard have cardEq:
"card (A∪C) = card A + card C"
"card (A∪C) = card B"
"card (A∪C) = V.dim"
by auto
from Abasis C cardEq have disj: "A∩C={}" by auto
from finA AinC C cardEq 62 have ACfin: "finite (A∪C)" and ACbasis: "V.basis (A∪C)"
by (auto intro!: V.dim_gen_is_basis)
have lm: "linear_map K V W T"..
txt ‹Let $C'$ be the image of $C$ under $T$. We will show $C'$ is a basis for $\text{im}(T)$.›
let ?C' = "T`C"
from Cfin have C'fin: "finite ?C'" by auto
from AinC C have cim: "?C'⊆imT" by (unfold im_def, auto)
txt ‹"There is a subtle detail: we first have to show $T$ is injective on $C$.›
txt ‹We establish that no nontrivial linear combination of $C$ can have image 0 under $T$,
because that would mean it is a linear combination of $A$, giving that $A\cup C$ is linearly dependent,
contradiction. We use this result in 2 ways: (1) if $T$ is not injective on $C$, then we obtain $v$, $w\in C$
such that $v-w$ is in the kernel, contradiction, (2) if $T(C)$ is linearly dependent,
taking the inverse image of that linear combination gives a linear combination of $C$ in the kernel,
contradiction. Hence $T$ is injective on $C$ and $T(C)$ is linearly independent.›
have lc_in_ker: "⋀d D v. ⟦D⊆C; d∈D→carrier K; T (V.module.lincomb d D) = 𝟬⇘W⇙;
v∈D; d v ≠𝟬⇘K⇙⟧⟹False"
proof -
fix d D v
assume D: "D⊆C" and d: "d∈D→carrier K" and T0: "T (V.module.lincomb d D) = 𝟬⇘W⇙"
and v: "v∈D" and dvnz: "d v ≠𝟬⇘K⇙"
from D Cfin have Dfin: "finite D" by (auto intro: finite_subset)
from D CinC have DinC: "D⊆carrier V" by auto
from T0 d Dfin DinC have lc_d: "V.module.lincomb d D∈kerT"
by (unfold ker_def, auto)
from lc_d spanA AinC have "∃a' A'. A'⊆A ∧ a'∈A'→carrier K ∧
V.module.lincomb a' A'= V.module.lincomb d D"
by (intro V.module.in_span, auto)
then obtain a' A' where a': "A'⊆A ∧ a'∈A'→carrier K ∧
V.module.lincomb d D = V.module.lincomb a' A'"
by metis
hence A'sub: "A'⊆A" and a'fun: "a'∈A'→carrier K"
and a'_lc:"V.module.lincomb d D = V.module.lincomb a' A'" by auto
from a' finA Dfin have A'fin: "finite (A')" by (auto intro: finite_subset)
from AinC A'sub have A'inC: "A'⊆carrier V" by auto
let ?e = "(λv. if v ∈ A' then a' v else ⊖⇘K⇙𝟭⇘K⇙⊗⇘K⇙ d v)"
from a'fun d have e_fun: "?e ∈ A' ∪ D → carrier K"
apply (unfold Pi_def)
by auto
A'fin Dfin (*finiteness*)
A'inC DinC (*in carrier*)
a'fun d e_fun (*coefficients valid*)
disj D A'sub (*A and C disjoint*)
have lccomp1:
"V.module.lincomb a' A' ⊕⇘V⇙ ⊖⇘K⇙𝟭⇘K⇙⊙⇘V⇙ V.module.lincomb d D =
V.module.lincomb (λv. if v∈A' then a' v else ⊖⇘K⇙𝟭⇘K⇙⊗⇘K⇙ d v) (A'∪D)"
apply (subst sym[OF V.module.lincomb_smult])
apply (simp_all)
apply (subst V.module.lincomb_union2)
by (auto)
A'fin (*finiteness*)
A'inC (*in carrier*)
a'fun (*coefficients valid*)
have lccomp2:
"V.module.lincomb a' A' ⊕⇘V⇙ ⊖⇘K⇙𝟭⇘K⇙⊙⇘V⇙ V.module.lincomb d D =
by (simp add: a'_lc
V.module.smult_minus_1 V.module.M.r_neg)
from lccomp1 lccomp2 have lc0: "V.module.lincomb (λv. if v∈A' then a' v else ⊖⇘K⇙𝟭⇘K⇙⊗⇘K⇙ d v) (A'∪D)
=𝟬⇘V⇙" by auto
from disj a' v D have v_nin: "v∉A'" by auto
from A'fin Dfin (*finiteness*)
A'inC DinC (*in carrier*)
e_fun d (*coefficients valid*)
A'sub D disj (*A' D are disjoint subsets*)
v dvnz (*d v is nonzero coefficient*)
have AC_ld: "V.module.lin_dep (A∪C)"
apply (intro V.module.lin_dep_crit[where ?A="A' ∪D" and
?S="A ∪C" and ?a="λv. if v ∈A' then a' v else ⊖⇘K⇙ 𝟭⇘K⇙ ⊗⇘K⇙ d v" and ?v="v"])
by (auto dest: integral)
from AC_ld ACbasis show False by (unfold V.basis_def, auto)
have C'_card: "inj_on T C" "card C = card ?C'"
proof -
show "inj_on T C"
proof (rule ccontr)
assume "¬inj_on T C"
then obtain v w where "v∈C" "w∈C" "v≠w" "T v = T w" by (unfold inj_on_def, auto)
from this CinC show False
apply (intro lc_in_ker[where ?D1="{v,w}" and ?d1="λx. if x =v then 𝟭⇘K⇙ else ⊖⇘K⇙ 𝟭⇘K⇙"
and ?v1="v"])
by (auto simp add: V.module.lincomb_def hom_sum ring_subset_carrier
W.module.smult_minus_1 r_neg T_im)
from this Cfin show "card C = card ?C'"
by (metis card_image)
let ?f="the_inv_into C T"
have f: "⋀x. x∈C ⟹ ?f (T x) = x" "⋀y. y∈?C' ⟹ T (?f y) = y"
apply (insert C'_card(1))
apply (metis the_inv_into_f_f)
by (metis f_the_inv_into_f)
(*We show C' is a basis for the image. First we show it is linearly independent.*)
have C'_li: "im.lin_indpt ?C'"
proof (rule ccontr)
assume Cld: "¬im.lin_indpt ?C'"
from Cld cim subs_im have CldW: "W.module.lin_dep ?C'"
apply (subst sym[OF W.module.span_li_not_depend(2)[where ?S="T `C" and ?N=" imT"]])
by auto
from C CldW have "∃c' v'. (c'∈ (?C'→carrier K)) ∧ (W.module.lincomb c' ?C' = 𝟬⇘W⇙)
∧ (v'∈?C') ∧ (c' v'≠ 𝟬⇘K⇙)" by (intro W.module.finite_lin_dep, auto)
then obtain c' v' where c': "(c'∈ (?C'→carrier K)) ∧ (W.module.lincomb c' ?C' = 𝟬⇘W⇙)
∧ (v'∈?C') ∧ (c' v'≠ 𝟬⇘K⇙)" by auto
hence c'fun: "(c'∈ (?C'→carrier K))" and c'lc: "(W.module.lincomb c' ?C' = 𝟬⇘W⇙)" and
v':"(v'∈?C')" and cvnz: "(c' v'≠ 𝟬⇘K⇙)" by auto
(*can't get c' directly with metis/auto with W.module.finite_lin_dep*)
txt ‹We take the inverse image of $C'$ under $T$ to get a linear combination of $C$ that is
in the kernel and hence a linear combination of $A$. This contradicts $A\cup C$ being linearly
let ?c="λv. c' (T v)"
from c'fun have c_fun: "?c∈ C→carrier K" by auto
from Cfin (*C finite*)
c_fun c'fun (*coefficients valid*)
C'_card (*bijective*)
CinC (*C in carrier*)
f (*inverse to T*)
c'lc (*lc c' = 0*)
have "T (V.module.lincomb ?c C) = 𝟬⇘W⇙"
apply (unfold V.module.lincomb_def W.module.lincomb_def)
apply (subst hom_sum, auto)
apply (simp cong: finsum_cong add: ring_subset_carrier coeff_in_ring)
apply (subst finsum_reindex[where ?f="λw. c' w ⊙⇘W⇙ w" and ?h="T" and ?A="C", THEN sym])
by auto
with f c'fun cvnz v' show False
by (intro lc_in_ker[where ?D1="C" and ?d1="?c" and ?v1="?f v'"], auto)
have C'_gen: "im.gen_set ?C'"
proof -
have C'_span: "span ?C' = imT"
proof (rule equalityI)
from cim subs_im show "W.module.span ?C' ⊆ imT"
by (intro span_is_subset, unfold subspace_def, auto)
show "imT⊆W.module.span ?C'"
proof (auto)
fix w
assume w: "w∈imT"
from this finA Cfin AinC CinC obtain v where v_inC: "v∈carrier V" and w_eq_T_v: "w= T v"
by (unfold im_def image_def, auto)
from finA Cfin AinC CinC v_inC ACgen have "∃a. a ∈ A∪C → carrier K∧ V.module.lincomb a (A∪C) = v"
by (intro V.module.finite_in_span, auto)
then obtain a where
a_fun: "a ∈ A∪C → carrier K" and
lc_a_v: "v= V.module.lincomb a (A∪C)"
by auto
let ?a'="λv. a (?f v)"
from finA Cfin AinC CinC a_fun disj Ainker f C'_card have Tv: "T v = W.module.lincomb ?a' ?C'"
apply (subst lc_a_v)
apply (subst V.module.lincomb_union, simp_all) (*Break up the union A∪C*)
apply (unfold lincomb_def V.module.lincomb_def)
apply (subst hom_sum, auto) (*Take T inside the sum over A*)
apply (simp add: subsetD coeff_in_ring
hom_sum (*Take T inside the sum over C*)
T_ker (*all terms become 0 because the vectors are in the kernel.*)
apply (subst finsum_reindex[where ?h="T" and ?f="λv. ?a' v ⊙⇘W⇙ v"], auto)
by (auto cong: finsum_cong simp add: coeff_in_ring ring_subset_carrier)
from a_fun f have a'_fun: "?a'∈?C' → carrier K" by auto
from C'fin CinC this w_eq_T_v a'_fun Tv show "w ∈ LinearCombinations.module.span K W (T ` C)"
by (subst finite_span, auto)
from this subs_im CinC show ?thesis
apply (subst span_li_not_depend(1))
by (unfold im_def subspace_def, auto)
from C'_li C'_gen C cim have C'_basis: "im.basis (T`C)"
by (unfold im.basis_def, auto)
have C_card_im: "card C = (vectorspace.dim K (W.vs imT))"
using C'_basis C'_card(2) C'fin im.dim_basis by auto
from finA Abasis have "ker.dim = card A" by (rule ker.dim_basis)
note * = this C_card_im cardEq
show "(vectorspace.dim K (W.vs imT)) + (vectorspace.dim K (V.vs kerT)) = V.dim" using * by auto
assume "T` (carrier V) = carrier W"
from * surj_imp_imT_carrier[OF this]
show "W.fin_dim" using C'_basis C'fin unfolding W.fin_dim_def im.basis_def by auto
theorem (in linear_map) rank_nullity:
assumes fd: "V.fin_dim"
shows "(vectorspace.dim K (W.vs imT)) + (vectorspace.dim K (V.vs kerT)) = V.dim"
by (rule rank_nullity_main[OF fd]) | {"url":"https://www.isa-afp.org/browser_info/current/AFP/Jordan_Normal_Form/VectorSpace.VectorSpace.html","timestamp":"2024-11-04T11:49:14Z","content_type":"application/xhtml+xml","content_length":"909943","record_id":"<urn:uuid:b014d338-8797-45b3-8549-96747dcecc5a>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00566.warc.gz"} |
An estimation algorithm for 2-D polynomial phase signals
We consider nonhomogeneous 2-D signals that can be represented by a constant modulus polynomial-phase model. A novel 2-D phase differencing operator is introduced and used to develop a
computationally efficient estimation algorithm for the parameters of this model. The operation of the algorithm is illustrated using an example.
ASJC Scopus subject areas
• Software
• Computer Graphics and Computer-Aided Design
Dive into the research topics of 'An estimation algorithm for 2-D polynomial phase signals'. Together they form a unique fingerprint. | {"url":"https://cris.bgu.ac.il/en/publications/an-estimation-algorithm-for-2-d-polynomial-phase-signals","timestamp":"2024-11-12T19:59:06Z","content_type":"text/html","content_length":"56118","record_id":"<urn:uuid:46163eda-165e-41cb-858f-2ddc2506f2ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00513.warc.gz"} |
How to calculate algebraic expansion of power of a binomial
How to calculate algebraic expansion of power of a binomial in Python
To solve the nth power of a binomial (a+b)^n o (a-b)^n in python, you can use the expand() function of the sympy module.
The parameter x is the binomial of power in symbolic or numerical form.
$$ (a+b)^n $$
The expand() function calculates the algebraic expansion of the binomial and returns it as output.
Note. If the binomial is also composed of variables x, y, z, they must be previously defined as symbols.
To calculate the power of the binomial
$$ (x-3)^4 $$
The variable x is defined as a symbol using the Symbol () function. The algebraic expression is assigned to the variable b.
The expand () function algebraically expands the power of the binomial.
from sympy import Symbol, expand
The function outputs the result in symbolic form
x**4 - 12*x**3 + 54*x**2 - 108*x + 81
The result is equivalent to the algebraic expression
$$ x^4 - 12x^3 + 54x^2 - 108x +81 $$
Report an error or share a suggestion to enhance this page | {"url":"https://how.okpedia.org/en/python/how-to-calculate-algebraic-expansion-of-power-of-a-binomial-in-python","timestamp":"2024-11-07T14:02:46Z","content_type":"text/html","content_length":"12970","record_id":"<urn:uuid:bdc74ce0-c503-4439-baa6-d5f97449c689>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00534.warc.gz"} |
Working with Limits
You want to insert a summation formula like "summation of s^k from k = 0 to n" at the cursor in a Writer text document.
You see the Math input window and the Elements pane on the left.
From the list on the upper part of the Elements pane, select the Operators item.
In the lower part of the Elements pane, click the Sum icon.
To enable lower and upper limits, click additionally the Upper and Lower Limits icon.
In the input window, the first placeholder or marker is selected, and you can start to enter the lower limit:
Press F4 to advance to the next marker, and enter the upper limit:
Press F4 to advance to the next marker, and enter the summand:
Now the formula is complete. Click into your text document outside the formula to leave the formula editor.
In the same way, you can enter an Integral formula with limits. When you click an icon from the Elements pane, the assigned text command is inserted in the input window. If you know the text
commands, you can enter the commands directly in the input window.
Click in the input window and enter the following line:
A small gap exists between f(x) and dx, which you can also enter using the Elements pane: select the Formats item from the list on the top, then the Small Gap icon.
If you don't like the font of the letters f and x, choose and select other fonts. Click the Default button to use the new fonts as default from now on.
If you need the formula within a line of text, the limits increase the line height. You can choose to place the limits besides the Sum or Integral symbol, which reduces the line height. | {"url":"https://help.libreoffice.org/latest/eo/text/smath/guide/limits.html","timestamp":"2024-11-13T06:25:52Z","content_type":"text/html","content_length":"13182","record_id":"<urn:uuid:e1cac4a5-d0b5-4d41-99e7-fef0e4c1db1a>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00363.warc.gz"} |
Simplifying the node tree using several Group Input nodes
Creating beautiful and impressive procedural shaders or geometric nodes in Blender requires building complex node trees consisting of a large number of nodes and many connections (links) between
them. After some time it becomes difficult to track where this or that link, stretched across the entire node tree “on three screens”, begins and where exactly it ends. A simple trick can help to
simplify the node tree a little and reduce the number of connections – using several copies of the Group Input node.
The secret is simple – all outputs on all copies of the Group Input node work exactly the same and produce the same values.
Therefore, instead of conducting a long link from a single Group Input node somewhere at the far end of a complex node tree, it is much simpler and more convenient to create a copy of the Group Input
node by pressing the shift + d key combination or simply adding another such node (shift + a – Group – Group Input) and place it in the right place.
This is a node tree with the single Group Input node and a long link:
It will work exactly the same as a node tree with two Group Input nodes:
However, in the second case, there is no need to create a long link.
For example, let’s discover a slightly more complex node tree. With the single Group Input node, it might look like this:
Or using several Group Input nodes – like this:
In the second case, the node tree obviously looks more accurate and easier to read.
This principle works in both Shader Nodes and Geometry Nodes in Blender.
0 Comment
Inline Feedbacks
View all comments | {"url":"https://b3d.interplanety.org/en/simplifying-the-node-tree-using-several-group-input-nodes/","timestamp":"2024-11-02T22:02:59Z","content_type":"text/html","content_length":"212044","record_id":"<urn:uuid:0b670e79-9890-439b-ba19-73a97571f2c2>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00058.warc.gz"} |
Power in AC Circuit in Grade 12 Physics - Alternating Currents | Online Notes Nepal
Power in AC Circuit
The power in the AC circuit is explained below:
AC circuits are usually three-phase for electrical distribution and electrical transmission purposes. Single-phase circuits are commonly used in our domestic supply system.
The total power of a three-phase AC circuit is equal to three times the single-phase power.
So if the power in a single phase of a three-phase system is ‘P’, then the total power of the three-phase system would be 3P (provided the three-phase system is perfectly balanced).
But if the three-phase system is not exactly balanced, then the total power of the system would be the sum of the power of individual phases.
Suppose, in a three-phase system, the power at R phase is PR, at Y phase is PY and at B phase is PB, then total power of the system would be
This is a simple scalar sum since power is a scalar quantity. This is the season if we consider only single-phase during calculating and analyzing of three-phase power, it is enough.
Let us consider, network A is electrically connected with network B as shown in the figure below:
Let us consider the expression of the voltage waveform of a single-phase system is:
Where V is the amplitude of the waveform, ω is the angular velocity of propagation of the wave.
Now, consider the current of the system is i(t) and this current has a phase difference from the voltage by an angle φ. That means the current wave propagates with φ radiant lag in respect of the
voltage. The voltage and current waveform can be represented graphically as shown below:
Now, let us plot the term P versus time,
It is seen from the graph that, the term P does not have any negative value. So, it will have a nonzero average value. It is sinusoidal with a frequency twice of system frequency. Let us now plot the
second term of the power equation, i.e. Q.
This is purely sinusoidal and has a zero average value. So from these two graphs, it is clear that P is the component of power in an AC circuit, which actually transported from network A to network
B. This power is consumed in network B as electric power.
Q, on the other hand, does not really flow from network A to network B. Rather it oscillates between networks A and B. This is also a component of power, actually flowing into and out of the
inductor, capacitor-like energy storage elements of the network.
Here, P is known as the real or active part of the power, and Q is known as the imaginary or reactive part of the power.
Hence, P is called real power or active power, and Q is called imaginary or active power. The unit of active power is Watt, whereas the unit of reactive power is Voltage Ampere Reactive or VAR. | {"url":"https://onlinenotesnepal.com/power-in-ac-circuit","timestamp":"2024-11-09T10:43:30Z","content_type":"text/html","content_length":"81627","record_id":"<urn:uuid:2a61ca8c-88a8-44a0-a60c-aea6525fb8c3>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00476.warc.gz"} |
Resolving Vectors: A Guide for HSC Physics Students
How to Resolve Vectors in Physics
This topic is part of the HSC Physics course under the section Motion on a Plane
HSC Physics Syllabus
• analyse vectors in one and two dimensions to:
– resolve a vector into two perpendicular components
– add two perpendicular vector components to obtain a single vector (ACSPH061)
How to Resolve Vectors
Vectors are used extensively in physics to describe the magnitude and direction of quantities such as displacement, velocity, and force. To solve problems involving vectors, it is often necessary to
resolve them into their components. In this article, we will discuss how to resolve vectors in physics and provide examples to help HSC physics students understand this concept.
What is Vector Resolution?
Vector resolution is the process of breaking down a vector into its components along two or more axes. These components can then be analysed separately using the laws of vector addition and
subtraction. Vector resolution is important in physics because it allows us to analyse the motion of an object in two or more dimensions.
Resolving Vectors into Components
To resolve a vector into its components, we need to know the magnitude and direction of the vector, as well as the angles between the vector and the axes of the coordinate system (e.g. north, east,
south and west).
When resolving a vector on a two-dimensional plane, the vector to be resolved is to become the hypotenuse of a right angled triangle, and the components become the two perpendicular sides.
Let's consider an example to illustrate this concept:
Example 1: An object is fired at 10 `m s^{-1}` at 40º above the horizontal. Resolve the force into its horizontal and vertical components.
Here's how to do it:
Step 1: Draw a diagram
First, we need to draw a diagram of the vector and the axes of the coordinate system, as shown below:
Step 2: Determine the magnitudes of the components
Next, we need to determine the magnitudes of the horizontal and vertical components using trigonometry. The horizontal component can be found using the cosine function, while the vertical component
can be found using the sine function:
Horizontal component = 10 cos θ = 10 cos 40° = 7.66 `m s^{-1}`
Vertical component = 10 sin θ = 10 sin 40° = 6.43 `m s^{-1}`
Therefore, the horizontal component is 7.66 m/s to the right, and the vertical component is 6.43 m/s upward.
Finding Resultant Vector
Once we have resolved vectors into their components, we can use the laws of vector addition to find the resultant vector. The resultant vector is the vector that represents the sum of the individual
The resultant vector is formed by drawing an arrow from the tail of one component to the head of the other.
The magnitude of the resultant vector can be calculated using Pythagoras' theorem. The direction of the resultant vector can be calculated using trigonometry.
Example 2: A boat is traveling at a speed of 10 m/s due north. There is a current flowing at a speed of 5 m/s due east. What is the boat's resultant velocity?
The resultant velocity considers the velocity of the boat as well as the current. To find the resultant vector from vector addition, you need to follow these steps:
Step 1: Draw a diagram
Draw a diagram of the vectors you want to add, making sure to label the magnitude and direction of each vector. You can use a scale to ensure the lengths of the vectors are proportional to their
The resultant vector is drawn such that is becomes the hypotenuse of a right angled triangle.
Step 2: Calculate the magnitude of the resultant vector
Use the Pythagorean theorem to find the magnitude of the resultant vector.
$$v^2 = 5^2 + 10^2$$
$$v = \sqrt{5^2+10^2}$$
$$v = 11.2 m s^{-1}$$
Step 3: Calculate the direction of the resultant vector.
Since velocity is a vector quantity, the direction is required in addition to the magnitude. The direction of the resultant vector can be found using trigonometry.
The angle between the resultant vector and the x-axis (east-west) is given by:
$$\theta = 85º$$ | {"url":"https://scienceready.com.au/pages/how-to-resolve-vectors-in-physics","timestamp":"2024-11-11T14:40:58Z","content_type":"text/html","content_length":"312612","record_id":"<urn:uuid:4d5293da-55bb-414e-9ea4-b4afb33dbd98>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00052.warc.gz"} |
MES: VanEck Vectors Gulf States Index ETF | Logical Invest
What do these metrics mean?
'Total return is the amount of value an investor earns from a security over a specific period, typically one year, when all distributions are reinvested. Total return is expressed as a percentage of
the amount invested. For example, a total return of 20% means the security increased by 20% of its original value due to a price increase, distribution of dividends (if a stock), coupons (if a bond)
or capital gains (if a fund). Total return is a strong measure of an investment’s overall performance.'
Which means for our asset as example:
• Compared with the benchmark SPY (109.2%) in the period of the last 5 years, the total return, or increase in value of % of VanEck Vectors Gulf States Index ETF is lower, thus worse.
• Compared with SPY (33.3%) in the period of the last 3 years, the total return of % is lower, thus worse.
'The compound annual growth rate isn't a true return rate, but rather a representational figure. It is essentially a number that describes the rate at which an investment would have grown if it had
grown the same rate every year and the profits were reinvested at the end of each year. In reality, this sort of performance is unlikely. However, CAGR can be used to smooth returns so that they may
be more easily understood when compared to alternative investments.'
Applying this definition to our asset in some examples:
• The compounded annual growth rate (CAGR) over 5 years of VanEck Vectors Gulf States Index ETF is %, which is lower, thus worse compared to the benchmark SPY (15.9%) in the same period.
• During the last 3 years, the compounded annual growth rate (CAGR) is %, which is smaller, thus worse than the value of 10.1% from the benchmark.
'Volatility is a statistical measure of the dispersion of returns for a given security or market index. Volatility can either be measured by using the standard deviation or variance between returns
from that same security or market index. Commonly, the higher the volatility, the riskier the security. In the securities markets, volatility is often associated with big swings in either direction.
For example, when the stock market rises and falls more than one percent over a sustained period of time, it is called a 'volatile' market.'
Using this definition on our asset we see for example:
• Looking at the historical 30 days volatility of % in the last 5 years of VanEck Vectors Gulf States Index ETF, we see it is relatively lower, thus better in comparison to the benchmark SPY
• Looking at 30 days standard deviation in of % in the period of the last 3 years, we see it is relatively lower, thus better in comparison to SPY (17.6%).
'Downside risk is the financial risk associated with losses. That is, it is the risk of the actual return being below the expected return, or the uncertainty about the magnitude of that difference.
Risk measures typically quantify the downside risk, whereas the standard deviation (an example of a deviation risk measure) measures both the upside and downside risk. Specifically, downside risk in
our definition is the semi-deviation, that is the standard deviation of all negative returns.'
Using this definition on our asset we see for example:
• The downside risk over 5 years of VanEck Vectors Gulf States Index ETF is %, which is lower, thus better compared to the benchmark SPY (14.9%) in the same period.
• During the last 3 years, the downside deviation is %, which is smaller, thus better than the value of 12.3% from the benchmark.
'The Sharpe ratio is the measure of risk-adjusted return of a financial portfolio. Sharpe ratio is a measure of excess portfolio return over the risk-free rate relative to its standard deviation.
Normally, the 90-day Treasury bill rate is taken as the proxy for risk-free rate. A portfolio with a higher Sharpe ratio is considered superior relative to its peers. The measure was named after
William F Sharpe, a Nobel laureate and professor of finance, emeritus at Stanford University.'
Using this definition on our asset we see for example:
• The risk / return profile (Sharpe) over 5 years of VanEck Vectors Gulf States Index ETF is , which is smaller, thus worse compared to the benchmark SPY (0.64) in the same period.
• Looking at ratio of return and volatility (Sharpe) in of in the period of the last 3 years, we see it is relatively smaller, thus worse in comparison to SPY (0.43).
'The Sortino ratio, a variation of the Sharpe ratio only factors in the downside, or negative volatility, rather than the total volatility used in calculating the Sharpe ratio. The theory behind the
Sortino variation is that upside volatility is a plus for the investment, and it, therefore, should not be included in the risk calculation. Therefore, the Sortino ratio takes upside volatility out
of the equation and uses only the downside standard deviation in its calculation instead of the total standard deviation that is used in calculating the Sharpe ratio.'
Applying this definition to our asset in some examples:
• The excess return divided by the downside deviation over 5 years of VanEck Vectors Gulf States Index ETF is , which is smaller, thus worse compared to the benchmark SPY (0.9) in the same period.
• Looking at ratio of annual return and downside deviation in of in the period of the last 3 years, we see it is relatively lower, thus worse in comparison to SPY (0.62).
'The Ulcer Index is a technical indicator that measures downside risk, in terms of both the depth and duration of price declines. The index increases in value as the price moves farther away from a
recent high and falls as the price rises to new highs. The indicator is usually calculated over a 14-day period, with the Ulcer Index showing the percentage drawdown a trader can expect from the high
over that period. The greater the value of the Ulcer Index, the longer it takes for a stock to get back to the former high.'
Which means for our asset as example:
• Looking at the Ulcer Index of in the last 5 years of VanEck Vectors Gulf States Index ETF, we see it is relatively lower, thus better in comparison to the benchmark SPY (9.32 )
• Compared with SPY (10 ) in the period of the last 3 years, the Ulcer Ratio of is lower, thus better.
'A maximum drawdown is the maximum loss from a peak to a trough of a portfolio, before a new peak is attained. Maximum Drawdown is an indicator of downside risk over a specified time period. It can
be used both as a stand-alone measure or as an input into other metrics such as 'Return over Maximum Drawdown' and the Calmar Ratio. Maximum Drawdown is expressed in percentage terms.'
Using this definition on our asset we see for example:
• Looking at the maximum reduction from previous high of days in the last 5 years of VanEck Vectors Gulf States Index ETF, we see it is relatively lower, thus worse in comparison to the benchmark
SPY (-33.7 days)
• During the last 3 years, the maximum reduction from previous high is days, which is lower, thus worse than the value of -24.5 days from the benchmark.
'The Drawdown Duration is the length of any peak to peak period, or the time between new equity highs. The Max Drawdown Duration is the worst (the maximum/longest) amount of time an investment has
seen between peaks (equity highs). Many assume Max DD Duration is the length of time between new highs during which the Max DD (magnitude) occurred. But that isn’t always the case. The Max DD
duration is the longest time between peaks, period. So it could be the time when the program also had its biggest peak to valley loss (and usually is, because the program needs a long time to recover
from the largest loss), but it doesn’t have to be'
Using this definition on our asset we see for example:
• The maximum time in days below previous high water mark over 5 years of VanEck Vectors Gulf States Index ETF is days, which is lower, thus better compared to the benchmark SPY (488 days) in the
same period.
• Looking at maximum days below previous high in of days in the period of the last 3 years, we see it is relatively lower, thus better in comparison to SPY (488 days).
'The Drawdown Duration is the length of any peak to peak period, or the time between new equity highs. The Avg Drawdown Duration is the average amount of time an investment has seen between peaks
(equity highs), or in other terms the average of time under water of all drawdowns. So in contrast to the Maximum duration it does not measure only one drawdown event but calculates the average of
Applying this definition to our asset in some examples:
• Looking at the average days below previous high of days in the last 5 years of VanEck Vectors Gulf States Index ETF, we see it is relatively lower, thus better in comparison to the benchmark SPY
(123 days)
• Looking at average time in days below previous high water mark in of days in the period of the last 3 years, we see it is relatively smaller, thus better in comparison to SPY (176 days). | {"url":"https://logical-invest.com/app/stock/mes/vaneck-vectors-gulf-states-index-etf","timestamp":"2024-11-11T06:25:25Z","content_type":"text/html","content_length":"60530","record_id":"<urn:uuid:72e1b825-1251-490c-b850-ddf472e8c607>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00696.warc.gz"} |
: Chapter VI: The Treaty of Croton
Chapter VI
ACCORDING to tradition that strange mystic, philosopher, and mathematician, Pythagoras, spent over a quarter of a century in study and travel in Egypt before founding his own great Brotherhood and
school at Croton, a Greek colony in Southern Italy. With his peculiar brand of mysticism we have no concern here, except to recall that it was a particularly wild one based on numbers, and that it
infected most of his thinking. But in spite of his flights into the clouds of pure verbalism, Pythagoras did three things of the first magnitude, any one of which is probably sufficient to ensure his
remembrance for as long as human beings can remember anything. These were: the first definitely recorded physical experiment in history; the invention of irrational numbers, on which. the whole vast
structure of modern mathematical analysis rests; and the first definitely recorded insistence upon proof for statements about numbers and geometrical figures. It is the last only of these which we
need to discuss in our pursuit of "truth" through the mazes of deductive reasoning. Pythagoras, dates (doubtful) are B.C. 569–496.
Let us go back for a moment to Egypt where, Pythagoras tells us, he learned much from the wise priests.
We saw that the Egyptian who found a consistently usable formula for the volume of a pyramid either used abstract reasoning subconsciously or was such a phenomenally good guesser that he needed no
reasoning. It was suggested that civilized human beings sooner or later must resort to abstract reasoning if their civilization is not to slip backward.
To bring out the point about the "common agreement" which we spoke of in connection with land surveying, let us restate an extremely simple problem in that practical science in a form which would
have appealed to a Greek, in particular to Pythagoras.
Here is the problem: how many square yards are there in a rectangular field which is 100 yards long and 50 yards broad? This is not quite hard enough for Pythagoras, so we generalize it: how many
square yards are there in a field which is L yards long and B yards broad, where L and B are any numbers?
If you answer 5000 to the first problem, you are right. That answer leads to consistency with other problems of the same kind. But this is not enough for Pythagoras. "Prove it," he demands, just like
one rude little boy to another who has used an offensive epithet. That seems easy: "Oh, you get the area of a rectangle by multiplying the length by the breadth—." Pythagoras interrupts: "Prove it."
A very stupid person would multiply 100 by 50 and proudly exhibit the correct answer 5000.
We have all done such things when we were at school and got stuck in arithmetic; we looked up the answer at the end of the book. But Pythagoras is not interested in the answer; what he wants to know
is how do you know that it is right when you get it?
When we understand what he is driving at we make a fussy attempt to recall what we were taught, get all hot and bothered, and finally fling back at him the Baconian answer: "Go and measure your
beastly field. Cut it up with stretched strings or vermicelli or anything you like into square yards and count them." "Do that for B and L," Pythagoras grins, and possibly you tell him to go to L
himself, for he has caught you in a trap from which you cannot escape by measurement, that is, by experiment, no matter if you have a million years to try. So here is a strikingly simple problem
completely beyond the reach of the "operational" method. The element of generality, or universality in the "any" of the problem puts it in a realm which is inaccessible to concrete experiment.
Pythagoras relents. Seeing that any B and any L are too hard, he draws a simple figure in the sand: a square with one of its diagonals. By an easy construction he then makes a rectangle with length
equal to the diagonal of the square and breadth equal to one side of the square. "Measure it," he says, handing you a thin thread. "If the thread is too coarse, you may use a spider's string. The
breadth of that rectangle is 12 inches. You measure the area and tell me how many square inches there are in the rectangle. When you give me the right answer, I'll give you all the gold in Greece.
But if you give me a wrong answer, I'll have you sewed up in a sack and pitched off that cliff at high tide. Something like that happened to one of our Brotherhood only last Monday."
Of course nobody would accept such odds as that from a wild-eyed mystic like Pythagoras. He might sew the gambler up anyway, win or lose.
Men who cannot give Pythagoras the answer to his childishly simple-looking problem can nevertheless tell us how long the universe will last, when time began and when it will end, and what God has in
store for us to pass away the ages through eternity. When we thoroughly understand what Pythagoras was talking about, and when we see into and through the machinery invented by Pythagoras and his
immediate successors for disposing of his problem, we shall be able to laugh in the prophets' faces. His problem is simply to find the area of that rectangle he drew on the sand by measurement, that
is, by experiment.
It cannot be done, and Pythagoras knew that it could not. The fact that the numerical measure of the diagonal of the square is "irrational" (not obtainable by dividing one whole number by another) is
the disconcerting fact which caused Pythagoras to abandon his sublime extrapolation that nature and (possibly) reason are based on the simple pattern of the whole numbers 1, 2, 3,.... The "universe
of geometrical lines" is, in this sense, not "rational;" "irrational" lengths can be humanly constructed.
But common sense tells us that the rectangle does have a definite area; we can see it with our eyes, if not with our minds. A rough approximation is 1.4 square feet, a little better 1.41, and so on,
indefinitely, that is, interminably. There is no end to the process. And if there is no end, is it likely that we shall be able to And an exact answer by experiment? In this simple problem we have
again stumbled across the infinite. Can any performable experiment continue without end? It begins to look as if that "common agreement" desired by all sane farmers on the way of measuring a field is
less simple than it seemed. If one man says that the correct answer is 1.41, and another 1.412, how is it to be decided which, if either, is right, or which has a closer approximation to the right
answer if there is one? How do we know that there is a "right" answer to the problem? It is fairly obvious that we do not, until we agree upon some set of conventions.
Now, all this is as old as the hills to anyone who has ever been through a carefully presented course in elementary school geometry. But mere familiarity is not enough to prevent some devout believer
in an abstract and eternal "truth," over and above our human conventions by which we reach agreements in geometry, as in everything else, from appealing to this "everlasting truth" for the right
answer. "There must be one answer which is right, and we can find that one because it is true."
They can call upon "truth" till they are out of breath and blue in the face; they will get no answer. That "still, small voice" which they expect to hear has nothing whatever to say about the area of
a rectangle. This may be hard doctrine, but unless our habits were completely perverted before we were seventeen by traditional teaching we shall see as we go on that it is saner doctrine than the
other. There is nothing new in this, and most of us have known it ever since we began thinking at all, however much some of us may have rebelled when we first realized it.
Pythagoras insisted upon proof. Although we cannot say what "truth" is in Pilate's sense, simply because it has no meaning, we can say with rigid exactness what proof is in a deductive system of
reasoning. That kind of system is the one which is relevant for the problem of the rectangle.
First, we lay down certain outright assumptions which we agree to accept without further argument. These are called our postulates (sometimes axioms). For example in geometry one postulate is: "The
whole is greater than any one of its parts." The postulates agreed upon may or may not have been suggested by experience, or by induction from a large number of experiments. However they may have
been suggested is wholly irrelevant in this question of proof.
To dispose of a possible objection here, the postulates are not always "obvious," or such that all sane beings could agree upon as being sensible. This objection harks back to the subconscious belief
in that mysterious "truth." To give this dogmatic denial some shadow of backing, I may state that the trivial example above about the whole and its parts was chosen deliberately. It works admirably
when we reason about a finite collection of things, say all the stars in all the nebulae reached by telescopes, or.all the human beings who have ever lived. But it does not work when we try to reason
about an infinity of things, say all the points on a straight line. There is nothing "obvious" about it, nor does it "necessarily" apply to such a simple thing as the entire universe. It was
indicated in an earlier chapter that the "whole-part" axiom fails for the infinite collection of all the common whole numbers. To repeat, because the point really is important for everything that is
to follow, postulates are out-and-out assumptions.
Having seen the necessity for agreeing upon a set of postulates before undertaking to prove anything, Pythagoras and his successors next laid down the completely arbitrary rule that a statement shall
be said to be proved when, and only when, the statement follows from the postulates by an application of the rules of logic. Nothing but these rules is to be injected into the process of "proof."
When the statement does follow as just described, we say that it has been deduced from the postulates. The process is called deduction, and the type of reasoning by which it proceeds is the abstract,
deductive reasoning already mentioned in connection with the Egyptians.
There are no historical grounds for saying indisputably that Pythagoras himself ever got as far as this cold, clear conception of proof. Indeed many of his speculations seem to show that he did not
recognize the complete arbitrariness of the postulates, particularly in geometry. He seems to have been still in the mystic stage, and he reasoned subconsciously. A musician does not have to
understand the theory of music in order to compose; harmonizing and the rest come after the composition has been conceived.
It is even doubtful whether any of the Greeks ever got as far as the above conception of proof. That conception seems to have been clearly grasped only in the Nineteenth Century. The Twentieth has
gone considerably ahead of this, as regards logic, but that part of the story does not come until almost the end. Pythagoras however undoubtedly was the first human being on record who recognized the
necessity for proof in order to enable men to reach common conclusions everywhere and always from the same set of data concerning numbers or geometrical figures. The suggestion he made was so simple,
so rational, that it seems inevitable. We are so used to it that we take it for granted, forgetting the chaos which reigned before Pythagoras lived.
If we are inclined to underestimate what he did, we need but think of the situation with regard to human problems—social, ethical, moral, economic, religious—at the present time. Is there any sign of
an agreement in sight? Is anyone ever likely to devise a set of rules for procedure in human problems on which almost all sane men can agree everywhere and in all times? Ask the Egyptians of 4241
B.C. To them the simpler task—simpler perhaps only because it has been done—of formulating such a set of rules for the material problems of their day may have looked as hopeless as our own task looks
to us.
Any man who can get the rest of mankind to accept a treaty on anything must have quite unusual powers of persuasion. Pythagoras seems to be the only man on record who almost succeeded. His actual
success however was so great that for long it completely overshadowed his partial failure. That partial failure is the significant item in the evolution—or history—of abstract thinking. The trouble
began when the pyramid cropped up again to bother the Greeks. But we must first glance at the rules of logic which have been mentioned several times but not yet sufficiently discussed. | {"url":"http://mathlair.allfunandgames.ca/searchfortruth6.php","timestamp":"2024-11-12T16:01:59Z","content_type":"text/html","content_length":"15537","record_id":"<urn:uuid:6fcbe35f-4a68-4cfb-876d-d1a46afbcb52>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00848.warc.gz"} |